Group #20245427 — subgroups

Subgroups — #20245427

Order 6016.

174 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 174 conjugacy classes of subgroups — too many to draw. Only its 49 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×#13093969	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#13093968	C₂	1	1	yes	maximal, characteristic, complemented
3	#15689498	C₂	1	1	yes	maximal, characteristic
4	#15689217	C₂	1	1	yes	maximal, characteristic, complemented
5	Dic₄₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	#15688923	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#357	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₁₈₈	C₂²	1	1	yes	characteristic
9	C₂×D₃₇₆	C₂²	1	1	yes	characteristic
10	C₂×#10625334	—	2	1	no	—
11	#13094020	—	2	1	no	—
12	#13093989	—	2	1	no	—
13	#13093969	—	2	2	no	—
14	#13093968	—	2	2	no	—
15	C₂×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
16	#13093947	—	2	1	no	—
17	C₈×Dic₄₇	C₂²	1	1	yes	characteristic
18	C₂×D₈×C₄₇	C₂²	1	1	yes	characteristic
19	C₂×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
20	#107×C₄₇	C₂²	1	1	yes	characteristic
21	C₂×D₄×C₄₇	—	2	1	no	—
22	C₂×C₂²⋊D₄₇	—	4	1	no	—
23	C₂×C₂²⋊D₄₇	—	4	1	no	—
24	C₂×D₁₈₈	—	2	1	no	—
25	C₂×Dic₉₄	—	2	1	no	—
26	D₈×C₄₇	—	2	1	no	—
27	D₈×C₄₇	—	2	1	no	—
28	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
29	C₄⋊C₄×C₄₇	—	2	1	no	—
30	#10625334	—	4	2	no	—
31	Dic₄₇⋊C₄	—	4	1	no	—
32	C₄×Dic₄₇	—	4	1	no	—
33	C₄×Dic₄₇	D₄	1	1	yes	characteristic
34	C₂×C₄₇⋊C₈	D₄	1	1	yes	characteristic
35	Dic₁₈₈	—	2	1	no	—
36	Dic₁₈₈	D₄	1	2	yes	complemented
37	D₃₇₆	—	2	1	no	—
38	D₃₇₆	—	2	1	no	—
39	C₁₆×C₄₇	—	2	1	no	—
40	C₄₇⋊C₁₆	—	2	1	no	—
41	C₂³×C₄₇	—	4	1	no	—
42	C₂²×D₄₇	—	4	1	no	—
43	D₄×C₄₇	—	4	1	no	—
44	D₄×C₄₇	—	4	1	no	—
45	C₂×C₄×C₄₇	—	4	1	no	—
46	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
47	C₂²⋊D₄₇	—	8	2	no	—
48	C₂²⋊D₄₇	—	8	2	no	—
49	C₂×Dic₄₇	—	4	1	no	—
50	C₂×Dic₄₇	D₈	1	1	yes	characteristic
51	C₂×Dic₄₇	D₈	1	1	yes	characteristic
52	D₁₈₈	—	4	1	no	—
53	D₁₈₈	—	4	1	no	—
54	Dic₉₄	—	2	2	no	—
55	Dic₉₄	—	4	1	no	—
56	C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
57	C₈×C₄₇	#41	1	1	yes	characteristic
58	C₄₇⋊C₈	—	2	1	no	—
59	C₂²×C₄₇	—	8	1	no	—
60	C₂²×C₄₇	—	8	1	no	—
61	C₂²×C₄₇	—	8	1	no	—
62	C₂²×C₄₇	C₂×D₈	1	1	yes	characteristic
63	C₂×D₄₇	—	8	1	no	—
64	C₂×D₄₇	—	8	1	no	—
65	C₂×D₄₇	—	8	1	no	—
66	C₄×C₄₇	—	8	1	no	—
67	C₄×C₄₇	#121	1	1	yes	characteristic
68	C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
69	Dic₄₇	—	2	1	no	—
70	Dic₄₇	—	4	2	no	—
71	Dic₄₇	SD₃₂	1	2	yes	complemented
72	#2207	—	47	1	no	maximal, Sylow
73	C₂×C₄₇	—	16	1	no	—
74	C₂×C₄₇	#459	1	1	yes	characteristic
75	C₂×C₄₇	C₂×SD₃₂	1	1	yes	characteristic
76	C₂×C₄₇	#509	1	1	yes	characteristic
77	D₄₇	—	16	1	no	—
78	C₂×SD₃₂	—	47	1	no	—
79	C₂×SD₃₂	—	47	1	no	—
80	#493	—	47	1	no	—
81	C₄×Q₁₆	—	47	1	no	—
82	C₄⋊C₁₆	—	47	1	no	—
83	#357	—	47	1	no	—
84	#357	D₄₇	1	1	yes	characteristic, complemented
85	C₄₇	#2207	1	1	yes	characteristic, complemented, Sylow
86	C₂×Q₁₆	—	47	1	no	—
87	C₂×D₈	—	47	1	no	—
88	C₂×D₈	—	94	1	no	—
89	C₂×D₈	C₂×D₄₇	1	1	yes	characteristic
90	#127	—	94	1	no	—
91	C₄×Q₈	—	94	1	no	—
92	SD₃₂	—	94	2	no	—
93	SD₃₂	—	94	2	no	—
94	C₂×C₁₆	—	47	1	no	—
95	C₂×C₁₆	C₂×D₄₇	1	1	yes	characteristic
96	#107	C₂×D₄₇	1	1	yes	characteristic
97	#103	—	94	1	no	—
98	C₄×C₈	—	47	1	no	—
99	C₂×Q₈	—	94	1	no	—
100	C₂×D₄	—	2	1	no	—
101	C₂×D₄	—	94	1	no	—
102	C₂×D₄	—	188	1	no	—
103	C₂×D₄	—	188	1	no	—
104	Q₁₆	—	47	2	no	—
105	Q₁₆	—	94	1	no	—
106	D₈	—	2	1	no	—
107	D₈	—	2	1	no	—
108	D₈	—	94	1	no	—
109	D₈	—	94	1	no	—
110	D₈	—	188	2	no	—
111	C₂×C₈	—	47	1	no	—
112	C₂×C₈	C₂²×D₄₇	1	1	yes	characteristic
113	C₄⋊C₄	—	2	1	no	—
114	C₄⋊C₄	—	188	1	no	—
115	C₄²	—	47	1	no	—
116	C₄²	—	188	1	no	—
117	C₁₆	—	2	1	no	—
118	C₁₆	—	94	1	no	—
119	C₂³	—	4	1	no	—
120	C₂³	—	188	1	no	—
121	Q₈	—	94	2	no	—
122	Q₈	—	188	1	no	—
123	D₄	—	4	1	no	—
124	D₄	—	4	1	no	—
125	D₄	—	376	2	no	—
126	D₄	—	376	2	no	—
127	D₄	—	188	1	no	—
128	D₄	—	188	1	no	—
129	C₂×C₄	—	4	1	no	—
130	C₂×C₄	—	47	1	no	—
131	C₂×C₄	—	47	1	no	—
132	C₂×C₄	—	188	1	no	—
133	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
134	C₈	—	94	1	no	—
135	C₈	#10625350	1	1	yes	characteristic
136	C₈	D₄×D₄₇	1	1	yes	characteristic
137	C₂²	—	8	1	no	—
138	C₂²	—	8	1	no	—
139	C₂²	—	8	1	no	—
140	C₂²	—	376	1	no	—
141	C₂²	—	376	1	no	—
142	C₂²	—	376	1	no	—
143	C₂²	D₈×D₄₇	1	1	yes	characteristic
144	C₄	—	8	1	no	—
145	C₄	—	47	2	no	—
146	C₄	—	94	1	no	—
147	C₄	—	188	2	no	—
148	C₄	#13093997	1	1	yes	characteristic
149	C₄	#13094051	1	1	yes	characteristic
150	C₂	—	16	1	no	—
151	C₂	—	752	1	no	—
152	C₂	#15689317	1	1	yes	characteristic
153	C₂	SD₃₂×D₄₇	1	1	yes	characteristic
154	C₂	#15689557	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13093969

C₂

yes

maximal, characteristic, complemented

C₂×#13093968

C₂

yes

maximal, characteristic, complemented

#15689498

C₂

yes

maximal, characteristic

#15689217

C₂

yes

maximal, characteristic, complemented

Dic₄₇⋊C₁₆

C₂

yes

maximal, characteristic, complemented

#15688923

C₂

yes

maximal, characteristic, complemented

C₄₇×#357

C₂

yes

maximal, characteristic, complemented

C₂×Dic₁₈₈

C₂²

yes

characteristic

C₂×D₃₇₆

C₂²

yes

—

—

—

—

—

C₂×C₄₇⋊C₁₆

C₂²

yes

characteristic

#13093947

—

C₈×Dic₄₇

C₂²

yes

characteristic

C₂×D₈×C₄₇

C₂²

yes

characteristic

C₂×C₁₆×C₄₇

C₂²

yes

characteristic

#107×C₄₇

C₂²

yes

characteristic

C₂×D₄×C₄₇

—

C₂×C₂²⋊D₄₇

—

C₂×C₂²⋊D₄₇

—

C₂×D₁₈₈

—

C₂×Dic₉₄

—

D₈×C₄₇

—

D₈×C₄₇

—

C₂×C₈×C₄₇

C₂³

yes

characteristic

C₄⋊C₄×C₄₇

—

#10625334

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

C₄×Dic₄₇

D₄

yes

characteristic

C₂×C₄₇⋊C₈

D₄

yes

characteristic

Dic₁₈₈

—

Dic₁₈₈

D₄

yes

complemented

D₃₇₆

—

D₃₇₆

—

C₁₆×C₄₇

—

C₄₇⋊C₁₆

—

C₂³×C₄₇

—

C₂²×D₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²⋊D₄₇

—

C₂²⋊D₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

D₈

yes

characteristic

C₂×Dic₄₇

D₈

yes

characteristic

D₁₈₈

—

D₁₈₈

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

C₂×D₄

yes

characteristic

C₈×C₄₇

#41

yes

characteristic

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×D₈

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

Dic₄₇

—

Dic₄₇

—

Dic₄₇

SD₃₂

yes

complemented

#2207

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

#459

yes

characteristic

C₂×C₄₇

C₂×SD₃₂

yes

characteristic

C₂×C₄₇

#509

yes

characteristic

D₄₇

—

C₂×SD₃₂

—

C₂×SD₃₂

—

#493

—

C₄×Q₁₆

—

C₄⋊C₁₆

—

#357

—

#357

D₄₇

yes

characteristic, complemented

C₄₇

#2207

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

C₂×D₄₇

yes

characteristic

#127

—

C₄×Q₈

—

SD₃₂

—

SD₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₄₇

yes

characteristic

#107

C₂×D₄₇

yes

characteristic

#103

—

C₄×C₈

—

C₂×Q₈

—

100

C₂×D₄

—

101

C₂×D₄

—

102

C₂×D₄

—

188

—

103

C₂×D₄

—

188

—

104

Q₁₆

—

105

Q₁₆

—

106

D₈

—

107

D₈

—

108

D₈

—

109

D₈

—

110

D₈

—

188

—

111

C₂×C₈

—

112

C₂×C₈

C₂²×D₄₇

yes

characteristic

113

C₄⋊C₄

—

114

C₄⋊C₄

—

188

—

115

C₄²

—

116

C₄²

—

188

—

117

C₁₆

—

118

C₁₆

—

119

C₂³

—

120

C₂³

—

188

—

121

Q₈

—

122

Q₈

—

188

—

123

D₄

—

124

D₄

—

125

D₄

—

376

—

126

D₄

—

376

—

127

D₄

—

188

—

128

D₄

—

188

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

188

—

133

C₂×C₄

D₄×D₄₇

yes

characteristic

134

C₈

—

135

C₈

#10625350

yes

characteristic

136

C₈

D₄×D₄₇

yes

characteristic

137

C₂²

—

138

C₂²

—

139

C₂²

—

140

C₂²

—

376

—

141

C₂²

—

376

—

142

C₂²

—

376

—

143

C₂²

D₈×D₄₇

yes

characteristic

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

—

188

—

148

C₄

#13093997

yes

characteristic

149

C₄

#13094051

yes

characteristic

150

C₂

—

151

C₂

—

752

—

152

C₂

#15689317

yes

characteristic

153

C₂

SD₃₂×D₄₇

yes

characteristic

154

C₂

#15689557

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₄₇
commutator	C₂×C₈×C₄₇	C₂³
Frattini	C₂×C₈	C₂²×D₄₇
Fitting	C₄₇×#357	C₂
socle	C₂²×C₄₇	C₂×D₈
hypercenter	#357	D₄₇
hypocenter	C₄₇	#2207
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #357

subgroup

G/H

center

C₂²

D₈×D₄₇

commutator

C₂×C₈×C₄₇

C₂³

Frattini

C₂×C₈

C₂²×D₄₇

Fitting

C₄₇×#357

C₂

socle

C₂²×C₄₇

C₂×D₈

D₄₇

C₄₇

G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #357

N	H ≅ G/N	type
1. C₂×#13093969	C₂	semidirect
2. C₂×#13093968	C₂	semidirect
4. #15689217	C₂	semidirect
5. Dic₄₇⋊C₁₆	C₂	semidirect
6. #15688923	C₂	semidirect
7. C₄₇×#357	C₂	semidirect
36. Dic₁₈₈	D₄	semidirect
71. Dic₄₇	SD₃₂	semidirect
84. #357	D₄₇	semidirect
85. C₄₇	#2207	semidirect

H ≅ G/N

type

1. C₂×#13093969

C₂

semidirect

2. C₂×#13093968

C₂

semidirect

4. #15689217

C₂

semidirect

5. Dic₄₇⋊C₁₆

C₂

semidirect

6. #15688923

C₂

semidirect

7. C₄₇×#357

C₂

semidirect

36. Dic₁₈₈

D₄

semidirect

71. Dic₄₇

SD₃₂

semidirect

84. #357

D₄₇

semidirect

85. C₄₇

#2207

semidirect

Subgroups — #20245427

Conjugacy classes of subgroups

Complements