Group #15689837 — subgroups

Subgroups — #15689837

Order 3008.

236 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 97 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	#13094011	C₂	1	2	yes	maximal, complemented
3	#13094010	C₂	1	2	yes	maximal, complemented
4	#13093996	C₂	1	2	yes	maximal, complemented
5	#13093995	C₂	1	1	yes	maximal, characteristic, complemented
6	D₄×Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×C₂²⋊D₄₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#13093983	C₂	1	1	yes	maximal, characteristic, complemented
9	#13093982	C₂	1	1	yes	maximal, characteristic, complemented
10	#13093980	C₂	1	2	yes	maximal, complemented
11	#121×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₄×C₄₇	C₂²	1	1	yes	characteristic
13	C₂×D₄×C₄₇	C₂²	1	2	yes	—
14	C₂²×C₄×C₄₇	C₂²	1	1	yes	characteristic
15	C₂×C₂²⋊D₄₇	C₂²	1	1	yes	characteristic
16	C₂×C₂²⋊D₄₇	C₂²	1	2	yes	—
17	C₂²×Dic₄₇	C₂²	1	2	yes	—
18	C₂×C₄×D₄₇	C₂²	1	1	yes	characteristic
19	C₂×Dic₉₄	C₂²	1	2	yes	—
20	C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
21	C₂²⋊C₄×C₄₇	C₂²	1	2	yes	—
22	C₂²⋊Dic₄₇	C₂²	1	2	yes	—
23	C₂²⋊Dic₄₇	C₂²	1	2	yes	—
24	C₂²⋊Dic₄₇	C₂²	1	1	yes	characteristic
25	C₂²⋊Dic₄₇	C₂²	1	2	yes	—
26	(C₂×D₄₇)⋊C₄	C₂²	1	2	yes	—
27	(C₂×D₄₇)⋊C₄	C₂²	1	1	yes	characteristic
28	C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
29	C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
30	Dic₄₇⋊C₄	C₂²	1	1	yes	characteristic
31	Dic₄₇⋊C₄	C₂²	1	2	yes	—
32	C₄×Dic₄₇	C₂²	1	2	yes	—
33	C₄×Dic₄₇	C₂²	1	2	yes	—
34	C₄×Dic₄₇	C₂²	1	1	yes	characteristic
35	C₂³×C₄₇	C₂³	1	1	yes	characteristic
36	C₂³×C₄₇	C₂³	1	2	yes	—
37	C₂²×D₄₇	C₂³	1	1	yes	characteristic
38	D₄×C₄₇	—	2	2	no	—
39	D₄×C₄₇	—	4	2	no	—
40	C₂×C₄×C₄₇	—	2	2	no	—
41	C₂×C₄×C₄₇	C₂³	1	1	yes	characteristic
42	C₂×C₄×C₄₇	C₂³	1	2	yes	—
43	C₂×C₄×C₄₇	C₂³	1	1	yes	characteristic
44	C₂²⋊D₄₇	—	2	2	no	—
45	C₂²⋊D₄₇	—	4	2	no	—
46	C₂×Dic₄₇	—	2	4	no	—
47	C₂×Dic₄₇	C₂³	1	1	yes	characteristic
48	C₂×Dic₄₇	C₂³	1	2	yes	—
49	C₂×Dic₄₇	C₂³	1	2	yes	—
50	C₂×Dic₄₇	C₂³	1	1	yes	characteristic
51	C₂×Dic₄₇	C₂³	1	1	yes	characteristic
52	C₄×D₄₇	—	2	2	no	—
53	Dic₉₄	—	4	2	no	—
54	C₂²×C₄₇	—	2	2	no	—
55	C₂²×C₄₇	—	2	1	no	—
56	C₂²×C₄₇	—	2	1	no	—
57	C₂²×C₄₇	—	2	2	no	—
58	C₂²×C₄₇	—	2	2	no	—
59	C₂²×C₄₇	—	2	1	no	—
60	C₂²×C₄₇	C₂⁴	1	1	yes	characteristic
61	C₂×D₄₇	—	2	1	no	—
62	C₂×D₄₇	—	2	1	no	—
63	C₂×D₄₇	—	2	1	no	—
64	C₄×C₄₇	—	2	1	no	—
65	C₄×C₄₇	—	2	2	no	—
66	C₄×C₄₇	#41	1	2	yes	—
67	Dic₄₇	—	2	1	no	—
68	Dic₄₇	—	2	2	no	—
69	Dic₄₇	—	2	1	no	—
70	Dic₄₇	—	2	2	no	—
71	Dic₄₇	#41	1	2	yes	—
72	C₂×C₄₇	—	4	2	no	—
73	C₂×C₄₇	—	4	1	no	—
74	C₂×C₄₇	C₂×#41	1	1	yes	characteristic
75	C₂×C₄₇	C₂×#41	1	1	yes	characteristic
76	C₂×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
77	D₄₇	—	4	1	no	—
78	#555	—	47	1	no	maximal, Sylow
79	C₄₇	#555	1	1	yes	characteristic, complemented, Sylow
80	C₄⋊Q₈	—	47	1	no	—
81	#124	—	47	2	no	—
82	#124	—	47	2	no	—
83	#121	—	47	1	no	—
84	#121	—	47	2	no	—
85	#121	D₄₇	1	1	yes	characteristic, complemented
86	C₄×D₄	—	47	1	no	—
87	C₄×D₄	—	47	1	no	—
88	#117	—	47	1	no	—
89	#117	—	47	1	no	—
90	#117	—	47	2	no	—
91	C₂×Q₈	—	47	2	no	—
92	C₂×D₄	—	47	1	no	—
93	C₂×D₄	—	47	2	no	—
94	C₂×D₄	C₂×D₄₇	1	1	yes	characteristic
95	C₂×D₄	C₂×D₄₇	1	2	yes	—
96	C₂²×C₄	—	47	2	no	—
97	C₂²×C₄	—	47	1	no	—
98	C₂²×C₄	C₂×D₄₇	1	1	yes	characteristic
99	C₄⋊C₄	—	47	1	no	—
100	C₄⋊C₄	—	47	2	no	—
101	C₄⋊C₄	—	47	1	no	—
102	C₄⋊C₄	—	47	1	no	—
103	C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
104	C₂²⋊C₄	—	47	2	no	—
105	C₂²⋊C₄	—	47	2	no	—
106	C₂²⋊C₄	—	47	1	no	—
107	C₂²⋊C₄	—	47	2	no	—
108	C₂²⋊C₄	—	47	2	no	—
109	C₂²⋊C₄	—	47	1	no	—
110	C₂²⋊C₄	C₂×D₄₇	1	2	yes	—
111	C₄²	—	47	2	no	—
112	C₄²	—	47	2	no	—
113	C₄²	—	47	1	no	—
114	C₂³	—	47	1	no	—
115	C₂³	C₂²×D₄₇	1	1	yes	characteristic
116	C₂³	C₂²×D₄₇	1	2	yes	—
117	Q₈	—	188	2	no	—
118	D₄	—	2	2	no	—
119	D₄	—	4	2	no	—
120	D₄	—	94	2	no	—
121	D₄	—	188	2	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	47	1	no	—
124	C₂×C₄	—	47	2	no	—
125	C₂×C₄	—	47	1	no	—
126	C₂×C₄	—	47	1	no	—
127	C₂×C₄	—	47	2	no	—
128	C₂×C₄	—	94	4	no	—
129	C₂×C₄	—	94	2	no	—
130	C₂×C₄	C₂²×D₄₇	1	1	yes	characteristic
131	C₂×C₄	C₂²×D₄₇	1	2	yes	—
132	C₂×C₄	C₂²×D₄₇	1	1	yes	characteristic
133	C₂²	—	2	2	no	—
134	C₂²	—	2	1	no	—
135	C₂²	—	2	1	no	—
136	C₂²	—	2	2	no	—
137	C₂²	—	2	1	no	—
138	C₂²	—	2	2	no	—
139	C₂²	—	94	1	no	—
140	C₂²	—	94	1	no	—
141	C₂²	—	94	1	no	—
142	C₂²	C₂³×D₄₇	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	2	2	no	—
145	C₄	—	47	2	no	—
146	C₄	—	94	2	no	—
147	C₄	—	94	1	no	—
148	C₄	—	94	2	no	—
149	C₄	—	94	1	no	—
150	C₄	#10625352	1	2	yes	—
151	C₂	—	4	2	no	—
152	C₂	—	4	1	no	—
153	C₂	—	188	1	no	—
154	C₂	#41×D₄₇	1	1	yes	characteristic
155	C₂	C₂×#10625352	1	1	yes	characteristic
156	C₂	#13094069	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13094023

C₂

yes

maximal, characteristic, complemented

#13094011

C₂

yes

maximal, complemented

#13094010

C₂

yes

maximal, complemented

#13093996

C₂

yes

maximal, complemented

#13093995

C₂

yes

maximal, characteristic, complemented

D₄×Dic₄₇

C₂

yes

maximal, characteristic, complemented

C₄×C₂²⋊D₄₇

C₂

yes

maximal, characteristic, complemented

#13093983

C₂

yes

maximal, characteristic, complemented

#13093982

C₂

yes

maximal, characteristic, complemented

#13093980

C₂

yes

maximal, complemented

#121×C₄₇

C₂

yes

maximal, characteristic, complemented

C₂×D₄×C₄₇

C₂²

yes

characteristic

C₂×D₄×C₄₇

C₂²

yes

—

C₂²×C₄×C₄₇

C₂²

yes

characteristic

C₂×C₂²⋊D₄₇

C₂²

yes

characteristic

C₂×C₂²⋊D₄₇

C₂²

yes

—

C₂²×Dic₄₇

C₂²

yes

—

C₂×C₄×D₄₇

C₂²

yes

characteristic

C₂×Dic₉₄

C₂²

yes

—

C₄⋊C₄×C₄₇

C₂²

yes

characteristic

C₂²⋊C₄×C₄₇

C₂²

yes

—

C₂²⋊Dic₄₇

C₂²

yes

—

C₂²⋊Dic₄₇

C₂²

yes

—

C₂²⋊Dic₄₇

C₂²

yes

characteristic

C₂²⋊Dic₄₇

C₂²

yes

—

(C₂×D₄₇)⋊C₄

C₂²

yes

—

(C₂×D₄₇)⋊C₄

C₂²

yes

characteristic

C₄⋊Dic₄₇

C₂²

yes

characteristic

C₄⋊Dic₄₇

C₂²

yes

characteristic

Dic₄₇⋊C₄

C₂²

yes

characteristic

Dic₄₇⋊C₄

C₂²

yes

—

C₄×Dic₄₇

C₂²

yes

—

C₄×Dic₄₇

C₂²

yes

—

C₄×Dic₄₇

C₂²

yes

characteristic

C₂³×C₄₇

C₂³

yes

characteristic

C₂³×C₄₇

C₂³

yes

—

C₂²×D₄₇

C₂³

yes

characteristic

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂³

yes

characteristic

C₂×C₄×C₄₇

C₂³

yes

—

C₂×C₄×C₄₇

C₂³

yes

characteristic

C₂²⋊D₄₇

—

C₂²⋊D₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

C₂³

yes

characteristic

C₂×Dic₄₇

C₂³

yes

—

C₂×Dic₄₇

C₂³

yes

—

C₂×Dic₄₇

C₂³

yes

characteristic

C₂×Dic₄₇

C₂³

yes

characteristic

C₄×D₄₇

—

Dic₉₄

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂⁴

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#41

yes

—

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

Dic₄₇

#41

yes

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

C₂×#41

yes

characteristic

C₂×C₄₇

C₂×#41

yes

characteristic

C₂×C₄₇

H(5,𝔽₂)

yes

characteristic

D₄₇

—

#555

—

maximal, Sylow

C₄₇

#555

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

—

—

—

—

D₄₇

yes

characteristic, complemented

C₄×D₄

—

C₄×D₄

—

#117

—

#117

—

#117

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×D₄₇

yes

characteristic

C₂×D₄

C₂×D₄₇

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄₇

yes

characteristic

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

C₂×D₄₇

yes

characteristic

104

C₂²⋊C₄

—

105

C₂²⋊C₄

—

106

C₂²⋊C₄

—

107

C₂²⋊C₄

—

108

C₂²⋊C₄

—

109

C₂²⋊C₄

—

110

C₂²⋊C₄

C₂×D₄₇

yes

—

111

C₄²

—

112

C₄²

—

113

C₄²

—

114

C₂³

—

115

C₂³

C₂²×D₄₇

yes

characteristic

116

C₂³

C₂²×D₄₇

yes

—

117

Q₈

—

188

—

118

D₄

—

119

D₄

—

120

D₄

—

121

D₄

—

188

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

C₂²×D₄₇

yes

characteristic

131

C₂×C₄

C₂²×D₄₇

yes

—

132

C₂×C₄

C₂²×D₄₇

yes

characteristic

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

—

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

C₂³×D₄₇

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

#10625352

yes

—

151

C₂

—

152

C₂

—

153

C₂

—

188

—

154

C₂

#41×D₄₇

yes

characteristic

155

C₂

C₂×#10625352

yes

characteristic

156

C₂

#13094069

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₄₇
commutator, socle, cosocle	C₂²×C₄₇	C₂⁴
Fitting	#121×C₄₇	C₂
hypercenter	#121	D₄₇
hypocenter	C₄₇	#555
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ #121

subgroup

G/H

center, Frattini

C₂²

C₂³×D₄₇

commutator, socle, cosocle

C₂²×C₄₇

C₂⁴

Fitting

#121×C₄₇

C₂

D₄₇

C₄₇

G ▷ C₂²×C₄₇ ▷ C₄₇

C₁ ◁ C₂² ◁ #121

N	H ≅ G/N	type
1. #13094023	C₂	semidirect
2. #13094011	C₂	semidirect
3. #13094010	C₂	semidirect
4. #13093996	C₂	semidirect
5. #13093995	C₂	semidirect
6. D₄×Dic₄₇	C₂	semidirect
7. C₄×C₂²⋊D₄₇	C₂	semidirect
8. #13093983	C₂	semidirect
9. #13093982	C₂	semidirect
10. #13093980	C₂	semidirect
11. #121×C₄₇	C₂	semidirect
79. C₄₇	#555	semidirect
85. #121	D₄₇	semidirect

H ≅ G/N

type

1. #13094023

C₂

semidirect

2. #13094011

C₂

semidirect

3. #13094010

C₂

semidirect

4. #13093996

C₂

semidirect

5. #13093995

C₂

semidirect

6. D₄×Dic₄₇

C₂

semidirect

7. C₄×C₂²⋊D₄₇

C₂

semidirect

8. #13093983

C₂

semidirect

9. #13093982

C₂

semidirect

10. #13093980

C₂

semidirect

11. #121×C₄₇

C₂

semidirect

79. C₄₇

#555

semidirect

85. #121

D₄₇

semidirect

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

subgroup	index
#13094023	2
#13094011	2
#13094010	2
#13093996	2
#13093995	2
D₄×Dic₄₇	2
C₄×C₂²⋊D₄₇	2
#13093983	2
#13093982	2
#13093980	2
#121×C₄₇	2
#555	47

subgroup

index

D₄×Dic₄₇

C₄×C₂²⋊D₄₇

#121×C₄₇

Subgroups — #15689837

Conjugacy classes of subgroups

Complements

Maximal subgroups