Group #21469277 — subgroups

Subgroups — C₃₁×#12518 #21469277

Order 7936.

330 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 110 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2248	C₂	1	2	yes	maximal, complemented
2	C₃₁×#2247	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2246	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1709	C₂	1	2	yes	maximal, complemented
5	C₈×C₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃₁×Q₃₂	—	2	2	no	—
7	C₂×C₃₁×SD₃₂	—	2	4	no	—
8	C₂×C₃₁×D₁₆	—	2	2	no	—
9	C₃₁×#494	C₂²	1	2	yes	—
10	C₃₁×#493	C₂²	1	2	yes	—
11	C₃₁×#486	—	2	4	no	—
12	C₄×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
13	C₄×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
14	C₈²×C₃₁	C₂²	1	1	yes	characteristic
15	C₂×Q₁₆×C₃₁	—	2	2	no	—
16	C₂×Q₁₆×C₃₁	—	2	2	no	—
17	C₂×D₈×C₃₁	—	2	2	no	—
18	C₂×D₈×C₃₁	—	2	2	no	—
19	C₃₁×C₄⋊Q₈	—	2	2	no	—
20	C₃₁×#127	—	2	2	no	—
21	C₃₁×Q₃₂	—	4	4	no	—
22	C₃₁×SD₃₂	—	4	8	no	—
23	C₃₁×D₁₆	—	4	4	no	—
24	C₂×C₁₆×C₃₁	D₄	1	2	yes	—
25	C₂×C₁₆×C₃₁	D₄	1	2	yes	—
26	C₃₁×#102	—	4	8	no	—
27	C₄×C₈×C₃₁	D₄	1	2	yes	—
28	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
29	C₂×Q₈×C₃₁	—	4	2	no	—
30	C₂×D₄×C₃₁	—	4	2	no	—
31	C₂×D₄×C₃₁	—	4	4	no	—
32	Q₁₆×C₃₁	—	4	4	no	—
33	Q₁₆×C₃₁	—	4	4	no	—
34	D₈×C₃₁	—	4	4	no	—
35	D₈×C₃₁	—	4	4	no	—
36	C₂×C₈×C₃₁	D₈	1	4	yes	—
37	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
38	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
39	C₄⋊C₄×C₃₁	—	4	4	no	—
40	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
41	C₁₆×C₃₁	D₈	1	4	yes	—
42	C₁₆×C₃₁	D₈	1	4	yes	—
43	#12518	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
44	C₂³×C₃₁	—	8	2	no	—
45	Q₈×C₃₁	—	8	2	no	—
46	Q₈×C₃₁	—	8	2	no	—
47	D₄×C₃₁	—	8	2	no	—
48	D₄×C₃₁	—	8	2	no	—
49	D₄×C₃₁	—	8	8	no	—
50	C₂×C₄×C₃₁	—	8	2	no	—
51	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
52	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
53	C₈×C₃₁	—	2	4	no	—
54	C₈×C₃₁	C₂×D₈	1	2	yes	—
55	C₈×C₃₁	C₂×D₈	1	2	yes	—
56	#2248	C₂×C₃₁	1	2	yes	complemented
57	#2247	C₂×C₃₁	1	1	yes	characteristic, complemented
58	#2246	C₂×C₃₁	1	1	yes	characteristic
59	#1709	C₂×C₃₁	1	2	yes	complemented
60	C₈×C₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
61	C₂²×C₃₁	—	16	2	no	—
62	C₂²×C₃₁	—	16	4	no	—
63	C₂²×C₃₁	#493	1	1	yes	characteristic
64	C₄×C₃₁	—	16	2	no	—
65	C₄×C₃₁	#493	1	1	yes	characteristic
66	C₄×C₃₁	#493	1	1	yes	characteristic
67	C₄×C₃₁	#508	1	4	yes	—
68	C₂×Q₃₂	—	2	2	no	—
69	C₂×SD₃₂	—	2	4	no	—
70	C₂×D₁₆	—	2	2	no	—
71	#494	C₂²×C₃₁	1	2	yes	—
72	#493	C₂²×C₃₁	1	2	yes	—
73	#486	—	2	4	no	—
74	C₄×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
75	C₄×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
76	C₈²	C₂²×C₃₁	1	1	yes	characteristic
77	C₂×C₃₁	—	32	2	no	—
78	C₂×C₃₁	#1712	1	1	yes	characteristic
79	C₂×C₃₁	#2249	1	2	yes	—
80	C₂×Q₁₆	—	2	2	no	—
81	C₂×Q₁₆	—	2	2	no	—
82	C₂×D₈	—	2	2	no	—
83	C₂×D₈	—	2	2	no	—
84	C₄⋊Q₈	—	2	2	no	—
85	#127	—	2	2	no	—
86	Q₃₂	—	4	4	no	—
87	SD₃₂	—	4	8	no	—
88	D₁₆	—	4	4	no	—
89	C₂×C₁₆	D₄×C₃₁	1	2	yes	—
90	C₂×C₁₆	D₄×C₃₁	1	2	yes	—
91	#102	—	4	8	no	—
92	C₄×C₈	D₄×C₃₁	1	2	yes	—
93	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
94	C₃₁	#12518	1	1	yes	characteristic, direct product factor, Sylow
95	C₂×Q₈	—	4	2	no	—
96	C₂×D₄	—	4	2	no	—
97	C₂×D₄	—	4	4	no	—
98	Q₁₆	—	4	4	no	—
99	Q₁₆	—	4	4	no	—
100	D₈	—	4	4	no	—
101	D₈	—	4	4	no	—
102	C₂×C₈	D₈×C₃₁	1	4	yes	—
103	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
104	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
105	C₄⋊C₄	—	4	4	no	—
106	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
107	C₁₆	D₈×C₃₁	1	4	yes	—
108	C₁₆	D₈×C₃₁	1	4	yes	—
109	C₂³	—	8	2	no	—
110	Q₈	—	8	2	no	—
111	Q₈	—	8	2	no	—
112	D₄	—	8	2	no	—
113	D₄	—	8	2	no	—
114	D₄	—	8	8	no	—
115	C₂×C₄	—	8	2	no	—
116	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
117	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
118	C₈	—	2	4	no	—
119	C₈	C₂×D₈×C₃₁	1	2	yes	—
120	C₈	C₂×D₈×C₃₁	1	2	yes	—
121	C₂²	—	16	2	no	—
122	C₂²	—	16	4	no	—
123	C₂²	C₃₁×#493	1	1	yes	characteristic
124	C₄	—	16	2	no	—
125	C₄	C₃₁×#493	1	1	yes	characteristic
126	C₄	C₃₁×#493	1	1	yes	characteristic
127	C₄	C₃₁×#508	1	4	yes	—
128	C₂	—	32	2	no	—
129	C₂	C₃₁×#1712	1	1	yes	characteristic
130	C₂	C₃₁×#2249	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃₁×#2248

C₂

yes

maximal, complemented

C₃₁×#2247

C₂

yes

maximal, characteristic, complemented

C₃₁×#2246

C₂

yes

maximal, characteristic

C₃₁×#1709

C₂

yes

maximal, complemented

C₈×C₁₆×C₃₁

C₂

yes

maximal, characteristic, complemented

C₂×C₃₁×Q₃₂

—

C₂×C₃₁×SD₃₂

—

C₂×C₃₁×D₁₆

—

C₃₁×#494

C₂²

yes

—

C₃₁×#493

C₂²

yes

—

C₃₁×#486

—

C₄×C₁₆×C₃₁

C₂²

yes

characteristic

C₄×C₁₆×C₃₁

C₂²

yes

characteristic

C₈²×C₃₁

C₂²

yes

characteristic

C₂×Q₁₆×C₃₁

—

C₂×Q₁₆×C₃₁

—

C₂×D₈×C₃₁

—

C₂×D₈×C₃₁

—

C₃₁×C₄⋊Q₈

—

C₃₁×#127

—

C₃₁×Q₃₂

—

C₃₁×SD₃₂

—

C₃₁×D₁₆

—

C₂×C₁₆×C₃₁

D₄

yes

—

C₂×C₁₆×C₃₁

D₄

yes

—

C₃₁×#102

—

C₄×C₈×C₃₁

D₄

yes

—

C₄×C₈×C₃₁

C₂³

yes

characteristic

C₂×Q₈×C₃₁

—

C₂×D₄×C₃₁

—

C₂×D₄×C₃₁

—

Q₁₆×C₃₁

—

Q₁₆×C₃₁

—

D₈×C₃₁

—

D₈×C₃₁

—

C₂×C₈×C₃₁

D₈

yes

—

C₂×C₈×C₃₁

C₂×D₄

yes

characteristic

C₂×C₈×C₃₁

C₂×D₄

yes

characteristic

C₄⋊C₄×C₃₁

—

C₄²×C₃₁

C₂×D₄

yes

characteristic

C₁₆×C₃₁

D₈

yes

—

C₁₆×C₃₁

D₈

yes

—

#12518

C₃₁

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₃₁

—

Q₈×C₃₁

—

Q₈×C₃₁

—

D₄×C₃₁

—

D₄×C₃₁

—

D₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

#127

yes

characteristic

C₂×C₄×C₃₁

C₂×D₈

yes

—

C₈×C₃₁

—

C₈×C₃₁

C₂×D₈

yes

—

C₈×C₃₁

C₂×D₈

yes

—

#2248

C₂×C₃₁

yes

complemented

#2247

C₂×C₃₁

yes

characteristic, complemented

#2246

C₂×C₃₁

yes

characteristic

#1709

C₂×C₃₁

yes

complemented

C₈×C₁₆

C₂×C₃₁

yes

characteristic, complemented

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

#493

yes

characteristic

C₄×C₃₁

—

C₄×C₃₁

yes

C₄×C₃₁

yes

C₄×C₃₁

yes

—

C₂×Q₃₂

—

C₂×SD₃₂

—

C₂×D₁₆

—

#494

C₂²×C₃₁

yes

—

#493

C₂²×C₃₁

yes

—

#486

—

C₄×C₁₆

C₂²×C₃₁

yes

characteristic

C₄×C₁₆

C₂²×C₃₁

yes

characteristic

C₈²

C₂²×C₃₁

yes

characteristic

C₂×C₃₁

—

C₂×C₃₁

#1712

yes

characteristic

C₂×C₃₁

#2249

yes

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

#127

—

Q₃₂

—

SD₃₂

—

D₁₆

—

C₂×C₁₆

D₄×C₃₁

yes

—

C₂×C₁₆

D₄×C₃₁

yes

—

#102

—

C₄×C₈

D₄×C₃₁

yes

—

C₄×C₈

C₂³×C₃₁

yes

characteristic

C₃₁

#12518

yes

characteristic, direct product factor, Sylow

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

Q₁₆

—

Q₁₆

—

100

D₈

—

101

D₈

—

102

C₂×C₈

D₈×C₃₁

yes

—

103

C₂×C₈

C₂×D₄×C₃₁

yes

characteristic

104

C₂×C₈

C₂×D₄×C₃₁

yes

characteristic

105

C₄⋊C₄

—

106

C₄²

C₂×D₄×C₃₁

yes

characteristic

107

C₁₆

D₈×C₃₁

yes

—

108

C₁₆

D₈×C₃₁

yes

—

109

C₂³

—

110

Q₈

—

111

Q₈

—

112

D₄

—

113

D₄

—

114

D₄

—

115

C₂×C₄

—

116

C₂×C₄

C₃₁×#127

yes

characteristic

117

C₂×C₄

C₂×D₈×C₃₁

yes

—

118

C₈

—

119

C₈

C₂×D₈×C₃₁

yes

—

120

C₈

C₂×D₈×C₃₁

yes

—

121

C₂²

—

122

C₂²

—

123

C₂²

C₃₁×#493

yes

characteristic

124

C₄

—

125

C₄

C₃₁×#493

yes

characteristic

126

C₄

C₃₁×#493

yes

characteristic

127

C₄

C₃₁×#508

yes

—

128

C₂

—

129

C₂

C₃₁×#1712

yes

characteristic

130

C₂

C₃₁×#2249

yes

—

	subgroup	G/H
center, socle	C₂²×C₃₁	#493
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₃₁
prime radical	C₃₁×#2246	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄²×C₃₁ ◁ C₈²×C₃₁ ◁ G

subgroup

G/H

center, socle

C₂²×C₃₁

#493

commutator, Frattini, cosocle

C₄×C₈

C₂³×C₃₁

prime radical

C₃₁×#2246

C₂

LCS

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₃₁ ◁ C₄²×C₃₁ ◁ C₈²×C₃₁ ◁ G

N	H ≅ G/N	type
1. C₃₁×#2248	C₂	semidirect
2. C₃₁×#2247	C₂	semidirect
4. C₃₁×#1709	C₂	semidirect
5. C₈×C₁₆×C₃₁	C₂	semidirect
43. #12518	C₃₁	direct (N × H)
56. #2248	C₂×C₃₁	semidirect
57. #2247	C₂×C₃₁	semidirect
59. #1709	C₂×C₃₁	semidirect
60. C₈×C₁₆	C₂×C₃₁	semidirect
94. C₃₁	#12518	direct (N × H)

H ≅ G/N

type

1. C₃₁×#2248

C₂

semidirect

2. C₃₁×#2247

C₂

semidirect

4. C₃₁×#1709

C₂

semidirect

5. C₈×C₁₆×C₃₁

C₂

semidirect

43. #12518

C₃₁

direct (N × H)

56. #2248

C₂×C₃₁

semidirect

57. #2247

C₂×C₃₁

semidirect

59. #1709

C₂×C₃₁

semidirect

60. C₈×C₁₆

C₂×C₃₁

semidirect

94. C₃₁

#12518

direct (N × H)

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

subgroup	index
C₃₁×#2248	2
C₃₁×#2247	2
C₃₁×#2246	2
C₃₁×#1709	2
C₈×C₁₆×C₃₁	2
#12518	31

subgroup

index

C₃₁×#2248

C₃₁×#2247

C₃₁×#2246

C₃₁×#1709

C₈×C₁₆×C₃₁

#12518

Subgroups — C₃₁×#12518 #21469277

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C31×#12518 #21469277

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#12518 #21469277