Subgroups — C₃₁×#60516 #21517275

Order 7936.

2142 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3449	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#3428	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3099	C₂	1	4	yes	maximal, complemented
4	C₃₁×#3098	C₂	1	4	yes	maximal, complemented
5	C₃₁×#3097	C₂	1	4	yes	maximal, complemented
6	C₃₁×#3096	C₂	1	4	yes	maximal, complemented
7	C₃₁×#3091	C₂	1	8	yes	maximal, complemented
8	C₃₁×#2911	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₃₁×#427	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#2896	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₃₁×#416	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#408	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×#41×C₃₁	C₂²	1	1	yes	characteristic
14	C₂²×#41×C₃₁	C₂²	1	1	yes	characteristic
15	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
16	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
17	C₃₁×#537	C₂²	1	4	yes	—
18	C₃₁×#536	—	2	1	no	—
19	C₃₁×#536	C₂²	1	2	yes	—
20	C₃₁×#535	—	2	1	no	—
21	C₃₁×#535	C₂²	1	2	yes	—
22	C₃₁×#534	—	2	2	no	—
23	C₃₁×#532	—	2	2	no	—
24	C₃₁×#525	—	2	2	no	—
25	C₃₁×#525	C₂²	1	4	yes	—
26	C₃₁×#520	C₂²	1	4	yes	—
27	C₃₁×#520	C₂²	1	4	yes	complemented
28	C₃₁×#519	—	2	1	no	—
29	C₃₁×#518	—	2	2	no	—
30	C₃₁×#518	—	2	1	no	—
31	C₄×#41×C₃₁	—	2	2	no	—
32	C₄×#41×C₃₁	—	2	2	no	—
33	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
34	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
35	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
36	C₂×C₃₁×#117	C₂²	1	4	yes	complemented
37	C₃₁×#485	C₂²	1	8	yes	complemented
38	C₃₁×#485	C₂²	1	8	yes	complemented
39	C₃₁×#484	C₂²	1	8	yes	complemented
40	C₃₁×#483	C₂²	1	8	yes	complemented
41	C₃₁×#482	C₂²	1	8	yes	complemented
42	C₃₁×#482	C₂²	1	8	yes	complemented
43	C₃₁×#481	C₂²	1	8	yes	complemented
44	C₃₁×#480	C₂²	1	8	yes	complemented
45	C₃₁×#428	C₂²	1	4	yes	complemented
46	C₃₁×#428	C₂²	1	4	yes	complemented
47	C₃₁×#427	C₂²	1	4	yes	complemented
48	C₃₁×#427	C₂²	1	4	yes	complemented
49	C₃₁×#427	C₂²	1	4	yes	complemented
50	C₂×C₃₁×#107	C₂²	1	2	yes	complemented
51	C₂×C₃₁×#106	C₂²	1	2	yes	complemented
52	C₃₁×#419	C₂²	1	2	yes	complemented
53	C₃₁×#418	C₂²	1	2	yes	complemented
54	C₃₁×#417	C₂²	1	4	yes	complemented
55	C₃₁×#416	C₂²	1	4	yes	—
56	C₃₁×#416	C₂²	1	4	yes	—
57	C₃₁×#416	C₂²	1	2	yes	complemented
58	C₃₁×#416	C₂²	1	2	yes	complemented
59	C₂×C₃₁×#103	C₂²	1	2	yes	complemented
60	C₂×C₃₁×#102	C₂²	1	2	yes	complemented
61	C₃₁×#408	C₂²	1	4	yes	—
62	C₃₁×#408	C₂²	1	2	yes	complemented
63	C₃₁×#408	C₂²	1	2	yes	complemented
64	C₂×C₃₁×C₂²⋊C₈	C₂²	1	4	yes	complemented
65	C₂×#41×C₃₁	—	2	1	no	—
66	C₂×#41×C₃₁	—	2	1	no	—
67	C₂×#41×C₃₁	—	2	4	no	—
68	C₂×#41×C₃₁	—	2	4	no	—
69	C₂×#41×C₃₁	—	2	4	no	—
70	C₂×#41×C₃₁	—	2	4	no	—
71	C₂×#41×C₃₁	D₄	1	2	yes	—
72	C₂×#41×C₃₁	D₄	1	2	yes	—
73	C₂×#41×C₃₁	C₂³	1	2	yes	—
74	C₂×#41×C₃₁	C₂³	1	2	yes	—
75	C₂×#41×C₃₁	C₂³	1	2	yes	—
76	C₂×#41×C₃₁	C₂³	1	2	yes	—
77	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
78	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
79	C₂²×D₄×C₃₁	—	2	1	no	—
80	C₂²×D₄×C₃₁	D₄	1	2	yes	—
81	C₂²×D₄×C₃₁	D₄	1	1	yes	characteristic
82	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
83	C₂³×C₄×C₃₁	—	2	1	no	—
84	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
85	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
86	C₂×M₁₆×C₃₁	—	2	4	no	—
87	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
88	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
89	C₂²×C₈×C₃₁	—	2	4	no	—
90	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
91	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
92	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
93	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
94	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
95	C₃₁×C₄⋊Q₈	—	2	2	no	—
96	C₃₁×#127	—	2	2	no	—
97	C₃₁×#124	—	2	2	no	—
98	C₃₁×#124	—	2	2	no	—
99	C₃₁×#123	—	2	4	no	—
100	C₃₁×#123	—	2	4	no	—
101	C₃₁×C₂²⋊Q₈	—	2	4	no	—
102	C₃₁×C₂²⋊Q₈	C₂³	1	8	yes	—
103	C₃₁×#121	—	2	4	no	—
104	C₃₁×#121	—	2	8	no	—
105	C₃₁×#121	C₂³	1	8	yes	—
106	C₃₁×#120	—	2	4	no	—
107	C₃₁×#120	—	2	4	no	—
108	C₄×Q₈×C₃₁	—	2	4	no	—
109	C₄×Q₈×C₃₁	—	2	4	no	—
110	C₄×D₄×C₃₁	—	2	8	no	—
111	C₄×D₄×C₃₁	—	2	4	no	—
112	C₄×D₄×C₃₁	—	2	8	no	—
113	C₄×D₄×C₃₁	—	2	4	no	—
114	C₄×D₄×C₃₁	—	2	8	no	—
115	C₃₁×#117	—	2	1	no	—
116	C₃₁×#117	—	2	1	no	—
117	C₃₁×#117	—	2	1	no	—
118	C₃₁×#117	—	2	1	no	—
119	C₃₁×#117	—	2	1	no	—
120	C₃₁×#117	—	2	1	no	—
121	C₃₁×#117	—	2	4	no	—
122	C₃₁×#117	—	2	4	no	—
123	C₃₁×#117	C₂³	1	2	yes	—
124	C₃₁×#117	C₂³	1	2	yes	—
125	C₃₁×#117	C₂³	1	2	yes	—
126	C₃₁×#117	C₂³	1	2	yes	—
127	C₃₁×#117	C₂³	1	2	yes	—
128	C₃₁×#117	C₂³	1	2	yes	—
129	C₃₁×#117	C₂³	1	8	yes	—
130	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
131	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
132	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
133	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
134	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
135	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
136	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
137	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
138	C₂×C₄²×C₃₁	—	2	1	no	—
139	C₂×C₄²×C₃₁	—	2	1	no	—
140	C₂×C₄²×C₃₁	—	2	1	no	—
141	C₂×C₄²×C₃₁	—	2	4	no	—
142	C₃₁×#107	C₂³	1	4	yes	—
143	C₃₁×#107	C₂³	1	4	yes	—
144	C₃₁×#107	C₂³	1	8	yes	—
145	C₃₁×#106	C₂³	1	4	yes	—
146	C₃₁×#106	C₂³	1	4	yes	—
147	C₃₁×#106	C₂³	1	8	yes	—
148	C₃₁×#103	C₂³	1	4	yes	—
149	C₃₁×#103	C₂³	1	4	yes	—
150	C₃₁×#103	C₂³	1	8	yes	—
151	C₃₁×#102	C₂³	1	4	yes	—
152	C₃₁×#102	C₂³	1	4	yes	—
153	C₃₁×#102	C₂³	1	8	yes	—
154	C₃₁×C₂²⋊C₈	C₂³	1	8	yes	—
155	C₃₁×C₂²⋊C₈	C₂³	1	8	yes	—
156	C₂⁴×C₃₁	—	2	1	no	—
157	C₂⁴×C₃₁	C₂×D₄	1	2	yes	—
158	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
159	#41×C₃₁	—	2	8	no	—
160	#41×C₃₁	—	2	8	no	—
161	#41×C₃₁	—	2	4	no	—
162	#41×C₃₁	—	2	4	no	—
163	#41×C₃₁	—	2	4	no	—
164	#41×C₃₁	—	2	4	no	—
165	#41×C₃₁	—	2	4	no	—
166	#41×C₃₁	—	2	4	no	—
167	#41×C₃₁	—	2	4	no	—
168	#41×C₃₁	—	2	4	no	—
169	#41×C₃₁	—	4	4	no	—
170	#41×C₃₁	—	4	4	no	—
171	C₂×Q₈×C₃₁	—	2	4	no	—
172	C₂×Q₈×C₃₁	—	2	1	no	—
173	C₂×Q₈×C₃₁	—	2	2	no	—
174	C₂×Q₈×C₃₁	—	2	2	no	—
175	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
176	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
177	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
178	C₂×D₄×C₃₁	—	2	8	no	—
179	C₂×D₄×C₃₁	—	2	2	no	—
180	C₂×D₄×C₃₁	—	2	2	no	—
181	C₂×D₄×C₃₁	—	2	8	no	—
182	C₂×D₄×C₃₁	—	2	4	no	—
183	C₂×D₄×C₃₁	—	2	1	no	—
184	C₂×D₄×C₃₁	—	2	2	no	—
185	C₂×D₄×C₃₁	—	2	2	no	—
186	C₂×D₄×C₃₁	—	4	4	no	—
187	C₂×D₄×C₃₁	C₂×D₄	1	4	yes	—
188	C₂×D₄×C₃₁	C₂×D₄	1	4	yes	—
189	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
190	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
191	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
192	C₂²×C₄×C₃₁	—	2	4	no	—
193	C₂²×C₄×C₃₁	—	2	4	no	—
194	C₂²×C₄×C₃₁	—	2	1	no	—
195	C₂²×C₄×C₃₁	—	2	8	no	—
196	C₂²×C₄×C₃₁	—	2	2	no	—
197	C₂²×C₄×C₃₁	—	2	2	no	—
198	C₂²×C₄×C₃₁	—	2	2	no	—
199	C₂²×C₄×C₃₁	—	2	2	no	—
200	C₂²×C₄×C₃₁	—	2	2	no	—
201	C₂²×C₄×C₃₁	—	4	4	no	—
202	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
203	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
204	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
205	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
206	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
207	C₂²×C₄×C₃₁	#41	1	4	yes	—
208	C₂²×C₄×C₃₁	#41	1	4	yes	—
209	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
210	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
211	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
212	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
213	M₁₆×C₃₁	—	2	4	no	—
214	M₁₆×C₃₁	—	2	4	no	—
215	C₂×C₈×C₃₁	—	2	4	no	—
216	C₂×C₈×C₃₁	—	2	4	no	—
217	C₂×C₈×C₃₁	—	2	2	no	—
218	C₂×C₈×C₃₁	—	2	2	no	—
219	C₂×C₈×C₃₁	—	2	4	no	—
220	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
221	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
222	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
223	C₄⋊C₄×C₃₁	—	2	8	no	—
224	C₄⋊C₄×C₃₁	—	2	2	no	—
225	C₄⋊C₄×C₃₁	—	2	2	no	—
226	C₄⋊C₄×C₃₁	—	2	2	no	—
227	C₄⋊C₄×C₃₁	—	2	8	no	—
228	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
229	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
230	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
231	C₂²⋊C₄×C₃₁	—	2	2	no	—
232	C₂²⋊C₄×C₃₁	—	2	2	no	—
233	C₂²⋊C₄×C₃₁	—	2	8	no	—
234	C₂²⋊C₄×C₃₁	—	2	2	no	—
235	C₂²⋊C₄×C₃₁	—	2	2	no	—
236	C₂²⋊C₄×C₃₁	—	2	2	no	—
237	C₂²⋊C₄×C₃₁	—	2	2	no	—
238	C₂²⋊C₄×C₃₁	—	2	8	no	—
239	C₂²⋊C₄×C₃₁	—	2	8	no	—
240	C₄²×C₃₁	—	2	8	no	—
241	C₄²×C₃₁	—	2	2	no	—
242	C₄²×C₃₁	—	2	2	no	—
243	C₄²×C₃₁	—	2	2	no	—
244	C₄²×C₃₁	—	2	2	no	—
245	C₄²×C₃₁	—	2	2	no	—
246	C₄²×C₃₁	—	2	2	no	—
247	#60516	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
248	C₂³×C₃₁	—	2	4	no	—
249	C₂³×C₃₁	—	2	2	no	—
250	C₂³×C₃₁	—	4	2	no	—
251	C₂³×C₃₁	—	4	4	no	—
252	C₂³×C₃₁	—	4	2	no	—
253	C₂³×C₃₁	—	8	2	no	—
254	C₂³×C₃₁	#123	1	4	yes	—
255	C₂³×C₃₁	C₂²×D₄	1	4	yes	—
256	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
257	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
258	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
259	Q₈×C₃₁	—	2	4	no	—
260	Q₈×C₃₁	—	2	4	no	—
261	Q₈×C₃₁	—	2	2	no	—
262	Q₈×C₃₁	—	2	2	no	—
263	Q₈×C₃₁	—	4	2	no	—
264	D₄×C₃₁	—	2	8	no	—
265	D₄×C₃₁	—	2	8	no	—
266	D₄×C₃₁	—	2	4	no	—
267	D₄×C₃₁	—	2	4	no	—
268	D₄×C₃₁	—	2	2	no	—
269	D₄×C₃₁	—	2	2	no	—
270	D₄×C₃₁	—	4	4	no	—
271	D₄×C₃₁	—	4	4	no	—
272	D₄×C₃₁	—	4	2	no	—
273	D₄×C₃₁	—	8	8	no	—
274	C₂×C₄×C₃₁	—	2	4	no	—
275	C₂×C₄×C₃₁	—	2	8	no	—
276	C₂×C₄×C₃₁	—	2	4	no	—
277	C₂×C₄×C₃₁	—	2	4	no	—
278	C₂×C₄×C₃₁	—	2	4	no	—
279	C₂×C₄×C₃₁	—	2	4	no	—
280	C₂×C₄×C₃₁	—	2	4	no	—
281	C₂×C₄×C₃₁	—	2	2	no	—
282	C₂×C₄×C₃₁	—	4	2	no	—
283	C₂×C₄×C₃₁	—	4	4	no	—
284	C₂×C₄×C₃₁	—	4	4	no	—
285	C₂×C₄×C₃₁	—	8	2	no	—
286	C₂×C₄×C₃₁	—	8	2	no	—
287	C₂×C₄×C₃₁	—	8	8	no	—
288	C₂×C₄×C₃₁	—	8	2	no	—
289	C₂×C₄×C₃₁	#123	1	4	yes	—
290	C₂×C₄×C₃₁	#123	1	4	yes	—
291	C₂×C₄×C₃₁	#123	1	4	yes	—
292	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
293	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
294	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
295	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
296	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
297	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
298	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
299	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
300	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
301	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
302	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
303	C₈×C₃₁	—	2	2	no	—
304	C₈×C₃₁	—	2	2	no	—
305	C₈×C₃₁	—	2	4	no	—
306	#3449	C₂×C₃₁	1	1	yes	characteristic
307	#3428	C₂×C₃₁	1	1	yes	characteristic, complemented
308	#3099	C₂×C₃₁	1	4	yes	complemented
309	#3098	C₂×C₃₁	1	4	yes	complemented
310	#3097	C₂×C₃₁	1	4	yes	complemented
311	#3096	C₂×C₃₁	1	4	yes	complemented
312	#3091	C₂×C₃₁	1	8	yes	complemented
313	#2911	C₂×C₃₁	1	1	yes	characteristic, complemented
314	C₂×#427	C₂×C₃₁	1	1	yes	characteristic, complemented
315	#2896	C₂×C₃₁	1	1	yes	characteristic, complemented
316	C₂×#416	C₂×C₃₁	1	1	yes	characteristic, complemented
317	C₂×#408	C₂×C₃₁	1	1	yes	characteristic, complemented
318	C₂²×C₃₁	—	2	2	no	—
319	C₂²×C₃₁	—	2	2	no	—
320	C₂²×C₃₁	—	2	2	no	—
321	C₂²×C₃₁	—	2	8	no	—
322	C₂²×C₃₁	—	2	4	no	—
323	C₂²×C₃₁	—	4	4	no	—
324	C₂²×C₃₁	—	4	4	no	—
325	C₂²×C₃₁	—	4	2	no	—
326	C₂²×C₃₁	—	8	4	no	—
327	C₂²×C₃₁	—	8	4	no	—
328	C₂²×C₃₁	C₂×#123	1	2	yes	—
329	C₂²×C₃₁	C₂×#123	1	2	yes	—
330	C₂²×C₃₁	C₂×#123	1	2	yes	—
331	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
332	C₄×C₃₁	—	2	4	no	—
333	C₄×C₃₁	—	4	4	no	—
334	C₄×C₃₁	—	4	4	no	—
335	C₄×C₃₁	—	4	4	no	—
336	C₄×C₃₁	—	4	2	no	—
337	C₄×C₃₁	C₂×#123	1	2	yes	—
338	C₄×C₃₁	C₂×#123	1	2	yes	—
339	C₄×C₃₁	C₂×#123	1	2	yes	—
340	C₄×C₃₁	C₂²×#41	1	1	yes	characteristic
341	C₄×C₃₁	C₂²×#41	1	1	yes	characteristic
342	C₂²×#41	C₂²×C₃₁	1	1	yes	characteristic
343	C₂²×#41	C₂²×C₃₁	1	1	yes	characteristic
344	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
345	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
346	#537	C₂²×C₃₁	1	4	yes	—
347	#536	—	2	1	no	—
348	#536	C₂²×C₃₁	1	2	yes	—
349	#535	—	2	1	no	—
350	#535	C₂²×C₃₁	1	2	yes	—
351	#534	—	2	2	no	—
352	#532	—	2	2	no	—
353	#525	—	2	2	no	—
354	#525	C₂²×C₃₁	1	4	yes	—
355	#520	C₂²×C₃₁	1	4	yes	—
356	#520	C₂²×C₃₁	1	4	yes	complemented
357	#519	—	2	1	no	—
358	#518	—	2	2	no	—
359	#518	—	2	1	no	—
360	C₄×#41	—	2	2	no	—
361	C₄×#41	—	2	2	no	—
362	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
363	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
364	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
365	C₂×#117	C₂²×C₃₁	1	4	yes	complemented
366	#485	C₂²×C₃₁	1	8	yes	complemented
367	#485	C₂²×C₃₁	1	8	yes	complemented
368	#484	C₂²×C₃₁	1	8	yes	complemented
369	#483	C₂²×C₃₁	1	8	yes	complemented
370	#482	C₂²×C₃₁	1	8	yes	complemented
371	#482	C₂²×C₃₁	1	8	yes	complemented
372	#481	C₂²×C₃₁	1	8	yes	complemented
373	#480	C₂²×C₃₁	1	8	yes	complemented
374	#428	C₂²×C₃₁	1	4	yes	complemented
375	#428	C₂²×C₃₁	1	4	yes	complemented
376	#427	C₂²×C₃₁	1	4	yes	complemented
377	#427	C₂²×C₃₁	1	4	yes	complemented
378	#427	C₂²×C₃₁	1	4	yes	complemented
379	C₂×#107	C₂²×C₃₁	1	2	yes	complemented
380	C₂×#106	C₂²×C₃₁	1	2	yes	complemented
381	#419	C₂²×C₃₁	1	2	yes	complemented
382	#418	C₂²×C₃₁	1	2	yes	complemented
383	#417	C₂²×C₃₁	1	4	yes	complemented
384	#416	C₂²×C₃₁	1	4	yes	—
385	#416	C₂²×C₃₁	1	4	yes	—
386	#416	C₂²×C₃₁	1	2	yes	complemented
387	#416	C₂²×C₃₁	1	2	yes	complemented
388	C₂×#103	C₂²×C₃₁	1	2	yes	complemented
389	C₂×#102	C₂²×C₃₁	1	2	yes	complemented
390	#408	C₂²×C₃₁	1	4	yes	—
391	#408	C₂²×C₃₁	1	2	yes	complemented
392	#408	C₂²×C₃₁	1	2	yes	complemented
393	C₂×C₂²⋊C₈	C₂²×C₃₁	1	4	yes	complemented
394	C₂×C₃₁	—	2	2	no	—
395	C₂×C₃₁	—	2	2	no	—
396	C₂×C₃₁	—	2	2	no	—
397	C₂×C₃₁	—	4	4	no	—
398	C₂×C₃₁	—	8	2	no	—
399	C₂×C₃₁	C₂²×#123	1	1	yes	characteristic
400	C₂×C₃₁	#3584	1	2	yes	—
401	C₂×#41	—	2	1	no	—
402	C₂×#41	—	2	1	no	—
403	C₂×#41	—	2	4	no	—
404	C₂×#41	—	2	4	no	—
405	C₂×#41	—	2	4	no	—
406	C₂×#41	—	2	4	no	—
407	C₂×#41	D₄×C₃₁	1	2	yes	—
408	C₂×#41	D₄×C₃₁	1	2	yes	—
409	C₂×#41	C₂³×C₃₁	1	2	yes	—
410	C₂×#41	C₂³×C₃₁	1	2	yes	—
411	C₂×#41	C₂³×C₃₁	1	2	yes	—
412	C₂×#41	C₂³×C₃₁	1	2	yes	—
413	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
414	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
415	C₂²×D₄	—	2	1	no	—
416	C₂²×D₄	D₄×C₃₁	1	2	yes	—
417	C₂²×D₄	D₄×C₃₁	1	1	yes	characteristic
418	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
419	C₂³×C₄	—	2	1	no	—
420	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
421	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
422	C₂×M₁₆	—	2	4	no	—
423	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
424	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
425	C₂²×C₈	—	2	4	no	—
426	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
427	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
428	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
429	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
430	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
431	C₄⋊Q₈	—	2	2	no	—
432	#127	—	2	2	no	—
433	#124	—	2	2	no	—
434	#124	—	2	2	no	—
435	#123	—	2	4	no	—
436	#123	—	2	4	no	—
437	C₂²⋊Q₈	—	2	4	no	—
438	C₂²⋊Q₈	C₂³×C₃₁	1	8	yes	—
439	#121	—	2	4	no	—
440	#121	—	2	8	no	—
441	#121	C₂³×C₃₁	1	8	yes	—
442	#120	—	2	4	no	—
443	#120	—	2	4	no	—
444	C₄×Q₈	—	2	4	no	—
445	C₄×Q₈	—	2	4	no	—
446	C₄×D₄	—	2	8	no	—
447	C₄×D₄	—	2	4	no	—
448	C₄×D₄	—	2	8	no	—
449	C₄×D₄	—	2	4	no	—
450	C₄×D₄	—	2	8	no	—
451	#117	—	2	1	no	—
452	#117	—	2	1	no	—
453	#117	—	2	1	no	—
454	#117	—	2	1	no	—
455	#117	—	2	1	no	—
456	#117	—	2	1	no	—
457	#117	—	2	4	no	—
458	#117	—	2	4	no	—
459	#117	C₂³×C₃₁	1	2	yes	—
460	#117	C₂³×C₃₁	1	2	yes	—
461	#117	C₂³×C₃₁	1	2	yes	—
462	#117	C₂³×C₃₁	1	2	yes	—
463	#117	C₂³×C₃₁	1	2	yes	—
464	#117	C₂³×C₃₁	1	2	yes	—
465	#117	C₂³×C₃₁	1	8	yes	—
466	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
467	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
468	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
469	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
470	C₂×C₂²⋊C₄	—	2	1	no	—
471	C₂×C₂²⋊C₄	—	2	1	no	—
472	C₂×C₂²⋊C₄	—	2	1	no	—
473	C₂×C₂²⋊C₄	—	2	4	no	—
474	C₂×C₄²	—	2	1	no	—
475	C₂×C₄²	—	2	1	no	—
476	C₂×C₄²	—	2	1	no	—
477	C₂×C₄²	—	2	4	no	—
478	#107	C₂³×C₃₁	1	4	yes	—
479	#107	C₂³×C₃₁	1	4	yes	—
480	#107	C₂³×C₃₁	1	8	yes	—
481	#106	C₂³×C₃₁	1	4	yes	—
482	#106	C₂³×C₃₁	1	4	yes	—
483	#106	C₂³×C₃₁	1	8	yes	—
484	#103	C₂³×C₃₁	1	4	yes	—
485	#103	C₂³×C₃₁	1	4	yes	—
486	#103	C₂³×C₃₁	1	8	yes	—
487	#102	C₂³×C₃₁	1	4	yes	—
488	#102	C₂³×C₃₁	1	4	yes	—
489	#102	C₂³×C₃₁	1	8	yes	—
490	C₂²⋊C₈	C₂³×C₃₁	1	8	yes	—
491	C₂²⋊C₈	C₂³×C₃₁	1	8	yes	—
492	C₃₁	#60516	1	1	yes	characteristic, direct product factor, Sylow
493	C₂⁴	—	2	1	no	—
494	C₂⁴	C₂×D₄×C₃₁	1	2	yes	—
495	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
496	#41	—	2	8	no	—
497	#41	—	2	8	no	—
498	#41	—	2	4	no	—
499	#41	—	2	4	no	—
500	#41	—	2	4	no	—
501	#41	—	2	4	no	—
502	#41	—	2	4	no	—
503	#41	—	2	4	no	—
504	#41	—	2	4	no	—
505	#41	—	2	4	no	—
506	#41	—	4	4	no	—
507	#41	—	4	4	no	—
508	C₂×Q₈	—	2	4	no	—
509	C₂×Q₈	—	2	1	no	—
510	C₂×Q₈	—	2	2	no	—
511	C₂×Q₈	—	2	2	no	—
512	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
513	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
514	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
515	C₂×D₄	—	2	8	no	—
516	C₂×D₄	—	2	2	no	—
517	C₂×D₄	—	2	2	no	—
518	C₂×D₄	—	2	8	no	—
519	C₂×D₄	—	2	4	no	—
520	C₂×D₄	—	2	1	no	—
521	C₂×D₄	—	2	2	no	—
522	C₂×D₄	—	2	2	no	—
523	C₂×D₄	—	4	4	no	—
524	C₂×D₄	C₂×D₄×C₃₁	1	4	yes	—
525	C₂×D₄	C₂×D₄×C₃₁	1	4	yes	—
526	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
527	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
528	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
529	C₂²×C₄	—	2	4	no	—
530	C₂²×C₄	—	2	4	no	—
531	C₂²×C₄	—	2	1	no	—
532	C₂²×C₄	—	2	8	no	—
533	C₂²×C₄	—	2	2	no	—
534	C₂²×C₄	—	2	2	no	—
535	C₂²×C₄	—	2	2	no	—
536	C₂²×C₄	—	2	2	no	—
537	C₂²×C₄	—	2	2	no	—
538	C₂²×C₄	—	4	4	no	—
539	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
540	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
541	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
542	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
543	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
544	C₂²×C₄	#41×C₃₁	1	4	yes	—
545	C₂²×C₄	#41×C₃₁	1	4	yes	—
546	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
547	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
548	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
549	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
550	M₁₆	—	2	4	no	—
551	M₁₆	—	2	4	no	—
552	C₂×C₈	—	2	4	no	—
553	C₂×C₈	—	2	4	no	—
554	C₂×C₈	—	2	2	no	—
555	C₂×C₈	—	2	2	no	—
556	C₂×C₈	—	2	4	no	—
557	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
558	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
559	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
560	C₄⋊C₄	—	2	8	no	—
561	C₄⋊C₄	—	2	2	no	—
562	C₄⋊C₄	—	2	2	no	—
563	C₄⋊C₄	—	2	2	no	—
564	C₄⋊C₄	—	2	8	no	—
565	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
566	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
567	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
568	C₂²⋊C₄	—	2	2	no	—
569	C₂²⋊C₄	—	2	2	no	—
570	C₂²⋊C₄	—	2	8	no	—
571	C₂²⋊C₄	—	2	2	no	—
572	C₂²⋊C₄	—	2	2	no	—
573	C₂²⋊C₄	—	2	2	no	—
574	C₂²⋊C₄	—	2	2	no	—
575	C₂²⋊C₄	—	2	8	no	—
576	C₂²⋊C₄	—	2	8	no	—
577	C₄²	—	2	2	no	—
578	C₄²	—	2	2	no	—
579	C₄²	—	2	2	no	—
580	C₄²	—	2	8	no	—
581	C₄²	—	2	2	no	—
582	C₄²	—	2	2	no	—
583	C₄²	—	2	2	no	—
584	C₂³	—	2	4	no	—
585	C₂³	—	2	2	no	—
586	C₂³	—	4	2	no	—
587	C₂³	—	4	4	no	—
588	C₂³	—	4	2	no	—
589	C₂³	—	8	2	no	—
590	C₂³	C₃₁×#123	1	4	yes	—
591	C₂³	C₂²×D₄×C₃₁	1	4	yes	—
592	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
593	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
594	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
595	Q₈	—	2	4	no	—
596	Q₈	—	2	4	no	—
597	Q₈	—	2	2	no	—
598	Q₈	—	2	2	no	—
599	Q₈	—	4	2	no	—
600	D₄	—	2	8	no	—
601	D₄	—	2	8	no	—
602	D₄	—	2	4	no	—
603	D₄	—	2	4	no	—
604	D₄	—	2	2	no	—
605	D₄	—	2	2	no	—
606	D₄	—	4	4	no	—
607	D₄	—	4	4	no	—
608	D₄	—	4	2	no	—
609	D₄	—	8	8	no	—
610	C₂×C₄	—	2	4	no	—
611	C₂×C₄	—	2	8	no	—
612	C₂×C₄	—	2	4	no	—
613	C₂×C₄	—	2	4	no	—
614	C₂×C₄	—	2	4	no	—
615	C₂×C₄	—	2	4	no	—
616	C₂×C₄	—	2	4	no	—
617	C₂×C₄	—	2	2	no	—
618	C₂×C₄	—	4	2	no	—
619	C₂×C₄	—	4	4	no	—
620	C₂×C₄	—	4	4	no	—
621	C₂×C₄	—	8	2	no	—
622	C₂×C₄	—	8	2	no	—
623	C₂×C₄	—	8	8	no	—
624	C₂×C₄	—	8	2	no	—
625	C₂×C₄	C₃₁×#123	1	4	yes	—
626	C₂×C₄	C₃₁×#123	1	4	yes	—
627	C₂×C₄	C₃₁×#123	1	4	yes	—
628	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
629	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
630	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
631	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
632	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
633	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
634	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
635	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
636	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
637	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
638	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
639	C₈	—	2	2	no	—
640	C₈	—	2	2	no	—
641	C₈	—	2	4	no	—
642	C₂²	—	2	2	no	—
643	C₂²	—	2	2	no	—
644	C₂²	—	2	2	no	—
645	C₂²	—	2	4	no	—
646	C₂²	—	2	8	no	—
647	C₂²	—	4	4	no	—
648	C₂²	—	4	2	no	—
649	C₂²	—	4	4	no	—
650	C₂²	—	8	4	no	—
651	C₂²	—	8	4	no	—
652	C₂²	C₂×C₃₁×#123	1	2	yes	—
653	C₂²	C₂×C₃₁×#123	1	2	yes	—
654	C₂²	C₂×C₃₁×#123	1	2	yes	—
655	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
656	C₄	—	2	4	no	—
657	C₄	—	4	2	no	—
658	C₄	—	4	4	no	—
659	C₄	—	4	4	no	—
660	C₄	—	4	4	no	—
661	C₄	C₂×C₃₁×#123	1	2	yes	—
662	C₄	C₂×C₃₁×#123	1	2	yes	—
663	C₄	C₂×C₃₁×#123	1	2	yes	—
664	C₄	C₂²×#41×C₃₁	1	1	yes	characteristic
665	C₄	C₂²×#41×C₃₁	1	1	yes	characteristic
666	C₂	—	2	2	no	—
667	C₂	—	2	2	no	—
668	C₂	—	2	2	no	—
669	C₂	—	4	4	no	—
670	C₂	—	8	2	no	—
671	C₂	C₂²×C₃₁×#123	1	1	yes	characteristic
672	C₂	C₃₁×#3584	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
socle	C₂²×C₃₁	C₂³×D₄
prime radical	C₃₁×#3449	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×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×#3097	C₂	semidirect
6. C₃₁×#3096	C₂	semidirect
7. C₃₁×#3091	C₂	semidirect
8. C₃₁×#2911	C₂	semidirect
9. C₂×C₃₁×#427	C₂	semidirect
10. C₃₁×#2896	C₂	semidirect
11. C₂×C₃₁×#416	C₂	semidirect
12. C₂×C₃₁×#408	C₂	semidirect
27. C₃₁×#520	C₂²	semidirect
36. C₂×C₃₁×#117	C₂²	semidirect
37. C₃₁×#485	C₂²	semidirect
38. C₃₁×#485	C₂²	semidirect
39. C₃₁×#484	C₂²	semidirect
40. C₃₁×#483	C₂²	semidirect
41. C₃₁×#482	C₂²	semidirect
42. C₃₁×#482	C₂²	semidirect
43. C₃₁×#481	C₂²	semidirect
44. C₃₁×#480	C₂²	semidirect
45. C₃₁×#428	C₂²	semidirect
46. C₃₁×#428	C₂²	semidirect
47. C₃₁×#427	C₂²	semidirect
48. C₃₁×#427	C₂²	semidirect
49. C₃₁×#427	C₂²	semidirect
50. C₂×C₃₁×#107	C₂²	semidirect
51. C₂×C₃₁×#106	C₂²	semidirect
52. C₃₁×#419	C₂²	semidirect
53. C₃₁×#418	C₂²	semidirect
54. C₃₁×#417	C₂²	semidirect
57. C₃₁×#416	C₂²	semidirect
58. C₃₁×#416	C₂²	semidirect
59. C₂×C₃₁×#103	C₂²	semidirect
60. C₂×C₃₁×#102	C₂²	semidirect
62. C₃₁×#408	C₂²	semidirect
63. C₃₁×#408	C₂²	semidirect
64. C₂×C₃₁×C₂²⋊C₈	C₂²	semidirect
247. #60516	C₃₁	direct (N × H)
307. #3428	C₂×C₃₁	semidirect
308. #3099	C₂×C₃₁	semidirect
309. #3098	C₂×C₃₁	semidirect
310. #3097	C₂×C₃₁	semidirect
311. #3096	C₂×C₃₁	semidirect
312. #3091	C₂×C₃₁	semidirect
313. #2911	C₂×C₃₁	semidirect
314. C₂×#427	C₂×C₃₁	semidirect
315. #2896	C₂×C₃₁	semidirect
316. C₂×#416	C₂×C₃₁	semidirect
317. C₂×#408	C₂×C₃₁	semidirect
356. #520	C₂²×C₃₁	semidirect
365. C₂×#117	C₂²×C₃₁	semidirect
366. #485	C₂²×C₃₁	semidirect
367. #485	C₂²×C₃₁	semidirect
368. #484	C₂²×C₃₁	semidirect
369. #483	C₂²×C₃₁	semidirect
370. #482	C₂²×C₃₁	semidirect
371. #482	C₂²×C₃₁	semidirect
372. #481	C₂²×C₃₁	semidirect
373. #480	C₂²×C₃₁	semidirect
374. #428	C₂²×C₃₁	semidirect
375. #428	C₂²×C₃₁	semidirect
376. #427	C₂²×C₃₁	semidirect
377. #427	C₂²×C₃₁	semidirect
378. #427	C₂²×C₃₁	semidirect
379. C₂×#107	C₂²×C₃₁	semidirect
380. C₂×#106	C₂²×C₃₁	semidirect
381. #419	C₂²×C₃₁	semidirect
382. #418	C₂²×C₃₁	semidirect
383. #417	C₂²×C₃₁	semidirect
386. #416	C₂²×C₃₁	semidirect
387. #416	C₂²×C₃₁	semidirect
388. C₂×#103	C₂²×C₃₁	semidirect
389. C₂×#102	C₂²×C₃₁	semidirect
391. #408	C₂²×C₃₁	semidirect
392. #408	C₂²×C₃₁	semidirect
393. C₂×C₂²⋊C₈	C₂²×C₃₁	semidirect
492. C₃₁	#60516	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×#3449	2
C₃₁×#3428	2
C₃₁×#3099	2
C₃₁×#3098	2
C₃₁×#3097	2
C₃₁×#3096	2
C₃₁×#3091	2
C₃₁×#2911	2
C₂×C₃₁×#427	2
C₃₁×#2896	2
C₂×C₃₁×#416	2
C₂×C₃₁×#408	2
#60516	31

Subgroups — C31×#60516 #21517275

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#60516 #21517275