Group #15689752 — subgroups

Subgroups — #15689752

Order 3008.

206 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 103 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	#13094023	C₂	1	2	yes	maximal, complemented
2	#13094022	C₂	1	2	yes	maximal, complemented
3	#13094014	C₂	1	2	yes	maximal, complemented
4	#13094014	C₂	1	2	yes	maximal, complemented
5	#13094003	C₂	1	1	yes	maximal, characteristic
6	#13094001	C₂	1	2	yes	maximal
7	#13093980	C₂	1	2	yes	maximal
8	C₂×Dic₄₇⋊C₄	C₂	1	1	yes	maximal, characteristic
9	C₂²⋊Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
10	C₂×Q₈×C₄₇	C₂²	1	1	yes	characteristic
11	C₂²×C₄×C₄₇	C₂²	1	1	yes	characteristic
12	C₂²×Dic₄₇	C₂²	1	1	yes	characteristic
13	C₂²×Dic₄₇	C₂²	1	1	yes	characteristic
14	C₂×Dic₉₄	C₂²	1	2	yes	—
15	C₂×Dic₉₄	C₂²	1	1	yes	characteristic
16	C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
17	C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
18	C₂²⋊C₄×C₄₇	C₂²	1	2	yes	—
19	C₂²⋊Dic₄₇	C₂²	1	2	yes	—
20	C₄⋊Dic₄₇	C₂²	1	2	yes	—
21	C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
22	Dic₄₇⋊C₄	C₂²	1	2	yes	—
23	Dic₄₇⋊C₄	C₂²	1	2	yes	—
24	Dic₄₇⋊C₄	C₂²	1	2	yes	—
25	Dic₄₇⋊C₄	C₂²	1	2	yes	—
26	Dic₄₇⋊C₄	C₂²	1	2	yes	—
27	Dic₄₇⋊C₄	C₂²	1	2	yes	—
28	Dic₄₇⋊C₄	C₂²	1	2	yes	—
29	C₄×Dic₄₇	C₂²	1	2	yes	—
30	C₄×Dic₄₇	C₂²	1	2	yes	—
31	C₂³×C₄₇	C₂³	1	1	yes	characteristic
32	Q₈×C₄₇	—	4	1	no	—
33	C₂×C₄×C₄₇	—	4	1	no	—
34	C₂×C₄×C₄₇	C₂³	1	2	yes	—
35	C₂×C₄×C₄₇	C₂³	1	2	yes	—
36	C₂×C₄×C₄₇	C₂³	1	1	yes	characteristic
37	C₂×C₄×C₄₇	C₂³	1	1	yes	characteristic
38	C₂×Dic₄₇	—	4	1	no	—
39	C₂×Dic₄₇	Q₈	1	4	yes	complemented
40	C₂×Dic₄₇	C₂³	1	2	yes	—
41	C₂×Dic₄₇	C₂³	1	2	yes	—
42	C₂×Dic₄₇	C₂³	1	2	yes	—
43	C₂×Dic₄₇	C₂³	1	2	yes	—
44	Dic₉₄	—	4	1	no	—
45	Dic₉₄	—	4	2	no	—
46	C₂²×C₄₇	—	2	1	no	—
47	C₂²×C₄₇	—	2	1	no	—
48	C₂²×C₄₇	C₂×Q₈	1	2	yes	—
49	C₂²×C₄₇	C₂⁴	1	1	yes	characteristic
50	C₄×C₄₇	—	2	1	no	—
51	C₄×C₄₇	—	2	2	no	—
52	C₄×C₄₇	—	2	1	no	—
53	C₄×C₄₇	—	2	2	no	—
54	Dic₄₇	—	2	2	no	—
55	Dic₄₇	—	2	2	no	—
56	Dic₄₇	—	2	2	no	—
57	Dic₄₇	C₂×Q₈	1	4	yes	—
58	C₂×C₄₇	—	2	2	no	—
59	C₂×C₄₇	C₂²×Q₈	1	1	yes	characteristic
60	C₂×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
61	C₂×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
62	#544	—	47	1	no	maximal, Sylow
63	C₄₇	#544	1	1	yes	characteristic, complemented, Sylow
64	C₄⋊Q₈	—	47	2	no	—
65	C₄⋊Q₈	—	47	2	no	—
66	#125	—	47	2	no	—
67	#125	—	47	2	no	—
68	C₂²⋊Q₈	—	47	1	no	—
69	C₂²⋊Q₈	—	47	2	no	—
70	C₂²⋊Q₈	D₄₇	1	1	yes	characteristic
71	#117	—	47	2	no	—
72	C₂×C₄⋊C₄	—	47	1	no	—
73	C₂×Q₈	—	47	1	no	—
74	C₂×Q₈	—	47	2	no	—
75	C₂×Q₈	C₂×D₄₇	1	1	yes	characteristic
76	C₂²×C₄	—	47	1	no	—
77	C₂²×C₄	—	47	1	no	—
78	C₂²×C₄	C₂×D₄₇	1	1	yes	characteristic
79	C₄⋊C₄	—	47	2	no	—
80	C₄⋊C₄	—	47	2	no	—
81	C₄⋊C₄	—	47	2	no	—
82	C₄⋊C₄	—	47	2	no	—
83	C₄⋊C₄	—	47	2	no	—
84	C₄⋊C₄	—	47	2	no	—
85	C₄⋊C₄	—	47	2	no	—
86	C₄⋊C₄	—	47	1	no	—
87	C₄⋊C₄	—	47	2	no	—
88	C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
89	C₄⋊C₄	C₂×D₄₇	1	2	yes	—
90	C₂²⋊C₄	—	47	2	no	—
91	C₂²⋊C₄	C₂×D₄₇	1	2	yes	—
92	C₄²	—	47	2	no	—
93	C₄²	—	47	2	no	—
94	C₂³	C₂²×D₄₇	1	1	yes	characteristic
95	Q₈	—	4	1	no	—
96	Q₈	—	188	1	no	—
97	Q₈	—	188	2	no	—
98	C₂×C₄	—	4	1	no	—
99	C₂×C₄	—	47	4	no	—
100	C₂×C₄	—	47	2	no	—
101	C₂×C₄	—	47	2	no	—
102	C₂×C₄	—	47	2	no	—
103	C₂×C₄	—	47	2	no	—
104	C₂×C₄	—	188	1	no	—
105	C₂×C₄	C₂²×D₄₇	1	2	yes	—
106	C₂×C₄	C₂²×D₄₇	1	2	yes	—
107	C₂×C₄	C₂²×D₄₇	1	1	yes	characteristic
108	C₂×C₄	C₂²×D₄₇	1	1	yes	characteristic
109	C₂²	—	2	1	no	—
110	C₂²	—	2	1	no	—
111	C₂²	Q₈×D₄₇	1	2	yes	—
112	C₂²	C₂³×D₄₇	1	1	yes	characteristic
113	C₄	—	2	1	no	—
114	C₄	—	2	1	no	—
115	C₄	—	2	2	no	—
116	C₄	—	2	2	no	—
117	C₄	—	47	4	no	—
118	C₄	—	94	2	no	—
119	C₄	—	94	2	no	—
120	C₄	—	94	2	no	—
121	C₂	—	2	2	no	—
122	C₂	C₂×Q₈×D₄₇	1	1	yes	characteristic
123	C₂	#13094069	1	1	yes	characteristic
124	C₂	#13094071	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13094023

C₂

yes

maximal, complemented

#13094022

C₂

yes

maximal, complemented

#13094014

C₂

yes

maximal, complemented

#13094014

C₂

yes

maximal, complemented

#13094003

C₂

yes

maximal, characteristic

#13094001

C₂

yes

maximal

#13093980

C₂

yes

maximal

C₂×Dic₄₇⋊C₄

C₂

yes

maximal, characteristic

C₂²⋊Q₈×C₄₇

C₂

yes

maximal, characteristic

C₂×Q₈×C₄₇

C₂²

yes

characteristic

C₂²×C₄×C₄₇

C₂²

yes

characteristic

C₂²×Dic₄₇

C₂²

yes

characteristic

C₂²×Dic₄₇

C₂²

yes

characteristic

C₂×Dic₉₄

C₂²

yes

—

C₂×Dic₉₄

C₂²

yes

characteristic

C₄⋊C₄×C₄₇

C₂²

yes

characteristic

C₄⋊C₄×C₄₇

C₂²

yes

—

C₂²⋊C₄×C₄₇

C₂²

yes

—

C₂²⋊Dic₄₇

C₂²

yes

—

C₄⋊Dic₄₇

C₂²

yes

—

C₄⋊Dic₄₇

C₂²

yes

characteristic

Dic₄₇⋊C₄

C₂²

yes

—

Dic₄₇⋊C₄

C₂²

yes

—

Dic₄₇⋊C₄

C₂²

yes

—

Dic₄₇⋊C₄

C₂²

yes

—

Dic₄₇⋊C₄

C₂²

yes

—

Dic₄₇⋊C₄

C₂²

yes

—

Dic₄₇⋊C₄

C₂²

yes

—

C₄×Dic₄₇

C₂²

yes

—

C₄×Dic₄₇

C₂²

yes

—

C₂³×C₄₇

C₂³

yes

characteristic

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂³

yes

—

C₂×C₄×C₄₇

C₂³

yes

—

C₂×C₄×C₄₇

C₂³

yes

characteristic

C₂×C₄×C₄₇

C₂³

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

Q₈

yes

complemented

C₂×Dic₄₇

C₂³

yes

—

C₂×Dic₄₇

C₂³

yes

—

C₂×Dic₄₇

C₂³

yes

—

C₂×Dic₄₇

C₂³

yes

—

Dic₉₄

—

Dic₉₄

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×Q₈

yes

—

C₂²×C₄₇

C₂⁴

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

Dic₄₇

C₂×Q₈

yes

—

C₂×C₄₇

—

C₂×C₄₇

C₂²×Q₈

yes

characteristic

C₂×C₄₇

H(5,𝔽₂)

yes

characteristic

C₂×C₄₇

2₋¹⁺⁴

yes

—

C₄₇

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₂²⋊Q₈

D₄₇

yes

characteristic

#117

—

C₂×C₄⋊C₄

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×D₄₇

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₄₇

yes

characteristic

C₄⋊C₄

C₂×D₄₇

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

C₂×D₄₇

yes

—

C₄²

—

C₄²

—

C₂³

C₂²×D₄₇

yes

characteristic

Q₈

—

Q₈

—

188

—

Q₈

—

188

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

188

—

105

C₂×C₄

C₂²×D₄₇

yes

—

106

C₂×C₄

C₂²×D₄₇

yes

—

107

C₂×C₄

C₂²×D₄₇

yes

characteristic

108

C₂×C₄

C₂²×D₄₇

yes

characteristic

109

C₂²

—

110

C₂²

—

111

C₂²

Q₈×D₄₇

yes

—

112

C₂²

C₂³×D₄₇

yes

characteristic

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₂

—

122

C₂

C₂×Q₈×D₄₇

yes

characteristic

123

C₂

#13094069

yes

characteristic

124

C₂

#13094071

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₄₇
commutator, socle, cosocle	C₂²×C₄₇	C₂⁴
Fitting	C₂²⋊Q₈×C₄₇	C₂
prime radical	C₂³×C₄₇	C₂³
hypercenter	C₂²⋊Q₈	D₄₇
hypocenter	C₄₇	#544
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₄₇

commutator, socle, cosocle

C₂²×C₄₇

C₂⁴

Fitting

C₂²⋊Q₈×C₄₇

C₂

prime radical

C₂³×C₄₇

C₂³

hypercenter

C₂²⋊Q₈

D₄₇

C₄₇

G ▷ C₂²×C₄₇ ▷ C₄₇

C₁ ◁ C₂² ◁ C₂²⋊Q₈

N	H ≅ G/N	type
1. #13094023	C₂	semidirect
2. #13094022	C₂	semidirect
3. #13094014	C₂	semidirect
4. #13094014	C₂	semidirect
39. C₂×Dic₄₇	Q₈	semidirect
63. C₄₇	#544	semidirect

H ≅ G/N

type

1. #13094023

C₂

semidirect

2. #13094022

C₂

semidirect

3. #13094014

C₂

semidirect

4. #13094014

C₂

semidirect

39. C₂×Dic₄₇

Q₈

semidirect

63. C₄₇

#544

semidirect

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

subgroup	index
#13094023	2
#13094022	2
#13094014	2
#13094003	2
#13094001	2
#13093980	2
C₂×Dic₄₇⋊C₄	2
C₂²⋊Q₈×C₄₇	2
#544	47

subgroup

index

C₂×Dic₄₇⋊C₄

C₂²⋊Q₈×C₄₇

#544

Subgroups — #15689752

Conjugacy classes of subgroups

Complements

Maximal subgroups