Subgroups — C₃₁×#25016 #21481775

Order 7936.

898 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#484	C₂	1	2	yes	maximal, complemented
2	C₃₁×#2538	C₂	1	1	yes	maximal, characteristic
3	D₄×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#2076	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2036	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1954	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#1947	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1852	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1845	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1803	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₃₁×#103	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
13	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₃₁	—	2	1	no	—
15	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
16	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
17	C₃₁×#484	—	2	8	no	—
18	C₂×C₃₁×#107	C₂²	1	2	yes	—
19	C₂×C₃₁×#107	C₂²	1	2	yes	—
20	C₂×C₃₁×#107	C₂²	1	2	yes	—
21	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×#103	C₂²	1	2	yes	—
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	2	yes	—
26	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
28	C₃₁×#396	—	2	2	no	—
29	C₃₁×#390	—	2	1	no	—
30	C₃₁×#389	—	2	1	no	—
31	C₃₁×#389	—	2	1	no	—
32	C₃₁×#385	—	2	2	no	—
33	C₃₁×#384	C₂²	1	1	yes	characteristic
34	C₃₁×#384	C₂²	1	1	yes	characteristic
35	C₃₁×#380	C₂²	1	2	yes	—
36	C₃₁×#380	C₂²	1	2	yes	—
37	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
38	C₃₁×#340	C₂²	1	2	yes	—
39	C₃₁×#340	C₂²	1	1	yes	characteristic
40	C₃₁×#340	C₂²	1	2	yes	—
41	C₃₁×#340	C₂²	1	1	yes	characteristic
42	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
43	C₂²×D₄×C₃₁	D₄	1	1	yes	characteristic
44	C₂³×C₄×C₃₁	—	2	2	no	—
45	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
46	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
47	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
48	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
49	C₃₁×C₂²⋊Q₈	—	2	4	no	—
50	C₃₁×C₂²⋊Q₈	—	4	2	no	—
51	C₄×Q₈×C₃₁	—	2	2	no	—
52	C₄×Q₈×C₃₁	—	2	2	no	—
53	C₄×D₄×C₃₁	—	2	2	no	—
54	C₄×D₄×C₃₁	—	2	2	no	—
55	C₄×D₄×C₃₁	—	4	4	no	—
56	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
57	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
58	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
59	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
60	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
61	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
62	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
63	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
64	C₂×C₄⋊C₄×C₃₁	—	4	2	no	—
65	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
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	2	yes	—
69	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
70	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
71	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
72	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
73	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
74	C₂×C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
75	C₂×C₄²×C₃₁	—	2	1	no	—
76	C₂×C₄²×C₃₁	—	2	1	no	—
77	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
78	C₃₁×#107	—	2	4	no	—
79	C₃₁×#107	—	2	4	no	—
80	C₃₁×#107	—	4	2	no	—
81	C₃₁×C₄⋊C₈	—	2	2	no	—
82	C₃₁×#103	—	2	4	no	—
83	C₃₁×#103	—	2	2	no	—
84	C₃₁×#103	D₄	1	4	yes	complemented
85	C₃₁×C₂²⋊C₈	—	2	4	no	—
86	C₃₁×C₂²⋊C₈	—	2	4	no	—
87	C₄×C₈×C₃₁	—	2	2	no	—
88	C₃₁×#95	—	2	2	no	—
89	C₃₁×#95	—	2	1	no	—
90	C₃₁×#95	—	2	2	no	—
91	C₂⁴×C₃₁	C₂×D₄	1	2	yes	—
92	C₂×Q₈×C₃₁	—	2	1	no	—
93	C₂×Q₈×C₃₁	—	2	1	no	—
94	C₂×Q₈×C₃₁	—	2	1	no	—
95	C₂×Q₈×C₃₁	—	2	1	no	—
96	C₂×Q₈×C₃₁	—	2	1	no	—
97	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
98	C₂×D₄×C₃₁	—	2	2	no	—
99	C₂×D₄×C₃₁	—	2	2	no	—
100	C₂×D₄×C₃₁	Q₁₆	1	4	yes	—
101	C₂²×C₄×C₃₁	—	2	1	no	—
102	C₂²×C₄×C₃₁	—	2	1	no	—
103	C₂²×C₄×C₃₁	—	2	2	no	—
104	C₂²×C₄×C₃₁	—	2	2	no	—
105	C₂²×C₄×C₃₁	—	2	2	no	—
106	C₂²×C₄×C₃₁	—	2	2	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	2	no	—
112	C₂²×C₄×C₃₁	—	2	2	no	—
113	C₂²×C₄×C₃₁	—	4	4	no	—
114	C₂²×C₄×C₃₁	—	4	4	no	—
115	C₂²×C₄×C₃₁	—	8	2	no	—
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	1	yes	characteristic
119	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
120	C₂×C₈×C₃₁	—	2	1	no	—
121	C₂×C₈×C₃₁	—	2	1	no	—
122	C₂×C₈×C₃₁	—	2	1	no	—
123	C₂×C₈×C₃₁	—	2	1	no	—
124	C₂×C₈×C₃₁	—	2	2	no	—
125	C₂×C₈×C₃₁	—	2	2	no	—
126	C₂×C₈×C₃₁	—	2	2	no	—
127	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
128	C₂×C₈×C₃₁	#41	1	2	yes	—
129	C₄⋊C₄×C₃₁	—	2	2	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	1	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	1	no	—
139	C₄⋊C₄×C₃₁	—	4	4	no	—
140	C₄⋊C₄×C₃₁	—	4	4	no	—
141	C₄⋊C₄×C₃₁	—	4	2	no	—
142	C₄⋊C₄×C₃₁	—	4	2	no	—
143	C₄⋊C₄×C₃₁	—	4	2	no	—
144	C₄⋊C₄×C₃₁	—	8	1	no	—
145	C₄⋊C₄×C₃₁	—	8	1	no	—
146	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
147	C₂²⋊C₄×C₃₁	—	2	4	no	—
148	C₂²⋊C₄×C₃₁	—	4	4	no	—
149	C₂²⋊C₄×C₃₁	—	4	4	no	—
150	C₂²⋊C₄×C₃₁	—	8	2	no	—
151	C₄²×C₃₁	—	4	2	no	—
152	C₄²×C₃₁	—	4	2	no	—
153	C₄²×C₃₁	#41	1	2	yes	—
154	C₄²×C₃₁	#41	1	2	yes	—
155	#25016	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
156	C₂³×C₃₁	—	2	2	no	—
157	C₂³×C₃₁	—	2	2	no	—
158	C₂³×C₃₁	—	2	2	no	—
159	C₂³×C₃₁	—	2	2	no	—
160	C₂³×C₃₁	—	2	2	no	—
161	C₂³×C₃₁	—	2	2	no	—
162	C₂³×C₃₁	C₂×Q₁₆	1	4	yes	—
163	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
164	Q₈×C₃₁	—	2	2	no	—
165	Q₈×C₃₁	—	2	2	no	—
166	Q₈×C₃₁	—	4	1	no	—
167	Q₈×C₃₁	—	4	1	no	—
168	D₄×C₃₁	—	2	8	no	—
169	C₂×C₄×C₃₁	—	2	1	no	—
170	C₂×C₄×C₃₁	—	2	1	no	—
171	C₂×C₄×C₃₁	—	2	2	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	—	4	1	no	—
177	C₂×C₄×C₃₁	—	4	1	no	—
178	C₂×C₄×C₃₁	—	4	1	no	—
179	C₂×C₄×C₃₁	—	4	1	no	—
180	C₂×C₄×C₃₁	—	4	2	no	—
181	C₂×C₄×C₃₁	—	4	2	no	—
182	C₂×C₄×C₃₁	—	4	2	no	—
183	C₂×C₄×C₃₁	—	4	2	no	—
184	C₂×C₄×C₃₁	—	4	1	no	—
185	C₂×C₄×C₃₁	—	4	2	no	—
186	C₂×C₄×C₃₁	—	4	2	no	—
187	C₂×C₄×C₃₁	—	4	1	no	—
188	C₂×C₄×C₃₁	—	4	1	no	—
189	C₂×C₄×C₃₁	—	4	2	no	—
190	C₂×C₄×C₃₁	—	4	2	no	—
191	C₂×C₄×C₃₁	—	4	1	no	—
192	C₂×C₄×C₃₁	—	4	2	no	—
193	C₂×C₄×C₃₁	—	4	2	no	—
194	C₂×C₄×C₃₁	—	4	1	no	—
195	C₂×C₄×C₃₁	—	4	1	no	—
196	C₂×C₄×C₃₁	—	8	8	no	—
197	C₂×C₄×C₃₁	#123	1	2	yes	—
198	C₂×C₄×C₃₁	#123	1	2	yes	—
199	C₂×C₄×C₃₁	C₂×Q₁₆	1	2	yes	—
200	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
201	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
202	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
203	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
204	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
205	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
206	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
207	C₈×C₃₁	—	2	2	no	—
208	C₈×C₃₁	—	2	2	no	—
209	C₈×C₃₁	—	4	2	no	—
210	C₂×#484	C₂×C₃₁	1	2	yes	complemented
211	#2538	C₂×C₃₁	1	1	yes	characteristic
212	D₄×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic
213	#2076	C₂×C₃₁	1	2	yes	complemented
214	#2036	C₂×C₃₁	1	2	yes	complemented
215	#1954	C₂×C₃₁	1	1	yes	characteristic
216	#1947	C₂×C₃₁	1	1	yes	characteristic, complemented
217	#1852	C₂×C₃₁	1	2	yes	complemented
218	#1845	C₂×C₃₁	1	1	yes	characteristic, complemented
219	#1803	C₂×C₃₁	1	1	yes	characteristic, complemented
220	C₄×#103	C₂×C₃₁	1	1	yes	characteristic, complemented
221	C₂²×C₃₁	—	2	4	no	—
222	C₂²×C₃₁	—	2	4	no	—
223	C₂²×C₃₁	—	2	4	no	—
224	C₂²×C₃₁	—	4	2	no	—
225	C₂²×C₃₁	—	4	2	no	—
226	C₂²×C₃₁	—	4	2	no	—
227	C₂²×C₃₁	—	4	2	no	—
228	C₂²×C₃₁	C₂×#123	1	1	yes	characteristic
229	C₂²×C₃₁	D₄²	1	1	yes	characteristic
230	C₂²×C₃₁	#547	1	1	yes	characteristic
231	C₂²×C₃₁	C₂²×Q₁₆	1	1	yes	characteristic
232	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
233	C₂²×C₃₁	#576	1	1	yes	characteristic
234	C₂²×C₃₁	#578	1	1	yes	characteristic
235	C₄×C₃₁	—	2	2	no	—
236	C₄×C₃₁	—	4	2	no	—
237	C₄×C₃₁	—	4	2	no	—
238	C₄×C₃₁	—	4	2	no	—
239	C₄×C₃₁	—	8	1	no	—
240	C₄×C₃₁	—	8	1	no	—
241	C₄×C₃₁	—	8	2	no	—
242	C₄×C₃₁	—	8	2	no	—
243	C₄×C₃₁	#484	1	4	yes	complemented
244	C₄×C₃₁	#532	1	1	yes	characteristic
245	C₄×C₃₁	#542	1	1	yes	characteristic
246	C₄×C₃₁	#547	1	1	yes	characteristic
247	C₄×C₃₁	#553	1	1	yes	characteristic
248	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
249	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
250	C₂×C₄×D₄	—	2	1	no	—
251	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
252	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
253	#484	—	2	8	no	—
254	C₂×#107	C₂²×C₃₁	1	2	yes	—
255	C₂×#107	C₂²×C₃₁	1	2	yes	—
256	C₂×#107	C₂²×C₃₁	1	2	yes	—
257	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
258	C₂×#103	C₂²×C₃₁	1	2	yes	—
259	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
260	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
261	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
262	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
263	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
264	#396	—	2	2	no	—
265	#390	—	2	1	no	—
266	#389	—	2	1	no	—
267	#389	—	2	1	no	—
268	#385	—	2	2	no	—
269	#384	C₂²×C₃₁	1	1	yes	characteristic
270	#384	C₂²×C₃₁	1	1	yes	characteristic
271	#380	C₂²×C₃₁	1	2	yes	—
272	#380	C₂²×C₃₁	1	2	yes	—
273	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
274	#340	C₂²×C₃₁	1	2	yes	—
275	#340	C₂²×C₃₁	1	1	yes	characteristic
276	#340	C₂²×C₃₁	1	2	yes	—
277	#340	C₂²×C₃₁	1	1	yes	characteristic
278	C₂×C₃₁	—	4	4	no	—
279	C₂×C₃₁	#2422	1	1	yes	characteristic
280	C₂×C₃₁	C₂×#484	1	1	yes	characteristic
281	C₂×C₃₁	#3098	1	1	yes	characteristic
282	C₂×C₃₁	#3274	1	1	yes	characteristic
283	C₂×C₃₁	D₄×Q₁₆	1	1	yes	characteristic
284	C₂×C₃₁	#3338	1	1	yes	characteristic
285	C₂×C₃₁	#3353	1	1	yes	characteristic
286	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
287	C₂²×D₄	D₄×C₃₁	1	1	yes	characteristic
288	C₂³×C₄	—	2	2	no	—
289	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
290	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
291	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
292	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
293	C₂²⋊Q₈	—	2	4	no	—
294	C₂²⋊Q₈	—	4	2	no	—
295	C₄×Q₈	—	2	2	no	—
296	C₄×Q₈	—	2	2	no	—
297	C₄×D₄	—	2	2	no	—
298	C₄×D₄	—	2	2	no	—
299	C₄×D₄	—	4	4	no	—
300	C₂×C₄⋊C₄	—	2	4	no	—
301	C₂×C₄⋊C₄	—	2	2	no	—
302	C₂×C₄⋊C₄	—	2	1	no	—
303	C₂×C₄⋊C₄	—	2	1	no	—
304	C₂×C₄⋊C₄	—	2	2	no	—
305	C₂×C₄⋊C₄	—	2	1	no	—
306	C₂×C₄⋊C₄	—	2	1	no	—
307	C₂×C₄⋊C₄	—	2	1	no	—
308	C₂×C₄⋊C₄	—	4	2	no	—
309	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
310	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
311	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
312	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
313	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
314	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
315	C₂×C₂²⋊C₄	—	2	2	no	—
316	C₂×C₂²⋊C₄	—	2	2	no	—
317	C₂×C₂²⋊C₄	—	2	2	no	—
318	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
319	C₂×C₄²	—	2	1	no	—
320	C₂×C₄²	—	2	1	no	—
321	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
322	#107	—	2	4	no	—
323	#107	—	2	4	no	—
324	#107	—	4	2	no	—
325	C₄⋊C₈	—	2	2	no	—
326	#103	—	2	4	no	—
327	#103	—	2	2	no	—
328	#103	D₄×C₃₁	1	4	yes	complemented
329	C₂²⋊C₈	—	2	4	no	—
330	C₂²⋊C₈	—	2	4	no	—
331	C₄×C₈	—	2	2	no	—
332	#95	—	2	1	no	—
333	#95	—	2	2	no	—
334	#95	—	2	2	no	—
335	C₃₁	#25016	1	1	yes	characteristic, direct product factor, Sylow
336	C₂⁴	C₂×D₄×C₃₁	1	2	yes	—
337	C₂×Q₈	—	2	1	no	—
338	C₂×Q₈	—	2	1	no	—
339	C₂×Q₈	—	2	1	no	—
340	C₂×Q₈	—	2	1	no	—
341	C₂×Q₈	—	2	1	no	—
342	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
343	C₂×D₄	—	2	2	no	—
344	C₂×D₄	—	2	2	no	—
345	C₂×D₄	Q₁₆×C₃₁	1	4	yes	—
346	C₂²×C₄	—	2	1	no	—
347	C₂²×C₄	—	2	1	no	—
348	C₂²×C₄	—	2	2	no	—
349	C₂²×C₄	—	2	2	no	—
350	C₂²×C₄	—	2	2	no	—
351	C₂²×C₄	—	2	2	no	—
352	C₂²×C₄	—	2	2	no	—
353	C₂²×C₄	—	2	2	no	—
354	C₂²×C₄	—	2	1	no	—
355	C₂²×C₄	—	2	1	no	—
356	C₂²×C₄	—	2	2	no	—
357	C₂²×C₄	—	2	2	no	—
358	C₂²×C₄	—	4	4	no	—
359	C₂²×C₄	—	4	4	no	—
360	C₂²×C₄	—	8	2	no	—
361	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
362	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
363	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
364	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
365	C₂×C₈	—	2	1	no	—
366	C₂×C₈	—	2	1	no	—
367	C₂×C₈	—	2	1	no	—
368	C₂×C₈	—	2	1	no	—
369	C₂×C₈	—	2	2	no	—
370	C₂×C₈	—	2	2	no	—
371	C₂×C₈	—	2	2	no	—
372	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
373	C₂×C₈	#41×C₃₁	1	2	yes	—
374	C₄⋊C₄	—	2	2	no	—
375	C₄⋊C₄	—	2	1	no	—
376	C₄⋊C₄	—	2	2	no	—
377	C₄⋊C₄	—	2	2	no	—
378	C₄⋊C₄	—	2	1	no	—
379	C₄⋊C₄	—	2	1	no	—
380	C₄⋊C₄	—	2	1	no	—
381	C₄⋊C₄	—	2	2	no	—
382	C₄⋊C₄	—	2	2	no	—
383	C₄⋊C₄	—	2	1	no	—
384	C₄⋊C₄	—	4	4	no	—
385	C₄⋊C₄	—	4	4	no	—
386	C₄⋊C₄	—	4	2	no	—
387	C₄⋊C₄	—	4	2	no	—
388	C₄⋊C₄	—	4	2	no	—
389	C₄⋊C₄	—	8	1	no	—
390	C₄⋊C₄	—	8	1	no	—
391	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
392	C₂²⋊C₄	—	2	4	no	—
393	C₂²⋊C₄	—	4	4	no	—
394	C₂²⋊C₄	—	4	4	no	—
395	C₂²⋊C₄	—	8	2	no	—
396	C₄²	—	4	2	no	—
397	C₄²	—	4	2	no	—
398	C₄²	#41×C₃₁	1	2	yes	—
399	C₄²	#41×C₃₁	1	2	yes	—
400	C₂³	—	2	2	no	—
401	C₂³	—	2	2	no	—
402	C₂³	—	2	2	no	—
403	C₂³	—	2	2	no	—
404	C₂³	—	2	2	no	—
405	C₂³	—	2	2	no	—
406	C₂³	C₂×Q₁₆×C₃₁	1	4	yes	—
407	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
408	Q₈	—	2	2	no	—
409	Q₈	—	2	2	no	—
410	Q₈	—	4	1	no	—
411	Q₈	—	4	1	no	—
412	D₄	—	2	8	no	—
413	C₂×C₄	—	2	1	no	—
414	C₂×C₄	—	2	1	no	—
415	C₂×C₄	—	2	2	no	—
416	C₂×C₄	—	2	2	no	—
417	C₂×C₄	—	2	2	no	—
418	C₂×C₄	—	2	2	no	—
419	C₂×C₄	—	2	2	no	—
420	C₂×C₄	—	4	1	no	—
421	C₂×C₄	—	4	1	no	—
422	C₂×C₄	—	4	1	no	—
423	C₂×C₄	—	4	1	no	—
424	C₂×C₄	—	4	2	no	—
425	C₂×C₄	—	4	2	no	—
426	C₂×C₄	—	4	2	no	—
427	C₂×C₄	—	4	2	no	—
428	C₂×C₄	—	4	1	no	—
429	C₂×C₄	—	4	2	no	—
430	C₂×C₄	—	4	2	no	—
431	C₂×C₄	—	4	1	no	—
432	C₂×C₄	—	4	1	no	—
433	C₂×C₄	—	4	2	no	—
434	C₂×C₄	—	4	2	no	—
435	C₂×C₄	—	4	1	no	—
436	C₂×C₄	—	4	2	no	—
437	C₂×C₄	—	4	2	no	—
438	C₂×C₄	—	4	1	no	—
439	C₂×C₄	—	4	1	no	—
440	C₂×C₄	—	8	8	no	—
441	C₂×C₄	C₃₁×#123	1	2	yes	—
442	C₂×C₄	C₃₁×#123	1	2	yes	—
443	C₂×C₄	C₂×Q₁₆×C₃₁	1	2	yes	—
444	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
445	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
446	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
447	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
448	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
449	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
450	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
451	C₈	—	2	2	no	—
452	C₈	—	2	2	no	—
453	C₈	—	4	2	no	—
454	C₂²	—	2	4	no	—
455	C₂²	—	2	4	no	—
456	C₂²	—	2	4	no	—
457	C₂²	—	4	2	no	—
458	C₂²	—	4	2	no	—
459	C₂²	—	4	2	no	—
460	C₂²	—	4	2	no	—
461	C₂²	C₂×C₃₁×#123	1	1	yes	characteristic
462	C₂²	D₄²×C₃₁	1	1	yes	characteristic
463	C₂²	C₃₁×#547	1	1	yes	characteristic
464	C₂²	C₂²×Q₁₆×C₃₁	1	1	yes	characteristic
465	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
466	C₂²	C₃₁×#576	1	1	yes	characteristic
467	C₂²	C₃₁×#578	1	1	yes	characteristic
468	C₄	—	2	2	no	—
469	C₄	—	4	2	no	—
470	C₄	—	4	2	no	—
471	C₄	—	4	2	no	—
472	C₄	—	8	1	no	—
473	C₄	—	8	1	no	—
474	C₄	—	8	2	no	—
475	C₄	—	8	2	no	—
476	C₄	C₃₁×#484	1	4	yes	complemented
477	C₄	C₃₁×#532	1	1	yes	characteristic
478	C₄	C₃₁×#542	1	1	yes	characteristic
479	C₄	C₃₁×#547	1	1	yes	characteristic
480	C₄	C₃₁×#553	1	1	yes	characteristic
481	C₂	—	4	4	no	—
482	C₂	C₃₁×#2422	1	1	yes	characteristic
483	C₂	C₂×C₃₁×#484	1	1	yes	characteristic
484	C₂	C₃₁×#3098	1	1	yes	characteristic
485	C₂	C₃₁×#3274	1	1	yes	characteristic
486	C₂	D₄×Q₁₆×C₃₁	1	1	yes	characteristic
487	C₂	C₃₁×#3338	1	1	yes	characteristic
488	C₂	C₃₁×#3353	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#484	C₂	semidirect
4. C₃₁×#2076	C₂	semidirect
5. C₃₁×#2036	C₂	semidirect
7. C₃₁×#1947	C₂	semidirect
8. C₃₁×#1852	C₂	semidirect
9. C₃₁×#1845	C₂	semidirect
10. C₃₁×#1803	C₂	semidirect
11. C₄×C₃₁×#103	C₂	semidirect
84. C₃₁×#103	D₄	semidirect
155. #25016	C₃₁	direct (N × H)
210. C₂×#484	C₂×C₃₁	semidirect
213. #2076	C₂×C₃₁	semidirect
214. #2036	C₂×C₃₁	semidirect
216. #1947	C₂×C₃₁	semidirect
217. #1852	C₂×C₃₁	semidirect
218. #1845	C₂×C₃₁	semidirect
219. #1803	C₂×C₃₁	semidirect
220. C₄×#103	C₂×C₃₁	semidirect
243. C₄×C₃₁	#484	semidirect
328. #103	D₄×C₃₁	semidirect
335. C₃₁	#25016	direct (N × H)
476. C₄	C₃₁×#484	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#484	2
C₃₁×#2538	2
D₄×C₄⋊C₄×C₃₁	2
C₃₁×#2076	2
C₃₁×#2036	2
C₃₁×#1954	2
C₃₁×#1947	2
C₃₁×#1852	2
C₃₁×#1845	2
C₃₁×#1803	2
C₄×C₃₁×#103	2
#25016	31

Subgroups — C31×#25016 #21481775

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#25016 #21481775