Group #20244177 — subgroups

Subgroups — #20244177

Order 6016.

184 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 79 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	#15689172	C₂	1	1	yes	maximal, characteristic
2	#15689171	C₂	1	1	yes	maximal, characteristic
3	C₂×#13093959	C₂	1	1	yes	maximal, characteristic
4	C₂×#13093959	C₂	1	1	yes	maximal, characteristic
5	#15689079	C₂	1	1	yes	maximal, characteristic
6	#15688893	C₂	1	1	yes	maximal, characteristic
7	C₄₇×#344	C₂	1	1	yes	maximal, characteristic
8	C₂×#10625329	C₂²	1	1	yes	characteristic
9	C₂×#10625329	C₂²	1	1	yes	characteristic
10	C₂×#10625329	C₂²	1	1	yes	characteristic
11	C₂²×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
12	#13093960	—	2	2	no	—
13	#13093959	—	2	2	no	—
14	#13093959	—	2	1	no	—
15	#13093959	—	2	1	no	—
16	#13093959	C₄	1	4	yes	complemented
17	#13093935	—	2	1	no	—
18	C₄×C₄₇⋊C₈	—	2	1	no	—
19	C₂×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
20	C₂×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
21	#117×C₄₇	C₂²	1	1	yes	characteristic
22	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
23	M₁₆×C₄₇	—	2	1	no	—
24	M₁₆×C₄₇	—	2	1	no	—
25	M₁₆×C₄₇	—	2	1	no	—
26	M₁₆×C₄₇	C₂×C₄	1	2	yes	—
27	C₂×C₈×C₄₇	—	2	1	no	—
28	C₂×C₈×C₄₇	—	2	1	no	—
29	C₄⋊C₄×C₄₇	—	2	1	no	—
30	C₂²⋊C₄×C₄₇	Q₈	1	2	yes	—
31	C₄²×C₄₇	—	2	1	no	—
32	#10625329	—	2	1	no	—
33	#10625329	—	2	1	no	—
34	#10625329	—	2	1	no	—
35	#10625329	—	2	1	no	—
36	#10625329	C₂×C₄	1	2	yes	—
37	#10625329	D₄	1	2	yes	—
38	C₂×C₄₇⋊C₈	—	2	1	no	—
39	C₂×C₄₇⋊C₈	—	2	1	no	—
40	C₂×C₄₇⋊C₈	—	2	1	no	—
41	C₂×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
42	C₂×C₄₇⋊C₈	D₄	1	1	yes	characteristic
43	C₂×C₄₇⋊C₈	D₄	1	1	yes	characteristic
44	C₂×C₄₇⋊C₈	Q₈	1	2	yes	—
45	C₂³×C₄₇	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄×C₄₇	—	2	2	no	—
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×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
52	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
53	C₈×C₄₇	—	2	2	no	—
54	C₈×C₄₇	—	4	1	no	—
55	C₄₇⋊C₈	—	2	2	no	—
56	C₄₇⋊C₈	—	2	2	no	—
57	C₄₇⋊C₈	—	4	1	no	—
58	C₄₇⋊C₈	C₄⋊C₄	1	2	yes	—
59	C₄₇⋊C₈	C₄⋊C₄	1	2	yes	—
60	C₂²×C₄₇	—	4	1	no	—
61	C₂²×C₄₇	C₄×Q₈	1	1	yes	characteristic
62	C₂²×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
63	C₂²×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
64	C₄×C₄₇	—	4	2	no	—
65	C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
67	C₄×C₄₇	#121	1	1	yes	characteristic
68	C₄×C₄₇	#125	1	1	yes	characteristic
69	#1953	—	47	1	no	maximal, Sylow
70	C₂×C₄₇	—	2	1	no	—
71	C₂×C₄₇	—	2	1	no	—
72	C₂×C₄₇	—	2	1	no	—
73	C₂×C₄₇	#389	1	1	yes	characteristic
74	#430	—	47	1	no	—
75	#430	—	47	1	no	—
76	C₂×#108	—	47	1	no	—
77	C₂×#108	—	47	1	no	—
78	#405	—	47	1	no	—
79	#344	—	47	1	no	—
80	#344	D₄₇	1	1	yes	characteristic
81	C₄₇	#1953	1	1	yes	characteristic, complemented, Sylow
82	C₂×M₁₆	—	47	1	no	—
83	C₂×M₁₆	—	47	1	no	—
84	C₂×M₁₆	—	47	1	no	—
85	C₂×M₁₆	C₂×D₄₇	1	1	yes	characteristic
86	C₂×M₁₆	C₂×D₄₇	1	1	yes	characteristic
87	C₂²×C₈	—	47	1	no	—
88	#117	C₂×D₄₇	1	1	yes	characteristic
89	#108	—	47	4	no	—
90	#108	—	94	2	no	—
91	#108	—	94	1	no	—
92	#108	—	94	1	no	—
93	#108	—	94	2	no	—
94	#97	—	94	1	no	—
95	C₄×C₈	—	94	1	no	—
96	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
97	M₁₆	—	2	1	no	—
98	M₁₆	—	2	1	no	—
99	M₁₆	—	2	1	no	—
100	M₁₆	—	47	2	no	—
101	M₁₆	—	47	2	no	—
102	M₁₆	—	94	1	no	—
103	M₁₆	—	94	1	no	—
104	M₁₆	—	94	1	no	—
105	M₁₆	—	94	1	no	—
106	M₁₆	C₄×D₄₇	1	2	yes	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	2	1	no	—
109	C₂×C₈	—	47	2	no	—
110	C₂×C₈	—	47	1	no	—
111	C₂×C₈	—	47	1	no	—
112	C₂×C₈	—	47	2	no	—
113	C₂×C₈	—	94	1	no	—
114	C₂×C₈	—	94	1	no	—
115	C₂×C₈	—	94	1	no	—
116	C₄⋊C₄	—	2	1	no	—
117	C₂²⋊C₄	Dic₉₄	1	2	yes	—
118	C₄²	—	2	1	no	—
119	C₂³	C₂×Dic₉₄	1	1	yes	characteristic
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
122	C₂×C₄	#10625350	1	1	yes	characteristic
123	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
124	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
125	C₂×C₄	#10625352	1	1	yes	characteristic
126	C₂×C₄	Q₈×D₄₇	1	1	yes	characteristic
127	C₈	—	2	2	no	—
128	C₈	—	4	1	no	—
129	C₈	—	47	2	no	—
130	C₈	—	47	2	no	—
131	C₈	—	94	2	no	—
132	C₈	—	94	2	no	—
133	C₈	—	188	1	no	—
134	C₂²	—	4	1	no	—
135	C₂²	C₄×Dic₉₄	1	1	yes	characteristic
136	C₂²	#13094001	1	1	yes	characteristic
137	C₂²	#13094023	1	1	yes	characteristic
138	C₄	—	4	2	no	—
139	C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
140	C₄	(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
141	C₄	#13093997	1	1	yes	characteristic
142	C₄	#13094014	1	1	yes	characteristic
143	C₂	—	2	1	no	—
144	C₂	—	2	1	no	—
145	C₂	—	2	1	no	—
146	C₂	#15689020	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689172

C₂

yes

maximal, characteristic

#15689171

C₂

yes

maximal, characteristic

C₂×#13093959

C₂

yes

maximal, characteristic

C₂×#13093959

C₂

yes

maximal, characteristic

#15689079

C₂

yes

maximal, characteristic

#15688893

C₂

yes

maximal, characteristic

C₄₇×#344

C₂

yes

maximal, characteristic

C₂×#10625329

C₂²

yes

characteristic

C₂×#10625329

C₂²

yes

characteristic

C₂×#10625329

C₂²

yes

characteristic

C₂²×C₄₇⋊C₈

C₂²

yes

—

—

—

—

C₄

yes

complemented

#13093935

—

C₄×C₄₇⋊C₈

—

C₂×M₁₆×C₄₇

C₂²

yes

characteristic

C₂×M₁₆×C₄₇

C₂²

yes

characteristic

#117×C₄₇

C₂²

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

characteristic

M₁₆×C₄₇

—

M₁₆×C₄₇

—

M₁₆×C₄₇

—

M₁₆×C₄₇

C₂×C₄

yes

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₄⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

Q₈

yes

—

C₄²×C₄₇

—

—

—

—

—

C₂×C₄

yes

—

#10625329

D₄

yes

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

C₂×C₄

yes

—

C₂×C₄₇⋊C₈

D₄

yes

characteristic

C₂×C₄₇⋊C₈

D₄

yes

characteristic

C₂×C₄₇⋊C₈

Q₈

yes

—

C₂³×C₄₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₄₇

—

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

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

C₄⋊C₄

yes

—

C₄₇⋊C₈

C₄⋊C₄

yes

—

C₂²×C₄₇

—

C₂²×C₄₇

C₄×Q₈

yes

characteristic

C₂²×C₄₇

C₂²⋊Q₈

yes

characteristic

C₂²×C₄₇

C₄⋊Q₈

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₄₇

C₄×D₄

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

yes

—

—

—

—

—

—

D₄₇

yes

characteristic

C₄₇

#1953

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₄₇

yes

characteristic

C₂×M₁₆

C₂×D₄₇

yes

characteristic

C₂²×C₈

—

#117

C₂×D₄₇

yes

—

—

—

—

—

—

C₄×C₈

—

C₂²×C₄

C₂²×D₄₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

100

M₁₆

—

101

M₁₆

—

102

M₁₆

—

103

M₁₆

—

104

M₁₆

—

105

M₁₆

—

106

M₁₆

C₄×D₄₇

yes

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₄⋊C₄

—

117

C₂²⋊C₄

Dic₉₄

yes

—

118

C₄²

—

119

C₂³

C₂×Dic₉₄

yes

characteristic

120

C₂×C₄

—

121

C₂×C₄

C₂×C₄×D₄₇

yes

characteristic

122

C₂×C₄

#10625350

yes

characteristic

123

C₂×C₄

D₄×D₄₇

yes

characteristic

124

C₂×C₄

D₄×D₄₇

yes

characteristic

125

C₂×C₄

#10625352

yes

characteristic

126

C₂×C₄

Q₈×D₄₇

yes

characteristic

127

C₈

—

128

C₈

—

129

C₈

—

130

C₈

—

131

C₈

—

132

C₈

—

133

C₈

—

188

—

134

C₂²

—

135

C₂²

C₄×Dic₉₄

yes

characteristic

136

C₂²

#13094001

yes

characteristic

137

C₂²

#13094023

yes

characteristic

138

C₄

—

139

C₄

C₄⋊C₄×D₄₇

yes

characteristic

140

C₄

(C₂²⋊D₄₇)⋊C₄

yes

characteristic

141

C₄

#13093997

yes

characteristic

142

C₄

#13094014

yes

characteristic

143

C₂

—

144

C₂

—

145

C₂

—

146

C₂

#15689020

yes

characteristic

	subgroup	G/H
center	C₄	#13093997
commutator	C₂×C₄×C₄₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₄₇
Fitting	C₄₇×#344	C₂
socle	C₂×C₄₇	#389
hypercenter	#344	D₄₇
hypocenter	C₄₇	#1953
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #344

subgroup

G/H

center

C₄

#13093997

commutator

C₂×C₄×C₄₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₄₇

Fitting

C₄₇×#344

C₂

C₂×C₄₇

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #344

N	H ≅ G/N	type
16. #13093959	C₄	semidirect
81. C₄₇	#1953	semidirect

H ≅ G/N

type

16. #13093959

C₄

semidirect

81. C₄₇

#1953

semidirect

Subgroups — #20244177

Conjugacy classes of subgroups

Complements