Group #14521646 — subgroups

Subgroups — #14521646

Order 1984.

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

normal subgroups only

This group has 218 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	C₂×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
2	#11758507	C₂	1	1	yes	maximal, characteristic
3	#11758506	C₂	1	2	yes	maximal, complemented
4	C₄×C₂²⋊D₃₁	C₂	1	2	yes	maximal, complemented
5	#11758501	C₂	1	1	yes	maximal, characteristic, complemented
6	#11758475	C₂	1	2	yes	maximal, complemented
7	#11758473	C₂	1	2	yes	maximal, complemented
8	#11758471	C₂	1	2	yes	maximal, complemented
9	#11758466	C₂	1	2	yes	maximal, complemented
10	C₂²×C₄×C₃₁	C₂²	1	1	yes	characteristic
11	C₂²×C₄×C₃₁	C₂²	1	1	yes	characteristic
12	C₂²×C₄×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊D₃₁	C₂²	1	2	yes	—
14	C₂×D₁₂₄	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₃₁	C₂²	1	2	yes	—
16	C₂×Dic₆₂	C₂²	1	1	yes	characteristic
17	C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
18	C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
19	C₂²⋊Dic₃₁	C₂²	1	2	yes	—
20	(C₂×D₃₁)⋊C₄	C₂²	1	2	yes	—
21	(C₂×D₃₁)⋊C₄	C₂²	1	2	yes	—
22	(C₂×D₃₁)⋊C₄	C₂²	1	2	yes	—
23	(C₂×D₃₁)⋊C₄	C₂²	1	2	yes	—
24	C₄⋊Dic₃₁	C₂²	1	1	yes	characteristic
25	C₄⋊Dic₃₁	C₂²	1	1	yes	characteristic
26	Dic₃₁⋊C₄	C₂²	1	2	yes	—
27	Dic₃₁⋊C₄	C₂²	1	2	yes	—
28	Dic₃₁⋊C₄	C₂²	1	2	yes	—
29	Dic₃₁⋊C₄	C₂²	1	2	yes	—
30	C₄×Dic₃₁	C₂²	1	2	yes	—
31	C₂³×C₃₁	C₂³	1	1	yes	characteristic
32	C₂²×D₃₁	C₂³	1	2	yes	—
33	C₂×C₄×C₃₁	—	2	2	no	—
34	C₂×C₄×C₃₁	—	4	1	no	—
35	C₂×C₄×C₃₁	—	4	1	no	—
36	C₂×C₄×C₃₁	C₂³	1	2	yes	—
37	C₂×C₄×C₃₁	C₂³	1	1	yes	characteristic
38	C₂×C₄×C₃₁	C₂³	1	2	yes	—
39	C₂×C₄×C₃₁	C₂³	1	1	yes	characteristic
40	C₂²⋊D₃₁	—	2	4	no	—
41	C₂×Dic₃₁	C₂³	1	2	yes	—
42	C₂×Dic₃₁	C₂³	1	2	yes	—
43	C₂×Dic₃₁	C₂³	1	2	yes	—
44	D₁₂₄	—	4	1	no	—
45	C₄×D₃₁	—	4	2	no	—
46	Dic₆₂	—	4	1	no	—
47	C₂²×C₃₁	—	2	1	no	—
48	C₂²×C₃₁	—	2	1	no	—
49	C₂²×C₃₁	#41	1	2	yes	—
50	C₂²×C₃₁	C₂⁴	1	1	yes	characteristic
51	C₂×D₃₁	—	2	2	no	—
52	C₂×D₃₁	—	2	2	no	—
53	C₂×D₃₁	—	2	2	no	—
54	C₄×C₃₁	—	2	2	no	—
55	C₄×C₃₁	—	2	1	no	—
56	C₄×C₃₁	—	2	2	no	—
57	C₄×C₃₁	—	2	1	no	—
58	Dic₃₁	—	2	2	no	—
59	Dic₃₁	—	2	2	no	—
60	Dic₃₁	—	2	2	no	—
61	#539	—	31	1	no	maximal, Sylow
62	C₂×C₃₁	—	2	2	no	—
63	C₂×C₃₁	C₂×#41	1	1	yes	characteristic
64	C₂×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
65	C₂×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
66	D₃₁	—	4	2	no	—
67	#126	—	31	2	no	—
68	#125	—	31	2	no	—
69	#123	—	31	2	no	—
70	#123	—	31	2	no	—
71	C₂²⋊Q₈	—	31	1	no	—
72	C₂²⋊Q₈	—	31	2	no	—
73	#121	—	31	1	no	—
74	C₄×D₄	—	31	2	no	—
75	C₂×C₄⋊C₄	D₃₁	1	1	yes	characteristic, complemented
76	C₃₁	#539	1	1	yes	characteristic, complemented, Sylow
77	C₂×Q₈	—	31	1	no	—
78	C₂×D₄	—	31	2	no	—
79	C₂×D₄	—	31	1	no	—
80	C₂²×C₄	—	31	2	no	—
81	C₂²×C₄	C₂×D₃₁	1	1	yes	characteristic
82	C₂²×C₄	C₂×D₃₁	1	1	yes	characteristic
83	C₂²×C₄	C₂×D₃₁	1	1	yes	characteristic
84	C₄⋊C₄	—	31	2	no	—
85	C₄⋊C₄	—	31	2	no	—
86	C₄⋊C₄	—	31	2	no	—
87	C₄⋊C₄	—	31	2	no	—
88	C₄⋊C₄	—	31	1	no	—
89	C₄⋊C₄	—	31	1	no	—
90	C₄⋊C₄	C₂×D₃₁	1	2	yes	—
91	C₄⋊C₄	C₂×D₃₁	1	2	yes	—
92	C₂²⋊C₄	—	31	2	no	—
93	C₂²⋊C₄	—	31	2	no	—
94	C₂²⋊C₄	—	31	2	no	—
95	C₂²⋊C₄	—	31	2	no	—
96	C₂²⋊C₄	—	31	2	no	—
97	C₄²	—	31	2	no	—
98	C₂³	—	31	2	no	—
99	C₂³	C₂²×D₃₁	1	1	yes	characteristic
100	Q₈	—	124	1	no	—
101	D₄	—	62	4	no	—
102	D₄	—	124	1	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	4	1	no	—
105	C₂×C₄	—	4	1	no	—
106	C₂×C₄	—	31	2	no	—
107	C₂×C₄	—	31	2	no	—
108	C₂×C₄	—	31	2	no	—
109	C₂×C₄	—	124	2	no	—
110	C₂×C₄	C₂²×D₃₁	1	2	yes	—
111	C₂×C₄	C₂²×D₃₁	1	2	yes	—
112	C₂×C₄	C₂²×D₃₁	1	1	yes	characteristic
113	C₂×C₄	C₂²×D₃₁	1	1	yes	characteristic
114	C₂²	—	2	1	no	—
115	C₂²	—	2	1	no	—
116	C₂²	—	62	2	no	—
117	C₂²	—	62	2	no	—
118	C₂²	—	62	2	no	—
119	C₂²	#92495	1	2	yes	—
120	C₂²	C₂³×D₃₁	1	1	yes	characteristic
121	C₄	—	2	2	no	—
122	C₄	—	2	2	no	—
123	C₄	—	2	1	no	—
124	C₄	—	2	1	no	—
125	C₄	—	62	2	no	—
126	C₄	—	62	2	no	—
127	C₄	—	62	2	no	—
128	C₂	—	2	2	no	—
129	C₂	—	124	2	no	—
130	C₂	C₂×#92495	1	1	yes	characteristic
131	C₂	#11758561	1	1	yes	characteristic
132	C₂	#11758564	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊C₄×C₃₁

C₂

yes

maximal, characteristic, complemented

#11758507

C₂

yes

maximal, characteristic

#11758506

C₂

yes

maximal, complemented

C₄×C₂²⋊D₃₁

C₂

yes

maximal, complemented

#11758501

C₂

yes

maximal, characteristic, complemented

#11758475

C₂

yes

maximal, complemented

#11758473

C₂

yes

maximal, complemented

#11758471

C₂

yes

maximal, complemented

#11758466

C₂

yes

maximal, complemented

C₂²×C₄×C₃₁

C₂²

yes

characteristic

C₂²×C₄×C₃₁

C₂²

yes

characteristic

C₂²×C₄×C₃₁

C₂²

yes

characteristic

C₂×C₂²⋊D₃₁

C₂²

yes

—

C₂×D₁₂₄

C₂²

yes

characteristic

C₂×C₄×D₃₁

C₂²

yes

—

C₂×Dic₆₂

C₂²

yes

characteristic

C₄⋊C₄×C₃₁

C₂²

yes

—

C₄⋊C₄×C₃₁

C₂²

yes

—

C₂²⋊Dic₃₁

C₂²

yes

—

(C₂×D₃₁)⋊C₄

C₂²

yes

—

(C₂×D₃₁)⋊C₄

C₂²

yes

—

(C₂×D₃₁)⋊C₄

C₂²

yes

—

(C₂×D₃₁)⋊C₄

C₂²

yes

—

C₄⋊Dic₃₁

C₂²

yes

characteristic

C₄⋊Dic₃₁

C₂²

yes

characteristic

Dic₃₁⋊C₄

C₂²

yes

—

Dic₃₁⋊C₄

C₂²

yes

—

Dic₃₁⋊C₄

C₂²

yes

—

Dic₃₁⋊C₄

C₂²

yes

—

C₄×Dic₃₁

C₂²

yes

—

C₂³×C₃₁

C₂³

yes

characteristic

C₂²×D₃₁

C₂³

yes

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂³

yes

—

C₂×C₄×C₃₁

C₂³

yes

characteristic

C₂×C₄×C₃₁

C₂³

yes

—

C₂×C₄×C₃₁

C₂³

yes

characteristic

C₂²⋊D₃₁

—

C₂×Dic₃₁

C₂³

yes

—

C₂×Dic₃₁

C₂³

yes

—

C₂×Dic₃₁

C₂³

yes

—

D₁₂₄

—

C₄×D₃₁

—

Dic₆₂

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

#41

yes

—

C₂²×C₃₁

C₂⁴

yes

characteristic

C₂×D₃₁

—

C₂×D₃₁

—

C₂×D₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

Dic₃₁

—

Dic₃₁

—

Dic₃₁

—

#539

—

maximal, Sylow

C₂×C₃₁

—

C₂×C₃₁

C₂×#41

yes

characteristic

C₂×C₃₁

H(5,𝔽₂)

yes

characteristic

C₂×C₃₁

2₋¹⁺⁴

yes

characteristic

D₃₁

—

—

—

—

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

#121

—

C₄×D₄

—

C₂×C₄⋊C₄

D₃₁

yes

characteristic, complemented

C₃₁

#539

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₃₁

yes

characteristic

C₂²×C₄

C₂×D₃₁

yes

characteristic

C₂²×C₄

C₂×D₃₁

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₃₁

yes

—

C₄⋊C₄

C₂×D₃₁

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₂³

—

C₂³

C₂²×D₃₁

yes

characteristic

100

Q₈

—

124

—

101

D₄

—

102

D₄

—

124

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

124

—

110

C₂×C₄

C₂²×D₃₁

yes

—

111

C₂×C₄

C₂²×D₃₁

yes

—

112

C₂×C₄

C₂²×D₃₁

yes

characteristic

113

C₂×C₄

C₂²×D₃₁

yes

characteristic

114

C₂²

—

115

C₂²

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

#92495

yes

—

120

C₂²

C₂³×D₃₁

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

128

C₂

—

129

C₂

—

124

—

130

C₂

C₂×#92495

yes

characteristic

131

C₂

#11758561

yes

characteristic

132

C₂

#11758564

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₁
commutator, socle, cosocle	C₂²×C₃₁	C₂⁴
Fitting	C₂×C₄⋊C₄×C₃₁	C₂
prime radical	#11758507	C₂
hypercenter	C₂×C₄⋊C₄	D₃₁
hypocenter	C₃₁	#539
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄

subgroup

G/H

center, Frattini

C₂²

C₂³×D₃₁

commutator, socle, cosocle

C₂²×C₃₁

C₂⁴

Fitting

C₂×C₄⋊C₄×C₃₁

C₂

prime radical

#11758507

C₂

hypercenter

C₂×C₄⋊C₄

D₃₁

C₃₁

G ▷ C₂²×C₃₁ ▷ C₃₁

C₁ ◁ C₂² ◁ C₂×C₄⋊C₄

N	H ≅ G/N	type
1. C₂×C₄⋊C₄×C₃₁	C₂	semidirect
3. #11758506	C₂	semidirect
4. C₄×C₂²⋊D₃₁	C₂	semidirect
5. #11758501	C₂	semidirect
6. #11758475	C₂	semidirect
7. #11758473	C₂	semidirect
8. #11758471	C₂	semidirect
9. #11758466	C₂	semidirect
75. C₂×C₄⋊C₄	D₃₁	semidirect
76. C₃₁	#539	semidirect

H ≅ G/N

type

1. C₂×C₄⋊C₄×C₃₁

C₂

semidirect

3. #11758506

C₂

semidirect

4. C₄×C₂²⋊D₃₁

C₂

semidirect

5. #11758501

C₂

semidirect

6. #11758475

C₂

semidirect

7. #11758473

C₂

semidirect

8. #11758471

C₂

semidirect

9. #11758466

C₂

semidirect

75. C₂×C₄⋊C₄

D₃₁

semidirect

76. C₃₁

#539

semidirect

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

subgroup	index
C₂×C₄⋊C₄×C₃₁	2
#11758507	2
#11758506	2
C₄×C₂²⋊D₃₁	2
#11758501	2
#11758475	2
#11758473	2
#11758471	2
#11758466	2
#539	31

subgroup

index

C₂×C₄⋊C₄×C₃₁

C₄×C₂²⋊D₃₁

Subgroups — #14521646

Conjugacy classes of subgroups

Complements

Maximal subgroups