Subgroups — C₃₁×#36720 #21493479

Order 7936.

2450 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#543	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃₁×#123	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2848	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2687	C₂	1	8	yes	maximal, complemented
5	C₃₁×#2680	C₂	1	8	yes	maximal, complemented
6	C₃₁×#2627	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2561	C₂	1	2	yes	maximal, complemented
8	C₃₁×#2502	C₂	1	2	yes	maximal, complemented
9	C₃₁×#2483	C₂	1	2	yes	maximal, complemented
10	C₃₁×#2405	C₂	1	2	yes	maximal
11	C₃₁×#2312	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂³×D₄×C₃₁	C₂²	1	1	yes	characteristic
13	C₂⁴×C₄×C₃₁	C₂²	1	1	yes	characteristic
14	C₃₁×#543	—	2	8	no	—
15	C₂×C₃₁×#123	—	2	8	no	—
16	C₂×C₃₁×#123	C₂²	1	8	yes	—
17	C₂×C₃₁×#123	C₂²	1	4	yes	—
18	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
19	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
20	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	4	yes	complemented
21	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	4	yes	complemented
22	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	4	yes	complemented
23	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
24	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
25	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
26	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
27	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
28	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
29	C₃₁×#400	C₂²	1	4	yes	complemented
30	C₃₁×#398	C₂²	1	8	yes	complemented
31	C₃₁×#398	C₂²	1	8	yes	complemented
32	C₃₁×#397	C₂²	1	4	yes	complemented
33	C₃₁×#397	C₂²	1	8	yes	complemented
34	C₃₁×#395	C₂²	1	4	yes	complemented
35	C₃₁×#394	C₂²	1	8	yes	—
36	C₃₁×#393	C₂²	1	4	yes	complemented
37	C₃₁×#391	C₂²	1	4	yes	complemented
38	C₃₁×#388	C₂²	1	8	yes	complemented
39	C₃₁×#388	C₂²	1	8	yes	complemented
40	C₃₁×#386	C₂²	1	4	yes	—
41	C₃₁×#385	C₂²	1	4	yes	—
42	C₃₁×#385	C₂²	1	4	yes	—
43	C₃₁×#385	C₂²	1	8	yes	complemented
44	C₃₁×#385	C₂²	1	4	yes	complemented
45	C₃₁×#384	C₂²	1	4	yes	complemented
46	C₃₁×#381	C₂²	1	4	yes	—
47	C₃₁×#381	C₂²	1	4	yes	complemented
48	C₃₁×#380	C₂²	1	2	yes	—
49	C₃₁×#380	C₂²	1	4	yes	complemented
50	C₃₁×#379	C₂²	1	2	yes	—
51	C₃₁×#379	C₂²	1	4	yes	—
52	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
53	C₂⁵×C₃₁	C₂³	1	1	yes	characteristic
54	C₂⁵×C₃₁	C₂³	1	1	yes	characteristic
55	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
56	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
57	C₂²×D₄×C₃₁	—	2	4	no	—
58	C₂²×D₄×C₃₁	—	4	4	no	—
59	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
60	C₂³×C₄×C₃₁	—	2	4	no	—
61	C₂³×C₄×C₃₁	—	4	4	no	—
62	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
63	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
64	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
65	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
66	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
67	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
68	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
69	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
70	C₃₁×#123	—	2	32	no	—
71	C₃₁×#123	—	4	8	no	—
72	C₃₁×C₂²⋊Q₈	—	2	8	no	—
73	C₃₁×C₂²⋊Q₈	—	2	8	no	—
74	C₃₁×C₂²⋊Q₈	—	2	16	no	—
75	C₃₁×C₂²⋊Q₈	—	2	16	no	—
76	C₃₁×C₂²⋊Q₈	—	4	8	no	—
77	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
78	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
79	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
80	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
81	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
82	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
83	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
84	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
85	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
86	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
87	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
88	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
89	C₂×C₂²⋊C₄×C₃₁	—	2	8	no	—
90	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
91	C₂×C₂²⋊C₄×C₃₁	D₄	1	8	yes	—
92	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
93	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
94	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
95	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
96	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
97	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
98	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
99	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
100	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
101	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
102	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
103	C₃₁×#95	C₂³	1	8	yes	—
104	C₃₁×#95	C₂³	1	4	yes	—
105	C₃₁×#95	C₂³	1	4	yes	—
106	C₃₁×#95	C₂³	1	8	yes	—
107	C₃₁×#95	C₂³	1	8	yes	—
108	C₃₁×#95	C₂³	1	8	yes	—
109	C₂⁴×C₃₁	—	2	2	no	—
110	C₂⁴×C₃₁	—	2	4	no	—
111	C₂⁴×C₃₁	—	2	4	no	—
112	C₂⁴×C₃₁	—	2	4	no	—
113	C₂⁴×C₃₁	—	4	2	no	—
114	C₂⁴×C₃₁	—	4	2	no	—
115	C₂⁴×C₃₁	—	4	2	no	—
116	C₂⁴×C₃₁	C₂×D₄	1	4	yes	—
117	C₂⁴×C₃₁	C₂⁴	1	2	yes	—
118	C₂⁴×C₃₁	C₂⁴	1	2	yes	—
119	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
120	C₂×Q₈×C₃₁	—	2	4	no	—
121	C₂×Q₈×C₃₁	—	2	4	no	—
122	C₂×Q₈×C₃₁	—	4	4	no	—
123	C₂×Q₈×C₃₁	—	4	4	no	—
124	C₂×D₄×C₃₁	—	2	8	no	—
125	C₂×D₄×C₃₁	—	2	8	no	—
126	C₂×D₄×C₃₁	—	4	4	no	—
127	C₂×D₄×C₃₁	—	4	16	no	—
128	C₂²×C₄×C₃₁	—	2	2	no	—
129	C₂²×C₄×C₃₁	—	2	2	no	—
130	C₂²×C₄×C₃₁	—	2	2	no	—
131	C₂²×C₄×C₃₁	—	2	8	no	—
132	C₂²×C₄×C₃₁	—	2	4	no	—
133	C₂²×C₄×C₃₁	—	2	4	no	—
134	C₂²×C₄×C₃₁	—	2	4	no	—
135	C₂²×C₄×C₃₁	—	2	8	no	—
136	C₂²×C₄×C₃₁	—	4	2	no	—
137	C₂²×C₄×C₃₁	—	4	4	no	—
138	C₂²×C₄×C₃₁	—	4	2	no	—
139	C₂²×C₄×C₃₁	—	4	2	no	—
140	C₂²×C₄×C₃₁	—	4	2	no	—
141	C₂²×C₄×C₃₁	—	4	4	no	—
142	C₂²×C₄×C₃₁	—	4	4	no	—
143	C₂²×C₄×C₃₁	—	4	16	no	—
144	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
145	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
146	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
147	C₂²×C₄×C₃₁	C₂⁴	1	8	yes	—
148	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
149	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
150	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
151	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
152	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
153	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
154	C₄⋊C₄×C₃₁	—	2	8	no	—
155	C₄⋊C₄×C₃₁	—	2	8	no	—
156	C₄⋊C₄×C₃₁	—	2	4	no	—
157	C₄⋊C₄×C₃₁	—	2	8	no	—
158	C₄⋊C₄×C₃₁	—	2	8	no	—
159	C₄⋊C₄×C₃₁	—	2	4	no	—
160	C₄⋊C₄×C₃₁	—	4	4	no	—
161	C₄⋊C₄×C₃₁	—	4	8	no	—
162	C₄⋊C₄×C₃₁	—	4	8	no	—
163	C₂²⋊C₄×C₃₁	—	2	8	no	—
164	C₂²⋊C₄×C₃₁	—	2	8	no	—
165	C₂²⋊C₄×C₃₁	—	2	8	no	—
166	C₂²⋊C₄×C₃₁	—	2	16	no	—
167	C₂²⋊C₄×C₃₁	—	2	8	no	—
168	C₂²⋊C₄×C₃₁	—	2	8	no	—
169	C₂²⋊C₄×C₃₁	—	4	4	no	—
170	C₂²⋊C₄×C₃₁	—	4	8	no	—
171	C₂²⋊C₄×C₃₁	—	4	4	no	—
172	C₂²⋊C₄×C₃₁	C₂×D₄	1	16	yes	—
173	C₄²×C₃₁	—	4	4	no	—
174	#36720	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
175	C₂³×C₃₁	—	2	2	no	—
176	C₂³×C₃₁	—	2	2	no	—
177	C₂³×C₃₁	—	2	8	no	—
178	C₂³×C₃₁	—	2	2	no	—
179	C₂³×C₃₁	—	2	4	no	—
180	C₂³×C₃₁	—	2	2	no	—
181	C₂³×C₃₁	—	2	4	no	—
182	C₂³×C₃₁	—	2	4	no	—
183	C₂³×C₃₁	—	2	4	no	—
184	C₂³×C₃₁	—	2	4	no	—
185	C₂³×C₃₁	—	2	8	no	—
186	C₂³×C₃₁	—	4	4	no	—
187	C₂³×C₃₁	—	4	4	no	—
188	C₂³×C₃₁	—	4	16	no	—
189	C₂³×C₃₁	—	4	4	no	—
190	C₂³×C₃₁	—	4	4	no	—
191	C₂³×C₃₁	—	8	8	no	—
192	C₂³×C₃₁	C₂²×D₄	1	4	yes	—
193	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
194	C₂³×C₃₁	H(5,𝔽₂)	1	4	yes	—
195	C₂³×C₃₁	2₋¹⁺⁴	1	4	yes	—
196	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
197	Q₈×C₃₁	—	8	4	no	—
198	Q₈×C₃₁	—	8	4	no	—
199	D₄×C₃₁	—	4	16	no	—
200	C₂×C₄×C₃₁	—	2	4	no	—
201	C₂×C₄×C₃₁	—	2	4	no	—
202	C₂×C₄×C₃₁	—	2	2	no	—
203	C₂×C₄×C₃₁	—	2	4	no	—
204	C₂×C₄×C₃₁	—	2	4	no	—
205	C₂×C₄×C₃₁	—	2	2	no	—
206	C₂×C₄×C₃₁	—	2	2	no	—
207	C₂×C₄×C₃₁	—	2	8	no	—
208	C₂×C₄×C₃₁	—	2	8	no	—
209	C₂×C₄×C₃₁	—	2	4	no	—
210	C₂×C₄×C₃₁	—	2	4	no	—
211	C₂×C₄×C₃₁	—	2	8	no	—
212	C₂×C₄×C₃₁	—	2	4	no	—
213	C₂×C₄×C₃₁	—	2	8	no	—
214	C₂×C₄×C₃₁	—	2	8	no	—
215	C₂×C₄×C₃₁	—	4	4	no	—
216	C₂×C₄×C₃₁	—	4	4	no	—
217	C₂×C₄×C₃₁	—	4	16	no	—
218	C₂×C₄×C₃₁	—	4	4	no	—
219	C₂×C₄×C₃₁	—	4	4	no	—
220	C₂×C₄×C₃₁	—	4	4	no	—
221	C₂×C₄×C₃₁	—	8	4	no	—
222	C₂×C₄×C₃₁	—	8	4	no	—
223	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
224	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
225	C₂×#543	C₂×C₃₁	1	1	yes	characteristic, complemented
226	C₂²×#123	C₂×C₃₁	1	1	yes	characteristic
227	#2848	C₂×C₃₁	1	2	yes	complemented
228	#2687	C₂×C₃₁	1	8	yes	complemented
229	#2680	C₂×C₃₁	1	8	yes	complemented
230	#2627	C₂×C₃₁	1	2	yes	complemented
231	#2561	C₂×C₃₁	1	2	yes	complemented
232	#2502	C₂×C₃₁	1	2	yes	complemented
233	#2483	C₂×C₃₁	1	2	yes	complemented
234	#2405	C₂×C₃₁	1	2	yes	—
235	#2312	C₂×C₃₁	1	1	yes	characteristic, complemented
236	C₂²×C₃₁	—	2	2	no	—
237	C₂²×C₃₁	—	2	2	no	—
238	C₂²×C₃₁	—	2	8	no	—
239	C₂²×C₃₁	—	2	4	no	—
240	C₂²×C₃₁	—	2	4	no	—
241	C₂²×C₃₁	—	2	4	no	—
242	C₂²×C₃₁	—	2	4	no	—
243	C₂²×C₃₁	—	2	8	no	—
244	C₂²×C₃₁	—	4	16	no	—
245	C₂²×C₃₁	—	4	2	no	—
246	C₂²×C₃₁	—	4	2	no	—
247	C₂²×C₃₁	—	4	4	no	—
248	C₂²×C₃₁	—	4	8	no	—
249	C₂²×C₃₁	—	8	8	no	—
250	C₂²×C₃₁	#536	1	4	yes	—
251	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
252	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	2	yes	—
253	C₂²×C₃₁	C₂×2₋¹⁺⁴	1	2	yes	—
254	C₂²×C₃₁	#585	1	2	yes	—
255	C₄×C₃₁	—	2	4	no	—
256	C₄×C₃₁	—	2	4	no	—
257	C₄×C₃₁	—	4	2	no	—
258	C₄×C₃₁	—	4	2	no	—
259	C₄×C₃₁	—	4	2	no	—
260	C₄×C₃₁	—	4	4	no	—
261	C₄×C₃₁	—	4	4	no	—
262	C₄×C₃₁	—	4	8	no	—
263	C₂³×D₄	C₂²×C₃₁	1	1	yes	characteristic
264	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
265	#543	—	2	8	no	—
266	C₂×#123	—	2	8	no	—
267	C₂×#123	C₂²×C₃₁	1	8	yes	—
268	C₂×#123	C₂²×C₃₁	1	4	yes	—
269	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
270	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
271	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	4	yes	complemented
272	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	4	yes	complemented
273	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	4	yes	complemented
274	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
275	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
276	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
277	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
278	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
279	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
280	#400	C₂²×C₃₁	1	4	yes	complemented
281	#398	C₂²×C₃₁	1	8	yes	complemented
282	#398	C₂²×C₃₁	1	8	yes	complemented
283	#397	C₂²×C₃₁	1	4	yes	complemented
284	#397	C₂²×C₃₁	1	8	yes	complemented
285	#395	C₂²×C₃₁	1	4	yes	complemented
286	#394	C₂²×C₃₁	1	8	yes	—
287	#393	C₂²×C₃₁	1	4	yes	complemented
288	#391	C₂²×C₃₁	1	4	yes	complemented
289	#388	C₂²×C₃₁	1	8	yes	complemented
290	#388	C₂²×C₃₁	1	8	yes	complemented
291	#386	C₂²×C₃₁	1	4	yes	—
292	#385	C₂²×C₃₁	1	4	yes	—
293	#385	C₂²×C₃₁	1	4	yes	—
294	#385	C₂²×C₃₁	1	8	yes	complemented
295	#385	C₂²×C₃₁	1	4	yes	complemented
296	#384	C₂²×C₃₁	1	4	yes	complemented
297	#381	C₂²×C₃₁	1	4	yes	—
298	#381	C₂²×C₃₁	1	4	yes	complemented
299	#380	C₂²×C₃₁	1	2	yes	—
300	#380	C₂²×C₃₁	1	4	yes	complemented
301	#379	C₂²×C₃₁	1	4	yes	—
302	#379	C₂²×C₃₁	1	2	yes	—
303	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
304	C₂×C₃₁	—	2	4	no	—
305	C₂×C₃₁	—	4	4	no	—
306	C₂×C₃₁	—	4	4	no	—
307	C₂×C₃₁	C₂×#536	1	1	yes	characteristic
308	C₂×C₃₁	#3484	1	1	yes	characteristic
309	C₂×C₃₁	#3485	1	1	yes	characteristic
310	C₂×C₃₁	#3535	1	4	yes	—
311	C₂⁵	C₂³×C₃₁	1	1	yes	characteristic
312	C₂⁵	C₂³×C₃₁	1	1	yes	characteristic
313	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
314	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
315	C₂²×D₄	—	2	4	no	—
316	C₂²×D₄	—	4	4	no	—
317	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
318	C₂³×C₄	—	2	4	no	—
319	C₂³×C₄	—	4	4	no	—
320	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
321	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
322	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
323	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
324	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
325	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
326	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
327	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
328	#123	—	2	32	no	—
329	#123	—	4	8	no	—
330	C₂²⋊Q₈	—	2	8	no	—
331	C₂²⋊Q₈	—	2	8	no	—
332	C₂²⋊Q₈	—	2	16	no	—
333	C₂²⋊Q₈	—	2	16	no	—
334	C₂²⋊Q₈	—	4	8	no	—
335	C₂×C₄⋊C₄	—	2	4	no	—
336	C₂×C₄⋊C₄	—	2	4	no	—
337	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
338	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
339	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
340	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
341	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
342	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
343	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
344	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
345	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
346	C₂×C₂²⋊C₄	—	2	4	no	—
347	C₂×C₂²⋊C₄	—	2	8	no	—
348	C₂×C₂²⋊C₄	—	2	4	no	—
349	C₂×C₂²⋊C₄	D₄×C₃₁	1	8	yes	—
350	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
351	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
352	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
353	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
354	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
355	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
356	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
357	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
358	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
359	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
360	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
361	#95	C₂³×C₃₁	1	4	yes	—
362	#95	C₂³×C₃₁	1	4	yes	—
363	#95	C₂³×C₃₁	1	8	yes	—
364	#95	C₂³×C₃₁	1	8	yes	—
365	#95	C₂³×C₃₁	1	8	yes	—
366	#95	C₂³×C₃₁	1	8	yes	—
367	C₃₁	#36720	1	1	yes	characteristic, direct product factor, Sylow
368	C₂⁴	—	2	2	no	—
369	C₂⁴	—	2	4	no	—
370	C₂⁴	—	2	4	no	—
371	C₂⁴	—	2	4	no	—
372	C₂⁴	—	4	2	no	—
373	C₂⁴	—	4	2	no	—
374	C₂⁴	—	4	2	no	—
375	C₂⁴	C₂×D₄×C₃₁	1	4	yes	—
376	C₂⁴	C₂⁴×C₃₁	1	2	yes	—
377	C₂⁴	C₂⁴×C₃₁	1	2	yes	—
378	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
379	C₂×Q₈	—	2	4	no	—
380	C₂×Q₈	—	2	4	no	—
381	C₂×Q₈	—	4	4	no	—
382	C₂×Q₈	—	4	4	no	—
383	C₂×D₄	—	2	8	no	—
384	C₂×D₄	—	2	8	no	—
385	C₂×D₄	—	4	4	no	—
386	C₂×D₄	—	4	16	no	—
387	C₂²×C₄	—	2	2	no	—
388	C₂²×C₄	—	2	2	no	—
389	C₂²×C₄	—	2	2	no	—
390	C₂²×C₄	—	2	8	no	—
391	C₂²×C₄	—	2	4	no	—
392	C₂²×C₄	—	2	4	no	—
393	C₂²×C₄	—	2	4	no	—
394	C₂²×C₄	—	2	8	no	—
395	C₂²×C₄	—	4	2	no	—
396	C₂²×C₄	—	4	4	no	—
397	C₂²×C₄	—	4	2	no	—
398	C₂²×C₄	—	4	2	no	—
399	C₂²×C₄	—	4	2	no	—
400	C₂²×C₄	—	4	4	no	—
401	C₂²×C₄	—	4	4	no	—
402	C₂²×C₄	—	4	16	no	—
403	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
404	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
405	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
406	C₂²×C₄	C₂⁴×C₃₁	1	8	yes	—
407	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
408	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
409	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
410	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
411	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
412	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
413	C₄⋊C₄	—	2	8	no	—
414	C₄⋊C₄	—	2	8	no	—
415	C₄⋊C₄	—	2	4	no	—
416	C₄⋊C₄	—	2	8	no	—
417	C₄⋊C₄	—	2	4	no	—
418	C₄⋊C₄	—	2	8	no	—
419	C₄⋊C₄	—	4	4	no	—
420	C₄⋊C₄	—	4	8	no	—
421	C₄⋊C₄	—	4	8	no	—
422	C₂²⋊C₄	—	2	8	no	—
423	C₂²⋊C₄	—	2	8	no	—
424	C₂²⋊C₄	—	2	8	no	—
425	C₂²⋊C₄	—	2	16	no	—
426	C₂²⋊C₄	—	2	8	no	—
427	C₂²⋊C₄	—	2	8	no	—
428	C₂²⋊C₄	—	4	4	no	—
429	C₂²⋊C₄	—	4	8	no	—
430	C₂²⋊C₄	—	4	4	no	—
431	C₂²⋊C₄	C₂×D₄×C₃₁	1	16	yes	—
432	C₄²	—	4	4	no	—
433	C₂³	—	2	2	no	—
434	C₂³	—	2	2	no	—
435	C₂³	—	2	8	no	—
436	C₂³	—	2	2	no	—
437	C₂³	—	2	4	no	—
438	C₂³	—	2	2	no	—
439	C₂³	—	2	4	no	—
440	C₂³	—	2	4	no	—
441	C₂³	—	2	4	no	—
442	C₂³	—	2	4	no	—
443	C₂³	—	2	8	no	—
444	C₂³	—	4	4	no	—
445	C₂³	—	4	4	no	—
446	C₂³	—	4	16	no	—
447	C₂³	—	4	4	no	—
448	C₂³	—	4	4	no	—
449	C₂³	—	8	8	no	—
450	C₂³	C₂²×D₄×C₃₁	1	4	yes	—
451	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
452	C₂³	C₃₁×H(5,𝔽₂)	1	4	yes	—
453	C₂³	C₃₁×2₋¹⁺⁴	1	4	yes	—
454	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
455	Q₈	—	8	4	no	—
456	Q₈	—	8	4	no	—
457	D₄	—	4	16	no	—
458	C₂×C₄	—	2	4	no	—
459	C₂×C₄	—	2	4	no	—
460	C₂×C₄	—	2	2	no	—
461	C₂×C₄	—	2	4	no	—
462	C₂×C₄	—	2	4	no	—
463	C₂×C₄	—	2	2	no	—
464	C₂×C₄	—	2	2	no	—
465	C₂×C₄	—	2	8	no	—
466	C₂×C₄	—	2	8	no	—
467	C₂×C₄	—	2	4	no	—
468	C₂×C₄	—	2	4	no	—
469	C₂×C₄	—	2	8	no	—
470	C₂×C₄	—	2	4	no	—
471	C₂×C₄	—	2	8	no	—
472	C₂×C₄	—	2	8	no	—
473	C₂×C₄	—	4	4	no	—
474	C₂×C₄	—	4	4	no	—
475	C₂×C₄	—	4	16	no	—
476	C₂×C₄	—	4	4	no	—
477	C₂×C₄	—	4	4	no	—
478	C₂×C₄	—	4	4	no	—
479	C₂×C₄	—	8	4	no	—
480	C₂×C₄	—	8	4	no	—
481	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
482	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
483	C₂²	—	2	2	no	—
484	C₂²	—	2	2	no	—
485	C₂²	—	2	8	no	—
486	C₂²	—	2	4	no	—
487	C₂²	—	2	4	no	—
488	C₂²	—	2	4	no	—
489	C₂²	—	2	4	no	—
490	C₂²	—	2	8	no	—
491	C₂²	—	4	16	no	—
492	C₂²	—	4	2	no	—
493	C₂²	—	4	2	no	—
494	C₂²	—	4	4	no	—
495	C₂²	—	4	8	no	—
496	C₂²	—	8	8	no	—
497	C₂²	C₃₁×#536	1	4	yes	—
498	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
499	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	2	yes	—
500	C₂²	C₂×C₃₁×2₋¹⁺⁴	1	2	yes	—
501	C₂²	C₃₁×#585	1	2	yes	—
502	C₄	—	2	4	no	—
503	C₄	—	2	4	no	—
504	C₄	—	4	2	no	—
505	C₄	—	4	2	no	—
506	C₄	—	4	2	no	—
507	C₄	—	4	4	no	—
508	C₄	—	4	4	no	—
509	C₄	—	4	8	no	—
510	C₂	—	2	4	no	—
511	C₂	—	4	4	no	—
512	C₂	—	4	4	no	—
513	C₂	C₂×C₃₁×#536	1	1	yes	characteristic
514	C₂	C₃₁×#3484	1	1	yes	characteristic
515	C₂	C₃₁×#3485	1	1	yes	characteristic
516	C₂	C₃₁×#3535	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₃₁
prime radical	C₂³×D₄×C₃₁	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#543	C₂	semidirect
3. C₃₁×#2848	C₂	semidirect
4. C₃₁×#2687	C₂	semidirect
5. C₃₁×#2680	C₂	semidirect
6. C₃₁×#2627	C₂	semidirect
7. C₃₁×#2561	C₂	semidirect
8. C₃₁×#2502	C₂	semidirect
9. C₃₁×#2483	C₂	semidirect
11. C₃₁×#2312	C₂	semidirect
20. C₂×C₃₁×C₂²⋊Q₈	C₂²	semidirect
21. C₂×C₃₁×C₂²⋊Q₈	C₂²	semidirect
22. C₂×C₃₁×C₂²⋊Q₈	C₂²	semidirect
29. C₃₁×#400	C₂²	semidirect
30. C₃₁×#398	C₂²	semidirect
31. C₃₁×#398	C₂²	semidirect
32. C₃₁×#397	C₂²	semidirect
33. C₃₁×#397	C₂²	semidirect
34. C₃₁×#395	C₂²	semidirect
36. C₃₁×#393	C₂²	semidirect
37. C₃₁×#391	C₂²	semidirect
38. C₃₁×#388	C₂²	semidirect
39. C₃₁×#388	C₂²	semidirect
43. C₃₁×#385	C₂²	semidirect
44. C₃₁×#385	C₂²	semidirect
45. C₃₁×#384	C₂²	semidirect
47. C₃₁×#381	C₂²	semidirect
49. C₃₁×#380	C₂²	semidirect
174. #36720	C₃₁	direct (N × H)
225. C₂×#543	C₂×C₃₁	semidirect
227. #2848	C₂×C₃₁	semidirect
228. #2687	C₂×C₃₁	semidirect
229. #2680	C₂×C₃₁	semidirect
230. #2627	C₂×C₃₁	semidirect
231. #2561	C₂×C₃₁	semidirect
232. #2502	C₂×C₃₁	semidirect
233. #2483	C₂×C₃₁	semidirect
235. #2312	C₂×C₃₁	semidirect
271. C₂×C₂²⋊Q₈	C₂²×C₃₁	semidirect
272. C₂×C₂²⋊Q₈	C₂²×C₃₁	semidirect
273. C₂×C₂²⋊Q₈	C₂²×C₃₁	semidirect
280. #400	C₂²×C₃₁	semidirect
281. #398	C₂²×C₃₁	semidirect
282. #398	C₂²×C₃₁	semidirect
283. #397	C₂²×C₃₁	semidirect
284. #397	C₂²×C₃₁	semidirect
285. #395	C₂²×C₃₁	semidirect
287. #393	C₂²×C₃₁	semidirect
288. #391	C₂²×C₃₁	semidirect
289. #388	C₂²×C₃₁	semidirect
290. #388	C₂²×C₃₁	semidirect
294. #385	C₂²×C₃₁	semidirect
295. #385	C₂²×C₃₁	semidirect
296. #384	C₂²×C₃₁	semidirect
298. #381	C₂²×C₃₁	semidirect
300. #380	C₂²×C₃₁	semidirect
367. C₃₁	#36720	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×#543	2
C₂²×C₃₁×#123	2
C₃₁×#2848	2
C₃₁×#2687	2
C₃₁×#2680	2
C₃₁×#2627	2
C₃₁×#2561	2
C₃₁×#2502	2
C₃₁×#2483	2
C₃₁×#2405	2
C₃₁×#2312	2
#36720	31

Subgroups — C31×#36720 #21493479

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#36720 #21493479