Group #20242002 — subgroups

Subgroups — #20242002

Order 6016.

212 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 73 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	#15689680	C₂	1	1	yes	maximal, characteristic
2	#15689180	C₂	1	1	yes	maximal, characteristic
3	#15688864	C₂	1	4	yes	maximal, complemented
4	C₂×#97×C₄₇	C₂	1	1	yes	maximal, characteristic
5	#13094024	C₄	1	1	yes	characteristic
6	#13094024	C₄	1	1	yes	characteristic
7	#13094016	C₂²	1	2	yes	—
8	#13094003	—	2	1	no	—
9	#13094003	—	2	1	no	—
10	C₄×Dic₉₄	—	2	1	no	—
11	C₄×Dic₉₄	—	2	1	no	—
12	#13093983	C₄	1	2	yes	—
13	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
14	C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
15	#13093935	C₂²	1	2	yes	—
16	C₂²×C₈×C₄₇	—	2	1	no	—
17	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
18	#97×C₄₇	—	2	1	no	—
19	#97×C₄₇	C₂²	1	2	yes	—
20	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
22	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
23	C₂×Dic₉₄	—	2	1	no	—
24	C₂×Dic₉₄	—	2	1	no	—
25	C₂×C₈×C₄₇	—	2	4	no	—
26	C₂×C₈×C₄₇	—	2	2	no	—
27	C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
28	C₄²×C₄₇	D₄	1	1	yes	characteristic
29	C₄²×C₄₇	D₄	1	1	yes	characteristic
30	C₄²×C₄₇	C₂³	1	1	yes	characteristic
31	C₂²⋊Dic₄₇	—	2	2	no	—
32	C₄⋊Dic₄₇	—	2	1	no	—
33	C₄⋊Dic₄₇	—	2	1	no	—
34	C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
35	C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
36	Dic₄₇⋊C₄	—	2	2	no	—
37	Dic₄₇⋊C₄	—	2	2	no	—
38	C₄×Dic₄₇	—	2	2	no	—
39	C₂×C₄₇⋊C₈	—	2	2	no	—
40	C₂³×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₄×C₄₇	—	2	2	no	—
42	C₂×C₄×C₄₇	—	2	2	no	—
43	C₂×C₄×C₄₇	—	4	1	no	—
44	C₂×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
50	C₂×Dic₄₇	—	2	2	no	—
51	C₂×Dic₄₇	—	2	2	no	—
52	Dic₉₄	—	4	2	no	—
53	Dic₉₄	—	4	2	no	—
54	C₈×C₄₇	—	2	4	no	—
55	C₄₇⋊C₈	—	4	2	no	—
56	C₂²×C₄₇	—	2	1	no	—
57	C₂²×C₄₇	—	2	1	no	—
58	C₂²×C₄₇	#101	1	2	yes	—
59	C₂²×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₄×C₄₇	—	2	1	no	—
61	C₄×C₄₇	—	2	1	no	—
62	C₄×C₄₇	—	2	1	no	—
63	C₄×C₄₇	—	2	1	no	—
64	C₄×C₄₇	—	2	1	no	—
65	C₄×C₄₇	#104	1	2	yes	—
66	Dic₄₇	—	4	2	no	—
67	Dic₄₇	—	4	2	no	—
68	#1531	—	47	1	no	maximal, Sylow
69	C₂×C₄₇	—	2	2	no	—
70	C₂×C₄₇	C₂×#101	1	1	yes	characteristic
71	C₂×C₄₇	C₂×#104	1	1	yes	characteristic
72	C₂×C₄₇	#421	1	1	yes	characteristic
73	#533	—	47	1	no	—
74	#432	—	47	1	no	—
75	C₂×#97	D₄₇	1	1	yes	characteristic
76	#330	—	47	4	no	—
77	C₄₇	#1531	1	1	yes	characteristic, complemented, Sylow
78	C₂²×C₈	—	2	1	no	—
79	C₄⋊Q₈	—	47	1	no	—
80	C₄⋊Q₈	—	47	1	no	—
81	#125	—	47	2	no	—
82	C₂²⋊Q₈	—	94	1	no	—
83	C₂²⋊Q₈	—	94	1	no	—
84	C₄×Q₈	—	94	1	no	—
85	C₄×Q₈	—	94	1	no	—
86	#117	—	47	2	no	—
87	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
88	C₄⋊C₈	—	94	1	no	—
89	C₂²⋊C₈	—	94	1	no	—
90	#97	—	2	1	no	—
91	#97	—	47	2	no	—
92	#97	C₂×D₄₇	1	2	yes	—
93	C₂×Q₈	—	94	1	no	—
94	C₂×Q₈	—	94	1	no	—
95	C₂²×C₄	C₄×D₄₇	1	1	yes	characteristic
96	C₂²×C₄	D₁₈₈	1	1	yes	characteristic
97	C₂²×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
98	C₂×C₈	—	2	4	no	—
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	—	94	2	no	—
101	C₄⋊C₄	—	47	2	no	—
102	C₄⋊C₄	—	47	2	no	—
103	C₄⋊C₄	—	94	1	no	—
104	C₄⋊C₄	—	94	1	no	—
105	C₄⋊C₄	—	94	2	no	—
106	C₄⋊C₄	—	94	2	no	—
107	C₂²⋊C₄	—	94	2	no	—
108	C₄²	—	94	2	no	—
109	C₄²	C₄×D₄₇	1	1	yes	characteristic
110	C₄²	D₁₈₈	1	1	yes	characteristic
111	C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
112	C₄²	C₂²×D₄₇	1	1	yes	characteristic
113	C₂³	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
114	Q₈	—	188	2	no	—
115	Q₈	—	188	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	4	1	no	—
119	C₂×C₄	—	94	2	no	—
120	C₂×C₄	—	94	2	no	—
121	C₂×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
122	C₂×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
123	C₂×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
124	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
125	C₂×C₄	C₂×D₁₈₈	1	1	yes	characteristic
126	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
127	C₈	—	2	4	no	—
128	C₈	—	188	2	no	—
129	C₂²	—	2	1	no	—
130	C₂²	—	2	1	no	—
131	C₂²	#13093942	1	2	yes	—
132	C₂²	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	2	1	no	—
135	C₄	—	2	1	no	—
136	C₄	—	2	1	no	—
137	C₄	—	2	1	no	—
138	C₄	—	188	2	no	—
139	C₄	—	188	2	no	—
140	C₄	#13093952	1	2	yes	—
141	C₂	—	2	2	no	—
142	C₂	C₂×#13093942	1	1	yes	characteristic
143	C₂	C₂×#13093952	1	1	yes	characteristic
144	C₂	#15689139	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689680

C₂

yes

maximal, characteristic

#15689180

C₂

yes

maximal, characteristic

#15688864

C₂

yes

maximal, complemented

C₂×#97×C₄₇

C₂

yes

maximal, characteristic

#13094024

C₄

yes

characteristic

#13094024

C₄

yes

characteristic

#13094016

C₂²

yes

—

#13094003

—

#13094003

—

C₄×Dic₉₄

—

C₄×Dic₉₄

—

#13093983

C₄

yes

—

C₄⋊(C₄₇⋊C₈)

—

C₂²⋊(C₄₇⋊C₈)

—

#13093935

C₂²

yes

—

C₂²×C₈×C₄₇

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

#97×C₄₇

—

#97×C₄₇

C₂²

yes

—

C₂²×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₄²×C₄₇

C₂×C₄

yes

characteristic

C₄²×C₄₇

D₄

yes

characteristic

C₄²×C₄₇

D₄

yes

characteristic

C₄²×C₄₇

C₂³

yes

characteristic

C₂²⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

C₂×C₄

yes

—

C₄⋊Dic₄₇

C₂×C₄

yes

—

Dic₄₇⋊C₄

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₂²⋊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×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#101

yes

—

C₂²×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#104

yes

—

Dic₄₇

—

Dic₄₇

—

#1531

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

—

C₂×#97

D₄₇

yes

characteristic

#330

—

C₄₇

#1531

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂×D₄₇

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

#97

—

#97

C₂×D₄₇

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

C₄×D₄₇

yes

characteristic

C₂²×C₄

D₁₈₈

yes

characteristic

C₂²×C₄

C₂²⋊D₄₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₂²⋊C₄

—

108

C₄²

—

109

C₄²

C₄×D₄₇

yes

characteristic

110

C₄²

D₁₈₈

yes

characteristic

111

C₄²

C₂²⋊D₄₇

yes

characteristic

112

C₄²

C₂²×D₄₇

yes

characteristic

113

C₂³

(C₂×D₄₇)⋊C₄

yes

characteristic

114

Q₈

—

188

—

115

Q₈

—

188

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

(C₂×D₄₇)⋊C₄

yes

characteristic

122

C₂×C₄

(C₂×D₄₇)⋊C₄

yes

characteristic

123

C₂×C₄

(C₂×D₄₇)⋊C₄

yes

characteristic

124

C₂×C₄

C₂×C₄×D₄₇

yes

characteristic

125

C₂×C₄

C₂×D₁₈₈

yes

characteristic

126

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

127

C₈

—

128

C₈

—

188

—

129

C₂²

—

130

C₂²

—

131

C₂²

#13093942

yes

—

132

C₂²

C₂×(C₂×D₄₇)⋊C₄

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

188

—

139

C₄

—

188

—

140

C₄

#13093952

yes

—

141

C₂

—

142

C₂

C₂×#13093942

yes

characteristic

143

C₂

C₂×#13093952

yes

characteristic

144

C₂

#15689139

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×(C₂×D₄₇)⋊C₄
commutator	C₂×C₄×C₄₇	C₂²×C₄
Frattini	C₄²	C₂²×D₄₇
Fitting	C₂×#97×C₄₇	C₂
socle	C₂²×C₄₇	C₂×C₂²⋊C₄
hypercenter	C₂×#97	D₄₇
hypocenter	C₄₇	#1531
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#97

subgroup

G/H

center

C₂²

C₂×(C₂×D₄₇)⋊C₄

commutator

C₂×C₄×C₄₇

C₂²×C₄

Frattini

C₄²

C₂²×D₄₇

Fitting

C₂×#97×C₄₇

C₂

socle

C₂²×C₄₇

C₂×C₂²⋊C₄

hypercenter

C₂×#97

D₄₇

hypocenter

C₄₇

#1531

LCS

G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#97

N	H ≅ G/N	type
3. #15688864	C₂	semidirect
77. C₄₇	#1531	semidirect

H ≅ G/N

type

3. #15688864

C₂

semidirect

77. C₄₇

#1531

semidirect

Subgroups — #20242002

Conjugacy classes of subgroups

Complements