Subgroups — C₃₁×#18616 #21475375

Order 7936.

964 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3506	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3214	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3182	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#3100	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1647	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1646	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1625	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1624	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1554	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1490	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1484	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#559	—	2	2	no	—
13	C₃₁×#556	—	2	1	no	—
14	C₃₁×#555	C₂²	1	2	yes	—
15	C₃₁×#553	—	2	2	no	—
16	C₃₁×#550	—	2	1	no	—
17	C₃₁×#547	C₂²	1	2	yes	—
18	C₃₁×#538	—	2	2	no	—
19	C₃₁×#535	—	2	1	no	—
20	C₃₁×#533	C₂²	1	1	yes	characteristic
21	C₃₁×#532	C₂²	1	1	yes	characteristic
22	C₃₁×#532	C₂²	1	1	yes	characteristic
23	C₃₁×#525	—	2	2	no	—
24	C₃₁×#520	—	2	1	no	—
25	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
26	C₃₁×#500	C₂²	1	1	yes	characteristic, complemented
27	C₃₁×#496	—	2	1	no	—
28	C₃₁×#492	C₂²	1	1	yes	characteristic, complemented
29	C₃₁×#489	—	2	1	no	—
30	C₃₁×#487	C₂²	1	1	yes	characteristic, complemented
31	C₃₁×#486	C₂²	1	1	yes	characteristic, complemented
32	C₃₁×#480	—	2	1	no	—
33	C₃₁×#474	C₂²	1	2	yes	—
34	C₃₁×#468	—	2	2	no	—
35	C₃₁×#461	C₂²	1	2	yes	complemented
36	C₃₁×#459	C₂²	1	2	yes	—
37	C₃₁×#459	C₂²	1	2	yes	complemented
38	C₃₁×#449	—	2	2	no	—
39	C₃₁×#447	—	2	1	no	—
40	C₃₁×#440	—	2	2	no	—
41	C₄×D₈×C₃₁	—	2	2	no	—
42	C₃₁×#432	C₂²	1	1	yes	characteristic
43	C₃₁×#431	C₂²	1	1	yes	characteristic
44	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
45	C₃₁×#333	C₂²	1	2	yes	complemented
46	C₃₁×#332	C₂²	1	2	yes	complemented
47	C₃₁×D₄⋊C₈	C₂²	1	2	yes	—
48	C₃₁×D₄⋊C₈	C₂²	1	2	yes	—
49	C₃₁×D₄⋊C₈	C₂²	1	2	yes	complemented
50	C₃₁×D₄⋊C₈	C₂²	1	2	yes	complemented
51	C₂×#41×C₃₁	—	2	2	no	—
52	C₂×#41×C₃₁	—	2	1	no	—
53	C₂×#41×C₃₁	—	2	1	no	—
54	C₂²×D₄×C₃₁	D₄	1	1	yes	characteristic
55	C₂³×C₄×C₃₁	—	2	1	no	—
56	C₂×SD₁₆×C₃₁	—	2	2	no	—
57	C₂×SD₁₆×C₃₁	—	2	2	no	—
58	C₂×D₈×C₃₁	—	2	2	no	—
59	C₂×D₈×C₃₁	—	2	2	no	—
60	C₂×M₁₆×C₃₁	—	2	1	no	—
61	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
62	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
63	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
64	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
65	C₃₁×#127	—	2	2	no	—
66	C₃₁×#127	D₄	1	1	yes	characteristic
67	C₃₁×#127	C₂³	1	1	yes	characteristic
68	C₃₁×#126	—	2	4	no	—
69	C₃₁×#125	—	2	1	no	—
70	C₃₁×#124	—	2	2	no	—
71	C₃₁×#124	—	2	4	no	—
72	C₃₁×#124	—	2	1	no	—
73	C₃₁×#124	—	2	1	no	—
74	C₃₁×#123	—	2	4	no	—
75	C₃₁×#123	—	2	4	no	—
76	C₃₁×C₂²⋊Q₈	—	2	1	no	—
77	C₃₁×C₂²⋊Q₈	—	2	2	no	—
78	C₃₁×C₂²⋊Q₈	D₄	1	2	yes	—
79	C₃₁×#121	—	2	4	no	—
80	C₃₁×#121	—	2	2	no	—
81	C₃₁×#121	—	2	2	no	—
82	C₃₁×#121	—	2	2	no	—
83	C₃₁×#121	—	2	1	no	—
84	C₃₁×#121	—	2	1	no	—
85	C₃₁×#121	—	2	1	no	—
86	C₃₁×#121	D₄	1	2	yes	—
87	C₃₁×#120	—	2	4	no	—
88	C₄×Q₈×C₃₁	—	2	2	no	—
89	C₄×Q₈×C₃₁	—	2	1	no	—
90	C₄×Q₈×C₃₁	D₄	1	2	yes	—
91	C₄×D₄×C₃₁	—	2	4	no	—
92	C₄×D₄×C₃₁	—	2	2	no	—
93	C₄×D₄×C₃₁	—	2	4	no	—
94	C₄×D₄×C₃₁	—	2	1	no	—
95	C₄×D₄×C₃₁	—	2	1	no	—
96	C₄×D₄×C₃₁	—	2	1	no	—
97	C₄×D₄×C₃₁	—	2	1	no	—
98	C₄×D₄×C₃₁	—	2	1	no	—
99	C₄×D₄×C₃₁	D₄	1	2	yes	—
100	C₄×D₄×C₃₁	C₂³	1	2	yes	—
101	C₄×D₄×C₃₁	C₂³	1	2	yes	—
102	C₃₁×#117	—	2	1	no	—
103	C₃₁×#117	—	2	2	no	—
104	C₃₁×#117	—	2	2	no	—
105	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
106	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
107	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
108	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
109	C₃₁×#107	—	2	2	no	—
110	C₃₁×#107	—	2	2	no	—
111	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
112	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
113	C₃₁×#103	—	2	2	no	—
114	C₃₁×#103	—	2	2	no	—
115	C₃₁×#102	—	2	2	no	—
116	C₃₁×#102	—	2	2	no	—
117	C₃₁×#102	—	2	2	no	—
118	C₃₁×#102	—	2	2	no	—
119	C₃₁×C₂²⋊C₈	—	2	1	no	—
120	C₃₁×C₂²⋊C₈	—	2	1	no	—
121	C₃₁×#97	—	2	1	no	—
122	C₃₁×#97	—	2	1	no	—
123	C₄×C₈×C₃₁	—	2	1	no	—
124	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
125	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
126	C₂⁴×C₃₁	—	2	1	no	—
127	#41×C₃₁	—	4	4	no	—
128	#41×C₃₁	—	8	2	no	—
129	#41×C₃₁	—	8	2	no	—
130	#41×C₃₁	—	8	2	no	—
131	C₂×Q₈×C₃₁	—	2	1	no	—
132	C₂×Q₈×C₃₁	—	2	1	no	—
133	C₂×Q₈×C₃₁	—	2	1	no	—
134	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
135	C₂×D₄×C₃₁	—	2	2	no	—
136	C₂×D₄×C₃₁	—	2	4	no	—
137	C₂×D₄×C₃₁	—	2	4	no	—
138	C₂×D₄×C₃₁	—	2	1	no	—
139	C₂×D₄×C₃₁	—	2	1	no	—
140	C₂×D₄×C₃₁	—	2	1	no	—
141	C₂×D₄×C₃₁	—	2	1	no	—
142	C₂×D₄×C₃₁	—	2	1	no	—
143	C₂×D₄×C₃₁	—	2	1	no	—
144	C₂×D₄×C₃₁	—	4	2	no	—
145	C₂×D₄×C₃₁	—	4	1	no	—
146	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
147	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
148	C₂²×C₄×C₃₁	—	2	4	no	—
149	C₂²×C₄×C₃₁	—	2	1	no	—
150	C₂²×C₄×C₃₁	—	2	2	no	—
151	C₂²×C₄×C₃₁	—	2	2	no	—
152	C₂²×C₄×C₃₁	—	2	2	no	—
153	C₂²×C₄×C₃₁	—	4	1	no	—
154	C₂²×C₄×C₃₁	—	8	2	no	—
155	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
156	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
157	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
158	SD₁₆×C₃₁	—	4	4	no	—
159	SD₁₆×C₃₁	—	8	2	no	—
160	D₈×C₃₁	—	4	4	no	—
161	D₈×C₃₁	—	8	2	no	—
162	M₁₆×C₃₁	—	4	2	no	—
163	C₂×C₈×C₃₁	—	2	1	no	—
164	C₂×C₈×C₃₁	—	2	2	no	—
165	C₂×C₈×C₃₁	—	2	1	no	—
166	C₂×C₈×C₃₁	—	2	2	no	—
167	C₄⋊C₄×C₃₁	—	2	4	no	—
168	C₄⋊C₄×C₃₁	—	2	1	no	—
169	C₄⋊C₄×C₃₁	—	2	2	no	—
170	C₄⋊C₄×C₃₁	—	2	1	no	—
171	C₄⋊C₄×C₃₁	—	2	2	no	—
172	C₄⋊C₄×C₃₁	—	2	1	no	—
173	C₄⋊C₄×C₃₁	—	2	1	no	—
174	C₄⋊C₄×C₃₁	—	2	1	no	—
175	C₄⋊C₄×C₃₁	—	2	1	no	—
176	C₄⋊C₄×C₃₁	—	2	1	no	—
177	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
178	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
179	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
180	C₂²⋊C₄×C₃₁	—	2	4	no	—
181	C₂²⋊C₄×C₃₁	—	2	4	no	—
182	C₂²⋊C₄×C₃₁	—	2	4	no	—
183	C₂²⋊C₄×C₃₁	—	2	2	no	—
184	C₂²⋊C₄×C₃₁	—	2	2	no	—
185	C₂²⋊C₄×C₃₁	—	2	2	no	—
186	C₂²⋊C₄×C₃₁	—	2	2	no	—
187	C₂²⋊C₄×C₃₁	—	2	2	no	—
188	C₂²⋊C₄×C₃₁	—	4	1	no	—
189	C₂²⋊C₄×C₃₁	—	4	1	no	—
190	C₄²×C₃₁	—	2	4	no	—
191	C₄²×C₃₁	—	2	2	no	—
192	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
193	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
194	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
195	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
196	#18616	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
197	C₂³×C₃₁	—	2	2	no	—
198	C₂³×C₃₁	—	2	2	no	—
199	C₂³×C₃₁	—	4	2	no	—
200	C₂³×C₃₁	—	4	1	no	—
201	C₂³×C₃₁	—	8	1	no	—
202	C₂³×C₃₁	—	8	1	no	—
203	C₂³×C₃₁	#120	1	1	yes	characteristic
204	Q₈×C₃₁	—	4	2	no	—
205	Q₈×C₃₁	—	4	2	no	—
206	Q₈×C₃₁	—	4	2	no	—
207	Q₈×C₃₁	—	4	2	no	—
208	D₄×C₃₁	—	2	2	no	—
209	D₄×C₃₁	—	2	2	no	—
210	D₄×C₃₁	—	2	2	no	—
211	D₄×C₃₁	—	2	2	no	—
212	D₄×C₃₁	—	4	2	no	—
213	D₄×C₃₁	—	4	2	no	—
214	D₄×C₃₁	—	4	2	no	—
215	D₄×C₃₁	—	4	4	no	—
216	D₄×C₃₁	—	4	2	no	—
217	D₄×C₃₁	—	4	2	no	—
218	D₄×C₃₁	—	8	1	no	—
219	D₄×C₃₁	—	8	4	no	—
220	D₄×C₃₁	—	8	4	no	—
221	D₄×C₃₁	—	8	2	no	—
222	C₂×C₄×C₃₁	—	2	2	no	—
223	C₂×C₄×C₃₁	—	2	2	no	—
224	C₂×C₄×C₃₁	—	2	2	no	—
225	C₂×C₄×C₃₁	—	2	2	no	—
226	C₂×C₄×C₃₁	—	2	2	no	—
227	C₂×C₄×C₃₁	—	2	2	no	—
228	C₂×C₄×C₃₁	—	4	1	no	—
229	C₂×C₄×C₃₁	—	4	4	no	—
230	C₂×C₄×C₃₁	—	4	2	no	—
231	C₂×C₄×C₃₁	—	4	2	no	—
232	C₂×C₄×C₃₁	—	4	1	no	—
233	C₂×C₄×C₃₁	—	8	2	no	—
234	C₂×C₄×C₃₁	—	8	2	no	—
235	C₂×C₄×C₃₁	—	8	1	no	—
236	C₂×C₄×C₃₁	—	8	4	no	—
237	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
238	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
239	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
240	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
241	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
242	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
243	C₈×C₃₁	—	2	2	no	—
244	C₈×C₃₁	—	4	1	no	—
245	C₈×C₃₁	—	4	2	no	—
246	C₈×C₃₁	—	4	2	no	—
247	#3506	C₂×C₃₁	1	1	yes	characteristic, complemented
248	#3214	C₂×C₃₁	1	1	yes	characteristic, complemented
249	#3182	C₂×C₃₁	1	1	yes	characteristic, complemented
250	#3100	C₂×C₃₁	1	1	yes	characteristic, complemented
251	#1647	C₂×C₃₁	1	2	yes	complemented
252	#1646	C₂×C₃₁	1	2	yes	complemented
253	#1625	C₂×C₃₁	1	2	yes	complemented
254	#1624	C₂×C₃₁	1	2	yes	complemented
255	#1554	C₂×C₃₁	1	1	yes	characteristic, complemented
256	#1490	C₂×C₃₁	1	1	yes	characteristic, complemented
257	#1484	C₂×C₃₁	1	1	yes	characteristic, complemented
258	C₂²×C₃₁	—	2	1	no	—
259	C₂²×C₃₁	—	2	1	no	—
260	C₂²×C₃₁	—	2	1	no	—
261	C₂²×C₃₁	—	4	2	no	—
262	C₂²×C₃₁	—	4	2	no	—
263	C₂²×C₃₁	—	4	2	no	—
264	C₂²×C₃₁	—	4	2	no	—
265	C₂²×C₃₁	—	4	2	no	—
266	C₂²×C₃₁	—	4	2	no	—
267	C₂²×C₃₁	—	8	4	no	—
268	C₂²×C₃₁	—	8	1	no	—
269	C₂²×C₃₁	—	8	1	no	—
270	C₂²×C₃₁	—	8	1	no	—
271	C₂²×C₃₁	C₂×#120	1	1	yes	characteristic
272	C₄×C₃₁	—	2	1	no	—
273	C₄×C₃₁	—	2	1	no	—
274	C₄×C₃₁	—	2	1	no	—
275	C₄×C₃₁	—	2	1	no	—
276	C₄×C₃₁	—	4	2	no	—
277	C₄×C₃₁	—	4	2	no	—
278	C₄×C₃₁	—	4	2	no	—
279	C₄×C₃₁	—	4	2	no	—
280	C₄×C₃₁	—	8	1	no	—
281	C₄×C₃₁	#575	1	2	yes	—
282	C₄×C₃₁	#576	1	2	yes	—
283	#559	—	2	2	no	—
284	#556	—	2	1	no	—
285	#555	C₂²×C₃₁	1	2	yes	—
286	#553	—	2	2	no	—
287	#550	—	2	1	no	—
288	#547	C₂²×C₃₁	1	2	yes	—
289	#538	—	2	2	no	—
290	#535	—	2	1	no	—
291	#533	C₂²×C₃₁	1	1	yes	characteristic
292	#532	C₂²×C₃₁	1	1	yes	characteristic
293	#532	C₂²×C₃₁	1	1	yes	characteristic
294	#525	—	2	2	no	—
295	#520	—	2	1	no	—
296	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
297	#500	C₂²×C₃₁	1	1	yes	characteristic, complemented
298	#496	—	2	1	no	—
299	#492	C₂²×C₃₁	1	1	yes	characteristic, complemented
300	#489	—	2	1	no	—
301	#487	C₂²×C₃₁	1	1	yes	characteristic, complemented
302	#486	C₂²×C₃₁	1	1	yes	characteristic, complemented
303	#480	—	2	1	no	—
304	#474	C₂²×C₃₁	1	2	yes	—
305	#468	—	2	2	no	—
306	#461	C₂²×C₃₁	1	2	yes	complemented
307	#459	C₂²×C₃₁	1	2	yes	—
308	#459	C₂²×C₃₁	1	2	yes	complemented
309	#449	—	2	2	no	—
310	#447	—	2	1	no	—
311	#440	—	2	2	no	—
312	C₄×D₈	—	2	2	no	—
313	#432	C₂²×C₃₁	1	1	yes	characteristic
314	#431	C₂²×C₃₁	1	1	yes	characteristic
315	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
316	#333	C₂²×C₃₁	1	2	yes	complemented
317	#332	C₂²×C₃₁	1	2	yes	complemented
318	D₄⋊C₈	C₂²×C₃₁	1	2	yes	—
319	D₄⋊C₈	C₂²×C₃₁	1	2	yes	—
320	D₄⋊C₈	C₂²×C₃₁	1	2	yes	complemented
321	D₄⋊C₈	C₂²×C₃₁	1	2	yes	complemented
322	C₂×C₃₁	—	4	1	no	—
323	C₂×C₃₁	—	8	2	no	—
324	C₂×C₃₁	—	8	2	no	—
325	C₂×C₃₁	—	16	1	no	—
326	C₂×C₃₁	#3005	1	1	yes	characteristic
327	C₂×C₃₁	#3008	1	1	yes	characteristic
328	C₂×C₃₁	#3018	1	1	yes	characteristic
329	C₂×#41	—	2	2	no	—
330	C₂×#41	—	2	1	no	—
331	C₂×#41	—	2	1	no	—
332	C₂²×D₄	D₄×C₃₁	1	1	yes	characteristic
333	C₂³×C₄	—	2	1	no	—
334	C₂×SD₁₆	—	2	2	no	—
335	C₂×SD₁₆	—	2	2	no	—
336	C₂×D₈	—	2	2	no	—
337	C₂×D₈	—	2	2	no	—
338	C₂×M₁₆	—	2	1	no	—
339	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
340	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
341	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
342	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
343	#127	—	2	2	no	—
344	#127	D₄×C₃₁	1	1	yes	characteristic
345	#127	C₂³×C₃₁	1	1	yes	characteristic
346	#126	—	2	4	no	—
347	#125	—	2	1	no	—
348	#124	—	2	2	no	—
349	#124	—	2	4	no	—
350	#124	—	2	1	no	—
351	#124	—	2	1	no	—
352	#123	—	2	4	no	—
353	#123	—	2	4	no	—
354	C₂²⋊Q₈	—	2	1	no	—
355	C₂²⋊Q₈	—	2	2	no	—
356	C₂²⋊Q₈	D₄×C₃₁	1	2	yes	—
357	#121	—	2	4	no	—
358	#121	—	2	2	no	—
359	#121	—	2	2	no	—
360	#121	—	2	2	no	—
361	#121	—	2	1	no	—
362	#121	—	2	1	no	—
363	#121	—	2	1	no	—
364	#121	D₄×C₃₁	1	2	yes	—
365	#120	—	2	4	no	—
366	C₄×Q₈	—	2	2	no	—
367	C₄×Q₈	—	2	1	no	—
368	C₄×Q₈	D₄×C₃₁	1	2	yes	—
369	C₄×D₄	—	2	4	no	—
370	C₄×D₄	—	2	2	no	—
371	C₄×D₄	—	2	4	no	—
372	C₄×D₄	—	2	1	no	—
373	C₄×D₄	—	2	1	no	—
374	C₄×D₄	—	2	1	no	—
375	C₄×D₄	—	2	1	no	—
376	C₄×D₄	—	2	1	no	—
377	C₄×D₄	D₄×C₃₁	1	2	yes	—
378	C₄×D₄	C₂³×C₃₁	1	2	yes	—
379	C₄×D₄	C₂³×C₃₁	1	2	yes	—
380	#117	—	2	1	no	—
381	#117	—	2	2	no	—
382	#117	—	2	2	no	—
383	C₂×C₄⋊C₄	—	2	2	no	—
384	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
385	C₂×C₂²⋊C₄	—	2	1	no	—
386	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
387	#107	—	2	2	no	—
388	#107	—	2	2	no	—
389	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
390	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
391	#103	—	2	2	no	—
392	#103	—	2	2	no	—
393	#102	—	2	2	no	—
394	#102	—	2	2	no	—
395	#102	—	2	2	no	—
396	#102	—	2	2	no	—
397	C₂²⋊C₈	—	2	1	no	—
398	C₂²⋊C₈	—	2	1	no	—
399	#97	—	2	1	no	—
400	#97	—	2	1	no	—
401	C₄×C₈	—	2	1	no	—
402	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
403	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
404	C₃₁	#18616	1	1	yes	characteristic, direct product factor, Sylow
405	C₂⁴	—	2	1	no	—
406	#41	—	4	4	no	—
407	#41	—	8	2	no	—
408	#41	—	8	2	no	—
409	#41	—	8	2	no	—
410	C₂×Q₈	—	2	1	no	—
411	C₂×Q₈	—	2	1	no	—
412	C₂×Q₈	—	2	1	no	—
413	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
414	C₂×D₄	—	2	2	no	—
415	C₂×D₄	—	2	4	no	—
416	C₂×D₄	—	2	4	no	—
417	C₂×D₄	—	2	1	no	—
418	C₂×D₄	—	2	1	no	—
419	C₂×D₄	—	2	1	no	—
420	C₂×D₄	—	2	1	no	—
421	C₂×D₄	—	2	1	no	—
422	C₂×D₄	—	2	1	no	—
423	C₂×D₄	—	4	2	no	—
424	C₂×D₄	—	4	1	no	—
425	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
426	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
427	C₂²×C₄	—	2	4	no	—
428	C₂²×C₄	—	2	1	no	—
429	C₂²×C₄	—	2	2	no	—
430	C₂²×C₄	—	2	2	no	—
431	C₂²×C₄	—	2	2	no	—
432	C₂²×C₄	—	4	1	no	—
433	C₂²×C₄	—	8	2	no	—
434	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
435	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
436	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
437	SD₁₆	—	4	4	no	—
438	SD₁₆	—	8	2	no	—
439	D₈	—	4	4	no	—
440	D₈	—	8	2	no	—
441	M₁₆	—	4	2	no	—
442	C₂×C₈	—	2	1	no	—
443	C₂×C₈	—	2	2	no	—
444	C₂×C₈	—	2	1	no	—
445	C₂×C₈	—	2	2	no	—
446	C₄⋊C₄	—	2	4	no	—
447	C₄⋊C₄	—	2	1	no	—
448	C₄⋊C₄	—	2	2	no	—
449	C₄⋊C₄	—	2	1	no	—
450	C₄⋊C₄	—	2	2	no	—
451	C₄⋊C₄	—	2	1	no	—
452	C₄⋊C₄	—	2	1	no	—
453	C₄⋊C₄	—	2	1	no	—
454	C₄⋊C₄	—	2	1	no	—
455	C₄⋊C₄	—	2	1	no	—
456	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
457	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
458	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
459	C₂²⋊C₄	—	2	4	no	—
460	C₂²⋊C₄	—	2	4	no	—
461	C₂²⋊C₄	—	2	4	no	—
462	C₂²⋊C₄	—	2	2	no	—
463	C₂²⋊C₄	—	2	2	no	—
464	C₂²⋊C₄	—	2	2	no	—
465	C₂²⋊C₄	—	2	2	no	—
466	C₂²⋊C₄	—	2	2	no	—
467	C₂²⋊C₄	—	4	1	no	—
468	C₂²⋊C₄	—	4	1	no	—
469	C₄²	—	2	4	no	—
470	C₄²	—	2	2	no	—
471	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
472	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
473	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
474	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
475	C₂³	—	2	2	no	—
476	C₂³	—	2	2	no	—
477	C₂³	—	4	2	no	—
478	C₂³	—	4	1	no	—
479	C₂³	—	8	1	no	—
480	C₂³	—	8	1	no	—
481	C₂³	C₃₁×#120	1	1	yes	characteristic
482	Q₈	—	4	2	no	—
483	Q₈	—	4	2	no	—
484	Q₈	—	4	2	no	—
485	Q₈	—	4	2	no	—
486	D₄	—	2	2	no	—
487	D₄	—	2	2	no	—
488	D₄	—	2	2	no	—
489	D₄	—	2	2	no	—
490	D₄	—	4	2	no	—
491	D₄	—	4	2	no	—
492	D₄	—	4	2	no	—
493	D₄	—	4	4	no	—
494	D₄	—	4	2	no	—
495	D₄	—	4	2	no	—
496	D₄	—	8	1	no	—
497	D₄	—	8	4	no	—
498	D₄	—	8	4	no	—
499	D₄	—	8	2	no	—
500	C₂×C₄	—	2	2	no	—
501	C₂×C₄	—	2	2	no	—
502	C₂×C₄	—	2	2	no	—
503	C₂×C₄	—	2	2	no	—
504	C₂×C₄	—	2	2	no	—
505	C₂×C₄	—	2	2	no	—
506	C₂×C₄	—	4	1	no	—
507	C₂×C₄	—	4	4	no	—
508	C₂×C₄	—	4	2	no	—
509	C₂×C₄	—	4	2	no	—
510	C₂×C₄	—	4	1	no	—
511	C₂×C₄	—	8	2	no	—
512	C₂×C₄	—	8	2	no	—
513	C₂×C₄	—	8	1	no	—
514	C₂×C₄	—	8	4	no	—
515	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
516	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
517	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
518	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
519	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
520	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
521	C₈	—	2	2	no	—
522	C₈	—	4	1	no	—
523	C₈	—	4	2	no	—
524	C₈	—	4	2	no	—
525	C₂²	—	2	1	no	—
526	C₂²	—	2	1	no	—
527	C₂²	—	2	1	no	—
528	C₂²	—	4	2	no	—
529	C₂²	—	4	2	no	—
530	C₂²	—	4	2	no	—
531	C₂²	—	4	2	no	—
532	C₂²	—	4	2	no	—
533	C₂²	—	4	2	no	—
534	C₂²	—	8	4	no	—
535	C₂²	—	8	1	no	—
536	C₂²	—	8	1	no	—
537	C₂²	—	8	1	no	—
538	C₂²	C₂×C₃₁×#120	1	1	yes	characteristic
539	C₄	—	2	1	no	—
540	C₄	—	2	1	no	—
541	C₄	—	2	1	no	—
542	C₄	—	2	1	no	—
543	C₄	—	4	2	no	—
544	C₄	—	4	2	no	—
545	C₄	—	4	2	no	—
546	C₄	—	4	2	no	—
547	C₄	—	8	1	no	—
548	C₄	C₃₁×#575	1	2	yes	—
549	C₄	C₃₁×#576	1	2	yes	—
550	C₂	—	4	1	no	—
551	C₂	—	8	2	no	—
552	C₂	—	8	2	no	—
553	C₂	—	16	1	no	—
554	C₂	C₃₁×#3005	1	1	yes	characteristic
555	C₂	C₃₁×#3008	1	1	yes	characteristic
556	C₂	C₃₁×#3018	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3506	C₂	semidirect
2. C₃₁×#3214	C₂	semidirect
3. C₃₁×#3182	C₂	semidirect
4. C₃₁×#3100	C₂	semidirect
5. C₃₁×#1647	C₂	semidirect
6. C₃₁×#1646	C₂	semidirect
7. C₃₁×#1625	C₂	semidirect
8. C₃₁×#1624	C₂	semidirect
9. C₃₁×#1554	C₂	semidirect
10. C₃₁×#1490	C₂	semidirect
11. C₃₁×#1484	C₂	semidirect
26. C₃₁×#500	C₂²	semidirect
28. C₃₁×#492	C₂²	semidirect
30. C₃₁×#487	C₂²	semidirect
31. C₃₁×#486	C₂²	semidirect
35. C₃₁×#461	C₂²	semidirect
37. C₃₁×#459	C₂²	semidirect
45. C₃₁×#333	C₂²	semidirect
46. C₃₁×#332	C₂²	semidirect
49. C₃₁×D₄⋊C₈	C₂²	semidirect
50. C₃₁×D₄⋊C₈	C₂²	semidirect
196. #18616	C₃₁	direct (N × H)
247. #3506	C₂×C₃₁	semidirect
248. #3214	C₂×C₃₁	semidirect
249. #3182	C₂×C₃₁	semidirect
250. #3100	C₂×C₃₁	semidirect
251. #1647	C₂×C₃₁	semidirect
252. #1646	C₂×C₃₁	semidirect
253. #1625	C₂×C₃₁	semidirect
254. #1624	C₂×C₃₁	semidirect
255. #1554	C₂×C₃₁	semidirect
256. #1490	C₂×C₃₁	semidirect
257. #1484	C₂×C₃₁	semidirect
297. #500	C₂²×C₃₁	semidirect
299. #492	C₂²×C₃₁	semidirect
301. #487	C₂²×C₃₁	semidirect
302. #486	C₂²×C₃₁	semidirect
306. #461	C₂²×C₃₁	semidirect
308. #459	C₂²×C₃₁	semidirect
316. #333	C₂²×C₃₁	semidirect
317. #332	C₂²×C₃₁	semidirect
320. D₄⋊C₈	C₂²×C₃₁	semidirect
321. D₄⋊C₈	C₂²×C₃₁	semidirect
404. C₃₁	#18616	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×#3506	2
C₃₁×#3214	2
C₃₁×#3182	2
C₃₁×#3100	2
C₃₁×#1647	2
C₃₁×#1646	2
C₃₁×#1625	2
C₃₁×#1624	2
C₃₁×#1554	2
C₃₁×#1490	2
C₃₁×#1484	2
#18616	31

Subgroups — C31×#18616 #21475375

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#18616 #21475375