Group #20242728 — subgroups

Subgroups — #20242728

Order 6016.

188 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 53 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	#15689495	C₂	1	1	yes	maximal, characteristic, complemented
2	#15689404	C₂	1	1	yes	maximal, characteristic, complemented
3	#15689315	C₂	1	1	yes	maximal, characteristic
4	#15689188	C₂	1	1	yes	maximal, characteristic, complemented
5	#15688873	C₂	1	1	yes	maximal, characteristic, complemented
6	#15688870	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#331	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10625335	—	2	1	no	—
9	C₂×#10625335	—	2	1	no	—
10	C₂×#10625334	—	2	1	no	—
11	C₂×C₈×D₄₇	—	2	1	no	—
12	#13094024	C₂²	1	1	yes	characteristic
13	C₄×D₁₈₈	C₂²	1	1	yes	characteristic
14	#13093958	—	2	1	no	—
15	C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
16	#13093946	—	2	1	no	—
17	#13093945	—	2	1	no	—
18	(C₂×D₄₇)⋊C₈	—	2	1	no	—
19	C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
20	#127×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×D₄×C₄₇	—	2	1	no	—
24	C₂×D₄×C₄₇	—	2	1	no	—
25	C₂×D₄×C₄₇	—	2	1	no	—
26	C₂×D₁₈₈	D₄	1	1	yes	characteristic
27	C₂×C₄×D₄₇	—	2	1	no	—
28	C₂×Dic₉₄	—	2	1	no	—
29	C₂×C₈×C₄₇	—	2	1	no	—
30	C₂×C₈×C₄₇	—	2	1	no	—
31	C₄²×C₄₇	C₂³	1	1	yes	characteristic
32	#10625335	—	4	2	no	—
33	#10625335	—	4	2	no	—
34	#10625334	—	4	2	no	—
35	(C₂×D₄₇)⋊C₄	—	2	1	no	—
36	C₄⋊Dic₄₇	—	2	1	no	—
37	C₄⋊Dic₄₇	—	2	1	no	—
38	C₄⋊Dic₄₇	D₄	1	1	yes	characteristic
39	C₂×C₄₇⋊C₈	—	2	1	no	—
40	C₂×C₄₇⋊C₈	D₄	1	1	yes	characteristic
41	C₂×C₄₇⋊C₈	D₄	1	1	yes	characteristic
42	C₈×D₄₇	—	4	2	no	—
43	C₂³×C₄₇	—	4	1	no	—
44	C₂²×D₄₇	—	2	1	no	—
45	D₄×C₄₇	—	4	1	no	—
46	D₄×C₄₇	—	4	1	no	—
47	D₄×C₄₇	—	4	1	no	—
48	D₄×C₄₇	—	4	1	no	—
49	D₄×C₄₇	—	8	1	no	—
50	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
53	C₂×Dic₄₇	—	2	1	no	—
54	C₂×Dic₄₇	—	4	1	no	—
55	D₁₈₈	—	2	1	no	—
56	D₁₈₈	SD₁₆	1	2	yes	complemented
57	C₄×D₄₇	—	4	1	no	—
58	C₄×D₄₇	—	4	1	no	—
59	Dic₉₄	—	4	1	no	—
60	Dic₉₄	—	4	1	no	—
61	C₈×C₄₇	—	4	1	no	—
62	C₈×C₄₇	—	4	1	no	—
63	C₄₇⋊C₈	—	2	1	no	—
64	C₄₇⋊C₈	—	4	1	no	—
65	C₄₇⋊C₈	D₈	1	2	yes	complemented
66	C₂²×C₄₇	—	8	1	no	—
67	C₂²×C₄₇	—	8	1	no	—
68	C₂²×C₄₇	—	8	1	no	—
69	C₂²×C₄₇	#121	1	1	yes	characteristic
70	C₂×D₄₇	—	2	2	no	—
71	C₂×D₄₇	—	4	1	no	—
72	C₂×D₄₇	—	4	1	no	—
73	C₄×C₄₇	—	2	1	no	—
74	C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
75	C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
76	C₄×C₄₇	#135	1	1	yes	characteristic
77	C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
78	Dic₄₇	—	4	1	no	—
79	Dic₄₇	—	8	1	no	—
80	#1665	—	47	1	no	maximal, Sylow
81	C₂×C₄₇	—	16	1	no	—
82	C₂×C₄₇	#459	1	1	yes	characteristic
83	C₂×C₄₇	#465	1	1	yes	characteristic
84	C₂×C₄₇	#471	1	1	yes	characteristic
85	D₄₇	—	4	2	no	—
86	#492	—	47	1	no	—
87	#474	—	47	1	no	—
88	#459	—	47	1	no	—
89	C₈×D₄	—	47	1	no	—
90	#334	—	47	1	no	—
91	#332	—	47	1	no	—
92	#331	D₄₇	1	1	yes	characteristic, complemented
93	C₄₇	#1665	1	1	yes	characteristic, complemented, Sylow
94	C₂×SD₁₆	—	94	1	no	—
95	C₂×SD₁₆	—	94	1	no	—
96	C₂×D₈	—	94	1	no	—
97	C₂²×C₈	—	94	1	no	—
98	C₄⋊Q₈	—	47	1	no	—
99	#127	C₂×D₄₇	1	1	yes	characteristic
100	C₄×D₄	—	47	1	no	—
101	#107	—	94	1	no	—
102	C₄⋊C₈	—	47	1	no	—
103	C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
104	C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
105	#102	—	94	1	no	—
106	#102	—	94	1	no	—
107	C₂²⋊C₈	—	94	1	no	—
108	C₄×C₈	—	47	1	no	—
109	C₂×Q₈	—	94	1	no	—
110	C₂×D₄	—	2	1	no	—
111	C₂×D₄	—	2	1	no	—
112	C₂×D₄	—	2	1	no	—
113	C₂×D₄	—	47	1	no	—
114	C₂²×C₄	—	94	1	no	—
115	SD₁₆	—	188	2	no	—
116	SD₁₆	—	188	2	no	—
117	D₈	—	188	2	no	—
118	C₂×C₈	—	2	1	no	—
119	C₂×C₈	—	2	1	no	—
120	C₂×C₈	—	47	1	no	—
121	C₂×C₈	—	47	1	no	—
122	C₂×C₈	—	94	1	no	—
123	C₂×C₈	—	188	2	no	—
124	C₄⋊C₄	—	47	1	no	—
125	C₄⋊C₄	—	94	1	no	—
126	C₄⋊C₄	—	94	1	no	—
127	C₂²⋊C₄	—	94	1	no	—
128	C₄²	C₂²×D₄₇	1	1	yes	characteristic
129	C₂³	—	4	1	no	—
130	C₂³	—	94	1	no	—
131	Q₈	—	188	1	no	—
132	Q₈	—	188	1	no	—
133	D₄	—	4	1	no	—
134	D₄	—	4	1	no	—
135	D₄	—	4	1	no	—
136	D₄	—	4	1	no	—
137	D₄	—	8	1	no	—
138	D₄	—	47	2	no	—
139	D₄	—	94	1	no	—
140	C₂×C₄	—	94	1	no	—
141	C₂×C₄	—	188	1	no	—
142	C₂×C₄	—	188	1	no	—
143	C₂×C₄	—	188	1	no	—
144	C₂×C₄	#10625350	1	1	yes	characteristic
145	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
146	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
147	C₈	—	4	1	no	—
148	C₈	—	4	1	no	—
149	C₈	—	47	2	no	—
150	C₈	—	94	1	no	—
151	C₈	—	188	1	no	—
152	C₂²	—	8	1	no	—
153	C₂²	—	8	1	no	—
154	C₂²	—	8	1	no	—
155	C₂²	—	94	2	no	—
156	C₂²	—	188	1	no	—
157	C₂²	—	188	1	no	—
158	C₂²	#13093997	1	1	yes	characteristic
159	C₄	—	2	1	no	—
160	C₄	—	376	1	no	—
161	C₄	—	188	1	no	—
162	C₄	D₈×D₄₇	1	1	yes	characteristic
163	C₄	SD₁₆×D₄₇	1	1	yes	characteristic
164	C₄	#13094043	1	1	yes	characteristic
165	C₄	#13094051	1	1	yes	characteristic
166	C₂	—	16	1	no	—
167	C₂	—	188	2	no	—
168	C₂	#15689317	1	1	yes	characteristic
169	C₂	#15689351	1	1	yes	characteristic
170	C₂	#15689385	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689495

C₂

yes

maximal, characteristic, complemented

#15689404

C₂

yes

maximal, characteristic, complemented

#15689315

C₂

yes

maximal, characteristic

#15689188

C₂

yes

maximal, characteristic, complemented

#15688873

C₂

yes

maximal, characteristic, complemented

#15688870

C₂

yes

maximal, characteristic, complemented

C₄₇×#331

C₂

yes

maximal, characteristic, complemented

C₂×#10625335

—

C₂×#10625335

—

C₂×#10625334

—

C₂×C₈×D₄₇

—

#13094024

C₂²

yes

characteristic

C₄×D₁₈₈

C₂²

yes

characteristic

#13093958

—

C₄⋊(C₄₇⋊C₈)

C₂²

yes

characteristic

#13093946

—

#13093945

—

(C₂×D₄₇)⋊C₈

—

C₄×C₄₇⋊C₈

C₂²

yes

characteristic

#127×C₄₇

C₂²

yes

characteristic

C₄⋊C₈×C₄₇

C₂²

yes

characteristic

C₄⋊C₈×C₄₇

C₂²

yes

characteristic

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₁₈₈

D₄

yes

characteristic

C₂×C₄×D₄₇

—

C₂×Dic₉₄

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₄²×C₄₇

C₂³

yes

—

—

—

(C₂×D₄₇)⋊C₄

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

D₄

yes

characteristic

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

D₄

yes

characteristic

C₂×C₄₇⋊C₈

D₄

yes

characteristic

C₈×D₄₇

—

C₂³×C₄₇

—

C₂²×D₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

#41

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

D₁₈₈

—

D₁₈₈

SD₁₆

yes

complemented

C₄×D₄₇

—

C₄×D₄₇

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

D₈

yes

complemented

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#121

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂×D₈

yes

characteristic

C₄×C₄₇

C₂×SD₁₆

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

Dic₄₇

—

Dic₄₇

—

#1665

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

D₄₇

—

#492

—

#474

—

#459

—

C₈×D₄

—

#334

—

#332

—

#331

D₄₇

yes

characteristic, complemented

C₄₇

#1665

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

C₂×SD₁₆

—

C₂×D₈

—

C₂²×C₈

—

C₄⋊Q₈

—

#127

C₂×D₄₇

yes

characteristic

100

C₄×D₄

—

101

#107

—

102

C₄⋊C₈

—

103

C₄⋊C₈

C₂×D₄₇

yes

characteristic

104

C₄⋊C₈

C₂×D₄₇

yes

characteristic

105

#102

—

106

#102

—

107

C₂²⋊C₈

—

108

C₄×C₈

—

109

C₂×Q₈

—

110

C₂×D₄

—

111

C₂×D₄

—

112

C₂×D₄

—

113

C₂×D₄

—

114

C₂²×C₄

—

115

SD₁₆

—

188

—

116

SD₁₆

—

188

—

117

D₈

—

188

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

188

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₂²⋊C₄

—

128

C₄²

C₂²×D₄₇

yes

characteristic

129

C₂³

—

130

C₂³

—

131

Q₈

—

188

—

132

Q₈

—

188

—

133

D₄

—

134

D₄

—

135

D₄

—

136

D₄

—

137

D₄

—

138

D₄

—

139

D₄

—

140

C₂×C₄

—

141

C₂×C₄

—

188

—

142

C₂×C₄

—

188

—

143

C₂×C₄

—

188

—

144

C₂×C₄

#10625350

yes

characteristic

145

C₂×C₄

D₄×D₄₇

yes

characteristic

146

C₂×C₄

D₄×D₄₇

yes

characteristic

147

C₈

—

148

C₈

—

149

C₈

—

150

C₈

—

151

C₈

—

188

—

152

C₂²

—

153

C₂²

—

154

C₂²

—

155

C₂²

—

156

C₂²

—

188

—

157

C₂²

—

188

—

158

C₂²

#13093997

yes

characteristic

159

C₄

—

160

C₄

—

376

—

161

C₄

—

188

—

162

C₄

D₈×D₄₇

yes

characteristic

163

C₄

SD₁₆×D₄₇

yes

characteristic

164

C₄

#13094043

yes

characteristic

165

C₄

#13094051

yes

characteristic

166

C₂

—

167

C₂

—

188

—

168

C₂

#15689317

yes

characteristic

169

C₂

#15689351

yes

characteristic

170

C₂

#15689385

yes

characteristic

	subgroup	G/H
center	C₂²	#13093997
commutator	C₄²×C₄₇	C₂³
Frattini	C₄²	C₂²×D₄₇
Fitting	C₄₇×#331	C₂
socle	C₂²×C₄₇	#121
hypercenter	#331	D₄₇
hypocenter	C₄₇	#1665
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #331

subgroup

G/H

center

C₂²

#13093997

commutator

C₄²×C₄₇

C₂³

Frattini

C₄²

C₂²×D₄₇

Fitting

C₄₇×#331

C₂

C₂²×C₄₇

D₄₇

C₄₇

G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #331

N	H ≅ G/N	type
1. #15689495	C₂	semidirect
2. #15689404	C₂	semidirect
4. #15689188	C₂	semidirect
5. #15688873	C₂	semidirect
6. #15688870	C₂	semidirect
7. C₄₇×#331	C₂	semidirect
56. D₁₈₈	SD₁₆	semidirect
65. C₄₇⋊C₈	D₈	semidirect
92. #331	D₄₇	semidirect
93. C₄₇	#1665	semidirect

H ≅ G/N

type

1. #15689495

C₂

semidirect

2. #15689404

C₂

semidirect

4. #15689188

C₂

semidirect

5. #15688873

C₂

semidirect

6. #15688870

C₂

semidirect

7. C₄₇×#331

C₂

semidirect

56. D₁₈₈

SD₁₆

semidirect

65. C₄₇⋊C₈

D₈

semidirect

92. #331

D₄₇

semidirect

93. C₄₇

#1665

semidirect

Subgroups — #20242728

Conjugacy classes of subgroups

Complements