Subgroups — C₃₁×#60521 #21517280

Order 7936.

2234 conjugacy classes of subgroups.

The subgroup lattice has 536 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	C₂×C₃₁×#520	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3098	C₂	1	6	yes	maximal, complemented
4	C₃₁×#3096	C₂	1	6	yes	maximal, complemented
5	C₃₁×#3094	C₂	1	3	yes	maximal, complemented
6	C₃₁×#3092	C₂	1	3	yes	maximal, complemented
7	C₃₁×#3091	C₂	1	3	yes	maximal, complemented
8	C₂×C₃₁×#480	C₂	1	3	yes	maximal, complemented
9	C₃₁×#2913	C₂	1	2	yes	maximal, complemented
10	C₃₁×#2897	C₂	1	2	yes	maximal, complemented
11	C₃₁×#2880	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×H(5,𝔽₂)	C₂²	1	1	yes	characteristic
13	C₂²×#41×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×#131	C₂²	1	2	yes	—
15	C₃₁×#550	—	2	1	no	—
16	C₃₁×#548	—	2	1	no	—
17	C₃₁×#546	—	2	3	no	—
18	D₄²×C₃₁	—	2	3	no	—
19	C₃₁×#537	C₂²	1	3	yes	—
20	C₃₁×#535	C₂²	1	3	yes	—
21	C₃₁×#525	C₂²	1	3	yes	—
22	C₂×C₃₁×#121	C₂²	1	3	yes	—
23	C₃₁×#520	—	2	3	no	—
24	C₃₁×#520	—	2	1	no	—
25	C₃₁×#520	C₂²	1	2	yes	—
26	C₃₁×#520	C₂²	1	1	yes	characteristic
27	C₃₁×#520	C₂²	1	6	yes	complemented
28	C₂×C₄×Q₈×C₃₁	—	2	1	no	—
29	C₂×C₄×D₄×C₃₁	—	2	3	no	—
30	C₂×C₃₁×#117	C₂²	1	3	yes	complemented
31	C₂²×C₄⋊C₄×C₃₁	C₂²	1	3	yes	complemented
32	C₃₁×#485	C₂²	1	6	yes	complemented
33	C₃₁×#484	C₂²	1	1	yes	characteristic, complemented
34	C₃₁×#484	C₂²	1	6	yes	complemented
35	C₃₁×#483	C₂²	1	3	yes	complemented
36	C₃₁×#482	C₂²	1	6	yes	complemented
37	C₃₁×#482	C₂²	1	12	yes	complemented
38	C₃₁×#481	C₂²	1	3	yes	complemented
39	C₃₁×#481	C₂²	1	6	yes	complemented
40	C₃₁×#480	C₂²	1	3	yes	complemented
41	C₃₁×#480	C₂²	1	6	yes	complemented
42	C₃₁×#480	C₂²	1	12	yes	complemented
43	C₃₁×#428	C₂²	1	6	yes	complemented
44	C₃₁×#428	C₂²	1	6	yes	complemented
45	C₃₁×#427	C₂²	1	6	yes	complemented
46	C₂×C₃₁×#107	C₂²	1	6	yes	complemented
47	C₃₁×#418	C₂²	1	6	yes	complemented
48	C₃₁×#417	C₂²	1	6	yes	complemented
49	C₃₁×#416	C₂²	1	6	yes	complemented
50	C₂×C₃₁×#102	C₂²	1	6	yes	complemented
51	C₃₁×#408	C₂²	1	6	yes	complemented
52	C₃₁×#407	C₂²	1	3	yes	complemented
53	C₂×C₃₁×C₂²⋊C₈	C₂²	1	3	yes	complemented
54	C₃₁×H(5,𝔽₂)	—	2	8	no	—
55	C₂×#41×C₃₁	—	2	8	no	—
56	C₂×#41×C₃₁	—	2	1	no	—
57	C₂×#41×C₃₁	D₄	1	3	yes	—
58	C₂×#41×C₃₁	D₄	1	1	yes	characteristic
59	C₂×#41×C₃₁	C₂³	1	3	yes	—
60	C₂×#41×C₃₁	C₂³	1	3	yes	—
61	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
62	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
63	C₂²×D₄×C₃₁	—	2	3	no	—
64	C₂²×D₄×C₃₁	—	2	3	no	—
65	C₂²×D₄×C₃₁	D₄	1	3	yes	—
66	C₂²×D₄×C₃₁	C₂³	1	3	yes	—
67	C₂³×C₄×C₃₁	—	2	3	no	—
68	C₂³×C₄×C₃₁	C₂³	1	3	yes	—
69	C₃₁×#131	—	2	8	no	—
70	C₂×M₁₆×C₃₁	C₂³	1	6	yes	—
71	C₂²×C₈×C₃₁	C₂³	1	6	yes	—
72	C₃₁×#127	—	2	3	no	—
73	C₃₁×#124	—	2	3	no	—
74	C₃₁×#123	—	2	3	no	—
75	C₃₁×C₂²⋊Q₈	—	2	3	no	—
76	C₃₁×C₂²⋊Q₈	C₂³	1	1	yes	characteristic
77	C₃₁×C₂²⋊Q₈	C₂³	1	3	yes	—
78	C₃₁×#121	—	2	6	no	—
79	C₃₁×#121	—	2	3	no	—
80	C₃₁×#121	—	2	3	no	—
81	C₃₁×#121	—	2	3	no	—
82	C₃₁×#121	C₂³	1	3	yes	—
83	C₃₁×#121	C₂³	1	6	yes	—
84	C₃₁×#121	C₂³	1	3	yes	—
85	C₃₁×#120	—	2	3	no	—
86	C₃₁×#120	—	2	6	no	—
87	C₄×Q₈×C₃₁	—	2	2	no	—
88	C₄×Q₈×C₃₁	—	2	6	no	—
89	C₄×Q₈×C₃₁	—	2	1	no	—
90	C₄×D₄×C₃₁	—	2	6	no	—
91	C₄×D₄×C₃₁	—	2	12	no	—
92	C₄×D₄×C₃₁	—	2	6	no	—
93	C₄×D₄×C₃₁	—	2	3	no	—
94	C₄×D₄×C₃₁	—	2	3	no	—
95	C₄×D₄×C₃₁	—	2	3	no	—
96	C₃₁×#117	—	2	3	no	—
97	C₃₁×#117	—	2	3	no	—
98	C₃₁×#117	C₂³	1	6	yes	—
99	C₃₁×#117	C₂³	1	3	yes	—
100	C₃₁×#117	C₂³	1	6	yes	—
101	C₂×C₄⋊C₄×C₃₁	—	2	3	no	—
102	C₂×C₄⋊C₄×C₃₁	—	2	3	no	—
103	C₂×C₄⋊C₄×C₃₁	—	2	3	no	—
104	C₂×C₄⋊C₄×C₃₁	C₂³	1	6	yes	—
105	C₂×C₄⋊C₄×C₃₁	C₂³	1	3	yes	—
106	C₂×C₄⋊C₄×C₃₁	C₂³	1	3	yes	—
107	C₂×C₄⋊C₄×C₃₁	C₂³	1	6	yes	—
108	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
109	C₂×C₂²⋊C₄×C₃₁	—	2	3	no	—
110	C₂×C₂²⋊C₄×C₃₁	—	2	3	no	—
111	C₂×C₄²×C₃₁	—	2	3	no	—
112	C₃₁×#107	C₂³	1	6	yes	—
113	C₃₁×#107	C₂³	1	12	yes	—
114	C₃₁×#107	C₂³	1	2	yes	—
115	C₃₁×#106	C₂³	1	6	yes	—
116	C₃₁×#106	C₂³	1	6	yes	—
117	C₃₁×#103	C₂³	1	2	yes	—
118	C₃₁×#103	C₂³	1	6	yes	—
119	C₃₁×#102	C₂³	1	6	yes	—
120	C₃₁×#102	C₂³	1	12	yes	—
121	C₃₁×#102	C₂³	1	6	yes	—
122	C₃₁×C₂²⋊C₈	C₂³	1	1	yes	characteristic
123	C₃₁×C₂²⋊C₈	C₂³	1	3	yes	—
124	C₃₁×C₂²⋊C₈	C₂³	1	6	yes	—
125	C₃₁×C₂²⋊C₈	C₂³	1	6	yes	—
126	C₂⁴×C₃₁	—	2	3	no	—
127	C₂⁴×C₃₁	C₂×D₄	1	3	yes	—
128	#41×C₃₁	—	2	4	no	—
129	#41×C₃₁	—	2	12	no	—
130	#41×C₃₁	—	2	12	no	—
131	#41×C₃₁	—	2	12	no	—
132	#41×C₃₁	—	4	4	no	—
133	#41×C₃₁	#41	1	8	yes	—
134	C₂×Q₈×C₃₁	—	2	4	no	—
135	C₂×Q₈×C₃₁	C₂×D₄	1	3	yes	—
136	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
137	C₂×Q₈×C₃₁	C₂⁴	1	1	yes	characteristic
138	C₂×D₄×C₃₁	—	2	12	no	—
139	C₂×D₄×C₃₁	—	2	3	no	—
140	C₂×D₄×C₃₁	—	2	3	no	—
141	C₂×D₄×C₃₁	—	2	3	no	—
142	C₂×D₄×C₃₁	—	2	6	no	—
143	C₂×D₄×C₃₁	—	2	3	no	—
144	C₂×D₄×C₃₁	—	2	12	no	—
145	C₂×D₄×C₃₁	—	4	12	no	—
146	C₂×D₄×C₃₁	—	8	6	no	—
147	C₂×D₄×C₃₁	C₂×D₄	1	6	yes	—
148	C₂×D₄×C₃₁	C₂×D₄	1	3	yes	—
149	C₂×D₄×C₃₁	C₂×D₄	1	3	yes	—
150	C₂×D₄×C₃₁	C₂⁴	1	3	yes	—
151	C₂²×C₄×C₃₁	—	2	12	no	—
152	C₂²×C₄×C₃₁	—	2	1	no	—
153	C₂²×C₄×C₃₁	—	2	3	no	—
154	C₂²×C₄×C₃₁	—	2	3	no	—
155	C₂²×C₄×C₃₁	—	2	6	no	—
156	C₂²×C₄×C₃₁	—	2	6	no	—
157	C₂²×C₄×C₃₁	—	2	3	no	—
158	C₂²×C₄×C₃₁	—	8	6	no	—
159	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
160	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
161	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
162	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
163	C₂²×C₄×C₃₁	C₂⁴	1	3	yes	—
164	C₂²×C₄×C₃₁	C₂⁴	1	3	yes	—
165	M₁₆×C₃₁	—	2	12	no	—
166	C₂×C₈×C₃₁	—	2	12	no	—
167	C₂×C₈×C₃₁	C₂⁴	1	6	yes	—
168	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
169	C₄⋊C₄×C₃₁	—	2	3	no	—
170	C₄⋊C₄×C₃₁	—	2	6	no	—
171	C₄⋊C₄×C₃₁	—	2	6	no	—
172	C₄⋊C₄×C₃₁	—	2	1	no	—
173	C₄⋊C₄×C₃₁	—	2	3	no	—
174	C₄⋊C₄×C₃₁	—	2	1	no	—
175	C₄⋊C₄×C₃₁	—	2	3	no	—
176	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
177	C₄⋊C₄×C₃₁	C₂⁴	1	6	yes	—
178	C₄⋊C₄×C₃₁	C₂⁴	1	1	yes	characteristic
179	C₄⋊C₄×C₃₁	C₂⁴	1	3	yes	—
180	C₂²⋊C₄×C₃₁	—	2	6	no	—
181	C₂²⋊C₄×C₃₁	—	2	6	no	—
182	C₂²⋊C₄×C₃₁	—	2	3	no	—
183	C₂²⋊C₄×C₃₁	—	2	1	no	—
184	C₂²⋊C₄×C₃₁	—	2	3	no	—
185	C₂²⋊C₄×C₃₁	—	2	3	no	—
186	C₂²⋊C₄×C₃₁	—	2	6	no	—
187	C₂²⋊C₄×C₃₁	—	2	3	no	—
188	C₄²×C₃₁	—	2	6	no	—
189	C₄²×C₃₁	—	2	6	no	—
190	C₄²×C₃₁	—	2	3	no	—
191	#60521	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
192	C₂³×C₃₁	—	2	6	no	—
193	C₂³×C₃₁	—	2	6	no	—
194	C₂³×C₃₁	—	2	3	no	—
195	C₂³×C₃₁	—	4	3	no	—
196	C₂³×C₃₁	—	4	6	no	—
197	C₂³×C₃₁	—	8	3	no	—
198	C₂³×C₃₁	—	8	3	no	—
199	C₂³×C₃₁	C₂²×D₄	1	3	yes	—
200	C₂³×C₃₁	C₂²×D₄	1	3	yes	—
201	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
202	Q₈×C₃₁	—	2	6	no	—
203	Q₈×C₃₁	—	2	2	no	—
204	Q₈×C₃₁	#123	1	4	yes	—
205	D₄×C₃₁	—	2	12	no	—
206	D₄×C₃₁	—	2	6	no	—
207	D₄×C₃₁	—	2	6	no	—
208	D₄×C₃₁	—	4	6	no	—
209	D₄×C₃₁	—	4	6	no	—
210	D₄×C₃₁	—	4	6	no	—
211	D₄×C₃₁	—	8	3	no	—
212	D₄×C₃₁	—	8	6	no	—
213	D₄×C₃₁	#123	1	12	yes	—
214	C₂×C₄×C₃₁	—	2	6	no	—
215	C₂×C₄×C₃₁	—	2	6	no	—
216	C₂×C₄×C₃₁	—	2	6	no	—
217	C₂×C₄×C₃₁	—	2	2	no	—
218	C₂×C₄×C₃₁	—	2	6	no	—
219	C₂×C₄×C₃₁	—	2	2	no	—
220	C₂×C₄×C₃₁	—	2	1	no	—
221	C₂×C₄×C₃₁	—	2	6	no	—
222	C₂×C₄×C₃₁	—	2	3	no	—
223	C₂×C₄×C₃₁	—	2	1	no	—
224	C₂×C₄×C₃₁	—	4	2	no	—
225	C₂×C₄×C₃₁	—	4	6	no	—
226	C₂×C₄×C₃₁	—	4	6	no	—
227	C₂×C₄×C₃₁	—	8	6	no	—
228	C₂×C₄×C₃₁	—	8	6	no	—
229	C₂×C₄×C₃₁	—	8	3	no	—
230	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
231	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
232	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
233	C₂×C₄×C₃₁	C₂×#41	1	12	yes	—
234	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
235	C₈×C₃₁	—	2	2	no	—
236	C₈×C₃₁	—	2	6	no	—
237	#3488	C₂×C₃₁	1	1	yes	characteristic
238	C₂×#520	C₂×C₃₁	1	1	yes	characteristic, complemented
239	#3098	C₂×C₃₁	1	6	yes	complemented
240	#3096	C₂×C₃₁	1	6	yes	complemented
241	#3094	C₂×C₃₁	1	3	yes	complemented
242	#3092	C₂×C₃₁	1	3	yes	complemented
243	#3091	C₂×C₃₁	1	3	yes	complemented
244	C₂×#480	C₂×C₃₁	1	3	yes	complemented
245	#2913	C₂×C₃₁	1	2	yes	complemented
246	#2897	C₂×C₃₁	1	2	yes	complemented
247	#2880	C₂×C₃₁	1	1	yes	characteristic, complemented
248	C₂²×C₃₁	—	2	1	no	—
249	C₂²×C₃₁	—	2	3	no	—
250	C₂²×C₃₁	—	2	1	no	—
251	C₂²×C₃₁	—	2	3	no	—
252	C₂²×C₃₁	—	2	3	no	—
253	C₂²×C₃₁	—	2	3	no	—
254	C₂²×C₃₁	—	2	3	no	—
255	C₂²×C₃₁	—	4	12	no	—
256	C₂²×C₃₁	—	4	3	no	—
257	C₂²×C₃₁	—	4	3	no	—
258	C₂²×C₃₁	—	4	3	no	—
259	C₂²×C₃₁	—	16	6	no	—
260	C₂²×C₃₁	C₂×#123	1	6	yes	—
261	C₂²×C₃₁	#576	1	2	yes	—
262	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
263	C₄×C₃₁	—	2	1	no	—
264	C₄×C₃₁	—	2	3	no	—
265	C₄×C₃₁	—	4	2	no	—
266	C₄×C₃₁	—	4	1	no	—
267	C₄×C₃₁	—	4	1	no	—
268	C₄×C₃₁	—	4	6	no	—
269	C₄×C₃₁	—	4	3	no	—
270	C₄×C₃₁	C₂×#123	1	6	yes	—
271	C₄×C₃₁	C₂²×#41	1	2	yes	—
272	C₂×H(5,𝔽₂)	C₂²×C₃₁	1	1	yes	characteristic
273	C₂²×#41	C₂²×C₃₁	1	1	yes	characteristic
274	C₂×#131	C₂²×C₃₁	1	2	yes	—
275	#550	—	2	1	no	—
276	#548	—	2	1	no	—
277	#546	—	2	3	no	—
278	D₄²	—	2	3	no	—
279	#537	C₂²×C₃₁	1	3	yes	—
280	#535	C₂²×C₃₁	1	3	yes	—
281	#525	C₂²×C₃₁	1	3	yes	—
282	C₂×#121	C₂²×C₃₁	1	3	yes	—
283	#520	—	2	3	no	—
284	#520	—	2	1	no	—
285	#520	C₂²×C₃₁	1	2	yes	—
286	#520	C₂²×C₃₁	1	1	yes	characteristic
287	#520	C₂²×C₃₁	1	6	yes	complemented
288	C₂×C₄×Q₈	—	2	1	no	—
289	C₂×C₄×D₄	—	2	3	no	—
290	C₂×#117	C₂²×C₃₁	1	3	yes	complemented
291	C₂²×C₄⋊C₄	C₂²×C₃₁	1	3	yes	complemented
292	#485	C₂²×C₃₁	1	6	yes	complemented
293	#484	C₂²×C₃₁	1	1	yes	characteristic, complemented
294	#484	C₂²×C₃₁	1	6	yes	complemented
295	#483	C₂²×C₃₁	1	3	yes	complemented
296	#482	C₂²×C₃₁	1	6	yes	complemented
297	#482	C₂²×C₃₁	1	12	yes	complemented
298	#481	C₂²×C₃₁	1	3	yes	complemented
299	#481	C₂²×C₃₁	1	6	yes	complemented
300	#480	C₂²×C₃₁	1	3	yes	complemented
301	#480	C₂²×C₃₁	1	6	yes	complemented
302	#480	C₂²×C₃₁	1	12	yes	complemented
303	#428	C₂²×C₃₁	1	6	yes	complemented
304	#428	C₂²×C₃₁	1	6	yes	complemented
305	#427	C₂²×C₃₁	1	6	yes	complemented
306	C₂×#107	C₂²×C₃₁	1	6	yes	complemented
307	#418	C₂²×C₃₁	1	6	yes	complemented
308	#417	C₂²×C₃₁	1	6	yes	complemented
309	#416	C₂²×C₃₁	1	6	yes	complemented
310	C₂×#102	C₂²×C₃₁	1	6	yes	complemented
311	#408	C₂²×C₃₁	1	6	yes	complemented
312	#407	C₂²×C₃₁	1	3	yes	complemented
313	C₂×C₂²⋊C₈	C₂²×C₃₁	1	3	yes	complemented
314	C₂×C₃₁	—	2	2	no	—
315	C₂×C₃₁	—	2	6	no	—
316	C₂×C₃₁	—	4	3	no	—
317	C₂×C₃₁	—	8	3	no	—
318	C₂×C₃₁	C₂²×#123	1	1	yes	characteristic
319	C₂×C₃₁	C₂×#576	1	1	yes	characteristic
320	C₂×C₃₁	#3586	1	1	yes	characteristic
321	H(5,𝔽₂)	—	2	8	no	—
322	C₂×#41	—	2	8	no	—
323	C₂×#41	—	2	1	no	—
324	C₂×#41	D₄×C₃₁	1	3	yes	—
325	C₂×#41	D₄×C₃₁	1	1	yes	characteristic
326	C₂×#41	C₂³×C₃₁	1	3	yes	—
327	C₂×#41	C₂³×C₃₁	1	3	yes	—
328	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
329	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
330	C₂²×D₄	—	2	3	no	—
331	C₂²×D₄	—	2	3	no	—
332	C₂²×D₄	D₄×C₃₁	1	3	yes	—
333	C₂²×D₄	C₂³×C₃₁	1	3	yes	—
334	C₂³×C₄	—	2	3	no	—
335	C₂³×C₄	C₂³×C₃₁	1	3	yes	—
336	#131	—	2	8	no	—
337	C₂×M₁₆	C₂³×C₃₁	1	6	yes	—
338	C₂²×C₈	C₂³×C₃₁	1	6	yes	—
339	#127	—	2	3	no	—
340	#124	—	2	3	no	—
341	#123	—	2	3	no	—
342	C₂²⋊Q₈	—	2	3	no	—
343	C₂²⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
344	C₂²⋊Q₈	C₂³×C₃₁	1	3	yes	—
345	#121	—	2	6	no	—
346	#121	—	2	3	no	—
347	#121	—	2	3	no	—
348	#121	—	2	3	no	—
349	#121	C₂³×C₃₁	1	3	yes	—
350	#121	C₂³×C₃₁	1	6	yes	—
351	#121	C₂³×C₃₁	1	3	yes	—
352	#120	—	2	3	no	—
353	#120	—	2	6	no	—
354	C₄×Q₈	—	2	2	no	—
355	C₄×Q₈	—	2	6	no	—
356	C₄×Q₈	—	2	1	no	—
357	C₄×D₄	—	2	6	no	—
358	C₄×D₄	—	2	12	no	—
359	C₄×D₄	—	2	6	no	—
360	C₄×D₄	—	2	3	no	—
361	C₄×D₄	—	2	3	no	—
362	C₄×D₄	—	2	3	no	—
363	#117	—	2	3	no	—
364	#117	—	2	3	no	—
365	#117	C₂³×C₃₁	1	6	yes	—
366	#117	C₂³×C₃₁	1	6	yes	—
367	#117	C₂³×C₃₁	1	3	yes	—
368	C₂×C₄⋊C₄	—	2	3	no	—
369	C₂×C₄⋊C₄	—	2	3	no	—
370	C₂×C₄⋊C₄	—	2	3	no	—
371	C₂×C₄⋊C₄	C₂³×C₃₁	1	3	yes	—
372	C₂×C₄⋊C₄	C₂³×C₃₁	1	6	yes	—
373	C₂×C₄⋊C₄	C₂³×C₃₁	1	6	yes	—
374	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
375	C₂×C₄⋊C₄	C₂³×C₃₁	1	3	yes	—
376	C₂×C₂²⋊C₄	—	2	3	no	—
377	C₂×C₂²⋊C₄	—	2	3	no	—
378	C₂×C₄²	—	2	3	no	—
379	#107	C₂³×C₃₁	1	6	yes	—
380	#107	C₂³×C₃₁	1	12	yes	—
381	#107	C₂³×C₃₁	1	2	yes	—
382	#106	C₂³×C₃₁	1	6	yes	—
383	#106	C₂³×C₃₁	1	6	yes	—
384	#103	C₂³×C₃₁	1	2	yes	—
385	#103	C₂³×C₃₁	1	6	yes	—
386	#102	C₂³×C₃₁	1	6	yes	—
387	#102	C₂³×C₃₁	1	12	yes	—
388	#102	C₂³×C₃₁	1	6	yes	—
389	C₂²⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
390	C₂²⋊C₈	C₂³×C₃₁	1	3	yes	—
391	C₂²⋊C₈	C₂³×C₃₁	1	6	yes	—
392	C₂²⋊C₈	C₂³×C₃₁	1	6	yes	—
393	C₃₁	#60521	1	1	yes	characteristic, direct product factor, Sylow
394	C₂⁴	—	2	3	no	—
395	C₂⁴	C₂×D₄×C₃₁	1	3	yes	—
396	#41	—	2	4	no	—
397	#41	—	2	12	no	—
398	#41	—	2	12	no	—
399	#41	—	2	12	no	—
400	#41	—	4	4	no	—
401	#41	#41×C₃₁	1	8	yes	—
402	C₂×Q₈	—	2	4	no	—
403	C₂×Q₈	C₂×D₄×C₃₁	1	3	yes	—
404	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
405	C₂×Q₈	C₂⁴×C₃₁	1	1	yes	characteristic
406	C₂×D₄	—	2	12	no	—
407	C₂×D₄	—	2	3	no	—
408	C₂×D₄	—	2	3	no	—
409	C₂×D₄	—	2	3	no	—
410	C₂×D₄	—	2	6	no	—
411	C₂×D₄	—	2	3	no	—
412	C₂×D₄	—	2	12	no	—
413	C₂×D₄	—	4	12	no	—
414	C₂×D₄	—	8	6	no	—
415	C₂×D₄	C₂×D₄×C₃₁	1	6	yes	—
416	C₂×D₄	C₂×D₄×C₃₁	1	3	yes	—
417	C₂×D₄	C₂×D₄×C₃₁	1	3	yes	—
418	C₂×D₄	C₂⁴×C₃₁	1	3	yes	—
419	C₂²×C₄	—	2	12	no	—
420	C₂²×C₄	—	2	1	no	—
421	C₂²×C₄	—	2	3	no	—
422	C₂²×C₄	—	2	3	no	—
423	C₂²×C₄	—	2	6	no	—
424	C₂²×C₄	—	2	6	no	—
425	C₂²×C₄	—	2	3	no	—
426	C₂²×C₄	—	8	6	no	—
427	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
428	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
429	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
430	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
431	C₂²×C₄	C₂⁴×C₃₁	1	3	yes	—
432	C₂²×C₄	C₂⁴×C₃₁	1	3	yes	—
433	M₁₆	—	2	12	no	—
434	C₂×C₈	—	2	12	no	—
435	C₂×C₈	C₂⁴×C₃₁	1	6	yes	—
436	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
437	C₄⋊C₄	—	2	3	no	—
438	C₄⋊C₄	—	2	6	no	—
439	C₄⋊C₄	—	2	6	no	—
440	C₄⋊C₄	—	2	1	no	—
441	C₄⋊C₄	—	2	3	no	—
442	C₄⋊C₄	—	2	1	no	—
443	C₄⋊C₄	—	2	3	no	—
444	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
445	C₄⋊C₄	C₂⁴×C₃₁	1	6	yes	—
446	C₄⋊C₄	C₂⁴×C₃₁	1	1	yes	characteristic
447	C₄⋊C₄	C₂⁴×C₃₁	1	3	yes	—
448	C₂²⋊C₄	—	2	6	no	—
449	C₂²⋊C₄	—	2	6	no	—
450	C₂²⋊C₄	—	2	3	no	—
451	C₂²⋊C₄	—	2	1	no	—
452	C₂²⋊C₄	—	2	3	no	—
453	C₂²⋊C₄	—	2	3	no	—
454	C₂²⋊C₄	—	2	6	no	—
455	C₂²⋊C₄	—	2	3	no	—
456	C₄²	—	2	6	no	—
457	C₄²	—	2	6	no	—
458	C₄²	—	2	3	no	—
459	C₂³	—	2	6	no	—
460	C₂³	—	2	6	no	—
461	C₂³	—	2	3	no	—
462	C₂³	—	4	3	no	—
463	C₂³	—	4	6	no	—
464	C₂³	—	8	3	no	—
465	C₂³	—	8	3	no	—
466	C₂³	C₂²×D₄×C₃₁	1	3	yes	—
467	C₂³	C₂²×D₄×C₃₁	1	3	yes	—
468	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
469	Q₈	—	2	6	no	—
470	Q₈	—	2	2	no	—
471	Q₈	C₃₁×#123	1	4	yes	—
472	D₄	—	2	12	no	—
473	D₄	—	2	6	no	—
474	D₄	—	2	6	no	—
475	D₄	—	4	6	no	—
476	D₄	—	4	6	no	—
477	D₄	—	4	6	no	—
478	D₄	—	8	3	no	—
479	D₄	—	8	6	no	—
480	D₄	C₃₁×#123	1	12	yes	—
481	C₂×C₄	—	2	6	no	—
482	C₂×C₄	—	2	6	no	—
483	C₂×C₄	—	2	6	no	—
484	C₂×C₄	—	2	2	no	—
485	C₂×C₄	—	2	6	no	—
486	C₂×C₄	—	2	2	no	—
487	C₂×C₄	—	2	1	no	—
488	C₂×C₄	—	2	6	no	—
489	C₂×C₄	—	2	3	no	—
490	C₂×C₄	—	2	1	no	—
491	C₂×C₄	—	4	2	no	—
492	C₂×C₄	—	4	6	no	—
493	C₂×C₄	—	4	6	no	—
494	C₂×C₄	—	8	6	no	—
495	C₂×C₄	—	8	6	no	—
496	C₂×C₄	—	8	3	no	—
497	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
498	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
499	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
500	C₂×C₄	C₂×#41×C₃₁	1	12	yes	—
501	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
502	C₈	—	2	2	no	—
503	C₈	—	2	6	no	—
504	C₂²	—	2	1	no	—
505	C₂²	—	2	3	no	—
506	C₂²	—	2	1	no	—
507	C₂²	—	2	3	no	—
508	C₂²	—	2	3	no	—
509	C₂²	—	2	3	no	—
510	C₂²	—	2	3	no	—
511	C₂²	—	4	12	no	—
512	C₂²	—	4	3	no	—
513	C₂²	—	4	3	no	—
514	C₂²	—	4	3	no	—
515	C₂²	—	16	6	no	—
516	C₂²	C₂×C₃₁×#123	1	6	yes	—
517	C₂²	C₃₁×#576	1	2	yes	—
518	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
519	C₄	—	2	1	no	—
520	C₄	—	2	3	no	—
521	C₄	—	4	1	no	—
522	C₄	—	4	6	no	—
523	C₄	—	4	3	no	—
524	C₄	—	4	2	no	—
525	C₄	—	4	1	no	—
526	C₄	C₂×C₃₁×#123	1	6	yes	—
527	C₄	C₂²×#41×C₃₁	1	2	yes	—
528	C₂	—	2	2	no	—
529	C₂	—	2	6	no	—
530	C₂	—	4	3	no	—
531	C₂	—	8	3	no	—
532	C₂	C₂²×C₃₁×#123	1	1	yes	characteristic
533	C₂	C₂×C₃₁×#576	1	1	yes	characteristic
534	C₂	C₃₁×#3586	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. C₂×C₃₁×#520	C₂	semidirect
3. C₃₁×#3098	C₂	semidirect
4. C₃₁×#3096	C₂	semidirect
5. C₃₁×#3094	C₂	semidirect
6. C₃₁×#3092	C₂	semidirect
7. C₃₁×#3091	C₂	semidirect
8. C₂×C₃₁×#480	C₂	semidirect
9. C₃₁×#2913	C₂	semidirect
10. C₃₁×#2897	C₂	semidirect
11. C₃₁×#2880	C₂	semidirect
27. C₃₁×#520	C₂²	semidirect
30. C₂×C₃₁×#117	C₂²	semidirect
31. C₂²×C₄⋊C₄×C₃₁	C₂²	semidirect
32. C₃₁×#485	C₂²	semidirect
33. C₃₁×#484	C₂²	semidirect
34. C₃₁×#484	C₂²	semidirect
35. C₃₁×#483	C₂²	semidirect
36. C₃₁×#482	C₂²	semidirect
37. C₃₁×#482	C₂²	semidirect
38. C₃₁×#481	C₂²	semidirect
39. C₃₁×#481	C₂²	semidirect
40. C₃₁×#480	C₂²	semidirect
41. C₃₁×#480	C₂²	semidirect
42. C₃₁×#480	C₂²	semidirect
43. C₃₁×#428	C₂²	semidirect
44. C₃₁×#428	C₂²	semidirect
45. C₃₁×#427	C₂²	semidirect
46. C₂×C₃₁×#107	C₂²	semidirect
47. C₃₁×#418	C₂²	semidirect
48. C₃₁×#417	C₂²	semidirect
49. C₃₁×#416	C₂²	semidirect
50. C₂×C₃₁×#102	C₂²	semidirect
51. C₃₁×#408	C₂²	semidirect
52. C₃₁×#407	C₂²	semidirect
53. C₂×C₃₁×C₂²⋊C₈	C₂²	semidirect
191. #60521	C₃₁	direct (N × H)
238. C₂×#520	C₂×C₃₁	semidirect
239. #3098	C₂×C₃₁	semidirect
240. #3096	C₂×C₃₁	semidirect
241. #3094	C₂×C₃₁	semidirect
242. #3092	C₂×C₃₁	semidirect
243. #3091	C₂×C₃₁	semidirect
244. C₂×#480	C₂×C₃₁	semidirect
245. #2913	C₂×C₃₁	semidirect
246. #2897	C₂×C₃₁	semidirect
247. #2880	C₂×C₃₁	semidirect
287. #520	C₂²×C₃₁	semidirect
290. C₂×#117	C₂²×C₃₁	semidirect
291. C₂²×C₄⋊C₄	C₂²×C₃₁	semidirect
292. #485	C₂²×C₃₁	semidirect
293. #484	C₂²×C₃₁	semidirect
294. #484	C₂²×C₃₁	semidirect
295. #483	C₂²×C₃₁	semidirect
296. #482	C₂²×C₃₁	semidirect
297. #482	C₂²×C₃₁	semidirect
298. #481	C₂²×C₃₁	semidirect
299. #481	C₂²×C₃₁	semidirect
300. #480	C₂²×C₃₁	semidirect
301. #480	C₂²×C₃₁	semidirect
302. #480	C₂²×C₃₁	semidirect
303. #428	C₂²×C₃₁	semidirect
304. #428	C₂²×C₃₁	semidirect
305. #427	C₂²×C₃₁	semidirect
306. C₂×#107	C₂²×C₃₁	semidirect
307. #418	C₂²×C₃₁	semidirect
308. #417	C₂²×C₃₁	semidirect
309. #416	C₂²×C₃₁	semidirect
310. C₂×#102	C₂²×C₃₁	semidirect
311. #408	C₂²×C₃₁	semidirect
312. #407	C₂²×C₃₁	semidirect
313. C₂×C₂²⋊C₈	C₂²×C₃₁	semidirect
393. C₃₁	#60521	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
C₂×C₃₁×#520	2
C₃₁×#3098	2
C₃₁×#3096	2
C₃₁×#3094	2
C₃₁×#3092	2
C₃₁×#3091	2
C₂×C₃₁×#480	2
C₃₁×#2913	2
C₃₁×#2897	2
C₃₁×#2880	2
#60521	31

Subgroups — C31×#60521 #21517280

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#60521 #21517280