Subgroups — C₃₁×#60523 #21517282

Order 7936.

2184 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3488	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#3428	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3099	C₂	1	3	yes	maximal, complemented
4	C₃₁×#3098	C₂	1	3	yes	maximal, complemented
5	C₃₁×#3096	C₂	1	3	yes	maximal, complemented
6	C₃₁×#3096	C₂	1	3	yes	maximal, complemented
7	C₃₁×#3094	C₂	1	3	yes	maximal, complemented
8	C₃₁×#3093	C₂	1	3	yes	maximal, complemented
9	C₃₁×#3091	C₂	1	3	yes	maximal, complemented
10	C₂×C₃₁×#482	C₂	1	3	yes	maximal, complemented
11	C₃₁×#2913	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#2912	C₂	1	1	yes	maximal, characteristic, complemented
13	C₃₁×#2897	C₂	1	1	yes	maximal, characteristic, complemented
14	C₃₁×#2897	C₂	1	1	yes	maximal, characteristic, complemented
15	C₃₁×#2880	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₃₁×H(5,𝔽₂)	C₂²	1	1	yes	characteristic
17	C₂²×#41×C₃₁	C₂²	1	1	yes	characteristic
18	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic
19	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic
20	C₃₁×#550	—	2	1	no	—
21	C₃₁×#548	—	2	1	no	—
22	C₃₁×#546	—	2	3	no	—
23	D₄²×C₃₁	—	2	3	no	—
24	C₃₁×#537	C₂²	1	3	yes	—
25	C₃₁×#535	C₂²	1	3	yes	—
26	C₃₁×#525	C₂²	1	3	yes	—
27	C₂×C₃₁×#121	C₂²	1	3	yes	—
28	C₃₁×#520	C₂²	1	1	yes	characteristic
29	C₃₁×#520	C₂²	1	1	yes	characteristic
30	C₃₁×#520	C₂²	1	1	yes	characteristic
31	C₃₁×#520	C₂²	1	3	yes	complemented
32	C₃₁×#520	C₂²	1	3	yes	complemented
33	C₃₁×#519	—	2	1	no	—
34	C₃₁×#518	—	2	3	no	—
35	C₄×#41×C₃₁	—	2	3	no	—
36	C₄×#41×C₃₁	—	2	1	no	—
37	C₂×C₃₁×#117	C₂²	1	3	yes	complemented
38	C₂×C₃₁×#117	C₂²	1	3	yes	complemented
39	C₃₁×#485	C₂²	1	1	yes	characteristic, complemented
40	C₃₁×#485	C₂²	1	3	yes	complemented
41	C₃₁×#485	C₂²	1	6	yes	complemented
42	C₃₁×#484	C₂²	1	3	yes	complemented
43	C₃₁×#483	C₂²	1	3	yes	complemented
44	C₃₁×#482	C₂²	1	3	yes	complemented
45	C₃₁×#482	C₂²	1	6	yes	complemented
46	C₃₁×#482	C₂²	1	6	yes	complemented
47	C₃₁×#482	C₂²	1	6	yes	complemented
48	C₃₁×#482	C₂²	1	3	yes	complemented
49	C₃₁×#482	C₂²	1	6	yes	complemented
50	C₃₁×#481	C₂²	1	3	yes	complemented
51	C₃₁×#481	C₂²	1	6	yes	complemented
52	C₃₁×#480	C₂²	1	3	yes	complemented
53	C₃₁×#480	C₂²	1	6	yes	complemented
54	C₃₁×#428	C₂²	1	3	yes	complemented
55	C₃₁×#428	C₂²	1	3	yes	complemented
56	C₃₁×#428	C₂²	1	3	yes	complemented
57	C₃₁×#428	C₂²	1	3	yes	complemented
58	C₃₁×#427	C₂²	1	3	yes	complemented
59	C₃₁×#427	C₂²	1	3	yes	complemented
60	C₂×C₃₁×#107	C₂²	1	3	yes	complemented
61	C₂×C₃₁×#106	C₂²	1	3	yes	complemented
62	C₃₁×#418	C₂²	1	3	yes	complemented
63	C₃₁×#418	C₂²	1	3	yes	complemented
64	C₃₁×#417	C₂²	1	3	yes	complemented
65	C₃₁×#417	C₂²	1	3	yes	complemented
66	C₃₁×#416	C₂²	1	3	yes	complemented
67	C₃₁×#416	C₂²	1	3	yes	complemented
68	C₂×C₃₁×#102	C₂²	1	3	yes	complemented
69	C₂×C₃₁×#102	C₂²	1	3	yes	complemented
70	C₃₁×#408	C₂²	1	3	yes	complemented
71	C₃₁×#408	C₂²	1	3	yes	complemented
72	C₃₁×#407	C₂²	1	3	yes	complemented
73	C₂×C₃₁×C₂²⋊C₈	C₂²	1	3	yes	complemented
74	C₃₁×H(5,𝔽₂)	—	2	4	no	—
75	C₃₁×H(5,𝔽₂)	—	2	4	no	—
76	C₂×#41×C₃₁	—	2	4	no	—
77	C₂×#41×C₃₁	—	2	4	no	—
78	C₂×#41×C₃₁	—	2	1	no	—
79	C₂×#41×C₃₁	D₄	1	3	yes	—
80	C₂×#41×C₃₁	D₄	1	1	yes	characteristic
81	C₂×#41×C₃₁	C₂³	1	3	yes	—
82	C₂×#41×C₃₁	C₂³	1	3	yes	—
83	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
84	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
85	C₂²×D₄×C₃₁	—	2	3	no	—
86	C₂²×D₄×C₃₁	—	2	3	no	—
87	C₂²×D₄×C₃₁	D₄	1	3	yes	—
88	C₂²×D₄×C₃₁	C₂³	1	3	yes	—
89	C₂³×C₄×C₃₁	C₂³	1	3	yes	—
90	C₃₁×#131	—	2	4	no	—
91	C₃₁×#131	—	2	4	no	—
92	C₂×M₁₆×C₃₁	C₂³	1	3	yes	—
93	C₂×M₁₆×C₃₁	C₂³	1	3	yes	—
94	C₂²×C₈×C₃₁	C₂³	1	3	yes	—
95	C₂²×C₈×C₃₁	C₂³	1	3	yes	—
96	C₃₁×#127	—	2	3	no	—
97	C₃₁×#124	—	2	3	no	—
98	C₃₁×#123	—	2	3	no	—
99	C₃₁×C₂²⋊Q₈	—	2	3	no	—
100	C₃₁×C₂²⋊Q₈	C₂³	1	3	yes	—
101	C₃₁×C₂²⋊Q₈	C₂³	1	1	yes	characteristic
102	C₃₁×#121	—	2	6	no	—
103	C₃₁×#121	—	2	3	no	—
104	C₃₁×#121	—	2	3	no	—
105	C₃₁×#121	—	2	3	no	—
106	C₃₁×#121	C₂³	1	6	yes	—
107	C₃₁×#121	C₂³	1	3	yes	—
108	C₃₁×#121	C₂³	1	3	yes	—
109	C₃₁×#120	—	2	3	no	—
110	C₃₁×#120	—	2	6	no	—
111	C₄×Q₈×C₃₁	—	2	1	no	—
112	C₄×Q₈×C₃₁	—	2	1	no	—
113	C₄×Q₈×C₃₁	—	2	3	no	—
114	C₄×Q₈×C₃₁	—	2	3	no	—
115	C₄×Q₈×C₃₁	—	2	1	no	—
116	C₄×D₄×C₃₁	—	2	3	no	—
117	C₄×D₄×C₃₁	—	2	3	no	—
118	C₄×D₄×C₃₁	—	2	6	no	—
119	C₄×D₄×C₃₁	—	2	6	no	—
120	C₄×D₄×C₃₁	—	2	3	no	—
121	C₄×D₄×C₃₁	—	2	3	no	—
122	C₄×D₄×C₃₁	—	2	3	no	—
123	C₄×D₄×C₃₁	—	2	3	no	—
124	C₄×D₄×C₃₁	—	2	3	no	—
125	C₃₁×#117	—	2	3	no	—
126	C₃₁×#117	—	2	3	no	—
127	C₃₁×#117	—	2	3	no	—
128	C₃₁×#117	—	2	3	no	—
129	C₃₁×#117	C₂³	1	3	yes	—
130	C₃₁×#117	C₂³	1	3	yes	—
131	C₃₁×#117	C₂³	1	3	yes	—
132	C₃₁×#117	C₂³	1	3	yes	—
133	C₃₁×#117	C₂³	1	1	yes	characteristic
134	C₃₁×#117	C₂³	1	6	yes	—
135	C₂×C₄⋊C₄×C₃₁	C₂³	1	3	yes	—
136	C₂×C₄⋊C₄×C₃₁	C₂³	1	3	yes	—
137	C₂×C₄⋊C₄×C₃₁	C₂³	1	3	yes	—
138	C₂×C₄⋊C₄×C₃₁	C₂³	1	3	yes	—
139	C₂×C₄⋊C₄×C₃₁	C₂³	1	3	yes	—
140	C₂×C₂²⋊C₄×C₃₁	—	2	3	no	—
141	C₂×C₂²⋊C₄×C₃₁	—	2	3	no	—
142	C₂×C₂²⋊C₄×C₃₁	—	2	3	no	—
143	C₂×C₄²×C₃₁	—	2	3	no	—
144	C₂×C₄²×C₃₁	—	2	3	no	—
145	C₃₁×#107	C₂³	1	3	yes	—
146	C₃₁×#107	C₂³	1	6	yes	—
147	C₃₁×#107	C₂³	1	1	yes	characteristic
148	C₃₁×#107	C₂³	1	3	yes	—
149	C₃₁×#107	C₂³	1	3	yes	—
150	C₃₁×#106	C₂³	1	3	yes	—
151	C₃₁×#106	C₂³	1	6	yes	—
152	C₃₁×#106	C₂³	1	1	yes	characteristic
153	C₃₁×#106	C₂³	1	3	yes	—
154	C₃₁×#106	C₂³	1	3	yes	—
155	C₃₁×#103	C₂³	1	1	yes	characteristic
156	C₃₁×#103	C₂³	1	1	yes	characteristic
157	C₃₁×#103	C₂³	1	3	yes	—
158	C₃₁×#103	C₂³	1	3	yes	—
159	C₃₁×#102	C₂³	1	3	yes	—
160	C₃₁×#102	C₂³	1	3	yes	—
161	C₃₁×#102	C₂³	1	6	yes	—
162	C₃₁×#102	C₂³	1	6	yes	—
163	C₃₁×#102	C₂³	1	3	yes	—
164	C₃₁×#102	C₂³	1	3	yes	—
165	C₃₁×C₂²⋊C₈	C₂³	1	1	yes	characteristic
166	C₃₁×C₂²⋊C₈	C₂³	1	3	yes	—
167	C₃₁×C₂²⋊C₈	C₂³	1	3	yes	—
168	C₃₁×C₂²⋊C₈	C₂³	1	3	yes	—
169	C₃₁×C₂²⋊C₈	C₂³	1	6	yes	—
170	C₂⁴×C₃₁	—	2	3	no	—
171	C₂⁴×C₃₁	C₂×D₄	1	3	yes	—
172	#41×C₃₁	—	2	4	no	—
173	#41×C₃₁	—	2	12	no	—
174	#41×C₃₁	—	2	6	no	—
175	#41×C₃₁	—	2	6	no	—
176	#41×C₃₁	—	2	6	no	—
177	#41×C₃₁	—	2	6	no	—
178	#41×C₃₁	—	4	4	no	—
179	#41×C₃₁	#41	1	4	yes	—
180	#41×C₃₁	#41	1	4	yes	—
181	C₂×Q₈×C₃₁	—	2	4	no	—
182	C₂×Q₈×C₃₁	C₂×D₄	1	3	yes	—
183	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
184	C₂×Q₈×C₃₁	C₂⁴	1	1	yes	characteristic
185	C₂×D₄×C₃₁	—	2	12	no	—
186	C₂×D₄×C₃₁	—	2	3	no	—
187	C₂×D₄×C₃₁	—	2	3	no	—
188	C₂×D₄×C₃₁	—	2	3	no	—
189	C₂×D₄×C₃₁	—	2	6	no	—
190	C₂×D₄×C₃₁	—	2	3	no	—
191	C₂×D₄×C₃₁	—	2	6	no	—
192	C₂×D₄×C₃₁	—	2	6	no	—
193	C₂×D₄×C₃₁	—	4	12	no	—
194	C₂×D₄×C₃₁	—	8	6	no	—
195	C₂×D₄×C₃₁	C₂×D₄	1	6	yes	—
196	C₂×D₄×C₃₁	C₂×D₄	1	3	yes	—
197	C₂×D₄×C₃₁	C₂×D₄	1	3	yes	—
198	C₂×D₄×C₃₁	C₂⁴	1	3	yes	—
199	C₂²×C₄×C₃₁	—	2	6	no	—
200	C₂²×C₄×C₃₁	—	2	6	no	—
201	C₂²×C₄×C₃₁	—	2	3	no	—
202	C₂²×C₄×C₃₁	—	2	3	no	—
203	C₂²×C₄×C₃₁	—	2	3	no	—
204	C₂²×C₄×C₃₁	—	2	3	no	—
205	C₂²×C₄×C₃₁	—	2	3	no	—
206	C₂²×C₄×C₃₁	—	2	3	no	—
207	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
208	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
209	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
210	C₂²×C₄×C₃₁	C₂⁴	1	3	yes	—
211	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
212	C₂²×C₄×C₃₁	C₂⁴	1	3	yes	—
213	M₁₆×C₃₁	—	2	6	no	—
214	M₁₆×C₃₁	—	2	6	no	—
215	C₂×C₈×C₃₁	—	2	6	no	—
216	C₂×C₈×C₃₁	—	2	6	no	—
217	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
218	C₂×C₈×C₃₁	C₂⁴	1	3	yes	—
219	C₂×C₈×C₃₁	C₂⁴	1	3	yes	—
220	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
221	C₄⋊C₄×C₃₁	—	2	3	no	—
222	C₄⋊C₄×C₃₁	—	2	3	no	—
223	C₄⋊C₄×C₃₁	—	2	3	no	—
224	C₄⋊C₄×C₃₁	—	2	3	no	—
225	C₄⋊C₄×C₃₁	—	2	3	no	—
226	C₄⋊C₄×C₃₁	—	2	1	no	—
227	C₄⋊C₄×C₃₁	—	2	3	no	—
228	C₄⋊C₄×C₃₁	C₂⁴	1	3	yes	—
229	C₄⋊C₄×C₃₁	C₂⁴	1	1	yes	characteristic
230	C₄⋊C₄×C₃₁	C₂⁴	1	3	yes	—
231	C₄⋊C₄×C₃₁	C₂⁴	1	1	yes	characteristic
232	C₄⋊C₄×C₃₁	C₂⁴	1	3	yes	—
233	C₄⋊C₄×C₃₁	C₂⁴	1	1	yes	characteristic
234	C₂²⋊C₄×C₃₁	—	2	3	no	—
235	C₂²⋊C₄×C₃₁	—	2	1	no	—
236	C₂²⋊C₄×C₃₁	—	2	6	no	—
237	C₂²⋊C₄×C₃₁	—	2	3	no	—
238	C₂²⋊C₄×C₃₁	—	2	3	no	—
239	C₂²⋊C₄×C₃₁	—	2	3	no	—
240	C₂²⋊C₄×C₃₁	—	2	1	no	—
241	C₂²⋊C₄×C₃₁	—	2	3	no	—
242	C₂²⋊C₄×C₃₁	—	2	3	no	—
243	C₂²⋊C₄×C₃₁	—	2	6	no	—
244	C₂²⋊C₄×C₃₁	—	2	3	no	—
245	C₄²×C₃₁	—	2	3	no	—
246	C₄²×C₃₁	—	2	1	no	—
247	C₄²×C₃₁	—	2	6	no	—
248	C₄²×C₃₁	—	2	3	no	—
249	C₄²×C₃₁	—	2	3	no	—
250	C₄²×C₃₁	—	2	3	no	—
251	#60523	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
252	C₂³×C₃₁	—	2	6	no	—
253	C₂³×C₃₁	—	2	3	no	—
254	C₂³×C₃₁	—	4	3	no	—
255	C₂³×C₃₁	—	4	3	no	—
256	C₂³×C₃₁	—	4	6	no	—
257	C₂³×C₃₁	—	8	3	no	—
258	C₂³×C₃₁	—	8	3	no	—
259	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
260	C₂³×C₃₁	C₂²×D₄	1	3	yes	—
261	C₂³×C₃₁	C₂²×D₄	1	3	yes	—
262	Q₈×C₃₁	—	2	6	no	—
263	Q₈×C₃₁	—	2	1	no	—
264	Q₈×C₃₁	—	2	1	no	—
265	Q₈×C₃₁	#123	1	4	yes	—
266	D₄×C₃₁	—	2	12	no	—
267	D₄×C₃₁	—	2	6	no	—
268	D₄×C₃₁	—	2	3	no	—
269	D₄×C₃₁	—	2	3	no	—
270	D₄×C₃₁	—	4	6	no	—
271	D₄×C₃₁	—	4	6	no	—
272	D₄×C₃₁	—	4	6	no	—
273	D₄×C₃₁	—	8	3	no	—
274	D₄×C₃₁	—	8	6	no	—
275	D₄×C₃₁	#123	1	12	yes	—
276	C₂×C₄×C₃₁	—	2	6	no	—
277	C₂×C₄×C₃₁	—	2	6	no	—
278	C₂×C₄×C₃₁	—	2	6	no	—
279	C₂×C₄×C₃₁	—	2	1	no	—
280	C₂×C₄×C₃₁	—	2	1	no	—
281	C₂×C₄×C₃₁	—	2	3	no	—
282	C₂×C₄×C₃₁	—	2	3	no	—
283	C₂×C₄×C₃₁	—	2	3	no	—
284	C₂×C₄×C₃₁	—	2	1	no	—
285	C₂×C₄×C₃₁	—	2	3	no	—
286	C₂×C₄×C₃₁	—	2	1	no	—
287	C₂×C₄×C₃₁	—	2	1	no	—
288	C₂×C₄×C₃₁	—	2	3	no	—
289	C₂×C₄×C₃₁	—	2	1	no	—
290	C₂×C₄×C₃₁	—	4	6	no	—
291	C₂×C₄×C₃₁	—	8	3	no	—
292	C₂×C₄×C₃₁	—	8	3	no	—
293	C₂×C₄×C₃₁	—	8	3	no	—
294	C₂×C₄×C₃₁	—	8	3	no	—
295	C₂×C₄×C₃₁	—	8	3	no	—
296	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
297	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
298	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
299	C₂×C₄×C₃₁	C₂×#41	1	6	yes	—
300	C₂×C₄×C₃₁	C₂×#41	1	6	yes	—
301	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
302	C₈×C₃₁	—	2	1	no	—
303	C₈×C₃₁	—	2	1	no	—
304	C₈×C₃₁	—	2	3	no	—
305	C₈×C₃₁	—	2	3	no	—
306	#3488	C₂×C₃₁	1	1	yes	characteristic
307	#3428	C₂×C₃₁	1	1	yes	characteristic, complemented
308	#3099	C₂×C₃₁	1	3	yes	complemented
309	#3098	C₂×C₃₁	1	3	yes	complemented
310	#3096	C₂×C₃₁	1	3	yes	complemented
311	#3096	C₂×C₃₁	1	3	yes	complemented
312	#3094	C₂×C₃₁	1	3	yes	complemented
313	#3093	C₂×C₃₁	1	3	yes	complemented
314	#3091	C₂×C₃₁	1	3	yes	complemented
315	C₂×#482	C₂×C₃₁	1	3	yes	complemented
316	#2913	C₂×C₃₁	1	1	yes	characteristic, complemented
317	#2912	C₂×C₃₁	1	1	yes	characteristic, complemented
318	#2897	C₂×C₃₁	1	1	yes	characteristic, complemented
319	#2897	C₂×C₃₁	1	1	yes	characteristic, complemented
320	#2880	C₂×C₃₁	1	1	yes	characteristic, complemented
321	C₂²×C₃₁	—	2	3	no	—
322	C₂²×C₃₁	—	2	3	no	—
323	C₂²×C₃₁	—	2	3	no	—
324	C₂²×C₃₁	—	2	1	no	—
325	C₂²×C₃₁	—	2	3	no	—
326	C₂²×C₃₁	—	2	1	no	—
327	C₂²×C₃₁	—	2	1	no	—
328	C₂²×C₃₁	—	2	3	no	—
329	C₂²×C₃₁	—	4	3	no	—
330	C₂²×C₃₁	—	4	3	no	—
331	C₂²×C₃₁	—	4	3	no	—
332	C₂²×C₃₁	—	8	6	no	—
333	C₂²×C₃₁	—	16	6	no	—
334	C₂²×C₃₁	C₂×#123	1	6	yes	—
335	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
336	C₄×C₃₁	—	2	1	no	—
337	C₄×C₃₁	—	2	3	no	—
338	C₄×C₃₁	—	4	3	no	—
339	C₄×C₃₁	—	4	1	no	—
340	C₄×C₃₁	—	4	1	no	—
341	C₄×C₃₁	—	4	3	no	—
342	C₄×C₃₁	—	4	1	no	—
343	C₄×C₃₁	—	4	3	no	—
344	C₄×C₃₁	—	4	1	no	—
345	C₄×C₃₁	C₂×#123	1	6	yes	—
346	C₄×C₃₁	C₂²×#41	1	1	yes	characteristic
347	C₄×C₃₁	C₂²×#41	1	1	yes	characteristic
348	C₂×H(5,𝔽₂)	C₂²×C₃₁	1	1	yes	characteristic
349	C₂²×#41	C₂²×C₃₁	1	1	yes	characteristic
350	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
351	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
352	#550	—	2	1	no	—
353	#548	—	2	1	no	—
354	#546	—	2	3	no	—
355	D₄²	—	2	3	no	—
356	#537	C₂²×C₃₁	1	3	yes	—
357	#535	C₂²×C₃₁	1	3	yes	—
358	#525	C₂²×C₃₁	1	3	yes	—
359	C₂×#121	C₂²×C₃₁	1	3	yes	—
360	#520	C₂²×C₃₁	1	1	yes	characteristic
361	#520	C₂²×C₃₁	1	1	yes	characteristic
362	#520	C₂²×C₃₁	1	1	yes	characteristic
363	#520	C₂²×C₃₁	1	3	yes	complemented
364	#520	C₂²×C₃₁	1	3	yes	complemented
365	#519	—	2	1	no	—
366	#518	—	2	3	no	—
367	C₄×#41	—	2	3	no	—
368	C₄×#41	—	2	1	no	—
369	C₂×#117	C₂²×C₃₁	1	3	yes	complemented
370	C₂×#117	C₂²×C₃₁	1	3	yes	complemented
371	#485	C₂²×C₃₁	1	1	yes	characteristic, complemented
372	#485	C₂²×C₃₁	1	3	yes	complemented
373	#485	C₂²×C₃₁	1	6	yes	complemented
374	#484	C₂²×C₃₁	1	3	yes	complemented
375	#483	C₂²×C₃₁	1	3	yes	complemented
376	#482	C₂²×C₃₁	1	3	yes	complemented
377	#482	C₂²×C₃₁	1	6	yes	complemented
378	#482	C₂²×C₃₁	1	6	yes	complemented
379	#482	C₂²×C₃₁	1	6	yes	complemented
380	#482	C₂²×C₃₁	1	3	yes	complemented
381	#482	C₂²×C₃₁	1	6	yes	complemented
382	#481	C₂²×C₃₁	1	3	yes	complemented
383	#481	C₂²×C₃₁	1	6	yes	complemented
384	#480	C₂²×C₃₁	1	3	yes	complemented
385	#480	C₂²×C₃₁	1	6	yes	complemented
386	#428	C₂²×C₃₁	1	3	yes	complemented
387	#428	C₂²×C₃₁	1	3	yes	complemented
388	#428	C₂²×C₃₁	1	3	yes	complemented
389	#428	C₂²×C₃₁	1	3	yes	complemented
390	#427	C₂²×C₃₁	1	3	yes	complemented
391	#427	C₂²×C₃₁	1	3	yes	complemented
392	C₂×#107	C₂²×C₃₁	1	3	yes	complemented
393	C₂×#106	C₂²×C₃₁	1	3	yes	complemented
394	#418	C₂²×C₃₁	1	3	yes	complemented
395	#418	C₂²×C₃₁	1	3	yes	complemented
396	#417	C₂²×C₃₁	1	3	yes	complemented
397	#417	C₂²×C₃₁	1	3	yes	complemented
398	#416	C₂²×C₃₁	1	3	yes	complemented
399	#416	C₂²×C₃₁	1	3	yes	complemented
400	C₂×#102	C₂²×C₃₁	1	3	yes	complemented
401	C₂×#102	C₂²×C₃₁	1	3	yes	complemented
402	#408	C₂²×C₃₁	1	3	yes	complemented
403	#408	C₂²×C₃₁	1	3	yes	complemented
404	#407	C₂²×C₃₁	1	3	yes	complemented
405	C₂×C₂²⋊C₈	C₂²×C₃₁	1	3	yes	complemented
406	C₂×C₃₁	—	2	6	no	—
407	C₂×C₃₁	—	4	3	no	—
408	C₂×C₃₁	—	4	1	no	—
409	C₂×C₃₁	—	8	3	no	—
410	C₂×C₃₁	C₂²×#123	1	1	yes	characteristic
411	C₂×C₃₁	#3584	1	1	yes	characteristic
412	C₂×C₃₁	#3585	1	1	yes	characteristic
413	H(5,𝔽₂)	—	2	4	no	—
414	H(5,𝔽₂)	—	2	4	no	—
415	C₂×#41	—	2	4	no	—
416	C₂×#41	—	2	4	no	—
417	C₂×#41	—	2	1	no	—
418	C₂×#41	D₄×C₃₁	1	3	yes	—
419	C₂×#41	D₄×C₃₁	1	1	yes	characteristic
420	C₂×#41	C₂³×C₃₁	1	3	yes	—
421	C₂×#41	C₂³×C₃₁	1	3	yes	—
422	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
423	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
424	C₂²×D₄	—	2	3	no	—
425	C₂²×D₄	—	2	3	no	—
426	C₂²×D₄	D₄×C₃₁	1	3	yes	—
427	C₂²×D₄	C₂³×C₃₁	1	3	yes	—
428	C₂³×C₄	C₂³×C₃₁	1	3	yes	—
429	#131	—	2	4	no	—
430	#131	—	2	4	no	—
431	C₂×M₁₆	C₂³×C₃₁	1	3	yes	—
432	C₂×M₁₆	C₂³×C₃₁	1	3	yes	—
433	C₂²×C₈	C₂³×C₃₁	1	3	yes	—
434	C₂²×C₈	C₂³×C₃₁	1	3	yes	—
435	#127	—	2	3	no	—
436	#124	—	2	3	no	—
437	#123	—	2	3	no	—
438	C₂²⋊Q₈	—	2	3	no	—
439	C₂²⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
440	C₂²⋊Q₈	C₂³×C₃₁	1	3	yes	—
441	#121	—	2	6	no	—
442	#121	—	2	3	no	—
443	#121	—	2	3	no	—
444	#121	—	2	3	no	—
445	#121	C₂³×C₃₁	1	3	yes	—
446	#121	C₂³×C₃₁	1	6	yes	—
447	#121	C₂³×C₃₁	1	3	yes	—
448	#120	—	2	3	no	—
449	#120	—	2	6	no	—
450	C₄×Q₈	—	2	1	no	—
451	C₄×Q₈	—	2	1	no	—
452	C₄×Q₈	—	2	3	no	—
453	C₄×Q₈	—	2	3	no	—
454	C₄×Q₈	—	2	1	no	—
455	C₄×D₄	—	2	3	no	—
456	C₄×D₄	—	2	3	no	—
457	C₄×D₄	—	2	6	no	—
458	C₄×D₄	—	2	6	no	—
459	C₄×D₄	—	2	3	no	—
460	C₄×D₄	—	2	3	no	—
461	C₄×D₄	—	2	3	no	—
462	C₄×D₄	—	2	3	no	—
463	C₄×D₄	—	2	3	no	—
464	#117	—	2	3	no	—
465	#117	—	2	3	no	—
466	#117	—	2	3	no	—
467	#117	—	2	3	no	—
468	#117	C₂³×C₃₁	1	3	yes	—
469	#117	C₂³×C₃₁	1	1	yes	characteristic
470	#117	C₂³×C₃₁	1	3	yes	—
471	#117	C₂³×C₃₁	1	3	yes	—
472	#117	C₂³×C₃₁	1	6	yes	—
473	#117	C₂³×C₃₁	1	3	yes	—
474	C₂×C₄⋊C₄	C₂³×C₃₁	1	3	yes	—
475	C₂×C₄⋊C₄	C₂³×C₃₁	1	3	yes	—
476	C₂×C₄⋊C₄	C₂³×C₃₁	1	3	yes	—
477	C₂×C₄⋊C₄	C₂³×C₃₁	1	3	yes	—
478	C₂×C₄⋊C₄	C₂³×C₃₁	1	3	yes	—
479	C₂×C₂²⋊C₄	—	2	3	no	—
480	C₂×C₂²⋊C₄	—	2	3	no	—
481	C₂×C₂²⋊C₄	—	2	3	no	—
482	C₂×C₄²	—	2	3	no	—
483	C₂×C₄²	—	2	3	no	—
484	#107	C₂³×C₃₁	1	3	yes	—
485	#107	C₂³×C₃₁	1	6	yes	—
486	#107	C₂³×C₃₁	1	1	yes	characteristic
487	#107	C₂³×C₃₁	1	3	yes	—
488	#107	C₂³×C₃₁	1	3	yes	—
489	#106	C₂³×C₃₁	1	3	yes	—
490	#106	C₂³×C₃₁	1	6	yes	—
491	#106	C₂³×C₃₁	1	1	yes	characteristic
492	#106	C₂³×C₃₁	1	3	yes	—
493	#106	C₂³×C₃₁	1	3	yes	—
494	#103	C₂³×C₃₁	1	1	yes	characteristic
495	#103	C₂³×C₃₁	1	1	yes	characteristic
496	#103	C₂³×C₃₁	1	3	yes	—
497	#103	C₂³×C₃₁	1	3	yes	—
498	#102	C₂³×C₃₁	1	3	yes	—
499	#102	C₂³×C₃₁	1	3	yes	—
500	#102	C₂³×C₃₁	1	6	yes	—
501	#102	C₂³×C₃₁	1	6	yes	—
502	#102	C₂³×C₃₁	1	3	yes	—
503	#102	C₂³×C₃₁	1	3	yes	—
504	C₂²⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
505	C₂²⋊C₈	C₂³×C₃₁	1	3	yes	—
506	C₂²⋊C₈	C₂³×C₃₁	1	3	yes	—
507	C₂²⋊C₈	C₂³×C₃₁	1	3	yes	—
508	C₂²⋊C₈	C₂³×C₃₁	1	6	yes	—
509	C₃₁	#60523	1	1	yes	characteristic, direct product factor, Sylow
510	C₂⁴	—	2	3	no	—
511	C₂⁴	C₂×D₄×C₃₁	1	3	yes	—
512	#41	—	2	4	no	—
513	#41	—	2	12	no	—
514	#41	—	2	6	no	—
515	#41	—	2	6	no	—
516	#41	—	2	6	no	—
517	#41	—	2	6	no	—
518	#41	—	4	4	no	—
519	#41	#41×C₃₁	1	4	yes	—
520	#41	#41×C₃₁	1	4	yes	—
521	C₂×Q₈	—	2	4	no	—
522	C₂×Q₈	C₂×D₄×C₃₁	1	3	yes	—
523	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
524	C₂×Q₈	C₂⁴×C₃₁	1	1	yes	characteristic
525	C₂×D₄	—	2	12	no	—
526	C₂×D₄	—	2	3	no	—
527	C₂×D₄	—	2	3	no	—
528	C₂×D₄	—	2	3	no	—
529	C₂×D₄	—	2	6	no	—
530	C₂×D₄	—	2	3	no	—
531	C₂×D₄	—	2	6	no	—
532	C₂×D₄	—	2	6	no	—
533	C₂×D₄	—	4	12	no	—
534	C₂×D₄	—	8	6	no	—
535	C₂×D₄	C₂×D₄×C₃₁	1	6	yes	—
536	C₂×D₄	C₂×D₄×C₃₁	1	3	yes	—
537	C₂×D₄	C₂×D₄×C₃₁	1	3	yes	—
538	C₂×D₄	C₂⁴×C₃₁	1	3	yes	—
539	C₂²×C₄	—	2	6	no	—
540	C₂²×C₄	—	2	6	no	—
541	C₂²×C₄	—	2	3	no	—
542	C₂²×C₄	—	2	3	no	—
543	C₂²×C₄	—	2	3	no	—
544	C₂²×C₄	—	2	3	no	—
545	C₂²×C₄	—	2	3	no	—
546	C₂²×C₄	—	2	3	no	—
547	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
548	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
549	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
550	C₂²×C₄	C₂⁴×C₃₁	1	3	yes	—
551	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
552	C₂²×C₄	C₂⁴×C₃₁	1	3	yes	—
553	M₁₆	—	2	6	no	—
554	M₁₆	—	2	6	no	—
555	C₂×C₈	—	2	6	no	—
556	C₂×C₈	—	2	6	no	—
557	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
558	C₂×C₈	C₂⁴×C₃₁	1	3	yes	—
559	C₂×C₈	C₂⁴×C₃₁	1	3	yes	—
560	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
561	C₄⋊C₄	—	2	3	no	—
562	C₄⋊C₄	—	2	3	no	—
563	C₄⋊C₄	—	2	3	no	—
564	C₄⋊C₄	—	2	3	no	—
565	C₄⋊C₄	—	2	3	no	—
566	C₄⋊C₄	—	2	1	no	—
567	C₄⋊C₄	—	2	3	no	—
568	C₄⋊C₄	C₂⁴×C₃₁	1	3	yes	—
569	C₄⋊C₄	C₂⁴×C₃₁	1	1	yes	characteristic
570	C₄⋊C₄	C₂⁴×C₃₁	1	3	yes	—
571	C₄⋊C₄	C₂⁴×C₃₁	1	1	yes	characteristic
572	C₄⋊C₄	C₂⁴×C₃₁	1	3	yes	—
573	C₄⋊C₄	C₂⁴×C₃₁	1	1	yes	characteristic
574	C₂²⋊C₄	—	2	3	no	—
575	C₂²⋊C₄	—	2	1	no	—
576	C₂²⋊C₄	—	2	6	no	—
577	C₂²⋊C₄	—	2	3	no	—
578	C₂²⋊C₄	—	2	3	no	—
579	C₂²⋊C₄	—	2	3	no	—
580	C₂²⋊C₄	—	2	1	no	—
581	C₂²⋊C₄	—	2	3	no	—
582	C₂²⋊C₄	—	2	3	no	—
583	C₂²⋊C₄	—	2	6	no	—
584	C₂²⋊C₄	—	2	3	no	—
585	C₄²	—	2	3	no	—
586	C₄²	—	2	1	no	—
587	C₄²	—	2	6	no	—
588	C₄²	—	2	3	no	—
589	C₄²	—	2	3	no	—
590	C₄²	—	2	3	no	—
591	C₂³	—	2	6	no	—
592	C₂³	—	2	3	no	—
593	C₂³	—	4	3	no	—
594	C₂³	—	4	3	no	—
595	C₂³	—	4	6	no	—
596	C₂³	—	8	3	no	—
597	C₂³	—	8	3	no	—
598	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
599	C₂³	C₂²×D₄×C₃₁	1	3	yes	—
600	C₂³	C₂²×D₄×C₃₁	1	3	yes	—
601	Q₈	—	2	6	no	—
602	Q₈	—	2	1	no	—
603	Q₈	—	2	1	no	—
604	Q₈	C₃₁×#123	1	4	yes	—
605	D₄	—	2	12	no	—
606	D₄	—	2	6	no	—
607	D₄	—	2	3	no	—
608	D₄	—	2	3	no	—
609	D₄	—	4	6	no	—
610	D₄	—	4	6	no	—
611	D₄	—	4	6	no	—
612	D₄	—	8	3	no	—
613	D₄	—	8	6	no	—
614	D₄	C₃₁×#123	1	12	yes	—
615	C₂×C₄	—	2	6	no	—
616	C₂×C₄	—	2	6	no	—
617	C₂×C₄	—	2	6	no	—
618	C₂×C₄	—	2	1	no	—
619	C₂×C₄	—	2	1	no	—
620	C₂×C₄	—	2	3	no	—
621	C₂×C₄	—	2	3	no	—
622	C₂×C₄	—	2	3	no	—
623	C₂×C₄	—	2	1	no	—
624	C₂×C₄	—	2	3	no	—
625	C₂×C₄	—	2	1	no	—
626	C₂×C₄	—	2	1	no	—
627	C₂×C₄	—	2	3	no	—
628	C₂×C₄	—	2	1	no	—
629	C₂×C₄	—	4	6	no	—
630	C₂×C₄	—	8	3	no	—
631	C₂×C₄	—	8	3	no	—
632	C₂×C₄	—	8	3	no	—
633	C₂×C₄	—	8	3	no	—
634	C₂×C₄	—	8	3	no	—
635	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
636	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
637	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
638	C₂×C₄	C₂×#41×C₃₁	1	6	yes	—
639	C₂×C₄	C₂×#41×C₃₁	1	6	yes	—
640	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
641	C₈	—	2	1	no	—
642	C₈	—	2	1	no	—
643	C₈	—	2	3	no	—
644	C₈	—	2	3	no	—
645	C₂²	—	2	3	no	—
646	C₂²	—	2	3	no	—
647	C₂²	—	2	1	no	—
648	C₂²	—	2	3	no	—
649	C₂²	—	2	3	no	—
650	C₂²	—	2	1	no	—
651	C₂²	—	2	3	no	—
652	C₂²	—	2	1	no	—
653	C₂²	—	4	3	no	—
654	C₂²	—	4	3	no	—
655	C₂²	—	4	3	no	—
656	C₂²	—	8	6	no	—
657	C₂²	—	16	6	no	—
658	C₂²	C₂×C₃₁×#123	1	6	yes	—
659	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
660	C₄	—	2	1	no	—
661	C₄	—	2	3	no	—
662	C₄	—	4	1	no	—
663	C₄	—	4	3	no	—
664	C₄	—	4	1	no	—
665	C₄	—	4	1	no	—
666	C₄	—	4	3	no	—
667	C₄	—	4	1	no	—
668	C₄	—	4	3	no	—
669	C₄	C₂×C₃₁×#123	1	6	yes	—
670	C₄	C₂²×#41×C₃₁	1	1	yes	characteristic
671	C₄	C₂²×#41×C₃₁	1	1	yes	characteristic
672	C₂	—	2	6	no	—
673	C₂	—	4	1	no	—
674	C₂	—	4	3	no	—
675	C₂	—	8	3	no	—
676	C₂	C₂²×C₃₁×#123	1	1	yes	characteristic
677	C₂	C₃₁×#3584	1	1	yes	characteristic
678	C₂	C₃₁×#3585	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
prime radical	C₃₁×#3488	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂²×#41×C₃₁ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₃₁×#3428	C₂	semidirect
3. C₃₁×#3099	C₂	semidirect
4. C₃₁×#3098	C₂	semidirect
5. C₃₁×#3096	C₂	semidirect
6. C₃₁×#3096	C₂	semidirect
7. C₃₁×#3094	C₂	semidirect
8. C₃₁×#3093	C₂	semidirect
9. C₃₁×#3091	C₂	semidirect
10. C₂×C₃₁×#482	C₂	semidirect
11. C₃₁×#2913	C₂	semidirect
12. C₃₁×#2912	C₂	semidirect
13. C₃₁×#2897	C₂	semidirect
14. C₃₁×#2897	C₂	semidirect
15. C₃₁×#2880	C₂	semidirect
31. C₃₁×#520	C₂²	semidirect
32. C₃₁×#520	C₂²	semidirect
37. C₂×C₃₁×#117	C₂²	semidirect
38. C₂×C₃₁×#117	C₂²	semidirect
39. C₃₁×#485	C₂²	semidirect
40. C₃₁×#485	C₂²	semidirect
41. C₃₁×#485	C₂²	semidirect
42. C₃₁×#484	C₂²	semidirect
43. C₃₁×#483	C₂²	semidirect
44. C₃₁×#482	C₂²	semidirect
45. C₃₁×#482	C₂²	semidirect
46. C₃₁×#482	C₂²	semidirect
47. C₃₁×#482	C₂²	semidirect
48. C₃₁×#482	C₂²	semidirect
49. C₃₁×#482	C₂²	semidirect
50. C₃₁×#481	C₂²	semidirect
51. C₃₁×#481	C₂²	semidirect
52. C₃₁×#480	C₂²	semidirect
53. C₃₁×#480	C₂²	semidirect
54. C₃₁×#428	C₂²	semidirect
55. C₃₁×#428	C₂²	semidirect
56. C₃₁×#428	C₂²	semidirect
57. C₃₁×#428	C₂²	semidirect
58. C₃₁×#427	C₂²	semidirect
59. C₃₁×#427	C₂²	semidirect
60. C₂×C₃₁×#107	C₂²	semidirect
61. C₂×C₃₁×#106	C₂²	semidirect
62. C₃₁×#418	C₂²	semidirect
63. C₃₁×#418	C₂²	semidirect
64. C₃₁×#417	C₂²	semidirect
65. C₃₁×#417	C₂²	semidirect
66. C₃₁×#416	C₂²	semidirect
67. C₃₁×#416	C₂²	semidirect
68. C₂×C₃₁×#102	C₂²	semidirect
69. C₂×C₃₁×#102	C₂²	semidirect
70. C₃₁×#408	C₂²	semidirect
71. C₃₁×#408	C₂²	semidirect
72. C₃₁×#407	C₂²	semidirect
73. C₂×C₃₁×C₂²⋊C₈	C₂²	semidirect
251. #60523	C₃₁	direct (N × H)
307. #3428	C₂×C₃₁	semidirect
308. #3099	C₂×C₃₁	semidirect
309. #3098	C₂×C₃₁	semidirect
310. #3096	C₂×C₃₁	semidirect
311. #3096	C₂×C₃₁	semidirect
312. #3094	C₂×C₃₁	semidirect
313. #3093	C₂×C₃₁	semidirect
314. #3091	C₂×C₃₁	semidirect
315. C₂×#482	C₂×C₃₁	semidirect
316. #2913	C₂×C₃₁	semidirect
317. #2912	C₂×C₃₁	semidirect
318. #2897	C₂×C₃₁	semidirect
319. #2897	C₂×C₃₁	semidirect
320. #2880	C₂×C₃₁	semidirect
363. #520	C₂²×C₃₁	semidirect
364. #520	C₂²×C₃₁	semidirect
369. C₂×#117	C₂²×C₃₁	semidirect
370. C₂×#117	C₂²×C₃₁	semidirect
371. #485	C₂²×C₃₁	semidirect
372. #485	C₂²×C₃₁	semidirect
373. #485	C₂²×C₃₁	semidirect
374. #484	C₂²×C₃₁	semidirect
375. #483	C₂²×C₃₁	semidirect
376. #482	C₂²×C₃₁	semidirect
377. #482	C₂²×C₃₁	semidirect
378. #482	C₂²×C₃₁	semidirect
379. #482	C₂²×C₃₁	semidirect
380. #482	C₂²×C₃₁	semidirect
381. #482	C₂²×C₃₁	semidirect
382. #481	C₂²×C₃₁	semidirect
383. #481	C₂²×C₃₁	semidirect
384. #480	C₂²×C₃₁	semidirect
385. #480	C₂²×C₃₁	semidirect
386. #428	C₂²×C₃₁	semidirect
387. #428	C₂²×C₃₁	semidirect
388. #428	C₂²×C₃₁	semidirect
389. #428	C₂²×C₃₁	semidirect
390. #427	C₂²×C₃₁	semidirect
391. #427	C₂²×C₃₁	semidirect
392. C₂×#107	C₂²×C₃₁	semidirect
393. C₂×#106	C₂²×C₃₁	semidirect
394. #418	C₂²×C₃₁	semidirect
395. #418	C₂²×C₃₁	semidirect
396. #417	C₂²×C₃₁	semidirect
397. #417	C₂²×C₃₁	semidirect
398. #416	C₂²×C₃₁	semidirect
399. #416	C₂²×C₃₁	semidirect
400. C₂×#102	C₂²×C₃₁	semidirect
401. C₂×#102	C₂²×C₃₁	semidirect
402. #408	C₂²×C₃₁	semidirect
403. #408	C₂²×C₃₁	semidirect
404. #407	C₂²×C₃₁	semidirect
405. C₂×C₂²⋊C₈	C₂²×C₃₁	semidirect
509. C₃₁	#60523	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃₁×#3488	2
C₃₁×#3428	2
C₃₁×#3099	2
C₃₁×#3098	2
C₃₁×#3096	2
C₃₁×#3094	2
C₃₁×#3093	2
C₃₁×#3091	2
C₂×C₃₁×#482	2
C₃₁×#2913	2
C₃₁×#2912	2
C₃₁×#2897	2
C₃₁×#2880	2
#60523	31

Subgroups — C31×#60523 #21517282

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#60523 #21517282