Group #14521657 — subgroups

Subgroups — #14521657

Order 1984.

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

normal subgroups only

This group has 192 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	C₄×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
2	#11758523	C₂	1	1	yes	maximal, characteristic
3	#11758521	C₂	1	1	yes	maximal, characteristic, complemented
4	#11758475	C₂	1	2	yes	maximal
5	#11758474	C₂	1	1	yes	maximal, characteristic
6	#11758467	C₂	1	1	yes	maximal, characteristic, complemented
7	#11758466	C₂	1	2	yes	maximal, complemented
8	#11758453	C₂	1	2	yes	maximal
9	#11758449	C₂	1	1	yes	maximal, characteristic
10	#11758447	C₂	1	2	yes	maximal, complemented
11	C₄×Dic₆₂	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₈×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₃₁	C₂²	1	1	yes	characteristic
14	C₂×Dic₆₂	C₂²	1	1	yes	characteristic
15	C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
16	C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
17	C₄²×C₃₁	C₂²	1	1	yes	characteristic
18	C₄²×C₃₁	C₂²	1	2	yes	—
19	(C₂×D₃₁)⋊C₄	C₂²	1	1	yes	characteristic
20	(C₂×D₃₁)⋊C₄	C₂²	1	1	yes	characteristic
21	(C₂×D₃₁)⋊C₄	C₂²	1	2	yes	—
22	(C₂×D₃₁)⋊C₄	C₂²	1	2	yes	—
23	C₄⋊Dic₃₁	C₂²	1	1	yes	characteristic
24	C₄⋊Dic₃₁	C₂²	1	2	yes	—
25	C₄⋊Dic₃₁	C₂²	1	2	yes	—
26	Dic₃₁⋊C₄	C₂²	1	2	yes	—
27	Dic₃₁⋊C₄	C₂²	1	1	yes	characteristic
28	Dic₃₁⋊C₄	C₂²	1	2	yes	—
29	Dic₃₁⋊C₄	C₂²	1	2	yes	—
30	Dic₃₁⋊C₄	C₂²	1	1	yes	characteristic
31	Dic₃₁⋊C₄	C₂²	1	2	yes	—
32	Dic₃₁⋊C₄	C₂²	1	2	yes	—
33	C₄×Dic₃₁	C₂²	1	2	yes	—
34	C₄×Dic₃₁	C₂²	1	1	yes	characteristic
35	C₂²×D₃₁	C₂³	1	1	yes	characteristic
36	Q₈×C₃₁	—	2	2	no	—
37	C₂×C₄×C₃₁	C₂³	1	1	yes	characteristic
38	C₂×C₄×C₃₁	C₂³	1	1	yes	characteristic
39	C₂×C₄×C₃₁	C₂³	1	2	yes	—
40	C₂×C₄×C₃₁	C₂³	1	2	yes	—
41	C₂×C₄×C₃₁	C₂³	1	1	yes	characteristic
42	C₂×Dic₃₁	C₂³	1	1	yes	characteristic
43	C₂×Dic₃₁	C₂³	1	2	yes	—
44	C₂×Dic₃₁	C₂³	1	1	yes	characteristic
45	C₂×Dic₃₁	C₂³	1	2	yes	—
46	C₂×Dic₃₁	C₂³	1	1	yes	characteristic
47	C₄×D₃₁	—	2	2	no	—
48	Dic₆₂	—	2	2	no	—
49	C₂²×C₃₁	C₂⁴	1	1	yes	characteristic
50	C₂×D₃₁	—	2	1	no	—
51	C₂×D₃₁	—	2	1	no	—
52	C₂×D₃₁	—	2	1	no	—
53	C₄×C₃₁	—	2	2	no	—
54	C₄×C₃₁	—	2	1	no	—
55	C₄×C₃₁	—	2	1	no	—
56	C₄×C₃₁	—	2	2	no	—
57	C₄×C₃₁	#41	1	2	yes	—
58	Dic₃₁	—	2	2	no	—
59	Dic₃₁	—	2	1	no	—
60	Dic₃₁	—	2	2	no	—
61	Dic₃₁	—	2	1	no	—
62	Dic₃₁	—	2	1	no	—
63	#541	—	31	1	no	maximal, Sylow
64	C₂×C₃₁	C₂×#41	1	1	yes	characteristic
65	C₂×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
66	C₂×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
67	D₃₁	—	4	1	no	—
68	C₄⋊Q₈	—	31	1	no	—
69	#126	—	31	2	no	—
70	#126	—	31	2	no	—
71	#125	—	31	2	no	—
72	#125	—	31	1	no	—
73	#125	—	31	2	no	—
74	C₂²⋊Q₈	—	31	1	no	—
75	C₂²⋊Q₈	—	31	1	no	—
76	C₄×Q₈	—	31	1	no	—
77	C₄×Q₈	D₃₁	1	1	yes	characteristic, complemented
78	#117	—	31	1	no	—
79	C₃₁	#541	1	1	yes	characteristic, complemented, Sylow
80	C₂×Q₈	—	31	1	no	—
81	C₂×Q₈	C₂×D₃₁	1	1	yes	characteristic
82	C₂²×C₄	—	31	1	no	—
83	C₄⋊C₄	—	31	1	no	—
84	C₄⋊C₄	—	31	1	no	—
85	C₄⋊C₄	—	31	2	no	—
86	C₄⋊C₄	—	31	2	no	—
87	C₄⋊C₄	—	31	2	no	—
88	C₄⋊C₄	—	31	2	no	—
89	C₄⋊C₄	—	31	2	no	—
90	C₄⋊C₄	—	31	1	no	—
91	C₄⋊C₄	—	31	2	no	—
92	C₄⋊C₄	—	31	2	no	—
93	C₄⋊C₄	C₂×D₃₁	1	1	yes	characteristic
94	C₄⋊C₄	C₂×D₃₁	1	2	yes	—
95	C₂²⋊C₄	—	31	1	no	—
96	C₂²⋊C₄	—	31	1	no	—
97	C₂²⋊C₄	—	31	2	no	—
98	C₂²⋊C₄	—	31	2	no	—
99	C₄²	—	31	1	no	—
100	C₄²	—	31	2	no	—
101	C₄²	C₂×D₃₁	1	1	yes	characteristic
102	C₄²	C₂×D₃₁	1	2	yes	—
103	C₂³	—	31	1	no	—
104	Q₈	—	2	2	no	—
105	Q₈	—	62	2	no	—
106	C₂×C₄	—	31	1	no	—
107	C₂×C₄	—	31	2	no	—
108	C₂×C₄	—	31	1	no	—
109	C₂×C₄	—	31	2	no	—
110	C₂×C₄	—	31	1	no	—
111	C₂×C₄	—	62	2	no	—
112	C₂×C₄	C₂²×D₃₁	1	1	yes	characteristic
113	C₂×C₄	C₂²×D₃₁	1	1	yes	characteristic
114	C₂×C₄	C₂²×D₃₁	1	2	yes	—
115	C₂×C₄	C₂²×D₃₁	1	2	yes	—
116	C₂×C₄	C₂²×D₃₁	1	1	yes	characteristic
117	C₂²	—	62	1	no	—
118	C₂²	—	62	1	no	—
119	C₂²	—	62	1	no	—
120	C₂²	C₂³×D₃₁	1	1	yes	characteristic
121	C₄	—	2	2	no	—
122	C₄	—	2	1	no	—
123	C₄	—	2	1	no	—
124	C₄	—	2	2	no	—
125	C₄	—	62	1	no	—
126	C₄	—	62	2	no	—
127	C₄	—	62	1	no	—
128	C₄	—	62	2	no	—
129	C₄	—	62	1	no	—
130	C₄	#92495	1	2	yes	—
131	C₂	—	124	1	no	—
132	C₂	C₂×#92495	1	1	yes	characteristic
133	C₂	#11758564	1	1	yes	characteristic
134	C₂	#11758567	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×Q₈×C₃₁

C₂

yes

maximal, characteristic, complemented

#11758523

C₂

yes

maximal, characteristic

#11758521

C₂

yes

maximal, characteristic, complemented

#11758475

C₂

yes

maximal

#11758474

C₂

yes

maximal, characteristic

#11758467

C₂

yes

maximal, characteristic, complemented

#11758466

C₂

yes

maximal, complemented

#11758453

C₂

yes

maximal

#11758449

C₂

yes

maximal, characteristic

#11758447

C₂

yes

maximal, complemented

C₄×Dic₆₂

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×C₃₁

C₂²

yes

characteristic

C₂×C₄×D₃₁

C₂²

yes

characteristic

C₂×Dic₆₂

C₂²

yes

characteristic

C₄⋊C₄×C₃₁

C₂²

yes

—

C₄⋊C₄×C₃₁

C₂²

yes

characteristic

C₄²×C₃₁

C₂²

yes

characteristic

C₄²×C₃₁

C₂²

yes

—

(C₂×D₃₁)⋊C₄

C₂²

yes

characteristic

(C₂×D₃₁)⋊C₄

C₂²

yes

characteristic

(C₂×D₃₁)⋊C₄

C₂²

yes

—

(C₂×D₃₁)⋊C₄

C₂²

yes

—

C₄⋊Dic₃₁

C₂²

yes

characteristic

C₄⋊Dic₃₁

C₂²

yes

—

C₄⋊Dic₃₁

C₂²

yes

—

Dic₃₁⋊C₄

C₂²

yes

—

Dic₃₁⋊C₄

C₂²

yes

characteristic

Dic₃₁⋊C₄

C₂²

yes

—

Dic₃₁⋊C₄

C₂²

yes

—

Dic₃₁⋊C₄

C₂²

yes

characteristic

Dic₃₁⋊C₄

C₂²

yes

—

Dic₃₁⋊C₄

C₂²

yes

—

C₄×Dic₃₁

C₂²

yes

—

C₄×Dic₃₁

C₂²

yes

characteristic

C₂²×D₃₁

C₂³

yes

characteristic

Q₈×C₃₁

—

C₂×C₄×C₃₁

C₂³

yes

characteristic

C₂×C₄×C₃₁

C₂³

yes

characteristic

C₂×C₄×C₃₁

C₂³

yes

—

C₂×C₄×C₃₁

C₂³

yes

—

C₂×C₄×C₃₁

C₂³

yes

characteristic

C₂×Dic₃₁

C₂³

yes

characteristic

C₂×Dic₃₁

C₂³

yes

—

C₂×Dic₃₁

C₂³

yes

characteristic

C₂×Dic₃₁

C₂³

yes

—

C₂×Dic₃₁

C₂³

yes

characteristic

C₄×D₃₁

—

Dic₆₂

—

C₂²×C₃₁

C₂⁴

yes

characteristic

C₂×D₃₁

—

C₂×D₃₁

—

C₂×D₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

#41

yes

—

Dic₃₁

—

Dic₃₁

—

Dic₃₁

—

Dic₃₁

—

Dic₃₁

—

#541

—

maximal, Sylow

C₂×C₃₁

C₂×#41

yes

characteristic

C₂×C₃₁

2₋¹⁺⁴

yes

characteristic

C₂×C₃₁

2₋¹⁺⁴

yes

characteristic

D₃₁

—

C₄⋊Q₈

—

—

—

—

—

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

D₃₁

yes

characteristic, complemented

#117

—

C₃₁

#541

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

C₂×D₃₁

yes

characteristic

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₃₁

yes

characteristic

C₄⋊C₄

C₂×D₃₁

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

100

C₄²

—

101

C₄²

C₂×D₃₁

yes

characteristic

102

C₄²

C₂×D₃₁

yes

—

103

C₂³

—

104

Q₈

—

105

Q₈

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

C₂²×D₃₁

yes

characteristic

113

C₂×C₄

C₂²×D₃₁

yes

characteristic

114

C₂×C₄

C₂²×D₃₁

yes

—

115

C₂×C₄

C₂²×D₃₁

yes

—

116

C₂×C₄

C₂²×D₃₁

yes

characteristic

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

C₂³×D₃₁

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

#92495

yes

—

131

C₂

—

124

—

132

C₂

C₂×#92495

yes

characteristic

133

C₂

#11758564

yes

characteristic

134

C₂

#11758567

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₁
commutator, socle, cosocle	C₂²×C₃₁	C₂⁴
Fitting	C₄×Q₈×C₃₁	C₂
prime radical	C₂²×D₃₁	C₂³
hypercenter	C₄×Q₈	D₃₁
hypocenter	C₃₁	#541
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₃₁

commutator, socle, cosocle

C₂²×C₃₁

C₂⁴

Fitting

C₄×Q₈×C₃₁

C₂

prime radical

C₂²×D₃₁

C₂³

hypercenter

C₄×Q₈

D₃₁

C₃₁

G ▷ C₂²×C₃₁ ▷ C₃₁

C₁ ◁ C₂² ◁ C₄×Q₈

N	H ≅ G/N	type
1. C₄×Q₈×C₃₁	C₂	semidirect
3. #11758521	C₂	semidirect
6. #11758467	C₂	semidirect
7. #11758466	C₂	semidirect
10. #11758447	C₂	semidirect
11. C₄×Dic₆₂	C₂	semidirect
77. C₄×Q₈	D₃₁	semidirect
79. C₃₁	#541	semidirect

H ≅ G/N

type

1. C₄×Q₈×C₃₁

C₂

semidirect

3. #11758521

C₂

semidirect

6. #11758467

C₂

semidirect

7. #11758466

C₂

semidirect

10. #11758447

C₂

semidirect

11. C₄×Dic₆₂

C₂

semidirect

77. C₄×Q₈

D₃₁

semidirect

79. C₃₁

#541

semidirect

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

subgroup	index
C₄×Q₈×C₃₁	2
#11758523	2
#11758521	2
#11758475	2
#11758474	2
#11758467	2
#11758466	2
#11758453	2
#11758449	2
#11758447	2
C₄×Dic₆₂	2
#541	31

subgroup

index

C₄×Q₈×C₃₁

C₄×Dic₆₂

Subgroups — #14521657

Conjugacy classes of subgroups

Complements

Maximal subgroups