Subgroups — C₃₁×#25650 #21482409

Order 7936.

1054 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#556	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#487	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#474	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#2546	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2042	C₂	1	4	yes	maximal
6	C₃₁×#1984	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1981	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1863	C₂	1	4	yes	maximal, complemented
9	C₄×C₃₁×#102	C₂	1	1	yes	maximal, characteristic
10	C₃₁×#556	—	2	4	no	—
11	C₃₁×#556	—	2	4	no	—
12	C₂×C₃₁×C₄⋊Q₈	—	2	1	no	—
13	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#126	—	2	2	no	—
16	C₂×C₃₁×#124	C₂²	1	2	yes	—
17	C₂×C₃₁×C₂²⋊Q₈	—	2	1	no	—
18	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
19	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
20	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×#117	—	2	1	no	—
22	C₃₁×#487	—	2	4	no	—
23	C₃₁×#474	—	2	4	no	—
24	C₂×C₃₁×#107	C₂²	1	2	yes	—
25	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
26	C₂×C₃₁×#103	C₂²	1	4	yes	—
27	C₂×C₃₁×#103	C₂²	1	4	yes	—
28	C₂×C₃₁×#102	C₂²	1	2	yes	—
29	C₂×C₃₁×#102	C₂²	1	2	yes	—
30	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
31	C₃₁×#398	—	2	2	no	—
32	C₃₁×#391	—	2	1	no	—
33	C₃₁×#389	C₂²	1	4	yes	—
34	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
35	C₃₁×#340	C₂²	1	2	yes	—
36	C₃₁×#336	C₂²	1	4	yes	—
37	C₂²×Q₈×C₃₁	—	2	1	no	—
38	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
39	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
40	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
41	C₂³×C₄×C₃₁	—	2	1	no	—
42	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
43	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
44	C₃₁×C₄⋊Q₈	—	2	2	no	—
45	C₃₁×C₄⋊Q₈	—	2	2	no	—
46	C₃₁×C₄⋊Q₈	—	2	2	no	—
47	C₃₁×C₄⋊Q₈	—	4	4	no	—
48	C₃₁×C₄⋊Q₈	D₄	1	4	yes	—
49	C₃₁×#126	—	2	16	no	—
50	C₃₁×#124	—	2	4	no	—
51	C₃₁×#124	—	2	4	no	—
52	C₃₁×C₂²⋊Q₈	—	2	8	no	—
53	C₄×Q₈×C₃₁	—	2	4	no	—
54	C₄×Q₈×C₃₁	—	2	4	no	—
55	C₄×Q₈×C₃₁	—	2	2	no	—
56	C₄×Q₈×C₃₁	—	2	2	no	—
57	C₄×D₄×C₃₁	—	2	2	no	—
58	C₄×D₄×C₃₁	D₄	1	4	yes	—
59	C₃₁×#117	—	2	8	no	—
60	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
61	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
62	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
63	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
64	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
65	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
66	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
67	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
68	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
69	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
70	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
71	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
72	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
73	C₂×C₄²×C₃₁	—	2	2	no	—
74	C₂×C₄²×C₃₁	—	2	1	no	—
75	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
76	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
77	C₃₁×#107	—	2	4	no	—
78	C₃₁×C₄⋊C₈	—	2	2	no	—
79	C₃₁×#103	—	2	8	no	—
80	C₃₁×#103	—	2	8	no	—
81	C₃₁×#102	—	2	4	no	—
82	C₃₁×#102	—	2	4	no	—
83	C₄×C₈×C₃₁	—	2	2	no	—
84	C₃₁×#95	—	2	1	no	—
85	C₃₁×#95	—	2	4	no	—
86	C₂⁴×C₃₁	—	2	1	no	—
87	C₂×Q₈×C₃₁	—	2	2	no	—
88	C₂×Q₈×C₃₁	—	2	2	no	—
89	C₂×Q₈×C₃₁	—	2	2	no	—
90	C₂×Q₈×C₃₁	—	2	2	no	—
91	C₂×Q₈×C₃₁	—	2	4	no	—
92	C₂×Q₈×C₃₁	—	2	4	no	—
93	C₂×Q₈×C₃₁	—	4	2	no	—
94	C₂×Q₈×C₃₁	—	4	2	no	—
95	C₂×Q₈×C₃₁	—	4	2	no	—
96	C₂×Q₈×C₃₁	C₂×D₄	1	4	yes	—
97	C₂×D₄×C₃₁	—	2	1	no	—
98	C₂×D₄×C₃₁	—	2	2	no	—
99	C₂×D₄×C₃₁	—	2	2	no	—
100	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
101	C₂²×C₄×C₃₁	—	2	2	no	—
102	C₂²×C₄×C₃₁	—	2	1	no	—
103	C₂²×C₄×C₃₁	—	2	2	no	—
104	C₂²×C₄×C₃₁	—	2	4	no	—
105	C₂²×C₄×C₃₁	—	2	2	no	—
106	C₂²×C₄×C₃₁	—	4	2	no	—
107	C₂²×C₄×C₃₁	—	4	2	no	—
108	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
109	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
110	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
111	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
112	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
113	C₂×C₈×C₃₁	—	2	2	no	—
114	C₂×C₈×C₃₁	—	2	2	no	—
115	C₂×C₈×C₃₁	—	2	2	no	—
116	C₂×C₈×C₃₁	—	2	4	no	—
117	C₂×C₈×C₃₁	—	2	2	no	—
118	C₄⋊C₄×C₃₁	—	2	8	no	—
119	C₄⋊C₄×C₃₁	—	2	4	no	—
120	C₄⋊C₄×C₃₁	—	2	4	no	—
121	C₄⋊C₄×C₃₁	—	2	2	no	—
122	C₄⋊C₄×C₃₁	—	2	4	no	—
123	C₄⋊C₄×C₃₁	—	2	1	no	—
124	C₄⋊C₄×C₃₁	—	2	2	no	—
125	C₄⋊C₄×C₃₁	—	2	2	no	—
126	C₄⋊C₄×C₃₁	—	2	2	no	—
127	C₄⋊C₄×C₃₁	—	4	8	no	—
128	C₄⋊C₄×C₃₁	—	4	2	no	—
129	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
130	C₄⋊C₄×C₃₁	#41	1	4	yes	—
131	C₂²⋊C₄×C₃₁	—	2	8	no	—
132	C₂²⋊C₄×C₃₁	—	2	8	no	—
133	C₂²⋊C₄×C₃₁	—	2	4	no	—
134	C₄²×C₃₁	—	2	8	no	—
135	C₄²×C₃₁	—	2	2	no	—
136	C₄²×C₃₁	—	2	2	no	—
137	C₄²×C₃₁	—	4	2	no	—
138	C₄²×C₃₁	C₂×D₄	1	2	yes	—
139	C₄²×C₃₁	C₂×D₄	1	2	yes	—
140	#25650	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
141	C₂³×C₃₁	—	2	2	no	—
142	C₂³×C₃₁	—	4	2	no	—
143	C₂³×C₃₁	—	4	1	no	—
144	C₂³×C₃₁	—	4	1	no	—
145	C₂³×C₃₁	—	4	2	no	—
146	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
147	Q₈×C₃₁	—	2	8	no	—
148	Q₈×C₃₁	—	4	2	no	—
149	Q₈×C₃₁	—	4	2	no	—
150	Q₈×C₃₁	—	4	4	no	—
151	Q₈×C₃₁	—	8	4	no	—
152	D₄×C₃₁	—	2	2	no	—
153	D₄×C₃₁	—	2	2	no	—
154	D₄×C₃₁	—	4	2	no	—
155	C₂×C₄×C₃₁	—	2	2	no	—
156	C₂×C₄×C₃₁	—	2	2	no	—
157	C₂×C₄×C₃₁	—	2	2	no	—
158	C₂×C₄×C₃₁	—	2	4	no	—
159	C₂×C₄×C₃₁	—	2	2	no	—
160	C₂×C₄×C₃₁	—	2	4	no	—
161	C₂×C₄×C₃₁	—	2	2	no	—
162	C₂×C₄×C₃₁	—	2	2	no	—
163	C₂×C₄×C₃₁	—	2	4	no	—
164	C₂×C₄×C₃₁	—	4	4	no	—
165	C₂×C₄×C₃₁	—	4	1	no	—
166	C₂×C₄×C₃₁	—	4	2	no	—
167	C₂×C₄×C₃₁	—	4	1	no	—
168	C₂×C₄×C₃₁	—	4	2	no	—
169	C₂×C₄×C₃₁	—	4	8	no	—
170	C₂×C₄×C₃₁	—	4	2	no	—
171	C₂×C₄×C₃₁	—	4	2	no	—
172	C₂×C₄×C₃₁	—	4	2	no	—
173	C₂×C₄×C₃₁	#120	1	2	yes	—
174	C₂×C₄×C₃₁	#120	1	2	yes	—
175	C₂×C₄×C₃₁	#135	1	2	yes	—
176	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
177	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
178	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
179	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
180	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
181	C₈×C₃₁	—	4	2	no	—
182	C₈×C₃₁	—	4	2	no	—
183	C₂×#556	C₂×C₃₁	1	1	yes	characteristic
184	C₂×#487	C₂×C₃₁	1	1	yes	characteristic, complemented
185	C₂×#474	C₂×C₃₁	1	1	yes	characteristic
186	#2546	C₂×C₃₁	1	1	yes	characteristic, complemented
187	#2042	C₂×C₃₁	1	4	yes	—
188	#1984	C₂×C₃₁	1	1	yes	characteristic, complemented
189	#1981	C₂×C₃₁	1	1	yes	characteristic, complemented
190	#1863	C₂×C₃₁	1	4	yes	complemented
191	C₄×#102	C₂×C₃₁	1	1	yes	characteristic
192	C₂²×C₃₁	—	4	2	no	—
193	C₂²×C₃₁	—	4	2	no	—
194	C₂²×C₃₁	—	4	2	no	—
195	C₂²×C₃₁	—	8	2	no	—
196	C₂²×C₃₁	—	8	2	no	—
197	C₂²×C₃₁	C₂×#120	1	1	yes	characteristic
198	C₂²×C₃₁	#548	1	2	yes	—
199	C₂²×C₃₁	C₂×#135	1	1	yes	characteristic
200	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
201	C₂²×C₃₁	#578	1	2	yes	—
202	C₄×C₃₁	—	2	2	no	—
203	C₄×C₃₁	—	4	4	no	—
204	C₄×C₃₁	—	4	4	no	—
205	C₄×C₃₁	—	4	2	no	—
206	C₄×C₃₁	—	4	2	no	—
207	C₄×C₃₁	—	4	2	no	—
208	C₄×C₃₁	—	8	2	no	—
209	C₄×C₃₁	#449	1	4	yes	—
210	C₄×C₃₁	#532	1	1	yes	characteristic
211	C₄×C₃₁	#536	1	1	yes	characteristic
212	C₄×C₃₁	#547	1	2	yes	—
213	#556	—	2	4	no	—
214	#556	—	2	4	no	—
215	C₂×C₄⋊Q₈	—	2	1	no	—
216	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
217	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
218	C₂×#126	—	2	2	no	—
219	C₂×#124	C₂²×C₃₁	1	2	yes	—
220	C₂×C₂²⋊Q₈	—	2	1	no	—
221	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
222	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
223	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
224	C₂×#117	—	2	1	no	—
225	#487	—	2	4	no	—
226	#474	—	2	4	no	—
227	C₂×#107	C₂²×C₃₁	1	2	yes	—
228	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
229	C₂×#103	C₂²×C₃₁	1	4	yes	—
230	C₂×#103	C₂²×C₃₁	1	4	yes	—
231	C₂×#102	C₂²×C₃₁	1	2	yes	—
232	C₂×#102	C₂²×C₃₁	1	2	yes	—
233	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
234	#398	—	2	2	no	—
235	#391	—	2	1	no	—
236	#389	C₂²×C₃₁	1	4	yes	—
237	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
238	#340	C₂²×C₃₁	1	2	yes	—
239	#336	C₂²×C₃₁	1	4	yes	—
240	C₂×C₃₁	—	8	2	no	—
241	C₂×C₃₁	#2432	1	1	yes	characteristic
242	C₂×C₃₁	C₂×#449	1	1	yes	characteristic
243	C₂×C₃₁	#3011	1	1	yes	characteristic
244	C₂×C₃₁	#3364	1	2	yes	—
245	C₂×C₃₁	#3367	1	2	yes	—
246	C₂²×Q₈	—	2	1	no	—
247	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
248	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
249	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
250	C₂³×C₄	—	2	1	no	—
251	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
252	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
253	C₄⋊Q₈	—	2	2	no	—
254	C₄⋊Q₈	—	2	2	no	—
255	C₄⋊Q₈	—	2	2	no	—
256	C₄⋊Q₈	—	4	4	no	—
257	C₄⋊Q₈	D₄×C₃₁	1	4	yes	—
258	#126	—	2	16	no	—
259	#124	—	2	4	no	—
260	#124	—	2	4	no	—
261	C₂²⋊Q₈	—	2	8	no	—
262	C₄×Q₈	—	2	4	no	—
263	C₄×Q₈	—	2	4	no	—
264	C₄×Q₈	—	2	2	no	—
265	C₄×Q₈	—	2	2	no	—
266	C₄×D₄	—	2	2	no	—
267	C₄×D₄	D₄×C₃₁	1	4	yes	—
268	#117	—	2	8	no	—
269	C₂×C₄⋊C₄	—	2	1	no	—
270	C₂×C₄⋊C₄	—	2	2	no	—
271	C₂×C₄⋊C₄	—	2	1	no	—
272	C₂×C₄⋊C₄	—	2	4	no	—
273	C₂×C₄⋊C₄	—	2	1	no	—
274	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
275	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
276	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
277	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
278	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
279	C₂×C₂²⋊C₄	—	2	2	no	—
280	C₂×C₂²⋊C₄	—	2	1	no	—
281	C₂×C₂²⋊C₄	—	2	2	no	—
282	C₂×C₄²	—	2	2	no	—
283	C₂×C₄²	—	2	1	no	—
284	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
285	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
286	#107	—	2	4	no	—
287	C₄⋊C₈	—	2	2	no	—
288	#103	—	2	8	no	—
289	#103	—	2	8	no	—
290	#102	—	2	4	no	—
291	#102	—	2	4	no	—
292	C₄×C₈	—	2	2	no	—
293	#95	—	2	1	no	—
294	#95	—	2	4	no	—
295	C₃₁	#25650	1	1	yes	characteristic, direct product factor, Sylow
296	C₂⁴	—	2	1	no	—
297	C₂×Q₈	—	2	2	no	—
298	C₂×Q₈	—	2	2	no	—
299	C₂×Q₈	—	2	2	no	—
300	C₂×Q₈	—	2	2	no	—
301	C₂×Q₈	—	2	4	no	—
302	C₂×Q₈	—	2	4	no	—
303	C₂×Q₈	—	4	2	no	—
304	C₂×Q₈	—	4	2	no	—
305	C₂×Q₈	—	4	2	no	—
306	C₂×Q₈	C₂×D₄×C₃₁	1	4	yes	—
307	C₂×D₄	—	2	1	no	—
308	C₂×D₄	—	2	2	no	—
309	C₂×D₄	—	2	2	no	—
310	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
311	C₂²×C₄	—	2	2	no	—
312	C₂²×C₄	—	2	1	no	—
313	C₂²×C₄	—	2	2	no	—
314	C₂²×C₄	—	2	4	no	—
315	C₂²×C₄	—	2	2	no	—
316	C₂²×C₄	—	4	2	no	—
317	C₂²×C₄	—	4	2	no	—
318	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
319	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
320	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
321	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
322	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
323	C₂×C₈	—	2	2	no	—
324	C₂×C₈	—	2	2	no	—
325	C₂×C₈	—	2	2	no	—
326	C₂×C₈	—	2	4	no	—
327	C₂×C₈	—	2	2	no	—
328	C₄⋊C₄	—	2	8	no	—
329	C₄⋊C₄	—	2	4	no	—
330	C₄⋊C₄	—	2	4	no	—
331	C₄⋊C₄	—	2	2	no	—
332	C₄⋊C₄	—	2	4	no	—
333	C₄⋊C₄	—	2	1	no	—
334	C₄⋊C₄	—	2	2	no	—
335	C₄⋊C₄	—	2	2	no	—
336	C₄⋊C₄	—	2	2	no	—
337	C₄⋊C₄	—	4	8	no	—
338	C₄⋊C₄	—	4	2	no	—
339	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
340	C₄⋊C₄	#41×C₃₁	1	4	yes	—
341	C₂²⋊C₄	—	2	8	no	—
342	C₂²⋊C₄	—	2	8	no	—
343	C₂²⋊C₄	—	2	4	no	—
344	C₄²	—	2	8	no	—
345	C₄²	—	2	2	no	—
346	C₄²	—	2	2	no	—
347	C₄²	—	4	2	no	—
348	C₄²	C₂×D₄×C₃₁	1	2	yes	—
349	C₄²	C₂×D₄×C₃₁	1	2	yes	—
350	C₂³	—	2	2	no	—
351	C₂³	—	4	2	no	—
352	C₂³	—	4	1	no	—
353	C₂³	—	4	1	no	—
354	C₂³	—	4	2	no	—
355	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
356	Q₈	—	2	8	no	—
357	Q₈	—	4	4	no	—
358	Q₈	—	4	2	no	—
359	Q₈	—	4	2	no	—
360	Q₈	—	8	4	no	—
361	D₄	—	2	2	no	—
362	D₄	—	2	2	no	—
363	D₄	—	4	2	no	—
364	C₂×C₄	—	2	2	no	—
365	C₂×C₄	—	2	2	no	—
366	C₂×C₄	—	2	2	no	—
367	C₂×C₄	—	2	4	no	—
368	C₂×C₄	—	2	2	no	—
369	C₂×C₄	—	2	4	no	—
370	C₂×C₄	—	2	2	no	—
371	C₂×C₄	—	2	2	no	—
372	C₂×C₄	—	2	4	no	—
373	C₂×C₄	—	4	4	no	—
374	C₂×C₄	—	4	1	no	—
375	C₂×C₄	—	4	2	no	—
376	C₂×C₄	—	4	1	no	—
377	C₂×C₄	—	4	2	no	—
378	C₂×C₄	—	4	8	no	—
379	C₂×C₄	—	4	2	no	—
380	C₂×C₄	—	4	2	no	—
381	C₂×C₄	—	4	2	no	—
382	C₂×C₄	C₃₁×#120	1	2	yes	—
383	C₂×C₄	C₃₁×#120	1	2	yes	—
384	C₂×C₄	C₃₁×#135	1	2	yes	—
385	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
386	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
387	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
388	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
389	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
390	C₈	—	4	2	no	—
391	C₈	—	4	2	no	—
392	C₂²	—	4	2	no	—
393	C₂²	—	4	2	no	—
394	C₂²	—	4	2	no	—
395	C₂²	—	8	2	no	—
396	C₂²	—	8	2	no	—
397	C₂²	C₂×C₃₁×#120	1	1	yes	characteristic
398	C₂²	C₃₁×#548	1	2	yes	—
399	C₂²	C₂×C₃₁×#135	1	1	yes	characteristic
400	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
401	C₂²	C₃₁×#578	1	2	yes	—
402	C₄	—	2	2	no	—
403	C₄	—	4	4	no	—
404	C₄	—	4	4	no	—
405	C₄	—	4	2	no	—
406	C₄	—	4	2	no	—
407	C₄	—	4	2	no	—
408	C₄	—	8	2	no	—
409	C₄	C₃₁×#449	1	4	yes	—
410	C₄	C₃₁×#532	1	1	yes	characteristic
411	C₄	C₃₁×#536	1	1	yes	characteristic
412	C₄	C₃₁×#547	1	2	yes	—
413	C₂	—	8	2	no	—
414	C₂	C₃₁×#2432	1	1	yes	characteristic
415	C₂	C₂×C₃₁×#449	1	1	yes	characteristic
416	C₂	C₃₁×#3011	1	1	yes	characteristic
417	C₂	C₃₁×#3364	1	2	yes	—
418	C₂	C₃₁×#3367	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂²×D₄×C₃₁	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄×Q₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#487	C₂	semidirect
4. C₃₁×#2546	C₂	semidirect
6. C₃₁×#1984	C₂	semidirect
7. C₃₁×#1981	C₂	semidirect
8. C₃₁×#1863	C₂	semidirect
140. #25650	C₃₁	direct (N × H)
184. C₂×#487	C₂×C₃₁	semidirect
186. #2546	C₂×C₃₁	semidirect
188. #1984	C₂×C₃₁	semidirect
189. #1981	C₂×C₃₁	semidirect
190. #1863	C₂×C₃₁	semidirect
295. C₃₁	#25650	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#556	2
C₂×C₃₁×#487	2
C₂×C₃₁×#474	2
C₃₁×#2546	2
C₃₁×#2042	2
C₃₁×#1984	2
C₃₁×#1981	2
C₃₁×#1863	2
C₄×C₃₁×#102	2
#25650	31

Subgroups — C31×#25650 #21482409

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#25650 #21482409