Subgroups — C₃₁×#19236 #21475995

Order 7936.

662 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3245	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#3227	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3141	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2935	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1703	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1697	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1683	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	C₂×C₃₁×#333	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1540	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#131	—	2	1	no	—
13	C₃₁×#532	C₂²	1	1	yes	characteristic
14	C₃₁×#532	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₃₁×#498	—	2	1	no	—
17	C₃₁×#489	—	2	2	no	—
18	C₃₁×#487	C₂²	1	2	yes	—
19	C₃₁×#486	—	2	1	no	—
20	C₃₁×#486	C₂²	1	2	yes	—
21	C₃₁×#481	—	2	2	no	—
22	C₃₁×#476	C₂²	1	2	yes	—
23	C₃₁×#476	C₂²	1	2	yes	—
24	C₃₁×#474	—	2	1	no	—
25	C₃₁×#474	C₂²	1	2	yes	—
26	C₃₁×#466	—	2	2	no	—
27	C₃₁×#459	—	2	1	no	—
28	C₃₁×#459	C₂²	1	2	yes	—
29	C₃₁×#436	C₂²	1	2	yes	—
30	C₃₁×#435	—	2	2	no	—
31	C₈×D₄×C₃₁	C₂²	1	2	yes	—
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×#417	—	2	1	no	—
36	C₃₁×#417	—	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	—
40	C₃₁×#334	C₂²	1	2	yes	—
41	C₃₁×#333	C₂²	1	2	yes	—
42	C₃₁×#333	C₂²	1	2	yes	—
43	C₃₁×#332	C₂²	1	1	yes	characteristic
44	C₃₁×#332	C₂²	1	1	yes	characteristic
45	C₃₁×#331	C₂²	1	1	yes	characteristic
46	C₃₁×#331	C₂²	1	1	yes	characteristic
47	C₂×#41×C₃₁	—	2	1	no	—
48	C₂×#41×C₃₁	—	2	1	no	—
49	C₂²×Q₈×C₃₁	—	2	1	no	—
50	C₂×D₈×C₃₁	—	2	2	no	—
51	C₃₁×#131	—	4	4	no	—
52	C₂×M₁₆×C₃₁	—	2	2	no	—
53	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
54	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
55	C₂²×C₈×C₃₁	—	2	1	no	—
56	C₂²×C₈×C₃₁	—	2	1	no	—
57	C₂²×C₈×C₃₁	—	2	2	no	—
58	C₃₁×C₄⋊Q₈	—	2	1	no	—
59	C₃₁×C₄⋊Q₈	—	2	1	no	—
60	C₃₁×C₄⋊Q₈	—	2	1	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	2	yes	—
64	C₃₁×#127	D₄	1	1	yes	characteristic
65	C₃₁×#127	C₂³	1	1	yes	characteristic
66	C₃₁×#124	—	2	1	no	—
67	C₃₁×#124	—	2	1	no	—
68	C₃₁×#121	—	2	1	no	—
69	C₃₁×#121	—	2	1	no	—
70	C₃₁×#121	—	2	1	no	—
71	C₃₁×#121	D₄	1	2	yes	—
72	C₄×D₄×C₃₁	—	2	1	no	—
73	C₄×D₄×C₃₁	—	2	1	no	—
74	C₄×D₄×C₃₁	—	2	1	no	—
75	C₄×D₄×C₃₁	C₂³	1	2	yes	—
76	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
77	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
78	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
79	C₃₁×#107	—	2	2	no	—
80	C₃₁×#106	—	2	2	no	—
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×#103	—	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	#41×C₃₁	—	4	2	no	—
104	#41×C₃₁	—	4	2	no	—
105	#41×C₃₁	—	4	2	no	—
106	#41×C₃₁	—	4	2	no	—
107	C₂×Q₈×C₃₁	—	2	1	no	—
108	C₂×Q₈×C₃₁	—	2	1	no	—
109	C₂×Q₈×C₃₁	—	2	2	no	—
110	C₂×Q₈×C₃₁	—	4	1	no	—
111	C₂×Q₈×C₃₁	—	8	2	no	—
112	C₂×D₄×C₃₁	—	2	2	no	—
113	C₂×D₄×C₃₁	—	2	1	no	—
114	C₂×D₄×C₃₁	—	2	1	no	—
115	C₂×D₄×C₃₁	—	2	1	no	—
116	C₂×D₄×C₃₁	—	2	1	no	—
117	C₂×D₄×C₃₁	—	2	1	no	—
118	C₂×D₄×C₃₁	—	4	1	no	—
119	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
120	C₂²×C₄×C₃₁	—	2	2	no	—
121	C₂²×C₄×C₃₁	—	4	1	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	D₈×C₃₁	—	8	2	no	—
127	M₁₆×C₃₁	—	2	2	no	—
128	M₁₆×C₃₁	—	2	2	no	—
129	M₁₆×C₃₁	—	4	4	no	—
130	C₂×C₈×C₃₁	—	2	1	no	—
131	C₂×C₈×C₃₁	—	2	1	no	—
132	C₂×C₈×C₃₁	—	2	1	no	—
133	C₂×C₈×C₃₁	—	2	1	no	—
134	C₂×C₈×C₃₁	—	2	2	no	—
135	C₂×C₈×C₃₁	—	4	4	no	—
136	C₂×C₈×C₃₁	—	4	2	no	—
137	C₂×C₈×C₃₁	—	8	1	no	—
138	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
139	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
140	C₄⋊C₄×C₃₁	—	2	1	no	—
141	C₄⋊C₄×C₃₁	—	2	2	no	—
142	C₄⋊C₄×C₃₁	—	2	2	no	—
143	C₄⋊C₄×C₃₁	—	2	1	no	—
144	C₄⋊C₄×C₃₁	—	2	1	no	—
145	C₄⋊C₄×C₃₁	—	4	1	no	—
146	C₄⋊C₄×C₃₁	—	4	1	no	—
147	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
148	C₂²⋊C₄×C₃₁	—	2	2	no	—
149	C₂²⋊C₄×C₃₁	—	2	2	no	—
150	C₂²⋊C₄×C₃₁	—	4	1	no	—
151	C₂²⋊C₄×C₃₁	—	4	1	no	—
152	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
153	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
154	C₄²×C₃₁	#41	1	1	yes	characteristic
155	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
156	#19236	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
157	C₂³×C₃₁	—	2	2	no	—
158	C₂³×C₃₁	—	4	1	no	—
159	C₂³×C₃₁	#121	1	1	yes	characteristic
160	Q₈×C₃₁	—	4	2	no	—
161	Q₈×C₃₁	—	4	1	no	—
162	Q₈×C₃₁	—	4	1	no	—
163	Q₈×C₃₁	—	8	2	no	—
164	Q₈×C₃₁	—	8	2	no	—
165	D₄×C₃₁	—	2	2	no	—
166	D₄×C₃₁	—	2	2	no	—
167	D₄×C₃₁	—	4	4	no	—
168	D₄×C₃₁	—	4	1	no	—
169	D₄×C₃₁	—	4	1	no	—
170	D₄×C₃₁	—	4	1	no	—
171	D₄×C₃₁	—	4	1	no	—
172	D₄×C₃₁	—	4	2	no	—
173	D₄×C₃₁	—	8	1	no	—
174	D₄×C₃₁	—	8	1	no	—
175	D₄×C₃₁	—	8	2	no	—
176	C₂×C₄×C₃₁	—	2	1	no	—
177	C₂×C₄×C₃₁	—	2	1	no	—
178	C₂×C₄×C₃₁	—	2	2	no	—
179	C₂×C₄×C₃₁	—	4	1	no	—
180	C₂×C₄×C₃₁	—	4	2	no	—
181	C₂×C₄×C₃₁	—	4	2	no	—
182	C₂×C₄×C₃₁	—	4	1	no	—
183	C₂×C₄×C₃₁	—	4	2	no	—
184	C₂×C₄×C₃₁	—	8	2	no	—
185	C₂×C₄×C₃₁	—	8	2	no	—
186	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
187	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
188	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
189	C₂×C₄×C₃₁	#135	1	2	yes	—
190	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
191	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
192	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
193	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
194	C₈×C₃₁	—	2	2	no	—
195	C₈×C₃₁	—	2	2	no	—
196	C₈×C₃₁	—	4	1	no	—
197	C₈×C₃₁	—	4	1	no	—
198	C₈×C₃₁	—	4	1	no	—
199	C₈×C₃₁	—	4	2	no	—
200	C₈×C₃₁	—	4	1	no	—
201	#3245	C₂×C₃₁	1	1	yes	characteristic
202	#3227	C₂×C₃₁	1	1	yes	characteristic, complemented
203	#3141	C₂×C₃₁	1	1	yes	characteristic, complemented
204	#2935	C₂×C₃₁	1	1	yes	characteristic, complemented
205	#1703	C₂×C₃₁	1	2	yes	complemented
206	#1697	C₂×C₃₁	1	2	yes	complemented
207	#1683	C₂×C₃₁	1	2	yes	complemented
208	#1677	C₂×C₃₁	1	2	yes	complemented
209	#1565	C₂×C₃₁	1	1	yes	characteristic, complemented
210	C₂×#333	C₂×C₃₁	1	1	yes	characteristic, complemented
211	#1540	C₂×C₃₁	1	1	yes	characteristic, complemented
212	C₂²×C₃₁	—	2	1	no	—
213	C₂²×C₃₁	—	2	1	no	—
214	C₂²×C₃₁	—	4	2	no	—
215	C₂²×C₃₁	—	4	2	no	—
216	C₂²×C₃₁	—	4	2	no	—
217	C₂²×C₃₁	—	8	1	no	—
218	C₂²×C₃₁	—	8	1	no	—
219	C₂²×C₃₁	—	8	1	no	—
220	C₂²×C₃₁	#463	1	2	yes	—
221	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
222	C₄×C₃₁	—	2	1	no	—
223	C₄×C₃₁	—	2	1	no	—
224	C₄×C₃₁	—	2	1	no	—
225	C₄×C₃₁	—	4	2	no	—
226	C₄×C₃₁	—	8	1	no	—
227	C₄×C₃₁	—	8	2	no	—
228	C₄×C₃₁	#463	1	2	yes	—
229	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
230	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
231	C₄×C₃₁	#577	1	1	yes	characteristic
232	C₄×C₃₁	#578	1	1	yes	characteristic
233	C₂×#131	—	2	1	no	—
234	#532	C₂²×C₃₁	1	1	yes	characteristic
235	#532	C₂²×C₃₁	1	1	yes	characteristic
236	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
237	#498	—	2	1	no	—
238	#489	—	2	2	no	—
239	#487	C₂²×C₃₁	1	2	yes	—
240	#486	—	2	1	no	—
241	#486	C₂²×C₃₁	1	2	yes	—
242	#481	—	2	2	no	—
243	#476	C₂²×C₃₁	1	2	yes	—
244	#476	C₂²×C₃₁	1	2	yes	—
245	#474	—	2	1	no	—
246	#474	C₂²×C₃₁	1	2	yes	—
247	#466	—	2	2	no	—
248	#459	—	2	1	no	—
249	#459	C₂²×C₃₁	1	2	yes	—
250	#436	C₂²×C₃₁	1	2	yes	—
251	#435	—	2	2	no	—
252	C₈×D₄	C₂²×C₃₁	1	2	yes	—
253	#431	C₂²×C₃₁	1	1	yes	characteristic
254	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
255	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
256	#417	—	2	1	no	—
257	#417	—	2	1	no	—
258	#408	—	2	1	no	—
259	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
260	#335	C₂²×C₃₁	1	2	yes	—
261	#334	C₂²×C₃₁	1	2	yes	—
262	#333	C₂²×C₃₁	1	2	yes	—
263	#333	C₂²×C₃₁	1	2	yes	—
264	#332	C₂²×C₃₁	1	1	yes	characteristic
265	#332	C₂²×C₃₁	1	1	yes	characteristic
266	#331	C₂²×C₃₁	1	1	yes	characteristic
267	#331	C₂²×C₃₁	1	1	yes	characteristic
268	C₂×C₃₁	—	2	2	no	—
269	C₂×C₃₁	—	8	2	no	—
270	C₂×C₃₁	—	16	1	no	—
271	C₂×C₃₁	C₂×#463	1	1	yes	characteristic
272	C₂×C₃₁	#3063	1	1	yes	characteristic
273	C₂×C₃₁	#3068	1	1	yes	characteristic
274	C₂×#41	—	2	1	no	—
275	C₂×#41	—	2	1	no	—
276	C₂²×Q₈	—	2	1	no	—
277	C₂×D₈	—	2	2	no	—
278	#131	—	4	4	no	—
279	C₂×M₁₆	—	2	2	no	—
280	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
281	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
282	C₂²×C₈	—	2	1	no	—
283	C₂²×C₈	—	2	1	no	—
284	C₂²×C₈	—	2	2	no	—
285	C₄⋊Q₈	—	2	1	no	—
286	C₄⋊Q₈	—	2	1	no	—
287	C₄⋊Q₈	—	2	1	no	—
288	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
289	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
290	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
291	#127	D₄×C₃₁	1	1	yes	characteristic
292	#127	C₂³×C₃₁	1	1	yes	characteristic
293	#124	—	2	1	no	—
294	#124	—	2	1	no	—
295	#121	—	2	1	no	—
296	#121	—	2	1	no	—
297	#121	—	2	1	no	—
298	#121	D₄×C₃₁	1	2	yes	—
299	C₄×D₄	—	2	1	no	—
300	C₄×D₄	—	2	1	no	—
301	C₄×D₄	—	2	1	no	—
302	C₄×D₄	C₂³×C₃₁	1	2	yes	—
303	C₂×C₄⋊C₄	—	2	1	no	—
304	C₂×C₄⋊C₄	—	2	1	no	—
305	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
306	#107	—	2	2	no	—
307	#106	—	2	2	no	—
308	#106	—	2	2	no	—
309	C₄⋊C₈	—	2	1	no	—
310	C₄⋊C₈	—	2	1	no	—
311	C₄⋊C₈	C₂³×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	1	yes	characteristic
315	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
316	#103	—	2	2	no	—
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	#102	—	2	2	no	—
324	C₂²⋊C₈	—	2	1	no	—
325	C₂²⋊C₈	—	2	2	no	—
326	C₂²⋊C₈	—	2	2	no	—
327	#97	D₄×C₃₁	1	2	yes	—
328	C₄×C₈	—	2	1	no	—
329	C₄×C₈	C₂³×C₃₁	1	2	yes	—
330	C₃₁	#19236	1	1	yes	characteristic, direct product factor, Sylow
331	#41	—	4	2	no	—
332	#41	—	4	2	no	—
333	#41	—	4	2	no	—
334	#41	—	4	2	no	—
335	C₂×Q₈	—	2	1	no	—
336	C₂×Q₈	—	2	1	no	—
337	C₂×Q₈	—	2	2	no	—
338	C₂×Q₈	—	4	1	no	—
339	C₂×Q₈	—	8	2	no	—
340	C₂×D₄	—	2	2	no	—
341	C₂×D₄	—	2	1	no	—
342	C₂×D₄	—	2	1	no	—
343	C₂×D₄	—	2	1	no	—
344	C₂×D₄	—	2	1	no	—
345	C₂×D₄	—	2	1	no	—
346	C₂×D₄	—	4	1	no	—
347	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
348	C₂²×C₄	—	2	2	no	—
349	C₂²×C₄	—	4	1	no	—
350	C₂²×C₄	—	4	1	no	—
351	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
352	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
353	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
354	D₈	—	8	2	no	—
355	M₁₆	—	2	2	no	—
356	M₁₆	—	2	2	no	—
357	M₁₆	—	4	4	no	—
358	C₂×C₈	—	2	1	no	—
359	C₂×C₈	—	2	1	no	—
360	C₂×C₈	—	2	1	no	—
361	C₂×C₈	—	2	1	no	—
362	C₂×C₈	—	2	2	no	—
363	C₂×C₈	—	4	4	no	—
364	C₂×C₈	—	4	2	no	—
365	C₂×C₈	—	8	1	no	—
366	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
367	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
368	C₄⋊C₄	—	2	1	no	—
369	C₄⋊C₄	—	2	2	no	—
370	C₄⋊C₄	—	2	2	no	—
371	C₄⋊C₄	—	2	1	no	—
372	C₄⋊C₄	—	2	1	no	—
373	C₄⋊C₄	—	4	1	no	—
374	C₄⋊C₄	—	4	1	no	—
375	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
376	C₂²⋊C₄	—	2	2	no	—
377	C₂²⋊C₄	—	2	2	no	—
378	C₂²⋊C₄	—	4	1	no	—
379	C₂²⋊C₄	—	4	1	no	—
380	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
381	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
382	C₄²	#41×C₃₁	1	1	yes	characteristic
383	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
384	C₂³	—	2	2	no	—
385	C₂³	—	4	1	no	—
386	C₂³	C₃₁×#121	1	1	yes	characteristic
387	Q₈	—	4	2	no	—
388	Q₈	—	4	1	no	—
389	Q₈	—	4	1	no	—
390	Q₈	—	8	2	no	—
391	Q₈	—	8	2	no	—
392	D₄	—	2	2	no	—
393	D₄	—	2	2	no	—
394	D₄	—	4	4	no	—
395	D₄	—	4	2	no	—
396	D₄	—	4	1	no	—
397	D₄	—	4	1	no	—
398	D₄	—	4	1	no	—
399	D₄	—	4	1	no	—
400	D₄	—	8	1	no	—
401	D₄	—	8	1	no	—
402	D₄	—	8	2	no	—
403	C₂×C₄	—	2	1	no	—
404	C₂×C₄	—	2	1	no	—
405	C₂×C₄	—	2	2	no	—
406	C₂×C₄	—	4	1	no	—
407	C₂×C₄	—	4	2	no	—
408	C₂×C₄	—	4	2	no	—
409	C₂×C₄	—	4	1	no	—
410	C₂×C₄	—	4	2	no	—
411	C₂×C₄	—	8	2	no	—
412	C₂×C₄	—	8	2	no	—
413	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
414	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
415	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
416	C₂×C₄	C₃₁×#135	1	2	yes	—
417	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
418	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
419	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
420	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
421	C₈	—	2	2	no	—
422	C₈	—	2	2	no	—
423	C₈	—	4	1	no	—
424	C₈	—	4	1	no	—
425	C₈	—	4	1	no	—
426	C₈	—	4	2	no	—
427	C₈	—	4	1	no	—
428	C₂²	—	2	1	no	—
429	C₂²	—	2	1	no	—
430	C₂²	—	4	2	no	—
431	C₂²	—	4	2	no	—
432	C₂²	—	4	2	no	—
433	C₂²	—	8	1	no	—
434	C₂²	—	8	1	no	—
435	C₂²	—	8	1	no	—
436	C₂²	C₃₁×#463	1	2	yes	—
437	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
438	C₄	—	2	1	no	—
439	C₄	—	2	1	no	—
440	C₄	—	2	1	no	—
441	C₄	—	4	2	no	—
442	C₄	—	8	1	no	—
443	C₄	—	8	2	no	—
444	C₄	C₃₁×#463	1	2	yes	—
445	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
446	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
447	C₄	C₃₁×#577	1	1	yes	characteristic
448	C₄	C₃₁×#578	1	1	yes	characteristic
449	C₂	—	2	2	no	—
450	C₂	—	8	2	no	—
451	C₂	—	16	1	no	—
452	C₂	C₂×C₃₁×#463	1	1	yes	characteristic
453	C₂	C₃₁×#3063	1	1	yes	characteristic
454	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×#3245	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
2. C₃₁×#3227	C₂	semidirect
3. C₃₁×#3141	C₂	semidirect
4. C₃₁×#2935	C₂	semidirect
5. C₃₁×#1703	C₂	semidirect
6. C₃₁×#1697	C₂	semidirect
7. C₃₁×#1683	C₂	semidirect
8. C₃₁×#1677	C₂	semidirect
9. C₃₁×#1565	C₂	semidirect
10. C₂×C₃₁×#333	C₂	semidirect
11. C₃₁×#1540	C₂	semidirect
156. #19236	C₃₁	direct (N × H)
202. #3227	C₂×C₃₁	semidirect
203. #3141	C₂×C₃₁	semidirect
204. #2935	C₂×C₃₁	semidirect
205. #1703	C₂×C₃₁	semidirect
206. #1697	C₂×C₃₁	semidirect
207. #1683	C₂×C₃₁	semidirect
208. #1677	C₂×C₃₁	semidirect
209. #1565	C₂×C₃₁	semidirect
210. C₂×#333	C₂×C₃₁	semidirect
211. #1540	C₂×C₃₁	semidirect
330. C₃₁	#19236	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×#3245	2
C₃₁×#3227	2
C₃₁×#3141	2
C₃₁×#2935	2
C₃₁×#1703	2
C₃₁×#1697	2
C₃₁×#1683	2
C₃₁×#1677	2
C₃₁×#1565	2
C₂×C₃₁×#333	2
C₃₁×#1540	2
#19236	31

Subgroups — C31×#19236 #21475995

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19236 #21475995