Subgroups — C₃₁×#8921 #21465680

Order 7936.

644 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2366	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1957	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1948	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1947	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₃₁×#331	C₂	1	1	yes	maximal, characteristic
6	C₃₁×#1279	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1279	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₄×C₃₁	—	2	1	no	—
10	C₂×C₄×D₄×C₃₁	—	2	1	no	—
11	C₂×C₃₁×#107	—	2	1	no	—
12	C₂×C₃₁×#106	—	2	1	no	—
13	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#102	—	2	2	no	—
16	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
17	C₃₁×#390	—	2	1	no	—
18	C₃₁×#390	—	2	1	no	—
19	C₃₁×#386	—	2	2	no	—
20	C₃₁×#384	C₂²	1	1	yes	characteristic
21	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
22	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
23	C₃₁×#340	—	2	1	no	—
24	C₃₁×#340	—	2	1	no	—
25	C₃₁×#331	C₄	1	4	yes	complemented
26	C₂²×D₄×C₃₁	—	2	1	no	—
27	C₂²×D₄×C₃₁	—	2	1	no	—
28	C₂²×D₄×C₃₁	D₄	1	2	yes	—
29	C₂³×C₄×C₃₁	—	2	2	no	—
30	C₂²×C₈×C₃₁	—	2	1	no	—
31	C₂²×C₈×C₃₁	—	2	1	no	—
32	C₂²×C₈×C₃₁	—	2	1	no	—
33	C₃₁×#127	—	2	1	no	—
34	C₃₁×#127	—	2	1	no	—
35	C₃₁×#127	C₂×C₄	1	2	yes	—
36	C₃₁×#127	Q₈	1	2	yes	—
37	C₄×D₄×C₃₁	—	4	2	no	—
38	C₄×D₄×C₃₁	—	4	2	no	—
39	C₄×D₄×C₃₁	—	4	2	no	—
40	C₄×D₄×C₃₁	—	4	2	no	—
41	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
42	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
43	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
44	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
45	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
46	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
47	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
48	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
49	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
50	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
51	C₂×C₄²×C₃₁	—	2	1	no	—
52	C₂×C₄²×C₃₁	—	2	1	no	—
53	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
54	C₃₁×#107	—	4	2	no	—
55	C₃₁×#106	—	4	2	no	—
56	C₃₁×C₄⋊C₈	—	2	1	no	—
57	C₃₁×C₄⋊C₈	—	2	1	no	—
58	C₃₁×C₄⋊C₈	C₂×C₄	1	2	yes	complemented
59	C₃₁×C₄⋊C₈	C₂×C₄	1	2	yes	complemented
60	C₃₁×#102	—	4	4	no	—
61	C₄×C₈×C₃₁	—	2	1	no	—
62	C₄×C₈×C₃₁	—	2	1	no	—
63	C₃₁×#95	—	2	1	no	—
64	C₃₁×#95	—	2	1	no	—
65	C₂⁴×C₃₁	—	2	2	no	—
66	C₂×D₄×C₃₁	—	2	2	no	—
67	C₂×D₄×C₃₁	—	2	2	no	—
68	C₂×D₄×C₃₁	—	2	2	no	—
69	C₂×D₄×C₃₁	—	2	2	no	—
70	C₂×D₄×C₃₁	—	2	2	no	—
71	C₂×D₄×C₃₁	—	4	2	no	—
72	C₂×D₄×C₃₁	—	4	2	no	—
73	C₂×D₄×C₃₁	—	4	1	no	—
74	C₂×D₄×C₃₁	—	4	1	no	—
75	C₂×D₄×C₃₁	—	4	1	no	—
76	C₂×D₄×C₃₁	—	4	1	no	—
77	C₂×D₄×C₃₁	—	4	1	no	—
78	C₂×D₄×C₃₁	—	4	1	no	—
79	C₂×D₄×C₃₁	—	4	1	no	—
80	C₂×D₄×C₃₁	—	4	1	no	—
81	C₂×D₄×C₃₁	C₄⋊C₄	1	4	yes	—
82	C₂²×C₄×C₃₁	—	2	1	no	—
83	C₂²×C₄×C₃₁	—	2	1	no	—
84	C₂²×C₄×C₃₁	—	2	1	no	—
85	C₂²×C₄×C₃₁	—	2	1	no	—
86	C₂²×C₄×C₃₁	—	4	4	no	—
87	C₂²×C₄×C₃₁	—	4	4	no	—
88	C₂²×C₄×C₃₁	—	4	1	no	—
89	C₂²×C₄×C₃₁	—	8	2	no	—
90	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₃₁	—	2	2	no	—
94	C₂×C₈×C₃₁	—	2	2	no	—
95	C₂×C₈×C₃₁	—	2	2	no	—
96	C₂×C₈×C₃₁	—	4	1	no	—
97	C₂×C₈×C₃₁	—	4	1	no	—
98	C₂×C₈×C₃₁	—	4	1	no	—
99	C₂×C₈×C₃₁	—	4	1	no	—
100	C₂×C₈×C₃₁	—	4	1	no	—
101	C₂×C₈×C₃₁	—	4	1	no	—
102	C₄⋊C₄×C₃₁	—	2	1	no	—
103	C₄⋊C₄×C₃₁	—	2	1	no	—
104	C₄⋊C₄×C₃₁	—	2	1	no	—
105	C₄⋊C₄×C₃₁	—	2	1	no	—
106	C₄⋊C₄×C₃₁	—	2	1	no	—
107	C₄⋊C₄×C₃₁	—	2	1	no	—
108	C₄⋊C₄×C₃₁	—	4	2	no	—
109	C₄⋊C₄×C₃₁	—	4	1	no	—
110	C₄⋊C₄×C₃₁	—	4	1	no	—
111	C₄⋊C₄×C₃₁	—	4	1	no	—
112	C₄⋊C₄×C₃₁	—	4	1	no	—
113	C₄⋊C₄×C₃₁	D₈	1	2	yes	—
114	C₄⋊C₄×C₃₁	SD₁₆	1	2	yes	—
115	C₂²⋊C₄×C₃₁	—	4	4	no	—
116	C₂²⋊C₄×C₃₁	—	4	4	no	—
117	C₂²⋊C₄×C₃₁	—	8	2	no	—
118	C₄²×C₃₁	—	4	1	no	—
119	C₄²×C₃₁	—	4	1	no	—
120	C₄²×C₃₁	—	4	1	no	—
121	C₄²×C₃₁	—	4	1	no	—
122	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
123	C₄²×C₃₁	C₂×Q₈	1	1	yes	characteristic
124	C₄²×C₃₁	#41	1	1	yes	characteristic
125	C₄²×C₃₁	#41	1	1	yes	characteristic
126	#8921	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
127	C₂³×C₃₁	—	2	4	no	—
128	C₂³×C₃₁	—	4	2	no	—
129	C₂³×C₃₁	—	4	2	no	—
130	C₂³×C₃₁	—	4	2	no	—
131	C₂³×C₃₁	—	4	2	no	—
132	C₂³×C₃₁	—	4	2	no	—
133	C₂³×C₃₁	—	4	2	no	—
134	C₂³×C₃₁	#120	1	1	yes	characteristic
135	D₄×C₃₁	—	4	4	no	—
136	D₄×C₃₁	—	4	2	no	—
137	D₄×C₃₁	—	4	2	no	—
138	D₄×C₃₁	—	4	2	no	—
139	D₄×C₃₁	—	4	2	no	—
140	D₄×C₃₁	—	4	2	no	—
141	D₄×C₃₁	—	4	2	no	—
142	D₄×C₃₁	—	4	2	no	—
143	D₄×C₃₁	—	4	2	no	—
144	D₄×C₃₁	—	8	2	no	—
145	C₂×C₄×C₃₁	—	2	1	no	—
146	C₂×C₄×C₃₁	—	2	1	no	—
147	C₂×C₄×C₃₁	—	2	2	no	—
148	C₂×C₄×C₃₁	—	2	2	no	—
149	C₂×C₄×C₃₁	—	4	1	no	—
150	C₂×C₄×C₃₁	—	4	1	no	—
151	C₂×C₄×C₃₁	—	4	1	no	—
152	C₂×C₄×C₃₁	—	4	1	no	—
153	C₂×C₄×C₃₁	—	4	1	no	—
154	C₂×C₄×C₃₁	—	4	1	no	—
155	C₂×C₄×C₃₁	—	4	1	no	—
156	C₂×C₄×C₃₁	—	4	1	no	—
157	C₂×C₄×C₃₁	—	4	1	no	—
158	C₂×C₄×C₃₁	—	4	1	no	—
159	C₂×C₄×C₃₁	—	8	8	no	—
160	C₂×C₄×C₃₁	—	8	1	no	—
161	C₂×C₄×C₃₁	—	8	1	no	—
162	C₂×C₄×C₃₁	—	8	1	no	—
163	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
164	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
165	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
166	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
167	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
168	C₂×C₄×C₃₁	#123	1	1	yes	characteristic
169	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
170	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
171	C₂×C₄×C₃₁	#135	1	1	yes	characteristic
172	C₂×C₄×C₃₁	#135	1	1	yes	characteristic
173	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
174	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
175	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
176	C₂×C₄×C₃₁	#137	1	1	yes	characteristic
177	C₈×C₃₁	—	4	2	no	—
178	C₈×C₃₁	—	4	2	no	—
179	C₈×C₃₁	—	4	2	no	—
180	#2366	C₂×C₃₁	1	1	yes	characteristic
181	#1957	C₂×C₃₁	1	1	yes	characteristic
182	#1948	C₂×C₃₁	1	1	yes	characteristic, complemented
183	#1947	C₂×C₃₁	1	1	yes	characteristic, complemented
184	C₂×#331	C₂×C₃₁	1	1	yes	characteristic
185	#1279	C₂×C₃₁	1	1	yes	characteristic, complemented
186	#1279	C₂×C₃₁	1	1	yes	characteristic, complemented
187	C₂²×C₃₁	—	4	4	no	—
188	C₂²×C₃₁	—	4	4	no	—
189	C₂²×C₃₁	—	4	4	no	—
190	C₂²×C₃₁	—	8	2	no	—
191	C₂²×C₃₁	—	8	2	no	—
192	C₂²×C₃₁	—	8	2	no	—
193	C₂²×C₃₁	—	8	2	no	—
194	C₂²×C₃₁	#385	1	1	yes	characteristic
195	C₂²×C₃₁	#447	1	1	yes	characteristic
196	C₂²×C₃₁	#449	1	1	yes	characteristic
197	C₂²×C₃₁	#450	1	1	yes	characteristic
198	C₂²×C₃₁	#452	1	1	yes	characteristic
199	C₂²×C₃₁	#453	1	1	yes	characteristic
200	C₂²×C₃₁	#455	1	1	yes	characteristic
201	C₄×C₃₁	—	2	1	no	—
202	C₄×C₃₁	—	2	1	no	—
203	C₄×C₃₁	—	4	2	no	—
204	C₄×C₃₁	—	4	2	no	—
205	C₄×C₃₁	—	8	1	no	—
206	C₄×C₃₁	—	8	1	no	—
207	C₄×C₃₁	—	16	1	no	—
208	C₄×C₃₁	C₄×D₈	1	1	yes	characteristic
209	C₄×C₃₁	C₄×SD₁₆	1	1	yes	characteristic
210	C₄×C₃₁	#440	1	1	yes	characteristic
211	C₄×C₃₁	#442	1	1	yes	characteristic
212	C₄×C₃₁	#474	1	1	yes	characteristic
213	C₄×C₃₁	#476	1	1	yes	characteristic
214	C₄×C₃₁	#478	1	1	yes	characteristic
215	C₄×C₃₁	#478	1	1	yes	characteristic
216	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
217	C₂×C₄×D₄	—	2	1	no	—
218	C₂×C₄×D₄	—	2	1	no	—
219	C₂×#107	—	2	1	no	—
220	C₂×#106	—	2	1	no	—
221	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
222	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
223	C₂×#102	—	2	2	no	—
224	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
225	#390	—	2	1	no	—
226	#390	—	2	1	no	—
227	#386	—	2	2	no	—
228	#384	C₂²×C₃₁	1	1	yes	characteristic
229	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
230	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
231	#340	—	2	1	no	—
232	#340	—	2	1	no	—
233	#331	C₄×C₃₁	1	4	yes	complemented
234	C₂×C₃₁	—	8	4	no	—
235	C₂×C₃₁	#1621	1	1	yes	characteristic
236	C₂×C₃₁	#1631	1	1	yes	characteristic
237	C₂×C₃₁	#1639	1	1	yes	characteristic
238	C₂×C₃₁	#1640	1	1	yes	characteristic
239	C₂×C₃₁	#1856	1	1	yes	characteristic
240	C₂×C₃₁	#1862	1	1	yes	characteristic
241	C₂×C₃₁	#1864	1	1	yes	characteristic
242	C₂²×D₄	—	2	1	no	—
243	C₂²×D₄	—	2	1	no	—
244	C₂²×D₄	D₄×C₃₁	1	2	yes	—
245	C₂³×C₄	—	2	2	no	—
246	C₂²×C₈	—	2	1	no	—
247	C₂²×C₈	—	2	1	no	—
248	C₂²×C₈	—	2	1	no	—
249	#127	—	2	1	no	—
250	#127	—	2	1	no	—
251	#127	C₂×C₄×C₃₁	1	2	yes	—
252	#127	Q₈×C₃₁	1	2	yes	—
253	C₄×D₄	—	4	2	no	—
254	C₄×D₄	—	4	2	no	—
255	C₄×D₄	—	4	2	no	—
256	C₄×D₄	—	4	2	no	—
257	C₂×C₄⋊C₄	—	2	1	no	—
258	C₂×C₄⋊C₄	—	2	1	no	—
259	C₂×C₄⋊C₄	—	2	1	no	—
260	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
261	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
262	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
263	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
264	C₂×C₂²⋊C₄	—	2	2	no	—
265	C₂×C₂²⋊C₄	—	2	2	no	—
266	C₂×C₂²⋊C₄	—	2	2	no	—
267	C₂×C₄²	—	2	1	no	—
268	C₂×C₄²	—	2	1	no	—
269	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
270	#107	—	4	2	no	—
271	#106	—	4	2	no	—
272	C₄⋊C₈	—	2	1	no	—
273	C₄⋊C₈	—	2	1	no	—
274	C₄⋊C₈	C₂×C₄×C₃₁	1	2	yes	complemented
275	C₄⋊C₈	C₂×C₄×C₃₁	1	2	yes	complemented
276	#102	—	4	4	no	—
277	C₄×C₈	—	2	1	no	—
278	C₄×C₈	—	2	1	no	—
279	#95	—	2	1	no	—
280	#95	—	2	1	no	—
281	C₃₁	#8921	1	1	yes	characteristic, direct product factor, Sylow
282	C₂⁴	—	2	2	no	—
283	C₂×D₄	—	2	2	no	—
284	C₂×D₄	—	2	2	no	—
285	C₂×D₄	—	2	2	no	—
286	C₂×D₄	—	2	2	no	—
287	C₂×D₄	—	2	2	no	—
288	C₂×D₄	—	4	2	no	—
289	C₂×D₄	—	4	2	no	—
290	C₂×D₄	—	4	1	no	—
291	C₂×D₄	—	4	1	no	—
292	C₂×D₄	—	4	1	no	—
293	C₂×D₄	—	4	1	no	—
294	C₂×D₄	—	4	1	no	—
295	C₂×D₄	—	4	1	no	—
296	C₂×D₄	—	4	1	no	—
297	C₂×D₄	—	4	1	no	—
298	C₂×D₄	C₄⋊C₄×C₃₁	1	4	yes	—
299	C₂²×C₄	—	2	1	no	—
300	C₂²×C₄	—	2	1	no	—
301	C₂²×C₄	—	2	1	no	—
302	C₂²×C₄	—	2	1	no	—
303	C₂²×C₄	—	4	4	no	—
304	C₂²×C₄	—	4	4	no	—
305	C₂²×C₄	—	4	1	no	—
306	C₂²×C₄	—	8	2	no	—
307	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
308	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
309	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
310	C₂×C₈	—	2	2	no	—
311	C₂×C₈	—	2	2	no	—
312	C₂×C₈	—	2	2	no	—
313	C₂×C₈	—	4	1	no	—
314	C₂×C₈	—	4	1	no	—
315	C₂×C₈	—	4	1	no	—
316	C₂×C₈	—	4	1	no	—
317	C₂×C₈	—	4	1	no	—
318	C₂×C₈	—	4	1	no	—
319	C₄⋊C₄	—	2	1	no	—
320	C₄⋊C₄	—	2	1	no	—
321	C₄⋊C₄	—	2	1	no	—
322	C₄⋊C₄	—	2	1	no	—
323	C₄⋊C₄	—	2	1	no	—
324	C₄⋊C₄	—	2	1	no	—
325	C₄⋊C₄	—	4	2	no	—
326	C₄⋊C₄	—	4	1	no	—
327	C₄⋊C₄	—	4	1	no	—
328	C₄⋊C₄	—	4	1	no	—
329	C₄⋊C₄	—	4	1	no	—
330	C₄⋊C₄	D₈×C₃₁	1	2	yes	—
331	C₄⋊C₄	SD₁₆×C₃₁	1	2	yes	—
332	C₂²⋊C₄	—	4	4	no	—
333	C₂²⋊C₄	—	4	4	no	—
334	C₂²⋊C₄	—	8	2	no	—
335	C₄²	—	4	1	no	—
336	C₄²	—	4	1	no	—
337	C₄²	—	4	1	no	—
338	C₄²	—	4	1	no	—
339	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
340	C₄²	C₂×Q₈×C₃₁	1	1	yes	characteristic
341	C₄²	#41×C₃₁	1	1	yes	characteristic
342	C₄²	#41×C₃₁	1	1	yes	characteristic
343	C₂³	—	2	4	no	—
344	C₂³	—	4	2	no	—
345	C₂³	—	4	2	no	—
346	C₂³	—	4	2	no	—
347	C₂³	—	4	2	no	—
348	C₂³	—	4	2	no	—
349	C₂³	—	4	2	no	—
350	C₂³	C₃₁×#120	1	1	yes	characteristic
351	D₄	—	4	4	no	—
352	D₄	—	4	2	no	—
353	D₄	—	4	2	no	—
354	D₄	—	4	2	no	—
355	D₄	—	4	2	no	—
356	D₄	—	4	2	no	—
357	D₄	—	4	2	no	—
358	D₄	—	4	2	no	—
359	D₄	—	4	2	no	—
360	D₄	—	8	2	no	—
361	C₂×C₄	—	2	1	no	—
362	C₂×C₄	—	2	1	no	—
363	C₂×C₄	—	2	2	no	—
364	C₂×C₄	—	2	2	no	—
365	C₂×C₄	—	4	1	no	—
366	C₂×C₄	—	4	1	no	—
367	C₂×C₄	—	4	1	no	—
368	C₂×C₄	—	4	1	no	—
369	C₂×C₄	—	4	1	no	—
370	C₂×C₄	—	4	1	no	—
371	C₂×C₄	—	4	1	no	—
372	C₂×C₄	—	4	1	no	—
373	C₂×C₄	—	4	1	no	—
374	C₂×C₄	—	4	1	no	—
375	C₂×C₄	—	8	8	no	—
376	C₂×C₄	—	8	1	no	—
377	C₂×C₄	—	8	1	no	—
378	C₂×C₄	—	8	1	no	—
379	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
380	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
381	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
382	C₂×C₄	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
383	C₂×C₄	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
384	C₂×C₄	C₃₁×#123	1	1	yes	characteristic
385	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
386	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
387	C₂×C₄	C₃₁×#135	1	1	yes	characteristic
388	C₂×C₄	C₃₁×#135	1	1	yes	characteristic
389	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
390	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
391	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
392	C₂×C₄	C₃₁×#137	1	1	yes	characteristic
393	C₈	—	4	2	no	—
394	C₈	—	4	2	no	—
395	C₈	—	4	2	no	—
396	C₂²	—	4	4	no	—
397	C₂²	—	4	4	no	—
398	C₂²	—	4	4	no	—
399	C₂²	—	8	2	no	—
400	C₂²	—	8	2	no	—
401	C₂²	—	8	2	no	—
402	C₂²	—	8	2	no	—
403	C₂²	C₃₁×#385	1	1	yes	characteristic
404	C₂²	C₃₁×#447	1	1	yes	characteristic
405	C₂²	C₃₁×#449	1	1	yes	characteristic
406	C₂²	C₃₁×#450	1	1	yes	characteristic
407	C₂²	C₃₁×#452	1	1	yes	characteristic
408	C₂²	C₃₁×#453	1	1	yes	characteristic
409	C₂²	C₃₁×#455	1	1	yes	characteristic
410	C₄	—	2	1	no	—
411	C₄	—	2	1	no	—
412	C₄	—	4	2	no	—
413	C₄	—	4	2	no	—
414	C₄	—	8	1	no	—
415	C₄	—	8	1	no	—
416	C₄	—	16	1	no	—
417	C₄	C₄×D₈×C₃₁	1	1	yes	characteristic
418	C₄	C₄×SD₁₆×C₃₁	1	1	yes	characteristic
419	C₄	C₃₁×#440	1	1	yes	characteristic
420	C₄	C₃₁×#442	1	1	yes	characteristic
421	C₄	C₃₁×#474	1	1	yes	characteristic
422	C₄	C₃₁×#476	1	1	yes	characteristic
423	C₄	C₃₁×#478	1	1	yes	characteristic
424	C₄	C₃₁×#478	1	1	yes	characteristic
425	C₂	—	8	4	no	—
426	C₂	C₃₁×#1621	1	1	yes	characteristic
427	C₂	C₃₁×#1631	1	1	yes	characteristic
428	C₂	C₃₁×#1639	1	1	yes	characteristic
429	C₂	C₃₁×#1640	1	1	yes	characteristic
430	C₂	C₃₁×#1856	1	1	yes	characteristic
431	C₂	C₃₁×#1862	1	1	yes	characteristic
432	C₂	C₃₁×#1864	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
3. C₃₁×#1948	C₂	semidirect
4. C₃₁×#1947	C₂	semidirect
6. C₃₁×#1279	C₂	semidirect
7. C₃₁×#1279	C₂	semidirect
25. C₃₁×#331	C₄	semidirect
58. C₃₁×C₄⋊C₈	C₂×C₄	semidirect
59. C₃₁×C₄⋊C₈	C₂×C₄	semidirect
126. #8921	C₃₁	direct (N × H)
182. #1948	C₂×C₃₁	semidirect
183. #1947	C₂×C₃₁	semidirect
185. #1279	C₂×C₃₁	semidirect
186. #1279	C₂×C₃₁	semidirect
233. #331	C₄×C₃₁	semidirect
274. C₄⋊C₈	C₂×C₄×C₃₁	semidirect
275. C₄⋊C₈	C₂×C₄×C₃₁	semidirect
281. C₃₁	#8921	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×#2366	2
C₃₁×#1957	2
C₃₁×#1948	2
C₃₁×#1947	2
C₂×C₃₁×#331	2
C₃₁×#1279	2
#8921	31

Subgroups — C31×#8921 #21465680

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#8921 #21465680