Subgroups — C₃₁×#10747 #21467506

Order 7936.

1214 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3428	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#413	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#408	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1742	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1729	C₂	1	2	yes	maximal, complemented
6	C₂²×#41×C₃₁	C₂²	1	1	yes	characteristic
7	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
8	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
9	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
10	C₃₁×#520	—	2	4	no	—
11	C₃₁×#519	—	2	1	no	—
12	C₃₁×#518	—	2	1	no	—
13	C₃₁×#518	C₄	1	4	yes	—
14	C₄×#41×C₃₁	—	2	4	no	—
15	C₂×C₃₁×#117	—	2	2	no	—
16	C₂×C₃₁×#117	C₂²	1	2	yes	—
17	C₃₁×#413	—	2	2	no	—
18	C₃₁×#413	C₄	1	4	yes	complemented
19	C₂×C₃₁×#101	—	2	1	no	—
20	C₂×C₃₁×#100	—	2	1	no	—
21	C₃₁×#408	—	2	2	no	—
22	C₃₁×#408	C₄	1	4	yes	complemented
23	C₂×C₃₁×C₂²⋊C₈	—	2	2	no	—
24	C₃₁×#343	—	2	4	no	—
25	C₃₁×#336	—	2	4	no	—
26	C₂×#41×C₃₁	—	2	2	no	—
27	C₂×#41×C₃₁	—	2	2	no	—
28	C₂×#41×C₃₁	—	2	2	no	—
29	C₂×#41×C₃₁	—	2	2	no	—
30	C₂×#41×C₃₁	C₂×C₄	1	2	yes	—
31	C₂×#41×C₃₁	C₂×C₄	1	2	yes	—
32	C₂×#41×C₃₁	D₄	1	2	yes	—
33	C₂×#41×C₃₁	Q₈	1	2	yes	—
34	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
35	C₂²×D₄×C₃₁	C₂×C₄	1	2	yes	—
36	C₂²×D₄×C₃₁	D₄	1	1	yes	characteristic
37	C₂³×C₄×C₃₁	D₄	1	2	yes	—
38	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
39	C₂×M₁₆×C₃₁	—	2	1	no	—
40	C₂×M₁₆×C₃₁	—	2	2	no	—
41	C₂×M₁₆×C₃₁	—	2	2	no	—
42	C₂×M₁₆×C₃₁	—	2	2	no	—
43	C₂×M₁₆×C₃₁	—	2	2	no	—
44	C₂×M₁₆×C₃₁	—	2	2	no	—
45	C₂×M₁₆×C₃₁	—	2	2	no	—
46	C₂×M₁₆×C₃₁	—	2	2	no	—
47	C₂×M₁₆×C₃₁	C₂×C₄	1	2	yes	complemented
48	C₂×M₁₆×C₃₁	C₂×C₄	1	4	yes	complemented
49	C₂²×C₈×C₃₁	—	2	1	no	—
50	C₂²×C₈×C₃₁	—	2	1	no	—
51	C₂²×C₈×C₃₁	—	2	1	no	—
52	C₂²×C₈×C₃₁	—	2	2	no	—
53	C₂²×C₈×C₃₁	—	2	2	no	—
54	C₂²×C₈×C₃₁	—	2	2	no	—
55	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	complemented
56	C₄×Q₈×C₃₁	—	2	8	no	—
57	C₄×D₄×C₃₁	—	2	8	no	—
58	C₄×D₄×C₃₁	—	2	16	no	—
59	C₃₁×#117	—	2	4	no	—
60	C₃₁×#117	—	2	4	no	—
61	C₃₁×#117	—	2	4	no	—
62	C₃₁×#117	—	2	2	no	—
63	C₃₁×#117	—	2	2	no	—
64	C₃₁×#117	—	2	2	no	—
65	C₃₁×#117	—	2	2	no	—
66	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
67	C₂×C₄⋊C₄×C₃₁	D₄	1	4	yes	—
68	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
69	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
70	C₂×C₄²×C₃₁	—	2	4	no	—
71	C₂×C₄²×C₃₁	D₄	1	4	yes	—
72	C₃₁×#101	—	2	4	no	—
73	C₃₁×#100	—	2	4	no	—
74	C₃₁×C₂²⋊C₈	—	2	8	no	—
75	C₂⁴×C₃₁	C₂²⋊C₄	1	2	yes	—
76	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
77	#41×C₃₁	—	2	4	no	—
78	#41×C₃₁	—	2	4	no	—
79	#41×C₃₁	—	2	4	no	—
80	#41×C₃₁	—	2	4	no	—
81	#41×C₃₁	—	2	4	no	—
82	#41×C₃₁	—	2	4	no	—
83	#41×C₃₁	—	2	4	no	—
84	#41×C₃₁	—	2	4	no	—
85	C₂×Q₈×C₃₁	—	2	2	no	—
86	C₂×Q₈×C₃₁	—	2	2	no	—
87	C₂×Q₈×C₃₁	C₂²⋊C₄	1	2	yes	—
88	C₂×Q₈×C₃₁	C₄⋊C₄	1	2	yes	—
89	C₂×D₄×C₃₁	—	2	4	no	—
90	C₂×D₄×C₃₁	—	2	4	no	—
91	C₂×D₄×C₃₁	—	2	2	no	—
92	C₂×D₄×C₃₁	—	2	2	no	—
93	C₂×D₄×C₃₁	C₄²	1	4	yes	—
94	C₂×D₄×C₃₁	C₂²⋊C₄	1	2	yes	—
95	C₂×D₄×C₃₁	C₄⋊C₄	1	4	yes	—
96	C₂×D₄×C₃₁	C₄⋊C₄	1	2	yes	—
97	C₂²×C₄×C₃₁	—	2	2	no	—
98	C₂²×C₄×C₃₁	—	2	2	no	—
99	C₂²×C₄×C₃₁	—	2	2	no	—
100	C₂²×C₄×C₃₁	—	2	2	no	—
101	C₂²×C₄×C₃₁	—	2	8	no	—
102	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
103	C₂²×C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
104	C₂²×C₄×C₃₁	C₂²⋊C₄	1	8	yes	—
105	C₂²×C₄×C₃₁	C₄⋊C₄	1	4	yes	—
106	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
107	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
108	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
110	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
111	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
112	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
113	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
114	C₂²×C₄×C₃₁	#41	1	2	yes	—
115	C₂²×C₄×C₃₁	#41	1	2	yes	—
116	M₁₆×C₃₁	—	2	4	no	—
117	M₁₆×C₃₁	—	2	4	no	—
118	M₁₆×C₃₁	—	2	4	no	—
119	M₁₆×C₃₁	—	4	2	no	—
120	M₁₆×C₃₁	—	4	2	no	—
121	M₁₆×C₃₁	—	4	2	no	—
122	C₂×C₈×C₃₁	—	2	2	no	—
123	C₂×C₈×C₃₁	—	2	2	no	—
124	C₂×C₈×C₃₁	—	2	4	no	—
125	C₂×C₈×C₃₁	—	2	4	no	—
126	C₂×C₈×C₃₁	—	4	2	no	—
127	C₂×C₈×C₃₁	—	4	1	no	—
128	C₂×C₈×C₃₁	—	4	1	no	—
129	C₂×C₈×C₃₁	—	4	2	no	—
130	C₂×C₈×C₃₁	—	4	2	no	—
131	C₂×C₈×C₃₁	—	4	1	no	—
132	C₂×C₈×C₃₁	—	4	1	no	—
133	C₄⋊C₄×C₃₁	—	2	8	no	—
134	C₄⋊C₄×C₃₁	—	2	4	no	—
135	C₄⋊C₄×C₃₁	—	2	4	no	—
136	C₂²⋊C₄×C₃₁	—	2	8	no	—
137	C₂²⋊C₄×C₃₁	—	2	4	no	—
138	C₂²⋊C₄×C₃₁	—	2	4	no	—
139	C₄²×C₃₁	—	2	8	no	—
140	C₄²×C₃₁	—	2	4	no	—
141	C₄²×C₃₁	—	2	4	no	—
142	#10747	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
143	C₂³×C₃₁	—	2	2	no	—
144	C₂³×C₃₁	—	2	2	no	—
145	C₂³×C₃₁	—	4	2	no	—
146	C₂³×C₃₁	—	4	4	no	—
147	C₂³×C₃₁	#95	1	4	yes	—
148	C₂³×C₃₁	C₂×C₄²	1	1	yes	characteristic
149	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
150	C₂³×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
151	C₂³×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
152	C₂³×C₃₁	#117	1	2	yes	—
153	Q₈×C₃₁	—	2	4	no	—
154	Q₈×C₃₁	—	2	4	no	—
155	D₄×C₃₁	—	2	8	no	—
156	D₄×C₃₁	—	2	8	no	—
157	D₄×C₃₁	—	2	4	no	—
158	D₄×C₃₁	—	2	4	no	—
159	C₂×C₄×C₃₁	—	2	2	no	—
160	C₂×C₄×C₃₁	—	2	2	no	—
161	C₂×C₄×C₃₁	—	2	4	no	—
162	C₂×C₄×C₃₁	—	2	4	no	—
163	C₂×C₄×C₃₁	—	2	4	no	—
164	C₂×C₄×C₃₁	—	2	4	no	—
165	C₂×C₄×C₃₁	—	2	8	no	—
166	C₂×C₄×C₃₁	—	4	8	no	—
167	C₂×C₄×C₃₁	—	8	8	no	—
168	C₂×C₄×C₃₁	#95	1	4	yes	—
169	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
170	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
171	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
172	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
173	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
174	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
175	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
176	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
177	C₂×C₄×C₃₁	#120	1	2	yes	—
178	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
179	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
180	C₂×C₄×C₃₁	#121	1	2	yes	—
181	C₂×C₄×C₃₁	#121	1	2	yes	—
182	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
183	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
184	C₂×C₄×C₃₁	#123	1	1	yes	characteristic
185	C₂×C₄×C₃₁	#123	1	1	yes	characteristic
186	C₂×C₄×C₃₁	#123	1	2	yes	—
187	C₈×C₃₁	—	4	2	no	—
188	C₈×C₃₁	—	4	2	no	—
189	C₈×C₃₁	—	4	4	no	—
190	#3428	C₂×C₃₁	1	1	yes	characteristic
191	C₂×#413	C₂×C₃₁	1	1	yes	characteristic
192	C₂×#408	C₂×C₃₁	1	1	yes	characteristic
193	#1742	C₂×C₃₁	1	2	yes	complemented
194	#1729	C₂×C₃₁	1	2	yes	complemented
195	C₂²×C₃₁	—	2	4	no	—
196	C₂²×C₃₁	—	2	2	no	—
197	C₂²×C₃₁	—	2	4	no	—
198	C₂²×C₃₁	—	4	1	no	—
199	C₂²×C₃₁	—	4	1	no	—
200	C₂²×C₃₁	—	4	2	no	—
201	C₂²×C₃₁	—	4	4	no	—
202	C₂²×C₃₁	—	4	4	no	—
203	C₂²×C₃₁	—	8	2	no	—
204	C₂²×C₃₁	C₂×#95	1	1	yes	characteristic
205	C₂²×C₃₁	C₄×C₂²⋊C₄	1	2	yes	—
206	C₂²×C₃₁	#379	1	1	yes	characteristic
207	C₂²×C₃₁	#380	1	2	yes	—
208	C₂²×C₃₁	#381	1	1	yes	characteristic
209	C₄×C₃₁	—	2	4	no	—
210	C₄×C₃₁	—	4	8	no	—
211	C₄×C₃₁	#385	1	1	yes	characteristic
212	C₄×C₃₁	#385	1	1	yes	characteristic
213	C₄×C₃₁	#385	1	2	yes	—
214	C₄×C₃₁	#386	1	1	yes	characteristic
215	C₄×C₃₁	#386	1	1	yes	characteristic
216	C₄×C₃₁	#386	1	2	yes	—
217	C₂²×#41	C₂²×C₃₁	1	1	yes	characteristic
218	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
219	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
220	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
221	#520	—	2	4	no	—
222	#519	—	2	1	no	—
223	#518	—	2	1	no	—
224	#518	C₄×C₃₁	1	4	yes	—
225	C₄×#41	—	2	4	no	—
226	C₂×#117	—	2	2	no	—
227	C₂×#117	C₂²×C₃₁	1	2	yes	—
228	#413	—	2	2	no	—
229	#413	C₄×C₃₁	1	4	yes	complemented
230	C₂×#101	—	2	1	no	—
231	C₂×#100	—	2	1	no	—
232	#408	—	2	2	no	—
233	#408	C₄×C₃₁	1	4	yes	complemented
234	C₂×C₂²⋊C₈	—	2	2	no	—
235	#343	—	2	4	no	—
236	#336	—	2	4	no	—
237	C₂×C₃₁	—	2	1	no	—
238	C₂×C₃₁	—	2	1	no	—
239	C₂×C₃₁	—	2	2	no	—
240	C₂×C₃₁	—	2	2	no	—
241	C₂×C₃₁	—	4	4	no	—
242	C₂×C₃₁	C₂²⋊#95	1	1	yes	characteristic
243	C₂×C₃₁	#1791	1	2	yes	—
244	C₂×#41	—	2	2	no	—
245	C₂×#41	—	2	2	no	—
246	C₂×#41	—	2	2	no	—
247	C₂×#41	—	2	2	no	—
248	C₂×#41	C₂×C₄×C₃₁	1	2	yes	—
249	C₂×#41	C₂×C₄×C₃₁	1	2	yes	—
250	C₂×#41	D₄×C₃₁	1	2	yes	—
251	C₂×#41	Q₈×C₃₁	1	2	yes	—
252	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
253	C₂²×D₄	C₂×C₄×C₃₁	1	2	yes	—
254	C₂²×D₄	D₄×C₃₁	1	1	yes	characteristic
255	C₂³×C₄	D₄×C₃₁	1	2	yes	—
256	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
257	C₂×M₁₆	—	2	1	no	—
258	C₂×M₁₆	—	2	2	no	—
259	C₂×M₁₆	—	2	2	no	—
260	C₂×M₁₆	—	2	2	no	—
261	C₂×M₁₆	—	2	2	no	—
262	C₂×M₁₆	—	2	2	no	—
263	C₂×M₁₆	—	2	2	no	—
264	C₂×M₁₆	—	2	2	no	—
265	C₂×M₁₆	C₂×C₄×C₃₁	1	2	yes	complemented
266	C₂×M₁₆	C₂×C₄×C₃₁	1	4	yes	complemented
267	C₂²×C₈	—	2	2	no	—
268	C₂²×C₈	—	2	1	no	—
269	C₂²×C₈	—	2	1	no	—
270	C₂²×C₈	—	2	2	no	—
271	C₂²×C₈	—	2	1	no	—
272	C₂²×C₈	—	2	2	no	—
273	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	complemented
274	C₄×Q₈	—	2	8	no	—
275	C₄×D₄	—	2	8	no	—
276	C₄×D₄	—	2	16	no	—
277	#117	—	2	4	no	—
278	#117	—	2	4	no	—
279	#117	—	2	4	no	—
280	#117	—	2	2	no	—
281	#117	—	2	2	no	—
282	#117	—	2	2	no	—
283	#117	—	2	2	no	—
284	C₂×C₄⋊C₄	—	2	4	no	—
285	C₂×C₄⋊C₄	D₄×C₃₁	1	4	yes	—
286	C₂×C₂²⋊C₄	—	2	4	no	—
287	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
288	C₂×C₄²	—	2	4	no	—
289	C₂×C₄²	D₄×C₃₁	1	4	yes	—
290	#101	—	2	4	no	—
291	#100	—	2	4	no	—
292	C₂²⋊C₈	—	2	8	no	—
293	C₃₁	#10747	1	1	yes	characteristic, direct product factor, Sylow
294	C₂⁴	C₂²⋊C₄×C₃₁	1	2	yes	—
295	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
296	#41	—	2	4	no	—
297	#41	—	2	4	no	—
298	#41	—	2	4	no	—
299	#41	—	2	4	no	—
300	#41	—	2	4	no	—
301	#41	—	2	4	no	—
302	#41	—	2	4	no	—
303	#41	—	2	4	no	—
304	C₂×Q₈	—	2	2	no	—
305	C₂×Q₈	—	2	2	no	—
306	C₂×Q₈	C₂²⋊C₄×C₃₁	1	2	yes	—
307	C₂×Q₈	C₄⋊C₄×C₃₁	1	2	yes	—
308	C₂×D₄	—	2	4	no	—
309	C₂×D₄	—	2	4	no	—
310	C₂×D₄	—	2	2	no	—
311	C₂×D₄	—	2	2	no	—
312	C₂×D₄	C₄²×C₃₁	1	4	yes	—
313	C₂×D₄	C₂²⋊C₄×C₃₁	1	2	yes	—
314	C₂×D₄	C₄⋊C₄×C₃₁	1	4	yes	—
315	C₂×D₄	C₄⋊C₄×C₃₁	1	2	yes	—
316	C₂²×C₄	—	2	2	no	—
317	C₂²×C₄	—	2	2	no	—
318	C₂²×C₄	—	2	2	no	—
319	C₂²×C₄	—	2	2	no	—
320	C₂²×C₄	—	2	8	no	—
321	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
322	C₂²×C₄	C₂²⋊C₄×C₃₁	1	4	yes	—
323	C₂²×C₄	C₂²⋊C₄×C₃₁	1	8	yes	—
324	C₂²×C₄	C₄⋊C₄×C₃₁	1	4	yes	—
325	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
326	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
327	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
328	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
329	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
330	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
331	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
332	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
333	C₂²×C₄	#41×C₃₁	1	2	yes	—
334	C₂²×C₄	#41×C₃₁	1	2	yes	—
335	M₁₆	—	2	4	no	—
336	M₁₆	—	2	4	no	—
337	M₁₆	—	2	4	no	—
338	M₁₆	—	4	2	no	—
339	M₁₆	—	4	2	no	—
340	M₁₆	—	4	2	no	—
341	C₂×C₈	—	2	2	no	—
342	C₂×C₈	—	2	2	no	—
343	C₂×C₈	—	2	4	no	—
344	C₂×C₈	—	2	4	no	—
345	C₂×C₈	—	4	1	no	—
346	C₂×C₈	—	4	2	no	—
347	C₂×C₈	—	4	1	no	—
348	C₂×C₈	—	4	2	no	—
349	C₂×C₈	—	4	2	no	—
350	C₂×C₈	—	4	1	no	—
351	C₂×C₈	—	4	1	no	—
352	C₄⋊C₄	—	2	8	no	—
353	C₄⋊C₄	—	2	4	no	—
354	C₄⋊C₄	—	2	4	no	—
355	C₂²⋊C₄	—	2	8	no	—
356	C₂²⋊C₄	—	2	4	no	—
357	C₂²⋊C₄	—	2	4	no	—
358	C₄²	—	2	8	no	—
359	C₄²	—	2	4	no	—
360	C₄²	—	2	4	no	—
361	C₂³	—	2	2	no	—
362	C₂³	—	2	2	no	—
363	C₂³	—	4	2	no	—
364	C₂³	—	4	4	no	—
365	C₂³	C₃₁×#95	1	4	yes	—
366	C₂³	C₂×C₄²×C₃₁	1	1	yes	characteristic
367	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
368	C₂³	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
369	C₂³	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
370	C₂³	C₃₁×#117	1	2	yes	—
371	Q₈	—	2	4	no	—
372	Q₈	—	2	4	no	—
373	D₄	—	2	8	no	—
374	D₄	—	2	8	no	—
375	D₄	—	2	4	no	—
376	D₄	—	2	4	no	—
377	C₂×C₄	—	2	2	no	—
378	C₂×C₄	—	2	2	no	—
379	C₂×C₄	—	2	4	no	—
380	C₂×C₄	—	2	4	no	—
381	C₂×C₄	—	2	4	no	—
382	C₂×C₄	—	2	4	no	—
383	C₂×C₄	—	2	8	no	—
384	C₂×C₄	—	4	8	no	—
385	C₂×C₄	—	8	8	no	—
386	C₂×C₄	C₃₁×#95	1	4	yes	—
387	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
388	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
389	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
390	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
391	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
392	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
393	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
394	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
395	C₂×C₄	C₃₁×#120	1	2	yes	—
396	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
397	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
398	C₂×C₄	C₃₁×#121	1	2	yes	—
399	C₂×C₄	C₃₁×#121	1	2	yes	—
400	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
401	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
402	C₂×C₄	C₃₁×#123	1	1	yes	characteristic
403	C₂×C₄	C₃₁×#123	1	1	yes	characteristic
404	C₂×C₄	C₃₁×#123	1	2	yes	—
405	C₈	—	4	2	no	—
406	C₈	—	4	2	no	—
407	C₈	—	4	4	no	—
408	C₂²	—	2	4	no	—
409	C₂²	—	2	2	no	—
410	C₂²	—	2	4	no	—
411	C₂²	—	4	1	no	—
412	C₂²	—	4	1	no	—
413	C₂²	—	4	2	no	—
414	C₂²	—	4	4	no	—
415	C₂²	—	4	4	no	—
416	C₂²	—	8	2	no	—
417	C₂²	C₂×C₃₁×#95	1	1	yes	characteristic
418	C₂²	C₄×C₂²⋊C₄×C₃₁	1	2	yes	—
419	C₂²	C₃₁×#379	1	1	yes	characteristic
420	C₂²	C₃₁×#380	1	2	yes	—
421	C₂²	C₃₁×#381	1	1	yes	characteristic
422	C₄	—	2	4	no	—
423	C₄	—	4	8	no	—
424	C₄	C₃₁×#385	1	1	yes	characteristic
425	C₄	C₃₁×#385	1	1	yes	characteristic
426	C₄	C₃₁×#385	1	2	yes	—
427	C₄	C₃₁×#386	1	1	yes	characteristic
428	C₄	C₃₁×#386	1	1	yes	characteristic
429	C₄	C₃₁×#386	1	2	yes	—
430	C₂	—	2	1	no	—
431	C₂	—	2	1	no	—
432	C₂	—	2	2	no	—
433	C₂	—	2	2	no	—
434	C₂	—	4	4	no	—
435	C₂	C₃₁×C₂²⋊#95	1	1	yes	characteristic
436	C₂	C₃₁×#1791	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₃₁	#120
commutator	C₂³	C₂×C₄²×C₃₁
Frattini, cosocle	C₂³×C₄	C₂³×C₃₁
socle	C₂²×C₃₁	#379
prime radical	C₂²×#41×C₃₁	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂³×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃₁×#1742	C₂	semidirect
5. C₃₁×#1729	C₂	semidirect
18. C₃₁×#413	C₄	semidirect
22. C₃₁×#408	C₄	semidirect
47. C₂×M₁₆×C₃₁	C₂×C₄	semidirect
48. C₂×M₁₆×C₃₁	C₂×C₄	semidirect
55. C₂²×C₈×C₃₁	C₂×C₄	semidirect
142. #10747	C₃₁	direct (N × H)
193. #1742	C₂×C₃₁	semidirect
194. #1729	C₂×C₃₁	semidirect
229. #413	C₄×C₃₁	semidirect
233. #408	C₄×C₃₁	semidirect
265. C₂×M₁₆	C₂×C₄×C₃₁	semidirect
266. C₂×M₁₆	C₂×C₄×C₃₁	semidirect
273. C₂²×C₈	C₂×C₄×C₃₁	semidirect
293. C₃₁	#10747	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×#3428	2
C₂×C₃₁×#413	2
C₂×C₃₁×#408	2
C₃₁×#1742	2
C₃₁×#1729	2
#10747	31

Subgroups — C31×#10747 #21467506

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#10747 #21467506