Subgroups — C₃₁×#8260 #21465019

Order 7936.

502 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₃₁×C₄⋊C₈	C₂	1	2	yes	maximal
2	C₄²×C₈×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#322	C₂	1	2	yes	maximal
4	C₃₁×#1273	C₂	1	2	yes	maximal
5	C₂×C₃₁×C₄⋊C₈	C₄	1	4	yes	—
6	C₂×C₄×C₈×C₃₁	C₄	1	2	yes	—
7	C₂×C₄×C₈×C₃₁	C₄	1	2	yes	—
8	C₂×C₄×C₈×C₃₁	C₂²	1	4	yes	—
9	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
11	C₄³×C₃₁	C₂²	1	1	yes	characteristic
12	C₃₁×#336	C₄	1	4	yes	—
13	C₃₁×#322	—	2	4	no	—
14	C₂²×C₈×C₃₁	C₂×C₄	1	8	yes	—
15	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
16	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
17	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
19	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
21	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
22	C₃₁×C₄⋊C₈	—	2	8	no	—
23	C₄×C₈×C₃₁	—	2	4	no	—
24	C₄×C₈×C₃₁	—	2	4	no	—
25	C₄×C₈×C₃₁	—	2	2	no	—
26	C₄×C₈×C₃₁	—	2	4	no	—
27	C₄×C₈×C₃₁	C₈	1	8	yes	complemented
28	C₄×C₈×C₃₁	D₄	1	2	yes	—
29	C₄×C₈×C₃₁	Q₈	1	2	yes	—
30	C₂²×C₄×C₃₁	C₄²	1	1	yes	characteristic
31	C₂²×C₄×C₃₁	C₄²	1	2	yes	—
32	C₂²×C₄×C₃₁	C₄²	1	1	yes	characteristic
33	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
35	C₂×C₈×C₃₁	—	2	2	no	—
36	C₂×C₈×C₃₁	—	2	2	no	—
37	C₂×C₈×C₃₁	—	2	8	no	—
38	C₂×C₈×C₃₁	—	2	8	no	—
39	C₂×C₈×C₃₁	—	2	8	no	—
40	C₂×C₈×C₃₁	C₄⋊C₄	1	4	yes	—
41	C₂×C₈×C₃₁	C₂×C₈	1	4	yes	—
42	C₂×C₈×C₃₁	C₂×C₈	1	4	yes	—
43	C₂×C₈×C₃₁	M₁₆	1	4	yes	—
44	C₄²×C₃₁	—	2	2	no	—
45	C₄²×C₃₁	—	2	4	no	—
46	C₄²×C₃₁	C₂×C₈	1	4	yes	—
47	C₄²×C₃₁	M₁₆	1	8	yes	—
48	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
49	C₄²×C₃₁	C₂×Q₈	1	1	yes	characteristic
50	C₄²×C₃₁	#41	1	2	yes	—
51	#8260	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
52	C₂³×C₃₁	C₂×C₄²	1	1	yes	characteristic
53	C₂×C₄×C₃₁	—	2	2	no	—
54	C₂×C₄×C₃₁	—	2	2	no	—
55	C₂×C₄×C₃₁	—	2	2	no	—
56	C₂×C₄×C₃₁	C₄×C₈	1	2	yes	—
57	C₂×C₄×C₃₁	C₄×C₈	1	2	yes	—
58	C₂×C₄×C₃₁	#97	1	4	yes	—
59	C₂×C₄×C₃₁	#97	1	4	yes	—
60	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₃₁	#117	1	1	yes	characteristic
62	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
63	C₂×C₄×C₃₁	C₄×Q₈	1	2	yes	—
64	C₂×C₄×C₃₁	C₂²×C₈	1	2	yes	—
65	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
66	C₂×C₄×C₃₁	C₂×M₁₆	1	4	yes	—
67	C₂×C₄×C₃₁	#131	1	4	yes	—
68	C₈×C₃₁	—	2	4	no	—
69	C₈×C₃₁	—	4	8	no	—
70	C₈×C₃₁	C₄⋊C₈	1	8	yes	complemented
71	C₄×C₄⋊C₈	C₂×C₃₁	1	2	yes	—
72	C₄²×C₈	C₂×C₃₁	1	1	yes	characteristic
73	C₂×#322	C₂×C₃₁	1	2	yes	—
74	#1273	C₂×C₃₁	1	2	yes	—
75	C₂²×C₃₁	C₄×C₄⋊C₄	1	1	yes	characteristic
76	C₂²×C₃₁	C₂×C₄×C₈	1	1	yes	characteristic
77	C₂²×C₃₁	C₂×#97	1	2	yes	—
78	C₂²×C₃₁	C₄×M₁₆	1	1	yes	characteristic
79	C₂²×C₃₁	#405	1	2	yes	—
80	C₄×C₃₁	—	2	2	no	—
81	C₄×C₃₁	—	2	4	no	—
82	C₄×C₃₁	#322	1	4	yes	complemented
83	C₄×C₃₁	C₂×C₄⋊C₈	1	2	yes	—
84	C₄×C₃₁	#431	1	2	yes	—
85	C₄×C₃₁	#436	1	4	yes	—
86	C₄×C₃₁	#446	1	4	yes	—
87	C₂×C₄⋊C₈	C₄×C₃₁	1	4	yes	—
88	C₂×C₄×C₈	C₄×C₃₁	1	2	yes	—
89	C₂×C₄×C₈	C₄×C₃₁	1	2	yes	—
90	C₂×C₄×C₈	C₂²×C₃₁	1	4	yes	—
91	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
92	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
93	C₄³	C₂²×C₃₁	1	1	yes	characteristic
94	#336	C₄×C₃₁	1	4	yes	—
95	#322	—	2	4	no	—
96	C₂×C₃₁	C₂×#322	1	1	yes	characteristic
97	C₂×C₃₁	#1450	1	1	yes	characteristic
98	C₂×C₃₁	#1451	1	2	yes	—
99	C₂×C₃₁	C₄×C₄⋊C₈	1	1	yes	characteristic
100	C₂×C₃₁	#1770	1	2	yes	—
101	C₂²×C₈	C₂×C₄×C₃₁	1	8	yes	—
102	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
103	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
104	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
105	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
106	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
107	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
108	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
109	C₄⋊C₈	—	2	8	no	—
110	C₄×C₈	—	2	4	no	—
111	C₄×C₈	—	2	4	no	—
112	C₄×C₈	—	2	2	no	—
113	C₄×C₈	—	2	4	no	—
114	C₄×C₈	C₈×C₃₁	1	8	yes	complemented
115	C₄×C₈	D₄×C₃₁	1	2	yes	—
116	C₄×C₈	Q₈×C₃₁	1	2	yes	—
117	C₃₁	#8260	1	1	yes	characteristic, direct product factor, Sylow
118	C₂²×C₄	C₄²×C₃₁	1	1	yes	characteristic
119	C₂²×C₄	C₄²×C₃₁	1	2	yes	—
120	C₂²×C₄	C₄²×C₃₁	1	1	yes	characteristic
121	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
122	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
123	C₂×C₈	—	2	2	no	—
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	—	2	8	no	—
126	C₂×C₈	—	2	8	no	—
127	C₂×C₈	—	2	8	no	—
128	C₂×C₈	C₄⋊C₄×C₃₁	1	4	yes	—
129	C₂×C₈	C₂×C₈×C₃₁	1	4	yes	—
130	C₂×C₈	C₂×C₈×C₃₁	1	4	yes	—
131	C₂×C₈	M₁₆×C₃₁	1	4	yes	—
132	C₄²	—	2	2	no	—
133	C₄²	—	2	4	no	—
134	C₄²	C₂×C₈×C₃₁	1	4	yes	—
135	C₄²	M₁₆×C₃₁	1	8	yes	—
136	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
137	C₄²	C₂×Q₈×C₃₁	1	1	yes	characteristic
138	C₄²	#41×C₃₁	1	2	yes	—
139	C₂³	C₂×C₄²×C₃₁	1	1	yes	characteristic
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	C₄×C₈×C₃₁	1	2	yes	—
144	C₂×C₄	C₄×C₈×C₃₁	1	2	yes	—
145	C₂×C₄	C₃₁×#97	1	4	yes	—
146	C₂×C₄	C₃₁×#97	1	4	yes	—
147	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
148	C₂×C₄	C₃₁×#117	1	1	yes	characteristic
149	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
150	C₂×C₄	C₄×Q₈×C₃₁	1	2	yes	—
151	C₂×C₄	C₂²×C₈×C₃₁	1	2	yes	—
152	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
153	C₂×C₄	C₂×M₁₆×C₃₁	1	4	yes	—
154	C₂×C₄	C₃₁×#131	1	4	yes	—
155	C₈	—	2	4	no	—
156	C₈	—	4	8	no	—
157	C₈	C₃₁×C₄⋊C₈	1	8	yes	complemented
158	C₂²	C₄×C₄⋊C₄×C₃₁	1	1	yes	characteristic
159	C₂²	C₂×C₄×C₈×C₃₁	1	1	yes	characteristic
160	C₂²	C₂×C₃₁×#97	1	2	yes	—
161	C₂²	C₄×M₁₆×C₃₁	1	1	yes	characteristic
162	C₂²	C₃₁×#405	1	2	yes	—
163	C₄	—	2	2	no	—
164	C₄	—	2	4	no	—
165	C₄	C₃₁×#322	1	4	yes	complemented
166	C₄	C₂×C₃₁×C₄⋊C₈	1	2	yes	—
167	C₄	C₃₁×#431	1	2	yes	—
168	C₄	C₃₁×#436	1	4	yes	—
169	C₄	C₃₁×#446	1	4	yes	—
170	C₂	C₂×C₃₁×#322	1	1	yes	characteristic
171	C₂	C₃₁×#1450	1	1	yes	characteristic
172	C₂	C₃₁×#1451	1	2	yes	—
173	C₂	C₄×C₃₁×C₄⋊C₈	1	1	yes	characteristic
174	C₂	C₃₁×#1770	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
27. C₄×C₈×C₃₁	C₈	semidirect
51. #8260	C₃₁	direct (N × H)
70. C₈×C₃₁	C₄⋊C₈	semidirect
82. C₄×C₃₁	#322	semidirect
114. C₄×C₈	C₈×C₃₁	semidirect
117. C₃₁	#8260	direct (N × H)
157. C₈	C₃₁×C₄⋊C₈	semidirect
165. C₄	C₃₁×#322	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₃₁×C₄⋊C₈	2
C₄²×C₈×C₃₁	2
C₂×C₃₁×#322	2
C₃₁×#1273	2
#8260	31

Subgroups — C31×#8260 #21465019

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#8260 #21465019