Group #20242862 — subgroups

Subgroups — #20242862

Order 6016.

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

normal subgroups only

This group has 168 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	#15689497	C₂	1	1	yes	maximal, characteristic
2	#15689323	C₂	1	1	yes	maximal, characteristic
3	#15689254	C₂	1	1	yes	maximal, characteristic, complemented
4	#15688875	C₂	1	1	yes	maximal, characteristic, complemented
5	#15688869	C₂	1	1	yes	maximal, characteristic, complemented
6	#15688866	C₂	1	1	yes	maximal, characteristic
7	C₄₇×#475	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10625336	—	2	1	no	—
9	C₂×#10625336	—	2	1	no	—
10	C₂×#10625336	—	2	1	no	—
11	#13094021	C₂²	1	1	yes	characteristic
12	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
13	C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
14	C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
15	#13093944	—	2	1	no	—
16	#13093935	—	2	1	no	—
17	C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
18	C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
19	C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
20	#106×C₄₇	—	2	1	no	—
21	C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
22	#103×C₄₇	—	2	1	no	—
23	C₂×Q₈×C₄₇	—	2	1	no	—
24	C₂×Q₈×C₄₇	D₄	1	1	yes	characteristic
25	C₂×D₁₈₈	—	2	1	no	—
26	C₂×D₁₈₈	—	2	1	no	—
27	C₂×D₁₈₈	—	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	C₄⋊C₄×C₄₇	—	2	1	no	—
32	C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
33	C₄²×C₄₇	—	2	1	no	—
34	C₄²×C₄₇	C₂³	1	1	yes	characteristic
35	#10625336	—	4	2	no	—
36	#10625336	—	4	2	no	—
37	#10625336	—	4	2	no	—
38	C₂×C₄₇⋊C₈	—	2	1	no	—
39	C₂×C₄₇⋊C₈	—	2	1	no	—
40	C₂×C₄₇⋊C₈	D₄	1	1	yes	characteristic
41	C₂×C₄₇⋊C₈	D₄	1	1	yes	characteristic
42	C₂²×D₄₇	—	4	1	no	—
43	Q₈×C₄₇	—	2	1	no	—
44	Q₈×C₄₇	—	2	1	no	—
45	Q₈×C₄₇	—	4	1	no	—
46	Q₈×C₄₇	—	4	1	no	—
47	C₂×C₄×C₄₇	—	2	1	no	—
48	C₂×C₄×C₄₇	—	2	1	no	—
49	C₂×C₄×C₄₇	—	4	1	no	—
50	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
53	D₁₈₈	—	4	1	no	—
54	D₁₈₈	—	4	1	no	—
55	D₁₈₈	—	4	1	no	—
56	D₁₈₈	—	4	1	no	—
57	D₁₈₈	—	8	1	no	—
58	C₈×C₄₇	—	4	1	no	—
59	C₄₇⋊C₈	—	2	1	no	—
60	C₄₇⋊C₈	—	4	1	no	—
61	C₄₇⋊C₈	—	4	1	no	—
62	C₄₇⋊C₈	SD₁₆	1	2	yes	complemented
63	C₂²×C₄₇	#121	1	1	yes	characteristic
64	C₂×D₄₇	—	8	1	no	—
65	C₂×D₄₇	—	8	1	no	—
66	C₂×D₄₇	—	8	1	no	—
67	C₄×C₄₇	—	2	1	no	—
68	C₄×C₄₇	—	4	1	no	—
69	C₄×C₄₇	—	4	1	no	—
70	C₄×C₄₇	—	8	1	no	—
71	C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
72	C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
73	C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
74	C₄×C₄₇	#137	1	1	yes	characteristic
75	#1686	—	47	1	no	maximal, Sylow
76	C₂×C₄₇	#460	1	1	yes	characteristic
77	C₂×C₄₇	#468	1	1	yes	characteristic
78	C₂×C₄₇	#471	1	1	yes	characteristic
79	D₄₇	—	16	1	no	—
80	#492	—	47	1	no	—
81	#475	D₄₇	1	1	yes	characteristic, complemented
82	#460	—	47	1	no	—
83	#446	—	47	1	no	—
84	#335	—	47	1	no	—
85	#332	—	47	1	no	—
86	#331	—	47	1	no	—
87	C₄₇	#1686	1	1	yes	characteristic, complemented, Sylow
88	C₂×SD₁₆	—	94	1	no	—
89	C₂×SD₁₆	—	94	1	no	—
90	C₂×SD₁₆	—	94	1	no	—
91	C₄⋊Q₈	C₂×D₄₇	1	1	yes	characteristic
92	#127	—	47	1	no	—
93	C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
94	#106	—	2	1	no	—
95	C₄⋊C₈	—	47	1	no	—
96	C₄⋊C₈	—	47	1	no	—
97	C₄⋊C₈	—	94	1	no	—
98	C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
99	#103	—	2	1	no	—
100	#102	—	94	1	no	—
101	#97	—	94	1	no	—
102	C₄×C₈	—	47	1	no	—
103	C₂×Q₈	—	2	1	no	—
104	C₂×Q₈	C₂²⋊D₄₇	1	1	yes	characteristic
105	C₂×D₄	—	94	1	no	—
106	C₂×D₄	—	94	1	no	—
107	C₂×D₄	—	94	1	no	—
108	SD₁₆	—	188	2	no	—
109	SD₁₆	—	188	2	no	—
110	SD₁₆	—	188	2	no	—
111	C₂×C₈	—	2	1	no	—
112	C₂×C₈	—	47	1	no	—
113	C₂×C₈	—	47	1	no	—
114	C₂×C₈	—	94	1	no	—
115	C₂×C₈	—	94	1	no	—
116	C₄⋊C₄	—	2	1	no	—
117	C₄⋊C₄	—	2	1	no	—
118	C₄⋊C₄	—	2	1	no	—
119	C₄⋊C₄	C₂²⋊D₄₇	1	1	yes	characteristic
120	C₄²	—	2	1	no	—
121	C₄²	C₂²×D₄₇	1	1	yes	characteristic
122	C₂³	—	188	1	no	—
123	Q₈	—	2	1	no	—
124	Q₈	—	2	1	no	—
125	Q₈	—	4	1	no	—
126	Q₈	—	4	1	no	—
127	D₄	—	376	1	no	—
128	D₄	—	188	1	no	—
129	D₄	—	188	1	no	—
130	D₄	—	188	1	no	—
131	D₄	—	188	1	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	4	1	no	—
135	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
136	C₂×C₄	#10625352	1	1	yes	characteristic
137	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
138	C₈	—	4	1	no	—
139	C₈	—	47	2	no	—
140	C₈	—	94	1	no	—
141	C₈	—	188	1	no	—
142	C₈	—	188	1	no	—
143	C₂²	—	376	1	no	—
144	C₂²	—	376	1	no	—
145	C₂²	—	376	1	no	—
146	C₂²	#13093996	1	1	yes	characteristic
147	C₄	—	2	1	no	—
148	C₄	—	4	1	no	—
149	C₄	—	4	1	no	—
150	C₄	—	8	1	no	—
151	C₄	SD₁₆×D₄₇	1	1	yes	characteristic
152	C₄	#13094049	1	1	yes	characteristic
153	C₄	#13094051	1	1	yes	characteristic
154	C₄	#13094055	1	1	yes	characteristic
155	C₂	—	752	1	no	—
156	C₂	#15689321	1	1	yes	characteristic
157	C₂	#15689371	1	1	yes	characteristic
158	C₂	#15689388	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689497

C₂

yes

maximal, characteristic

#15689323

C₂

yes

maximal, characteristic

#15689254

C₂

yes

maximal, characteristic, complemented

#15688875

C₂

yes

maximal, characteristic, complemented

#15688869

C₂

yes

maximal, characteristic, complemented

#15688866

C₂

yes

maximal, characteristic

C₄₇×#475

C₂

yes

maximal, characteristic, complemented

—

—

—

C₂²

yes

characteristic

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₄₇⋊C₈)

C₂²

yes

characteristic

#13093944

—

#13093935

—

C₄×C₄₇⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₄₇

C₂²

yes

characteristic

C₄×Q₈×C₄₇

C₂²

yes

characteristic

#106×C₄₇

—

C₄⋊C₈×C₄₇

C₂²

yes

characteristic

#103×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

D₄

yes

characteristic

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×C₈×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

D₄

yes

characteristic

C₄²×C₄₇

—

C₄²×C₄₇

C₂³

yes

—

—

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

D₄

yes

characteristic

C₂×C₄₇⋊C₈

D₄

yes

characteristic

C₂²×D₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

#41

yes

characteristic

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

SD₁₆

yes

complemented

C₂²×C₄₇

#121

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂×SD₁₆

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

D₄₇

—

#492

—

#475

D₄₇

yes

characteristic, complemented

—

—

—

—

—

C₄₇

#1686

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

C₂×SD₁₆

—

C₂×SD₁₆

—

C₄⋊Q₈

C₂×D₄₇

yes

characteristic

#127

—

C₄×Q₈

C₂×D₄₇

yes

characteristic

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₄₇

yes

characteristic

#103

—

100

#102

—

101

#97

—

102

C₄×C₈

—

103

C₂×Q₈

—

104

C₂×Q₈

C₂²⋊D₄₇

yes

characteristic

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂×D₄

—

108

SD₁₆

—

188

—

109

SD₁₆

—

188

—

110

SD₁₆

—

188

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

C₂²⋊D₄₇

yes

characteristic

120

C₄²

—

121

C₄²

C₂²×D₄₇

yes

characteristic

122

C₂³

—

188

—

123

Q₈

—

124

Q₈

—

125

Q₈

—

126

Q₈

—

127

D₄

—

376

—

128

D₄

—

188

—

129

D₄

—

188

—

130

D₄

—

188

—

131

D₄

—

188

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

D₄×D₄₇

yes

characteristic

136

C₂×C₄

#10625352

yes

characteristic

137

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

138

C₈

—

139

C₈

—

140

C₈

—

141

C₈

—

188

—

142

C₈

—

188

—

143

C₂²

—

376

—

144

C₂²

—

376

—

145

C₂²

—

376

—

146

C₂²

#13093996

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

SD₁₆×D₄₇

yes

characteristic

152

C₄

#13094049

yes

characteristic

153

C₄

#13094051

yes

characteristic

154

C₄

#13094055

yes

characteristic

155

C₂

—

752

—

156

C₂

#15689321

yes

characteristic

157

C₂

#15689371

yes

characteristic

158

C₂

#15689388

yes

characteristic

	subgroup	G/H
center	C₂²	#13093996
commutator	C₄²×C₄₇	C₂³
Frattini	C₄²	C₂²×D₄₇
Fitting	C₄₇×#475	C₂
socle	C₂²×C₄₇	#121
hypercenter	#475	D₄₇
hypocenter	C₄₇	#1686
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #475

subgroup

G/H

center

C₂²

#13093996

commutator

C₄²×C₄₇

C₂³

Frattini

C₄²

C₂²×D₄₇

Fitting

C₄₇×#475

C₂

C₂²×C₄₇

D₄₇

C₄₇

G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #475

N	H ≅ G/N	type
3. #15689254	C₂	semidirect
4. #15688875	C₂	semidirect
5. #15688869	C₂	semidirect
7. C₄₇×#475	C₂	semidirect
62. C₄₇⋊C₈	SD₁₆	semidirect
81. #475	D₄₇	semidirect
87. C₄₇	#1686	semidirect

H ≅ G/N

type

3. #15689254

C₂

semidirect

4. #15688875

C₂

semidirect

5. #15688869

C₂

semidirect

7. C₄₇×#475

C₂

semidirect

62. C₄₇⋊C₈

SD₁₆

semidirect

81. #475

D₄₇

semidirect

87. C₄₇

#1686

semidirect

Subgroups — #20242862

Conjugacy classes of subgroups

Complements