Subgroups — C₃₁×#8835 #21465594

Order 7936.

482 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2972	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1954	C₂	1	1	yes	maximal, characteristic
3	C₄×C₃₁×#100	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1565	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1278	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
7	C₈×D₄×C₃₁	—	2	2	no	—
8	C₈×D₄×C₃₁	—	2	2	no	—
9	C₃₁×#431	C₂²	1	2	yes	—
10	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₃₁×#100	C₄	1	1	yes	characteristic
12	C₂×C₃₁×#100	C₄	1	1	yes	characteristic
13	C₂×C₃₁×C₂²⋊C₈	—	2	2	no	—
14	C₂×C₃₁×C₂²⋊C₈	—	2	2	no	—
15	C₄×M₁₆×C₃₁	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
17	C₃₁×#343	C₄	1	2	yes	—
18	C₃₁×#335	—	2	2	no	—
19	C₃₁×#334	—	2	2	no	—
20	C₂²×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₃₁	D₄	1	2	yes	—
22	C₂×M₁₆×C₃₁	C₂×C₄	1	2	yes	—
23	C₂×M₁₆×C₃₁	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₃₁	—	2	1	no	—
25	C₂²×C₈×C₃₁	—	2	2	no	—
26	C₂²×C₈×C₃₁	—	2	2	no	—
27	C₂²×C₈×C₃₁	—	2	1	no	—
28	C₄×D₄×C₃₁	—	2	2	no	—
29	C₄×D₄×C₃₁	—	2	2	no	—
30	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
31	C₂×C₂²⋊C₄×C₃₁	Q₈	1	2	yes	—
32	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
33	C₃₁×C₄⋊C₈	—	2	2	no	—
34	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
35	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
36	C₃₁×#100	—	2	2	no	—
37	C₃₁×#100	C₈	1	4	yes	complemented
38	C₃₁×C₂²⋊C₈	—	2	4	no	—
39	C₃₁×C₂²⋊C₈	—	2	2	no	—
40	C₃₁×C₂²⋊C₈	—	4	4	no	—
41	C₃₁×#97	—	2	2	no	—
42	C₄×C₈×C₃₁	—	2	2	no	—
43	C₄×C₈×C₃₁	—	2	2	no	—
44	C₄×C₈×C₃₁	—	2	2	no	—
45	C₂⁴×C₃₁	C₄⋊C₄	1	2	yes	—
46	C₂×D₄×C₃₁	—	2	2	no	—
47	C₂×D₄×C₃₁	—	2	2	no	—
48	C₂×D₄×C₃₁	C₂×C₈	1	2	yes	—
49	C₂×D₄×C₃₁	M₁₆	1	2	yes	—
50	C₂²×C₄×C₃₁	—	2	2	no	—
51	C₂²×C₄×C₃₁	—	2	2	no	—
52	C₂²×C₄×C₃₁	—	4	2	no	—
53	C₂²×C₄×C₃₁	—	4	2	no	—
54	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
55	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
58	M₁₆×C₃₁	—	2	2	no	—
59	M₁₆×C₃₁	—	2	2	no	—
60	M₁₆×C₃₁	—	2	2	no	—
61	M₁₆×C₃₁	C₂×C₈	1	4	yes	complemented
62	C₂×C₈×C₃₁	—	2	2	no	—
63	C₂×C₈×C₃₁	—	2	2	no	—
64	C₂×C₈×C₃₁	—	2	2	no	—
65	C₂×C₈×C₃₁	—	2	2	no	—
66	C₂×C₈×C₃₁	—	2	2	no	—
67	C₂×C₈×C₃₁	—	4	1	no	—
68	C₂×C₈×C₃₁	—	4	1	no	—
69	C₂×C₈×C₃₁	—	4	2	no	—
70	C₂×C₈×C₃₁	—	4	1	no	—
71	C₂×C₈×C₃₁	—	8	2	no	—
72	C₂×C₈×C₃₁	—	8	2	no	—
73	C₄⋊C₄×C₃₁	—	2	2	no	—
74	C₂²⋊C₄×C₃₁	—	2	4	no	—
75	C₄²×C₃₁	C₂×D₄	1	2	yes	—
76	C₄²×C₃₁	#41	1	2	yes	—
77	#8835	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
78	C₂³×C₃₁	—	2	2	no	—
79	C₂³×C₃₁	—	2	2	no	—
80	C₂³×C₃₁	—	4	2	no	—
81	C₂³×C₃₁	—	4	2	no	—
82	C₂³×C₃₁	C₄⋊C₈	1	4	yes	—
83	C₂³×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
84	D₄×C₃₁	—	2	4	no	—
85	D₄×C₃₁	—	2	4	no	—
86	C₂×C₄×C₃₁	—	2	1	no	—
87	C₂×C₄×C₃₁	—	2	1	no	—
88	C₂×C₄×C₃₁	—	2	2	no	—
89	C₂×C₄×C₃₁	—	4	1	no	—
90	C₂×C₄×C₃₁	—	4	2	no	—
91	C₂×C₄×C₃₁	—	4	2	no	—
92	C₂×C₄×C₃₁	—	4	2	no	—
93	C₂×C₄×C₃₁	—	8	2	no	—
94	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
95	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
96	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
97	C₂×C₄×C₃₁	#123	1	1	yes	characteristic
98	C₂×C₄×C₃₁	C₂²×C₈	1	1	yes	characteristic
99	C₂×C₄×C₃₁	C₂×M₁₆	1	1	yes	characteristic
100	C₂×C₄×C₃₁	#131	1	2	yes	—
101	C₈×C₃₁	—	2	4	no	—
102	C₈×C₃₁	—	4	2	no	—
103	C₈×C₃₁	—	4	2	no	—
104	C₈×C₃₁	—	4	2	no	—
105	C₈×C₃₁	—	4	2	no	—
106	C₈×C₃₁	—	8	1	no	—
107	#2972	C₂×C₃₁	1	1	yes	characteristic
108	#1954	C₂×C₃₁	1	1	yes	characteristic
109	C₄×#100	C₂×C₃₁	1	1	yes	characteristic
110	#1565	C₂×C₃₁	1	2	yes	complemented
111	#1278	C₂×C₃₁	1	2	yes	complemented
112	C₂²×C₃₁	—	2	1	no	—
113	C₂²×C₃₁	—	2	1	no	—
114	C₂²×C₃₁	—	2	4	no	—
115	C₂²×C₃₁	—	4	4	no	—
116	C₂²×C₃₁	—	4	2	no	—
117	C₂²×C₃₁	—	4	2	no	—
118	C₂²×C₃₁	—	8	2	no	—
119	C₂²×C₃₁	#385	1	1	yes	characteristic
120	C₂²×C₃₁	C₂×C₄⋊C₈	1	1	yes	characteristic
121	C₂²×C₃₁	#424	1	1	yes	characteristic
122	C₄×C₃₁	—	2	1	no	—
123	C₄×C₃₁	—	2	2	no	—
124	C₄×C₃₁	—	4	2	no	—
125	C₄×C₃₁	C₈×D₄	1	2	yes	—
126	C₄×C₃₁	#435	1	2	yes	—
127	C₄×C₃₁	#453	1	1	yes	characteristic
128	C₄×C₃₁	#455	1	1	yes	characteristic
129	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
130	C₈×D₄	—	2	2	no	—
131	C₈×D₄	—	2	2	no	—
132	#431	C₂²×C₃₁	1	2	yes	—
133	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
134	C₂×#100	C₄×C₃₁	1	1	yes	characteristic
135	C₂×#100	C₄×C₃₁	1	1	yes	characteristic
136	C₂×C₂²⋊C₈	—	2	2	no	—
137	C₂×C₂²⋊C₈	—	2	2	no	—
138	C₄×M₁₆	C₂²×C₃₁	1	2	yes	—
139	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
140	#343	C₄×C₃₁	1	2	yes	—
141	#335	—	2	2	no	—
142	#334	—	2	2	no	—
143	C₂×C₃₁	—	2	1	no	—
144	C₂×C₃₁	—	2	1	no	—
145	C₂×C₃₁	—	4	4	no	—
146	C₂×C₃₁	#1850	1	1	yes	characteristic
147	C₂×C₃₁	#1856	1	1	yes	characteristic
148	C₂×C₃₁	#1858	1	1	yes	characteristic
149	C₂²×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
150	C₂³×C₄	D₄×C₃₁	1	2	yes	—
151	C₂×M₁₆	C₂×C₄×C₃₁	1	2	yes	—
152	C₂×M₁₆	C₂×C₄×C₃₁	1	2	yes	—
153	C₂²×C₈	—	2	1	no	—
154	C₂²×C₈	—	2	2	no	—
155	C₂²×C₈	—	2	2	no	—
156	C₂²×C₈	—	2	1	no	—
157	C₄×D₄	—	2	2	no	—
158	C₄×D₄	—	2	2	no	—
159	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
160	C₂×C₂²⋊C₄	Q₈×C₃₁	1	2	yes	—
161	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
162	C₄⋊C₈	—	2	2	no	—
163	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
164	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
165	#100	—	2	2	no	—
166	#100	C₈×C₃₁	1	4	yes	complemented
167	C₂²⋊C₈	—	2	4	no	—
168	C₂²⋊C₈	—	2	2	no	—
169	C₂²⋊C₈	—	4	4	no	—
170	#97	—	2	2	no	—
171	C₄×C₈	—	2	2	no	—
172	C₄×C₈	—	2	2	no	—
173	C₄×C₈	—	2	2	no	—
174	C₃₁	#8835	1	1	yes	characteristic, direct product factor, Sylow
175	C₂⁴	C₄⋊C₄×C₃₁	1	2	yes	—
176	C₂×D₄	—	2	2	no	—
177	C₂×D₄	—	2	2	no	—
178	C₂×D₄	C₂×C₈×C₃₁	1	2	yes	—
179	C₂×D₄	M₁₆×C₃₁	1	2	yes	—
180	C₂²×C₄	—	2	2	no	—
181	C₂²×C₄	—	2	2	no	—
182	C₂²×C₄	—	4	2	no	—
183	C₂²×C₄	—	4	2	no	—
184	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
185	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
186	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
187	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
188	M₁₆	—	2	2	no	—
189	M₁₆	—	2	2	no	—
190	M₁₆	—	2	2	no	—
191	M₁₆	C₂×C₈×C₃₁	1	4	yes	complemented
192	C₂×C₈	—	2	2	no	—
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	4	1	no	—
198	C₂×C₈	—	4	1	no	—
199	C₂×C₈	—	4	2	no	—
200	C₂×C₈	—	4	1	no	—
201	C₂×C₈	—	8	2	no	—
202	C₂×C₈	—	8	2	no	—
203	C₄⋊C₄	—	2	2	no	—
204	C₂²⋊C₄	—	2	4	no	—
205	C₄²	C₂×D₄×C₃₁	1	2	yes	—
206	C₄²	#41×C₃₁	1	2	yes	—
207	C₂³	—	2	2	no	—
208	C₂³	—	2	2	no	—
209	C₂³	—	4	2	no	—
210	C₂³	—	4	2	no	—
211	C₂³	C₃₁×C₄⋊C₈	1	4	yes	—
212	C₂³	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
213	D₄	—	2	4	no	—
214	D₄	—	2	4	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	4	1	no	—
219	C₂×C₄	—	4	2	no	—
220	C₂×C₄	—	4	2	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	—	8	2	no	—
223	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
224	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
225	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
226	C₂×C₄	C₃₁×#123	1	1	yes	characteristic
227	C₂×C₄	C₂²×C₈×C₃₁	1	1	yes	characteristic
228	C₂×C₄	C₂×M₁₆×C₃₁	1	1	yes	characteristic
229	C₂×C₄	C₃₁×#131	1	2	yes	—
230	C₈	—	2	4	no	—
231	C₈	—	4	2	no	—
232	C₈	—	4	2	no	—
233	C₈	—	4	2	no	—
234	C₈	—	4	2	no	—
235	C₈	—	8	1	no	—
236	C₂²	—	2	1	no	—
237	C₂²	—	2	1	no	—
238	C₂²	—	2	4	no	—
239	C₂²	—	4	4	no	—
240	C₂²	—	4	2	no	—
241	C₂²	—	4	2	no	—
242	C₂²	—	8	2	no	—
243	C₂²	C₃₁×#385	1	1	yes	characteristic
244	C₂²	C₂×C₃₁×C₄⋊C₈	1	1	yes	characteristic
245	C₂²	C₃₁×#424	1	1	yes	characteristic
246	C₄	—	2	1	no	—
247	C₄	—	2	2	no	—
248	C₄	—	4	2	no	—
249	C₄	C₈×D₄×C₃₁	1	2	yes	—
250	C₄	C₃₁×#435	1	2	yes	—
251	C₄	C₃₁×#453	1	1	yes	characteristic
252	C₄	C₃₁×#455	1	1	yes	characteristic
253	C₂	—	2	1	no	—
254	C₂	—	2	1	no	—
255	C₂	—	4	4	no	—
256	C₂	C₃₁×#1850	1	1	yes	characteristic
257	C₂	C₃₁×#1856	1	1	yes	characteristic
258	C₂	C₃₁×#1858	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	#120
commutator	C₂×C₄	C₂²×C₈×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
socle	C₂²×C₃₁	#385
prime radical	C₂²×D₄×C₃₁	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃₁×#1565	C₂	semidirect
5. C₃₁×#1278	C₂	semidirect
37. C₃₁×#100	C₈	semidirect
61. M₁₆×C₃₁	C₂×C₈	semidirect
77. #8835	C₃₁	direct (N × H)
110. #1565	C₂×C₃₁	semidirect
111. #1278	C₂×C₃₁	semidirect
166. #100	C₈×C₃₁	semidirect
174. C₃₁	#8835	direct (N × H)
191. M₁₆	C₂×C₈×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#2972	2
C₃₁×#1954	2
C₄×C₃₁×#100	2
C₃₁×#1565	2
C₃₁×#1278	2
#8835	31

Subgroups — C31×#8835 #21465594

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#8835 #21465594