Group #15689733 — subgroups

Subgroups — #15689733

Order 3008.

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	#13094022	C₂	1	1	yes	maximal, characteristic, complemented
2	#13094017	C₂	1	2	yes	maximal
3	#13094017	C₂	1	2	yes	maximal
4	#13094015	C₂	1	1	yes	maximal, characteristic, complemented
5	#13094015	C₂	1	2	yes	maximal, complemented
6	#13094014	C₂	1	2	yes	maximal, complemented
7	#13094003	C₂	1	1	yes	maximal, characteristic
8	Q₈×Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#13093989	C₂	1	1	yes	maximal, characteristic, complemented
10	#13093983	C₂	1	1	yes	maximal, characteristic
11	C₂²⋊Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
12	C₂×Q₈×C₄₇	C₂²	1	1	yes	characteristic
13	C₂²×C₄×C₄₇	C₂²	1	1	yes	characteristic
14	C₂×Dic₉₄	C₂²	1	1	yes	characteristic
15	C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
16	C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
17	C₂²⋊C₄×C₄₇	C₂²	1	2	yes	—
18	C₂²⋊Dic₄₇	C₂²	1	1	yes	characteristic
19	C₂²⋊Dic₄₇	C₂²	1	1	yes	characteristic
20	C₂²⋊Dic₄₇	C₂²	1	2	yes	—
21	C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
22	C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
23	C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
24	C₄⋊Dic₄₇	C₂²	1	2	yes	—
25	C₄⋊Dic₄₇	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	Dic₄₇⋊C₄	C₂²	1	2	yes	—
30	Dic₄₇⋊C₄	C₂²	1	2	yes	—
31	C₄×Dic₄₇	C₂²	1	1	yes	characteristic
32	C₄×Dic₄₇	C₂²	1	1	yes	characteristic
33	C₄×Dic₄₇	C₂²	1	2	yes	—
34	C₄×Dic₄₇	C₂²	1	2	yes	—
35	C₂³×C₄₇	C₂³	1	1	yes	characteristic
36	Q₈×C₄₇	—	2	2	no	—
37	C₂×C₄×C₄₇	—	2	2	no	—
38	C₂×C₄×C₄₇	C₂³	1	2	yes	—
39	C₂×C₄×C₄₇	C₂³	1	2	yes	—
40	C₂×C₄×C₄₇	C₂³	1	1	yes	characteristic
41	C₂×C₄×C₄₇	C₂³	1	1	yes	characteristic
42	C₂×Dic₄₇	C₂³	1	1	yes	characteristic
43	C₂×Dic₄₇	C₂³	1	1	yes	characteristic
44	C₂×Dic₄₇	C₂³	1	1	yes	characteristic
45	C₂×Dic₄₇	C₂³	1	2	yes	—
46	C₂×Dic₄₇	C₂³	1	1	yes	characteristic
47	C₂×Dic₄₇	C₂³	1	2	yes	—
48	Dic₉₄	—	2	2	no	—
49	C₂²×C₄₇	—	2	1	no	—
50	C₂²×C₄₇	—	2	1	no	—
51	C₂²×C₄₇	—	2	1	no	—
52	C₂²×C₄₇	C₂⁴	1	1	yes	characteristic
53	C₄×C₄₇	—	2	1	no	—
54	C₄×C₄₇	—	2	2	no	—
55	C₄×C₄₇	—	2	2	no	—
56	C₄×C₄₇	#41	1	2	yes	—
57	Dic₄₇	—	2	2	no	—
58	Dic₄₇	—	2	1	no	—
59	Dic₄₇	—	2	1	no	—
60	Dic₄₇	—	2	1	no	—
61	Dic₄₇	—	2	1	no	—
62	Dic₄₇	—	2	2	no	—
63	C₂×C₄₇	—	4	1	no	—
64	C₂×C₄₇	C₂×#41	1	1	yes	characteristic
65	C₂×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
66	C₂×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
67	#541	—	47	1	no	maximal, Sylow
68	C₄₇	#541	1	1	yes	characteristic, complemented, Sylow
69	C₄⋊Q₈	—	47	1	no	—
70	#126	—	47	2	no	—
71	#126	—	47	2	no	—
72	#125	—	47	2	no	—
73	#125	—	47	1	no	—
74	#125	—	47	2	no	—
75	C₂²⋊Q₈	—	47	1	no	—
76	C₂²⋊Q₈	D₄₇	1	1	yes	characteristic
77	C₄×Q₈	—	47	1	no	—
78	C₄×Q₈	—	47	1	no	—
79	#117	—	47	1	no	—
80	C₂×Q₈	—	47	1	no	—
81	C₂×Q₈	C₂×D₄₇	1	1	yes	characteristic
82	C₂²×C₄	C₂×D₄₇	1	1	yes	characteristic
83	C₄⋊C₄	—	47	2	no	—
84	C₄⋊C₄	—	47	2	no	—
85	C₄⋊C₄	—	47	2	no	—
86	C₄⋊C₄	—	47	2	no	—
87	C₄⋊C₄	—	47	2	no	—
88	C₄⋊C₄	—	47	1	no	—
89	C₄⋊C₄	—	47	1	no	—
90	C₄⋊C₄	—	47	1	no	—
91	C₄⋊C₄	—	47	2	no	—
92	C₄⋊C₄	—	47	2	no	—
93	C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
94	C₄⋊C₄	C₂×D₄₇	1	2	yes	—
95	C₂²⋊C₄	—	47	1	no	—
96	C₂²⋊C₄	—	47	1	no	—
97	C₂²⋊C₄	—	47	2	no	—
98	C₂²⋊C₄	C₂×D₄₇	1	2	yes	—
99	C₄²	—	47	1	no	—
100	C₄²	—	47	1	no	—
101	C₄²	—	47	2	no	—
102	C₄²	—	47	2	no	—
103	C₂³	C₂²×D₄₇	1	1	yes	characteristic
104	Q₈	—	2	2	no	—
105	Q₈	—	94	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	47	1	no	—
108	C₂×C₄	—	47	1	no	—
109	C₂×C₄	—	47	1	no	—
110	C₂×C₄	—	47	1	no	—
111	C₂×C₄	—	47	2	no	—
112	C₂×C₄	—	47	2	no	—
113	C₂×C₄	C₂²×D₄₇	1	2	yes	—
114	C₂×C₄	C₂²×D₄₇	1	2	yes	—
115	C₂×C₄	C₂²×D₄₇	1	1	yes	characteristic
116	C₂×C₄	C₂²×D₄₇	1	1	yes	characteristic
117	C₂²	—	2	1	no	—
118	C₂²	—	2	1	no	—
119	C₂²	—	2	1	no	—
120	C₂²	C₂³×D₄₇	1	1	yes	characteristic
121	C₄	—	2	1	no	—
122	C₄	—	2	2	no	—
123	C₄	—	2	2	no	—
124	C₄	—	94	2	no	—
125	C₄	—	94	1	no	—
126	C₄	—	94	1	no	—
127	C₄	—	94	1	no	—
128	C₄	—	94	1	no	—
129	C₄	—	94	2	no	—
130	C₄	#10625352	1	2	yes	—
131	C₂	—	4	1	no	—
132	C₂	C₂×#10625352	1	1	yes	characteristic
133	C₂	#13094071	1	1	yes	characteristic
134	C₂	#13094072	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13094022

C₂

yes

maximal, characteristic, complemented

#13094017

C₂

yes

maximal

#13094017

C₂

yes

maximal

#13094015

C₂

yes

maximal, characteristic, complemented

#13094015

C₂

yes

maximal, complemented

#13094014

C₂

yes

maximal, complemented

#13094003

C₂

yes

maximal, characteristic

Q₈×Dic₄₇

C₂

yes

maximal, characteristic, complemented

#13093989

C₂

yes

maximal, characteristic, complemented

#13093983

C₂

yes

maximal, characteristic

C₂²⋊Q₈×C₄₇

C₂

yes

maximal, characteristic

C₂×Q₈×C₄₇

C₂²

yes

characteristic

C₂²×C₄×C₄₇

C₂²

yes

characteristic

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

characteristic

C₂²⋊Dic₄₇

C₂²

yes

characteristic

C₂²⋊Dic₄₇

C₂²

yes

—

C₄⋊Dic₄₇

C₂²

yes

characteristic

C₄⋊Dic₄₇

C₂²

yes

characteristic

C₄⋊Dic₄₇

C₂²

yes

characteristic

C₄⋊Dic₄₇

C₂²

yes

—

C₄⋊Dic₄₇

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

characteristic

C₄×Dic₄₇

C₂²

yes

characteristic

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

C₂³

yes

characteristic

C₂×Dic₄₇

C₂³

yes

characteristic

C₂×Dic₄₇

C₂³

yes

characteristic

C₂×Dic₄₇

C₂³

yes

—

C₂×Dic₄₇

C₂³

yes

characteristic

C₂×Dic₄₇

C₂³

yes

—

Dic₉₄

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂⁴

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#41

yes

—

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

C₂×C₄₇

—

C₂×C₄₇

C₂×#41

yes

characteristic

C₂×C₄₇

2₋¹⁺⁴

yes

characteristic

C₂×C₄₇

2₋¹⁺⁴

yes

—

C₄₇

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

—

—

—

—

—

C₂²⋊Q₈

—

C₂²⋊Q₈

D₄₇

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×Q₈

—

C₂×Q₈

C₂×D₄₇

yes

characteristic

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

—

C₄⋊C₄

C₂×D₄₇

yes

characteristic

C₄⋊C₄

C₂×D₄₇

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

C₂×D₄₇

yes

—

C₄²

—

100

C₄²

—

101

C₄²

—

102

C₄²

—

103

C₂³

C₂²×D₄₇

yes

characteristic

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

—

113

C₂×C₄

C₂²×D₄₇

yes

—

114

C₂×C₄

C₂²×D₄₇

yes

—

115

C₂×C₄

C₂²×D₄₇

yes

characteristic

116

C₂×C₄

C₂²×D₄₇

yes

characteristic

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

C₂³×D₄₇

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

#10625352

yes

—

131

C₂

—

132

C₂

C₂×#10625352

yes

characteristic

133

C₂

#13094071

yes

characteristic

134

C₂

#13094072

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	#541
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. #13094022	C₂	semidirect
4. #13094015	C₂	semidirect
5. #13094015	C₂	semidirect
6. #13094014	C₂	semidirect
8. Q₈×Dic₄₇	C₂	semidirect
9. #13093989	C₂	semidirect
68. C₄₇	#541	semidirect

H ≅ G/N

type

1. #13094022

C₂

semidirect

4. #13094015

C₂

semidirect

5. #13094015

C₂

semidirect

6. #13094014

C₂

semidirect

8. Q₈×Dic₄₇

C₂

semidirect

9. #13093989

C₂

semidirect

68. C₄₇

#541

semidirect

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

subgroup	index
#13094022	2
#13094017	2
#13094015	2
#13094014	2
#13094003	2
Q₈×Dic₄₇	2
#13093989	2
#13093983	2
C₂²⋊Q₈×C₄₇	2
#541	47

subgroup

index

Q₈×Dic₄₇

C₂²⋊Q₈×C₄₇

Subgroups — #15689733

Conjugacy classes of subgroups

Complements

Maximal subgroups