Group #20245482 — subgroups

Subgroups — #20245482

Order 6016.

164 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 164 conjugacy classes of subgroups — too many to draw. Only its 55 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	C₂×#13093968	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₄₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#15689364	C₂	1	1	yes	maximal, characteristic, complemented
4	#15688950	C₂	1	1	yes	maximal, characteristic, complemented
5	#15688926	C₂	1	1	yes	maximal, characteristic, complemented
6	#15688924	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#466	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₁₈₈	C₂²	1	1	yes	characteristic
9	#13094003	—	2	1	no	—
10	#13093968	—	2	2	no	—
11	C₂×C₄₇⋊C₁₆	—	2	2	no	—
12	C₂×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
14	#13093958	C₂²	1	1	yes	characteristic
15	#13093949	—	2	1	no	—
16	C₂×D₈×C₄₇	C₂²	1	1	yes	characteristic
17	C₂²×C₈×C₄₇	C₂²	1	1	yes	characteristic
18	#121×C₄₇	—	2	1	no	—
19	#107×C₄₇	C₂²	1	1	yes	characteristic
20	#102×C₄₇	—	2	1	no	—
21	C₂×D₄×C₄₇	—	2	1	no	—
22	C₂×D₄×C₄₇	—	4	1	no	—
23	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
24	C₂×Dic₉₄	—	2	1	no	—
25	D₈×C₄₇	—	2	1	no	—
26	D₈×C₄₇	—	2	1	no	—
27	C₂×C₈×C₄₇	—	2	1	no	—
28	C₂×C₈×C₄₇	—	2	1	no	—
29	C₂×C₈×C₄₇	D₄	1	1	yes	characteristic
30	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
31	C₄⋊C₄×C₄₇	—	2	1	no	—
32	C₂²⋊C₄×C₄₇	—	4	1	no	—
33	C₂²⋊Dic₄₇	—	4	1	no	—
34	C₄⋊Dic₄₇	—	2	1	no	—
35	Dic₄₇⋊C₄	—	4	1	no	—
36	Dic₁₈₈	—	2	1	no	—
37	Dic₁₈₈	—	2	1	no	—
38	C₄₇⋊C₁₆	—	2	1	no	—
39	C₄₇⋊C₁₆	D₄	1	2	yes	complemented
40	C₂³×C₄₇	—	4	1	no	—
41	C₂³×C₄₇	D₈	1	1	yes	characteristic
42	D₄×C₄₇	—	4	1	no	—
43	D₄×C₄₇	—	4	1	no	—
44	D₄×C₄₇	—	8	2	no	—
45	C₂×C₄×C₄₇	—	2	1	no	—
46	C₂×C₄×C₄₇	—	2	1	no	—
47	C₂×C₄×C₄₇	—	4	1	no	—
48	C₂×C₄×C₄₇	D₈	1	1	yes	characteristic
49	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
50	C₂×Dic₄₇	—	4	1	no	—
51	C₂×Dic₄₇	—	4	1	no	—
52	Dic₉₄	—	4	1	no	—
53	Dic₉₄	—	4	1	no	—
54	C₈×C₄₇	—	2	1	no	—
55	C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
56	C₈×C₄₇	#41	1	1	yes	characteristic
57	C₂²×C₄₇	—	2	1	no	—
58	C₂²×C₄₇	—	2	1	no	—
59	C₂²×C₄₇	—	8	1	no	—
60	C₂²×C₄₇	—	8	1	no	—
61	C₂²×C₄₇	—	8	1	no	—
62	C₂²×C₄₇	SD₃₂	1	2	yes	complemented
63	C₂²×C₄₇	C₂×D₈	1	1	yes	characteristic
64	C₄×C₄₇	—	2	1	no	—
65	C₄×C₄₇	—	8	1	no	—
66	C₄×C₄₇	#121	1	1	yes	characteristic
67	C₄×C₄₇	#135	1	1	yes	characteristic
68	Dic₄₇	—	8	1	no	—
69	Dic₄₇	—	8	1	no	—
70	#2217	—	47	1	no	maximal, Sylow
71	C₂×C₄₇	—	2	2	no	—
72	C₂×C₄₇	—	16	1	no	—
73	C₂×C₄₇	#466	1	1	yes	characteristic
74	C₂×C₄₇	C₂×SD₃₂	1	1	yes	characteristic
75	C₂×C₄₇	#508	1	1	yes	characteristic
76	C₂×SD₃₂	—	47	1	no	—
77	C₂²×C₁₆	—	47	1	no	—
78	#467	—	47	1	no	—
79	#466	D₄₇	1	1	yes	characteristic
80	#367	—	47	1	no	—
81	#358	—	47	1	no	—
82	#357	—	47	1	no	—
83	C₄₇	#2217	1	1	yes	characteristic, complemented, Sylow
84	C₂×Q₁₆	—	47	1	no	—
85	C₂×D₈	C₂×D₄₇	1	1	yes	characteristic
86	C₂²×C₈	C₂×D₄₇	1	1	yes	characteristic
87	C₂²⋊Q₈	—	94	1	no	—
88	#121	—	2	1	no	—
89	SD₃₂	—	94	2	no	—
90	C₂×C₁₆	—	47	1	no	—
91	C₂×C₁₆	—	47	1	no	—
92	C₂×C₁₆	—	94	2	no	—
93	#107	—	47	1	no	—
94	#107	C₂×D₄₇	1	1	yes	characteristic
95	#103	—	94	1	no	—
96	#102	—	2	1	no	—
97	C₂×Q₈	—	94	1	no	—
98	C₂×D₄	—	2	1	no	—
99	C₂×D₄	—	4	1	no	—
100	C₂²×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
101	Q₁₆	—	94	1	no	—
102	Q₁₆	—	94	1	no	—
103	D₈	—	2	1	no	—
104	D₈	—	2	1	no	—
105	C₂×C₈	—	2	1	no	—
106	C₂×C₈	—	2	1	no	—
107	C₂×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
108	C₂×C₈	C₂²×D₄₇	1	1	yes	characteristic
109	C₄⋊C₄	—	2	1	no	—
110	C₄⋊C₄	—	94	1	no	—
111	C₄⋊C₄	—	188	1	no	—
112	C₂²⋊C₄	—	4	1	no	—
113	C₂²⋊C₄	—	188	1	no	—
114	C₁₆	—	47	2	no	—
115	C₁₆	—	94	1	no	—
116	C₂³	—	4	1	no	—
117	C₂³	#10625334	1	1	yes	characteristic
118	Q₈	—	188	1	no	—
119	Q₈	—	188	1	no	—
120	D₄	—	4	1	no	—
121	D₄	—	4	1	no	—
122	D₄	—	8	2	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	—	188	1	no	—
127	C₂×C₄	—	188	1	no	—
128	C₂×C₄	#10625334	1	1	yes	characteristic
129	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
130	C₈	—	2	1	no	—
131	C₈	D₄×D₄₇	1	1	yes	characteristic
132	C₈	#10625352	1	1	yes	characteristic
133	C₂²	—	2	1	no	—
134	C₂²	—	2	1	no	—
135	C₂²	—	8	1	no	—
136	C₂²	—	8	1	no	—
137	C₂²	—	8	1	no	—
138	C₂²	#13093968	1	2	yes	complemented
139	C₂²	C₂×#10625334	1	1	yes	characteristic
140	C₄	—	2	1	no	—
141	C₄	—	8	1	no	—
142	C₄	—	376	1	no	—
143	C₄	—	376	1	no	—
144	C₄	#13093996	1	1	yes	characteristic
145	C₄	#13094044	1	1	yes	characteristic
146	C₂	—	2	2	no	—
147	C₂	—	16	1	no	—
148	C₂	#15689357	1	1	yes	characteristic
149	C₂	C₂×#13093968	1	1	yes	characteristic
150	C₂	#15689553	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13093968

C₂

yes

maximal, characteristic, complemented

C₂²×C₄₇⋊C₁₆

C₂

yes

maximal, characteristic, complemented

#15689364

C₂

yes

maximal, characteristic, complemented

#15688950

C₂

yes

maximal, characteristic, complemented

#15688926

C₂

yes

maximal, characteristic, complemented

#15688924

C₂

yes

maximal, characteristic, complemented

C₄₇×#466

C₂

yes

maximal, characteristic

C₂×Dic₁₈₈

C₂²

yes

characteristic

#13094003

—

#13093968

—

C₂×C₄₇⋊C₁₆

—

C₂×C₄₇⋊C₁₆

C₂²

yes

characteristic

C₂×C₄₇⋊C₁₆

C₂²

yes

characteristic

#13093958

C₂²

yes

characteristic

#13093949

—

C₂×D₈×C₄₇

C₂²

yes

characteristic

C₂²×C₈×C₄₇

C₂²

yes

characteristic

#121×C₄₇

—

#107×C₄₇

C₂²

yes

characteristic

#102×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂×Dic₉₄

—

D₈×C₄₇

—

D₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

D₄

yes

characteristic

C₂×C₈×C₄₇

C₂³

yes

characteristic

C₄⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₂²⋊Dic₄₇

—

C₄⋊Dic₄₇

—

Dic₄₇⋊C₄

—

Dic₁₈₈

—

Dic₁₈₈

—

C₄₇⋊C₁₆

—

C₄₇⋊C₁₆

D₄

yes

complemented

C₂³×C₄₇

—

C₂³×C₄₇

D₈

yes

characteristic

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

D₈

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

—

C₈×C₄₇

C₂×D₄

yes

characteristic

C₈×C₄₇

#41

yes

characteristic

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

SD₃₂

yes

complemented

C₂²×C₄₇

C₂×D₈

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

Dic₄₇

—

Dic₄₇

—

#2217

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

#466

yes

characteristic

C₂×C₄₇

C₂×SD₃₂

yes

characteristic

C₂×C₄₇

#508

yes

characteristic

C₂×SD₃₂

—

C₂²×C₁₆

—

#467

—

#466

D₄₇

yes

—

—

—

C₄₇

#2217

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×D₈

C₂×D₄₇

yes

characteristic

C₂²×C₈

C₂×D₄₇

yes

characteristic

C₂²⋊Q₈

—

#121

—

SD₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#107

—

#107

C₂×D₄₇

yes

characteristic

#103

—

#102

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

100

C₂²×C₄

C₂²⋊D₄₇

yes

characteristic

101

Q₁₆

—

102

Q₁₆

—

103

D₈

—

104

D₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

C₂²⋊D₄₇

yes

characteristic

108

C₂×C₈

C₂²×D₄₇

yes

characteristic

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

188

—

112

C₂²⋊C₄

—

113

C₂²⋊C₄

—

188

—

114

C₁₆

—

115

C₁₆

—

116

C₂³

—

117

C₂³

#10625334

yes

characteristic

118

Q₈

—

188

—

119

Q₈

—

188

—

120

D₄

—

121

D₄

—

122

D₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

188

—

127

C₂×C₄

—

188

—

128

C₂×C₄

#10625334

yes

characteristic

129

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

130

C₈

—

131

C₈

D₄×D₄₇

yes

characteristic

132

C₈

#10625352

yes

characteristic

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

#13093968

yes

complemented

139

C₂²

C₂×#10625334

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

—

376

—

143

C₄

—

376

—

144

C₄

#13093996

yes

characteristic

145

C₄

#13094044

yes

characteristic

146

C₂

—

147

C₂

—

148

C₂

#15689357

yes

characteristic

149

C₂

C₂×#13093968

yes

characteristic

150

C₂

#15689553

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#10625334
commutator	C₂×C₈×C₄₇	C₂³
Frattini	C₂×C₈	C₂²×D₄₇
Fitting	C₄₇×#466	C₂
socle	C₂²×C₄₇	C₂×D₈
hypercenter	#466	D₄₇
hypocenter	C₄₇	#2217
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #466

subgroup

G/H

center

C₂²

C₂×#10625334

commutator

C₂×C₈×C₄₇

C₂³

Frattini

C₂×C₈

C₂²×D₄₇

Fitting

C₄₇×#466

C₂

socle

C₂²×C₄₇

C₂×D₈

D₄₇

C₄₇

G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #466

N	H ≅ G/N	type
1. C₂×#13093968	C₂	semidirect
2. C₂²×C₄₇⋊C₁₆	C₂	semidirect
3. #15689364	C₂	semidirect
4. #15688950	C₂	semidirect
5. #15688926	C₂	semidirect
6. #15688924	C₂	semidirect
39. C₄₇⋊C₁₆	D₄	semidirect
62. C₂²×C₄₇	SD₃₂	semidirect
83. C₄₇	#2217	semidirect
138. C₂²	#13093968	semidirect

H ≅ G/N

type

1. C₂×#13093968

C₂

semidirect

2. C₂²×C₄₇⋊C₁₆

C₂

semidirect

3. #15689364

C₂

semidirect

4. #15688950

C₂

semidirect

5. #15688926

C₂

semidirect

6. #15688924

C₂

semidirect

39. C₄₇⋊C₁₆

D₄

semidirect

62. C₂²×C₄₇

SD₃₂

semidirect

83. C₄₇

#2217

semidirect

138. C₂²

#13093968

semidirect

Subgroups — #20245482

Conjugacy classes of subgroups

Complements