Subgroups — C₃₁×#35089 #21491848

Order 7936.

3238 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#534	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃₁×#123	C₂	1	2	yes	maximal, complemented
3	C₂²×C₃₁×#117	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2854	C₂	1	4	yes	maximal, complemented
5	C₃₁×#2618	C₂	1	4	yes	maximal, complemented
6	C₃₁×#2413	C₂	1	8	yes	maximal, complemented
7	C₃₁×#2411	C₂	1	8	yes	maximal, complemented
8	C₃₁×#2315	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#2308	C₂	1	2	yes	maximal, complemented
10	C₂³×D₄×C₃₁	C₂²	1	2	yes	—
11	C₂⁴×C₄×C₃₁	C₂²	1	1	yes	characteristic
12	C₃₁×#534	—	2	8	no	—
13	C₂×C₃₁×#123	—	2	16	no	—
14	C₂×C₃₁×#123	C₂²	1	8	yes	—
15	C₂×C₃₁×#123	C₂²	1	4	yes	—
16	C₂×C₃₁×#123	C₂²	1	4	yes	complemented
17	C₂×C₃₁×#123	C₂²	1	4	yes	complemented
18	C₂×C₃₁×#121	C₂²	1	4	yes	—
19	C₂×C₃₁×#120	C₂²	1	4	yes	—
20	C₂×C₃₁×#117	—	2	8	no	—
21	C₂×C₃₁×#117	C₂²	1	8	yes	complemented
22	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	complemented
23	C₂²×C₄⋊C₄×C₃₁	C₂²	1	4	yes	complemented
24	C₂²×C₂²⋊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	4	yes	complemented
28	C₂²×C₄²×C₃₁	C₂²	1	2	yes	complemented
29	C₃₁×#398	C₂²	1	8	yes	complemented
30	C₃₁×#396	C₂²	1	16	yes	complemented
31	C₃₁×#394	C₂²	1	8	yes	complemented
32	C₃₁×#390	C₂²	1	8	yes	complemented
33	C₃₁×#389	C₂²	1	8	yes	complemented
34	C₃₁×#388	C₂²	1	16	yes	complemented
35	C₃₁×#386	C₂²	1	8	yes	—
36	C₃₁×#386	C₂²	1	8	yes	complemented
37	C₃₁×#385	C₂²	1	8	yes	complemented
38	C₃₁×#381	C₂²	1	4	yes	complemented
39	C₃₁×#380	C₂²	1	4	yes	complemented
40	C₃₁×#379	C₂²	1	4	yes	complemented
41	C₂⁵×C₃₁	C₂³	1	2	yes	—
42	C₂⁵×C₃₁	C₂³	1	1	yes	characteristic
43	C₂²×D₄×C₃₁	—	2	8	no	—
44	C₂²×D₄×C₃₁	—	2	8	no	—
45	C₂²×D₄×C₃₁	—	2	8	no	—
46	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
47	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
48	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
49	C₂³×C₄×C₃₁	—	2	4	no	—
50	C₂³×C₄×C₃₁	—	2	4	no	—
51	C₂³×C₄×C₃₁	D₄	1	8	yes	complemented
52	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
53	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
54	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
55	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
56	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
57	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
58	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
59	C₃₁×#123	—	2	32	no	—
60	C₃₁×#123	—	2	16	no	—
61	C₃₁×#123	—	2	16	no	—
62	C₃₁×#123	—	2	16	no	—
63	C₃₁×#121	—	2	16	no	—
64	C₃₁×#120	—	2	16	no	—
65	C₃₁×#117	—	2	32	no	—
66	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
67	C₂×C₄⋊C₄×C₃₁	—	2	16	no	—
68	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
69	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
70	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
71	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
72	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
73	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
74	C₂×C₂²⋊C₄×C₃₁	—	2	8	no	—
75	C₂×C₂²⋊C₄×C₃₁	—	2	16	no	—
76	C₂×C₂²⋊C₄×C₃₁	D₄	1	8	yes	—
77	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
78	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
79	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
80	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
81	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
82	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
83	C₂×C₂²⋊C₄×C₃₁	C₂³	1	16	yes	—
84	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
85	C₂×C₄²×C₃₁	—	2	8	no	—
86	C₂×C₄²×C₃₁	C₂³	1	8	yes	—
87	C₃₁×#95	C₂³	1	16	yes	—
88	C₃₁×#95	C₂³	1	8	yes	—
89	C₂⁴×C₃₁	—	2	4	no	—
90	C₂⁴×C₃₁	—	2	4	no	—
91	C₂⁴×C₃₁	—	2	4	no	—
92	C₂⁴×C₃₁	—	4	4	no	—
93	C₂⁴×C₃₁	—	4	8	no	—
94	C₂⁴×C₃₁	C₂×D₄	1	4	yes	—
95	C₂⁴×C₃₁	C₂×D₄	1	4	yes	—
96	C₂⁴×C₃₁	#41	1	4	yes	—
97	C₂⁴×C₃₁	C₂⁴	1	4	yes	—
98	C₂⁴×C₃₁	C₂⁴	1	2	yes	—
99	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
100	C₂×D₄×C₃₁	—	2	8	no	—
101	C₂×D₄×C₃₁	—	2	8	no	—
102	C₂×D₄×C₃₁	—	2	8	no	—
103	C₂×D₄×C₃₁	—	2	8	no	—
104	C₂×D₄×C₃₁	—	2	8	no	—
105	C₂×D₄×C₃₁	—	2	8	no	—
106	C₂×D₄×C₃₁	—	2	8	no	—
107	C₂×D₄×C₃₁	—	4	8	no	—
108	C₂×D₄×C₃₁	—	4	32	no	—
109	C₂²×C₄×C₃₁	—	2	32	no	—
110	C₂²×C₄×C₃₁	—	2	2	no	—
111	C₂²×C₄×C₃₁	—	2	2	no	—
112	C₂²×C₄×C₃₁	—	2	8	no	—
113	C₂²×C₄×C₃₁	—	2	8	no	—
114	C₂²×C₄×C₃₁	—	2	2	no	—
115	C₂²×C₄×C₃₁	—	2	2	no	—
116	C₂²×C₄×C₃₁	—	2	4	no	—
117	C₂²×C₄×C₃₁	—	2	8	no	—
118	C₂²×C₄×C₃₁	—	2	4	no	—
119	C₂²×C₄×C₃₁	—	2	4	no	—
120	C₂²×C₄×C₃₁	—	2	4	no	—
121	C₂²×C₄×C₃₁	—	4	4	no	—
122	C₂²×C₄×C₃₁	—	4	4	no	—
123	C₂²×C₄×C₃₁	—	4	8	no	—
124	C₂²×C₄×C₃₁	C₂×D₄	1	16	yes	—
125	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
126	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
127	C₂²×C₄×C₃₁	C₂×D₄	1	8	yes	—
128	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
129	C₂²×C₄×C₃₁	C₂⁴	1	8	yes	—
130	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
131	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
132	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
133	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
134	C₄⋊C₄×C₃₁	—	2	16	no	—
135	C₄⋊C₄×C₃₁	—	2	16	no	—
136	C₄⋊C₄×C₃₁	—	2	16	no	—
137	C₄⋊C₄×C₃₁	—	2	8	no	—
138	C₄⋊C₄×C₃₁	—	4	8	no	—
139	C₂²⋊C₄×C₃₁	—	2	8	no	—
140	C₂²⋊C₄×C₃₁	—	2	8	no	—
141	C₂²⋊C₄×C₃₁	—	2	8	no	—
142	C₂²⋊C₄×C₃₁	—	2	16	no	—
143	C₂²⋊C₄×C₃₁	—	2	16	no	—
144	C₂²⋊C₄×C₃₁	—	2	16	no	—
145	C₂²⋊C₄×C₃₁	—	4	16	no	—
146	C₂²⋊C₄×C₃₁	C₂×D₄	1	16	yes	—
147	C₄²×C₃₁	—	2	16	no	—
148	#35089	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
149	C₂³×C₃₁	—	2	4	no	—
150	C₂³×C₃₁	—	2	8	no	—
151	C₂³×C₃₁	—	2	4	no	—
152	C₂³×C₃₁	—	2	4	no	—
153	C₂³×C₃₁	—	2	4	no	—
154	C₂³×C₃₁	—	2	16	no	—
155	C₂³×C₃₁	—	4	8	no	—
156	C₂³×C₃₁	—	4	8	no	—
157	C₂³×C₃₁	—	4	8	no	—
158	C₂³×C₃₁	—	4	16	no	—
159	C₂³×C₃₁	—	4	8	no	—
160	C₂³×C₃₁	—	8	8	no	—
161	C₂³×C₃₁	—	8	8	no	—
162	C₂³×C₃₁	#121	1	16	yes	—
163	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
164	C₂³×C₃₁	C₂²×D₄	1	4	yes	—
165	C₂³×C₃₁	C₂²×D₄	1	4	yes	—
166	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
167	C₂³×C₃₁	C₂×#41	1	4	yes	—
168	C₂³×C₃₁	C₂×#41	1	2	yes	—
169	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
170	D₄×C₃₁	—	4	16	no	—
171	D₄×C₃₁	—	4	16	no	—
172	D₄×C₃₁	—	8	8	no	—
173	C₂×C₄×C₃₁	—	2	32	no	—
174	C₂×C₄×C₃₁	—	2	8	no	—
175	C₂×C₄×C₃₁	—	2	8	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	8	no	—
181	C₂×C₄×C₃₁	—	2	4	no	—
182	C₂×C₄×C₃₁	—	2	8	no	—
183	C₂×C₄×C₃₁	—	2	4	no	—
184	C₂×C₄×C₃₁	—	2	4	no	—
185	C₂×C₄×C₃₁	—	2	4	no	—
186	C₂×C₄×C₃₁	—	2	8	no	—
187	C₂×C₄×C₃₁	—	2	16	no	—
188	C₂×C₄×C₃₁	—	4	8	no	—
189	C₂×C₄×C₃₁	—	4	16	no	—
190	C₂×C₄×C₃₁	—	4	8	no	—
191	C₂×C₄×C₃₁	—	4	8	no	—
192	C₂×C₄×C₃₁	C₂²×D₄	1	8	yes	—
193	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
194	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
195	C₂×#534	C₂×C₃₁	1	1	yes	characteristic
196	C₂²×#123	C₂×C₃₁	1	2	yes	complemented
197	C₂²×#117	C₂×C₃₁	1	1	yes	characteristic, complemented
198	#2854	C₂×C₃₁	1	4	yes	complemented
199	#2618	C₂×C₃₁	1	4	yes	complemented
200	#2413	C₂×C₃₁	1	8	yes	complemented
201	#2411	C₂×C₃₁	1	8	yes	complemented
202	#2315	C₂×C₃₁	1	1	yes	characteristic, complemented
203	#2308	C₂×C₃₁	1	2	yes	complemented
204	C₂²×C₃₁	—	2	8	no	—
205	C₂²×C₃₁	—	2	4	no	—
206	C₂²×C₃₁	—	2	4	no	—
207	C₂²×C₃₁	—	2	8	no	—
208	C₂²×C₃₁	—	2	8	no	—
209	C₂²×C₃₁	—	2	4	no	—
210	C₂²×C₃₁	—	4	16	no	—
211	C₂²×C₃₁	—	4	4	no	—
212	C₂²×C₃₁	—	4	8	no	—
213	C₂²×C₃₁	—	4	8	no	—
214	C₂²×C₃₁	—	4	8	no	—
215	C₂²×C₃₁	—	8	16	no	—
216	C₂²×C₃₁	C₂×#121	1	8	yes	—
217	C₂²×C₃₁	C₂×#121	1	4	yes	—
218	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
219	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
220	C₂²×C₃₁	C₂²×#41	1	1	yes	characteristic
221	C₂²×C₃₁	#585	1	4	yes	—
222	C₄×C₃₁	—	2	4	no	—
223	C₄×C₃₁	—	2	4	no	—
224	C₄×C₃₁	—	2	8	no	—
225	C₄×C₃₁	—	4	4	no	—
226	C₄×C₃₁	—	4	8	no	—
227	C₄×C₃₁	—	4	4	no	—
228	C₂³×D₄	C₂²×C₃₁	1	2	yes	—
229	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
230	#534	—	2	8	no	—
231	C₂×#123	—	2	16	no	—
232	C₂×#123	C₂²×C₃₁	1	8	yes	—
233	C₂×#123	C₂²×C₃₁	1	4	yes	—
234	C₂×#123	C₂²×C₃₁	1	4	yes	complemented
235	C₂×#123	C₂²×C₃₁	1	4	yes	complemented
236	C₂×#121	C₂²×C₃₁	1	4	yes	—
237	C₂×#120	C₂²×C₃₁	1	4	yes	—
238	C₂×#117	—	2	8	no	—
239	C₂×#117	C₂²×C₃₁	1	8	yes	complemented
240	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	complemented
241	C₂²×C₄⋊C₄	C₂²×C₃₁	1	4	yes	complemented
242	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
243	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
244	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
245	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	complemented
246	C₂²×C₄²	C₂²×C₃₁	1	2	yes	complemented
247	#398	C₂²×C₃₁	1	8	yes	complemented
248	#396	C₂²×C₃₁	1	16	yes	complemented
249	#394	C₂²×C₃₁	1	8	yes	complemented
250	#390	C₂²×C₃₁	1	8	yes	complemented
251	#389	C₂²×C₃₁	1	8	yes	complemented
252	#388	C₂²×C₃₁	1	16	yes	complemented
253	#386	C₂²×C₃₁	1	8	yes	—
254	#386	C₂²×C₃₁	1	8	yes	complemented
255	#385	C₂²×C₃₁	1	8	yes	complemented
256	#381	C₂²×C₃₁	1	4	yes	complemented
257	#380	C₂²×C₃₁	1	4	yes	complemented
258	#379	C₂²×C₃₁	1	4	yes	complemented
259	C₂×C₃₁	—	2	8	no	—
260	C₂×C₃₁	—	2	4	no	—
261	C₂×C₃₁	—	8	4	no	—
262	C₂×C₃₁	C₂²×#121	1	1	yes	characteristic
263	C₂×C₃₁	#3449	1	2	yes	—
264	C₂×C₃₁	#3485	1	2	yes	—
265	C₂×C₃₁	#3491	1	2	yes	—
266	C₂⁵	C₂³×C₃₁	1	2	yes	—
267	C₂⁵	C₂³×C₃₁	1	1	yes	characteristic
268	C₂²×D₄	—	2	8	no	—
269	C₂²×D₄	—	2	8	no	—
270	C₂²×D₄	—	2	8	no	—
271	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
272	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
273	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
274	C₂³×C₄	—	2	4	no	—
275	C₂³×C₄	—	2	4	no	—
276	C₂³×C₄	D₄×C₃₁	1	8	yes	complemented
277	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
278	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
279	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
280	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
281	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
282	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
283	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
284	#123	—	2	32	no	—
285	#123	—	2	16	no	—
286	#123	—	2	16	no	—
287	#123	—	2	16	no	—
288	#121	—	2	16	no	—
289	#120	—	2	16	no	—
290	#117	—	2	32	no	—
291	C₂×C₄⋊C₄	—	2	8	no	—
292	C₂×C₄⋊C₄	—	2	16	no	—
293	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
294	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
295	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
296	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
297	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
298	C₂×C₂²⋊C₄	—	2	4	no	—
299	C₂×C₂²⋊C₄	—	2	8	no	—
300	C₂×C₂²⋊C₄	—	2	16	no	—
301	C₂×C₂²⋊C₄	D₄×C₃₁	1	8	yes	—
302	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
303	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
304	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
305	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
306	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
307	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
308	C₂×C₂²⋊C₄	C₂³×C₃₁	1	16	yes	—
309	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
310	C₂×C₄²	—	2	8	no	—
311	C₂×C₄²	C₂³×C₃₁	1	8	yes	—
312	#95	C₂³×C₃₁	1	8	yes	—
313	#95	C₂³×C₃₁	1	16	yes	—
314	C₃₁	#35089	1	1	yes	characteristic, direct product factor, Sylow
315	C₂⁴	—	2	4	no	—
316	C₂⁴	—	2	4	no	—
317	C₂⁴	—	2	4	no	—
318	C₂⁴	—	4	4	no	—
319	C₂⁴	—	4	8	no	—
320	C₂⁴	C₂×D₄×C₃₁	1	4	yes	—
321	C₂⁴	C₂×D₄×C₃₁	1	4	yes	—
322	C₂⁴	#41×C₃₁	1	4	yes	—
323	C₂⁴	C₂⁴×C₃₁	1	4	yes	—
324	C₂⁴	C₂⁴×C₃₁	1	2	yes	—
325	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
326	C₂×D₄	—	2	8	no	—
327	C₂×D₄	—	2	8	no	—
328	C₂×D₄	—	2	8	no	—
329	C₂×D₄	—	2	8	no	—
330	C₂×D₄	—	2	8	no	—
331	C₂×D₄	—	2	8	no	—
332	C₂×D₄	—	2	8	no	—
333	C₂×D₄	—	4	8	no	—
334	C₂×D₄	—	4	32	no	—
335	C₂²×C₄	—	2	32	no	—
336	C₂²×C₄	—	2	2	no	—
337	C₂²×C₄	—	2	2	no	—
338	C₂²×C₄	—	2	8	no	—
339	C₂²×C₄	—	2	8	no	—
340	C₂²×C₄	—	2	2	no	—
341	C₂²×C₄	—	2	2	no	—
342	C₂²×C₄	—	2	4	no	—
343	C₂²×C₄	—	2	8	no	—
344	C₂²×C₄	—	2	4	no	—
345	C₂²×C₄	—	2	4	no	—
346	C₂²×C₄	—	2	4	no	—
347	C₂²×C₄	—	4	4	no	—
348	C₂²×C₄	—	4	4	no	—
349	C₂²×C₄	—	4	8	no	—
350	C₂²×C₄	C₂×D₄×C₃₁	1	16	yes	—
351	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
352	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
353	C₂²×C₄	C₂×D₄×C₃₁	1	8	yes	—
354	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
355	C₂²×C₄	C₂⁴×C₃₁	1	8	yes	—
356	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
357	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
358	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
359	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
360	C₄⋊C₄	—	2	16	no	—
361	C₄⋊C₄	—	2	16	no	—
362	C₄⋊C₄	—	2	16	no	—
363	C₄⋊C₄	—	2	8	no	—
364	C₄⋊C₄	—	4	8	no	—
365	C₂²⋊C₄	—	2	8	no	—
366	C₂²⋊C₄	—	2	8	no	—
367	C₂²⋊C₄	—	2	8	no	—
368	C₂²⋊C₄	—	2	16	no	—
369	C₂²⋊C₄	—	2	16	no	—
370	C₂²⋊C₄	—	2	16	no	—
371	C₂²⋊C₄	—	4	16	no	—
372	C₂²⋊C₄	C₂×D₄×C₃₁	1	16	yes	—
373	C₄²	—	2	16	no	—
374	C₂³	—	2	4	no	—
375	C₂³	—	2	8	no	—
376	C₂³	—	2	4	no	—
377	C₂³	—	2	4	no	—
378	C₂³	—	2	4	no	—
379	C₂³	—	2	16	no	—
380	C₂³	—	4	8	no	—
381	C₂³	—	4	8	no	—
382	C₂³	—	4	8	no	—
383	C₂³	—	4	16	no	—
384	C₂³	—	4	8	no	—
385	C₂³	—	8	8	no	—
386	C₂³	—	8	8	no	—
387	C₂³	C₃₁×#121	1	16	yes	—
388	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
389	C₂³	C₂²×D₄×C₃₁	1	4	yes	—
390	C₂³	C₂²×D₄×C₃₁	1	4	yes	—
391	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
392	C₂³	C₂×#41×C₃₁	1	4	yes	—
393	C₂³	C₂×#41×C₃₁	1	2	yes	—
394	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
395	D₄	—	4	16	no	—
396	D₄	—	4	16	no	—
397	D₄	—	8	8	no	—
398	C₂×C₄	—	2	32	no	—
399	C₂×C₄	—	2	8	no	—
400	C₂×C₄	—	2	8	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	8	no	—
406	C₂×C₄	—	2	4	no	—
407	C₂×C₄	—	2	8	no	—
408	C₂×C₄	—	2	4	no	—
409	C₂×C₄	—	2	4	no	—
410	C₂×C₄	—	2	4	no	—
411	C₂×C₄	—	2	8	no	—
412	C₂×C₄	—	2	16	no	—
413	C₂×C₄	—	4	8	no	—
414	C₂×C₄	—	4	16	no	—
415	C₂×C₄	—	4	8	no	—
416	C₂×C₄	—	4	8	no	—
417	C₂×C₄	C₂²×D₄×C₃₁	1	8	yes	—
418	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
419	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
420	C₂²	—	2	8	no	—
421	C₂²	—	2	4	no	—
422	C₂²	—	2	4	no	—
423	C₂²	—	2	8	no	—
424	C₂²	—	2	8	no	—
425	C₂²	—	2	4	no	—
426	C₂²	—	4	16	no	—
427	C₂²	—	4	4	no	—
428	C₂²	—	4	8	no	—
429	C₂²	—	4	8	no	—
430	C₂²	—	4	8	no	—
431	C₂²	—	8	16	no	—
432	C₂²	C₂×C₃₁×#121	1	8	yes	—
433	C₂²	C₂×C₃₁×#121	1	4	yes	—
434	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
435	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
436	C₂²	C₂²×#41×C₃₁	1	1	yes	characteristic
437	C₂²	C₃₁×#585	1	4	yes	—
438	C₄	—	2	4	no	—
439	C₄	—	2	4	no	—
440	C₄	—	2	8	no	—
441	C₄	—	4	4	no	—
442	C₄	—	4	8	no	—
443	C₄	—	4	4	no	—
444	C₂	—	2	8	no	—
445	C₂	—	2	4	no	—
446	C₂	—	8	4	no	—
447	C₂	C₂²×C₃₁×#121	1	1	yes	characteristic
448	C₂	C₃₁×#3449	1	2	yes	—
449	C₂	C₃₁×#3485	1	2	yes	—
450	C₂	C₃₁×#3491	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₃₁
prime radical	C₂×C₃₁×#534	C₂
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
2. C₂²×C₃₁×#123	C₂	semidirect
3. C₂²×C₃₁×#117	C₂	semidirect
4. C₃₁×#2854	C₂	semidirect
5. C₃₁×#2618	C₂	semidirect
6. C₃₁×#2413	C₂	semidirect
7. C₃₁×#2411	C₂	semidirect
8. C₃₁×#2315	C₂	semidirect
9. C₃₁×#2308	C₂	semidirect
16. C₂×C₃₁×#123	C₂²	semidirect
17. C₂×C₃₁×#123	C₂²	semidirect
21. C₂×C₃₁×#117	C₂²	semidirect
22. C₂²×C₄⋊C₄×C₃₁	C₂²	semidirect
23. C₂²×C₄⋊C₄×C₃₁	C₂²	semidirect
27. C₂²×C₂²⋊C₄×C₃₁	C₂²	semidirect
28. C₂²×C₄²×C₃₁	C₂²	semidirect
29. C₃₁×#398	C₂²	semidirect
30. C₃₁×#396	C₂²	semidirect
31. C₃₁×#394	C₂²	semidirect
32. C₃₁×#390	C₂²	semidirect
33. C₃₁×#389	C₂²	semidirect
34. C₃₁×#388	C₂²	semidirect
36. C₃₁×#386	C₂²	semidirect
37. C₃₁×#385	C₂²	semidirect
38. C₃₁×#381	C₂²	semidirect
39. C₃₁×#380	C₂²	semidirect
40. C₃₁×#379	C₂²	semidirect
51. C₂³×C₄×C₃₁	D₄	semidirect
148. #35089	C₃₁	direct (N × H)
196. C₂²×#123	C₂×C₃₁	semidirect
197. C₂²×#117	C₂×C₃₁	semidirect
198. #2854	C₂×C₃₁	semidirect
199. #2618	C₂×C₃₁	semidirect
200. #2413	C₂×C₃₁	semidirect
201. #2411	C₂×C₃₁	semidirect
202. #2315	C₂×C₃₁	semidirect
203. #2308	C₂×C₃₁	semidirect
234. C₂×#123	C₂²×C₃₁	semidirect
235. C₂×#123	C₂²×C₃₁	semidirect
239. C₂×#117	C₂²×C₃₁	semidirect
240. C₂²×C₄⋊C₄	C₂²×C₃₁	semidirect
241. C₂²×C₄⋊C₄	C₂²×C₃₁	semidirect
245. C₂²×C₂²⋊C₄	C₂²×C₃₁	semidirect
246. C₂²×C₄²	C₂²×C₃₁	semidirect
247. #398	C₂²×C₃₁	semidirect
248. #396	C₂²×C₃₁	semidirect
249. #394	C₂²×C₃₁	semidirect
250. #390	C₂²×C₃₁	semidirect
251. #389	C₂²×C₃₁	semidirect
252. #388	C₂²×C₃₁	semidirect
254. #386	C₂²×C₃₁	semidirect
255. #385	C₂²×C₃₁	semidirect
256. #381	C₂²×C₃₁	semidirect
257. #380	C₂²×C₃₁	semidirect
258. #379	C₂²×C₃₁	semidirect
276. C₂³×C₄	D₄×C₃₁	semidirect
314. C₃₁	#35089	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×#534	2
C₂²×C₃₁×#123	2
C₂²×C₃₁×#117	2
C₃₁×#2854	2
C₃₁×#2618	2
C₃₁×#2413	2
C₃₁×#2411	2
C₃₁×#2315	2
C₃₁×#2308	2
#35089	31

Subgroups — C31×#35089 #21491848

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#35089 #21491848