Subgroups — C₃₁×#61797 #21518556

Order 7936.

1802 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3549	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3512	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3348	C₂	1	4	yes	maximal, complemented
4	C₃₁×#3347	C₂	1	4	yes	maximal, complemented
5	C₃₁×#3318	C₂	1	4	yes	maximal, complemented
6	C₃₁×#3310	C₂	1	4	yes	maximal, complemented
7	C₃₁×#3234	C₂	1	2	yes	maximal, complemented
8	C₃₁×#3198	C₂	1	4	yes	maximal, complemented
9	C₃₁×#3115	C₂	1	2	yes	maximal, complemented
10	C₃₁×#3096	C₂	1	4	yes	maximal, complemented
11	C₃₁×#2980	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×H(3,𝔽₄)	—	2	2	no	—
13	C₃₁×#555	—	2	1	no	—
14	C₃₁×#555	C₂²	1	4	yes	—
15	C₃₁×#553	—	2	4	no	—
16	C₃₁×#553	C₂²	1	4	yes	complemented
17	C₃₁×#552	C₂²	1	4	yes	complemented
18	D₄²×C₃₁	—	2	1	no	—
19	D₄²×C₃₁	C₂²	1	4	yes	—
20	C₃₁×#541	—	2	1	no	—
21	C₃₁×#540	—	2	1	no	—
22	C₃₁×#537	C₂²	1	2	yes	complemented
23	C₃₁×#535	—	2	4	no	—
24	C₃₁×#535	C₂²	1	4	yes	—
25	C₃₁×#532	C₂²	1	2	yes	—
26	C₃₁×#532	C₂²	1	1	yes	characteristic
27	C₃₁×#529	—	2	2	no	—
28	C₂×C₃₁×#125	C₂²	1	1	yes	characteristic, complemented
29	C₃₁×#520	C₂²	1	2	yes	complemented
30	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic, complemented
31	C₃₁×#501	C₂²	1	4	yes	complemented
32	C₃₁×#490	C₂²	1	4	yes	—
33	C₃₁×#485	C₂²	1	4	yes	complemented
34	C₃₁×#482	C₂²	1	4	yes	complemented
35	C₃₁×#482	C₂²	1	4	yes	complemented
36	C₃₁×#482	C₂²	1	4	yes	complemented
37	C₃₁×#482	C₂²	1	4	yes	complemented
38	C₃₁×#481	C₂²	1	4	yes	complemented
39	C₃₁×#481	C₂²	1	4	yes	complemented
40	C₃₁×#480	C₂²	1	4	yes	complemented
41	C₃₁×#480	C₂²	1	4	yes	complemented
42	C₃₁×#478	C₂²	1	4	yes	complemented
43	C₃₁×#478	C₂²	1	4	yes	complemented
44	C₃₁×#475	C₂²	1	2	yes	complemented
45	C₃₁×#474	C₂²	1	2	yes	complemented
46	C₃₁×#469	C₂²	1	4	yes	complemented
47	C₃₁×#468	C₂²	1	4	yes	complemented
48	C₃₁×#466	C₂²	1	4	yes	complemented
49	C₃₁×#465	C₂²	1	4	yes	complemented
50	C₃₁×#460	C₂²	1	2	yes	—
51	C₃₁×#459	C₂²	1	2	yes	—
52	C₃₁×#450	C₂²	1	4	yes	complemented
53	C₃₁×#449	C₂²	1	4	yes	complemented
54	C₃₁×#447	C₂²	1	4	yes	complemented
55	C₃₁×#442	C₂²	1	4	yes	complemented
56	C₃₁×#440	C₂²	1	4	yes	complemented
57	C₃₁×#435	C₂²	1	4	yes	complemented
58	C₃₁×#435	C₂²	1	4	yes	complemented
59	C₃₁×#432	C₂²	1	2	yes	complemented
60	C₃₁×#428	C₂²	1	4	yes	complemented
61	C₃₁×#427	C₂²	1	4	yes	complemented
62	C₃₁×#418	C₂²	1	4	yes	complemented
63	C₂×C₃₁×#102	C₂²	1	4	yes	complemented
64	C₃₁×#408	C₂²	1	4	yes	complemented
65	C₂×#41×C₃₁	—	2	2	no	—
66	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
67	C₂×#41×C₃₁	C₂³	1	2	yes	—
68	C₂²×D₄×C₃₁	—	2	4	no	—
69	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
70	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
71	C₂×D₈×C₃₁	C₂³	1	4	yes	—
72	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
73	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
74	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
75	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
76	C₃₁×#127	—	2	4	no	—
77	C₃₁×#127	D₄	1	1	yes	characteristic
78	C₃₁×#127	C₂³	1	2	yes	—
79	C₃₁×#126	—	2	4	no	—
80	C₃₁×#126	—	2	4	no	—
81	C₃₁×#125	—	2	4	no	—
82	C₃₁×#125	—	2	1	no	—
83	C₃₁×#125	—	2	1	no	—
84	C₃₁×#125	C₂³	1	4	yes	—
85	C₃₁×#124	—	2	4	no	—
86	C₃₁×#124	—	2	4	no	—
87	C₃₁×#124	—	2	4	no	—
88	C₃₁×#124	—	2	2	no	—
89	C₃₁×#124	D₄	1	2	yes	—
90	C₃₁×#123	—	2	4	no	—
91	C₃₁×#123	—	2	4	no	—
92	C₃₁×C₂²⋊Q₈	—	2	2	no	—
93	C₃₁×C₂²⋊Q₈	C₂³	1	4	yes	—
94	C₃₁×#121	—	2	2	no	—
95	C₃₁×#121	—	2	4	no	—
96	C₃₁×#121	—	2	4	no	—
97	C₃₁×#121	—	2	4	no	—
98	C₃₁×#121	—	2	2	no	—
99	C₃₁×#121	—	2	4	no	—
100	C₃₁×#121	D₄	1	4	yes	—
101	C₃₁×#121	C₂³	1	4	yes	—
102	C₃₁×#121	C₂³	1	4	yes	—
103	C₃₁×#121	C₂³	1	4	yes	—
104	C₃₁×#121	C₂³	1	4	yes	—
105	C₃₁×#120	—	2	4	no	—
106	C₃₁×#120	—	2	4	no	—
107	C₃₁×#120	—	2	4	no	—
108	C₃₁×#120	—	2	4	no	—
109	C₄×Q₈×C₃₁	—	2	2	no	—
110	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
111	C₄×D₄×C₃₁	—	2	2	no	—
112	C₄×D₄×C₃₁	—	2	4	no	—
113	C₄×D₄×C₃₁	—	2	2	no	—
114	C₄×D₄×C₃₁	—	2	4	no	—
115	C₄×D₄×C₃₁	—	2	2	no	—
116	C₄×D₄×C₃₁	C₂³	1	4	yes	—
117	C₄×D₄×C₃₁	C₂³	1	2	yes	—
118	C₄×D₄×C₃₁	C₂³	1	4	yes	—
119	C₃₁×#117	—	2	4	no	—
120	C₃₁×#117	—	2	4	no	—
121	C₃₁×#117	—	2	1	no	—
122	C₃₁×#117	C₂³	1	4	yes	—
123	C₃₁×#117	C₂³	1	4	yes	—
124	C₃₁×#117	C₂³	1	1	yes	characteristic
125	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
126	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
127	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
128	C₂×C₄²×C₃₁	—	2	1	no	—
129	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
130	C₃₁×#107	C₂³	1	4	yes	—
131	C₃₁×#107	C₂³	1	4	yes	—
132	C₃₁×#107	C₂³	1	4	yes	—
133	C₃₁×#106	C₂³	1	4	yes	—
134	C₃₁×#106	C₂³	1	4	yes	—
135	C₃₁×#106	C₂³	1	4	yes	—
136	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
137	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
138	C₃₁×#103	C₂³	1	4	yes	—
139	C₃₁×#102	C₂³	1	4	yes	—
140	C₃₁×#102	C₂³	1	4	yes	—
141	C₃₁×#102	C₂³	1	4	yes	—
142	C₃₁×#102	C₂³	1	4	yes	—
143	C₃₁×#102	C₂³	1	4	yes	—
144	C₃₁×#102	C₂³	1	4	yes	—
145	C₃₁×#102	C₂³	1	4	yes	—
146	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
147	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
148	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
149	C₃₁×#97	C₂³	1	4	yes	—
150	C₂⁴×C₃₁	—	2	4	no	—
151	#41×C₃₁	—	4	2	no	—
152	#41×C₃₁	—	4	2	no	—
153	#41×C₃₁	—	4	2	no	—
154	#41×C₃₁	—	8	4	no	—
155	C₂×Q₈×C₃₁	—	2	2	no	—
156	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
157	C₂×Q₈×C₃₁	C₂⁴	1	1	yes	characteristic
158	C₂×D₄×C₃₁	—	2	1	no	—
159	C₂×D₄×C₃₁	—	2	4	no	—
160	C₂×D₄×C₃₁	—	2	4	no	—
161	C₂×D₄×C₃₁	—	2	4	no	—
162	C₂×D₄×C₃₁	—	2	4	no	—
163	C₂×D₄×C₃₁	—	2	2	no	—
164	C₂×D₄×C₃₁	—	2	2	no	—
165	C₂×D₄×C₃₁	—	2	4	no	—
166	C₂×D₄×C₃₁	—	4	4	no	—
167	C₂×D₄×C₃₁	—	4	4	no	—
168	C₂×D₄×C₃₁	—	8	8	no	—
169	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
170	C₂×D₄×C₃₁	C₂×D₄	1	4	yes	—
171	C₂×D₄×C₃₁	C₂×D₄	1	4	yes	—
172	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
173	C₂×D₄×C₃₁	C₂⁴	1	1	yes	characteristic
174	C₂²×C₄×C₃₁	—	2	2	no	—
175	C₂²×C₄×C₃₁	—	2	4	no	—
176	C₂²×C₄×C₃₁	—	2	1	no	—
177	C₂²×C₄×C₃₁	—	2	4	no	—
178	C₂²×C₄×C₃₁	—	2	2	no	—
179	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
180	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
181	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
182	SD₁₆×C₃₁	—	4	4	no	—
183	D₈×C₃₁	—	4	4	no	—
184	M₁₆×C₃₁	—	4	4	no	—
185	C₂×C₈×C₃₁	—	4	4	no	—
186	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
187	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
188	C₄⋊C₄×C₃₁	—	2	4	no	—
189	C₄⋊C₄×C₃₁	—	2	4	no	—
190	C₄⋊C₄×C₃₁	—	2	2	no	—
191	C₄⋊C₄×C₃₁	—	2	4	no	—
192	C₄⋊C₄×C₃₁	—	2	2	no	—
193	C₄⋊C₄×C₃₁	—	2	2	no	—
194	C₄⋊C₄×C₃₁	—	2	4	no	—
195	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
196	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
197	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
198	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
199	C₂²⋊C₄×C₃₁	—	2	4	no	—
200	C₂²⋊C₄×C₃₁	—	2	1	no	—
201	C₂²⋊C₄×C₃₁	—	2	4	no	—
202	C₂²⋊C₄×C₃₁	—	2	4	no	—
203	C₂²⋊C₄×C₃₁	—	2	2	no	—
204	C₂²⋊C₄×C₃₁	—	2	4	no	—
205	C₂²⋊C₄×C₃₁	—	2	4	no	—
206	C₂²⋊C₄×C₃₁	—	2	4	no	—
207	C₂²⋊C₄×C₃₁	—	2	4	no	—
208	C₂²⋊C₄×C₃₁	—	2	4	no	—
209	C₂²⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
210	C₂²⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
211	C₄²×C₃₁	—	2	2	no	—
212	C₄²×C₃₁	—	2	1	no	—
213	C₄²×C₃₁	—	2	4	no	—
214	C₄²×C₃₁	—	2	4	no	—
215	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
216	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
217	C₄²×C₃₁	C₂⁴	1	2	yes	—
218	#61797	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
219	C₂³×C₃₁	—	2	1	no	—
220	C₂³×C₃₁	—	2	4	no	—
221	C₂³×C₃₁	—	8	4	no	—
222	C₂³×C₃₁	—	8	4	no	—
223	C₂³×C₃₁	—	8	4	no	—
224	C₂³×C₃₁	C₂²×D₄	1	4	yes	—
225	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
226	Q₈×C₃₁	—	2	2	no	—
227	Q₈×C₃₁	—	4	2	no	—
228	Q₈×C₃₁	—	8	2	no	—
229	D₄×C₃₁	—	2	2	no	—
230	D₄×C₃₁	—	2	4	no	—
231	D₄×C₃₁	—	2	4	no	—
232	D₄×C₃₁	—	4	4	no	—
233	D₄×C₃₁	—	4	2	no	—
234	D₄×C₃₁	—	4	4	no	—
235	D₄×C₃₁	—	4	4	no	—
236	D₄×C₃₁	—	4	2	no	—
237	D₄×C₃₁	—	8	4	no	—
238	D₄×C₃₁	—	8	2	no	—
239	D₄×C₃₁	—	8	4	no	—
240	D₄×C₃₁	—	8	4	no	—
241	D₄×C₃₁	—	8	4	no	—
242	D₄×C₃₁	—	8	4	no	—
243	C₂×C₄×C₃₁	—	2	4	no	—
244	C₂×C₄×C₃₁	—	2	4	no	—
245	C₂×C₄×C₃₁	—	2	2	no	—
246	C₂×C₄×C₃₁	—	2	2	no	—
247	C₂×C₄×C₃₁	—	2	4	no	—
248	C₂×C₄×C₃₁	—	2	1	no	—
249	C₂×C₄×C₃₁	—	2	4	no	—
250	C₂×C₄×C₃₁	—	4	2	no	—
251	C₂×C₄×C₃₁	—	4	8	no	—
252	C₂×C₄×C₃₁	—	4	2	no	—
253	C₂×C₄×C₃₁	—	8	2	no	—
254	C₂×C₄×C₃₁	—	8	4	no	—
255	C₂×C₄×C₃₁	—	8	2	no	—
256	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
257	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
258	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
259	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
260	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
261	C₈×C₃₁	—	2	4	no	—
262	C₈×C₃₁	—	2	4	no	—
263	#3549	C₂×C₃₁	1	1	yes	characteristic, complemented
264	#3512	C₂×C₃₁	1	1	yes	characteristic, complemented
265	#3348	C₂×C₃₁	1	4	yes	complemented
266	#3347	C₂×C₃₁	1	4	yes	complemented
267	#3318	C₂×C₃₁	1	4	yes	complemented
268	#3310	C₂×C₃₁	1	4	yes	complemented
269	#3234	C₂×C₃₁	1	2	yes	complemented
270	#3198	C₂×C₃₁	1	4	yes	complemented
271	#3115	C₂×C₃₁	1	2	yes	complemented
272	#3096	C₂×C₃₁	1	4	yes	complemented
273	#2980	C₂×C₃₁	1	1	yes	characteristic, complemented
274	C₂²×C₃₁	—	2	2	no	—
275	C₂²×C₃₁	—	2	4	no	—
276	C₂²×C₃₁	—	2	4	no	—
277	C₂²×C₃₁	—	2	1	no	—
278	C₂²×C₃₁	—	2	4	no	—
279	C₂²×C₃₁	—	4	4	no	—
280	C₂²×C₃₁	—	4	2	no	—
281	C₂²×C₃₁	—	4	4	no	—
282	C₂²×C₃₁	—	4	1	no	—
283	C₂²×C₃₁	—	4	4	no	—
284	C₂²×C₃₁	—	16	8	no	—
285	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
286	C₄×C₃₁	—	2	2	no	—
287	C₄×C₃₁	—	2	2	no	—
288	C₄×C₃₁	—	2	1	no	—
289	C₄×C₃₁	—	2	4	no	—
290	C₄×C₃₁	—	4	4	no	—
291	C₄×C₃₁	—	4	2	no	—
292	C₄×C₃₁	—	4	4	no	—
293	C₄×C₃₁	—	4	1	no	—
294	C₄×C₃₁	#585	1	2	yes	—
295	H(3,𝔽₄)	—	2	2	no	—
296	#555	—	2	1	no	—
297	#555	C₂²×C₃₁	1	4	yes	—
298	#553	—	2	4	no	—
299	#553	C₂²×C₃₁	1	4	yes	complemented
300	#552	C₂²×C₃₁	1	4	yes	complemented
301	D₄²	—	2	1	no	—
302	D₄²	C₂²×C₃₁	1	4	yes	—
303	#541	—	2	1	no	—
304	#540	—	2	1	no	—
305	#537	C₂²×C₃₁	1	2	yes	complemented
306	#535	—	2	4	no	—
307	#535	C₂²×C₃₁	1	4	yes	—
308	#532	C₂²×C₃₁	1	2	yes	—
309	#532	C₂²×C₃₁	1	1	yes	characteristic
310	#529	—	2	2	no	—
311	C₂×#125	C₂²×C₃₁	1	1	yes	characteristic, complemented
312	#520	C₂²×C₃₁	1	2	yes	complemented
313	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic, complemented
314	#501	C₂²×C₃₁	1	4	yes	complemented
315	#490	C₂²×C₃₁	1	4	yes	—
316	#485	C₂²×C₃₁	1	4	yes	complemented
317	#482	C₂²×C₃₁	1	4	yes	complemented
318	#482	C₂²×C₃₁	1	4	yes	complemented
319	#482	C₂²×C₃₁	1	4	yes	complemented
320	#482	C₂²×C₃₁	1	4	yes	complemented
321	#481	C₂²×C₃₁	1	4	yes	complemented
322	#481	C₂²×C₃₁	1	4	yes	complemented
323	#480	C₂²×C₃₁	1	4	yes	complemented
324	#480	C₂²×C₃₁	1	4	yes	complemented
325	#478	C₂²×C₃₁	1	4	yes	complemented
326	#478	C₂²×C₃₁	1	4	yes	complemented
327	#475	C₂²×C₃₁	1	2	yes	complemented
328	#474	C₂²×C₃₁	1	2	yes	complemented
329	#469	C₂²×C₃₁	1	4	yes	complemented
330	#468	C₂²×C₃₁	1	4	yes	complemented
331	#466	C₂²×C₃₁	1	4	yes	complemented
332	#465	C₂²×C₃₁	1	4	yes	complemented
333	#460	C₂²×C₃₁	1	2	yes	—
334	#459	C₂²×C₃₁	1	2	yes	—
335	#450	C₂²×C₃₁	1	4	yes	complemented
336	#449	C₂²×C₃₁	1	4	yes	complemented
337	#447	C₂²×C₃₁	1	4	yes	complemented
338	#442	C₂²×C₃₁	1	4	yes	complemented
339	#440	C₂²×C₃₁	1	4	yes	complemented
340	#435	C₂²×C₃₁	1	4	yes	complemented
341	#435	C₂²×C₃₁	1	4	yes	complemented
342	#432	C₂²×C₃₁	1	2	yes	complemented
343	#428	C₂²×C₃₁	1	4	yes	complemented
344	#427	C₂²×C₃₁	1	4	yes	complemented
345	#418	C₂²×C₃₁	1	4	yes	complemented
346	C₂×#102	C₂²×C₃₁	1	4	yes	complemented
347	#408	C₂²×C₃₁	1	4	yes	complemented
348	C₂×C₃₁	—	4	1	no	—
349	C₂×C₃₁	—	4	4	no	—
350	C₂×C₃₁	—	8	1	no	—
351	C₂×C₃₁	—	8	4	no	—
352	C₂×C₃₁	#3485	1	1	yes	characteristic
353	C₂×C₃₁	#3585	1	2	yes	—
354	C₂×#41	—	2	2	no	—
355	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
356	C₂×#41	C₂³×C₃₁	1	2	yes	—
357	C₂²×D₄	—	2	4	no	—
358	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
359	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
360	C₂×D₈	C₂³×C₃₁	1	4	yes	—
361	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
362	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
363	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
364	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
365	#127	—	2	4	no	—
366	#127	D₄×C₃₁	1	1	yes	characteristic
367	#127	C₂³×C₃₁	1	2	yes	—
368	#126	—	2	4	no	—
369	#126	—	2	4	no	—
370	#125	—	2	4	no	—
371	#125	—	2	1	no	—
372	#125	—	2	1	no	—
373	#125	C₂³×C₃₁	1	4	yes	—
374	#124	—	2	4	no	—
375	#124	—	2	4	no	—
376	#124	—	2	4	no	—
377	#124	—	2	2	no	—
378	#124	D₄×C₃₁	1	2	yes	—
379	#123	—	2	4	no	—
380	#123	—	2	4	no	—
381	C₂²⋊Q₈	—	2	2	no	—
382	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
383	#121	—	2	2	no	—
384	#121	—	2	4	no	—
385	#121	—	2	4	no	—
386	#121	—	2	4	no	—
387	#121	—	2	2	no	—
388	#121	—	2	4	no	—
389	#121	D₄×C₃₁	1	4	yes	—
390	#121	C₂³×C₃₁	1	4	yes	—
391	#121	C₂³×C₃₁	1	4	yes	—
392	#121	C₂³×C₃₁	1	4	yes	—
393	#121	C₂³×C₃₁	1	4	yes	—
394	#120	—	2	4	no	—
395	#120	—	2	4	no	—
396	#120	—	2	4	no	—
397	#120	—	2	4	no	—
398	C₄×Q₈	—	2	2	no	—
399	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
400	C₄×D₄	—	2	2	no	—
401	C₄×D₄	—	2	4	no	—
402	C₄×D₄	—	2	2	no	—
403	C₄×D₄	—	2	4	no	—
404	C₄×D₄	—	2	2	no	—
405	C₄×D₄	C₂³×C₃₁	1	2	yes	—
406	C₄×D₄	C₂³×C₃₁	1	4	yes	—
407	C₄×D₄	C₂³×C₃₁	1	4	yes	—
408	#117	—	2	4	no	—
409	#117	—	2	4	no	—
410	#117	—	2	1	no	—
411	#117	C₂³×C₃₁	1	1	yes	characteristic
412	#117	C₂³×C₃₁	1	4	yes	—
413	#117	C₂³×C₃₁	1	4	yes	—
414	C₂×C₄⋊C₄	—	2	2	no	—
415	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
416	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
417	C₂×C₄²	—	2	1	no	—
418	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
419	#107	C₂³×C₃₁	1	4	yes	—
420	#107	C₂³×C₃₁	1	4	yes	—
421	#107	C₂³×C₃₁	1	4	yes	—
422	#106	C₂³×C₃₁	1	4	yes	—
423	#106	C₂³×C₃₁	1	4	yes	—
424	#106	C₂³×C₃₁	1	4	yes	—
425	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
426	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
427	#103	C₂³×C₃₁	1	4	yes	—
428	#102	C₂³×C₃₁	1	4	yes	—
429	#102	C₂³×C₃₁	1	4	yes	—
430	#102	C₂³×C₃₁	1	4	yes	—
431	#102	C₂³×C₃₁	1	4	yes	—
432	#102	C₂³×C₃₁	1	4	yes	—
433	#102	C₂³×C₃₁	1	4	yes	—
434	#102	C₂³×C₃₁	1	4	yes	—
435	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
436	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
437	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
438	#97	C₂³×C₃₁	1	4	yes	—
439	C₃₁	#61797	1	1	yes	characteristic, direct product factor, Sylow
440	C₂⁴	—	2	4	no	—
441	#41	—	4	2	no	—
442	#41	—	4	2	no	—
443	#41	—	4	2	no	—
444	#41	—	8	4	no	—
445	C₂×Q₈	—	2	2	no	—
446	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
447	C₂×Q₈	C₂⁴×C₃₁	1	1	yes	characteristic
448	C₂×D₄	—	2	1	no	—
449	C₂×D₄	—	2	4	no	—
450	C₂×D₄	—	2	4	no	—
451	C₂×D₄	—	2	4	no	—
452	C₂×D₄	—	2	4	no	—
453	C₂×D₄	—	2	2	no	—
454	C₂×D₄	—	2	2	no	—
455	C₂×D₄	—	2	4	no	—
456	C₂×D₄	—	4	4	no	—
457	C₂×D₄	—	4	4	no	—
458	C₂×D₄	—	8	8	no	—
459	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
460	C₂×D₄	C₂×D₄×C₃₁	1	4	yes	—
461	C₂×D₄	C₂×D₄×C₃₁	1	4	yes	—
462	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
463	C₂×D₄	C₂⁴×C₃₁	1	1	yes	characteristic
464	C₂²×C₄	—	2	2	no	—
465	C₂²×C₄	—	2	4	no	—
466	C₂²×C₄	—	2	1	no	—
467	C₂²×C₄	—	2	4	no	—
468	C₂²×C₄	—	2	2	no	—
469	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
470	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
471	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
472	SD₁₆	—	4	4	no	—
473	D₈	—	4	4	no	—
474	M₁₆	—	4	4	no	—
475	C₂×C₈	—	4	4	no	—
476	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
477	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
478	C₄⋊C₄	—	2	4	no	—
479	C₄⋊C₄	—	2	4	no	—
480	C₄⋊C₄	—	2	2	no	—
481	C₄⋊C₄	—	2	4	no	—
482	C₄⋊C₄	—	2	2	no	—
483	C₄⋊C₄	—	2	2	no	—
484	C₄⋊C₄	—	2	4	no	—
485	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
486	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
487	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
488	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
489	C₂²⋊C₄	—	2	4	no	—
490	C₂²⋊C₄	—	2	1	no	—
491	C₂²⋊C₄	—	2	4	no	—
492	C₂²⋊C₄	—	2	4	no	—
493	C₂²⋊C₄	—	2	2	no	—
494	C₂²⋊C₄	—	2	4	no	—
495	C₂²⋊C₄	—	2	4	no	—
496	C₂²⋊C₄	—	2	4	no	—
497	C₂²⋊C₄	—	2	4	no	—
498	C₂²⋊C₄	—	2	4	no	—
499	C₂²⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
500	C₂²⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
501	C₄²	—	2	2	no	—
502	C₄²	—	2	1	no	—
503	C₄²	—	2	4	no	—
504	C₄²	—	2	4	no	—
505	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
506	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
507	C₄²	C₂⁴×C₃₁	1	2	yes	—
508	C₂³	—	2	1	no	—
509	C₂³	—	2	4	no	—
510	C₂³	—	8	4	no	—
511	C₂³	—	8	4	no	—
512	C₂³	—	8	4	no	—
513	C₂³	C₂²×D₄×C₃₁	1	4	yes	—
514	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
515	Q₈	—	2	2	no	—
516	Q₈	—	4	2	no	—
517	Q₈	—	8	2	no	—
518	D₄	—	2	2	no	—
519	D₄	—	2	4	no	—
520	D₄	—	2	4	no	—
521	D₄	—	4	4	no	—
522	D₄	—	4	2	no	—
523	D₄	—	4	4	no	—
524	D₄	—	4	4	no	—
525	D₄	—	4	2	no	—
526	D₄	—	8	4	no	—
527	D₄	—	8	2	no	—
528	D₄	—	8	4	no	—
529	D₄	—	8	4	no	—
530	D₄	—	8	4	no	—
531	D₄	—	8	4	no	—
532	C₂×C₄	—	2	4	no	—
533	C₂×C₄	—	2	4	no	—
534	C₂×C₄	—	2	2	no	—
535	C₂×C₄	—	2	2	no	—
536	C₂×C₄	—	2	4	no	—
537	C₂×C₄	—	2	1	no	—
538	C₂×C₄	—	2	4	no	—
539	C₂×C₄	—	4	2	no	—
540	C₂×C₄	—	4	8	no	—
541	C₂×C₄	—	4	2	no	—
542	C₂×C₄	—	8	2	no	—
543	C₂×C₄	—	8	4	no	—
544	C₂×C₄	—	8	2	no	—
545	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
546	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
547	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
548	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
549	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
550	C₈	—	2	4	no	—
551	C₈	—	2	4	no	—
552	C₂²	—	2	1	no	—
553	C₂²	—	2	4	no	—
554	C₂²	—	2	2	no	—
555	C₂²	—	2	4	no	—
556	C₂²	—	2	4	no	—
557	C₂²	—	4	1	no	—
558	C₂²	—	4	4	no	—
559	C₂²	—	4	4	no	—
560	C₂²	—	4	2	no	—
561	C₂²	—	4	4	no	—
562	C₂²	—	16	8	no	—
563	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
564	C₄	—	2	1	no	—
565	C₄	—	2	4	no	—
566	C₄	—	2	2	no	—
567	C₄	—	2	2	no	—
568	C₄	—	4	1	no	—
569	C₄	—	4	4	no	—
570	C₄	—	4	2	no	—
571	C₄	—	4	4	no	—
572	C₄	C₃₁×#585	1	2	yes	—
573	C₂	—	4	4	no	—
574	C₂	—	4	1	no	—
575	C₂	—	8	4	no	—
576	C₂	—	8	1	no	—
577	C₂	C₃₁×#3485	1	1	yes	characteristic
578	C₂	C₃₁×#3585	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
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
1. C₃₁×#3549	C₂	semidirect
2. C₃₁×#3512	C₂	semidirect
3. C₃₁×#3348	C₂	semidirect
4. C₃₁×#3347	C₂	semidirect
5. C₃₁×#3318	C₂	semidirect
6. C₃₁×#3310	C₂	semidirect
7. C₃₁×#3234	C₂	semidirect
8. C₃₁×#3198	C₂	semidirect
9. C₃₁×#3115	C₂	semidirect
10. C₃₁×#3096	C₂	semidirect
11. C₃₁×#2980	C₂	semidirect
16. C₃₁×#553	C₂²	semidirect
17. C₃₁×#552	C₂²	semidirect
22. C₃₁×#537	C₂²	semidirect
28. C₂×C₃₁×#125	C₂²	semidirect
29. C₃₁×#520	C₂²	semidirect
30. C₄×#41×C₃₁	C₂²	semidirect
31. C₃₁×#501	C₂²	semidirect
33. C₃₁×#485	C₂²	semidirect
34. C₃₁×#482	C₂²	semidirect
35. C₃₁×#482	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×#478	C₂²	semidirect
43. C₃₁×#478	C₂²	semidirect
44. C₃₁×#475	C₂²	semidirect
45. C₃₁×#474	C₂²	semidirect
46. C₃₁×#469	C₂²	semidirect
47. C₃₁×#468	C₂²	semidirect
48. C₃₁×#466	C₂²	semidirect
49. C₃₁×#465	C₂²	semidirect
52. C₃₁×#450	C₂²	semidirect
53. C₃₁×#449	C₂²	semidirect
54. C₃₁×#447	C₂²	semidirect
55. C₃₁×#442	C₂²	semidirect
56. C₃₁×#440	C₂²	semidirect
57. C₃₁×#435	C₂²	semidirect
58. C₃₁×#435	C₂²	semidirect
59. C₃₁×#432	C₂²	semidirect
60. C₃₁×#428	C₂²	semidirect
61. C₃₁×#427	C₂²	semidirect
62. C₃₁×#418	C₂²	semidirect
63. C₂×C₃₁×#102	C₂²	semidirect
64. C₃₁×#408	C₂²	semidirect
218. #61797	C₃₁	direct (N × H)
263. #3549	C₂×C₃₁	semidirect
264. #3512	C₂×C₃₁	semidirect
265. #3348	C₂×C₃₁	semidirect
266. #3347	C₂×C₃₁	semidirect
267. #3318	C₂×C₃₁	semidirect
268. #3310	C₂×C₃₁	semidirect
269. #3234	C₂×C₃₁	semidirect
270. #3198	C₂×C₃₁	semidirect
271. #3115	C₂×C₃₁	semidirect
272. #3096	C₂×C₃₁	semidirect
273. #2980	C₂×C₃₁	semidirect
299. #553	C₂²×C₃₁	semidirect
300. #552	C₂²×C₃₁	semidirect
305. #537	C₂²×C₃₁	semidirect
311. C₂×#125	C₂²×C₃₁	semidirect
312. #520	C₂²×C₃₁	semidirect
313. C₄×#41	C₂²×C₃₁	semidirect
314. #501	C₂²×C₃₁	semidirect
316. #485	C₂²×C₃₁	semidirect
317. #482	C₂²×C₃₁	semidirect
318. #482	C₂²×C₃₁	semidirect
319. #482	C₂²×C₃₁	semidirect
320. #482	C₂²×C₃₁	semidirect
321. #481	C₂²×C₃₁	semidirect
322. #481	C₂²×C₃₁	semidirect
323. #480	C₂²×C₃₁	semidirect
324. #480	C₂²×C₃₁	semidirect
325. #478	C₂²×C₃₁	semidirect
326. #478	C₂²×C₃₁	semidirect
327. #475	C₂²×C₃₁	semidirect
328. #474	C₂²×C₃₁	semidirect
329. #469	C₂²×C₃₁	semidirect
330. #468	C₂²×C₃₁	semidirect
331. #466	C₂²×C₃₁	semidirect
332. #465	C₂²×C₃₁	semidirect
335. #450	C₂²×C₃₁	semidirect
336. #449	C₂²×C₃₁	semidirect
337. #447	C₂²×C₃₁	semidirect
338. #442	C₂²×C₃₁	semidirect
339. #440	C₂²×C₃₁	semidirect
340. #435	C₂²×C₃₁	semidirect
341. #435	C₂²×C₃₁	semidirect
342. #432	C₂²×C₃₁	semidirect
343. #428	C₂²×C₃₁	semidirect
344. #427	C₂²×C₃₁	semidirect
345. #418	C₂²×C₃₁	semidirect
346. C₂×#102	C₂²×C₃₁	semidirect
347. #408	C₂²×C₃₁	semidirect
439. C₃₁	#61797	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×#3549	2
C₃₁×#3512	2
C₃₁×#3348	2
C₃₁×#3347	2
C₃₁×#3318	2
C₃₁×#3310	2
C₃₁×#3234	2
C₃₁×#3198	2
C₃₁×#3115	2
C₃₁×#3096	2
C₃₁×#2980	2
#61797	31

Subgroups — C31×#61797 #21518556

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#61797 #21518556