Group #20242187 — subgroups

Subgroups — #20242187

Order 6016.

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

normal subgroups only

This group has 180 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	#15689247	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#10625325	C₂	1	1	yes	maximal, characteristic
3	#15689188	C₂	1	1	yes	maximal, characteristic
4	#15688853	C₂	1	1	yes	maximal, characteristic, complemented
5	#15688850	C₂	1	1	yes	maximal, characteristic
6	C₈×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#334	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10625325	C₄	1	1	yes	characteristic
9	C₂×C₈×D₄₇	—	2	1	no	—
10	C₄×Dic₉₄	C₂²	1	1	yes	characteristic
11	C₄×D₁₈₈	C₂²	1	1	yes	characteristic
12	#13093956	C₄	1	1	yes	characteristic
13	Dic₄₇⋊C₈	—	2	1	no	—
14	#13093949	C₄	1	1	yes	characteristic
15	#13093946	C₄	1	1	yes	characteristic
16	(C₂×D₄₇)⋊C₈	—	2	1	no	—
17	C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
18	C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
19	C₈×Dic₄₇	—	2	1	no	—
20	C₄⋊C₈×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×D₁₈₈	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×D₄₇	—	2	1	no	—
25	C₂×Dic₉₄	C₂×C₄	1	1	yes	characteristic
26	C₂×C₈×C₄₇	—	2	1	no	—
27	C₂×C₈×C₄₇	—	2	1	no	—
28	C₂×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
29	C₂×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
30	C₄²×C₄₇	C₂³	1	1	yes	characteristic
31	(C₂×D₄₇)⋊C₄	—	2	1	no	—
32	C₄⋊Dic₄₇	C₂×C₄	1	1	yes	characteristic
33	C₄⋊Dic₄₇	C₂×C₄	1	1	yes	characteristic
34	Dic₄₇⋊C₄	—	2	1	no	—
35	C₄×Dic₄₇	—	2	1	no	—
36	C₂×C₄₇⋊C₈	—	2	1	no	—
37	C₂×C₄₇⋊C₈	D₄	1	1	yes	characteristic
38	C₂×C₄₇⋊C₈	D₄	1	1	yes	characteristic
39	#10625325	C₈	1	4	yes	complemented
40	C₈×D₄₇	—	4	2	no	—
41	C₂²×D₄₇	—	2	1	no	—
42	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
45	C₂×Dic₄₇	—	2	1	no	—
46	C₂×Dic₄₇	—	2	1	no	—
47	C₂×Dic₄₇	—	2	1	no	—
48	D₁₈₈	—	2	1	no	—
49	D₁₈₈	C₂×C₈	1	2	yes	—
50	C₄×D₄₇	—	4	1	no	—
51	C₄×D₄₇	—	4	1	no	—
52	Dic₉₄	—	2	1	no	—
53	Dic₉₄	C₂×C₈	1	2	yes	complemented
54	C₈×C₄₇	—	2	1	no	—
55	C₈×C₄₇	—	4	1	no	—
56	C₈×C₄₇	—	4	1	no	—
57	C₈×C₄₇	C₂×C₈	1	2	yes	complemented
58	C₄₇⋊C₈	—	2	2	no	—
59	C₄₇⋊C₈	—	2	1	no	—
60	C₄₇⋊C₈	SD₁₆	1	2	yes	complemented
61	C₂²×C₄₇	C₄×D₄	1	1	yes	characteristic
62	C₂×D₄₇	—	2	2	no	—
63	C₂×D₄₇	—	4	1	no	—
64	C₂×D₄₇	—	4	1	no	—
65	C₄×C₄₇	—	2	1	no	—
66	C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
67	C₄×C₄₇	#131	1	1	yes	characteristic
68	C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
69	C₄×C₄₇	#135	1	1	yes	characteristic
70	Dic₄₇	—	2	2	no	—
71	Dic₄₇	—	4	1	no	—
72	Dic₄₇	—	4	1	no	—
73	C₈×SD₁₆	—	47	1	no	maximal, Sylow
74	C₂×C₄₇	C₈×D₄	1	1	yes	characteristic
75	C₂×C₄₇	C₄×SD₁₆	1	1	yes	characteristic
76	C₂×C₄₇	#443	1	1	yes	characteristic
77	D₄₇	—	4	2	no	—
78	C₈×Q₈	—	47	1	no	—
79	C₄×SD₁₆	—	47	1	no	—
80	C₈×D₄	—	47	1	no	—
81	#334	D₄₇	1	1	yes	characteristic, complemented
82	Q₈⋊C₈	—	47	1	no	—
83	D₄⋊C₈	—	47	1	no	—
84	C₈²	—	47	1	no	—
85	C₄₇	C₈×SD₁₆	1	1	yes	characteristic, complemented, Sylow
86	C₂×SD₁₆	—	47	1	no	—
87	C₂²×C₈	—	94	1	no	—
88	C₄×Q₈	—	47	1	no	—
89	C₄×D₄	—	47	1	no	—
90	#106	—	47	1	no	—
91	C₄⋊C₈	—	94	1	no	—
92	C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
93	C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
94	#103	—	47	1	no	—
95	#102	—	47	1	no	—
96	C₂²⋊C₈	—	94	1	no	—
97	C₄×C₈	—	47	1	no	—
98	C₄×C₈	—	47	1	no	—
99	C₄×C₈	—	94	1	no	—
100	C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
101	C₂×Q₈	—	47	1	no	—
102	C₂×D₄	—	47	1	no	—
103	C₂²×C₄	—	94	1	no	—
104	SD₁₆	—	47	4	no	—
105	C₂×C₈	—	2	1	no	—
106	C₂×C₈	—	2	1	no	—
107	C₂×C₈	—	47	1	no	—
108	C₂×C₈	—	47	1	no	—
109	C₂×C₈	—	94	1	no	—
110	C₂×C₈	—	188	2	no	—
111	C₂×C₈	C₄×D₄₇	1	1	yes	characteristic
112	C₂×C₈	C₄×D₄₇	1	1	yes	characteristic
113	C₄⋊C₄	—	47	1	no	—
114	C₄⋊C₄	—	47	1	no	—
115	C₄⋊C₄	—	94	1	no	—
116	C₂²⋊C₄	—	94	1	no	—
117	C₄²	—	94	1	no	—
118	C₄²	C₂²×D₄₇	1	1	yes	characteristic
119	C₂³	—	94	1	no	—
120	Q₈	—	47	2	no	—
121	Q₈	—	94	1	no	—
122	D₄	—	47	2	no	—
123	D₄	—	94	1	no	—
124	C₂×C₄	—	94	1	no	—
125	C₂×C₄	—	94	1	no	—
126	C₂×C₄	—	94	1	no	—
127	C₂×C₄	—	188	1	no	—
128	C₂×C₄	—	188	1	no	—
129	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
130	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
131	C₂×C₄	#10625354	1	1	yes	characteristic
132	C₈	—	2	1	no	—
133	C₈	—	4	1	no	—
134	C₈	—	4	1	no	—
135	C₈	—	47	2	no	—
136	C₈	—	94	2	no	—
137	C₈	—	94	1	no	—
138	C₈	C₈×D₄₇	1	2	yes	complemented
139	C₂²	—	94	2	no	—
140	C₂²	—	188	1	no	—
141	C₂²	—	188	1	no	—
142	C₂²	#13093986	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	94	2	no	—
145	C₄	—	188	1	no	—
146	C₄	—	188	1	no	—
147	C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
148	C₄	#13094032	1	1	yes	characteristic
149	C₄	SD₁₆×D₄₇	1	1	yes	characteristic
150	C₄	#13094045	1	1	yes	characteristic
151	C₂	—	188	2	no	—
152	C₂	#15689188	1	1	yes	characteristic
153	C₂	#15689210	1	1	yes	characteristic
154	C₂	#15689239	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689247

C₂

yes

maximal, characteristic, complemented

C₄×#10625325

C₂

yes

maximal, characteristic

#15689188

C₂

yes

maximal, characteristic

#15688853

C₂

yes

maximal, characteristic, complemented

#15688850

C₂

yes

maximal, characteristic

C₈×C₄₇⋊C₈

C₂

yes

maximal, characteristic, complemented

C₄₇×#334

C₂

yes

maximal, characteristic, complemented

C₂×#10625325

C₄

yes

characteristic

C₂×C₈×D₄₇

—

C₄×Dic₉₄

C₂²

yes

characteristic

C₄×D₁₈₈

C₂²

yes

characteristic

#13093956

C₄

yes

characteristic

Dic₄₇⋊C₈

—

#13093949

C₄

yes

characteristic

#13093946

C₄

yes

characteristic

(C₂×D₄₇)⋊C₈

—

C₄×C₄₇⋊C₈

C₂²

yes

characteristic

C₄×C₄₇⋊C₈

C₂²

yes

characteristic

C₈×Dic₄₇

—

C₄⋊C₈×C₄₇

C₂²

yes

characteristic

C₄⋊C₈×C₄₇

C₂²

yes

characteristic

C₄×C₈×C₄₇

C₂²

yes

characteristic

C₂×D₁₈₈

C₂×C₄

yes

characteristic

C₂×C₄×D₄₇

—

C₂×Dic₉₄

C₂×C₄

yes

characteristic

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₂×C₄

yes

characteristic

C₂×C₈×C₄₇

C₂×C₄

yes

characteristic

C₄²×C₄₇

C₂³

yes

characteristic

(C₂×D₄₇)⋊C₄

—

C₄⋊Dic₄₇

C₂×C₄

yes

characteristic

C₄⋊Dic₄₇

C₂×C₄

yes

characteristic

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

D₄

yes

characteristic

C₂×C₄₇⋊C₈

D₄

yes

characteristic

#10625325

C₈

yes

complemented

C₈×D₄₇

—

C₂²×D₄₇

—

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

#41

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

D₁₈₈

—

D₁₈₈

C₂×C₈

yes

—

C₄×D₄₇

—

C₄×D₄₇

—

Dic₉₄

—

Dic₉₄

C₂×C₈

yes

complemented

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

C₂×C₈

yes

complemented

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

SD₁₆

yes

complemented

C₂²×C₄₇

C₄×D₄

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂²×C₈

yes

characteristic

C₄×C₄₇

#131

yes

characteristic

C₄×C₄₇

C₂×SD₁₆

yes

characteristic

C₄×C₄₇

#135

yes

characteristic

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

C₈×SD₁₆

—

maximal, Sylow

C₂×C₄₇

C₈×D₄

yes

characteristic

C₂×C₄₇

C₄×SD₁₆

yes

characteristic

C₂×C₄₇

#443

yes

characteristic

D₄₇

—

C₈×Q₈

—

C₄×SD₁₆

—

C₈×D₄

—

#334

D₄₇

yes

characteristic, complemented

Q₈⋊C₈

—

D₄⋊C₈

—

C₈²

—

C₄₇

C₈×SD₁₆

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

C₂²×C₈

—

C₄×Q₈

—

C₄×D₄

—

#106

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₄₇

yes

characteristic

C₄⋊C₈

C₂×D₄₇

yes

characteristic

#103

—

#102

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

100

C₄×C₈

C₂×D₄₇

yes

characteristic

101

C₂×Q₈

—

102

C₂×D₄

—

103

C₂²×C₄

—

104

SD₁₆

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

188

—

111

C₂×C₈

C₄×D₄₇

yes

characteristic

112

C₂×C₈

C₄×D₄₇

yes

characteristic

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₂²⋊C₄

—

117

C₄²

—

118

C₄²

C₂²×D₄₇

yes

characteristic

119

C₂³

—

120

Q₈

—

121

Q₈

—

122

D₄

—

123

D₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

188

—

128

C₂×C₄

—

188

—

129

C₂×C₄

C₂×C₄×D₄₇

yes

characteristic

130

C₂×C₄

D₄×D₄₇

yes

characteristic

131

C₂×C₄

#10625354

yes

characteristic

132

C₈

—

133

C₈

—

134

C₈

—

135

C₈

—

136

C₈

—

137

C₈

—

138

C₈

C₈×D₄₇

yes

complemented

139

C₂²

—

140

C₂²

—

188

—

141

C₂²

—

188

—

142

C₂²

#13093986

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

188

—

146

C₄

—

188

—

147

C₄

C₂×C₈×D₄₇

yes

characteristic

148

C₄

#13094032

yes

characteristic

149

C₄

SD₁₆×D₄₇

yes

characteristic

150

C₄

#13094045

yes

characteristic

151

C₂

—

188

—

152

C₂

#15689188

yes

characteristic

153

C₂

#15689210

yes

characteristic

154

C₂

#15689239

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₄₇
commutator	C₄×C₄₇	C₂²×C₈
Frattini	C₄²	C₂²×D₄₇
Fitting	C₄₇×#334	C₂
socle	C₂²×C₄₇	C₄×D₄
hypercenter	#334	D₄₇
hypocenter	C₄₇	C₈×SD₁₆
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ #334

subgroup

G/H

center

C₂×C₄

D₄×D₄₇

commutator

C₄×C₄₇

C₂²×C₈

Frattini

C₄²

C₂²×D₄₇

Fitting

C₄₇×#334

C₂

socle

C₂²×C₄₇

C₄×D₄

hypercenter

#334

D₄₇

hypocenter

C₄₇

C₈×SD₁₆

LCS

G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ #334

N	H ≅ G/N	type
1. #15689247	C₂	semidirect
4. #15688853	C₂	semidirect
6. C₈×C₄₇⋊C₈	C₂	semidirect
7. C₄₇×#334	C₂	semidirect
39. #10625325	C₈	semidirect
53. Dic₉₄	C₂×C₈	semidirect
57. C₈×C₄₇	C₂×C₈	semidirect
60. C₄₇⋊C₈	SD₁₆	semidirect
81. #334	D₄₇	semidirect
85. C₄₇	C₈×SD₁₆	semidirect
138. C₈	C₈×D₄₇	semidirect

H ≅ G/N

type

1. #15689247

C₂

semidirect

4. #15688853

C₂

semidirect

6. C₈×C₄₇⋊C₈

C₂

semidirect

7. C₄₇×#334

C₂

semidirect

39. #10625325

C₈

semidirect

53. Dic₉₄

C₂×C₈

semidirect

57. C₈×C₄₇

C₂×C₈

semidirect

60. C₄₇⋊C₈

SD₁₆

semidirect

81. #334

D₄₇

semidirect

85. C₄₇

C₈×SD₁₆

semidirect

138. C₈

C₈×D₄₇

semidirect

Subgroups — #20242187

Conjugacy classes of subgroups

Complements