Subgroups — C₃₁×#19352 #21476111

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×#3239	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3217	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#492	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2966	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1686	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1685	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1666	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1665	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1568	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1556	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1555	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×SD₁₆×C₃₁	—	2	1	no	—
13	C₂²×SD₁₆×C₃₁	—	2	1	no	—
14	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
16	C₃₁×#529	C₂²	1	1	yes	characteristic
17	C₃₁×#501	—	2	1	no	—
18	C₃₁×#496	—	2	1	no	—
19	C₃₁×#492	—	2	1	no	—
20	C₃₁×#492	—	2	1	no	—
21	C₃₁×#492	C₂²	1	2	yes	—
22	C₃₁×#492	C₂²	1	2	yes	—
23	C₃₁×#490	—	2	1	no	—
24	C₃₁×#489	—	2	1	no	—
25	C₃₁×#484	—	2	1	no	—
26	C₃₁×#481	—	2	1	no	—
27	C₃₁×#475	C₂²	1	2	yes	complemented
28	C₃₁×#474	C₂²	1	2	yes	complemented
29	C₃₁×#460	C₂²	1	2	yes	—
30	C₃₁×#459	C₂²	1	2	yes	—
31	C₃₁×#450	—	2	1	no	—
32	C₃₁×#447	—	2	1	no	—
33	C₃₁×#446	C₂²	1	1	yes	characteristic, complemented
34	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
35	C₃₁×#436	C₂²	1	1	yes	characteristic, complemented
36	C₃₁×#435	—	2	1	no	—
37	C₈×D₄×C₃₁	—	2	1	no	—
38	C₈×D₄×C₃₁	C₂²	1	1	yes	characteristic, complemented
39	C₃₁×#433	—	2	1	no	—
40	C₃₁×#432	C₂²	1	1	yes	characteristic
41	C₃₁×#432	C₂²	1	1	yes	characteristic
42	C₃₁×#432	C₂²	1	1	yes	characteristic
43	C₃₁×#405	—	2	1	no	—
44	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
45	C₃₁×#335	C₂²	1	2	yes	complemented
46	C₃₁×#334	C₂²	1	2	yes	complemented
47	C₃₁×#332	C₂²	1	2	yes	complemented
48	C₃₁×#332	C₂²	1	2	yes	complemented
49	C₃₁×#331	C₂²	1	2	yes	—
50	C₃₁×#331	C₂²	1	2	yes	—
51	C₂²×Q₈×C₃₁	—	2	1	no	—
52	C₂²×D₄×C₃₁	—	2	1	no	—
53	C₂²×D₄×C₃₁	—	2	1	no	—
54	C₂²×D₄×C₃₁	—	2	1	no	—
55	C₂×SD₁₆×C₃₁	—	2	2	no	—
56	C₂×SD₁₆×C₃₁	—	2	2	no	—
57	C₂×SD₁₆×C₃₁	—	2	2	no	—
58	C₂×SD₁₆×C₃₁	—	2	2	no	—
59	C₂×SD₁₆×C₃₁	—	2	2	no	—
60	C₂×SD₁₆×C₃₁	—	4	4	no	—
61	C₂×SD₁₆×C₃₁	—	4	4	no	—
62	C₂×D₈×C₃₁	—	2	2	no	—
63	C₂×M₁₆×C₃₁	—	2	1	no	—
64	C₂²×C₈×C₃₁	—	2	1	no	—
65	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
66	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
67	C₃₁×C₄⋊Q₈	—	2	1	no	—
68	C₃₁×C₄⋊Q₈	—	2	1	no	—
69	C₃₁×C₄⋊Q₈	—	2	1	no	—
70	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
71	C₃₁×#127	—	2	1	no	—
72	C₃₁×#127	—	2	1	no	—
73	C₃₁×#127	—	2	1	no	—
74	C₃₁×#127	C₂³	1	2	yes	—
75	C₃₁×#126	—	2	1	no	—
76	C₃₁×#125	D₄	1	1	yes	characteristic
77	C₃₁×#124	D₄	1	1	yes	characteristic
78	C₃₁×#123	—	2	1	no	—
79	C₃₁×C₂²⋊Q₈	D₄	1	1	yes	characteristic
80	C₃₁×#121	D₄	1	1	yes	characteristic
81	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
82	C₄×D₄×C₃₁	—	2	1	no	—
83	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
84	C₃₁×#117	—	2	1	no	—
85	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
86	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
87	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
88	C₃₁×#107	—	2	2	no	—
89	C₃₁×#106	—	2	2	no	—
90	C₃₁×C₄⋊C₈	—	2	1	no	—
91	C₃₁×C₄⋊C₈	—	2	1	no	—
92	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
93	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
94	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
95	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
96	C₃₁×#103	—	2	2	no	—
97	C₃₁×#102	—	2	2	no	—
98	C₃₁×#102	—	2	2	no	—
99	C₃₁×#102	—	2	2	no	—
100	C₃₁×C₂²⋊C₈	—	2	1	no	—
101	C₃₁×C₂²⋊C₈	—	2	1	no	—
102	C₃₁×C₂²⋊C₈	—	2	1	no	—
103	C₃₁×C₂²⋊C₈	—	2	1	no	—
104	C₃₁×C₂²⋊C₈	—	2	1	no	—
105	C₃₁×#97	—	2	1	no	—
106	C₃₁×#97	—	2	1	no	—
107	C₃₁×#97	—	2	1	no	—
108	C₃₁×#97	—	2	1	no	—
109	C₄×C₈×C₃₁	—	2	1	no	—
110	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
111	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
112	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
113	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
114	C₂⁴×C₃₁	—	4	1	no	—
115	C₂×Q₈×C₃₁	—	2	2	no	—
116	C₂×Q₈×C₃₁	—	4	2	no	—
117	C₂×Q₈×C₃₁	—	4	2	no	—
118	C₂×Q₈×C₃₁	—	4	1	no	—
119	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
120	C₂×D₄×C₃₁	—	2	1	no	—
121	C₂×D₄×C₃₁	—	2	2	no	—
122	C₂×D₄×C₃₁	—	2	2	no	—
123	C₂×D₄×C₃₁	—	2	2	no	—
124	C₂×D₄×C₃₁	—	4	2	no	—
125	C₂×D₄×C₃₁	—	4	2	no	—
126	C₂×D₄×C₃₁	—	4	1	no	—
127	C₂×D₄×C₃₁	—	4	1	no	—
128	C₂×D₄×C₃₁	—	4	1	no	—
129	C₂×D₄×C₃₁	—	8	2	no	—
130	C₂×D₄×C₃₁	—	8	2	no	—
131	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
132	C₂²×C₄×C₃₁	—	2	1	no	—
133	C₂²×C₄×C₃₁	—	4	1	no	—
134	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
135	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
136	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
137	SD₁₆×C₃₁	—	4	4	no	—
138	SD₁₆×C₃₁	—	4	4	no	—
139	SD₁₆×C₃₁	—	4	4	no	—
140	SD₁₆×C₃₁	—	4	4	no	—
141	SD₁₆×C₃₁	—	8	2	no	—
142	D₈×C₃₁	—	8	2	no	—
143	M₁₆×C₃₁	—	4	2	no	—
144	C₂×C₈×C₃₁	—	2	2	no	—
145	C₂×C₈×C₃₁	—	2	2	no	—
146	C₂×C₈×C₃₁	—	2	1	no	—
147	C₂×C₈×C₃₁	—	2	1	no	—
148	C₂×C₈×C₃₁	—	2	2	no	—
149	C₂×C₈×C₃₁	—	2	2	no	—
150	C₂×C₈×C₃₁	—	4	2	no	—
151	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
152	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
153	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
154	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
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	—	2	2	no	—
160	C₄⋊C₄×C₃₁	—	2	2	no	—
161	C₄⋊C₄×C₃₁	—	4	1	no	—
162	C₄⋊C₄×C₃₁	—	4	1	no	—
163	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
164	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
165	C₂²⋊C₄×C₃₁	—	2	1	no	—
166	C₂²⋊C₄×C₃₁	—	2	1	no	—
167	C₂²⋊C₄×C₃₁	—	2	1	no	—
168	C₂²⋊C₄×C₃₁	—	2	1	no	—
169	C₂²⋊C₄×C₃₁	—	2	1	no	—
170	C₄²×C₃₁	—	2	1	no	—
171	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
172	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
173	C₄²×C₃₁	#41	1	1	yes	characteristic
174	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
175	#19352	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
176	C₂³×C₃₁	—	2	1	no	—
177	C₂³×C₃₁	—	4	2	no	—
178	C₂³×C₃₁	—	8	2	no	—
179	C₂³×C₃₁	—	8	2	no	—
180	C₂³×C₃₁	—	8	2	no	—
181	C₂³×C₃₁	#121	1	1	yes	characteristic
182	Q₈×C₃₁	—	2	2	no	—
183	Q₈×C₃₁	—	4	2	no	—
184	Q₈×C₃₁	—	4	2	no	—
185	Q₈×C₃₁	—	8	2	no	—
186	D₄×C₃₁	—	2	2	no	—
187	D₄×C₃₁	—	4	2	no	—
188	D₄×C₃₁	—	4	2	no	—
189	D₄×C₃₁	—	4	2	no	—
190	D₄×C₃₁	—	4	2	no	—
191	D₄×C₃₁	—	4	2	no	—
192	D₄×C₃₁	—	8	2	no	—
193	D₄×C₃₁	—	8	2	no	—
194	D₄×C₃₁	—	8	2	no	—
195	D₄×C₃₁	—	8	2	no	—
196	C₂×C₄×C₃₁	—	2	2	no	—
197	C₂×C₄×C₃₁	—	2	1	no	—
198	C₂×C₄×C₃₁	—	2	1	no	—
199	C₂×C₄×C₃₁	—	2	2	no	—
200	C₂×C₄×C₃₁	—	2	1	no	—
201	C₂×C₄×C₃₁	—	2	1	no	—
202	C₂×C₄×C₃₁	—	2	1	no	—
203	C₂×C₄×C₃₁	—	4	1	no	—
204	C₂×C₄×C₃₁	—	4	1	no	—
205	C₂×C₄×C₃₁	—	4	2	no	—
206	C₂×C₄×C₃₁	—	8	2	no	—
207	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
208	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
209	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
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	—	4	1	no	—
216	C₈×C₃₁	—	4	1	no	—
217	C₈×C₃₁	—	4	2	no	—
218	C₈×C₃₁	—	4	2	no	—
219	C₈×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
220	C₈×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
221	#3239	C₂×C₃₁	1	1	yes	characteristic, complemented
222	#3217	C₂×C₃₁	1	1	yes	characteristic
223	C₂×#492	C₂×C₃₁	1	1	yes	characteristic, complemented
224	#2966	C₂×C₃₁	1	1	yes	characteristic, complemented
225	#1686	C₂×C₃₁	1	2	yes	complemented
226	#1685	C₂×C₃₁	1	2	yes	complemented
227	#1666	C₂×C₃₁	1	2	yes	complemented
228	#1665	C₂×C₃₁	1	2	yes	complemented
229	#1568	C₂×C₃₁	1	1	yes	characteristic, complemented
230	#1556	C₂×C₃₁	1	1	yes	characteristic, complemented
231	#1555	C₂×C₃₁	1	1	yes	characteristic, complemented
232	C₂²×C₃₁	—	2	1	no	—
233	C₂²×C₃₁	—	2	1	no	—
234	C₂²×C₃₁	—	4	1	no	—
235	C₂²×C₃₁	—	4	1	no	—
236	C₂²×C₃₁	—	4	1	no	—
237	C₂²×C₃₁	—	8	2	no	—
238	C₂²×C₃₁	—	8	2	no	—
239	C₂²×C₃₁	—	8	2	no	—
240	C₂²×C₃₁	—	16	4	no	—
241	C₂²×C₃₁	#471	1	2	yes	—
242	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
243	C₄×C₃₁	—	2	1	no	—
244	C₄×C₃₁	—	2	1	no	—
245	C₄×C₃₁	—	2	1	no	—
246	C₄×C₃₁	—	2	1	no	—
247	C₄×C₃₁	—	4	1	no	—
248	C₄×C₃₁	—	4	1	no	—
249	C₄×C₃₁	—	4	1	no	—
250	C₄×C₃₁	—	8	2	no	—
251	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
252	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
253	C₄×C₃₁	#577	1	2	yes	—
254	C₂²×SD₁₆	—	2	1	no	—
255	C₂²×SD₁₆	—	2	1	no	—
256	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
257	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
258	#529	C₂²×C₃₁	1	1	yes	characteristic
259	#501	—	2	1	no	—
260	#496	—	2	1	no	—
261	#492	—	2	1	no	—
262	#492	—	2	1	no	—
263	#492	C₂²×C₃₁	1	2	yes	—
264	#492	C₂²×C₃₁	1	2	yes	—
265	#490	—	2	1	no	—
266	#489	—	2	1	no	—
267	#484	—	2	1	no	—
268	#481	—	2	1	no	—
269	#475	C₂²×C₃₁	1	2	yes	complemented
270	#474	C₂²×C₃₁	1	2	yes	complemented
271	#460	C₂²×C₃₁	1	2	yes	—
272	#459	C₂²×C₃₁	1	2	yes	—
273	#450	—	2	1	no	—
274	#447	—	2	1	no	—
275	#446	C₂²×C₃₁	1	1	yes	characteristic, complemented
276	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
277	#436	C₂²×C₃₁	1	1	yes	characteristic, complemented
278	#435	—	2	1	no	—
279	C₈×D₄	—	2	1	no	—
280	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic, complemented
281	#433	—	2	1	no	—
282	#432	C₂²×C₃₁	1	1	yes	characteristic
283	#432	C₂²×C₃₁	1	1	yes	characteristic
284	#432	C₂²×C₃₁	1	1	yes	characteristic
285	#405	—	2	1	no	—
286	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
287	#335	C₂²×C₃₁	1	2	yes	complemented
288	#334	C₂²×C₃₁	1	2	yes	complemented
289	#332	C₂²×C₃₁	1	2	yes	complemented
290	#332	C₂²×C₃₁	1	2	yes	complemented
291	#331	C₂²×C₃₁	1	2	yes	—
292	#331	C₂²×C₃₁	1	2	yes	—
293	C₂×C₃₁	—	2	2	no	—
294	C₂×C₃₁	—	8	1	no	—
295	C₂×C₃₁	—	16	2	no	—
296	C₂×C₃₁	#3038	1	1	yes	characteristic
297	C₂×C₃₁	#3057	1	1	yes	characteristic
298	C₂×C₃₁	C₂×#471	1	1	yes	characteristic
299	C₂²×Q₈	—	2	1	no	—
300	C₂²×D₄	—	2	1	no	—
301	C₂²×D₄	—	2	1	no	—
302	C₂²×D₄	—	2	1	no	—
303	C₂×SD₁₆	—	2	2	no	—
304	C₂×SD₁₆	—	2	2	no	—
305	C₂×SD₁₆	—	2	2	no	—
306	C₂×SD₁₆	—	2	2	no	—
307	C₂×SD₁₆	—	2	2	no	—
308	C₂×SD₁₆	—	4	4	no	—
309	C₂×SD₁₆	—	4	4	no	—
310	C₂×D₈	—	2	2	no	—
311	C₂×M₁₆	—	2	1	no	—
312	C₂²×C₈	—	2	1	no	—
313	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
314	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
315	C₄⋊Q₈	—	2	1	no	—
316	C₄⋊Q₈	—	2	1	no	—
317	C₄⋊Q₈	—	2	1	no	—
318	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
319	#127	—	2	1	no	—
320	#127	—	2	1	no	—
321	#127	—	2	1	no	—
322	#127	C₂³×C₃₁	1	2	yes	—
323	#126	—	2	1	no	—
324	#125	D₄×C₃₁	1	1	yes	characteristic
325	#124	D₄×C₃₁	1	1	yes	characteristic
326	#123	—	2	1	no	—
327	C₂²⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
328	#121	D₄×C₃₁	1	1	yes	characteristic
329	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
330	C₄×D₄	—	2	1	no	—
331	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
332	#117	—	2	1	no	—
333	C₂×C₄⋊C₄	—	2	1	no	—
334	C₂×C₄⋊C₄	—	2	1	no	—
335	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
336	#107	—	2	2	no	—
337	#106	—	2	2	no	—
338	C₄⋊C₈	—	2	1	no	—
339	C₄⋊C₈	—	2	1	no	—
340	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
341	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
342	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
343	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
344	#103	—	2	2	no	—
345	#102	—	2	2	no	—
346	#102	—	2	2	no	—
347	#102	—	2	2	no	—
348	C₂²⋊C₈	—	2	1	no	—
349	C₂²⋊C₈	—	2	1	no	—
350	C₂²⋊C₈	—	2	1	no	—
351	C₂²⋊C₈	—	2	1	no	—
352	C₂²⋊C₈	—	2	1	no	—
353	#97	—	2	1	no	—
354	#97	—	2	1	no	—
355	#97	—	2	1	no	—
356	#97	—	2	1	no	—
357	C₄×C₈	—	2	1	no	—
358	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
359	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
360	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
361	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
362	C₃₁	#19352	1	1	yes	characteristic, direct product factor, Sylow
363	C₂⁴	—	4	1	no	—
364	C₂×Q₈	—	2	2	no	—
365	C₂×Q₈	—	4	2	no	—
366	C₂×Q₈	—	4	2	no	—
367	C₂×Q₈	—	4	1	no	—
368	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
369	C₂×D₄	—	2	1	no	—
370	C₂×D₄	—	2	2	no	—
371	C₂×D₄	—	2	2	no	—
372	C₂×D₄	—	2	2	no	—
373	C₂×D₄	—	4	2	no	—
374	C₂×D₄	—	4	2	no	—
375	C₂×D₄	—	4	1	no	—
376	C₂×D₄	—	4	1	no	—
377	C₂×D₄	—	4	1	no	—
378	C₂×D₄	—	8	2	no	—
379	C₂×D₄	—	8	2	no	—
380	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
381	C₂²×C₄	—	2	1	no	—
382	C₂²×C₄	—	4	1	no	—
383	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
384	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
385	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
386	SD₁₆	—	4	4	no	—
387	SD₁₆	—	4	4	no	—
388	SD₁₆	—	4	4	no	—
389	SD₁₆	—	4	4	no	—
390	SD₁₆	—	8	2	no	—
391	D₈	—	8	2	no	—
392	M₁₆	—	4	2	no	—
393	C₂×C₈	—	2	2	no	—
394	C₂×C₈	—	2	2	no	—
395	C₂×C₈	—	2	1	no	—
396	C₂×C₈	—	2	1	no	—
397	C₂×C₈	—	2	2	no	—
398	C₂×C₈	—	2	2	no	—
399	C₂×C₈	—	4	2	no	—
400	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
401	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
402	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
403	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
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	—	2	2	no	—
409	C₄⋊C₄	—	2	2	no	—
410	C₄⋊C₄	—	4	1	no	—
411	C₄⋊C₄	—	4	1	no	—
412	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
413	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
414	C₂²⋊C₄	—	2	1	no	—
415	C₂²⋊C₄	—	2	1	no	—
416	C₂²⋊C₄	—	2	1	no	—
417	C₂²⋊C₄	—	2	1	no	—
418	C₂²⋊C₄	—	2	1	no	—
419	C₄²	—	2	1	no	—
420	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
421	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
422	C₄²	#41×C₃₁	1	1	yes	characteristic
423	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
424	C₂³	—	2	1	no	—
425	C₂³	—	4	2	no	—
426	C₂³	—	8	2	no	—
427	C₂³	—	8	2	no	—
428	C₂³	—	8	2	no	—
429	C₂³	C₃₁×#121	1	1	yes	characteristic
430	Q₈	—	2	2	no	—
431	Q₈	—	4	2	no	—
432	Q₈	—	4	2	no	—
433	Q₈	—	8	2	no	—
434	D₄	—	2	2	no	—
435	D₄	—	4	2	no	—
436	D₄	—	4	2	no	—
437	D₄	—	4	2	no	—
438	D₄	—	4	2	no	—
439	D₄	—	4	2	no	—
440	D₄	—	8	2	no	—
441	D₄	—	8	2	no	—
442	D₄	—	8	2	no	—
443	D₄	—	8	2	no	—
444	C₂×C₄	—	2	2	no	—
445	C₂×C₄	—	2	1	no	—
446	C₂×C₄	—	2	1	no	—
447	C₂×C₄	—	2	2	no	—
448	C₂×C₄	—	2	1	no	—
449	C₂×C₄	—	2	1	no	—
450	C₂×C₄	—	2	1	no	—
451	C₂×C₄	—	4	1	no	—
452	C₂×C₄	—	4	1	no	—
453	C₂×C₄	—	4	2	no	—
454	C₂×C₄	—	8	2	no	—
455	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
456	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
457	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
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	—	4	1	no	—
464	C₈	—	4	1	no	—
465	C₈	—	4	2	no	—
466	C₈	—	4	2	no	—
467	C₈	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
468	C₈	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
469	C₂²	—	2	1	no	—
470	C₂²	—	2	1	no	—
471	C₂²	—	4	1	no	—
472	C₂²	—	4	1	no	—
473	C₂²	—	4	1	no	—
474	C₂²	—	8	2	no	—
475	C₂²	—	8	2	no	—
476	C₂²	—	8	2	no	—
477	C₂²	—	16	4	no	—
478	C₂²	C₃₁×#471	1	2	yes	—
479	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
480	C₄	—	2	1	no	—
481	C₄	—	2	1	no	—
482	C₄	—	2	1	no	—
483	C₄	—	2	1	no	—
484	C₄	—	4	1	no	—
485	C₄	—	4	1	no	—
486	C₄	—	4	1	no	—
487	C₄	—	8	2	no	—
488	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
489	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
490	C₄	C₃₁×#577	1	2	yes	—
491	C₂	—	2	2	no	—
492	C₂	—	8	1	no	—
493	C₂	—	16	2	no	—
494	C₂	C₃₁×#3038	1	1	yes	characteristic
495	C₂	C₃₁×#3057	1	1	yes	characteristic
496	C₂	C₂×C₃₁×#471	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₃₁×#3217	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×#3239	C₂	semidirect
3. C₂×C₃₁×#492	C₂	semidirect
4. C₃₁×#2966	C₂	semidirect
5. C₃₁×#1686	C₂	semidirect
6. C₃₁×#1685	C₂	semidirect
7. C₃₁×#1666	C₂	semidirect
8. C₃₁×#1665	C₂	semidirect
9. C₃₁×#1568	C₂	semidirect
10. C₃₁×#1556	C₂	semidirect
11. C₃₁×#1555	C₂	semidirect
27. C₃₁×#475	C₂²	semidirect
28. C₃₁×#474	C₂²	semidirect
33. C₃₁×#446	C₂²	semidirect
34. C₈×Q₈×C₃₁	C₂²	semidirect
35. C₃₁×#436	C₂²	semidirect
38. C₈×D₄×C₃₁	C₂²	semidirect
45. C₃₁×#335	C₂²	semidirect
46. C₃₁×#334	C₂²	semidirect
47. C₃₁×#332	C₂²	semidirect
48. C₃₁×#332	C₂²	semidirect
175. #19352	C₃₁	direct (N × H)
221. #3239	C₂×C₃₁	semidirect
223. C₂×#492	C₂×C₃₁	semidirect
224. #2966	C₂×C₃₁	semidirect
225. #1686	C₂×C₃₁	semidirect
226. #1685	C₂×C₃₁	semidirect
227. #1666	C₂×C₃₁	semidirect
228. #1665	C₂×C₃₁	semidirect
229. #1568	C₂×C₃₁	semidirect
230. #1556	C₂×C₃₁	semidirect
231. #1555	C₂×C₃₁	semidirect
269. #475	C₂²×C₃₁	semidirect
270. #474	C₂²×C₃₁	semidirect
275. #446	C₂²×C₃₁	semidirect
276. C₈×Q₈	C₂²×C₃₁	semidirect
277. #436	C₂²×C₃₁	semidirect
280. C₈×D₄	C₂²×C₃₁	semidirect
287. #335	C₂²×C₃₁	semidirect
288. #334	C₂²×C₃₁	semidirect
289. #332	C₂²×C₃₁	semidirect
290. #332	C₂²×C₃₁	semidirect
362. C₃₁	#19352	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×#3239	2
C₃₁×#3217	2
C₂×C₃₁×#492	2
C₃₁×#2966	2
C₃₁×#1686	2
C₃₁×#1685	2
C₃₁×#1666	2
C₃₁×#1665	2
C₃₁×#1568	2
C₃₁×#1556	2
C₃₁×#1555	2
#19352	31

Subgroups — C31×#19352 #21476111

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19352 #21476111