Subgroups — C₃₁×#61286 #21518045

Order 7936.

1754 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3507	C₂	1	2	yes	maximal
2	C₃₁×#3356	C₂	1	4	yes	maximal, complemented
3	C₃₁×#3352	C₂	1	4	yes	maximal, complemented
4	C₃₁×#3342	C₂	1	2	yes	maximal, complemented
5	C₃₁×#3338	C₂	1	2	yes	maximal, complemented
6	C₃₁×#3336	C₂	1	4	yes	maximal, complemented
7	C₃₁×#3160	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#3097	C₂	1	4	yes	maximal, complemented
9	C₃₁×#3078	C₂	1	2	yes	maximal, complemented
10	C₃₁×#3059	C₂	1	2	yes	maximal, complemented
11	C₂×C₄×Q₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#2931	C₂	1	1	yes	maximal, characteristic
13	C₃₁×#2913	C₂	1	2	yes	maximal, complemented
14	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#131	C₂²	1	2	yes	—
16	C₃₁×#559	—	2	2	no	—
17	C₃₁×#557	—	2	2	no	—
18	C₃₁×#556	—	2	4	no	—
19	C₃₁×#556	C₂²	1	4	yes	—
20	C₃₁×#554	C₂²	1	4	yes	—
21	C₃₁×#552	C₂²	1	8	yes	—
22	C₃₁×#539	—	2	4	no	—
23	C₃₁×#536	C₂²	1	4	yes	—
24	C₃₁×#533	C₂²	1	2	yes	—
25	C₃₁×#532	C₂²	1	1	yes	characteristic
26	C₃₁×#529	—	2	4	no	—
27	C₃₁×#520	C₂²	1	4	yes	—
28	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
29	C₃₁×#500	C₂²	1	1	yes	characteristic
30	C₃₁×#500	C₂²	1	1	yes	characteristic
31	C₃₁×#499	C₂²	1	2	yes	—
32	C₃₁×#499	C₂²	1	2	yes	—
33	C₃₁×#498	C₂²	1	2	yes	—
34	C₃₁×#485	C₂²	1	8	yes	—
35	C₃₁×#485	C₂²	1	8	yes	—
36	C₃₁×#484	C₂²	1	8	yes	—
37	C₃₁×#483	C₂²	1	8	yes	—
38	C₃₁×#479	C₂²	1	8	yes	—
39	C₃₁×#477	C₂²	1	4	yes	—
40	C₃₁×#475	C₂²	1	4	yes	—
41	C₃₁×#470	C₂²	1	8	yes	—
42	C₃₁×#467	C₂²	1	4	yes	—
43	C₃₁×#467	C₂²	1	4	yes	—
44	C₄×Q₁₆×C₃₁	C₂²	1	2	yes	—
45	C₄×Q₁₆×C₃₁	C₂²	1	2	yes	—
46	C₄×Q₁₆×C₃₁	C₂²	1	2	yes	—
47	C₄×Q₁₆×C₃₁	C₂²	1	2	yes	—
48	C₃₁×#436	C₂²	1	2	yes	—
49	C₃₁×#436	C₂²	1	2	yes	—
50	C₈×D₄×C₃₁	C₂²	1	2	yes	—
51	C₈×D₄×C₃₁	C₂²	1	2	yes	—
52	C₃₁×#428	C₂²	1	4	yes	—
53	C₃₁×#428	C₂²	1	8	yes	—
54	C₃₁×#427	C₂²	1	4	yes	—
55	C₃₁×#427	C₂²	1	2	yes	—
56	C₃₁×#427	C₂²	1	2	yes	—
57	C₂×C₃₁×#107	C₂²	1	4	yes	—
58	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
59	C₃₁×#423	C₂²	1	1	yes	characteristic
60	C₃₁×#419	C₂²	1	4	yes	—
61	C₂×C₃₁×#103	C₂²	1	2	yes	—
62	C₃₁×#408	C₂²	1	2	yes	—
63	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
64	C₂×#41×C₃₁	C₂³	1	2	yes	—
65	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
66	C₂×Q₁₆×C₃₁	—	2	4	no	—
67	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
68	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
69	C₃₁×#131	—	2	8	no	—
70	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
71	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
72	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
73	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
74	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
75	C₃₁×C₄⋊Q₈	—	2	4	no	—
76	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
77	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
78	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
79	C₃₁×#127	D₄	1	1	yes	characteristic
80	C₃₁×#126	—	2	8	no	—
81	C₃₁×#126	—	2	8	no	—
82	C₃₁×#125	—	2	4	no	—
83	C₃₁×#125	—	2	8	no	—
84	C₃₁×#125	C₂³	1	4	yes	—
85	C₃₁×#124	—	2	4	no	—
86	C₃₁×#124	—	2	4	no	—
87	C₃₁×#124	D₄	1	2	yes	—
88	C₃₁×#123	—	2	8	no	—
89	C₃₁×#123	—	2	8	no	—
90	C₃₁×C₂²⋊Q₈	—	2	2	no	—
91	C₃₁×C₂²⋊Q₈	—	2	2	no	—
92	C₃₁×C₂²⋊Q₈	—	2	8	no	—
93	C₃₁×C₂²⋊Q₈	C₂³	1	8	yes	—
94	C₃₁×C₂²⋊Q₈	C₂³	1	8	yes	—
95	C₃₁×#121	D₄	1	2	yes	—
96	C₃₁×#121	D₄	1	2	yes	—
97	C₄×Q₈×C₃₁	—	2	4	no	—
98	C₄×Q₈×C₃₁	—	2	4	no	—
99	C₄×Q₈×C₃₁	—	2	2	no	—
100	C₄×Q₈×C₃₁	—	2	2	no	—
101	C₄×Q₈×C₃₁	—	2	2	no	—
102	C₄×Q₈×C₃₁	—	2	2	no	—
103	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
104	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
105	C₄×D₄×C₃₁	—	2	8	no	—
106	C₄×D₄×C₃₁	—	2	4	no	—
107	C₄×D₄×C₃₁	—	2	4	no	—
108	C₄×D₄×C₃₁	C₂³	1	2	yes	—
109	C₄×D₄×C₃₁	C₂³	1	2	yes	—
110	C₃₁×#117	—	2	8	no	—
111	C₃₁×#117	C₂³	1	4	yes	—
112	C₃₁×#117	C₂³	1	8	yes	—
113	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
114	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
115	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
116	C₂×C₄²×C₃₁	—	2	2	no	—
117	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
118	C₃₁×#107	C₂³	1	2	yes	—
119	C₃₁×#107	C₂³	1	2	yes	—
120	C₃₁×#107	C₂³	1	2	yes	—
121	C₃₁×#107	C₂³	1	8	yes	—
122	C₃₁×#107	C₂³	1	2	yes	—
123	C₃₁×#107	C₂³	1	2	yes	—
124	C₃₁×#107	C₂³	1	2	yes	—
125	C₃₁×#106	C₂³	1	4	yes	—
126	C₃₁×#106	C₂³	1	8	yes	—
127	C₃₁×#106	C₂³	1	4	yes	—
128	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
129	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
130	C₃₁×#103	C₂³	1	4	yes	—
131	C₃₁×#103	C₂³	1	4	yes	—
132	C₃₁×#103	C₂³	1	8	yes	—
133	C₃₁×#103	C₂³	1	8	yes	—
134	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
135	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
136	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
137	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
138	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
139	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
140	#41×C₃₁	—	2	4	no	—
141	#41×C₃₁	—	2	4	no	—
142	C₂×Q₈×C₃₁	—	2	2	no	—
143	C₂×Q₈×C₃₁	—	2	2	no	—
144	C₂×Q₈×C₃₁	—	2	2	no	—
145	C₂×Q₈×C₃₁	—	2	4	no	—
146	C₂×Q₈×C₃₁	—	2	4	no	—
147	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
148	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
149	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
150	C₂×D₄×C₃₁	C₂×D₄	1	4	yes	—
151	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
152	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
153	C₂²×C₄×C₃₁	—	2	2	no	—
154	C₂²×C₄×C₃₁	—	2	2	no	—
155	C₂²×C₄×C₃₁	—	2	8	no	—
156	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
157	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
158	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
159	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
160	Q₁₆×C₃₁	—	2	4	no	—
161	Q₁₆×C₃₁	—	2	4	no	—
162	M₁₆×C₃₁	—	2	4	no	—
163	M₁₆×C₃₁	—	2	8	no	—
164	C₂×C₈×C₃₁	—	2	2	no	—
165	C₂×C₈×C₃₁	—	2	8	no	—
166	C₂×C₈×C₃₁	#41	1	4	yes	—
167	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
168	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
169	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
170	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
171	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
172	C₄⋊C₄×C₃₁	—	2	4	no	—
173	C₄⋊C₄×C₃₁	—	2	4	no	—
174	C₄⋊C₄×C₃₁	—	2	2	no	—
175	C₄⋊C₄×C₃₁	—	2	8	no	—
176	C₄⋊C₄×C₃₁	—	2	4	no	—
177	C₄⋊C₄×C₃₁	—	2	8	no	—
178	C₄⋊C₄×C₃₁	—	2	2	no	—
179	C₄⋊C₄×C₃₁	—	2	4	no	—
180	C₄⋊C₄×C₃₁	—	2	2	no	—
181	C₄⋊C₄×C₃₁	—	2	8	no	—
182	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
183	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
184	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
185	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
186	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
187	C₂²⋊C₄×C₃₁	—	2	8	no	—
188	C₂²⋊C₄×C₃₁	—	2	4	no	—
189	C₂²⋊C₄×C₃₁	—	2	8	no	—
190	C₂²⋊C₄×C₃₁	—	2	8	no	—
191	C₂²⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
192	C₂²⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
193	C₄²×C₃₁	—	2	4	no	—
194	C₄²×C₃₁	—	2	4	no	—
195	C₄²×C₃₁	—	2	8	no	—
196	C₄²×C₃₁	—	2	4	no	—
197	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
198	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
199	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
200	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
201	#61286	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
202	C₂³×C₃₁	C₂²×D₄	1	4	yes	—
203	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
204	Q₈×C₃₁	—	2	4	no	—
205	Q₈×C₃₁	—	2	4	no	—
206	Q₈×C₃₁	—	2	4	no	—
207	Q₈×C₃₁	—	4	4	no	—
208	Q₈×C₃₁	—	4	4	no	—
209	Q₈×C₃₁	—	8	2	no	—
210	D₄×C₃₁	—	2	8	no	—
211	D₄×C₃₁	—	2	4	no	—
212	D₄×C₃₁	—	2	4	no	—
213	D₄×C₃₁	—	4	2	no	—
214	C₂×C₄×C₃₁	—	2	2	no	—
215	C₂×C₄×C₃₁	—	2	4	no	—
216	C₂×C₄×C₃₁	—	2	4	no	—
217	C₂×C₄×C₃₁	—	2	4	no	—
218	C₂×C₄×C₃₁	—	2	4	no	—
219	C₂×C₄×C₃₁	—	2	4	no	—
220	C₂×C₄×C₃₁	—	2	4	no	—
221	C₂×C₄×C₃₁	—	4	1	no	—
222	C₂×C₄×C₃₁	—	4	4	no	—
223	C₂×C₄×C₃₁	—	8	2	no	—
224	C₂×C₄×C₃₁	—	8	8	no	—
225	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
226	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
227	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
228	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
229	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
230	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
231	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
232	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
233	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
234	C₈×C₃₁	—	2	4	no	—
235	C₈×C₃₁	—	2	1	no	—
236	C₈×C₃₁	—	2	1	no	—
237	C₈×C₃₁	C₂×#41	1	2	yes	—
238	C₈×C₃₁	C₂×#41	1	2	yes	—
239	#3507	C₂×C₃₁	1	2	yes	—
240	#3356	C₂×C₃₁	1	4	yes	complemented
241	#3352	C₂×C₃₁	1	4	yes	complemented
242	#3342	C₂×C₃₁	1	2	yes	complemented
243	#3338	C₂×C₃₁	1	2	yes	complemented
244	#3336	C₂×C₃₁	1	4	yes	complemented
245	#3160	C₂×C₃₁	1	1	yes	characteristic, complemented
246	#3097	C₂×C₃₁	1	4	yes	complemented
247	#3078	C₂×C₃₁	1	2	yes	complemented
248	#3059	C₂×C₃₁	1	2	yes	complemented
249	C₂×C₄×Q₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
250	#2931	C₂×C₃₁	1	1	yes	characteristic
251	#2913	C₂×C₃₁	1	2	yes	complemented
252	C₂²×C₃₁	—	2	1	no	—
253	C₂²×C₃₁	—	2	1	no	—
254	C₂²×C₃₁	—	2	4	no	—
255	C₂²×C₃₁	—	2	4	no	—
256	C₂²×C₃₁	—	2	4	no	—
257	C₂²×C₃₁	#547	1	2	yes	—
258	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
259	C₄×C₃₁	—	2	1	no	—
260	C₄×C₃₁	—	2	1	no	—
261	C₄×C₃₁	—	2	4	no	—
262	C₄×C₃₁	—	2	1	no	—
263	C₄×C₃₁	—	4	4	no	—
264	C₄×C₃₁	—	4	4	no	—
265	C₄×C₃₁	—	4	4	no	—
266	C₄×C₃₁	—	4	4	no	—
267	C₄×C₃₁	#547	1	2	yes	—
268	C₄×C₃₁	#578	1	2	yes	—
269	C₄×C₃₁	C₂²×#41	1	1	yes	characteristic
270	C₄×C₃₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
271	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
272	C₂×#131	C₂²×C₃₁	1	2	yes	—
273	#559	—	2	2	no	—
274	#557	—	2	2	no	—
275	#556	—	2	4	no	—
276	#556	C₂²×C₃₁	1	4	yes	—
277	#554	C₂²×C₃₁	1	4	yes	—
278	#552	C₂²×C₃₁	1	8	yes	—
279	#539	—	2	4	no	—
280	#536	C₂²×C₃₁	1	4	yes	—
281	#533	C₂²×C₃₁	1	2	yes	—
282	#532	C₂²×C₃₁	1	1	yes	characteristic
283	#529	—	2	4	no	—
284	#520	C₂²×C₃₁	1	4	yes	—
285	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
286	#500	C₂²×C₃₁	1	1	yes	characteristic
287	#500	C₂²×C₃₁	1	1	yes	characteristic
288	#499	C₂²×C₃₁	1	2	yes	—
289	#499	C₂²×C₃₁	1	2	yes	—
290	#498	C₂²×C₃₁	1	2	yes	—
291	#485	C₂²×C₃₁	1	8	yes	—
292	#485	C₂²×C₃₁	1	8	yes	—
293	#484	C₂²×C₃₁	1	8	yes	—
294	#483	C₂²×C₃₁	1	8	yes	—
295	#479	C₂²×C₃₁	1	8	yes	—
296	#477	C₂²×C₃₁	1	4	yes	—
297	#475	C₂²×C₃₁	1	4	yes	—
298	#470	C₂²×C₃₁	1	8	yes	—
299	#467	C₂²×C₃₁	1	4	yes	—
300	#467	C₂²×C₃₁	1	4	yes	—
301	C₄×Q₁₆	C₂²×C₃₁	1	2	yes	—
302	C₄×Q₁₆	C₂²×C₃₁	1	2	yes	—
303	C₄×Q₁₆	C₂²×C₃₁	1	2	yes	—
304	C₄×Q₁₆	C₂²×C₃₁	1	2	yes	—
305	#436	C₂²×C₃₁	1	2	yes	—
306	#436	C₂²×C₃₁	1	2	yes	—
307	C₈×D₄	C₂²×C₃₁	1	2	yes	—
308	C₈×D₄	C₂²×C₃₁	1	2	yes	—
309	#428	C₂²×C₃₁	1	4	yes	—
310	#428	C₂²×C₃₁	1	8	yes	—
311	#427	C₂²×C₃₁	1	4	yes	—
312	#427	C₂²×C₃₁	1	2	yes	—
313	#427	C₂²×C₃₁	1	2	yes	—
314	C₂×#107	C₂²×C₃₁	1	4	yes	—
315	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
316	#423	C₂²×C₃₁	1	1	yes	characteristic
317	#419	C₂²×C₃₁	1	4	yes	—
318	C₂×#103	C₂²×C₃₁	1	2	yes	—
319	#408	C₂²×C₃₁	1	2	yes	—
320	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
321	C₂×C₃₁	—	2	2	no	—
322	C₂×C₃₁	—	4	4	no	—
323	C₂×C₃₁	C₂×#547	1	1	yes	characteristic
324	C₂×C₃₁	C₂×#578	1	1	yes	characteristic
325	C₂×C₃₁	#3584	1	1	yes	characteristic
326	C₂×#41	C₂³×C₃₁	1	2	yes	—
327	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
328	C₂×Q₁₆	—	2	4	no	—
329	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
330	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
331	#131	—	2	8	no	—
332	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
333	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
334	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
335	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
336	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
337	C₄⋊Q₈	—	2	4	no	—
338	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
339	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
340	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
341	#127	D₄×C₃₁	1	1	yes	characteristic
342	#126	—	2	8	no	—
343	#126	—	2	8	no	—
344	#125	—	2	4	no	—
345	#125	—	2	8	no	—
346	#125	C₂³×C₃₁	1	4	yes	—
347	#124	—	2	4	no	—
348	#124	—	2	4	no	—
349	#124	D₄×C₃₁	1	2	yes	—
350	#123	—	2	8	no	—
351	#123	—	2	8	no	—
352	C₂²⋊Q₈	—	2	2	no	—
353	C₂²⋊Q₈	—	2	2	no	—
354	C₂²⋊Q₈	—	2	8	no	—
355	C₂²⋊Q₈	C₂³×C₃₁	1	8	yes	—
356	C₂²⋊Q₈	C₂³×C₃₁	1	8	yes	—
357	#121	D₄×C₃₁	1	2	yes	—
358	#121	D₄×C₃₁	1	2	yes	—
359	C₄×Q₈	—	2	4	no	—
360	C₄×Q₈	—	2	4	no	—
361	C₄×Q₈	—	2	2	no	—
362	C₄×Q₈	—	2	2	no	—
363	C₄×Q₈	—	2	2	no	—
364	C₄×Q₈	—	2	2	no	—
365	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
366	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
367	C₄×D₄	—	2	8	no	—
368	C₄×D₄	—	2	4	no	—
369	C₄×D₄	—	2	4	no	—
370	C₄×D₄	C₂³×C₃₁	1	2	yes	—
371	C₄×D₄	C₂³×C₃₁	1	2	yes	—
372	#117	—	2	8	no	—
373	#117	C₂³×C₃₁	1	4	yes	—
374	#117	C₂³×C₃₁	1	8	yes	—
375	C₂×C₄⋊C₄	—	2	2	no	—
376	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
377	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
378	C₂×C₄²	—	2	2	no	—
379	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
380	#107	C₂³×C₃₁	1	2	yes	—
381	#107	C₂³×C₃₁	1	2	yes	—
382	#107	C₂³×C₃₁	1	2	yes	—
383	#107	C₂³×C₃₁	1	8	yes	—
384	#107	C₂³×C₃₁	1	2	yes	—
385	#107	C₂³×C₃₁	1	2	yes	—
386	#107	C₂³×C₃₁	1	2	yes	—
387	#106	C₂³×C₃₁	1	4	yes	—
388	#106	C₂³×C₃₁	1	8	yes	—
389	#106	C₂³×C₃₁	1	4	yes	—
390	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
391	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
392	#103	C₂³×C₃₁	1	4	yes	—
393	#103	C₂³×C₃₁	1	4	yes	—
394	#103	C₂³×C₃₁	1	8	yes	—
395	#103	C₂³×C₃₁	1	8	yes	—
396	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
397	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
398	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
399	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
400	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
401	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
402	C₃₁	#61286	1	1	yes	characteristic, direct product factor, Sylow
403	#41	—	2	4	no	—
404	#41	—	2	4	no	—
405	C₂×Q₈	—	2	2	no	—
406	C₂×Q₈	—	2	2	no	—
407	C₂×Q₈	—	2	2	no	—
408	C₂×Q₈	—	2	4	no	—
409	C₂×Q₈	—	2	4	no	—
410	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
411	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
412	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
413	C₂×D₄	C₂×D₄×C₃₁	1	4	yes	—
414	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
415	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
416	C₂²×C₄	—	2	2	no	—
417	C₂²×C₄	—	2	2	no	—
418	C₂²×C₄	—	2	8	no	—
419	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
420	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
421	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
422	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
423	Q₁₆	—	2	4	no	—
424	Q₁₆	—	2	4	no	—
425	M₁₆	—	2	4	no	—
426	M₁₆	—	2	8	no	—
427	C₂×C₈	—	2	2	no	—
428	C₂×C₈	—	2	8	no	—
429	C₂×C₈	#41×C₃₁	1	4	yes	—
430	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
431	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
432	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
433	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
434	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
435	C₄⋊C₄	—	2	4	no	—
436	C₄⋊C₄	—	2	4	no	—
437	C₄⋊C₄	—	2	2	no	—
438	C₄⋊C₄	—	2	8	no	—
439	C₄⋊C₄	—	2	4	no	—
440	C₄⋊C₄	—	2	8	no	—
441	C₄⋊C₄	—	2	2	no	—
442	C₄⋊C₄	—	2	4	no	—
443	C₄⋊C₄	—	2	2	no	—
444	C₄⋊C₄	—	2	8	no	—
445	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
446	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
447	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
448	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
449	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
450	C₂²⋊C₄	—	2	8	no	—
451	C₂²⋊C₄	—	2	4	no	—
452	C₂²⋊C₄	—	2	8	no	—
453	C₂²⋊C₄	—	2	8	no	—
454	C₂²⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
455	C₂²⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
456	C₄²	—	2	4	no	—
457	C₄²	—	2	4	no	—
458	C₄²	—	2	8	no	—
459	C₄²	—	2	4	no	—
460	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
461	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
462	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
463	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
464	C₂³	C₂²×D₄×C₃₁	1	4	yes	—
465	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
466	Q₈	—	2	4	no	—
467	Q₈	—	2	4	no	—
468	Q₈	—	2	4	no	—
469	Q₈	—	4	4	no	—
470	Q₈	—	4	4	no	—
471	Q₈	—	8	2	no	—
472	D₄	—	2	8	no	—
473	D₄	—	2	4	no	—
474	D₄	—	2	4	no	—
475	D₄	—	4	2	no	—
476	C₂×C₄	—	2	2	no	—
477	C₂×C₄	—	2	4	no	—
478	C₂×C₄	—	2	4	no	—
479	C₂×C₄	—	2	4	no	—
480	C₂×C₄	—	2	4	no	—
481	C₂×C₄	—	2	4	no	—
482	C₂×C₄	—	2	4	no	—
483	C₂×C₄	—	4	1	no	—
484	C₂×C₄	—	4	4	no	—
485	C₂×C₄	—	8	2	no	—
486	C₂×C₄	—	8	8	no	—
487	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
488	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
489	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
490	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
491	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
492	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
493	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
494	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
495	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
496	C₈	—	2	4	no	—
497	C₈	—	2	1	no	—
498	C₈	—	2	1	no	—
499	C₈	C₂×#41×C₃₁	1	2	yes	—
500	C₈	C₂×#41×C₃₁	1	2	yes	—
501	C₂²	—	2	1	no	—
502	C₂²	—	2	1	no	—
503	C₂²	—	2	4	no	—
504	C₂²	—	2	4	no	—
505	C₂²	—	2	4	no	—
506	C₂²	C₃₁×#547	1	2	yes	—
507	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
508	C₄	—	2	4	no	—
509	C₄	—	2	1	no	—
510	C₄	—	2	1	no	—
511	C₄	—	2	1	no	—
512	C₄	—	4	4	no	—
513	C₄	—	4	4	no	—
514	C₄	—	4	4	no	—
515	C₄	—	4	4	no	—
516	C₄	C₃₁×#547	1	2	yes	—
517	C₄	C₃₁×#578	1	2	yes	—
518	C₄	C₂²×#41×C₃₁	1	1	yes	characteristic
519	C₄	C₂×C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
520	C₂	—	2	2	no	—
521	C₂	—	4	4	no	—
522	C₂	C₂×C₃₁×#547	1	1	yes	characteristic
523	C₂	C₂×C₃₁×#578	1	1	yes	characteristic
524	C₂	C₃₁×#3584	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
prime radical	C₃₁×#532	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₃₁×#532 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#3356	C₂	semidirect
3. C₃₁×#3352	C₂	semidirect
4. C₃₁×#3342	C₂	semidirect
5. C₃₁×#3338	C₂	semidirect
6. C₃₁×#3336	C₂	semidirect
7. C₃₁×#3160	C₂	semidirect
8. C₃₁×#3097	C₂	semidirect
9. C₃₁×#3078	C₂	semidirect
10. C₃₁×#3059	C₂	semidirect
11. C₂×C₄×Q₁₆×C₃₁	C₂	semidirect
13. C₃₁×#2913	C₂	semidirect
201. #61286	C₃₁	direct (N × H)
240. #3356	C₂×C₃₁	semidirect
241. #3352	C₂×C₃₁	semidirect
242. #3342	C₂×C₃₁	semidirect
243. #3338	C₂×C₃₁	semidirect
244. #3336	C₂×C₃₁	semidirect
245. #3160	C₂×C₃₁	semidirect
246. #3097	C₂×C₃₁	semidirect
247. #3078	C₂×C₃₁	semidirect
248. #3059	C₂×C₃₁	semidirect
249. C₂×C₄×Q₁₆	C₂×C₃₁	semidirect
251. #2913	C₂×C₃₁	semidirect
402. C₃₁	#61286	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×#3507	2
C₃₁×#3356	2
C₃₁×#3352	2
C₃₁×#3342	2
C₃₁×#3338	2
C₃₁×#3336	2
C₃₁×#3160	2
C₃₁×#3097	2
C₃₁×#3078	2
C₃₁×#3059	2
C₂×C₄×Q₁₆×C₃₁	2
C₃₁×#2931	2
C₃₁×#2913	2
#61286	31

Subgroups — C31×#61286 #21518045

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#61286 #21518045