Subgroups — C₃₁×#19279 #21476038

Order 7936.

712 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3167	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3077	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3036	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃₁×#436	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1699	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1697	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1691	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1685	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1567	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1550	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1547	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×AGL(1,ℤ/8)	—	2	1	no	—
13	C₂²×M₁₆×C₃₁	—	2	1	no	—
14	C₃₁×#533	C₂²	1	1	yes	characteristic
15	C₃₁×#532	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
17	C₃₁×#498	C₂²	1	1	yes	characteristic, complemented
18	C₃₁×#492	C₂²	1	1	yes	characteristic, complemented
19	C₃₁×#487	C₂²	1	1	yes	characteristic, complemented
20	C₃₁×#486	C₂²	1	1	yes	characteristic, complemented
21	C₃₁×#478	—	2	2	no	—
22	C₃₁×#476	C₂²	1	2	yes	—
23	C₃₁×#474	C₂²	1	2	yes	—
24	C₃₁×#465	—	2	1	no	—
25	C₃₁×#465	—	2	1	no	—
26	C₃₁×#463	—	2	2	no	—
27	C₃₁×#461	C₂²	1	2	yes	—
28	C₃₁×#459	C₂²	1	2	yes	—
29	C₄×SD₁₆×C₃₁	—	2	1	no	—
30	C₄×SD₁₆×C₃₁	—	2	1	no	—
31	C₃₁×#436	—	2	1	no	—
32	C₃₁×#436	—	2	1	no	—
33	C₃₁×#436	C₂²	1	2	yes	—
34	C₃₁×#436	C₂²	1	2	yes	—
35	C₃₁×#428	—	2	1	no	—
36	C₃₁×#423	C₂²	1	1	yes	characteristic
37	C₃₁×#423	C₂²	1	1	yes	characteristic
38	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
39	C₃₁×#418	—	2	1	no	—
40	C₃₁×#417	—	2	1	no	—
41	C₂×C₃₁×C₂²⋊C₈	—	2	1	no	—
42	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
43	C₃₁×#335	C₂²	1	2	yes	complemented
44	C₃₁×#335	C₂²	1	2	yes	complemented
45	C₃₁×#333	C₂²	1	2	yes	complemented
46	C₃₁×#332	C₂²	1	2	yes	complemented
47	C₃₁×#332	C₂²	1	2	yes	complemented
48	C₃₁×#331	C₂²	1	2	yes	complemented
49	C₂×#41×C₃₁	—	2	1	no	—
50	C₂²×D₄×C₃₁	D₄	1	1	yes	characteristic
51	C₂³×C₄×C₃₁	—	2	1	no	—
52	C₃₁×AGL(1,ℤ/8)	—	4	4	no	—
53	C₂×SD₁₆×C₃₁	—	2	2	no	—
54	C₂×SD₁₆×C₃₁	—	2	1	no	—
55	C₂×SD₁₆×C₃₁	—	2	1	no	—
56	C₂×D₈×C₃₁	—	2	2	no	—
57	C₂×M₁₆×C₃₁	—	2	1	no	—
58	C₂×M₁₆×C₃₁	—	2	1	no	—
59	C₂×M₁₆×C₃₁	—	2	2	no	—
60	C₂×M₁₆×C₃₁	—	2	2	no	—
61	C₂×M₁₆×C₃₁	—	4	4	no	—
62	C₂²×C₈×C₃₁	—	2	1	no	—
63	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
64	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
65	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
66	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
67	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
68	C₃₁×#127	C₂³	1	1	yes	characteristic
69	C₃₁×#125	—	2	1	no	—
70	C₃₁×#124	—	2	1	no	—
71	C₃₁×C₂²⋊Q₈	—	2	1	no	—
72	C₃₁×C₂²⋊Q₈	—	2	1	no	—
73	C₃₁×#121	—	2	1	no	—
74	C₃₁×#121	—	2	1	no	—
75	C₄×Q₈×C₃₁	—	2	1	no	—
76	C₄×Q₈×C₃₁	—	2	1	no	—
77	C₄×D₄×C₃₁	—	2	1	no	—
78	C₄×D₄×C₃₁	—	2	1	no	—
79	C₄×D₄×C₃₁	—	2	1	no	—
80	C₄×D₄×C₃₁	—	2	1	no	—
81	C₄×D₄×C₃₁	D₄	1	2	yes	—
82	C₄×D₄×C₃₁	C₂³	1	2	yes	—
83	C₃₁×#117	—	2	1	no	—
84	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
85	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
86	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
87	C₃₁×#107	—	2	2	no	—
88	C₃₁×#106	—	2	1	no	—
89	C₃₁×#106	—	2	1	no	—
90	C₃₁×#106	—	2	2	no	—
91	C₃₁×C₄⋊C₈	—	2	1	no	—
92	C₃₁×C₄⋊C₈	—	2	1	no	—
93	C₃₁×C₄⋊C₈	—	2	1	no	—
94	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
95	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
96	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
97	C₃₁×#103	—	2	1	no	—
98	C₃₁×#103	—	2	1	no	—
99	C₃₁×#103	—	2	2	no	—
100	C₃₁×#102	—	2	2	no	—
101	C₃₁×#102	—	2	2	no	—
102	C₃₁×#102	—	2	1	no	—
103	C₃₁×#102	—	2	1	no	—
104	C₃₁×#102	—	2	2	no	—
105	C₃₁×C₂²⋊C₈	—	2	2	no	—
106	C₃₁×C₂²⋊C₈	—	2	2	no	—
107	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
108	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
109	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
110	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
111	C₂⁴×C₃₁	—	2	1	no	—
112	#41×C₃₁	—	4	2	no	—
113	#41×C₃₁	—	4	2	no	—
114	C₂×Q₈×C₃₁	—	2	1	no	—
115	C₂×Q₈×C₃₁	—	2	1	no	—
116	C₂×Q₈×C₃₁	—	2	1	no	—
117	C₂×D₄×C₃₁	—	2	1	no	—
118	C₂×D₄×C₃₁	—	2	2	no	—
119	C₂×D₄×C₃₁	—	2	2	no	—
120	C₂×D₄×C₃₁	—	2	1	no	—
121	C₂×D₄×C₃₁	—	2	1	no	—
122	C₂×D₄×C₃₁	—	2	1	no	—
123	C₂×D₄×C₃₁	—	4	1	no	—
124	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
125	C₂²×C₄×C₃₁	—	2	1	no	—
126	C₂²×C₄×C₃₁	—	2	2	no	—
127	C₂²×C₄×C₃₁	—	2	2	no	—
128	C₂²×C₄×C₃₁	—	4	2	no	—
129	C₂²×C₄×C₃₁	—	4	2	no	—
130	C₂²×C₄×C₃₁	—	4	1	no	—
131	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
132	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
133	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
134	SD₁₆×C₃₁	—	4	2	no	—
135	SD₁₆×C₃₁	—	4	4	no	—
136	SD₁₆×C₃₁	—	4	2	no	—
137	D₈×C₃₁	—	4	4	no	—
138	M₁₆×C₃₁	—	4	4	no	—
139	M₁₆×C₃₁	—	4	2	no	—
140	M₁₆×C₃₁	—	4	2	no	—
141	M₁₆×C₃₁	—	4	4	no	—
142	C₂×C₈×C₃₁	—	2	2	no	—
143	C₂×C₈×C₃₁	—	2	2	no	—
144	C₂×C₈×C₃₁	—	2	2	no	—
145	C₂×C₈×C₃₁	—	2	2	no	—
146	C₂×C₈×C₃₁	—	4	1	no	—
147	C₂×C₈×C₃₁	—	4	2	no	—
148	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
149	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
150	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
151	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
152	C₄⋊C₄×C₃₁	—	2	1	no	—
153	C₄⋊C₄×C₃₁	—	2	1	no	—
154	C₄⋊C₄×C₃₁	—	2	1	no	—
155	C₄⋊C₄×C₃₁	—	2	1	no	—
156	C₄⋊C₄×C₃₁	—	2	1	no	—
157	C₄⋊C₄×C₃₁	—	2	1	no	—
158	C₄⋊C₄×C₃₁	—	2	1	no	—
159	C₄⋊C₄×C₃₁	—	4	1	no	—
160	C₄⋊C₄×C₃₁	—	4	1	no	—
161	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
162	C₂²⋊C₄×C₃₁	—	2	2	no	—
163	C₂²⋊C₄×C₃₁	—	2	2	no	—
164	C₂²⋊C₄×C₃₁	—	4	1	no	—
165	C₂²⋊C₄×C₃₁	—	4	1	no	—
166	C₂²⋊C₄×C₃₁	—	4	1	no	—
167	C₄²×C₃₁	—	4	1	no	—
168	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
169	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
170	C₄²×C₃₁	#41	1	1	yes	characteristic
171	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
172	#19279	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
173	C₂³×C₃₁	—	2	2	no	—
174	C₂³×C₃₁	—	4	2	no	—
175	C₂³×C₃₁	—	4	2	no	—
176	C₂³×C₃₁	—	4	2	no	—
177	C₂³×C₃₁	—	4	1	no	—
178	C₂³×C₃₁	#121	1	1	yes	characteristic
179	Q₈×C₃₁	—	4	1	no	—
180	Q₈×C₃₁	—	4	1	no	—
181	Q₈×C₃₁	—	4	1	no	—
182	Q₈×C₃₁	—	4	1	no	—
183	Q₈×C₃₁	—	8	1	no	—
184	D₄×C₃₁	—	2	2	no	—
185	D₄×C₃₁	—	2	2	no	—
186	D₄×C₃₁	—	4	1	no	—
187	D₄×C₃₁	—	4	1	no	—
188	D₄×C₃₁	—	4	1	no	—
189	D₄×C₃₁	—	4	1	no	—
190	D₄×C₃₁	—	4	2	no	—
191	D₄×C₃₁	—	8	1	no	—
192	D₄×C₃₁	—	8	2	no	—
193	C₂×C₄×C₃₁	—	2	1	no	—
194	C₂×C₄×C₃₁	—	2	1	no	—
195	C₂×C₄×C₃₁	—	2	2	no	—
196	C₂×C₄×C₃₁	—	4	1	no	—
197	C₂×C₄×C₃₁	—	4	4	no	—
198	C₂×C₄×C₃₁	—	4	2	no	—
199	C₂×C₄×C₃₁	—	4	2	no	—
200	C₂×C₄×C₃₁	—	4	2	no	—
201	C₂×C₄×C₃₁	—	4	1	no	—
202	C₂×C₄×C₃₁	—	4	1	no	—
203	C₂×C₄×C₃₁	—	4	1	no	—
204	C₂×C₄×C₃₁	—	8	2	no	—
205	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
206	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
207	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
208	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
209	C₂×C₄×C₃₁	#137	1	2	yes	—
210	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
211	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
212	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
213	C₈×C₃₁	—	2	1	no	—
214	C₈×C₃₁	—	2	1	no	—
215	C₈×C₃₁	—	2	1	no	—
216	C₈×C₃₁	—	4	2	no	—
217	C₈×C₃₁	—	4	2	no	—
218	C₈×C₃₁	—	4	2	no	—
219	C₈×C₃₁	AGL(1,ℤ/8)	1	2	yes	complemented
220	#3167	C₂×C₃₁	1	1	yes	characteristic, complemented
221	#3077	C₂×C₃₁	1	1	yes	characteristic, complemented
222	#3036	C₂×C₃₁	1	1	yes	characteristic
223	C₂×#436	C₂×C₃₁	1	1	yes	characteristic, complemented
224	#1699	C₂×C₃₁	1	2	yes	complemented
225	#1697	C₂×C₃₁	1	2	yes	complemented
226	#1691	C₂×C₃₁	1	2	yes	complemented
227	#1685	C₂×C₃₁	1	2	yes	complemented
228	#1567	C₂×C₃₁	1	1	yes	characteristic, complemented
229	#1550	C₂×C₃₁	1	1	yes	characteristic, complemented
230	#1547	C₂×C₃₁	1	1	yes	characteristic, complemented
231	C₂²×C₃₁	—	2	1	no	—
232	C₂²×C₃₁	—	2	1	no	—
233	C₂²×C₃₁	—	4	2	no	—
234	C₂²×C₃₁	—	4	2	no	—
235	C₂²×C₃₁	—	4	2	no	—
236	C₂²×C₃₁	—	8	4	no	—
237	C₂²×C₃₁	—	8	1	no	—
238	C₂²×C₃₁	—	8	1	no	—
239	C₂²×C₃₁	—	8	1	no	—
240	C₂²×C₃₁	#468	1	2	yes	—
241	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
242	C₄×C₃₁	—	2	1	no	—
243	C₄×C₃₁	—	2	1	no	—
244	C₄×C₃₁	—	2	1	no	—
245	C₄×C₃₁	—	4	2	no	—
246	C₄×C₃₁	—	8	1	no	—
247	C₄×C₃₁	—	8	1	no	—
248	C₄×C₃₁	—	8	1	no	—
249	C₄×C₃₁	#468	1	2	yes	—
250	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
251	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
252	C₄×C₃₁	C₂×#137	1	1	yes	characteristic
253	C₄×C₃₁	#575	1	1	yes	characteristic
254	C₂×AGL(1,ℤ/8)	—	2	1	no	—
255	C₂²×M₁₆	—	2	1	no	—
256	#533	C₂²×C₃₁	1	1	yes	characteristic
257	#532	C₂²×C₃₁	1	1	yes	characteristic
258	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
259	#498	C₂²×C₃₁	1	1	yes	characteristic, complemented
260	#492	C₂²×C₃₁	1	1	yes	characteristic, complemented
261	#487	C₂²×C₃₁	1	1	yes	characteristic, complemented
262	#486	C₂²×C₃₁	1	1	yes	characteristic, complemented
263	#478	—	2	2	no	—
264	#476	C₂²×C₃₁	1	2	yes	—
265	#474	C₂²×C₃₁	1	2	yes	—
266	#465	—	2	1	no	—
267	#465	—	2	1	no	—
268	#463	—	2	2	no	—
269	#461	C₂²×C₃₁	1	2	yes	—
270	#459	C₂²×C₃₁	1	2	yes	—
271	C₄×SD₁₆	—	2	1	no	—
272	C₄×SD₁₆	—	2	1	no	—
273	#436	—	2	1	no	—
274	#436	—	2	1	no	—
275	#436	C₂²×C₃₁	1	2	yes	—
276	#436	C₂²×C₃₁	1	2	yes	—
277	#428	—	2	1	no	—
278	#423	C₂²×C₃₁	1	1	yes	characteristic
279	#423	C₂²×C₃₁	1	1	yes	characteristic
280	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
281	#418	—	2	1	no	—
282	#417	—	2	1	no	—
283	C₂×C₂²⋊C₈	—	2	1	no	—
284	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
285	#335	C₂²×C₃₁	1	2	yes	complemented
286	#335	C₂²×C₃₁	1	2	yes	complemented
287	#333	C₂²×C₃₁	1	2	yes	complemented
288	#332	C₂²×C₃₁	1	2	yes	complemented
289	#332	C₂²×C₃₁	1	2	yes	complemented
290	#331	C₂²×C₃₁	1	2	yes	complemented
291	C₂×C₃₁	—	2	2	no	—
292	C₂×C₃₁	—	8	2	no	—
293	C₂×C₃₁	—	16	1	no	—
294	C₂×C₃₁	#3036	1	1	yes	characteristic
295	C₂×C₃₁	C₂×#468	1	1	yes	characteristic
296	C₂×C₃₁	#3067	1	1	yes	characteristic
297	C₂×#41	—	2	1	no	—
298	C₂²×D₄	D₄×C₃₁	1	1	yes	characteristic
299	C₂³×C₄	—	2	1	no	—
300	AGL(1,ℤ/8)	—	4	4	no	—
301	C₂×SD₁₆	—	2	2	no	—
302	C₂×SD₁₆	—	2	1	no	—
303	C₂×SD₁₆	—	2	1	no	—
304	C₂×D₈	—	2	2	no	—
305	C₂×M₁₆	—	2	1	no	—
306	C₂×M₁₆	—	2	1	no	—
307	C₂×M₁₆	—	2	2	no	—
308	C₂×M₁₆	—	2	2	no	—
309	C₂×M₁₆	—	4	4	no	—
310	C₂²×C₈	—	2	1	no	—
311	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
312	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
313	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
314	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
315	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
316	#127	C₂³×C₃₁	1	1	yes	characteristic
317	#125	—	2	1	no	—
318	#124	—	2	1	no	—
319	C₂²⋊Q₈	—	2	1	no	—
320	C₂²⋊Q₈	—	2	1	no	—
321	#121	—	2	1	no	—
322	#121	—	2	1	no	—
323	C₄×Q₈	—	2	1	no	—
324	C₄×Q₈	—	2	1	no	—
325	C₄×D₄	—	2	1	no	—
326	C₄×D₄	—	2	1	no	—
327	C₄×D₄	—	2	1	no	—
328	C₄×D₄	—	2	1	no	—
329	C₄×D₄	D₄×C₃₁	1	2	yes	—
330	C₄×D₄	C₂³×C₃₁	1	2	yes	—
331	#117	—	2	1	no	—
332	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
333	C₂×C₂²⋊C₄	—	2	1	no	—
334	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
335	#107	—	2	2	no	—
336	#106	—	2	1	no	—
337	#106	—	2	1	no	—
338	#106	—	2	2	no	—
339	C₄⋊C₈	—	2	1	no	—
340	C₄⋊C₈	—	2	1	no	—
341	C₄⋊C₈	—	2	1	no	—
342	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
343	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
344	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
345	#103	—	2	1	no	—
346	#103	—	2	1	no	—
347	#103	—	2	2	no	—
348	#102	—	2	2	no	—
349	#102	—	2	2	no	—
350	#102	—	2	1	no	—
351	#102	—	2	1	no	—
352	#102	—	2	2	no	—
353	C₂²⋊C₈	—	2	2	no	—
354	C₂²⋊C₈	—	2	2	no	—
355	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
356	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
357	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
358	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
359	C₃₁	#19279	1	1	yes	characteristic, direct product factor, Sylow
360	C₂⁴	—	2	1	no	—
361	#41	—	4	2	no	—
362	#41	—	4	2	no	—
363	C₂×Q₈	—	2	1	no	—
364	C₂×Q₈	—	2	1	no	—
365	C₂×Q₈	—	2	1	no	—
366	C₂×D₄	—	2	1	no	—
367	C₂×D₄	—	2	2	no	—
368	C₂×D₄	—	2	2	no	—
369	C₂×D₄	—	2	1	no	—
370	C₂×D₄	—	2	1	no	—
371	C₂×D₄	—	2	1	no	—
372	C₂×D₄	—	4	1	no	—
373	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
374	C₂²×C₄	—	2	1	no	—
375	C₂²×C₄	—	2	2	no	—
376	C₂²×C₄	—	2	2	no	—
377	C₂²×C₄	—	4	2	no	—
378	C₂²×C₄	—	4	2	no	—
379	C₂²×C₄	—	4	1	no	—
380	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
381	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
382	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
383	SD₁₆	—	4	2	no	—
384	SD₁₆	—	4	4	no	—
385	SD₁₆	—	4	2	no	—
386	D₈	—	4	4	no	—
387	M₁₆	—	4	4	no	—
388	M₁₆	—	4	2	no	—
389	M₁₆	—	4	2	no	—
390	M₁₆	—	4	4	no	—
391	C₂×C₈	—	2	2	no	—
392	C₂×C₈	—	2	2	no	—
393	C₂×C₈	—	2	2	no	—
394	C₂×C₈	—	2	2	no	—
395	C₂×C₈	—	4	1	no	—
396	C₂×C₈	—	4	2	no	—
397	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
398	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
399	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
400	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
401	C₄⋊C₄	—	2	1	no	—
402	C₄⋊C₄	—	2	1	no	—
403	C₄⋊C₄	—	2	1	no	—
404	C₄⋊C₄	—	2	1	no	—
405	C₄⋊C₄	—	2	1	no	—
406	C₄⋊C₄	—	2	1	no	—
407	C₄⋊C₄	—	2	1	no	—
408	C₄⋊C₄	—	4	1	no	—
409	C₄⋊C₄	—	4	1	no	—
410	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
411	C₂²⋊C₄	—	2	2	no	—
412	C₂²⋊C₄	—	2	2	no	—
413	C₂²⋊C₄	—	4	1	no	—
414	C₂²⋊C₄	—	4	1	no	—
415	C₂²⋊C₄	—	4	1	no	—
416	C₄²	—	4	1	no	—
417	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
418	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
419	C₄²	#41×C₃₁	1	1	yes	characteristic
420	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
421	C₂³	—	2	2	no	—
422	C₂³	—	4	2	no	—
423	C₂³	—	4	2	no	—
424	C₂³	—	4	2	no	—
425	C₂³	—	4	1	no	—
426	C₂³	C₃₁×#121	1	1	yes	characteristic
427	Q₈	—	4	1	no	—
428	Q₈	—	4	1	no	—
429	Q₈	—	4	1	no	—
430	Q₈	—	4	1	no	—
431	Q₈	—	8	1	no	—
432	D₄	—	2	2	no	—
433	D₄	—	2	2	no	—
434	D₄	—	4	2	no	—
435	D₄	—	4	1	no	—
436	D₄	—	4	1	no	—
437	D₄	—	4	1	no	—
438	D₄	—	4	1	no	—
439	D₄	—	8	1	no	—
440	D₄	—	8	2	no	—
441	C₂×C₄	—	2	1	no	—
442	C₂×C₄	—	2	1	no	—
443	C₂×C₄	—	2	2	no	—
444	C₂×C₄	—	4	1	no	—
445	C₂×C₄	—	4	4	no	—
446	C₂×C₄	—	4	2	no	—
447	C₂×C₄	—	4	2	no	—
448	C₂×C₄	—	4	2	no	—
449	C₂×C₄	—	4	1	no	—
450	C₂×C₄	—	4	1	no	—
451	C₂×C₄	—	4	1	no	—
452	C₂×C₄	—	8	2	no	—
453	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
454	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
455	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
456	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
457	C₂×C₄	C₃₁×#137	1	2	yes	—
458	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
459	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
460	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
461	C₈	—	2	1	no	—
462	C₈	—	2	1	no	—
463	C₈	—	2	1	no	—
464	C₈	—	4	2	no	—
465	C₈	—	4	2	no	—
466	C₈	—	4	2	no	—
467	C₈	C₃₁×AGL(1,ℤ/8)	1	2	yes	complemented
468	C₂²	—	2	1	no	—
469	C₂²	—	2	1	no	—
470	C₂²	—	4	2	no	—
471	C₂²	—	4	2	no	—
472	C₂²	—	4	2	no	—
473	C₂²	—	8	4	no	—
474	C₂²	—	8	1	no	—
475	C₂²	—	8	1	no	—
476	C₂²	—	8	1	no	—
477	C₂²	C₃₁×#468	1	2	yes	—
478	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
479	C₄	—	2	1	no	—
480	C₄	—	2	1	no	—
481	C₄	—	2	1	no	—
482	C₄	—	4	2	no	—
483	C₄	—	8	1	no	—
484	C₄	—	8	1	no	—
485	C₄	—	8	1	no	—
486	C₄	C₃₁×#468	1	2	yes	—
487	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
488	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
489	C₄	C₂×C₃₁×#137	1	1	yes	characteristic
490	C₄	C₃₁×#575	1	1	yes	characteristic
491	C₂	—	2	2	no	—
492	C₂	—	8	2	no	—
493	C₂	—	16	1	no	—
494	C₂	C₃₁×#3036	1	1	yes	characteristic
495	C₂	C₂×C₃₁×#468	1	1	yes	characteristic
496	C₂	C₃₁×#3067	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₃₁×#3036	C₂
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×#3167	C₂	semidirect
2. C₃₁×#3077	C₂	semidirect
4. C₂×C₃₁×#436	C₂	semidirect
5. C₃₁×#1699	C₂	semidirect
6. C₃₁×#1697	C₂	semidirect
7. C₃₁×#1691	C₂	semidirect
8. C₃₁×#1685	C₂	semidirect
9. C₃₁×#1567	C₂	semidirect
10. C₃₁×#1550	C₂	semidirect
11. C₃₁×#1547	C₂	semidirect
17. C₃₁×#498	C₂²	semidirect
18. C₃₁×#492	C₂²	semidirect
19. C₃₁×#487	C₂²	semidirect
20. C₃₁×#486	C₂²	semidirect
43. C₃₁×#335	C₂²	semidirect
44. C₃₁×#335	C₂²	semidirect
45. C₃₁×#333	C₂²	semidirect
46. C₃₁×#332	C₂²	semidirect
47. C₃₁×#332	C₂²	semidirect
48. C₃₁×#331	C₂²	semidirect
172. #19279	C₃₁	direct (N × H)
219. C₈×C₃₁	AGL(1,ℤ/8)	semidirect
220. #3167	C₂×C₃₁	semidirect
221. #3077	C₂×C₃₁	semidirect
223. C₂×#436	C₂×C₃₁	semidirect
224. #1699	C₂×C₃₁	semidirect
225. #1697	C₂×C₃₁	semidirect
226. #1691	C₂×C₃₁	semidirect
227. #1685	C₂×C₃₁	semidirect
228. #1567	C₂×C₃₁	semidirect
229. #1550	C₂×C₃₁	semidirect
230. #1547	C₂×C₃₁	semidirect
259. #498	C₂²×C₃₁	semidirect
260. #492	C₂²×C₃₁	semidirect
261. #487	C₂²×C₃₁	semidirect
262. #486	C₂²×C₃₁	semidirect
285. #335	C₂²×C₃₁	semidirect
286. #335	C₂²×C₃₁	semidirect
287. #333	C₂²×C₃₁	semidirect
288. #332	C₂²×C₃₁	semidirect
289. #332	C₂²×C₃₁	semidirect
290. #331	C₂²×C₃₁	semidirect
359. C₃₁	#19279	direct (N × H)
467. C₈	C₃₁×AGL(1,ℤ/8)	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃₁×#3167	2
C₃₁×#3077	2
C₃₁×#3036	2
C₂×C₃₁×#436	2
C₃₁×#1699	2
C₃₁×#1697	2
C₃₁×#1691	2
C₃₁×#1685	2
C₃₁×#1567	2
C₃₁×#1550	2
C₃₁×#1547	2
#19279	31

Subgroups — C31×#19279 #21476038

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19279 #21476038