Subgroups — C₃₁×#13528 #21470287

Order 7936.

378 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#367	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#366	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#357	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃₁×#357	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1970	C₂	1	1	yes	maximal, characteristic
6	C₃₁×#1846	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1374	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
9	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
10	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
11	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
12	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×#106	—	2	1	no	—
15	C₂×C₃₁×#102	—	2	1	no	—
16	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
17	C₃₁×#390	—	2	1	no	—
18	C₃₁×#382	—	2	1	no	—
19	C₃₁×#367	—	2	2	no	—
20	C₃₁×#366	—	2	2	no	—
21	C₃₁×#357	—	2	2	no	—
22	C₃₁×#357	—	2	2	no	—
23	C₂²×D₄×C₃₁	—	2	1	no	—
24	C₂×D₈×C₃₁	—	2	1	no	—
25	C₂×D₈×C₃₁	—	2	1	no	—
26	C₂×D₈×C₃₁	—	2	1	no	—
27	C₂×D₈×C₃₁	—	2	1	no	—
28	C₂×D₈×C₃₁	—	2	1	no	—
29	C₂×D₈×C₃₁	Q₈	1	2	yes	—
30	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
32	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
33	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
34	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
35	C₂×C₂²⋊C₄×C₃₁	—	4	2	no	—
36	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
37	C₂×C₁₆×C₃₁	—	2	1	no	—
38	C₂×C₁₆×C₃₁	—	2	1	no	—
39	C₂×C₁₆×C₃₁	—	2	1	no	—
40	C₂×C₁₆×C₃₁	—	2	1	no	—
41	C₂×C₁₆×C₃₁	D₄	1	2	yes	—
42	C₂×C₁₆×C₃₁	D₄	1	2	yes	—
43	C₃₁×#107	—	2	1	no	—
44	C₃₁×#107	—	2	1	no	—
45	C₃₁×#107	—	2	1	no	—
46	C₃₁×#107	—	2	1	no	—
47	C₃₁×#107	—	2	1	no	—
48	C₃₁×#107	—	2	1	no	—
49	C₃₁×#106	—	4	2	no	—
50	C₃₁×#102	—	4	2	no	—
51	C₄×C₈×C₃₁	—	2	1	no	—
52	C₄×C₈×C₃₁	—	2	1	no	—
53	C₃₁×#95	—	4	2	no	—
54	C₂⁴×C₃₁	—	4	1	no	—
55	C₂×D₄×C₃₁	—	2	2	no	—
56	C₂×D₄×C₃₁	—	4	1	no	—
57	C₂×D₄×C₃₁	—	4	1	no	—
58	C₂×D₄×C₃₁	—	4	1	no	—
59	C₂×D₄×C₃₁	—	4	1	no	—
60	C₂×D₄×C₃₁	—	4	1	no	—
61	C₂²×C₄×C₃₁	—	4	1	no	—
62	C₂²×C₄×C₃₁	—	4	1	no	—
63	C₂²×C₄×C₃₁	—	4	1	no	—
64	C₂²×C₄×C₃₁	D₈	1	2	yes	—
65	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
66	D₈×C₃₁	—	2	2	no	—
67	D₈×C₃₁	—	2	2	no	—
68	D₈×C₃₁	—	4	1	no	—
69	D₈×C₃₁	—	4	1	no	—
70	C₂×C₈×C₃₁	—	2	1	no	—
71	C₂×C₈×C₃₁	—	2	1	no	—
72	C₂×C₈×C₃₁	D₈	1	2	yes	—
73	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₃₁	C₂×Q₈	1	1	yes	characteristic
76	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
77	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
78	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
79	C₄⋊C₄×C₃₁	—	4	1	no	—
80	C₄⋊C₄×C₃₁	—	4	1	no	—
81	C₄⋊C₄×C₃₁	—	4	1	no	—
82	C₄⋊C₄×C₃₁	—	4	1	no	—
83	C₄⋊C₄×C₃₁	—	4	1	no	—
84	C₄⋊C₄×C₃₁	—	4	1	no	—
85	C₂²⋊C₄×C₃₁	—	8	4	no	—
86	C₄²×C₃₁	—	2	1	no	—
87	C₄²×C₃₁	—	2	1	no	—
88	C₁₆×C₃₁	—	2	2	no	—
89	C₁₆×C₃₁	—	2	2	no	—
90	#13528	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
91	C₂³×C₃₁	—	4	2	no	—
92	C₂³×C₃₁	—	8	1	no	—
93	C₂³×C₃₁	—	8	1	no	—
94	C₂³×C₃₁	—	8	2	no	—
95	C₂³×C₃₁	—	8	2	no	—
96	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
97	D₄×C₃₁	—	4	2	no	—
98	D₄×C₃₁	—	4	2	no	—
99	D₄×C₃₁	—	8	1	no	—
100	D₄×C₃₁	—	8	1	no	—
101	C₂×C₄×C₃₁	—	2	2	no	—
102	C₂×C₄×C₃₁	—	2	2	no	—
103	C₂×C₄×C₃₁	—	2	2	no	—
104	C₂×C₄×C₃₁	—	8	1	no	—
105	C₂×C₄×C₃₁	—	8	1	no	—
106	C₂×C₄×C₃₁	—	8	1	no	—
107	C₂×C₄×C₃₁	—	8	2	no	—
108	C₂×C₄×C₃₁	—	8	2	no	—
109	C₂×C₄×C₃₁	—	8	2	no	—
110	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
111	C₂×C₄×C₃₁	#123	1	1	yes	characteristic
112	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
113	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
114	C₂×C₄×C₃₁	#137	1	1	yes	characteristic
115	C₂×C₄×C₃₁	#137	1	1	yes	characteristic
116	C₈×C₃₁	—	2	2	no	—
117	C₈×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
118	C₈×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
119	C₈×C₃₁	#123	1	1	yes	characteristic
120	C₈×C₃₁	#123	1	1	yes	characteristic
121	C₂×#367	C₂×C₃₁	1	1	yes	characteristic, complemented
122	C₂×#366	C₂×C₃₁	1	1	yes	characteristic, complemented
123	C₂×#357	C₂×C₃₁	1	1	yes	characteristic
124	C₂×#357	C₂×C₃₁	1	1	yes	characteristic
125	#1970	C₂×C₃₁	1	1	yes	characteristic
126	#1846	C₂×C₃₁	1	1	yes	characteristic, complemented
127	#1374	C₂×C₃₁	1	1	yes	characteristic, complemented
128	C₂²×C₃₁	—	8	2	no	—
129	C₂²×C₃₁	—	8	4	no	—
130	C₂²×C₃₁	—	16	2	no	—
131	C₂²×C₃₁	—	16	1	no	—
132	C₂²×C₃₁	—	16	1	no	—
133	C₂²×C₃₁	#474	1	1	yes	characteristic
134	C₂²×C₃₁	#480	1	1	yes	characteristic
135	C₂²×C₃₁	#493	1	1	yes	characteristic
136	C₂²×C₃₁	#508	1	2	yes	—
137	C₂²×C₃₁	#509	1	2	yes	—
138	C₄×C₃₁	—	4	2	no	—
139	C₄×C₃₁	—	16	1	no	—
140	C₄×C₃₁	—	16	1	no	—
141	C₄×C₃₁	—	16	1	no	—
142	C₄×C₃₁	#399	1	1	yes	characteristic
143	C₄×C₃₁	#474	1	1	yes	characteristic
144	C₄×C₃₁	#480	1	1	yes	characteristic
145	C₄×C₃₁	#497	1	1	yes	characteristic
146	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
147	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
148	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
149	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
150	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
151	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
152	C₂×#106	—	2	1	no	—
153	C₂×#102	—	2	1	no	—
154	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
155	#390	—	2	1	no	—
156	#382	—	2	1	no	—
157	#367	—	2	2	no	—
158	#366	—	2	2	no	—
159	#357	—	2	2	no	—
160	#357	—	2	2	no	—
161	C₂×C₃₁	—	16	2	no	—
162	C₂×C₃₁	#2093	1	1	yes	characteristic
163	C₂×C₃₁	#2228	1	2	yes	—
164	C₂×C₃₁	#2234	1	2	yes	—
165	C₂×C₃₁	#2249	1	1	yes	characteristic
166	C₂×C₃₁	#2250	1	1	yes	characteristic
167	C₂²×D₄	—	2	1	no	—
168	C₂×D₈	—	2	1	no	—
169	C₂×D₈	—	2	1	no	—
170	C₂×D₈	—	2	1	no	—
171	C₂×D₈	—	2	1	no	—
172	C₂×D₈	—	2	1	no	—
173	C₂×D₈	Q₈×C₃₁	1	2	yes	—
174	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
175	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
176	C₂×C₄⋊C₄	—	2	1	no	—
177	C₂×C₄⋊C₄	—	2	1	no	—
178	C₂×C₄⋊C₄	—	2	1	no	—
179	C₂×C₂²⋊C₄	—	4	2	no	—
180	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
181	C₂×C₁₆	—	2	1	no	—
182	C₂×C₁₆	—	2	1	no	—
183	C₂×C₁₆	—	2	1	no	—
184	C₂×C₁₆	—	2	1	no	—
185	C₂×C₁₆	D₄×C₃₁	1	2	yes	—
186	C₂×C₁₆	D₄×C₃₁	1	2	yes	—
187	#107	—	2	1	no	—
188	#107	—	2	1	no	—
189	#107	—	2	1	no	—
190	#107	—	2	1	no	—
191	#107	—	2	1	no	—
192	#107	—	2	1	no	—
193	#106	—	4	2	no	—
194	#102	—	4	2	no	—
195	C₄×C₈	—	2	1	no	—
196	C₄×C₈	—	2	1	no	—
197	#95	—	4	2	no	—
198	C₃₁	#13528	1	1	yes	characteristic, direct product factor, Sylow
199	C₂⁴	—	4	1	no	—
200	C₂×D₄	—	2	2	no	—
201	C₂×D₄	—	4	1	no	—
202	C₂×D₄	—	4	1	no	—
203	C₂×D₄	—	4	1	no	—
204	C₂×D₄	—	4	1	no	—
205	C₂×D₄	—	4	1	no	—
206	C₂²×C₄	—	4	1	no	—
207	C₂²×C₄	—	4	1	no	—
208	C₂²×C₄	—	4	1	no	—
209	C₂²×C₄	D₈×C₃₁	1	2	yes	—
210	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
211	D₈	—	2	2	no	—
212	D₈	—	2	2	no	—
213	D₈	—	4	1	no	—
214	D₈	—	4	1	no	—
215	C₂×C₈	—	2	1	no	—
216	C₂×C₈	—	2	1	no	—
217	C₂×C₈	D₈×C₃₁	1	2	yes	—
218	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
219	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
220	C₂×C₈	C₂×Q₈×C₃₁	1	1	yes	characteristic
221	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
222	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
223	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
224	C₄⋊C₄	—	4	1	no	—
225	C₄⋊C₄	—	4	1	no	—
226	C₄⋊C₄	—	4	1	no	—
227	C₄⋊C₄	—	4	1	no	—
228	C₄⋊C₄	—	4	1	no	—
229	C₄⋊C₄	—	4	1	no	—
230	C₂²⋊C₄	—	8	4	no	—
231	C₄²	—	2	1	no	—
232	C₄²	—	2	1	no	—
233	C₁₆	—	2	2	no	—
234	C₁₆	—	2	2	no	—
235	C₂³	—	4	2	no	—
236	C₂³	—	8	1	no	—
237	C₂³	—	8	1	no	—
238	C₂³	—	8	2	no	—
239	C₂³	—	8	2	no	—
240	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
241	D₄	—	4	2	no	—
242	D₄	—	4	2	no	—
243	D₄	—	8	1	no	—
244	D₄	—	8	1	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	8	1	no	—
249	C₂×C₄	—	8	1	no	—
250	C₂×C₄	—	8	1	no	—
251	C₂×C₄	—	8	2	no	—
252	C₂×C₄	—	8	2	no	—
253	C₂×C₄	—	8	2	no	—
254	C₂×C₄	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
255	C₂×C₄	C₃₁×#123	1	1	yes	characteristic
256	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
257	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
258	C₂×C₄	C₃₁×#137	1	1	yes	characteristic
259	C₂×C₄	C₃₁×#137	1	1	yes	characteristic
260	C₈	—	2	2	no	—
261	C₈	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
262	C₈	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
263	C₈	C₃₁×#123	1	1	yes	characteristic
264	C₈	C₃₁×#123	1	1	yes	characteristic
265	C₂²	—	8	2	no	—
266	C₂²	—	8	4	no	—
267	C₂²	—	16	2	no	—
268	C₂²	—	16	1	no	—
269	C₂²	—	16	1	no	—
270	C₂²	C₃₁×#474	1	1	yes	characteristic
271	C₂²	C₃₁×#480	1	1	yes	characteristic
272	C₂²	C₃₁×#493	1	1	yes	characteristic
273	C₂²	C₃₁×#508	1	2	yes	—
274	C₂²	C₃₁×#509	1	2	yes	—
275	C₄	—	4	2	no	—
276	C₄	—	16	1	no	—
277	C₄	—	16	1	no	—
278	C₄	—	16	1	no	—
279	C₄	C₃₁×#399	1	1	yes	characteristic
280	C₄	C₃₁×#474	1	1	yes	characteristic
281	C₄	C₃₁×#480	1	1	yes	characteristic
282	C₄	C₃₁×#497	1	1	yes	characteristic
283	C₂	—	16	2	no	—
284	C₂	C₃₁×#2093	1	1	yes	characteristic
285	C₂	C₃₁×#2228	1	2	yes	—
286	C₂	C₃₁×#2234	1	2	yes	—
287	C₂	C₃₁×#2249	1	1	yes	characteristic
288	C₂	C₃₁×#2250	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₃₁
prime radical	C₂²×D₈×C₃₁	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#367	C₂	semidirect
2. C₂×C₃₁×#366	C₂	semidirect
6. C₃₁×#1846	C₂	semidirect
7. C₃₁×#1374	C₂	semidirect
90. #13528	C₃₁	direct (N × H)
121. C₂×#367	C₂×C₃₁	semidirect
122. C₂×#366	C₂×C₃₁	semidirect
126. #1846	C₂×C₃₁	semidirect
127. #1374	C₂×C₃₁	semidirect
198. C₃₁	#13528	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×#367	2
C₂×C₃₁×#366	2
C₂×C₃₁×#357	2
C₃₁×#1970	2
C₃₁×#1846	2
C₃₁×#1374	2
#13528	31

Subgroups — C31×#13528 #21470287

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#13528 #21470287