Group #20242752 — subgroups

Subgroups — #20242752

Order 6016.

172 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 53 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	#15689502	C₂	1	1	yes	maximal, characteristic, complemented
2	#15689404	C₂	1	1	yes	maximal, characteristic
3	#15689400	C₂	1	1	yes	maximal, characteristic
4	#15689188	C₂	1	1	yes	maximal, characteristic
5	#15688875	C₂	1	1	yes	maximal, characteristic, complemented
6	#15688870	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#332	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10625337	—	2	1	no	—
9	C₂×#10625337	—	2	1	no	—
10	C₂×C₈×D₄₇	—	2	1	no	—
11	#13094024	C₂²	1	1	yes	characteristic
12	C₄×D₁₈₈	C₂²	1	1	yes	characteristic
13	#13093958	—	2	1	no	—
14	#13093957	—	2	1	no	—
15	C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
16	#13093946	—	2	1	no	—
17	#13093944	—	2	1	no	—
18	(C₂×D₄₇)⋊C₈	—	2	1	no	—
19	C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
20	C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
23	C₂×Q₈×C₄₇	—	2	1	no	—
24	C₂×D₁₈₈	D₄	1	1	yes	characteristic
25	C₂×C₄×D₄₇	—	2	1	no	—
26	C₂×Dic₉₄	—	2	1	no	—
27	C₂×C₈×C₄₇	—	2	1	no	—
28	C₂×C₈×C₄₇	—	2	1	no	—
29	C₄⋊C₄×C₄₇	—	2	1	no	—
30	C₄⋊C₄×C₄₇	—	2	1	no	—
31	C₄²×C₄₇	C₂³	1	1	yes	characteristic
32	#10625337	—	4	2	no	—
33	#10625337	—	4	2	no	—
34	(C₂×D₄₇)⋊C₄	—	2	1	no	—
35	C₄⋊Dic₄₇	—	2	1	no	—
36	C₄⋊Dic₄₇	—	2	1	no	—
37	C₄⋊Dic₄₇	D₄	1	1	yes	characteristic
38	C₂×C₄₇⋊C₈	—	2	1	no	—
39	C₂×C₄₇⋊C₈	D₄	1	1	yes	characteristic
40	C₂×C₄₇⋊C₈	D₄	1	1	yes	characteristic
41	C₈×D₄₇	—	4	2	no	—
42	C₂²×D₄₇	—	2	1	no	—
43	Q₈×C₄₇	—	4	1	no	—
44	Q₈×C₄₇	—	4	1	no	—
45	C₂×C₄×C₄₇	—	4	1	no	—
46	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
49	C₂×Dic₄₇	—	2	1	no	—
50	C₂×Dic₄₇	—	4	1	no	—
51	D₁₈₈	—	2	1	no	—
52	D₁₈₈	Q₁₆	1	2	yes	complemented
53	C₄×D₄₇	—	4	1	no	—
54	C₄×D₄₇	—	4	1	no	—
55	Dic₉₄	—	4	1	no	—
56	Dic₉₄	—	4	1	no	—
57	C₈×C₄₇	—	4	1	no	—
58	C₈×C₄₇	—	4	1	no	—
59	C₄₇⋊C₈	—	2	1	no	—
60	C₄₇⋊C₈	—	4	1	no	—
61	C₄₇⋊C₈	D₈	1	2	yes	—
62	C₂²×C₄₇	#121	1	1	yes	characteristic
63	C₂×D₄₇	—	2	2	no	—
64	C₂×D₄₇	—	4	1	no	—
65	C₂×D₄₇	—	4	1	no	—
66	C₄×C₄₇	—	2	1	no	—
67	C₄×C₄₇	—	8	1	no	—
68	C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
69	C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
70	C₄×C₄₇	#135	1	1	yes	characteristic
71	C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
72	Dic₄₇	—	4	1	no	—
73	Dic₄₇	—	8	1	no	—
74	#1669	—	47	1	no	maximal, Sylow
75	C₂×C₄₇	#459	1	1	yes	characteristic
76	C₂×C₄₇	#467	1	1	yes	characteristic
77	C₂×C₄₇	#473	1	1	yes	characteristic
78	D₄₇	—	4	2	no	—
79	#494	—	47	1	no	—
80	#474	—	47	1	no	—
81	#474	—	47	1	no	—
82	C₈×D₄	—	47	1	no	—
83	#335	—	47	1	no	—
84	#332	—	47	1	no	—
85	#332	D₄₇	1	1	yes	characteristic, complemented
86	C₄₇	#1669	1	1	yes	characteristic, complemented, Sylow
87	C₂×Q₁₆	—	94	1	no	—
88	C₂×Q₁₆	—	94	1	no	—
89	C₂²×C₈	—	94	1	no	—
90	C₄⋊Q₈	—	47	1	no	—
91	C₄⋊Q₈	C₂×D₄₇	1	1	yes	characteristic
92	C₄×D₄	—	47	1	no	—
93	#107	—	94	1	no	—
94	#107	—	94	1	no	—
95	C₄⋊C₈	—	47	1	no	—
96	C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
97	C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
98	#102	—	94	1	no	—
99	#102	—	94	1	no	—
100	C₂²⋊C₈	—	94	1	no	—
101	C₄×C₈	—	47	1	no	—
102	C₂×Q₈	—	2	1	no	—
103	C₂×Q₈	—	94	1	no	—
104	C₂×D₄	—	47	1	no	—
105	C₂²×C₄	—	94	1	no	—
106	Q₁₆	—	188	2	no	—
107	Q₁₆	—	188	2	no	—
108	C₂×C₈	—	2	1	no	—
109	C₂×C₈	—	2	1	no	—
110	C₂×C₈	—	47	1	no	—
111	C₂×C₈	—	47	1	no	—
112	C₂×C₈	—	94	1	no	—
113	C₂×C₈	—	188	2	no	—
114	C₄⋊C₄	—	2	1	no	—
115	C₄⋊C₄	—	2	1	no	—
116	C₄⋊C₄	—	47	1	no	—
117	C₄⋊C₄	—	94	1	no	—
118	C₄⋊C₄	—	94	1	no	—
119	C₂²⋊C₄	—	94	1	no	—
120	C₄²	C₂²×D₄₇	1	1	yes	characteristic
121	C₂³	—	94	1	no	—
122	Q₈	—	4	1	no	—
123	Q₈	—	4	1	no	—
124	Q₈	—	188	1	no	—
125	Q₈	—	188	1	no	—
126	D₄	—	47	2	no	—
127	D₄	—	94	1	no	—
128	C₂×C₄	—	4	1	no	—
129	C₂×C₄	—	94	1	no	—
130	C₂×C₄	—	188	1	no	—
131	C₂×C₄	—	188	1	no	—
132	C₂×C₄	—	188	1	no	—
133	C₂×C₄	#10625350	1	1	yes	characteristic
134	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
135	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
136	C₈	—	4	1	no	—
137	C₈	—	4	1	no	—
138	C₈	—	47	2	no	—
139	C₈	—	94	1	no	—
140	C₈	—	188	1	no	—
141	C₂²	—	94	2	no	—
142	C₂²	—	188	1	no	—
143	C₂²	—	188	1	no	—
144	C₂²	#13093997	1	1	yes	characteristic
145	C₄	—	2	1	no	—
146	C₄	—	8	1	no	—
147	C₄	—	376	1	no	—
148	C₄	—	188	1	no	—
149	C₄	D₈×D₄₇	1	1	yes	characteristic
150	C₄	Q₁₆×D₄₇	1	1	yes	characteristic
151	C₄	#13094045	1	1	yes	characteristic
152	C₄	#13094051	1	1	yes	characteristic
153	C₂	—	188	2	no	—
154	C₂	#15689317	1	1	yes	characteristic
155	C₂	#15689363	1	1	yes	characteristic
156	C₂	#15689397	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689502

C₂

yes

maximal, characteristic, complemented

#15689404

C₂

yes

maximal, characteristic

#15689400

C₂

yes

maximal, characteristic

#15689188

C₂

yes

maximal, characteristic

#15688875

C₂

yes

maximal, characteristic, complemented

#15688870

C₂

yes

maximal, characteristic, complemented

C₄₇×#332

C₂

yes

maximal, characteristic, complemented

C₂×#10625337

—

C₂×#10625337

—

C₂×C₈×D₄₇

—

#13094024

C₂²

yes

characteristic

C₄×D₁₈₈

C₂²

yes

characteristic

#13093958

—

#13093957

—

C₄⋊(C₄₇⋊C₈)

C₂²

yes

characteristic

#13093946

—

#13093944

—

(C₂×D₄₇)⋊C₈

—

C₄×C₄₇⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₄₇

C₂²

yes

characteristic

C₄⋊C₈×C₄₇

C₂²

yes

characteristic

C₄⋊C₈×C₄₇

C₂²

yes

characteristic

C₂×Q₈×C₄₇

—

C₂×D₁₈₈

D₄

yes

characteristic

C₂×C₄×D₄₇

—

C₂×Dic₉₄

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

C₂³

yes

characteristic

#10625337

—

#10625337

—

(C₂×D₄₇)⋊C₄

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

D₄

yes

characteristic

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

D₄

yes

characteristic

C₂×C₄₇⋊C₈

D₄

yes

characteristic

C₈×D₄₇

—

C₂²×D₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

#41

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

D₁₈₈

—

D₁₈₈

Q₁₆

yes

complemented

C₄×D₄₇

—

C₄×D₄₇

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

D₈

yes

—

C₂²×C₄₇

#121

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂×D₈

yes

characteristic

C₄×C₄₇

C₂×Q₁₆

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

Dic₄₇

—

Dic₄₇

—

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

D₄₇

—

#494

—

#474

—

#474

—

C₈×D₄

—

#335

—

#332

—

#332

D₄₇

yes

characteristic, complemented

C₄₇

#1669

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₄₇

yes

characteristic

C₄×D₄

—

#107

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₄₇

yes

characteristic

C₄⋊C₈

C₂×D₄₇

yes

characteristic

#102

—

#102

—

100

C₂²⋊C₈

—

101

C₄×C₈

—

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂×D₄

—

105

C₂²×C₄

—

106

Q₁₆

—

188

—

107

Q₁₆

—

188

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

188

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₂²⋊C₄

—

120

C₄²

C₂²×D₄₇

yes

characteristic

121

C₂³

—

122

Q₈

—

123

Q₈

—

124

Q₈

—

188

—

125

Q₈

—

188

—

126

D₄

—

127

D₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

188

—

131

C₂×C₄

—

188

—

132

C₂×C₄

—

188

—

133

C₂×C₄

#10625350

yes

characteristic

134

C₂×C₄

D₄×D₄₇

yes

characteristic

135

C₂×C₄

D₄×D₄₇

yes

characteristic

136

C₈

—

137

C₈

—

138

C₈

—

139

C₈

—

140

C₈

—

188

—

141

C₂²

—

142

C₂²

—

188

—

143

C₂²

—

188

—

144

C₂²

#13093997

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

376

—

148

C₄

—

188

—

149

C₄

D₈×D₄₇

yes

characteristic

150

C₄

Q₁₆×D₄₇

yes

characteristic

151

C₄

#13094045

yes

characteristic

152

C₄

#13094051

yes

characteristic

153

C₂

—

188

—

154

C₂

#15689317

yes

characteristic

155

C₂

#15689363

yes

characteristic

156

C₂

#15689397

yes

characteristic

	subgroup	G/H
center	C₂²	#13093997
commutator	C₄²×C₄₇	C₂³
Frattini	C₄²	C₂²×D₄₇
Fitting	C₄₇×#332	C₂
socle	C₂²×C₄₇	#121
hypercenter	#332	D₄₇
hypocenter	C₄₇	#1669
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #332

subgroup

G/H

center

C₂²

#13093997

commutator

C₄²×C₄₇

C₂³

Frattini

C₄²

C₂²×D₄₇

Fitting

C₄₇×#332

C₂

C₂²×C₄₇

D₄₇

C₄₇

G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #332

N	H ≅ G/N	type
1. #15689502	C₂	semidirect
5. #15688875	C₂	semidirect
6. #15688870	C₂	semidirect
7. C₄₇×#332	C₂	semidirect
52. D₁₈₈	Q₁₆	semidirect
85. #332	D₄₇	semidirect
86. C₄₇	#1669	semidirect

H ≅ G/N

type

1. #15689502

C₂

semidirect

5. #15688875

C₂

semidirect

6. #15688870

C₂

semidirect

7. C₄₇×#332

C₂

semidirect

52. D₁₈₈

Q₁₆

semidirect

85. #332

D₄₇

semidirect

86. C₄₇

#1669

semidirect

Subgroups — #20242752

Conjugacy classes of subgroups

Complements