Group #14521783 — subgroups

Subgroups — #14521783

Order 1984.

236 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 95 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×#123	C₂	1	1	yes	maximal, characteristic, complemented
2	#11758517	C₂	1	1	yes	maximal, characteristic
3	#11758512	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×C₂²⋊D₃₁	C₂	1	2	yes	maximal, complemented
5	#11758471	C₂	1	2	yes	maximal, complemented
6	#11758464	C₂	1	2	yes	maximal, complemented
7	#11758462	C₂	1	2	yes	maximal, complemented
8	#11758462	C₂	1	2	yes	maximal, complemented
9	(C₂²⋊D₃₁)⋊C₄	C₂	1	2	yes	maximal, complemented
10	C₂×D₄×C₃₁	C₂²	1	1	yes	characteristic
11	C₂²×C₄×C₃₁	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊D₃₁	C₂²	1	2	yes	—
13	C₂×C₂²⋊D₃₁	C₂²	1	2	yes	—
14	C₂²×Dic₃₁	C₂²	1	1	yes	characteristic
15	C₂×D₁₂₄	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₃₁	C₂²	1	2	yes	—
17	C₂×Dic₆₂	C₂²	1	2	yes	—
18	C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
19	C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
20	C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
21	C₂²⋊Dic₃₁	C₂²	1	1	yes	characteristic
22	C₂²⋊Dic₃₁	C₂²	1	2	yes	—
23	(C₂×D₃₁)⋊C₄	C₂²	1	2	yes	—
24	(C₂×D₃₁)⋊C₄	C₂²	1	2	yes	—
25	(C₂×D₃₁)⋊C₄	C₂²	1	2	yes	—
26	Dic₃₁⋊C₄	C₂²	1	2	yes	—
27	Dic₃₁⋊C₄	C₂²	1	2	yes	—
28	C₄×Dic₃₁	C₂²	1	2	yes	—
29	C₄×Dic₃₁	C₂²	1	2	yes	—
30	C₄×Dic₃₁	C₂²	1	1	yes	characteristic
31	C₂³×C₃₁	C₂³	1	1	yes	characteristic
32	C₂³×C₃₁	C₂³	1	1	yes	characteristic
33	C₂²×D₃₁	C₂³	1	2	yes	—
34	D₄×C₃₁	—	4	1	no	—
35	C₂×C₄×C₃₁	—	4	1	no	—
36	C₂×C₄×C₃₁	C₂³	1	1	yes	characteristic
37	C₂×C₄×C₃₁	C₂³	1	2	yes	—
38	C₂×C₄×C₃₁	C₂³	1	2	yes	—
39	C₂²⋊D₃₁	—	2	4	no	—
40	C₂²⋊D₃₁	—	2	4	no	—
41	C₂×Dic₃₁	—	4	1	no	—
42	C₂×Dic₃₁	C₂³	1	2	yes	—
43	C₂×Dic₃₁	C₂³	1	2	yes	—
44	C₂×Dic₃₁	C₂³	1	2	yes	—
45	D₁₂₄	—	4	1	no	—
46	C₄×D₃₁	—	4	2	no	—
47	Dic₆₂	—	2	4	no	—
48	C₂²×C₃₁	—	2	2	no	—
49	C₂²×C₃₁	—	2	2	no	—
50	C₂²×C₃₁	—	2	1	no	—
51	C₂²×C₃₁	—	2	1	no	—
52	C₂²×C₃₁	C₂⁴	1	1	yes	characteristic
53	C₂×D₃₁	—	2	2	no	—
54	C₂×D₃₁	—	2	2	no	—
55	C₂×D₃₁	—	2	2	no	—
56	C₄×C₃₁	—	2	2	no	—
57	C₄×C₃₁	—	2	2	no	—
58	C₄×C₃₁	—	2	1	no	—
59	Dic₃₁	—	2	2	no	—
60	Dic₃₁	—	2	2	no	—
61	Dic₃₁	#41	1	4	yes	—
62	#559	—	31	1	no	maximal, Sylow
63	C₂×C₃₁	—	4	1	no	—
64	C₂×C₃₁	—	4	1	no	—
65	C₂×C₃₁	C₂×#41	1	2	yes	—
66	C₂×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
67	D₃₁	—	4	2	no	—
68	#127	—	31	1	no	—
69	#124	—	31	2	no	—
70	#124	—	31	2	no	—
71	#123	—	31	1	no	—
72	#123	—	31	2	no	—
73	#123	D₃₁	1	1	yes	characteristic, complemented
74	C₄×Q₈	—	31	2	no	—
75	C₄×D₄	—	31	2	no	—
76	C₄×D₄	—	31	2	no	—
77	C₃₁	#559	1	1	yes	characteristic, complemented, Sylow
78	C₂×Q₈	—	31	2	no	—
79	C₂×D₄	—	31	2	no	—
80	C₂×D₄	—	31	2	no	—
81	C₂×D₄	—	31	1	no	—
82	C₂×D₄	C₂×D₃₁	1	1	yes	characteristic
83	C₂²×C₄	—	31	1	no	—
84	C₂²×C₄	—	31	2	no	—
85	C₂²×C₄	C₂×D₃₁	1	1	yes	characteristic
86	C₄⋊C₄	—	31	2	no	—
87	C₄⋊C₄	—	31	2	no	—
88	C₄⋊C₄	C₂×D₃₁	1	2	yes	—
89	C₂²⋊C₄	—	31	2	no	—
90	C₂²⋊C₄	—	31	1	no	—
91	C₂²⋊C₄	—	31	2	no	—
92	C₂²⋊C₄	—	31	2	no	—
93	C₂²⋊C₄	—	31	2	no	—
94	C₂²⋊C₄	C₂×D₃₁	1	2	yes	—
95	C₂²⋊C₄	C₂×D₃₁	1	1	yes	characteristic
96	C₄²	—	31	2	no	—
97	C₄²	—	31	2	no	—
98	C₄²	—	31	1	no	—
99	C₂³	—	31	2	no	—
100	C₂³	C₂²×D₃₁	1	1	yes	characteristic
101	C₂³	C₂²×D₃₁	1	1	yes	characteristic
102	Q₈	—	62	4	no	—
103	D₄	—	4	1	no	—
104	D₄	—	62	4	no	—
105	D₄	—	62	4	no	—
106	D₄	—	124	1	no	—
107	C₂×C₄	—	4	1	no	—
108	C₂×C₄	—	31	2	no	—
109	C₂×C₄	—	31	2	no	—
110	C₂×C₄	—	31	2	no	—
111	C₂×C₄	—	124	2	no	—
112	C₂×C₄	—	124	1	no	—
113	C₂×C₄	C₂²×D₃₁	1	2	yes	—
114	C₂×C₄	C₂²×D₃₁	1	2	yes	—
115	C₂×C₄	C₂²×D₃₁	1	1	yes	characteristic
116	C₂²	—	2	1	no	—
117	C₂²	—	2	1	no	—
118	C₂²	—	2	2	no	—
119	C₂²	—	2	2	no	—
120	C₂²	—	62	2	no	—
121	C₂²	—	62	2	no	—
122	C₂²	—	62	2	no	—
123	C₂²	C₂³×D₃₁	1	1	yes	characteristic
124	C₄	—	2	2	no	—
125	C₄	—	2	1	no	—
126	C₄	—	2	2	no	—
127	C₄	—	31	4	no	—
128	C₄	—	62	2	no	—
129	C₄	—	62	2	no	—
130	C₂	—	4	1	no	—
131	C₂	—	4	1	no	—
132	C₂	—	124	2	no	—
133	C₂	#11758561	1	1	yes	characteristic
134	C₂	#41×D₃₁	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃₁×#123

C₂

yes

maximal, characteristic, complemented

#11758517

C₂

yes

maximal, characteristic

#11758512

C₂

yes

maximal, characteristic, complemented

C₄×C₂²⋊D₃₁

C₂

yes

maximal, complemented

#11758471

C₂

yes

maximal, complemented

#11758464

C₂

yes

maximal, complemented

#11758462

C₂

yes

maximal, complemented

#11758462

C₂

yes

maximal, complemented

(C₂²⋊D₃₁)⋊C₄

C₂

yes

maximal, complemented

C₂×D₄×C₃₁

C₂²

yes

characteristic

C₂²×C₄×C₃₁

C₂²

yes

characteristic

C₂×C₂²⋊D₃₁

C₂²

yes

—

C₂×C₂²⋊D₃₁

C₂²

yes

—

C₂²×Dic₃₁

C₂²

yes

characteristic

C₂×D₁₂₄

C₂²

yes

characteristic

C₂×C₄×D₃₁

C₂²

yes

—

C₂×Dic₆₂

C₂²

yes

—

C₄⋊C₄×C₃₁

C₂²

yes

—

C₂²⋊C₄×C₃₁

C₂²

yes

—

C₂²⋊C₄×C₃₁

C₂²

yes

characteristic

C₂²⋊Dic₃₁

C₂²

yes

characteristic

C₂²⋊Dic₃₁

C₂²

yes

—

(C₂×D₃₁)⋊C₄

C₂²

yes

—

(C₂×D₃₁)⋊C₄

C₂²

yes

—

(C₂×D₃₁)⋊C₄

C₂²

yes

—

Dic₃₁⋊C₄

C₂²

yes

—

Dic₃₁⋊C₄

C₂²

yes

—

C₄×Dic₃₁

C₂²

yes

—

C₄×Dic₃₁

C₂²

yes

—

C₄×Dic₃₁

C₂²

yes

characteristic

C₂³×C₃₁

C₂³

yes

characteristic

C₂³×C₃₁

C₂³

yes

characteristic

C₂²×D₃₁

C₂³

yes

—

D₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂³

yes

characteristic

C₂×C₄×C₃₁

C₂³

yes

—

C₂×C₄×C₃₁

C₂³

yes

—

C₂²⋊D₃₁

—

C₂²⋊D₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

C₂³

yes

—

C₂×Dic₃₁

C₂³

yes

—

C₂×Dic₃₁

C₂³

yes

—

D₁₂₄

—

C₄×D₃₁

—

Dic₆₂

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

C₂⁴

yes

characteristic

C₂×D₃₁

—

C₂×D₃₁

—

C₂×D₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

Dic₃₁

—

Dic₃₁

—

Dic₃₁

#41

yes

—

#559

—

maximal, Sylow

C₂×C₃₁

—

C₂×C₃₁

—

C₂×C₃₁

C₂×#41

yes

—

C₂×C₃₁

H(5,𝔽₂)

yes

characteristic

D₃₁

—

—

—

—

—

—

D₃₁

yes

characteristic, complemented

C₄×Q₈

—

C₄×D₄

—

C₄×D₄

—

C₃₁

#559

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×D₃₁

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₃₁

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₃₁

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

C₂×D₃₁

yes

—

C₂²⋊C₄

C₂×D₃₁

yes

characteristic

C₄²

—

C₄²

—

C₄²

—

C₂³

—

100

C₂³

C₂²×D₃₁

yes

characteristic

101

C₂³

C₂²×D₃₁

yes

characteristic

102

Q₈

—

103

D₄

—

104

D₄

—

105

D₄

—

106

D₄

—

124

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

124

—

112

C₂×C₄

—

124

—

113

C₂×C₄

C₂²×D₃₁

yes

—

114

C₂×C₄

C₂²×D₃₁

yes

—

115

C₂×C₄

C₂²×D₃₁

yes

characteristic

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

—

123

C₂²

C₂³×D₃₁

yes

characteristic

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

—

130

C₂

—

131

C₂

—

132

C₂

—

124

—

133

C₂

#11758561

yes

characteristic

134

C₂

#41×D₃₁

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₁
commutator, socle, cosocle	C₂²×C₃₁	C₂⁴
Fitting	C₃₁×#123	C₂
prime radical	#11758517	C₂
hypercenter	#123	D₃₁
hypocenter	C₃₁	#559
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ #123

subgroup

G/H

center, Frattini

C₂²

C₂³×D₃₁

commutator, socle, cosocle

C₂²×C₃₁

C₂⁴

Fitting

C₃₁×#123

C₂

C₂

D₃₁

C₃₁

G ▷ C₂²×C₃₁ ▷ C₃₁

C₁ ◁ C₂² ◁ #123

N	H ≅ G/N	type
1. C₃₁×#123	C₂	semidirect
3. #11758512	C₂	semidirect
4. C₄×C₂²⋊D₃₁	C₂	semidirect
5. #11758471	C₂	semidirect
6. #11758464	C₂	semidirect
7. #11758462	C₂	semidirect
8. #11758462	C₂	semidirect
9. (C₂²⋊D₃₁)⋊C₄	C₂	semidirect
73. #123	D₃₁	semidirect
77. C₃₁	#559	semidirect

H ≅ G/N

type

1. C₃₁×#123

C₂

semidirect

3. #11758512

C₂

semidirect

4. C₄×C₂²⋊D₃₁

C₂

semidirect

5. #11758471

C₂

semidirect

6. #11758464

C₂

semidirect

7. #11758462

C₂

semidirect

8. #11758462

C₂

semidirect

9. (C₂²⋊D₃₁)⋊C₄

C₂

semidirect

73. #123

D₃₁

semidirect

77. C₃₁

#559

semidirect

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

subgroup	index
C₃₁×#123	2
#11758517	2
#11758512	2
C₄×C₂²⋊D₃₁	2
#11758471	2
#11758464	2
#11758462	2
(C₂²⋊D₃₁)⋊C₄	2
#559	31

subgroup

index

C₃₁×#123

C₄×C₂²⋊D₃₁

(C₂²⋊D₃₁)⋊C₄

Subgroups — #14521783

Conjugacy classes of subgroups

Complements

Maximal subgroups