Subgroups — C₃₁×#22577 #21479336

Order 7936.

978 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#416	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2347	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2341	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1934	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1926	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1865	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1825	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1807	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1788	C₂	1	1	yes	maximal, characteristic
10	C₄×C₃₁×#103	C₂	1	2	yes	maximal, complemented
11	C₈×C₂²⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#41×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	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×#106	C₂²	1	2	yes	—
17	C₃₁×#416	C₄	1	8	yes	complemented
18	C₂×C₃₁×#103	C₂²	1	2	yes	—
19	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×#102	C₂²	1	2	yes	—
22	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
23	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
24	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
26	C₃₁×#390	—	2	2	no	—
27	C₃₁×#390	C₂²	1	2	yes	—
28	C₃₁×#389	C₂²	1	2	yes	—
29	C₃₁×#388	—	2	2	no	—
30	C₃₁×#383	—	2	1	no	—
31	C₃₁×#380	—	2	1	no	—
32	C₃₁×#380	C₂²	1	1	yes	characteristic
33	C₃₁×#380	C₂²	1	1	yes	characteristic
34	C₄×C₄⋊C₄×C₃₁	—	2	1	no	—
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	C₂×#41×C₃₁	—	2	2	no	—
43	C₂×#41×C₃₁	—	2	4	no	—
44	C₂×#41×C₃₁	—	4	2	no	—
45	C₂×#41×C₃₁	C₂×C₄	1	4	yes	—
46	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
47	C₂²×D₄×C₃₁	—	2	1	no	—
48	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
49	C₂³×C₄×C₃₁	—	2	1	no	—
50	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₃₁	—	4	2	no	—
52	C₂²×C₈×C₃₁	C₂×C₄	1	4	yes	—
53	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
54	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
55	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
56	C₄×Q₈×C₃₁	—	2	4	no	—
57	C₄×Q₈×C₃₁	—	4	2	no	—
58	C₃₁×#117	—	2	2	no	—
59	C₃₁×#117	C₂×C₄	1	4	yes	—
60	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
61	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
62	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
63	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
64	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
65	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
66	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
67	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
68	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
69	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
70	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
71	C₂×C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
72	C₂×C₄²×C₃₁	—	2	1	no	—
73	C₂×C₄²×C₃₁	—	2	2	no	—
74	C₂×C₄²×C₃₁	—	2	2	no	—
75	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
76	C₃₁×#106	—	4	2	no	—
77	C₃₁×#103	—	2	4	no	—
78	C₃₁×#103	C₂×C₄	1	4	yes	—
79	C₃₁×#103	C₂×C₄	1	4	yes	—
80	C₃₁×#102	—	4	2	no	—
81	C₃₁×#102	C₂×C₄	1	4	yes	—
82	C₃₁×#102	C₂×C₄	1	4	yes	—
83	C₃₁×C₂²⋊C₈	—	2	2	no	—
84	C₄×C₈×C₃₁	—	4	2	no	—
85	C₃₁×#95	—	2	1	no	—
86	C₃₁×#95	—	2	1	no	—
87	C₃₁×#95	—	2	2	no	—
88	C₃₁×#95	—	2	2	no	—
89	C₃₁×#95	D₄	1	1	yes	characteristic
90	C₃₁×#95	D₄	1	1	yes	characteristic
91	C₂⁴×C₃₁	—	2	1	no	—
92	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
93	#41×C₃₁	—	2	8	no	—
94	#41×C₃₁	—	4	4	no	—
95	#41×C₃₁	—	4	8	no	—
96	C₂×Q₈×C₃₁	—	2	1	no	—
97	C₂×Q₈×C₃₁	—	2	2	no	—
98	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
99	C₂×Q₈×C₃₁	#41	1	4	yes	—
100	C₂×D₄×C₃₁	—	2	4	no	—
101	C₂×D₄×C₃₁	—	2	2	no	—
102	C₂×D₄×C₃₁	—	2	1	no	—
103	C₂×D₄×C₃₁	—	2	2	no	—
104	C₂×D₄×C₃₁	—	4	4	no	—
105	C₂×D₄×C₃₁	C₂²×C₄	1	2	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	1	no	—
110	C₂²×C₄×C₃₁	—	2	1	no	—
111	C₂²×C₄×C₃₁	—	2	4	no	—
112	C₂²×C₄×C₃₁	—	2	2	no	—
113	C₂²×C₄×C₃₁	—	2	2	no	—
114	C₂²×C₄×C₃₁	—	4	4	no	—
115	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
116	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
117	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
118	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
119	C₂²×C₄×C₃₁	#41	1	2	yes	—
120	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
121	C₂×C₈×C₃₁	—	2	2	no	—
122	C₂×C₈×C₃₁	—	2	2	no	—
123	C₂×C₈×C₃₁	—	2	2	no	—
124	C₂×C₈×C₃₁	—	2	2	no	—
125	C₂×C₈×C₃₁	—	2	2	no	—
126	C₂×C₈×C₃₁	—	4	4	no	—
127	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
128	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
129	C₄⋊C₄×C₃₁	—	2	1	no	—
130	C₄⋊C₄×C₃₁	—	2	1	no	—
131	C₄⋊C₄×C₃₁	—	2	2	no	—
132	C₄⋊C₄×C₃₁	—	2	2	no	—
133	C₄⋊C₄×C₃₁	—	2	2	no	—
134	C₄⋊C₄×C₃₁	—	2	2	no	—
135	C₄⋊C₄×C₃₁	—	4	4	no	—
136	C₄⋊C₄×C₃₁	—	4	4	no	—
137	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
138	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
139	C₂²⋊C₄×C₃₁	—	2	4	no	—
140	C₂²⋊C₄×C₃₁	—	2	4	no	—
141	C₂²⋊C₄×C₃₁	—	4	4	no	—
142	C₂²⋊C₄×C₃₁	—	8	2	no	—
143	C₂²⋊C₄×C₃₁	—	8	1	no	—
144	C₂²⋊C₄×C₃₁	—	8	1	no	—
145	C₄²×C₃₁	—	2	4	no	—
146	C₄²×C₃₁	—	2	2	no	—
147	C₄²×C₃₁	—	2	2	no	—
148	C₄²×C₃₁	—	4	4	no	—
149	C₄²×C₃₁	—	8	2	no	—
150	#22577	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
151	C₂³×C₃₁	—	2	2	no	—
152	C₂³×C₃₁	—	2	2	no	—
153	C₂³×C₃₁	—	2	1	no	—
154	C₂³×C₃₁	—	2	1	no	—
155	C₂³×C₃₁	—	2	2	no	—
156	C₂³×C₃₁	—	4	2	no	—
157	C₂³×C₃₁	—	4	2	no	—
158	C₂³×C₃₁	—	4	1	no	—
159	C₂³×C₃₁	—	4	1	no	—
160	C₂³×C₃₁	C₄×D₄	1	2	yes	—
161	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
162	Q₈×C₃₁	—	2	2	no	—
163	Q₈×C₃₁	—	2	4	no	—
164	Q₈×C₃₁	#117	1	4	yes	—
165	D₄×C₃₁	—	2	2	no	—
166	D₄×C₃₁	—	2	2	no	—
167	D₄×C₃₁	—	4	8	no	—
168	D₄×C₃₁	—	4	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	4	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	—	2	2	no	—
182	C₂×C₄×C₃₁	—	4	8	no	—
183	C₂×C₄×C₃₁	—	4	2	no	—
184	C₂×C₄×C₃₁	—	4	1	no	—
185	C₂×C₄×C₃₁	—	4	1	no	—
186	C₂×C₄×C₃₁	—	4	2	no	—
187	C₂×C₄×C₃₁	—	4	2	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	C₄×D₄	1	2	yes	—
193	C₂×C₄×C₃₁	#135	1	4	yes	—
194	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
195	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
196	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
197	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
198	C₈×C₃₁	—	2	2	no	—
199	C₈×C₃₁	—	2	2	no	—
200	C₈×C₃₁	—	4	2	no	—
201	C₂×#416	C₂×C₃₁	1	1	yes	characteristic
202	#2347	C₂×C₃₁	1	1	yes	characteristic, complemented
203	#2341	C₂×C₃₁	1	1	yes	characteristic
204	#1934	C₂×C₃₁	1	2	yes	complemented
205	#1926	C₂×C₃₁	1	2	yes	complemented
206	#1865	C₂×C₃₁	1	2	yes	complemented
207	#1825	C₂×C₃₁	1	1	yes	characteristic, complemented
208	#1807	C₂×C₃₁	1	1	yes	characteristic, complemented
209	#1788	C₂×C₃₁	1	1	yes	characteristic
210	C₄×#103	C₂×C₃₁	1	2	yes	complemented
211	C₈×C₂²⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
212	C₂²×C₃₁	—	2	2	no	—
213	C₂²×C₃₁	—	2	2	no	—
214	C₂²×C₃₁	—	2	2	no	—
215	C₂²×C₃₁	—	4	2	no	—
216	C₂²×C₃₁	—	4	2	no	—
217	C₂²×C₃₁	—	4	2	no	—
218	C₂²×C₃₁	—	4	2	no	—
219	C₂²×C₃₁	—	4	2	no	—
220	C₂²×C₃₁	—	8	2	no	—
221	C₂²×C₃₁	—	8	2	no	—
222	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
223	C₂²×C₃₁	#546	1	2	yes	—
224	C₂²×C₃₁	C₂×#135	1	2	yes	—
225	C₂²×C₃₁	#577	1	1	yes	characteristic
226	C₂²×C₃₁	#577	1	1	yes	characteristic
227	C₄×C₃₁	—	2	2	no	—
228	C₄×C₃₁	—	2	4	no	—
229	C₄×C₃₁	—	2	4	no	—
230	C₄×C₃₁	—	4	2	no	—
231	C₄×C₃₁	—	4	2	no	—
232	C₄×C₃₁	—	4	2	no	—
233	C₄×C₃₁	—	8	2	no	—
234	C₄×C₃₁	—	8	2	no	—
235	C₄×C₃₁	C₂×#117	1	1	yes	characteristic
236	C₄×C₃₁	#518	1	1	yes	characteristic
237	C₄×C₃₁	#551	1	2	yes	—
238	C₂²×#41	C₂²×C₃₁	1	1	yes	characteristic
239	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
240	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
241	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
242	C₂×#106	C₂²×C₃₁	1	2	yes	—
243	#416	C₄×C₃₁	1	8	yes	complemented
244	C₂×#103	C₂²×C₃₁	1	2	yes	—
245	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
246	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
247	C₂×#102	C₂²×C₃₁	1	2	yes	—
248	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
249	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
250	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
251	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
252	#390	—	2	2	no	—
253	#390	C₂²×C₃₁	1	2	yes	—
254	#389	C₂²×C₃₁	1	2	yes	—
255	#388	—	2	2	no	—
256	#383	—	2	1	no	—
257	#380	—	2	1	no	—
258	#380	C₂²×C₃₁	1	1	yes	characteristic
259	#380	C₂²×C₃₁	1	1	yes	characteristic
260	C₄×C₄⋊C₄	—	2	1	no	—
261	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
262	C₄×C₂²⋊C₄	—	2	1	no	—
263	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
264	#340	C₂²×C₃₁	1	2	yes	—
265	#340	C₂²×C₃₁	1	2	yes	—
266	#340	C₂²×C₃₁	1	2	yes	—
267	#336	C₂²×C₃₁	1	2	yes	—
268	C₂×C₃₁	—	4	2	no	—
269	C₂×C₃₁	—	8	2	no	—
270	C₂×C₃₁	#2342	1	1	yes	characteristic
271	C₂×C₃₁	C₄×#135	1	1	yes	characteristic
272	C₂×C₃₁	#2952	1	1	yes	characteristic
273	C₂×C₃₁	#3301	1	2	yes	—
274	C₂×C₃₁	#3304	1	2	yes	—
275	C₂×#41	—	2	2	no	—
276	C₂×#41	—	2	4	no	—
277	C₂×#41	—	4	2	no	—
278	C₂×#41	C₂×C₄×C₃₁	1	4	yes	—
279	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
280	C₂²×D₄	—	2	1	no	—
281	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
282	C₂³×C₄	—	2	1	no	—
283	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
284	C₂²×C₈	—	4	2	no	—
285	C₂²×C₈	C₂×C₄×C₃₁	1	4	yes	—
286	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
287	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
288	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
289	C₄×Q₈	—	2	4	no	—
290	C₄×Q₈	—	4	2	no	—
291	#117	—	2	2	no	—
292	#117	C₂×C₄×C₃₁	1	4	yes	—
293	C₂×C₄⋊C₄	—	2	2	no	—
294	C₂×C₄⋊C₄	—	2	2	no	—
295	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
296	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
297	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
298	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
299	C₂×C₂²⋊C₄	—	2	1	no	—
300	C₂×C₂²⋊C₄	—	2	2	no	—
301	C₂×C₂²⋊C₄	—	2	1	no	—
302	C₂×C₂²⋊C₄	—	2	1	no	—
303	C₂×C₂²⋊C₄	—	2	2	no	—
304	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
305	C₂×C₄²	—	2	1	no	—
306	C₂×C₄²	—	2	2	no	—
307	C₂×C₄²	—	2	2	no	—
308	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
309	#106	—	4	2	no	—
310	#103	—	2	4	no	—
311	#103	C₂×C₄×C₃₁	1	4	yes	—
312	#103	C₂×C₄×C₃₁	1	4	yes	—
313	#102	—	4	2	no	—
314	#102	C₂×C₄×C₃₁	1	4	yes	—
315	#102	C₂×C₄×C₃₁	1	4	yes	—
316	C₂²⋊C₈	—	2	2	no	—
317	C₄×C₈	—	4	2	no	—
318	#95	—	2	1	no	—
319	#95	—	2	1	no	—
320	#95	—	2	2	no	—
321	#95	—	2	2	no	—
322	#95	D₄×C₃₁	1	1	yes	characteristic
323	#95	D₄×C₃₁	1	1	yes	characteristic
324	C₃₁	#22577	1	1	yes	characteristic, direct product factor, Sylow
325	C₂⁴	—	2	1	no	—
326	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
327	#41	—	2	8	no	—
328	#41	—	4	4	no	—
329	#41	—	4	8	no	—
330	C₂×Q₈	—	2	1	no	—
331	C₂×Q₈	—	2	2	no	—
332	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
333	C₂×Q₈	#41×C₃₁	1	4	yes	—
334	C₂×D₄	—	2	4	no	—
335	C₂×D₄	—	2	2	no	—
336	C₂×D₄	—	2	1	no	—
337	C₂×D₄	—	2	2	no	—
338	C₂×D₄	—	4	4	no	—
339	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
340	C₂²×C₄	—	2	1	no	—
341	C₂²×C₄	—	2	2	no	—
342	C₂²×C₄	—	2	2	no	—
343	C₂²×C₄	—	2	1	no	—
344	C₂²×C₄	—	2	1	no	—
345	C₂²×C₄	—	2	4	no	—
346	C₂²×C₄	—	2	2	no	—
347	C₂²×C₄	—	2	2	no	—
348	C₂²×C₄	—	4	4	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	1	yes	characteristic
352	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
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	1	no	—
364	C₄⋊C₄	—	2	1	no	—
365	C₄⋊C₄	—	2	2	no	—
366	C₄⋊C₄	—	2	2	no	—
367	C₄⋊C₄	—	2	2	no	—
368	C₄⋊C₄	—	2	2	no	—
369	C₄⋊C₄	—	4	4	no	—
370	C₄⋊C₄	—	4	4	no	—
371	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
372	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
373	C₂²⋊C₄	—	2	4	no	—
374	C₂²⋊C₄	—	2	4	no	—
375	C₂²⋊C₄	—	4	4	no	—
376	C₂²⋊C₄	—	8	2	no	—
377	C₂²⋊C₄	—	8	1	no	—
378	C₂²⋊C₄	—	8	1	no	—
379	C₄²	—	2	4	no	—
380	C₄²	—	2	2	no	—
381	C₄²	—	2	2	no	—
382	C₄²	—	4	4	no	—
383	C₄²	—	8	2	no	—
384	C₂³	—	2	2	no	—
385	C₂³	—	2	2	no	—
386	C₂³	—	2	1	no	—
387	C₂³	—	2	1	no	—
388	C₂³	—	2	2	no	—
389	C₂³	—	4	2	no	—
390	C₂³	—	4	2	no	—
391	C₂³	—	4	1	no	—
392	C₂³	—	4	1	no	—
393	C₂³	C₄×D₄×C₃₁	1	2	yes	—
394	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
395	Q₈	—	2	2	no	—
396	Q₈	—	2	4	no	—
397	Q₈	C₃₁×#117	1	4	yes	—
398	D₄	—	2	2	no	—
399	D₄	—	2	2	no	—
400	D₄	—	4	8	no	—
401	D₄	—	4	2	no	—
402	C₂×C₄	—	2	2	no	—
403	C₂×C₄	—	2	2	no	—
404	C₂×C₄	—	2	4	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	4	no	—
410	C₂×C₄	—	2	2	no	—
411	C₂×C₄	—	2	2	no	—
412	C₂×C₄	—	2	2	no	—
413	C₂×C₄	—	2	2	no	—
414	C₂×C₄	—	2	2	no	—
415	C₂×C₄	—	4	8	no	—
416	C₂×C₄	—	4	2	no	—
417	C₂×C₄	—	4	1	no	—
418	C₂×C₄	—	4	1	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	—	4	2	no	—
424	C₂×C₄	—	4	2	no	—
425	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
426	C₂×C₄	C₃₁×#135	1	4	yes	—
427	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
428	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
429	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
430	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
431	C₈	—	2	2	no	—
432	C₈	—	2	2	no	—
433	C₈	—	4	2	no	—
434	C₂²	—	2	2	no	—
435	C₂²	—	2	2	no	—
436	C₂²	—	2	2	no	—
437	C₂²	—	4	2	no	—
438	C₂²	—	4	2	no	—
439	C₂²	—	4	2	no	—
440	C₂²	—	4	2	no	—
441	C₂²	—	4	2	no	—
442	C₂²	—	8	2	no	—
443	C₂²	—	8	2	no	—
444	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
445	C₂²	C₃₁×#546	1	2	yes	—
446	C₂²	C₂×C₃₁×#135	1	2	yes	—
447	C₂²	C₃₁×#577	1	1	yes	characteristic
448	C₂²	C₃₁×#577	1	1	yes	characteristic
449	C₄	—	2	2	no	—
450	C₄	—	2	4	no	—
451	C₄	—	2	4	no	—
452	C₄	—	4	2	no	—
453	C₄	—	4	2	no	—
454	C₄	—	4	2	no	—
455	C₄	—	8	2	no	—
456	C₄	—	8	2	no	—
457	C₄	C₂×C₃₁×#117	1	1	yes	characteristic
458	C₄	C₃₁×#518	1	1	yes	characteristic
459	C₄	C₃₁×#551	1	2	yes	—
460	C₂	—	4	2	no	—
461	C₂	—	8	2	no	—
462	C₂	C₃₁×#2342	1	1	yes	characteristic
463	C₂	C₄×C₃₁×#135	1	1	yes	characteristic
464	C₂	C₃₁×#2952	1	1	yes	characteristic
465	C₂	C₃₁×#3301	1	2	yes	—
466	C₂	C₃₁×#3304	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	C₂²×#41×C₃₁	C₂²
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
2. C₃₁×#2347	C₂	semidirect
4. C₃₁×#1934	C₂	semidirect
5. C₃₁×#1926	C₂	semidirect
6. C₃₁×#1865	C₂	semidirect
7. C₃₁×#1825	C₂	semidirect
8. C₃₁×#1807	C₂	semidirect
10. C₄×C₃₁×#103	C₂	semidirect
11. C₈×C₂²⋊C₄×C₃₁	C₂	semidirect
17. C₃₁×#416	C₄	semidirect
150. #22577	C₃₁	direct (N × H)
202. #2347	C₂×C₃₁	semidirect
204. #1934	C₂×C₃₁	semidirect
205. #1926	C₂×C₃₁	semidirect
206. #1865	C₂×C₃₁	semidirect
207. #1825	C₂×C₃₁	semidirect
208. #1807	C₂×C₃₁	semidirect
210. C₄×#103	C₂×C₃₁	semidirect
211. C₈×C₂²⋊C₄	C₂×C₃₁	semidirect
243. #416	C₄×C₃₁	semidirect
324. C₃₁	#22577	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×#416	2
C₃₁×#2347	2
C₃₁×#2341	2
C₃₁×#1934	2
C₃₁×#1926	2
C₃₁×#1865	2
C₃₁×#1825	2
C₃₁×#1807	2
C₃₁×#1788	2
C₄×C₃₁×#103	2
C₈×C₂²⋊C₄×C₃₁	2
#22577	31

Subgroups — C31×#22577 #21479336

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22577 #21479336