Subgroups — C₃₁×#20647 #21477406

Order 7936.

1450 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	C₂²×C₄×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#423	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#419	C₂	1	2	yes	maximal
4	C₃₁×#2285	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1805	C₂	1	4	yes	maximal, complemented
6	C₃₁×#1803	C₂	1	4	yes	maximal, complemented
7	C₃₁×#1736	C₂	1	2	yes	maximal
8	C₂³×Q₈×C₃₁	C₂²	1	1	yes	characteristic
9	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
10	C₂×C₄×Q₈×C₃₁	—	2	4	no	—
11	C₂×C₄×Q₈×C₃₁	—	2	1	no	—
12	C₂×C₄×Q₈×C₃₁	—	2	1	no	—
13	C₂×C₄×Q₈×C₃₁	C₄	1	8	yes	complemented
14	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
15	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
16	C₂×C₃₁×#117	C₂²	1	2	yes	—
17	C₂²×C₄⋊C₄×C₃₁	—	2	1	no	—
18	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
19	C₂²×C₄²×C₃₁	—	2	1	no	—
20	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
21	C₃₁×#423	—	2	4	no	—
22	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
23	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
24	C₃₁×#419	—	2	8	no	—
25	C₂×C₃₁×#103	C₂²	1	8	yes	—
26	C₃₁×#384	C₂²	1	2	yes	—
27	C₃₁×#382	C₂²	1	2	yes	—
28	C₃₁×#380	—	2	1	no	—
29	C₃₁×#380	—	2	1	no	—
30	C₄×C₄⋊C₄×C₃₁	—	2	1	no	—
31	C₄×C₄⋊C₄×C₃₁	—	2	1	no	—
32	C₃₁×#340	C₂²	1	8	yes	—
33	C₂²×Q₈×C₃₁	—	2	2	no	—
34	C₂²×Q₈×C₃₁	—	2	1	no	—
35	C₂²×Q₈×C₃₁	—	2	2	no	—
36	C₂²×Q₈×C₃₁	—	2	2	no	—
37	C₂²×Q₈×C₃₁	C₂×C₄	1	4	yes	—
38	C₂²×Q₈×C₃₁	D₄	1	4	yes	—
39	C₂²×Q₈×C₃₁	Q₈	1	4	yes	—
40	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
41	C₂³×C₄×C₃₁	—	2	1	no	—
42	C₂³×C₄×C₃₁	—	2	1	no	—
43	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
44	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
45	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
46	C₂×M₁₆×C₃₁	—	2	4	no	—
47	C₂×M₁₆×C₃₁	—	2	4	no	—
48	C₂×M₁₆×C₃₁	—	2	4	no	—
49	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
50	C₄×Q₈×C₃₁	—	2	8	no	—
51	C₄×Q₈×C₃₁	—	2	8	no	—
52	C₄×Q₈×C₃₁	—	2	4	no	—
53	C₄×Q₈×C₃₁	—	2	4	no	—
54	C₄×Q₈×C₃₁	C₂×C₄	1	8	yes	—
55	C₄×Q₈×C₃₁	C₂×C₄	1	8	yes	—
56	C₃₁×#117	—	2	4	no	—
57	C₃₁×#117	—	2	4	no	—
58	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
59	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
60	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
61	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
62	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
63	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
64	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
65	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
66	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
67	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
68	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
69	C₂×C₄²×C₃₁	—	2	8	no	—
70	C₂×C₄²×C₃₁	—	2	2	no	—
71	C₂×C₄²×C₃₁	—	2	2	no	—
72	C₂×C₄²×C₃₁	—	2	2	no	—
73	C₂×C₄²×C₃₁	C₂×C₄	1	4	yes	—
74	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
75	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
76	C₂×C₄²×C₃₁	D₄	1	4	yes	—
77	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
78	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
79	C₃₁×C₄⋊C₈	—	2	4	no	—
80	C₃₁×C₄⋊C₈	—	2	4	no	—
81	C₃₁×#103	—	2	16	no	—
82	C₃₁×#95	—	2	2	no	—
83	C₃₁×#95	—	2	2	no	—
84	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
85	C₂×Q₈×C₃₁	—	2	4	no	—
86	C₂×Q₈×C₃₁	—	2	8	no	—
87	C₂×Q₈×C₃₁	—	2	4	no	—
88	C₂×Q₈×C₃₁	—	2	8	no	—
89	C₂×Q₈×C₃₁	—	2	4	no	—
90	C₂×Q₈×C₃₁	—	2	8	no	—
91	C₂×Q₈×C₃₁	—	2	2	no	—
92	C₂×Q₈×C₃₁	—	2	2	no	—
93	C₂×Q₈×C₃₁	—	2	2	no	—
94	C₂×Q₈×C₃₁	C₄⋊C₄	1	16	yes	complemented
95	C₂×Q₈×C₃₁	C₂²×C₄	1	4	yes	—
96	C₂×Q₈×C₃₁	C₂×D₄	1	4	yes	—
97	C₂×Q₈×C₃₁	C₂×Q₈	1	4	yes	—
98	C₂²×C₄×C₃₁	—	2	4	no	—
99	C₂²×C₄×C₃₁	—	2	4	no	—
100	C₂²×C₄×C₃₁	—	2	4	no	—
101	C₂²×C₄×C₃₁	—	2	2	no	—
102	C₂²×C₄×C₃₁	—	2	4	no	—
103	C₂²×C₄×C₃₁	—	2	4	no	—
104	C₂²×C₄×C₃₁	—	2	2	no	—
105	C₂²×C₄×C₃₁	—	2	4	no	—
106	C₂²×C₄×C₃₁	—	4	2	no	—
107	C₂²×C₄×C₃₁	—	4	4	no	—
108	C₂²×C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
109	C₂²×C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
110	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
111	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
112	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
114	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
115	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
116	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
117	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
118	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
119	C₂²×C₄×C₃₁	#41	1	2	yes	—
120	C₂²×C₄×C₃₁	#41	1	2	yes	—
121	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
122	M₁₆×C₃₁	—	4	8	no	—
123	C₂×C₈×C₃₁	—	2	4	no	—
124	C₂×C₈×C₃₁	—	2	4	no	—
125	C₂×C₈×C₃₁	—	2	4	no	—
126	C₄⋊C₄×C₃₁	—	2	8	no	—
127	C₄⋊C₄×C₃₁	—	2	8	no	—
128	C₄⋊C₄×C₃₁	—	2	4	no	—
129	C₄⋊C₄×C₃₁	—	2	2	no	—
130	C₄⋊C₄×C₃₁	—	2	4	no	—
131	C₄⋊C₄×C₃₁	—	2	4	no	—
132	C₄⋊C₄×C₃₁	—	4	4	no	—
133	C₄⋊C₄×C₃₁	—	4	4	no	—
134	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
135	C₂²⋊C₄×C₃₁	—	4	4	no	—
136	C₄²×C₃₁	—	2	8	no	—
137	C₄²×C₃₁	—	2	8	no	—
138	C₄²×C₃₁	—	4	4	no	—
139	C₄²×C₃₁	C₂²⋊C₄	1	4	yes	—
140	C₄²×C₃₁	C₂²⋊C₄	1	4	yes	—
141	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
142	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
143	C₄²×C₃₁	C₂×D₄	1	2	yes	—
144	C₄²×C₃₁	C₂×D₄	1	2	yes	—
145	#20647	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
146	C₂³×C₃₁	—	2	2	no	—
147	C₂³×C₃₁	—	2	2	no	—
148	C₂³×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
149	C₂³×C₃₁	#121	1	2	yes	—
150	C₂³×C₃₁	C₂²⋊Q₈	1	2	yes	—
151	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
152	Q₈×C₃₁	—	2	4	no	—
153	Q₈×C₃₁	—	2	4	no	—
154	Q₈×C₃₁	—	2	4	no	—
155	Q₈×C₃₁	—	2	16	no	—
156	Q₈×C₃₁	C₂×C₄⋊C₄	1	8	yes	—
157	C₂×C₄×C₃₁	—	2	4	no	—
158	C₂×C₄×C₃₁	—	2	4	no	—
159	C₂×C₄×C₃₁	—	2	4	no	—
160	C₂×C₄×C₃₁	—	2	2	no	—
161	C₂×C₄×C₃₁	—	2	2	no	—
162	C₂×C₄×C₃₁	—	2	2	no	—
163	C₂×C₄×C₃₁	—	2	16	no	—
164	C₂×C₄×C₃₁	—	2	8	no	—
165	C₂×C₄×C₃₁	—	2	4	no	—
166	C₂×C₄×C₃₁	—	4	4	no	—
167	C₂×C₄×C₃₁	—	4	8	no	—
168	C₂×C₄×C₃₁	—	4	4	no	—
169	C₂×C₄×C₃₁	—	4	4	no	—
170	C₂×C₄×C₃₁	—	4	4	no	—
171	C₂×C₄×C₃₁	—	4	4	no	—
172	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
173	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
174	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
175	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
176	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
177	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
178	C₂×C₄×C₃₁	#117	1	4	yes	—
179	C₂×C₄×C₃₁	#121	1	4	yes	—
180	C₂×C₄×C₃₁	#121	1	2	yes	—
181	C₂×C₄×C₃₁	C₂²⋊Q₈	1	4	yes	—
182	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
183	C₂×C₄×C₃₁	#137	1	4	yes	—
184	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
185	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
186	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
187	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
188	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
189	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
190	C₈×C₃₁	—	4	4	no	—
191	C₂²×C₄×Q₈	C₂×C₃₁	1	1	yes	characteristic
192	C₂×#423	C₂×C₃₁	1	1	yes	characteristic
193	C₂×#419	C₂×C₃₁	1	2	yes	—
194	#2285	C₂×C₃₁	1	1	yes	characteristic
195	#1805	C₂×C₃₁	1	4	yes	complemented
196	#1803	C₂×C₃₁	1	4	yes	complemented
197	#1736	C₂×C₃₁	1	2	yes	—
198	C₂²×C₃₁	—	2	4	no	—
199	C₂²×C₃₁	—	2	2	no	—
200	C₂²×C₃₁	—	2	2	no	—
201	C₂²×C₃₁	—	2	4	no	—
202	C₂²×C₃₁	#380	1	4	yes	—
203	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
204	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
205	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
206	C₂²×C₃₁	C₂×#137	1	2	yes	—
207	C₂²×C₃₁	#575	1	2	yes	—
208	C₄×C₃₁	—	2	2	no	—
209	C₄×C₃₁	—	2	2	no	—
210	C₄×C₃₁	—	2	2	no	—
211	C₄×C₃₁	—	2	8	no	—
212	C₄×C₃₁	—	4	4	no	—
213	C₄×C₃₁	—	8	4	no	—
214	C₄×C₃₁	#380	1	4	yes	—
215	C₄×C₃₁	#419	1	4	yes	complemented
216	C₄×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
217	C₄×C₃₁	C₂×#117	1	1	yes	characteristic
218	C₄×C₃₁	C₂×#121	1	1	yes	characteristic
219	C₄×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
220	C₂³×Q₈	C₂²×C₃₁	1	1	yes	characteristic
221	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
222	C₂×C₄×Q₈	—	2	4	no	—
223	C₂×C₄×Q₈	—	2	1	no	—
224	C₂×C₄×Q₈	—	2	1	no	—
225	C₂×C₄×Q₈	C₄×C₃₁	1	8	yes	complemented
226	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
227	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
228	C₂×#117	C₂²×C₃₁	1	2	yes	—
229	C₂²×C₄⋊C₄	—	2	1	no	—
230	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
231	C₂²×C₄²	—	2	1	no	—
232	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
233	#423	—	2	4	no	—
234	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
235	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
236	#419	—	2	8	no	—
237	C₂×#103	C₂²×C₃₁	1	8	yes	—
238	#384	C₂²×C₃₁	1	2	yes	—
239	#382	C₂²×C₃₁	1	2	yes	—
240	#380	—	2	1	no	—
241	#380	—	2	1	no	—
242	C₄×C₄⋊C₄	—	2	1	no	—
243	C₄×C₄⋊C₄	—	2	1	no	—
244	#340	C₂²×C₃₁	1	8	yes	—
245	C₂×C₃₁	—	2	4	no	—
246	C₂×C₃₁	C₂×#380	1	1	yes	characteristic
247	C₂×C₃₁	C₂×#419	1	1	yes	characteristic
248	C₂×C₃₁	#2896	1	1	yes	characteristic
249	C₂×C₃₁	#3037	1	2	yes	—
250	C₂×C₃₁	#3078	1	2	yes	—
251	C₂²×Q₈	—	2	2	no	—
252	C₂²×Q₈	—	2	1	no	—
253	C₂²×Q₈	—	2	2	no	—
254	C₂²×Q₈	—	2	2	no	—
255	C₂²×Q₈	C₂×C₄×C₃₁	1	4	yes	—
256	C₂²×Q₈	D₄×C₃₁	1	4	yes	—
257	C₂²×Q₈	Q₈×C₃₁	1	4	yes	—
258	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
259	C₂³×C₄	—	2	1	no	—
260	C₂³×C₄	—	2	1	no	—
261	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
262	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
263	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
264	C₂×M₁₆	—	2	4	no	—
265	C₂×M₁₆	—	2	4	no	—
266	C₂×M₁₆	—	2	4	no	—
267	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
268	C₄×Q₈	—	2	8	no	—
269	C₄×Q₈	—	2	8	no	—
270	C₄×Q₈	—	2	4	no	—
271	C₄×Q₈	—	2	4	no	—
272	C₄×Q₈	C₂×C₄×C₃₁	1	8	yes	—
273	C₄×Q₈	C₂×C₄×C₃₁	1	8	yes	—
274	#117	—	2	4	no	—
275	#117	—	2	4	no	—
276	C₂×C₄⋊C₄	—	2	8	no	—
277	C₂×C₄⋊C₄	—	2	2	no	—
278	C₂×C₄⋊C₄	—	2	2	no	—
279	C₂×C₄⋊C₄	—	2	1	no	—
280	C₂×C₄⋊C₄	—	2	2	no	—
281	C₂×C₄⋊C₄	—	2	2	no	—
282	C₂×C₄⋊C₄	—	2	2	no	—
283	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
284	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
285	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
286	C₂×C₂²⋊C₄	—	2	2	no	—
287	C₂×C₄²	—	2	8	no	—
288	C₂×C₄²	—	2	2	no	—
289	C₂×C₄²	—	2	2	no	—
290	C₂×C₄²	—	2	2	no	—
291	C₂×C₄²	C₂×C₄×C₃₁	1	4	yes	—
292	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
293	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
294	C₂×C₄²	D₄×C₃₁	1	4	yes	—
295	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
296	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
297	C₄⋊C₈	—	2	4	no	—
298	C₄⋊C₈	—	2	4	no	—
299	#103	—	2	16	no	—
300	#95	—	2	2	no	—
301	#95	—	2	2	no	—
302	C₃₁	#20647	1	1	yes	characteristic, direct product factor, Sylow
303	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
304	C₂×Q₈	—	2	4	no	—
305	C₂×Q₈	—	2	8	no	—
306	C₂×Q₈	—	2	4	no	—
307	C₂×Q₈	—	2	8	no	—
308	C₂×Q₈	—	2	4	no	—
309	C₂×Q₈	—	2	8	no	—
310	C₂×Q₈	—	2	2	no	—
311	C₂×Q₈	—	2	2	no	—
312	C₂×Q₈	—	2	2	no	—
313	C₂×Q₈	C₄⋊C₄×C₃₁	1	16	yes	complemented
314	C₂×Q₈	C₂²×C₄×C₃₁	1	4	yes	—
315	C₂×Q₈	C₂×D₄×C₃₁	1	4	yes	—
316	C₂×Q₈	C₂×Q₈×C₃₁	1	4	yes	—
317	C₂²×C₄	—	2	4	no	—
318	C₂²×C₄	—	2	4	no	—
319	C₂²×C₄	—	2	4	no	—
320	C₂²×C₄	—	2	2	no	—
321	C₂²×C₄	—	2	4	no	—
322	C₂²×C₄	—	2	4	no	—
323	C₂²×C₄	—	2	2	no	—
324	C₂²×C₄	—	2	4	no	—
325	C₂²×C₄	—	4	2	no	—
326	C₂²×C₄	—	4	4	no	—
327	C₂²×C₄	C₂²⋊C₄×C₃₁	1	4	yes	—
328	C₂²×C₄	C₂²⋊C₄×C₃₁	1	4	yes	—
329	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
330	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
331	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
332	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
333	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
334	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
335	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
336	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
337	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
338	C₂²×C₄	#41×C₃₁	1	2	yes	—
339	C₂²×C₄	#41×C₃₁	1	2	yes	—
340	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
341	M₁₆	—	4	8	no	—
342	C₂×C₈	—	2	4	no	—
343	C₂×C₈	—	2	4	no	—
344	C₂×C₈	—	2	4	no	—
345	C₄⋊C₄	—	2	8	no	—
346	C₄⋊C₄	—	2	8	no	—
347	C₄⋊C₄	—	2	4	no	—
348	C₄⋊C₄	—	2	2	no	—
349	C₄⋊C₄	—	2	4	no	—
350	C₄⋊C₄	—	2	4	no	—
351	C₄⋊C₄	—	4	4	no	—
352	C₄⋊C₄	—	4	4	no	—
353	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
354	C₂²⋊C₄	—	4	4	no	—
355	C₄²	—	2	8	no	—
356	C₄²	—	2	8	no	—
357	C₄²	—	4	4	no	—
358	C₄²	C₂²⋊C₄×C₃₁	1	4	yes	—
359	C₄²	C₂²⋊C₄×C₃₁	1	4	yes	—
360	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
361	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
362	C₄²	C₂×D₄×C₃₁	1	2	yes	—
363	C₄²	C₂×D₄×C₃₁	1	2	yes	—
364	C₂³	—	2	2	no	—
365	C₂³	—	2	2	no	—
366	C₂³	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
367	C₂³	C₃₁×#121	1	2	yes	—
368	C₂³	C₃₁×C₂²⋊Q₈	1	2	yes	—
369	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
370	Q₈	—	2	4	no	—
371	Q₈	—	2	4	no	—
372	Q₈	—	2	4	no	—
373	Q₈	—	2	16	no	—
374	Q₈	C₂×C₄⋊C₄×C₃₁	1	8	yes	—
375	C₂×C₄	—	2	4	no	—
376	C₂×C₄	—	2	4	no	—
377	C₂×C₄	—	2	4	no	—
378	C₂×C₄	—	2	2	no	—
379	C₂×C₄	—	2	2	no	—
380	C₂×C₄	—	2	2	no	—
381	C₂×C₄	—	2	16	no	—
382	C₂×C₄	—	2	8	no	—
383	C₂×C₄	—	2	4	no	—
384	C₂×C₄	—	4	4	no	—
385	C₂×C₄	—	4	8	no	—
386	C₂×C₄	—	4	4	no	—
387	C₂×C₄	—	4	4	no	—
388	C₂×C₄	—	4	4	no	—
389	C₂×C₄	—	4	4	no	—
390	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
391	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
392	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
393	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
394	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
395	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
396	C₂×C₄	C₃₁×#117	1	4	yes	—
397	C₂×C₄	C₃₁×#121	1	4	yes	—
398	C₂×C₄	C₃₁×#121	1	2	yes	—
399	C₂×C₄	C₃₁×C₂²⋊Q₈	1	4	yes	—
400	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
401	C₂×C₄	C₃₁×#137	1	4	yes	—
402	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
403	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
404	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
405	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
406	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
407	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
408	C₈	—	4	4	no	—
409	C₂²	—	2	4	no	—
410	C₂²	—	2	2	no	—
411	C₂²	—	2	2	no	—
412	C₂²	—	2	4	no	—
413	C₂²	C₃₁×#380	1	4	yes	—
414	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
415	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
416	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
417	C₂²	C₂×C₃₁×#137	1	2	yes	—
418	C₂²	C₃₁×#575	1	2	yes	—
419	C₄	—	2	2	no	—
420	C₄	—	2	2	no	—
421	C₄	—	2	2	no	—
422	C₄	—	2	8	no	—
423	C₄	—	4	4	no	—
424	C₄	—	8	4	no	—
425	C₄	C₃₁×#380	1	4	yes	—
426	C₄	C₃₁×#419	1	4	yes	complemented
427	C₄	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
428	C₄	C₂×C₃₁×#117	1	1	yes	characteristic
429	C₄	C₂×C₃₁×#121	1	1	yes	characteristic
430	C₄	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
431	C₂	—	2	4	no	—
432	C₂	C₂×C₃₁×#380	1	1	yes	characteristic
433	C₂	C₂×C₃₁×#419	1	1	yes	characteristic
434	C₂	C₃₁×#2896	1	1	yes	characteristic
435	C₂	C₃₁×#3037	1	2	yes	—
436	C₂	C₃₁×#3078	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂⁴×C₃₁	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
5. C₃₁×#1805	C₂	semidirect
6. C₃₁×#1803	C₂	semidirect
13. C₂×C₄×Q₈×C₃₁	C₄	semidirect
94. C₂×Q₈×C₃₁	C₄⋊C₄	semidirect
145. #20647	C₃₁	direct (N × H)
195. #1805	C₂×C₃₁	semidirect
196. #1803	C₂×C₃₁	semidirect
215. C₄×C₃₁	#419	semidirect
225. C₂×C₄×Q₈	C₄×C₃₁	semidirect
302. C₃₁	#20647	direct (N × H)
313. C₂×Q₈	C₄⋊C₄×C₃₁	semidirect
426. C₄	C₃₁×#419	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×Q₈×C₃₁	2
C₂×C₃₁×#423	2
C₂×C₃₁×#419	2
C₃₁×#2285	2
C₃₁×#1805	2
C₃₁×#1803	2
C₃₁×#1736	2
#20647	31

Subgroups — C31×#20647 #21477406

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#20647 #21477406