Subgroups — C₃₁×#19566 #21476325

Order 7936.

648 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3254	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3208	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3205	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#2986	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1698	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1688	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1678	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1668	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1574	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1545	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1543	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#533	C₂²	1	1	yes	characteristic
13	C₃₁×#532	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
15	C₃₁×#493	C₂²	1	2	yes	—
16	C₃₁×#492	—	2	1	no	—
17	C₃₁×#486	—	2	1	no	—
18	C₃₁×#486	C₂²	1	2	yes	—
19	C₃₁×#477	C₂²	1	2	yes	complemented
20	C₃₁×#474	—	2	1	no	—
21	C₃₁×#468	—	2	2	no	—
22	C₃₁×#461	—	2	1	no	—
23	C₃₁×#460	C₂²	1	2	yes	—
24	C₃₁×#460	C₂²	1	2	yes	—
25	C₃₁×#460	C₂²	1	2	yes	complemented
26	C₃₁×#459	—	2	1	no	—
27	C₃₁×#459	—	2	1	no	—
28	C₃₁×#450	—	2	2	no	—
29	C₃₁×#447	—	2	2	no	—
30	C₃₁×#446	—	2	1	no	—
31	C₃₁×#446	C₂²	1	2	yes	complemented
32	C₈×Q₈×C₃₁	—	2	1	no	—
33	C₈×Q₈×C₃₁	C₂²	1	2	yes	complemented
34	C₃₁×#433	—	2	2	no	—
35	C₃₁×#431	C₂²	1	1	yes	characteristic
36	C₃₁×#431	C₂²	1	1	yes	characteristic
37	C₃₁×#431	C₂²	1	1	yes	characteristic
38	C₃₁×#423	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×#331	C₂²	1	2	yes	—
44	C₃₁×#331	C₂²	1	2	yes	—
45	C₃₁×#331	C₂²	1	1	yes	characteristic, complemented
46	C₃₁×#331	C₂²	1	1	yes	characteristic, complemented
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×SD₁₆×C₃₁	—	2	2	no	—
53	C₂×SD₁₆×C₃₁	—	2	2	no	—
54	C₂×D₈×C₃₁	—	2	2	no	—
55	C₂×D₈×C₃₁	—	2	2	no	—
56	C₂×M₁₆×C₃₁	—	2	1	no	—
57	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
58	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
59	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
60	C₃₁×#127	—	2	1	no	—
61	C₃₁×#127	—	2	1	no	—
62	C₃₁×#127	—	2	1	no	—
63	C₃₁×#127	C₂³	1	2	yes	—
64	C₃₁×#127	C₂³	1	1	yes	characteristic
65	C₃₁×#125	—	2	1	no	—
66	C₃₁×#124	—	2	1	no	—
67	C₃₁×C₂²⋊Q₈	—	2	1	no	—
68	C₃₁×C₂²⋊Q₈	D₄	1	2	yes	—
69	C₃₁×#121	—	2	1	no	—
70	C₃₁×#121	—	2	1	no	—
71	C₄×Q₈×C₃₁	—	2	1	no	—
72	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
73	C₄×D₄×C₃₁	—	2	1	no	—
74	C₄×D₄×C₃₁	—	2	1	no	—
75	C₃₁×#117	—	2	1	no	—
76	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
77	C₃₁×#107	—	2	2	no	—
78	C₃₁×C₄⋊C₈	—	2	2	no	—
79	C₃₁×C₄⋊C₈	—	2	2	no	—
80	C₃₁×C₄⋊C₈	—	2	1	no	—
81	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
82	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
83	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
84	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
85	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
86	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
87	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
88	C₃₁×#103	—	2	2	no	—
89	C₃₁×#102	—	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×C₂²⋊C₈	—	2	1	no	—
95	C₃₁×C₂²⋊C₈	—	2	1	no	—
96	C₃₁×C₂²⋊C₈	D₄	1	2	yes	—
97	C₃₁×#97	—	2	2	no	—
98	C₄×C₈×C₃₁	—	2	2	no	—
99	C₄×C₈×C₃₁	—	2	1	no	—
100	C₄×C₈×C₃₁	—	2	1	no	—
101	C₄×C₈×C₃₁	C₂³	1	2	yes	—
102	C₂⁴×C₃₁	—	4	1	no	—
103	#41×C₃₁	—	8	2	no	—
104	C₂×Q₈×C₃₁	—	2	1	no	—
105	C₂×Q₈×C₃₁	—	2	1	no	—
106	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
107	C₂×D₄×C₃₁	—	2	1	no	—
108	C₂×D₄×C₃₁	—	2	2	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	—	4	1	no	—
114	C₂×D₄×C₃₁	—	4	1	no	—
115	C₂×D₄×C₃₁	—	4	1	no	—
116	C₂×D₄×C₃₁	—	4	1	no	—
117	C₂×D₄×C₃₁	—	8	1	no	—
118	C₂×D₄×C₃₁	—	8	1	no	—
119	C₂×D₄×C₃₁	—	8	1	no	—
120	C₂×D₄×C₃₁	—	8	1	no	—
121	C₂×D₄×C₃₁	—	8	2	no	—
122	C₂²×C₄×C₃₁	—	4	1	no	—
123	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
124	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
125	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
126	SD₁₆×C₃₁	—	8	2	no	—
127	SD₁₆×C₃₁	—	8	2	no	—
128	SD₁₆×C₃₁	—	8	2	no	—
129	D₈×C₃₁	—	8	2	no	—
130	D₈×C₃₁	—	8	2	no	—
131	M₁₆×C₃₁	—	8	1	no	—
132	C₂×C₈×C₃₁	—	2	1	no	—
133	C₂×C₈×C₃₁	—	2	1	no	—
134	C₂×C₈×C₃₁	—	2	1	no	—
135	C₂×C₈×C₃₁	—	2	1	no	—
136	C₂×C₈×C₃₁	—	2	2	no	—
137	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
138	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
139	C₄⋊C₄×C₃₁	—	2	1	no	—
140	C₄⋊C₄×C₃₁	—	2	1	no	—
141	C₄⋊C₄×C₃₁	—	2	1	no	—
142	C₄⋊C₄×C₃₁	—	2	2	no	—
143	C₄⋊C₄×C₃₁	—	2	2	no	—
144	C₄⋊C₄×C₃₁	—	2	1	no	—
145	C₄⋊C₄×C₃₁	—	2	1	no	—
146	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
147	C₂²⋊C₄×C₃₁	—	2	2	no	—
148	C₂²⋊C₄×C₃₁	—	4	1	no	—
149	C₂²⋊C₄×C₃₁	—	4	1	no	—
150	C₄²×C₃₁	—	2	2	no	—
151	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
152	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
153	C₄²×C₃₁	#41	1	1	yes	characteristic
154	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
155	#19566	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
156	C₂³×C₃₁	—	4	1	no	—
157	C₂³×C₃₁	—	4	2	no	—
158	C₂³×C₃₁	—	8	2	no	—
159	C₂³×C₃₁	—	8	2	no	—
160	C₂³×C₃₁	—	8	2	no	—
161	C₂³×C₃₁	#121	1	1	yes	characteristic
162	Q₈×C₃₁	—	2	2	no	—
163	Q₈×C₃₁	—	2	2	no	—
164	Q₈×C₃₁	—	4	2	no	—
165	Q₈×C₃₁	—	8	1	no	—
166	D₄×C₃₁	—	4	1	no	—
167	D₄×C₃₁	—	4	2	no	—
168	D₄×C₃₁	—	4	1	no	—
169	D₄×C₃₁	—	4	1	no	—
170	D₄×C₃₁	—	4	2	no	—
171	D₄×C₃₁	—	4	2	no	—
172	D₄×C₃₁	—	4	2	no	—
173	D₄×C₃₁	—	4	1	no	—
174	D₄×C₃₁	—	8	2	no	—
175	D₄×C₃₁	—	8	1	no	—
176	D₄×C₃₁	—	8	2	no	—
177	D₄×C₃₁	—	8	2	no	—
178	D₄×C₃₁	—	8	2	no	—
179	D₄×C₃₁	—	8	2	no	—
180	C₂×C₄×C₃₁	—	2	1	no	—
181	C₂×C₄×C₃₁	—	2	1	no	—
182	C₂×C₄×C₃₁	—	2	1	no	—
183	C₂×C₄×C₃₁	—	2	1	no	—
184	C₂×C₄×C₃₁	—	2	2	no	—
185	C₂×C₄×C₃₁	—	2	2	no	—
186	C₂×C₄×C₃₁	—	2	2	no	—
187	C₂×C₄×C₃₁	—	4	1	no	—
188	C₂×C₄×C₃₁	—	8	2	no	—
189	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
190	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
191	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
192	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
193	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
194	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
195	C₈×C₃₁	—	2	2	no	—
196	C₈×C₃₁	—	2	2	no	—
197	C₈×C₃₁	—	4	2	no	—
198	C₈×C₃₁	—	4	1	no	—
199	C₈×C₃₁	—	4	1	no	—
200	C₈×C₃₁	—	4	1	no	—
201	C₈×C₃₁	—	4	1	no	—
202	#3254	C₂×C₃₁	1	1	yes	characteristic, complemented
203	#3208	C₂×C₃₁	1	1	yes	characteristic, complemented
204	#3205	C₂×C₃₁	1	1	yes	characteristic
205	#2986	C₂×C₃₁	1	1	yes	characteristic, complemented
206	#1698	C₂×C₃₁	1	2	yes	complemented
207	#1688	C₂×C₃₁	1	2	yes	complemented
208	#1678	C₂×C₃₁	1	2	yes	complemented
209	#1668	C₂×C₃₁	1	2	yes	complemented
210	#1574	C₂×C₃₁	1	1	yes	characteristic, complemented
211	#1545	C₂×C₃₁	1	1	yes	characteristic, complemented
212	#1543	C₂×C₃₁	1	1	yes	characteristic, complemented
213	C₂²×C₃₁	—	2	1	no	—
214	C₂²×C₃₁	—	2	1	no	—
215	C₂²×C₃₁	—	2	1	no	—
216	C₂²×C₃₁	—	8	2	no	—
217	C₂²×C₃₁	—	8	1	no	—
218	C₂²×C₃₁	—	8	2	no	—
219	C₂²×C₃₁	—	8	1	no	—
220	C₂²×C₃₁	—	8	1	no	—
221	C₂²×C₃₁	—	8	2	no	—
222	C₂²×C₃₁	—	16	4	no	—
223	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
224	C₄×C₃₁	—	2	1	no	—
225	C₄×C₃₁	—	2	1	no	—
226	C₄×C₃₁	—	2	1	no	—
227	C₄×C₃₁	—	4	2	no	—
228	C₄×C₃₁	—	4	2	no	—
229	C₄×C₃₁	—	8	1	no	—
230	C₄×C₃₁	#472	1	2	yes	—
231	C₄×C₃₁	#576	1	1	yes	characteristic
232	C₄×C₃₁	#577	1	1	yes	characteristic
233	C₄×C₃₁	#577	1	1	yes	characteristic
234	C₄×C₃₁	#577	1	1	yes	characteristic
235	#533	C₂²×C₃₁	1	1	yes	characteristic
236	#532	C₂²×C₃₁	1	1	yes	characteristic
237	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
238	#493	C₂²×C₃₁	1	2	yes	—
239	#492	—	2	1	no	—
240	#486	—	2	1	no	—
241	#486	C₂²×C₃₁	1	2	yes	—
242	#477	C₂²×C₃₁	1	2	yes	complemented
243	#474	—	2	1	no	—
244	#468	—	2	2	no	—
245	#461	—	2	1	no	—
246	#460	C₂²×C₃₁	1	2	yes	—
247	#460	C₂²×C₃₁	1	2	yes	—
248	#460	C₂²×C₃₁	1	2	yes	complemented
249	#459	—	2	1	no	—
250	#459	—	2	1	no	—
251	#450	—	2	2	no	—
252	#447	—	2	2	no	—
253	#446	—	2	1	no	—
254	#446	C₂²×C₃₁	1	2	yes	complemented
255	C₈×Q₈	—	2	1	no	—
256	C₈×Q₈	C₂²×C₃₁	1	2	yes	complemented
257	#433	—	2	2	no	—
258	#431	C₂²×C₃₁	1	1	yes	characteristic
259	#431	C₂²×C₃₁	1	1	yes	characteristic
260	#431	C₂²×C₃₁	1	1	yes	characteristic
261	#423	C₂²×C₃₁	1	1	yes	characteristic
262	#335	C₂²×C₃₁	1	2	yes	complemented
263	#334	C₂²×C₃₁	1	2	yes	complemented
264	#333	C₂²×C₃₁	1	1	yes	characteristic, complemented
265	#333	C₂²×C₃₁	1	1	yes	characteristic, complemented
266	#331	C₂²×C₃₁	1	2	yes	—
267	#331	C₂²×C₃₁	1	2	yes	—
268	#331	C₂²×C₃₁	1	1	yes	characteristic, complemented
269	#331	C₂²×C₃₁	1	1	yes	characteristic, complemented
270	C₂×C₃₁	—	4	1	no	—
271	C₂×C₃₁	—	16	1	no	—
272	C₂×C₃₁	—	16	2	no	—
273	C₂×C₃₁	#3043	1	1	yes	characteristic
274	C₂×C₃₁	#3062	1	1	yes	characteristic
275	C₂×C₃₁	C₂×#472	1	1	yes	characteristic
276	C₂×#41	—	2	1	no	—
277	C₂²×D₄	—	2	1	no	—
278	C₂²×D₄	—	2	1	no	—
279	C₂²×D₄	—	2	1	no	—
280	C₂×SD₁₆	—	2	2	no	—
281	C₂×SD₁₆	—	2	2	no	—
282	C₂×SD₁₆	—	2	2	no	—
283	C₂×D₈	—	2	2	no	—
284	C₂×D₈	—	2	2	no	—
285	C₂×M₁₆	—	2	1	no	—
286	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
287	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
288	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
289	#127	—	2	1	no	—
290	#127	—	2	1	no	—
291	#127	—	2	1	no	—
292	#127	C₂³×C₃₁	1	2	yes	—
293	#127	C₂³×C₃₁	1	1	yes	characteristic
294	#125	—	2	1	no	—
295	#124	—	2	1	no	—
296	C₂²⋊Q₈	—	2	1	no	—
297	C₂²⋊Q₈	D₄×C₃₁	1	2	yes	—
298	#121	—	2	1	no	—
299	#121	—	2	1	no	—
300	C₄×Q₈	—	2	1	no	—
301	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
302	C₄×D₄	—	2	1	no	—
303	C₄×D₄	—	2	1	no	—
304	#117	—	2	1	no	—
305	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
306	#107	—	2	2	no	—
307	C₄⋊C₈	—	2	2	no	—
308	C₄⋊C₈	—	2	2	no	—
309	C₄⋊C₈	—	2	1	no	—
310	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
311	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
312	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
313	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
314	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
315	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
316	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
317	#103	—	2	2	no	—
318	#102	—	2	2	no	—
319	#102	—	2	2	no	—
320	#102	—	2	2	no	—
321	#102	—	2	2	no	—
322	#102	—	2	2	no	—
323	C₂²⋊C₈	—	2	1	no	—
324	C₂²⋊C₈	—	2	1	no	—
325	C₂²⋊C₈	D₄×C₃₁	1	2	yes	—
326	#97	—	2	2	no	—
327	C₄×C₈	—	2	2	no	—
328	C₄×C₈	—	2	1	no	—
329	C₄×C₈	—	2	1	no	—
330	C₄×C₈	C₂³×C₃₁	1	2	yes	—
331	C₃₁	#19566	1	1	yes	characteristic, direct product factor, Sylow
332	C₂⁴	—	4	1	no	—
333	#41	—	8	2	no	—
334	C₂×Q₈	—	2	1	no	—
335	C₂×Q₈	—	2	1	no	—
336	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
337	C₂×D₄	—	2	1	no	—
338	C₂×D₄	—	2	2	no	—
339	C₂×D₄	—	2	2	no	—
340	C₂×D₄	—	2	1	no	—
341	C₂×D₄	—	2	1	no	—
342	C₂×D₄	—	2	2	no	—
343	C₂×D₄	—	4	1	no	—
344	C₂×D₄	—	4	1	no	—
345	C₂×D₄	—	4	1	no	—
346	C₂×D₄	—	4	1	no	—
347	C₂×D₄	—	8	1	no	—
348	C₂×D₄	—	8	1	no	—
349	C₂×D₄	—	8	1	no	—
350	C₂×D₄	—	8	1	no	—
351	C₂×D₄	—	8	2	no	—
352	C₂²×C₄	—	4	1	no	—
353	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
354	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
355	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
356	SD₁₆	—	8	2	no	—
357	SD₁₆	—	8	2	no	—
358	SD₁₆	—	8	2	no	—
359	D₈	—	8	2	no	—
360	D₈	—	8	2	no	—
361	M₁₆	—	8	1	no	—
362	C₂×C₈	—	2	1	no	—
363	C₂×C₈	—	2	1	no	—
364	C₂×C₈	—	2	1	no	—
365	C₂×C₈	—	2	1	no	—
366	C₂×C₈	—	2	2	no	—
367	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
368	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
369	C₄⋊C₄	—	2	1	no	—
370	C₄⋊C₄	—	2	1	no	—
371	C₄⋊C₄	—	2	1	no	—
372	C₄⋊C₄	—	2	2	no	—
373	C₄⋊C₄	—	2	2	no	—
374	C₄⋊C₄	—	2	1	no	—
375	C₄⋊C₄	—	2	1	no	—
376	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
377	C₂²⋊C₄	—	2	2	no	—
378	C₂²⋊C₄	—	4	1	no	—
379	C₂²⋊C₄	—	4	1	no	—
380	C₄²	—	2	2	no	—
381	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
382	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
383	C₄²	#41×C₃₁	1	1	yes	characteristic
384	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
385	C₂³	—	4	1	no	—
386	C₂³	—	4	2	no	—
387	C₂³	—	8	2	no	—
388	C₂³	—	8	2	no	—
389	C₂³	—	8	2	no	—
390	C₂³	C₃₁×#121	1	1	yes	characteristic
391	Q₈	—	2	2	no	—
392	Q₈	—	2	2	no	—
393	Q₈	—	4	2	no	—
394	Q₈	—	8	1	no	—
395	D₄	—	4	1	no	—
396	D₄	—	4	2	no	—
397	D₄	—	4	1	no	—
398	D₄	—	4	1	no	—
399	D₄	—	4	2	no	—
400	D₄	—	4	2	no	—
401	D₄	—	4	2	no	—
402	D₄	—	4	1	no	—
403	D₄	—	8	2	no	—
404	D₄	—	8	1	no	—
405	D₄	—	8	2	no	—
406	D₄	—	8	2	no	—
407	D₄	—	8	2	no	—
408	D₄	—	8	2	no	—
409	C₂×C₄	—	2	2	no	—
410	C₂×C₄	—	2	1	no	—
411	C₂×C₄	—	2	1	no	—
412	C₂×C₄	—	2	1	no	—
413	C₂×C₄	—	2	1	no	—
414	C₂×C₄	—	2	2	no	—
415	C₂×C₄	—	2	2	no	—
416	C₂×C₄	—	4	1	no	—
417	C₂×C₄	—	8	2	no	—
418	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
419	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
420	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
421	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
422	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
423	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
424	C₈	—	2	2	no	—
425	C₈	—	2	2	no	—
426	C₈	—	4	2	no	—
427	C₈	—	4	1	no	—
428	C₈	—	4	1	no	—
429	C₈	—	4	1	no	—
430	C₈	—	4	1	no	—
431	C₂²	—	2	1	no	—
432	C₂²	—	2	1	no	—
433	C₂²	—	2	1	no	—
434	C₂²	—	8	2	no	—
435	C₂²	—	8	1	no	—
436	C₂²	—	8	2	no	—
437	C₂²	—	8	1	no	—
438	C₂²	—	8	1	no	—
439	C₂²	—	8	2	no	—
440	C₂²	—	16	4	no	—
441	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
442	C₄	—	2	1	no	—
443	C₄	—	2	1	no	—
444	C₄	—	2	1	no	—
445	C₄	—	4	2	no	—
446	C₄	—	4	2	no	—
447	C₄	—	8	1	no	—
448	C₄	C₃₁×#472	1	2	yes	—
449	C₄	C₃₁×#576	1	1	yes	characteristic
450	C₄	C₃₁×#577	1	1	yes	characteristic
451	C₄	C₃₁×#577	1	1	yes	characteristic
452	C₄	C₃₁×#577	1	1	yes	characteristic
453	C₂	—	4	1	no	—
454	C₂	—	16	2	no	—
455	C₂	—	16	1	no	—
456	C₂	C₃₁×#3043	1	1	yes	characteristic
457	C₂	C₃₁×#3062	1	1	yes	characteristic
458	C₂	C₂×C₃₁×#472	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×#3254	C₂	semidirect
2. C₃₁×#3208	C₂	semidirect
4. C₃₁×#2986	C₂	semidirect
5. C₃₁×#1698	C₂	semidirect
6. C₃₁×#1688	C₂	semidirect
7. C₃₁×#1678	C₂	semidirect
8. C₃₁×#1668	C₂	semidirect
9. C₃₁×#1574	C₂	semidirect
10. C₃₁×#1545	C₂	semidirect
11. C₃₁×#1543	C₂	semidirect
19. C₃₁×#477	C₂²	semidirect
25. C₃₁×#460	C₂²	semidirect
31. C₃₁×#446	C₂²	semidirect
33. C₈×Q₈×C₃₁	C₂²	semidirect
39. C₃₁×#335	C₂²	semidirect
40. C₃₁×#334	C₂²	semidirect
41. C₃₁×#333	C₂²	semidirect
42. C₃₁×#333	C₂²	semidirect
45. C₃₁×#331	C₂²	semidirect
46. C₃₁×#331	C₂²	semidirect
155. #19566	C₃₁	direct (N × H)
202. #3254	C₂×C₃₁	semidirect
203. #3208	C₂×C₃₁	semidirect
205. #2986	C₂×C₃₁	semidirect
206. #1698	C₂×C₃₁	semidirect
207. #1688	C₂×C₃₁	semidirect
208. #1678	C₂×C₃₁	semidirect
209. #1668	C₂×C₃₁	semidirect
210. #1574	C₂×C₃₁	semidirect
211. #1545	C₂×C₃₁	semidirect
212. #1543	C₂×C₃₁	semidirect
242. #477	C₂²×C₃₁	semidirect
248. #460	C₂²×C₃₁	semidirect
254. #446	C₂²×C₃₁	semidirect
256. C₈×Q₈	C₂²×C₃₁	semidirect
262. #335	C₂²×C₃₁	semidirect
263. #334	C₂²×C₃₁	semidirect
264. #333	C₂²×C₃₁	semidirect
265. #333	C₂²×C₃₁	semidirect
268. #331	C₂²×C₃₁	semidirect
269. #331	C₂²×C₃₁	semidirect
331. C₃₁	#19566	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×#3254	2
C₃₁×#3208	2
C₃₁×#3205	2
C₃₁×#2986	2
C₃₁×#1698	2
C₃₁×#1688	2
C₃₁×#1678	2
C₃₁×#1668	2
C₃₁×#1574	2
C₃₁×#1545	2
C₃₁×#1543	2
#19566	31

Subgroups — C31×#19566 #21476325

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19566 #21476325