Subgroups — C₃₁×#8979 #21465738

Order 7936.

660 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#460	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#1982	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1957	C₂	1	1	yes	maximal, characteristic
5	C₂×C₃₁×Q₈⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1296	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1279	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×SD₁₆×C₃₁	—	2	1	no	—
9	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
10	C₂×C₄×Q₈×C₃₁	—	2	1	no	—
11	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
12	C₃₁×#460	—	2	2	no	—
13	C₃₁×#460	C₄	1	4	yes	complemented
14	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×#102	—	2	1	no	—
17	C₂×C₃₁×#102	—	2	2	no	—
18	C₂×C₃₁×#102	—	2	1	no	—
19	C₂×C₃₁×#102	—	2	1	no	—
20	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
21	C₃₁×#389	—	2	1	no	—
22	C₃₁×#389	—	2	2	no	—
23	C₃₁×#384	—	2	1	no	—
24	C₃₁×#384	C₂²	1	1	yes	characteristic
25	C₄×C₄⋊C₄×C₃₁	—	2	1	no	—
26	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
27	C₃₁×Q₈⋊C₈	—	2	2	no	—
28	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
29	C₂²×D₄×C₃₁	—	2	1	no	—
30	C₂²×D₄×C₃₁	—	2	1	no	—
31	C₂²×D₄×C₃₁	—	2	1	no	—
32	C₂×SD₁₆×C₃₁	—	2	4	no	—
33	C₂×SD₁₆×C₃₁	—	4	2	no	—
34	C₂×SD₁₆×C₃₁	—	4	2	no	—
35	C₂²×C₈×C₃₁	—	2	1	no	—
36	C₂²×C₈×C₃₁	—	2	1	no	—
37	C₂²×C₈×C₃₁	—	2	1	no	—
38	C₃₁×#127	—	2	1	no	—
39	C₃₁×#127	—	2	1	no	—
40	C₃₁×#127	—	2	1	no	—
41	C₃₁×#127	C₂×C₄	1	2	yes	—
42	C₄×Q₈×C₃₁	—	2	1	no	—
43	C₄×Q₈×C₃₁	—	2	1	no	—
44	C₄×Q₈×C₃₁	—	4	2	no	—
45	C₄×Q₈×C₃₁	—	4	2	no	—
46	C₄×Q₈×C₃₁	C₂×C₄	1	2	yes	—
47	C₄×Q₈×C₃₁	D₄	1	2	yes	—
48	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
49	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
50	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
51	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
52	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
53	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
54	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
55	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
56	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
57	C₂×C₄²×C₃₁	—	2	1	no	—
58	C₂×C₄²×C₃₁	—	2	2	no	—
59	C₂×C₄²×C₃₁	—	2	1	no	—
60	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
61	C₃₁×C₄⋊C₈	—	2	1	no	—
62	C₃₁×C₄⋊C₈	—	2	1	no	—
63	C₃₁×C₄⋊C₈	—	2	1	no	—
64	C₃₁×C₄⋊C₈	C₂×C₄	1	2	yes	—
65	C₃₁×#102	—	2	4	no	—
66	C₃₁×#102	—	4	2	no	—
67	C₃₁×#102	—	4	4	no	—
68	C₃₁×#102	—	4	2	no	—
69	C₄×C₈×C₃₁	—	2	1	no	—
70	C₄×C₈×C₃₁	—	2	1	no	—
71	C₃₁×#95	—	2	2	no	—
72	C₃₁×#95	—	2	1	no	—
73	C₂⁴×C₃₁	—	4	1	no	—
74	C₂×Q₈×C₃₁	—	2	1	no	—
75	C₂×Q₈×C₃₁	—	2	1	no	—
76	C₂×Q₈×C₃₁	—	2	1	no	—
77	C₂×Q₈×C₃₁	C₂²⋊C₄	1	2	yes	—
78	C₂×Q₈×C₃₁	D₈	1	2	yes	—
79	C₂×Q₈×C₃₁	SD₁₆	1	2	yes	—
80	C₂×D₄×C₃₁	—	2	2	no	—
81	C₂×D₄×C₃₁	—	2	2	no	—
82	C₂×D₄×C₃₁	—	2	2	no	—
83	C₂×D₄×C₃₁	—	4	2	no	—
84	C₂×D₄×C₃₁	—	4	2	no	—
85	C₂×D₄×C₃₁	—	4	1	no	—
86	C₂×D₄×C₃₁	—	4	1	no	—
87	C₂×D₄×C₃₁	—	4	1	no	—
88	C₂×D₄×C₃₁	—	4	1	no	—
89	C₂×D₄×C₃₁	—	4	1	no	—
90	C₂×D₄×C₃₁	—	4	1	no	—
91	C₂×D₄×C₃₁	—	4	1	no	—
92	C₂×D₄×C₃₁	—	4	1	no	—
93	C₂×D₄×C₃₁	—	4	1	no	—
94	C₂×D₄×C₃₁	—	4	1	no	—
95	C₂×D₄×C₃₁	—	4	1	no	—
96	C₂²×C₄×C₃₁	—	2	1	no	—
97	C₂²×C₄×C₃₁	—	2	1	no	—
98	C₂²×C₄×C₃₁	—	2	1	no	—
99	C₂²×C₄×C₃₁	—	2	1	no	—
100	C₂²×C₄×C₃₁	—	2	2	no	—
101	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
102	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
104	SD₁₆×C₃₁	—	4	8	no	—
105	C₂×C₈×C₃₁	—	2	2	no	—
106	C₂×C₈×C₃₁	—	2	2	no	—
107	C₂×C₈×C₃₁	—	4	1	no	—
108	C₂×C₈×C₃₁	—	4	1	no	—
109	C₂×C₈×C₃₁	—	4	1	no	—
110	C₂×C₈×C₃₁	—	4	1	no	—
111	C₂×C₈×C₃₁	—	4	1	no	—
112	C₂×C₈×C₃₁	—	4	1	no	—
113	C₂×C₈×C₃₁	—	4	1	no	—
114	C₄⋊C₄×C₃₁	—	2	2	no	—
115	C₄⋊C₄×C₃₁	—	2	2	no	—
116	C₄⋊C₄×C₃₁	—	2	1	no	—
117	C₄⋊C₄×C₃₁	—	2	1	no	—
118	C₄⋊C₄×C₃₁	—	2	1	no	—
119	C₄⋊C₄×C₃₁	—	2	1	no	—
120	C₄⋊C₄×C₃₁	—	4	4	no	—
121	C₄⋊C₄×C₃₁	—	4	2	no	—
122	C₄⋊C₄×C₃₁	—	4	1	no	—
123	C₄⋊C₄×C₃₁	—	4	1	no	—
124	C₄⋊C₄×C₃₁	—	4	1	no	—
125	C₄⋊C₄×C₃₁	—	4	1	no	—
126	C₄⋊C₄×C₃₁	—	8	2	no	—
127	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
128	C₄⋊C₄×C₃₁	SD₁₆	1	2	yes	complemented
129	C₄²×C₃₁	—	2	2	no	—
130	C₄²×C₃₁	—	2	2	no	—
131	C₄²×C₃₁	—	4	4	no	—
132	C₄²×C₃₁	—	4	1	no	—
133	C₄²×C₃₁	—	4	1	no	—
134	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
135	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
136	C₄²×C₃₁	#41	1	1	yes	characteristic
137	C₄²×C₃₁	#41	1	1	yes	characteristic
138	#8979	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
139	C₂³×C₃₁	—	4	2	no	—
140	C₂³×C₃₁	—	8	1	no	—
141	C₂³×C₃₁	—	8	1	no	—
142	C₂³×C₃₁	—	8	1	no	—
143	C₂³×C₃₁	—	8	1	no	—
144	C₂³×C₃₁	—	8	1	no	—
145	C₂³×C₃₁	—	8	1	no	—
146	C₂³×C₃₁	#120	1	1	yes	characteristic
147	Q₈×C₃₁	—	2	2	no	—
148	Q₈×C₃₁	—	2	2	no	—
149	Q₈×C₃₁	—	2	2	no	—
150	Q₈×C₃₁	#102	1	4	yes	complemented
151	D₄×C₃₁	—	4	4	no	—
152	D₄×C₃₁	—	4	2	no	—
153	D₄×C₃₁	—	4	2	no	—
154	D₄×C₃₁	—	4	2	no	—
155	D₄×C₃₁	—	4	2	no	—
156	D₄×C₃₁	—	8	2	no	—
157	D₄×C₃₁	—	8	1	no	—
158	D₄×C₃₁	—	8	1	no	—
159	D₄×C₃₁	—	8	1	no	—
160	D₄×C₃₁	—	8	1	no	—
161	C₂×C₄×C₃₁	—	2	2	no	—
162	C₂×C₄×C₃₁	—	2	2	no	—
163	C₂×C₄×C₃₁	—	2	2	no	—
164	C₂×C₄×C₃₁	—	2	1	no	—
165	C₂×C₄×C₃₁	—	2	1	no	—
166	C₂×C₄×C₃₁	—	2	2	no	—
167	C₂×C₄×C₃₁	—	2	2	no	—
168	C₂×C₄×C₃₁	—	4	1	no	—
169	C₂×C₄×C₃₁	—	4	1	no	—
170	C₂×C₄×C₃₁	—	4	1	no	—
171	C₂×C₄×C₃₁	—	4	1	no	—
172	C₂×C₄×C₃₁	—	4	2	no	—
173	C₂×C₄×C₃₁	—	4	1	no	—
174	C₂×C₄×C₃₁	—	4	1	no	—
175	C₂×C₄×C₃₁	—	4	2	no	—
176	C₂×C₄×C₃₁	—	4	2	no	—
177	C₂×C₄×C₃₁	—	4	1	no	—
178	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
179	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
180	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
181	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
182	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
183	C₂×C₄×C₃₁	#123	1	1	yes	characteristic
184	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
185	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
186	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
187	C₂×C₄×C₃₁	#135	1	1	yes	characteristic
188	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
189	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
190	C₂×C₄×C₃₁	#137	1	1	yes	characteristic
191	C₂×C₄×C₃₁	#137	1	1	yes	characteristic
192	C₈×C₃₁	—	4	2	no	—
193	C₈×C₃₁	—	4	2	no	—
194	C₈×C₃₁	—	8	1	no	—
195	C₂×#460	C₂×C₃₁	1	1	yes	characteristic
196	Q₈×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
197	#1982	C₂×C₃₁	1	1	yes	characteristic
198	#1957	C₂×C₃₁	1	1	yes	characteristic
199	C₂×Q₈⋊C₈	C₂×C₃₁	1	1	yes	characteristic, complemented
200	#1296	C₂×C₃₁	1	1	yes	characteristic, complemented
201	#1279	C₂×C₃₁	1	1	yes	characteristic, complemented
202	C₂²×C₃₁	—	8	2	no	—
203	C₂²×C₃₁	—	8	2	no	—
204	C₂²×C₃₁	—	8	2	no	—
205	C₂²×C₃₁	—	16	1	no	—
206	C₂²×C₃₁	—	16	1	no	—
207	C₂²×C₃₁	—	16	1	no	—
208	C₂²×C₃₁	—	16	1	no	—
209	C₂²×C₃₁	#386	1	1	yes	characteristic
210	C₂²×C₃₁	#447	1	1	yes	characteristic
211	C₂²×C₃₁	#448	1	1	yes	characteristic
212	C₂²×C₃₁	#448	1	1	yes	characteristic
213	C₂²×C₃₁	#449	1	1	yes	characteristic
214	C₂²×C₃₁	#453	1	1	yes	characteristic
215	C₂²×C₃₁	#454	1	1	yes	characteristic
216	C₄×C₃₁	—	2	4	no	—
217	C₄×C₃₁	—	2	1	no	—
218	C₄×C₃₁	—	2	1	no	—
219	C₄×C₃₁	—	4	2	no	—
220	C₄×C₃₁	—	4	2	no	—
221	C₄×C₃₁	—	8	1	no	—
222	C₄×C₃₁	—	8	2	no	—
223	C₄×C₃₁	C₂×#102	1	1	yes	characteristic
224	C₄×C₃₁	#417	1	1	yes	characteristic
225	C₄×C₃₁	C₄×SD₁₆	1	1	yes	characteristic
226	C₄×C₃₁	#440	1	1	yes	characteristic
227	C₄×C₃₁	#460	1	1	yes	characteristic
228	C₄×C₃₁	#464	1	1	yes	characteristic
229	C₄×C₃₁	#480	1	1	yes	characteristic
230	C₄×C₃₁	#481	1	1	yes	characteristic
231	C₂²×SD₁₆	—	2	1	no	—
232	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
233	C₂×C₄×Q₈	—	2	1	no	—
234	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
235	#460	—	2	2	no	—
236	#460	C₄×C₃₁	1	4	yes	complemented
237	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
238	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
239	C₂×#102	—	2	1	no	—
240	C₂×#102	—	2	2	no	—
241	C₂×#102	—	2	1	no	—
242	C₂×#102	—	2	1	no	—
243	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
244	#389	—	2	1	no	—
245	#389	—	2	2	no	—
246	#384	—	2	1	no	—
247	#384	C₂²×C₃₁	1	1	yes	characteristic
248	C₄×C₄⋊C₄	—	2	1	no	—
249	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
250	Q₈⋊C₈	—	2	2	no	—
251	C₂×C₃₁	—	16	2	no	—
252	C₂×C₃₁	#1621	1	1	yes	characteristic
253	C₂×C₃₁	#1623	1	1	yes	characteristic
254	C₂×C₃₁	#1626	1	1	yes	characteristic
255	C₂×C₃₁	#1627	1	1	yes	characteristic
256	C₂×C₃₁	#1874	1	1	yes	characteristic
257	C₂×C₃₁	#1880	1	1	yes	characteristic
258	C₂×C₃₁	#1883	1	1	yes	characteristic
259	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
260	C₂²×D₄	—	2	1	no	—
261	C₂²×D₄	—	2	1	no	—
262	C₂²×D₄	—	2	1	no	—
263	C₂×SD₁₆	—	2	4	no	—
264	C₂×SD₁₆	—	4	2	no	—
265	C₂×SD₁₆	—	4	2	no	—
266	C₂²×C₈	—	2	1	no	—
267	C₂²×C₈	—	2	1	no	—
268	C₂²×C₈	—	2	1	no	—
269	#127	—	2	1	no	—
270	#127	—	2	1	no	—
271	#127	—	2	1	no	—
272	#127	C₂×C₄×C₃₁	1	2	yes	—
273	C₄×Q₈	—	2	1	no	—
274	C₄×Q₈	—	2	1	no	—
275	C₄×Q₈	—	4	2	no	—
276	C₄×Q₈	—	4	2	no	—
277	C₄×Q₈	C₂×C₄×C₃₁	1	2	yes	—
278	C₄×Q₈	D₄×C₃₁	1	2	yes	—
279	C₂×C₄⋊C₄	—	2	1	no	—
280	C₂×C₄⋊C₄	—	2	2	no	—
281	C₂×C₄⋊C₄	—	2	1	no	—
282	C₂×C₄⋊C₄	—	2	1	no	—
283	C₂×C₄⋊C₄	—	2	2	no	—
284	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
285	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
286	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
287	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
288	C₂×C₄²	—	2	1	no	—
289	C₂×C₄²	—	2	2	no	—
290	C₂×C₄²	—	2	1	no	—
291	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
292	C₄⋊C₈	—	2	1	no	—
293	C₄⋊C₈	—	2	1	no	—
294	C₄⋊C₈	—	2	1	no	—
295	C₄⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
296	#102	—	2	4	no	—
297	#102	—	4	2	no	—
298	#102	—	4	4	no	—
299	#102	—	4	2	no	—
300	C₄×C₈	—	2	1	no	—
301	C₄×C₈	—	2	1	no	—
302	#95	—	2	2	no	—
303	#95	—	2	1	no	—
304	C₃₁	#8979	1	1	yes	characteristic, direct product factor, Sylow
305	C₂⁴	—	4	1	no	—
306	C₂×Q₈	—	2	1	no	—
307	C₂×Q₈	—	2	1	no	—
308	C₂×Q₈	—	2	1	no	—
309	C₂×Q₈	C₂²⋊C₄×C₃₁	1	2	yes	—
310	C₂×Q₈	D₈×C₃₁	1	2	yes	—
311	C₂×Q₈	SD₁₆×C₃₁	1	2	yes	—
312	C₂×D₄	—	2	2	no	—
313	C₂×D₄	—	2	2	no	—
314	C₂×D₄	—	2	2	no	—
315	C₂×D₄	—	4	2	no	—
316	C₂×D₄	—	4	2	no	—
317	C₂×D₄	—	4	1	no	—
318	C₂×D₄	—	4	1	no	—
319	C₂×D₄	—	4	1	no	—
320	C₂×D₄	—	4	1	no	—
321	C₂×D₄	—	4	1	no	—
322	C₂×D₄	—	4	1	no	—
323	C₂×D₄	—	4	1	no	—
324	C₂×D₄	—	4	1	no	—
325	C₂×D₄	—	4	1	no	—
326	C₂×D₄	—	4	1	no	—
327	C₂×D₄	—	4	1	no	—
328	C₂²×C₄	—	2	1	no	—
329	C₂²×C₄	—	2	1	no	—
330	C₂²×C₄	—	2	1	no	—
331	C₂²×C₄	—	2	1	no	—
332	C₂²×C₄	—	2	2	no	—
333	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
334	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
335	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
336	SD₁₆	—	4	8	no	—
337	C₂×C₈	—	2	2	no	—
338	C₂×C₈	—	2	2	no	—
339	C₂×C₈	—	4	1	no	—
340	C₂×C₈	—	4	1	no	—
341	C₂×C₈	—	4	1	no	—
342	C₂×C₈	—	4	1	no	—
343	C₂×C₈	—	4	1	no	—
344	C₂×C₈	—	4	1	no	—
345	C₂×C₈	—	4	1	no	—
346	C₄⋊C₄	—	2	2	no	—
347	C₄⋊C₄	—	2	2	no	—
348	C₄⋊C₄	—	2	1	no	—
349	C₄⋊C₄	—	2	1	no	—
350	C₄⋊C₄	—	2	1	no	—
351	C₄⋊C₄	—	2	1	no	—
352	C₄⋊C₄	—	4	2	no	—
353	C₄⋊C₄	—	4	4	no	—
354	C₄⋊C₄	—	4	1	no	—
355	C₄⋊C₄	—	4	1	no	—
356	C₄⋊C₄	—	4	1	no	—
357	C₄⋊C₄	—	4	1	no	—
358	C₄⋊C₄	—	8	2	no	—
359	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
360	C₄⋊C₄	SD₁₆×C₃₁	1	2	yes	complemented
361	C₄²	—	2	2	no	—
362	C₄²	—	2	2	no	—
363	C₄²	—	4	4	no	—
364	C₄²	—	4	1	no	—
365	C₄²	—	4	1	no	—
366	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
367	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
368	C₄²	#41×C₃₁	1	1	yes	characteristic
369	C₄²	#41×C₃₁	1	1	yes	characteristic
370	C₂³	—	4	2	no	—
371	C₂³	—	8	1	no	—
372	C₂³	—	8	1	no	—
373	C₂³	—	8	1	no	—
374	C₂³	—	8	1	no	—
375	C₂³	—	8	1	no	—
376	C₂³	—	8	1	no	—
377	C₂³	C₃₁×#120	1	1	yes	characteristic
378	Q₈	—	2	2	no	—
379	Q₈	—	2	2	no	—
380	Q₈	—	2	2	no	—
381	Q₈	C₃₁×#102	1	4	yes	complemented
382	D₄	—	4	4	no	—
383	D₄	—	4	2	no	—
384	D₄	—	4	2	no	—
385	D₄	—	4	2	no	—
386	D₄	—	4	2	no	—
387	D₄	—	8	2	no	—
388	D₄	—	8	1	no	—
389	D₄	—	8	1	no	—
390	D₄	—	8	1	no	—
391	D₄	—	8	1	no	—
392	C₂×C₄	—	2	2	no	—
393	C₂×C₄	—	2	2	no	—
394	C₂×C₄	—	2	1	no	—
395	C₂×C₄	—	2	1	no	—
396	C₂×C₄	—	2	2	no	—
397	C₂×C₄	—	2	2	no	—
398	C₂×C₄	—	2	2	no	—
399	C₂×C₄	—	4	1	no	—
400	C₂×C₄	—	4	1	no	—
401	C₂×C₄	—	4	1	no	—
402	C₂×C₄	—	4	1	no	—
403	C₂×C₄	—	4	2	no	—
404	C₂×C₄	—	4	1	no	—
405	C₂×C₄	—	4	1	no	—
406	C₂×C₄	—	4	2	no	—
407	C₂×C₄	—	4	2	no	—
408	C₂×C₄	—	4	1	no	—
409	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
410	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
411	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
412	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
413	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
414	C₂×C₄	C₃₁×#123	1	1	yes	characteristic
415	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
416	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
417	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
418	C₂×C₄	C₃₁×#135	1	1	yes	characteristic
419	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
420	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
421	C₂×C₄	C₃₁×#137	1	1	yes	characteristic
422	C₂×C₄	C₃₁×#137	1	1	yes	characteristic
423	C₈	—	4	2	no	—
424	C₈	—	4	2	no	—
425	C₈	—	8	1	no	—
426	C₂²	—	8	2	no	—
427	C₂²	—	8	2	no	—
428	C₂²	—	8	2	no	—
429	C₂²	—	16	1	no	—
430	C₂²	—	16	1	no	—
431	C₂²	—	16	1	no	—
432	C₂²	—	16	1	no	—
433	C₂²	C₃₁×#386	1	1	yes	characteristic
434	C₂²	C₃₁×#447	1	1	yes	characteristic
435	C₂²	C₃₁×#448	1	1	yes	characteristic
436	C₂²	C₃₁×#448	1	1	yes	characteristic
437	C₂²	C₃₁×#449	1	1	yes	characteristic
438	C₂²	C₃₁×#453	1	1	yes	characteristic
439	C₂²	C₃₁×#454	1	1	yes	characteristic
440	C₄	—	2	4	no	—
441	C₄	—	2	1	no	—
442	C₄	—	2	1	no	—
443	C₄	—	4	2	no	—
444	C₄	—	4	2	no	—
445	C₄	—	8	1	no	—
446	C₄	—	8	2	no	—
447	C₄	C₂×C₃₁×#102	1	1	yes	characteristic
448	C₄	C₃₁×#417	1	1	yes	characteristic
449	C₄	C₄×SD₁₆×C₃₁	1	1	yes	characteristic
450	C₄	C₃₁×#440	1	1	yes	characteristic
451	C₄	C₃₁×#460	1	1	yes	characteristic
452	C₄	C₃₁×#464	1	1	yes	characteristic
453	C₄	C₃₁×#480	1	1	yes	characteristic
454	C₄	C₃₁×#481	1	1	yes	characteristic
455	C₂	—	16	2	no	—
456	C₂	C₃₁×#1621	1	1	yes	characteristic
457	C₂	C₃₁×#1623	1	1	yes	characteristic
458	C₂	C₃₁×#1626	1	1	yes	characteristic
459	C₂	C₃₁×#1627	1	1	yes	characteristic
460	C₂	C₃₁×#1874	1	1	yes	characteristic
461	C₂	C₃₁×#1880	1	1	yes	characteristic
462	C₂	C₃₁×#1883	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	#120
commutator	C₄²	C₂²×C₄×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
prime radical	C₂×C₃₁×#127	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. Q₈×C₄⋊C₄×C₃₁	C₂	semidirect
5. C₂×C₃₁×Q₈⋊C₈	C₂	semidirect
6. C₃₁×#1296	C₂	semidirect
7. C₃₁×#1279	C₂	semidirect
13. C₃₁×#460	C₄	semidirect
128. C₄⋊C₄×C₃₁	SD₁₆	semidirect
138. #8979	C₃₁	direct (N × H)
150. Q₈×C₃₁	#102	semidirect
196. Q₈×C₄⋊C₄	C₂×C₃₁	semidirect
199. C₂×Q₈⋊C₈	C₂×C₃₁	semidirect
200. #1296	C₂×C₃₁	semidirect
201. #1279	C₂×C₃₁	semidirect
236. #460	C₄×C₃₁	semidirect
304. C₃₁	#8979	direct (N × H)
360. C₄⋊C₄	SD₁₆×C₃₁	semidirect
381. Q₈	C₃₁×#102	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#460	2
Q₈×C₄⋊C₄×C₃₁	2
C₃₁×#1982	2
C₃₁×#1957	2
C₂×C₃₁×Q₈⋊C₈	2
C₃₁×#1296	2
C₃₁×#1279	2
#8979	31

Subgroups — C31×#8979 #21465738

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#8979 #21465738