Subgroups — C₃₁×#19234 #21475993

Order 7936.

704 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3246	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3205	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#3138	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2935	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1701	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1697	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1681	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1677	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1565	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1542	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₃₁×#331	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#131	—	2	1	no	—
13	C₃₁×#533	C₂²	1	1	yes	characteristic
14	C₃₁×#532	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
16	C₃₁×#493	—	2	1	no	—
17	C₃₁×#490	—	2	2	no	—
18	C₃₁×#486	—	2	1	no	—
19	C₃₁×#486	C₂²	1	2	yes	—
20	C₃₁×#486	C₂²	1	2	yes	—
21	C₃₁×#476	C₂²	1	2	yes	complemented
22	C₃₁×#475	—	2	1	no	—
23	C₃₁×#465	—	2	2	no	—
24	C₃₁×#461	C₂²	1	2	yes	complemented
25	C₃₁×#460	—	2	1	no	—
26	C₃₁×#459	C₂²	1	2	yes	—
27	C₃₁×#459	C₂²	1	2	yes	—
28	C₃₁×#447	—	2	2	no	—
29	C₃₁×#436	C₂²	1	2	yes	complemented
30	C₃₁×#435	—	2	2	no	—
31	C₈×D₄×C₃₁	C₂²	1	2	yes	complemented
32	C₃₁×#431	C₂²	1	1	yes	characteristic
33	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
34	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
35	C₃₁×#418	—	2	1	no	—
36	C₃₁×#418	—	2	1	no	—
37	C₃₁×#408	—	2	1	no	—
38	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
39	C₃₁×#335	C₂²	1	2	yes	complemented
40	C₃₁×#334	C₂²	1	2	yes	complemented
41	C₃₁×#333	C₂²	1	1	yes	characteristic, complemented
42	C₃₁×#333	C₂²	1	1	yes	characteristic, complemented
43	C₃₁×#332	C₂²	1	1	yes	characteristic, complemented
44	C₃₁×#332	C₂²	1	1	yes	characteristic, complemented
45	C₃₁×#331	C₂²	1	2	yes	—
46	C₃₁×#331	C₂²	1	2	yes	—
47	C₂×#41×C₃₁	—	2	1	no	—
48	C₂²×D₄×C₃₁	—	2	1	no	—
49	C₂²×D₄×C₃₁	—	2	1	no	—
50	C₂²×D₄×C₃₁	—	2	1	no	—
51	C₂×SD₁₆×C₃₁	—	2	2	no	—
52	C₂×D₈×C₃₁	—	2	2	no	—
53	C₂×D₈×C₃₁	—	2	2	no	—
54	C₃₁×#131	—	4	4	no	—
55	C₂×M₁₆×C₃₁	—	2	2	no	—
56	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
57	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
58	C₂²×C₈×C₃₁	—	2	1	no	—
59	C₂²×C₈×C₃₁	—	2	1	no	—
60	C₂²×C₈×C₃₁	—	2	2	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×#127	—	2	1	no	—
65	C₃₁×#127	—	2	1	no	—
66	C₃₁×#127	—	2	1	no	—
67	C₃₁×#127	D₄	1	1	yes	characteristic
68	C₃₁×#127	C₂³	1	2	yes	—
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	D₄	1	2	yes	—
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	C₂³	1	2	yes	—
79	C₃₁×#117	—	2	1	no	—
80	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
81	C₃₁×#106	—	2	2	no	—
82	C₃₁×C₄⋊C₈	—	2	1	no	—
83	C₃₁×C₄⋊C₈	—	2	1	no	—
84	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
85	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
86	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
87	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
88	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
89	C₃₁×#103	—	2	2	no	—
90	C₃₁×#102	—	2	2	no	—
91	C₃₁×#102	—	2	2	no	—
92	C₃₁×#102	—	2	2	no	—
93	C₃₁×#102	—	2	2	no	—
94	C₃₁×#102	—	2	2	no	—
95	C₃₁×#102	—	2	2	no	—
96	C₃₁×#102	—	2	2	no	—
97	C₃₁×C₂²⋊C₈	—	2	1	no	—
98	C₃₁×C₂²⋊C₈	—	2	2	no	—
99	C₃₁×C₂²⋊C₈	—	2	2	no	—
100	C₃₁×#97	D₄	1	2	yes	—
101	C₄×C₈×C₃₁	—	2	1	no	—
102	C₄×C₈×C₃₁	C₂³	1	2	yes	—
103	C₂⁴×C₃₁	—	4	1	no	—
104	#41×C₃₁	—	4	2	no	—
105	#41×C₃₁	—	4	2	no	—
106	C₂×Q₈×C₃₁	—	2	1	no	—
107	C₂×Q₈×C₃₁	—	2	1	no	—
108	C₂×Q₈×C₃₁	—	2	1	no	—
109	C₂×D₄×C₃₁	—	2	2	no	—
110	C₂×D₄×C₃₁	—	2	1	no	—
111	C₂×D₄×C₃₁	—	2	1	no	—
112	C₂×D₄×C₃₁	—	2	2	no	—
113	C₂×D₄×C₃₁	—	2	2	no	—
114	C₂×D₄×C₃₁	—	2	2	no	—
115	C₂×D₄×C₃₁	—	4	1	no	—
116	C₂×D₄×C₃₁	—	4	1	no	—
117	C₂×D₄×C₃₁	—	4	1	no	—
118	C₂×D₄×C₃₁	—	4	2	no	—
119	C₂×D₄×C₃₁	—	4	2	no	—
120	C₂×D₄×C₃₁	—	8	1	no	—
121	C₂×D₄×C₃₁	—	8	1	no	—
122	C₂×D₄×C₃₁	—	8	2	no	—
123	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
124	C₂²×C₄×C₃₁	—	2	2	no	—
125	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
126	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
127	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
128	SD₁₆×C₃₁	—	8	2	no	—
129	D₈×C₃₁	—	8	2	no	—
130	D₈×C₃₁	—	8	2	no	—
131	M₁₆×C₃₁	—	2	2	no	—
132	M₁₆×C₃₁	—	2	2	no	—
133	M₁₆×C₃₁	—	4	4	no	—
134	C₂×C₈×C₃₁	—	2	1	no	—
135	C₂×C₈×C₃₁	—	2	1	no	—
136	C₂×C₈×C₃₁	—	2	1	no	—
137	C₂×C₈×C₃₁	—	2	1	no	—
138	C₂×C₈×C₃₁	—	2	2	no	—
139	C₂×C₈×C₃₁	—	4	4	no	—
140	C₂×C₈×C₃₁	—	4	2	no	—
141	C₂×C₈×C₃₁	—	8	1	no	—
142	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
143	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
144	C₄⋊C₄×C₃₁	—	2	1	no	—
145	C₄⋊C₄×C₃₁	—	2	1	no	—
146	C₄⋊C₄×C₃₁	—	2	1	no	—
147	C₄⋊C₄×C₃₁	—	2	1	no	—
148	C₄⋊C₄×C₃₁	—	2	1	no	—
149	C₄⋊C₄×C₃₁	—	4	1	no	—
150	C₄⋊C₄×C₃₁	—	4	1	no	—
151	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
152	C₂²⋊C₄×C₃₁	—	2	2	no	—
153	C₂²⋊C₄×C₃₁	—	2	2	no	—
154	C₂²⋊C₄×C₃₁	—	4	1	no	—
155	C₄²×C₃₁	—	4	1	no	—
156	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
157	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
158	C₄²×C₃₁	#41	1	1	yes	characteristic
159	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
160	#19234	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
161	C₂³×C₃₁	—	2	2	no	—
162	C₂³×C₃₁	—	4	2	no	—
163	C₂³×C₃₁	—	8	2	no	—
164	C₂³×C₃₁	—	8	2	no	—
165	C₂³×C₃₁	—	8	2	no	—
166	C₂³×C₃₁	#121	1	1	yes	characteristic
167	Q₈×C₃₁	—	4	2	no	—
168	Q₈×C₃₁	—	4	1	no	—
169	Q₈×C₃₁	—	4	1	no	—
170	Q₈×C₃₁	—	8	1	no	—
171	D₄×C₃₁	—	2	2	no	—
172	D₄×C₃₁	—	2	2	no	—
173	D₄×C₃₁	—	4	4	no	—
174	D₄×C₃₁	—	4	2	no	—
175	D₄×C₃₁	—	4	2	no	—
176	D₄×C₃₁	—	4	2	no	—
177	D₄×C₃₁	—	4	2	no	—
178	D₄×C₃₁	—	4	2	no	—
179	D₄×C₃₁	—	8	1	no	—
180	D₄×C₃₁	—	8	2	no	—
181	D₄×C₃₁	—	8	2	no	—
182	D₄×C₃₁	—	8	2	no	—
183	D₄×C₃₁	—	8	2	no	—
184	C₂×C₄×C₃₁	—	2	1	no	—
185	C₂×C₄×C₃₁	—	2	1	no	—
186	C₂×C₄×C₃₁	—	2	2	no	—
187	C₂×C₄×C₃₁	—	4	1	no	—
188	C₂×C₄×C₃₁	—	4	2	no	—
189	C₂×C₄×C₃₁	—	4	2	no	—
190	C₂×C₄×C₃₁	—	4	1	no	—
191	C₂×C₄×C₃₁	—	4	1	no	—
192	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
193	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
194	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
195	C₂×C₄×C₃₁	#135	1	2	yes	—
196	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
197	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
198	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
199	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
200	C₈×C₃₁	—	2	2	no	—
201	C₈×C₃₁	—	2	2	no	—
202	C₈×C₃₁	—	4	1	no	—
203	C₈×C₃₁	—	4	1	no	—
204	C₈×C₃₁	—	4	2	no	—
205	C₈×C₃₁	—	4	1	no	—
206	C₈×C₃₁	—	4	1	no	—
207	#3246	C₂×C₃₁	1	1	yes	characteristic, complemented
208	#3205	C₂×C₃₁	1	1	yes	characteristic
209	#3138	C₂×C₃₁	1	1	yes	characteristic, complemented
210	#2935	C₂×C₃₁	1	1	yes	characteristic, complemented
211	#1701	C₂×C₃₁	1	2	yes	complemented
212	#1697	C₂×C₃₁	1	2	yes	complemented
213	#1681	C₂×C₃₁	1	2	yes	complemented
214	#1677	C₂×C₃₁	1	2	yes	complemented
215	#1565	C₂×C₃₁	1	1	yes	characteristic, complemented
216	#1542	C₂×C₃₁	1	1	yes	characteristic, complemented
217	C₂×#331	C₂×C₃₁	1	1	yes	characteristic, complemented
218	C₂²×C₃₁	—	2	1	no	—
219	C₂²×C₃₁	—	2	1	no	—
220	C₂²×C₃₁	—	4	2	no	—
221	C₂²×C₃₁	—	4	2	no	—
222	C₂²×C₃₁	—	4	2	no	—
223	C₂²×C₃₁	—	8	2	no	—
224	C₂²×C₃₁	—	8	2	no	—
225	C₂²×C₃₁	—	8	2	no	—
226	C₂²×C₃₁	—	16	4	no	—
227	C₂²×C₃₁	#463	1	2	yes	—
228	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
229	C₄×C₃₁	—	2	1	no	—
230	C₄×C₃₁	—	2	1	no	—
231	C₄×C₃₁	—	2	1	no	—
232	C₄×C₃₁	—	4	2	no	—
233	C₄×C₃₁	—	8	1	no	—
234	C₄×C₃₁	—	8	1	no	—
235	C₄×C₃₁	#463	1	2	yes	—
236	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
237	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
238	C₄×C₃₁	#576	1	1	yes	characteristic
239	C₄×C₃₁	#577	1	1	yes	characteristic
240	C₂×#131	—	2	1	no	—
241	#533	C₂²×C₃₁	1	1	yes	characteristic
242	#532	C₂²×C₃₁	1	1	yes	characteristic
243	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
244	#493	—	2	1	no	—
245	#490	—	2	2	no	—
246	#486	—	2	1	no	—
247	#486	C₂²×C₃₁	1	2	yes	—
248	#486	C₂²×C₃₁	1	2	yes	—
249	#476	C₂²×C₃₁	1	2	yes	complemented
250	#475	—	2	1	no	—
251	#465	—	2	2	no	—
252	#461	C₂²×C₃₁	1	2	yes	complemented
253	#460	—	2	1	no	—
254	#459	C₂²×C₃₁	1	2	yes	—
255	#459	C₂²×C₃₁	1	2	yes	—
256	#447	—	2	2	no	—
257	#436	C₂²×C₃₁	1	2	yes	complemented
258	#435	—	2	2	no	—
259	C₈×D₄	C₂²×C₃₁	1	2	yes	complemented
260	#431	C₂²×C₃₁	1	1	yes	characteristic
261	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
262	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
263	#418	—	2	1	no	—
264	#418	—	2	1	no	—
265	#408	—	2	1	no	—
266	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
267	#335	C₂²×C₃₁	1	2	yes	complemented
268	#334	C₂²×C₃₁	1	2	yes	complemented
269	#333	C₂²×C₃₁	1	1	yes	characteristic, complemented
270	#333	C₂²×C₃₁	1	1	yes	characteristic, complemented
271	#332	C₂²×C₃₁	1	1	yes	characteristic, complemented
272	#332	C₂²×C₃₁	1	1	yes	characteristic, complemented
273	#331	C₂²×C₃₁	1	2	yes	—
274	#331	C₂²×C₃₁	1	2	yes	—
275	C₂×C₃₁	—	2	2	no	—
276	C₂×C₃₁	—	8	2	no	—
277	C₂×C₃₁	—	16	2	no	—
278	C₂×C₃₁	C₂×#463	1	1	yes	characteristic
279	C₂×C₃₁	#3062	1	1	yes	characteristic
280	C₂×C₃₁	#3068	1	1	yes	characteristic
281	C₂×#41	—	2	1	no	—
282	C₂²×D₄	—	2	1	no	—
283	C₂²×D₄	—	2	1	no	—
284	C₂²×D₄	—	2	1	no	—
285	C₂×SD₁₆	—	2	2	no	—
286	C₂×D₈	—	2	2	no	—
287	C₂×D₈	—	2	2	no	—
288	#131	—	4	4	no	—
289	C₂×M₁₆	—	2	2	no	—
290	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
291	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
292	C₂²×C₈	—	2	1	no	—
293	C₂²×C₈	—	2	1	no	—
294	C₂²×C₈	—	2	2	no	—
295	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
296	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
297	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
298	#127	—	2	1	no	—
299	#127	—	2	1	no	—
300	#127	—	2	1	no	—
301	#127	D₄×C₃₁	1	1	yes	characteristic
302	#127	C₂³×C₃₁	1	2	yes	—
303	#125	—	2	1	no	—
304	#124	—	2	1	no	—
305	C₂²⋊Q₈	—	2	1	no	—
306	C₂²⋊Q₈	—	2	1	no	—
307	#121	—	2	1	no	—
308	#121	D₄×C₃₁	1	2	yes	—
309	C₄×Q₈	—	2	1	no	—
310	C₄×Q₈	—	2	1	no	—
311	C₄×D₄	—	2	1	no	—
312	C₄×D₄	C₂³×C₃₁	1	2	yes	—
313	#117	—	2	1	no	—
314	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
315	#106	—	2	2	no	—
316	C₄⋊C₈	—	2	1	no	—
317	C₄⋊C₈	—	2	1	no	—
318	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
319	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
320	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
321	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
322	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
323	#103	—	2	2	no	—
324	#102	—	2	2	no	—
325	#102	—	2	2	no	—
326	#102	—	2	2	no	—
327	#102	—	2	2	no	—
328	#102	—	2	2	no	—
329	#102	—	2	2	no	—
330	#102	—	2	2	no	—
331	C₂²⋊C₈	—	2	1	no	—
332	C₂²⋊C₈	—	2	2	no	—
333	C₂²⋊C₈	—	2	2	no	—
334	#97	D₄×C₃₁	1	2	yes	—
335	C₄×C₈	—	2	1	no	—
336	C₄×C₈	C₂³×C₃₁	1	2	yes	—
337	C₃₁	#19234	1	1	yes	characteristic, direct product factor, Sylow
338	C₂⁴	—	4	1	no	—
339	#41	—	4	2	no	—
340	#41	—	4	2	no	—
341	C₂×Q₈	—	2	1	no	—
342	C₂×Q₈	—	2	1	no	—
343	C₂×Q₈	—	2	1	no	—
344	C₂×D₄	—	2	2	no	—
345	C₂×D₄	—	2	1	no	—
346	C₂×D₄	—	2	1	no	—
347	C₂×D₄	—	2	2	no	—
348	C₂×D₄	—	2	2	no	—
349	C₂×D₄	—	2	2	no	—
350	C₂×D₄	—	4	1	no	—
351	C₂×D₄	—	4	1	no	—
352	C₂×D₄	—	4	1	no	—
353	C₂×D₄	—	4	2	no	—
354	C₂×D₄	—	4	2	no	—
355	C₂×D₄	—	8	1	no	—
356	C₂×D₄	—	8	1	no	—
357	C₂×D₄	—	8	2	no	—
358	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
359	C₂²×C₄	—	2	2	no	—
360	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
361	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
362	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
363	SD₁₆	—	8	2	no	—
364	D₈	—	8	2	no	—
365	D₈	—	8	2	no	—
366	M₁₆	—	2	2	no	—
367	M₁₆	—	2	2	no	—
368	M₁₆	—	4	4	no	—
369	C₂×C₈	—	2	1	no	—
370	C₂×C₈	—	2	1	no	—
371	C₂×C₈	—	2	1	no	—
372	C₂×C₈	—	2	1	no	—
373	C₂×C₈	—	2	2	no	—
374	C₂×C₈	—	4	4	no	—
375	C₂×C₈	—	4	2	no	—
376	C₂×C₈	—	8	1	no	—
377	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
378	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
379	C₄⋊C₄	—	2	1	no	—
380	C₄⋊C₄	—	2	1	no	—
381	C₄⋊C₄	—	2	1	no	—
382	C₄⋊C₄	—	2	1	no	—
383	C₄⋊C₄	—	2	1	no	—
384	C₄⋊C₄	—	4	1	no	—
385	C₄⋊C₄	—	4	1	no	—
386	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
387	C₂²⋊C₄	—	2	2	no	—
388	C₂²⋊C₄	—	2	2	no	—
389	C₂²⋊C₄	—	4	1	no	—
390	C₄²	—	4	1	no	—
391	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
392	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
393	C₄²	#41×C₃₁	1	1	yes	characteristic
394	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
395	C₂³	—	2	2	no	—
396	C₂³	—	4	2	no	—
397	C₂³	—	8	2	no	—
398	C₂³	—	8	2	no	—
399	C₂³	—	8	2	no	—
400	C₂³	C₃₁×#121	1	1	yes	characteristic
401	Q₈	—	4	2	no	—
402	Q₈	—	4	1	no	—
403	Q₈	—	4	1	no	—
404	Q₈	—	8	1	no	—
405	D₄	—	2	2	no	—
406	D₄	—	2	2	no	—
407	D₄	—	4	4	no	—
408	D₄	—	4	2	no	—
409	D₄	—	4	2	no	—
410	D₄	—	4	2	no	—
411	D₄	—	4	2	no	—
412	D₄	—	4	2	no	—
413	D₄	—	8	1	no	—
414	D₄	—	8	2	no	—
415	D₄	—	8	2	no	—
416	D₄	—	8	2	no	—
417	D₄	—	8	2	no	—
418	C₂×C₄	—	2	1	no	—
419	C₂×C₄	—	2	1	no	—
420	C₂×C₄	—	2	2	no	—
421	C₂×C₄	—	4	1	no	—
422	C₂×C₄	—	4	2	no	—
423	C₂×C₄	—	4	2	no	—
424	C₂×C₄	—	4	1	no	—
425	C₂×C₄	—	4	1	no	—
426	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
427	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
428	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
429	C₂×C₄	C₃₁×#135	1	2	yes	—
430	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
431	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
432	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
433	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
434	C₈	—	2	2	no	—
435	C₈	—	2	2	no	—
436	C₈	—	4	1	no	—
437	C₈	—	4	1	no	—
438	C₈	—	4	2	no	—
439	C₈	—	4	1	no	—
440	C₈	—	4	1	no	—
441	C₂²	—	2	1	no	—
442	C₂²	—	2	1	no	—
443	C₂²	—	4	2	no	—
444	C₂²	—	4	2	no	—
445	C₂²	—	4	2	no	—
446	C₂²	—	8	2	no	—
447	C₂²	—	8	2	no	—
448	C₂²	—	8	2	no	—
449	C₂²	—	16	4	no	—
450	C₂²	C₃₁×#463	1	2	yes	—
451	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
452	C₄	—	2	1	no	—
453	C₄	—	2	1	no	—
454	C₄	—	2	1	no	—
455	C₄	—	4	2	no	—
456	C₄	—	8	1	no	—
457	C₄	—	8	1	no	—
458	C₄	C₃₁×#463	1	2	yes	—
459	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
460	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
461	C₄	C₃₁×#576	1	1	yes	characteristic
462	C₄	C₃₁×#577	1	1	yes	characteristic
463	C₂	—	2	2	no	—
464	C₂	—	8	2	no	—
465	C₂	—	16	2	no	—
466	C₂	C₂×C₃₁×#463	1	1	yes	characteristic
467	C₂	C₃₁×#3062	1	1	yes	characteristic
468	C₂	C₃₁×#3068	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₃₁×#3205	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×#3246	C₂	semidirect
3. C₃₁×#3138	C₂	semidirect
4. C₃₁×#2935	C₂	semidirect
5. C₃₁×#1701	C₂	semidirect
6. C₃₁×#1697	C₂	semidirect
7. C₃₁×#1681	C₂	semidirect
8. C₃₁×#1677	C₂	semidirect
9. C₃₁×#1565	C₂	semidirect
10. C₃₁×#1542	C₂	semidirect
11. C₂×C₃₁×#331	C₂	semidirect
21. C₃₁×#476	C₂²	semidirect
24. C₃₁×#461	C₂²	semidirect
29. C₃₁×#436	C₂²	semidirect
31. C₈×D₄×C₃₁	C₂²	semidirect
39. C₃₁×#335	C₂²	semidirect
40. C₃₁×#334	C₂²	semidirect
41. C₃₁×#333	C₂²	semidirect
42. C₃₁×#333	C₂²	semidirect
43. C₃₁×#332	C₂²	semidirect
44. C₃₁×#332	C₂²	semidirect
160. #19234	C₃₁	direct (N × H)
207. #3246	C₂×C₃₁	semidirect
209. #3138	C₂×C₃₁	semidirect
210. #2935	C₂×C₃₁	semidirect
211. #1701	C₂×C₃₁	semidirect
212. #1697	C₂×C₃₁	semidirect
213. #1681	C₂×C₃₁	semidirect
214. #1677	C₂×C₃₁	semidirect
215. #1565	C₂×C₃₁	semidirect
216. #1542	C₂×C₃₁	semidirect
217. C₂×#331	C₂×C₃₁	semidirect
249. #476	C₂²×C₃₁	semidirect
252. #461	C₂²×C₃₁	semidirect
257. #436	C₂²×C₃₁	semidirect
259. C₈×D₄	C₂²×C₃₁	semidirect
267. #335	C₂²×C₃₁	semidirect
268. #334	C₂²×C₃₁	semidirect
269. #333	C₂²×C₃₁	semidirect
270. #333	C₂²×C₃₁	semidirect
271. #332	C₂²×C₃₁	semidirect
272. #332	C₂²×C₃₁	semidirect
337. C₃₁	#19234	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×#3246	2
C₃₁×#3205	2
C₃₁×#3138	2
C₃₁×#2935	2
C₃₁×#1701	2
C₃₁×#1697	2
C₃₁×#1681	2
C₃₁×#1677	2
C₃₁×#1565	2
C₃₁×#1542	2
C₂×C₃₁×#331	2
#19234	31

Subgroups — C31×#19234 #21475993

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19234 #21475993