Subgroups — C₃₁×#22563 #21479322

Order 7936.

990 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	C₂²×C₃₁×#103	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2344	C₂	1	1	yes	maximal, characteristic
3	Q₈×C₂²⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1935	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1928	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1863	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1823	C₂	1	1	yes	maximal, characteristic
8	C₃₁×#1808	C₂	1	1	yes	maximal, characteristic
9	C₃₁×#1787	C₂	1	1	yes	maximal, characteristic
10	C₃₁×#1763	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1752	C₂	1	1	yes	maximal, characteristic
12	C₂³×Q₈×C₃₁	C₂²	1	1	yes	characteristic
13	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
15	C₂×C₃₁×#117	—	2	1	no	—
16	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×#107	C₂²	1	2	yes	—
18	C₂×C₃₁×#103	C₄	1	8	yes	complemented
19	C₂×C₃₁×#103	C₂²	1	2	yes	—
20	C₂×C₃₁×#103	C₂²	1	2	yes	—
21	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
23	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
24	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
25	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
26	C₂×C₃₁×#97	C₂²	1	2	yes	—
27	C₃₁×#391	—	2	2	no	—
28	C₃₁×#391	C₂²	1	2	yes	—
29	C₃₁×#389	C₂²	1	2	yes	—
30	C₃₁×#387	—	2	2	no	—
31	C₃₁×#381	—	2	1	no	—
32	C₃₁×#380	—	2	1	no	—
33	C₃₁×#380	C₂²	1	1	yes	characteristic
34	C₃₁×#380	C₂²	1	1	yes	characteristic
35	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
36	C₄×C₂²⋊C₄×C₃₁	—	2	1	no	—
37	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
38	C₃₁×#340	C₂²	1	2	yes	—
39	C₃₁×#340	C₂²	1	2	yes	—
40	C₃₁×#340	C₂²	1	2	yes	—
41	C₃₁×#336	C₂²	1	2	yes	—
42	C₂²×Q₈×C₃₁	—	2	1	no	—
43	C₂²×Q₈×C₃₁	—	2	2	no	—
44	C₂²×Q₈×C₃₁	—	2	4	no	—
45	C₂²×Q₈×C₃₁	—	4	2	no	—
46	C₂²×Q₈×C₃₁	C₂×C₄	1	4	yes	—
47	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
48	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
49	C₂³×C₄×C₃₁	—	2	1	no	—
50	C₂³×C₄×C₃₁	—	2	1	no	—
51	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₃₁	—	4	2	no	—
53	C₂²×C₈×C₃₁	C₂×C₄	1	4	yes	—
54	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
55	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
56	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
57	C₄×Q₈×C₃₁	—	2	4	no	—
58	C₄×Q₈×C₃₁	—	4	2	no	—
59	C₃₁×#117	—	4	4	no	—
60	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
61	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
62	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
63	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
64	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
65	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
66	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
67	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
68	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
69	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
70	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
71	C₂×C₂²⋊C₄×C₃₁	D₄	1	1	yes	characteristic
72	C₂×C₂²⋊C₄×C₃₁	D₄	1	1	yes	characteristic
73	C₂×C₄²×C₃₁	—	2	2	no	—
74	C₂×C₄²×C₃₁	—	2	2	no	—
75	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
76	C₃₁×#107	—	4	2	no	—
77	C₃₁×#103	—	2	4	no	—
78	C₃₁×#103	—	4	2	no	—
79	C₃₁×#103	C₂×C₄	1	4	yes	—
80	C₃₁×#103	C₂×C₄	1	4	yes	—
81	C₃₁×#103	C₂×C₄	1	4	yes	—
82	C₃₁×#103	C₂×C₄	1	4	yes	—
83	C₃₁×C₂²⋊C₈	—	2	2	no	—
84	C₃₁×#97	—	4	2	no	—
85	C₃₁×#95	—	2	2	no	—
86	C₃₁×#95	—	2	2	no	—
87	C₃₁×#95	—	2	2	no	—
88	C₃₁×#95	—	2	1	no	—
89	C₃₁×#95	—	2	1	no	—
90	C₃₁×#95	—	2	2	no	—
91	C₃₁×#95	D₄	1	2	yes	—
92	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
93	C₂×Q₈×C₃₁	—	2	4	no	—
94	C₂×Q₈×C₃₁	—	2	1	no	—
95	C₂×Q₈×C₃₁	—	2	1	no	—
96	C₂×Q₈×C₃₁	—	2	2	no	—
97	C₂×Q₈×C₃₁	—	2	8	no	—
98	C₂×Q₈×C₃₁	—	2	2	no	—
99	C₂×Q₈×C₃₁	—	2	2	no	—
100	C₂×Q₈×C₃₁	—	4	4	no	—
101	C₂×Q₈×C₃₁	—	4	4	no	—
102	C₂×Q₈×C₃₁	—	4	8	no	—
103	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
104	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
105	C₂×Q₈×C₃₁	#41	1	4	yes	—
106	C₂²×C₄×C₃₁	—	2	1	no	—
107	C₂²×C₄×C₃₁	—	2	2	no	—
108	C₂²×C₄×C₃₁	—	2	2	no	—
109	C₂²×C₄×C₃₁	—	2	4	no	—
110	C₂²×C₄×C₃₁	—	2	4	no	—
111	C₂²×C₄×C₃₁	—	2	1	no	—
112	C₂²×C₄×C₃₁	—	2	1	no	—
113	C₂²×C₄×C₃₁	—	2	1	no	—
114	C₂²×C₄×C₃₁	—	2	2	no	—
115	C₂²×C₄×C₃₁	—	2	2	no	—
116	C₂²×C₄×C₃₁	—	4	4	no	—
117	C₂²×C₄×C₃₁	—	4	2	no	—
118	C₂²×C₄×C₃₁	—	4	2	no	—
119	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
120	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
121	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
122	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
123	C₂²×C₄×C₃₁	#41	1	2	yes	—
124	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
125	C₂×C₈×C₃₁	—	2	2	no	—
126	C₂×C₈×C₃₁	—	2	2	no	—
127	C₂×C₈×C₃₁	—	2	2	no	—
128	C₂×C₈×C₃₁	—	2	2	no	—
129	C₂×C₈×C₃₁	—	2	2	no	—
130	C₂×C₈×C₃₁	—	4	4	no	—
131	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
132	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
133	C₄⋊C₄×C₃₁	—	2	4	no	—
134	C₄⋊C₄×C₃₁	—	2	1	no	—
135	C₄⋊C₄×C₃₁	—	2	1	no	—
136	C₄⋊C₄×C₃₁	—	2	2	no	—
137	C₄⋊C₄×C₃₁	—	2	2	no	—
138	C₄⋊C₄×C₃₁	—	2	2	no	—
139	C₄⋊C₄×C₃₁	—	2	2	no	—
140	C₄⋊C₄×C₃₁	—	4	4	no	—
141	C₄⋊C₄×C₃₁	—	4	4	no	—
142	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
143	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
144	C₂²⋊C₄×C₃₁	—	2	2	no	—
145	C₂²⋊C₄×C₃₁	—	2	2	no	—
146	C₂²⋊C₄×C₃₁	—	4	2	no	—
147	C₂²⋊C₄×C₃₁	—	4	2	no	—
148	C₄²×C₃₁	—	2	2	no	—
149	C₄²×C₃₁	—	2	2	no	—
150	C₄²×C₃₁	—	4	4	no	—
151	C₄²×C₃₁	—	4	4	no	—
152	#22563	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
153	C₂³×C₃₁	—	2	2	no	—
154	C₂³×C₃₁	—	2	2	no	—
155	C₂³×C₃₁	C₄×D₄	1	2	yes	—
156	C₂³×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
157	C₂³×C₃₁	#137	1	2	yes	—
158	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
159	Q₈×C₃₁	—	2	2	no	—
160	Q₈×C₃₁	—	2	4	no	—
161	Q₈×C₃₁	—	2	2	no	—
162	Q₈×C₃₁	—	2	2	no	—
163	Q₈×C₃₁	—	4	8	no	—
164	Q₈×C₃₁	—	4	2	no	—
165	Q₈×C₃₁	#117	1	4	yes	complemented
166	C₂×C₄×C₃₁	—	2	2	no	—
167	C₂×C₄×C₃₁	—	2	4	no	—
168	C₂×C₄×C₃₁	—	2	2	no	—
169	C₂×C₄×C₃₁	—	2	2	no	—
170	C₂×C₄×C₃₁	—	2	2	no	—
171	C₂×C₄×C₃₁	—	2	4	no	—
172	C₂×C₄×C₃₁	—	2	2	no	—
173	C₂×C₄×C₃₁	—	2	2	no	—
174	C₂×C₄×C₃₁	—	2	2	no	—
175	C₂×C₄×C₃₁	—	2	2	no	—
176	C₂×C₄×C₃₁	—	2	2	no	—
177	C₂×C₄×C₃₁	—	2	2	no	—
178	C₂×C₄×C₃₁	—	2	2	no	—
179	C₂×C₄×C₃₁	—	2	2	no	—
180	C₂×C₄×C₃₁	—	2	2	no	—
181	C₂×C₄×C₃₁	—	4	8	no	—
182	C₂×C₄×C₃₁	—	4	2	no	—
183	C₂×C₄×C₃₁	—	4	1	no	—
184	C₂×C₄×C₃₁	—	4	1	no	—
185	C₂×C₄×C₃₁	—	4	1	no	—
186	C₂×C₄×C₃₁	—	4	1	no	—
187	C₂×C₄×C₃₁	—	4	8	no	—
188	C₂×C₄×C₃₁	—	4	2	no	—
189	C₂×C₄×C₃₁	—	4	2	no	—
190	C₂×C₄×C₃₁	—	4	2	no	—
191	C₂×C₄×C₃₁	—	4	2	no	—
192	C₂×C₄×C₃₁	—	4	2	no	—
193	C₂×C₄×C₃₁	—	4	2	no	—
194	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
195	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
196	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
197	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
198	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
199	C₈×C₃₁	—	2	2	no	—
200	C₈×C₃₁	—	2	2	no	—
201	C₈×C₃₁	—	4	2	no	—
202	C₂²×#103	C₂×C₃₁	1	1	yes	characteristic
203	#2344	C₂×C₃₁	1	1	yes	characteristic
204	Q₈×C₂²⋊C₄	C₂×C₃₁	1	1	yes	characteristic
205	#1935	C₂×C₃₁	1	2	yes	complemented
206	#1928	C₂×C₃₁	1	2	yes	complemented
207	#1863	C₂×C₃₁	1	2	yes	complemented
208	#1823	C₂×C₃₁	1	1	yes	characteristic
209	#1808	C₂×C₃₁	1	1	yes	characteristic
210	#1787	C₂×C₃₁	1	1	yes	characteristic
211	#1763	C₂×C₃₁	1	2	yes	complemented
212	#1752	C₂×C₃₁	1	1	yes	characteristic
213	C₂²×C₃₁	—	2	2	no	—
214	C₂²×C₃₁	—	2	2	no	—
215	C₂²×C₃₁	—	2	4	no	—
216	C₂²×C₃₁	—	2	4	no	—
217	C₂²×C₃₁	#440	1	4	yes	—
218	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
219	C₂²×C₃₁	#546	1	2	yes	—
220	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
221	C₂²×C₃₁	C₂×#137	1	1	yes	characteristic
222	C₂²×C₃₁	#578	1	2	yes	—
223	C₄×C₃₁	—	2	2	no	—
224	C₄×C₃₁	—	2	4	no	—
225	C₄×C₃₁	—	4	2	no	—
226	C₄×C₃₁	—	4	2	no	—
227	C₄×C₃₁	—	4	2	no	—
228	C₄×C₃₁	—	4	2	no	—
229	C₄×C₃₁	—	4	2	no	—
230	C₄×C₃₁	—	8	2	no	—
231	C₄×C₃₁	—	8	2	no	—
232	C₄×C₃₁	C₂×#117	1	1	yes	characteristic
233	C₄×C₃₁	#518	1	1	yes	characteristic
234	C₄×C₃₁	#551	1	2	yes	—
235	C₂³×Q₈	C₂²×C₃₁	1	1	yes	characteristic
236	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
237	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
238	C₂×#117	—	2	1	no	—
239	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
240	C₂×#107	C₂²×C₃₁	1	2	yes	—
241	C₂×#103	C₄×C₃₁	1	8	yes	complemented
242	C₂×#103	C₂²×C₃₁	1	2	yes	—
243	C₂×#103	C₂²×C₃₁	1	2	yes	—
244	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
245	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
246	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
247	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
248	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
249	C₂×#97	C₂²×C₃₁	1	2	yes	—
250	#391	—	2	2	no	—
251	#391	C₂²×C₃₁	1	2	yes	—
252	#389	C₂²×C₃₁	1	2	yes	—
253	#387	—	2	2	no	—
254	#381	—	2	1	no	—
255	#380	—	2	1	no	—
256	#380	C₂²×C₃₁	1	1	yes	characteristic
257	#380	C₂²×C₃₁	1	1	yes	characteristic
258	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
259	C₄×C₂²⋊C₄	—	2	1	no	—
260	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
261	#340	C₂²×C₃₁	1	2	yes	—
262	#340	C₂²×C₃₁	1	2	yes	—
263	#340	C₂²×C₃₁	1	2	yes	—
264	#336	C₂²×C₃₁	1	2	yes	—
265	C₂×C₃₁	—	2	4	no	—
266	C₂×C₃₁	#2342	1	1	yes	characteristic
267	C₂×C₃₁	C₂×#440	1	1	yes	characteristic
268	C₂×C₃₁	#2950	1	1	yes	characteristic
269	C₂×C₃₁	#3308	1	2	yes	—
270	C₂×C₃₁	#3312	1	2	yes	—
271	C₂²×Q₈	—	2	1	no	—
272	C₂²×Q₈	—	2	2	no	—
273	C₂²×Q₈	—	2	4	no	—
274	C₂²×Q₈	—	4	2	no	—
275	C₂²×Q₈	C₂×C₄×C₃₁	1	4	yes	—
276	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
277	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
278	C₂³×C₄	—	2	1	no	—
279	C₂³×C₄	—	2	1	no	—
280	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
281	C₂²×C₈	—	4	2	no	—
282	C₂²×C₈	C₂×C₄×C₃₁	1	4	yes	—
283	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
284	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
285	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
286	C₄×Q₈	—	2	4	no	—
287	C₄×Q₈	—	4	2	no	—
288	#117	—	4	4	no	—
289	C₂×C₄⋊C₄	—	2	1	no	—
290	C₂×C₄⋊C₄	—	2	2	no	—
291	C₂×C₄⋊C₄	—	2	2	no	—
292	C₂×C₄⋊C₄	—	2	2	no	—
293	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
294	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
295	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
296	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
297	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
298	C₂×C₂²⋊C₄	—	2	1	no	—
299	C₂×C₂²⋊C₄	—	2	1	no	—
300	C₂×C₂²⋊C₄	D₄×C₃₁	1	1	yes	characteristic
301	C₂×C₂²⋊C₄	D₄×C₃₁	1	1	yes	characteristic
302	C₂×C₄²	—	2	2	no	—
303	C₂×C₄²	—	2	2	no	—
304	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
305	#107	—	4	2	no	—
306	#103	—	2	4	no	—
307	#103	—	4	2	no	—
308	#103	C₂×C₄×C₃₁	1	4	yes	—
309	#103	C₂×C₄×C₃₁	1	4	yes	—
310	#103	C₂×C₄×C₃₁	1	4	yes	—
311	#103	C₂×C₄×C₃₁	1	4	yes	—
312	C₂²⋊C₈	—	2	2	no	—
313	#97	—	4	2	no	—
314	#95	—	2	2	no	—
315	#95	—	2	2	no	—
316	#95	—	2	2	no	—
317	#95	—	2	1	no	—
318	#95	—	2	1	no	—
319	#95	—	2	2	no	—
320	#95	D₄×C₃₁	1	2	yes	—
321	C₃₁	#22563	1	1	yes	characteristic, direct product factor, Sylow
322	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
323	C₂×Q₈	—	2	4	no	—
324	C₂×Q₈	—	2	1	no	—
325	C₂×Q₈	—	2	1	no	—
326	C₂×Q₈	—	2	2	no	—
327	C₂×Q₈	—	2	8	no	—
328	C₂×Q₈	—	2	2	no	—
329	C₂×Q₈	—	2	2	no	—
330	C₂×Q₈	—	4	4	no	—
331	C₂×Q₈	—	4	4	no	—
332	C₂×Q₈	—	4	8	no	—
333	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
334	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
335	C₂×Q₈	#41×C₃₁	1	4	yes	—
336	C₂²×C₄	—	2	1	no	—
337	C₂²×C₄	—	2	2	no	—
338	C₂²×C₄	—	2	2	no	—
339	C₂²×C₄	—	2	4	no	—
340	C₂²×C₄	—	2	4	no	—
341	C₂²×C₄	—	2	1	no	—
342	C₂²×C₄	—	2	1	no	—
343	C₂²×C₄	—	2	1	no	—
344	C₂²×C₄	—	2	2	no	—
345	C₂²×C₄	—	2	2	no	—
346	C₂²×C₄	—	4	4	no	—
347	C₂²×C₄	—	4	2	no	—
348	C₂²×C₄	—	4	2	no	—
349	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
350	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
351	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
352	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
353	C₂²×C₄	#41×C₃₁	1	2	yes	—
354	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
355	C₂×C₈	—	2	2	no	—
356	C₂×C₈	—	2	2	no	—
357	C₂×C₈	—	2	2	no	—
358	C₂×C₈	—	2	2	no	—
359	C₂×C₈	—	2	2	no	—
360	C₂×C₈	—	4	4	no	—
361	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
362	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
363	C₄⋊C₄	—	2	4	no	—
364	C₄⋊C₄	—	2	1	no	—
365	C₄⋊C₄	—	2	1	no	—
366	C₄⋊C₄	—	2	2	no	—
367	C₄⋊C₄	—	2	2	no	—
368	C₄⋊C₄	—	2	2	no	—
369	C₄⋊C₄	—	2	2	no	—
370	C₄⋊C₄	—	4	4	no	—
371	C₄⋊C₄	—	4	4	no	—
372	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
373	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
374	C₂²⋊C₄	—	2	2	no	—
375	C₂²⋊C₄	—	2	2	no	—
376	C₂²⋊C₄	—	4	2	no	—
377	C₂²⋊C₄	—	4	2	no	—
378	C₄²	—	2	2	no	—
379	C₄²	—	2	2	no	—
380	C₄²	—	4	4	no	—
381	C₄²	—	4	4	no	—
382	C₂³	—	2	2	no	—
383	C₂³	—	2	2	no	—
384	C₂³	C₄×D₄×C₃₁	1	2	yes	—
385	C₂³	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
386	C₂³	C₃₁×#137	1	2	yes	—
387	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
388	Q₈	—	2	2	no	—
389	Q₈	—	2	4	no	—
390	Q₈	—	2	2	no	—
391	Q₈	—	2	2	no	—
392	Q₈	—	4	8	no	—
393	Q₈	—	4	2	no	—
394	Q₈	C₃₁×#117	1	4	yes	complemented
395	C₂×C₄	—	2	2	no	—
396	C₂×C₄	—	2	4	no	—
397	C₂×C₄	—	2	2	no	—
398	C₂×C₄	—	2	2	no	—
399	C₂×C₄	—	2	2	no	—
400	C₂×C₄	—	2	4	no	—
401	C₂×C₄	—	2	2	no	—
402	C₂×C₄	—	2	2	no	—
403	C₂×C₄	—	2	2	no	—
404	C₂×C₄	—	2	2	no	—
405	C₂×C₄	—	2	2	no	—
406	C₂×C₄	—	2	2	no	—
407	C₂×C₄	—	2	2	no	—
408	C₂×C₄	—	2	2	no	—
409	C₂×C₄	—	2	2	no	—
410	C₂×C₄	—	4	8	no	—
411	C₂×C₄	—	4	8	no	—
412	C₂×C₄	—	4	2	no	—
413	C₂×C₄	—	4	1	no	—
414	C₂×C₄	—	4	1	no	—
415	C₂×C₄	—	4	1	no	—
416	C₂×C₄	—	4	1	no	—
417	C₂×C₄	—	4	2	no	—
418	C₂×C₄	—	4	2	no	—
419	C₂×C₄	—	4	2	no	—
420	C₂×C₄	—	4	2	no	—
421	C₂×C₄	—	4	2	no	—
422	C₂×C₄	—	4	2	no	—
423	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
424	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
425	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
426	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
427	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
428	C₈	—	2	2	no	—
429	C₈	—	2	2	no	—
430	C₈	—	4	2	no	—
431	C₂²	—	2	2	no	—
432	C₂²	—	2	2	no	—
433	C₂²	—	2	4	no	—
434	C₂²	—	2	4	no	—
435	C₂²	C₃₁×#440	1	4	yes	—
436	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
437	C₂²	C₃₁×#546	1	2	yes	—
438	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
439	C₂²	C₂×C₃₁×#137	1	1	yes	characteristic
440	C₂²	C₃₁×#578	1	2	yes	—
441	C₄	—	2	2	no	—
442	C₄	—	2	4	no	—
443	C₄	—	4	2	no	—
444	C₄	—	4	2	no	—
445	C₄	—	4	2	no	—
446	C₄	—	4	2	no	—
447	C₄	—	4	2	no	—
448	C₄	—	8	2	no	—
449	C₄	—	8	2	no	—
450	C₄	C₂×C₃₁×#117	1	1	yes	characteristic
451	C₄	C₃₁×#518	1	1	yes	characteristic
452	C₄	C₃₁×#551	1	2	yes	—
453	C₂	—	2	4	no	—
454	C₂	C₃₁×#2342	1	1	yes	characteristic
455	C₂	C₂×C₃₁×#440	1	1	yes	characteristic
456	C₂	C₃₁×#2950	1	1	yes	characteristic
457	C₂	C₃₁×#3308	1	2	yes	—
458	C₂	C₃₁×#3312	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 ◁ C₄×C₂²⋊C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃₁×#1935	C₂	semidirect
5. C₃₁×#1928	C₂	semidirect
6. C₃₁×#1863	C₂	semidirect
10. C₃₁×#1763	C₂	semidirect
18. C₂×C₃₁×#103	C₄	semidirect
152. #22563	C₃₁	direct (N × H)
165. Q₈×C₃₁	#117	semidirect
205. #1935	C₂×C₃₁	semidirect
206. #1928	C₂×C₃₁	semidirect
207. #1863	C₂×C₃₁	semidirect
211. #1763	C₂×C₃₁	semidirect
241. C₂×#103	C₄×C₃₁	semidirect
321. C₃₁	#22563	direct (N × H)
394. Q₈	C₃₁×#117	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃₁×#103	2
C₃₁×#2344	2
Q₈×C₂²⋊C₄×C₃₁	2
C₃₁×#1935	2
C₃₁×#1928	2
C₃₁×#1863	2
C₃₁×#1823	2
C₃₁×#1808	2
C₃₁×#1787	2
C₃₁×#1763	2
C₃₁×#1752	2
#22563	31

Subgroups — C31×#22563 #21479322

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22563 #21479322