Group #20244164 — subgroups

Subgroups — #20244164

Order 6016.

200 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 101 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×#13093960	C₂	1	2	yes	maximal
2	#15689147	C₂	1	1	yes	maximal, characteristic
3	C₄×#10625329	C₂	1	1	yes	maximal, characteristic
4	#15688888	C₂	1	2	yes	maximal
5	C₂×C₄⋊C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#10625329	C₂²	1	2	yes	—
7	C₂×#10625329	C₂²	1	1	yes	characteristic
8	C₂×#10625329	C₂²	1	1	yes	characteristic
9	#13093960	—	2	4	no	—
10	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
11	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
12	#13093935	—	2	1	no	—
13	C₄×C₄₇⋊C₈	—	2	1	no	—
14	C₂²×C₈×C₄₇	C₂²	1	2	yes	—
15	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₄₇	C₄	1	2	yes	—
17	C₄⋊C₈×C₄₇	C₄	1	2	yes	—
18	C₂²×C₄×C₄₇	Q₈	1	1	yes	characteristic
19	C₂²×C₄×C₄₇	Q₈	1	1	yes	characteristic
20	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
21	C₂×C₈×C₄₇	—	2	2	no	—
22	C₂×C₈×C₄₇	—	2	2	no	—
23	C₂×C₈×C₄₇	C₂×C₄	1	4	yes	—
24	C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
25	C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
26	C₄²×C₄₇	D₄	1	1	yes	characteristic
27	C₄²×C₄₇	D₄	1	1	yes	characteristic
28	#10625329	—	2	1	no	—
29	#10625329	—	2	1	no	—
30	#10625329	—	2	2	no	—
31	#10625329	—	2	2	no	—
32	#10625329	D₄	1	2	yes	—
33	#10625329	Q₈	1	2	yes	—
34	C₂×C₄₇⋊C₈	—	2	1	no	—
35	C₂×C₄₇⋊C₈	—	2	1	no	—
36	C₂×C₄₇⋊C₈	—	2	2	no	—
37	C₂³×C₄₇	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₄×C₄₇	—	2	2	no	—
39	C₂×C₄×C₄₇	—	4	1	no	—
40	C₂×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
49	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
50	C₈×C₄₇	—	2	4	no	—
51	C₄₇⋊C₈	—	2	2	no	—
52	C₄₇⋊C₈	—	2	2	no	—
53	C₄₇⋊C₈	—	4	2	no	—
54	C₂²×C₄₇	—	2	1	no	—
55	C₂²×C₄₇	—	2	1	no	—
56	C₂²×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₄₇	#125	1	1	yes	characteristic
58	C₂²×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
59	C₄×C₄₇	—	2	1	no	—
60	C₄×C₄₇	—	2	1	no	—
61	C₄×C₄₇	—	2	1	no	—
62	C₄×C₄₇	#108	1	2	yes	complemented
63	C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
64	C₄×C₄₇	C₄×Q₈	1	1	yes	characteristic
65	C₄×C₄₇	#121	1	1	yes	characteristic
66	C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
67	#1950	—	47	1	no	maximal, Sylow
68	C₂×C₄₇	—	2	1	no	—
69	C₂×C₄₇	—	2	1	no	—
70	C₂×C₄₇	#389	1	1	yes	characteristic
71	C₂×C₄₇	C₂×#108	1	1	yes	characteristic
72	C₂×C₄₇	#430	1	1	yes	characteristic
73	C₂×#108	—	47	2	no	—
74	#423	—	47	1	no	—
75	C₂×C₄⋊C₈	D₄₇	1	1	yes	characteristic
76	C₄×M₁₆	—	47	1	no	—
77	#341	—	47	2	no	—
78	C₄₇	#1950	1	1	yes	characteristic, complemented, Sylow
79	C₂×M₁₆	—	47	1	no	—
80	C₂×M₁₆	—	47	1	no	—
81	C₂×M₁₆	—	47	2	no	—
82	C₂²×C₈	C₂×D₄₇	1	2	yes	—
83	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
84	#108	—	94	4	no	—
85	C₄⋊C₈	—	94	1	no	—
86	C₄⋊C₈	—	94	1	no	—
87	C₄⋊C₈	Dic₄₇	1	2	yes	—
88	C₄⋊C₈	Dic₄₇	1	2	yes	—
89	#97	—	94	1	no	—
90	C₄×C₈	—	94	1	no	—
91	C₂²×C₄	Dic₉₄	1	1	yes	characteristic
92	C₂²×C₄	Dic₉₄	1	1	yes	characteristic
93	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
94	M₁₆	—	47	2	no	—
95	M₁₆	—	47	2	no	—
96	M₁₆	—	94	1	no	—
97	M₁₆	—	94	1	no	—
98	M₁₆	—	94	2	no	—
99	M₁₆	—	94	2	no	—
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	—	2	2	no	—
102	C₂×C₈	—	94	1	no	—
103	C₂×C₈	—	94	1	no	—
104	C₂×C₈	—	94	2	no	—
105	C₂×C₈	C₂×Dic₄₇	1	4	yes	—
106	C₄²	D₁₈₈	1	1	yes	characteristic
107	C₄²	D₁₈₈	1	1	yes	characteristic
108	C₄²	C₂×Dic₄₇	1	1	yes	characteristic
109	C₄²	C₂×Dic₄₇	1	1	yes	characteristic
110	C₂³	C₂×Dic₉₄	1	1	yes	characteristic
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	4	1	no	—
113	C₂×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
114	C₂×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
115	C₂×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
116	C₂×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
117	C₂×C₄	C₂×D₁₈₈	1	1	yes	characteristic
118	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
119	C₂×C₄	#10625352	1	1	yes	characteristic
120	C₂×C₄	Q₈×D₄₇	1	1	yes	characteristic
121	C₂×C₄	#10625354	1	1	yes	characteristic
122	C₂×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
123	C₈	—	2	4	no	—
124	C₈	—	94	2	no	—
125	C₈	—	94	2	no	—
126	C₈	—	188	2	no	—
127	C₂²	—	2	1	no	—
128	C₂²	—	2	1	no	—
129	C₂²	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
130	C₂²	#13094015	1	1	yes	characteristic
131	C₂²	#13094023	1	1	yes	characteristic
132	C₄	—	2	1	no	—
133	C₄	—	2	1	no	—
134	C₄	—	2	1	no	—
135	C₄	#13093960	1	2	yes	complemented
136	C₄	D₄×Dic₄₇	1	1	yes	characteristic
137	C₄	Q₈×Dic₄₇	1	1	yes	characteristic
138	C₄	#13093999	1	1	yes	characteristic
139	C₄	#13094004	1	1	yes	characteristic
140	C₂	—	2	1	no	—
141	C₂	—	2	1	no	—
142	C₂	#15689022	1	1	yes	characteristic
143	C₂	C₂×#13093960	1	1	yes	characteristic
144	C₂	#15689172	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13093960

C₂

yes

maximal

#15689147

C₂

yes

maximal, characteristic

C₄×#10625329

C₂

yes

maximal, characteristic

#15688888

C₂

yes

maximal

C₂×C₄⋊C₈×C₄₇

C₂

yes

maximal, characteristic

C₂×#10625329

C₂²

yes

—

C₂×#10625329

C₂²

yes

characteristic

C₂×#10625329

C₂²

yes

characteristic

#13093960

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

—

#13093935

—

C₄×C₄₇⋊C₈

—

C₂²×C₈×C₄₇

C₂²

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₄⋊C₈×C₄₇

C₄

yes

—

C₄⋊C₈×C₄₇

C₄

yes

—

C₂²×C₄×C₄₇

Q₈

yes

characteristic

C₂²×C₄×C₄₇

Q₈

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₂×C₄

yes

—

C₄²×C₄₇

C₂×C₄

yes

characteristic

C₄²×C₄₇

C₂×C₄

yes

characteristic

C₄²×C₄₇

D₄

yes

characteristic

C₄²×C₄₇

D₄

yes

—

—

—

—

D₄

yes

—

#10625329

Q₈

yes

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×Q₈

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₄₇

#125

yes

characteristic

C₂²×C₄₇

C₄⋊Q₈

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#108

yes

complemented

C₄×C₄₇

C₄×D₄

yes

characteristic

C₄×C₄₇

C₄×Q₈

yes

characteristic

C₄×C₄₇

#121

yes

characteristic

C₄×C₄₇

C₂²⋊Q₈

yes

characteristic

#1950

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

—

C₂×C₄⋊C₈

D₄₇

yes

characteristic

C₄×M₁₆

—

#341

—

C₄₇

#1950

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

C₂×D₄₇

yes

—

C₂×C₄²

C₂×D₄₇

yes

characteristic

#108

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

Dic₄₇

yes

—

C₄⋊C₈

Dic₄₇

yes

—

#97

—

C₄×C₈

—

C₂²×C₄

Dic₉₄

yes

characteristic

C₂²×C₄

Dic₉₄

yes

characteristic

C₂²×C₄

C₂²×D₄₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

C₂×Dic₄₇

yes

—

106

C₄²

D₁₈₈

yes

characteristic

107

C₄²

D₁₈₈

yes

characteristic

108

C₄²

C₂×Dic₄₇

yes

characteristic

109

C₄²

C₂×Dic₄₇

yes

characteristic

110

C₂³

C₂×Dic₉₄

yes

characteristic

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

C₄⋊Dic₄₇

yes

characteristic

114

C₂×C₄

C₄⋊Dic₄₇

yes

characteristic

115

C₂×C₄

C₄⋊Dic₄₇

yes

characteristic

116

C₂×C₄

C₄⋊Dic₄₇

yes

characteristic

117

C₂×C₄

C₂×D₁₈₈

yes

characteristic

118

C₂×C₄

D₄×D₄₇

yes

characteristic

119

C₂×C₄

#10625352

yes

characteristic

120

C₂×C₄

Q₈×D₄₇

yes

characteristic

121

C₂×C₄

#10625354

yes

characteristic

122

C₂×C₄

C₂²×Dic₄₇

yes

characteristic

123

C₈

—

124

C₈

—

125

C₈

—

126

C₈

—

188

—

127

C₂²

—

128

C₂²

—

129

C₂²

C₂×C₄⋊Dic₄₇

yes

characteristic

130

C₂²

#13094015

yes

characteristic

131

C₂²

#13094023

yes

characteristic

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

#13093960

yes

complemented

136

C₄

D₄×Dic₄₇

yes

characteristic

137

C₄

Q₈×Dic₄₇

yes

characteristic

138

C₄

#13093999

yes

characteristic

139

C₄

#13094004

yes

characteristic

140

C₂

—

141

C₂

—

142

C₂

#15689022

yes

characteristic

143

C₂

C₂×#13093960

yes

characteristic

144

C₂

#15689172

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₄₇
Fitting	C₂×C₄⋊C₈×C₄₇	C₂
socle	C₂²×C₄₇	C₂×C₄⋊C₄
hypercenter	C₂×C₄⋊C₈	D₄₇
hypocenter	C₄₇	#1950
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂×C₄

C₂×D₁₈₈

commutator

C₂×C₄×C₄₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₄₇

Fitting

C₂×C₄⋊C₈×C₄₇

C₂

socle

C₂²×C₄₇

C₂×C₄⋊C₄

hypercenter

C₂×C₄⋊C₈

D₄₇

hypocenter

C₄₇

#1950

LCS

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
62. C₄×C₄₇	#108	semidirect
78. C₄₇	#1950	semidirect
135. C₄	#13093960	semidirect

H ≅ G/N

type

62. C₄×C₄₇

#108

semidirect

78. C₄₇

#1950

semidirect

135. C₄

#13093960

semidirect

Subgroups — #20244164

Conjugacy classes of subgroups

Complements