Group #15689746 — subgroups

Subgroups — #15689746

Order 3008.

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

normal subgroups only

This group has 216 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	#13094023	C₂	1	1	yes	maximal, characteristic, complemented
2	#13094019	C₂	1	2	yes	maximal, complemented
3	#13094013	C₂	1	1	yes	maximal, characteristic, complemented
4	#13094010	C₂	1	2	yes	maximal, complemented
5	#13094006	C₂	1	2	yes	maximal, complemented
6	#13094004	C₂	1	2	yes	maximal, complemented
7	#13093998	C₂	1	1	yes	maximal, characteristic
8	Q₈×Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#13093986	C₂	1	1	yes	maximal, characteristic, complemented
10	#13093983	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²⋊Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₈×C₄₇	C₂²	1	1	yes	characteristic
13	C₂²×C₄×C₄₇	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊D₄₇	C₂²	1	2	yes	—
15	C₂×D₁₈₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₄₇	C₂²	1	2	yes	—
17	C₂×Dic₉₄	C₂²	1	2	yes	—
18	C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
19	C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
20	C₂²⋊C₄×C₄₇	C₂²	1	2	yes	—
21	C₂²⋊Dic₄₇	C₂²	1	1	yes	characteristic
22	C₂²⋊Dic₄₇	C₂²	1	1	yes	characteristic
23	(C₂×D₄₇)⋊C₄	C₂²	1	2	yes	—
24	(C₂×D₄₇)⋊C₄	C₂²	1	2	yes	—
25	(C₂×D₄₇)⋊C₄	C₂²	1	2	yes	—
26	(C₂×D₄₇)⋊C₄	C₂²	1	2	yes	—
27	C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
28	C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
29	C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
30	C₄⋊Dic₄₇	C₂²	1	2	yes	—
31	Dic₄₇⋊C₄	C₂²	1	2	yes	—
32	C₄×Dic₄₇	C₂²	1	1	yes	characteristic
33	C₄×Dic₄₇	C₂²	1	1	yes	characteristic
34	C₄×Dic₄₇	C₂²	1	2	yes	—
35	C₂³×C₄₇	C₂³	1	1	yes	characteristic
36	C₂²×D₄₇	C₂³	1	2	yes	—
37	Q₈×C₄₇	—	2	2	no	—
38	C₂×C₄×C₄₇	—	2	2	no	—
39	C₂×C₄×C₄₇	C₂³	1	2	yes	—
40	C₂×C₄×C₄₇	C₂³	1	2	yes	—
41	C₂×C₄×C₄₇	C₂³	1	1	yes	characteristic
42	C₂×C₄×C₄₇	C₂³	1	1	yes	characteristic
43	C₂²⋊D₄₇	—	4	2	no	—
44	C₂×Dic₄₇	C₂³	1	1	yes	characteristic
45	C₂×Dic₄₇	C₂³	1	1	yes	characteristic
46	C₂×Dic₄₇	C₂³	1	2	yes	—
47	C₂×Dic₄₇	C₂³	1	1	yes	characteristic
48	C₂×Dic₄₇	C₂³	1	1	yes	characteristic
49	D₁₈₈	—	2	2	no	—
50	C₄×D₄₇	—	4	2	no	—
51	Dic₉₄	—	4	2	no	—
52	C₂²×C₄₇	—	2	1	no	—
53	C₂²×C₄₇	—	2	1	no	—
54	C₂²×C₄₇	—	2	1	no	—
55	C₂²×C₄₇	C₂⁴	1	1	yes	characteristic
56	C₂×D₄₇	—	2	2	no	—
57	C₂×D₄₇	—	2	2	no	—
58	C₂×D₄₇	—	2	2	no	—
59	C₄×C₄₇	—	2	1	no	—
60	C₄×C₄₇	—	2	2	no	—
61	C₄×C₄₇	—	2	2	no	—
62	C₄×C₄₇	#41	1	2	yes	—
63	Dic₄₇	—	2	1	no	—
64	Dic₄₇	—	2	2	no	—
65	Dic₄₇	—	2	1	no	—
66	Dic₄₇	—	2	1	no	—
67	Dic₄₇	—	2	1	no	—
68	C₂×C₄₇	—	4	1	no	—
69	C₂×C₄₇	C₂×#41	1	1	yes	characteristic
70	C₂×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
71	C₂×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
72	D₄₇	—	4	2	no	—
73	#542	—	47	1	no	maximal, Sylow
74	C₄₇	#542	1	1	yes	characteristic, complemented, Sylow
75	C₄⋊Q₈	—	47	1	no	—
76	#126	—	47	2	no	—
77	#124	—	47	1	no	—
78	#124	—	47	2	no	—
79	#123	—	47	2	no	—
80	C₂²⋊Q₈	—	47	2	no	—
81	C₂²⋊Q₈	D₄₇	1	1	yes	characteristic, complemented
82	#121	—	47	1	no	—
83	C₄×Q₈	—	47	1	no	—
84	C₄×D₄	—	47	1	no	—
85	#117	—	47	1	no	—
86	C₂×Q₈	—	47	2	no	—
87	C₂×Q₈	C₂×D₄₇	1	1	yes	characteristic
88	C₂×D₄	—	47	2	no	—
89	C₂×D₄	—	47	1	no	—
90	C₂²×C₄	—	47	2	no	—
91	C₂²×C₄	C₂×D₄₇	1	1	yes	characteristic
92	C₄⋊C₄	—	47	2	no	—
93	C₄⋊C₄	—	47	1	no	—
94	C₄⋊C₄	—	47	1	no	—
95	C₄⋊C₄	—	47	1	no	—
96	C₄⋊C₄	—	47	2	no	—
97	C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
98	C₄⋊C₄	C₂×D₄₇	1	2	yes	—
99	C₂²⋊C₄	—	47	1	no	—
100	C₂²⋊C₄	—	47	1	no	—
101	C₂²⋊C₄	—	47	2	no	—
102	C₂²⋊C₄	—	47	2	no	—
103	C₂²⋊C₄	—	47	2	no	—
104	C₂²⋊C₄	—	47	2	no	—
105	C₂²⋊C₄	C₂×D₄₇	1	2	yes	—
106	C₄²	—	47	1	no	—
107	C₄²	—	47	1	no	—
108	C₄²	—	47	2	no	—
109	C₂³	—	47	2	no	—
110	C₂³	C₂²×D₄₇	1	1	yes	characteristic
111	Q₈	—	2	2	no	—
112	Q₈	—	188	2	no	—
113	D₄	—	94	2	no	—
114	D₄	—	188	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	47	1	no	—
117	C₂×C₄	—	47	1	no	—
118	C₂×C₄	—	47	1	no	—
119	C₂×C₄	—	47	1	no	—
120	C₂×C₄	—	47	2	no	—
121	C₂×C₄	—	188	2	no	—
122	C₂×C₄	C₂²×D₄₇	1	2	yes	—
123	C₂×C₄	C₂²×D₄₇	1	1	yes	characteristic
124	C₂×C₄	C₂²×D₄₇	1	2	yes	—
125	C₂×C₄	C₂²×D₄₇	1	1	yes	characteristic
126	C₂²	—	2	1	no	—
127	C₂²	—	2	1	no	—
128	C₂²	—	2	1	no	—
129	C₂²	—	94	2	no	—
130	C₂²	—	94	2	no	—
131	C₂²	—	94	2	no	—
132	C₂²	C₂³×D₄₇	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	2	2	no	—
135	C₄	—	2	2	no	—
136	C₄	—	94	1	no	—
137	C₄	—	94	2	no	—
138	C₄	—	94	1	no	—
139	C₄	—	94	1	no	—
140	C₄	—	94	1	no	—
141	C₄	#10625352	1	2	yes	—
142	C₂	—	4	1	no	—
143	C₂	—	188	2	no	—
144	C₂	C₂×#10625352	1	1	yes	characteristic
145	C₂	#13094070	1	1	yes	characteristic
146	C₂	#13094072	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13094023

C₂

yes

maximal, characteristic, complemented

#13094019

C₂

yes

maximal, complemented

#13094013

C₂

yes

maximal, characteristic, complemented

#13094010

C₂

yes

maximal, complemented

#13094006

C₂

yes

maximal, complemented

#13094004

C₂

yes

maximal, complemented

#13093998

C₂

yes

maximal, characteristic

Q₈×Dic₄₇

C₂

yes

maximal, characteristic, complemented

#13093986

C₂

yes

maximal, characteristic, complemented

#13093983

C₂

yes

maximal, characteristic, complemented

C₂²⋊Q₈×C₄₇

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×C₄₇

C₂²

yes

characteristic

C₂²×C₄×C₄₇

C₂²

yes

characteristic

C₂×C₂²⋊D₄₇

C₂²

yes

—

C₂×D₁₈₈

C₂²

yes

characteristic

C₂×C₄×D₄₇

C₂²

yes

—

C₂×Dic₉₄

C₂²

yes

—

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

(C₂×D₄₇)⋊C₄

C₂²

yes

—

(C₂×D₄₇)⋊C₄

C₂²

yes

—

(C₂×D₄₇)⋊C₄

C₂²

yes

—

(C₂×D₄₇)⋊C₄

C₂²

yes

—

C₄⋊Dic₄₇

C₂²

yes

characteristic

C₄⋊Dic₄₇

C₂²

yes

characteristic

C₄⋊Dic₄₇

C₂²

yes

characteristic

C₄⋊Dic₄₇

C₂²

yes

—

Dic₄₇⋊C₄

C₂²

yes

—

C₄×Dic₄₇

C₂²

yes

characteristic

C₄×Dic₄₇

C₂²

yes

characteristic

C₄×Dic₄₇

C₂²

yes

—

C₂³×C₄₇

C₂³

yes

characteristic

C₂²×D₄₇

C₂³

yes

—

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

C₂²⋊D₄₇

—

C₂×Dic₄₇

C₂³

yes

characteristic

C₂×Dic₄₇

C₂³

yes

characteristic

C₂×Dic₄₇

C₂³

yes

—

C₂×Dic₄₇

C₂³

yes

characteristic

C₂×Dic₄₇

C₂³

yes

characteristic

D₁₈₈

—

C₄×D₄₇

—

Dic₉₄

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂⁴

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#41

yes

—

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

C₂×C₄₇

—

C₂×C₄₇

C₂×#41

yes

characteristic

C₂×C₄₇

H(5,𝔽₂)

yes

characteristic

C₂×C₄₇

2₋¹⁺⁴

yes

characteristic

D₄₇

—

#542

—

maximal, Sylow

C₄₇

#542

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

—

—

—

—

C₂²⋊Q₈

—

C₂²⋊Q₈

D₄₇

yes

characteristic, complemented

#121

—

C₄×Q₈

—

C₄×D₄

—

#117

—

C₂×Q₈

—

C₂×Q₈

C₂×D₄₇

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₄₇

yes

characteristic

C₄⋊C₄

C₂×D₄₇

yes

—

C₂²⋊C₄

—

100

C₂²⋊C₄

—

101

C₂²⋊C₄

—

102

C₂²⋊C₄

—

103

C₂²⋊C₄

—

104

C₂²⋊C₄

—

105

C₂²⋊C₄

C₂×D₄₇

yes

—

106

C₄²

—

107

C₄²

—

108

C₄²

—

109

C₂³

—

110

C₂³

C₂²×D₄₇

yes

characteristic

111

Q₈

—

112

Q₈

—

188

—

113

D₄

—

114

D₄

—

188

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

188

—

122

C₂×C₄

C₂²×D₄₇

yes

—

123

C₂×C₄

C₂²×D₄₇

yes

characteristic

124

C₂×C₄

C₂²×D₄₇

yes

—

125

C₂×C₄

C₂²×D₄₇

yes

characteristic

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

C₂³×D₄₇

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

#10625352

yes

—

142

C₂

—

143

C₂

—

188

—

144

C₂

C₂×#10625352

yes

characteristic

145

C₂

#13094070

yes

characteristic

146

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	#13093998	C₂
hypercenter	C₂²⋊Q₈	D₄₇
hypocenter	C₄₇	#542
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

#13093998

C₂

hypercenter

C₂²⋊Q₈

D₄₇

C₄₇

G ▷ C₂²×C₄₇ ▷ C₄₇

C₁ ◁ C₂² ◁ C₂²⋊Q₈

N	H ≅ G/N	type
1. #13094023	C₂	semidirect
2. #13094019	C₂	semidirect
3. #13094013	C₂	semidirect
4. #13094010	C₂	semidirect
5. #13094006	C₂	semidirect
6. #13094004	C₂	semidirect
8. Q₈×Dic₄₇	C₂	semidirect
9. #13093986	C₂	semidirect
10. #13093983	C₂	semidirect
11. C₂²⋊Q₈×C₄₇	C₂	semidirect
74. C₄₇	#542	semidirect
81. C₂²⋊Q₈	D₄₇	semidirect

H ≅ G/N

type

1. #13094023

C₂

semidirect

2. #13094019

C₂

semidirect

3. #13094013

C₂

semidirect

4. #13094010

C₂

semidirect

5. #13094006

C₂

semidirect

6. #13094004

C₂

semidirect

8. Q₈×Dic₄₇

C₂

semidirect

9. #13093986

C₂

semidirect

10. #13093983

C₂

semidirect

11. C₂²⋊Q₈×C₄₇

C₂

semidirect

74. C₄₇

#542

semidirect

81. C₂²⋊Q₈

D₄₇

semidirect

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

subgroup	index
#13094023	2
#13094019	2
#13094013	2
#13094010	2
#13094006	2
#13094004	2
#13093998	2
Q₈×Dic₄₇	2
#13093986	2
#13093983	2
C₂²⋊Q₈×C₄₇	2
#542	47

subgroup

index

Q₈×Dic₄₇

C₂²⋊Q₈×C₄₇

Subgroups — #15689746

Conjugacy classes of subgroups

Complements

Maximal subgroups