Group #20252105 — subgroups

Subgroups — #20252105

Order 6016.

484 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 181 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×#13094032	C₂	1	2	yes	maximal, complemented
2	#15689673	C₂	1	1	yes	maximal, characteristic
3	#15689193	C₂	1	8	yes	maximal, complemented
4	#15689091	C₂	1	2	yes	maximal, complemented
5	C₂×#13093935	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄₇×#423	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#10625350	C₂²	1	2	yes	—
8	#13094032	—	2	8	no	—
9	C₂×#10625324	C₂²	1	4	yes	—
10	C₂²×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
11	C₂²×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₈×D₄₇	C₂²	1	4	yes	—
13	#13094024	C₄	1	1	yes	characteristic
14	#13094021	C₄	1	1	yes	characteristic
15	#13094012	C₄	1	2	yes	—
16	#13093998	C₄	1	4	yes	—
17	C₄×D₁₈₈	C₂²	1	4	yes	—
18	(C₂×D₄₇)⋊C₈	C₂²	1	8	yes	—
19	#13093935	C₂²	1	4	yes	—
20	C₂×M₁₆×C₄₇	C₂²	1	2	yes	—
21	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₄₇	C₂²	1	4	yes	—
23	C₂²×C₄×C₄₇	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
25	C₂×C₂²⋊D₄₇	C₂×C₄	1	4	yes	—
26	#10625350	—	2	4	no	—
27	#10625350	—	2	4	no	—
28	C₂×D₁₈₈	C₂×C₄	1	2	yes	—
29	C₂×D₁₈₈	C₂×C₄	1	4	yes	—
30	C₂×C₄×D₄₇	C₂³	1	4	yes	—
31	C₂×Dic₉₄	C₂×C₄	1	2	yes	—
32	M₁₆×C₄₇	—	2	4	no	—
33	C₂×C₈×C₄₇	C₂³	1	4	yes	—
34	C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
35	C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
36	C₄²×C₄₇	C₂³	1	2	yes	—
37	(C₂×D₄₇)⋊C₄	C₂×C₄	1	8	yes	—
38	C₄⋊Dic₄₇	C₂×C₄	1	4	yes	—
39	C₂×C₄₇⋊C₈	—	2	2	no	—
40	C₂×C₄₇⋊C₈	D₄	1	4	yes	complemented
41	C₂×C₄₇⋊C₈	C₂³	1	2	yes	—
42	C₂×C₄₇⋊C₈	C₂³	1	2	yes	—
43	#10625324	—	2	8	no	—
44	C₈×D₄₇	—	2	8	no	—
45	C₂³×C₄₇	C₂²×C₄	1	1	yes	characteristic
46	C₂²×D₄₇	C₂²×C₄	1	4	yes	—
47	C₂×C₄×C₄₇	—	4	2	no	—
48	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
49	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
50	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₄₇	#41	1	2	yes	—
53	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
54	C₂²⋊D₄₇	—	2	8	no	—
55	C₂×Dic₄₇	C₂²×C₄	1	4	yes	—
56	D₁₈₈	—	2	4	no	—
57	D₁₈₈	—	4	4	no	—
58	C₄×D₄₇	—	2	4	no	—
59	C₄×D₄₇	—	2	4	no	—
60	Dic₉₄	—	2	4	no	—
61	C₈×C₄₇	—	2	4	no	—
62	C₄₇⋊C₈	—	2	2	no	—
63	C₄₇⋊C₈	C₂×D₄	1	4	yes	—
64	C₂²×C₄₇	—	2	2	no	—
65	C₂²×C₄₇	C₄×D₄	1	2	yes	—
66	C₂²×C₄₇	C₂³×C₄	1	1	yes	characteristic
67	C₂×D₄₇	—	2	4	no	—
68	C₂×D₄₇	—	2	8	no	—
69	C₄×C₄₇	—	2	2	no	—
70	C₄×C₄₇	—	2	2	no	—
71	C₄×C₄₇	C₄×D₄	1	2	yes	—
72	C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
73	C₄×C₄₇	C₂×#41	1	1	yes	characteristic
74	Dic₄₇	—	2	4	no	—
75	#2932	—	47	1	no	maximal, Sylow
76	C₂×C₄₇	—	2	2	no	—
77	C₂×C₄₇	C₂×C₄×D₄	1	1	yes	characteristic
78	C₂×C₄₇	#568	1	2	yes	—
79	D₄₇	—	4	4	no	—
80	C₂×#131	—	47	2	no	—
81	#532	—	47	1	no	—
82	#435	—	47	8	no	—
83	#423	D₄₇	1	1	yes	characteristic, complemented
84	#408	—	47	2	no	—
85	C₂×#97	—	47	1	no	—
86	C₄₇	#2932	1	1	yes	characteristic, complemented, Sylow
87	C₂×#41	—	47	2	no	—
88	#131	—	94	8	no	—
89	C₂×M₁₆	—	47	4	no	—
90	C₂×M₁₆	C₂×D₄₇	1	2	yes	—
91	C₂²×C₈	—	47	1	no	—
92	C₂²×C₈	—	47	1	no	—
93	C₂²×C₈	—	47	4	no	—
94	C₄⋊Q₈	—	47	1	no	—
95	#127	—	47	1	no	—
96	#124	—	47	2	no	—
97	#121	—	47	4	no	—
98	C₄×D₄	—	47	4	no	—
99	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
100	C₄⋊C₈	C₂×D₄₇	1	4	yes	—
101	C₂²⋊C₈	—	47	8	no	—
102	#97	—	47	4	no	—
103	#41	—	94	4	no	—
104	#41	—	94	4	no	—
105	C₂×Q₈	—	47	2	no	—
106	C₂×D₄	—	47	4	no	—
107	C₂×D₄	—	47	2	no	—
108	C₂×D₄	—	47	4	no	—
109	C₂²×C₄	—	47	4	no	—
110	C₂²×C₄	C₄×D₄₇	1	2	yes	—
111	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
112	M₁₆	—	2	4	no	—
113	M₁₆	—	94	8	no	—
114	C₂×C₈	—	47	4	no	—
115	C₂×C₈	—	47	2	no	—
116	C₂×C₈	—	47	2	no	—
117	C₂×C₈	—	94	8	no	—
118	C₂×C₈	—	94	2	no	—
119	C₂×C₈	C₂²×D₄₇	1	4	yes	—
120	C₄⋊C₄	—	47	4	no	—
121	C₂²⋊C₄	—	47	8	no	—
122	C₄²	C₄×D₄₇	1	1	yes	characteristic
123	C₄²	C₄×D₄₇	1	1	yes	characteristic
124	C₄²	C₂²×D₄₇	1	2	yes	—
125	C₂³	—	47	4	no	—
126	C₂³	C₂×C₄×D₄₇	1	1	yes	characteristic
127	Q₈	—	94	4	no	—
128	D₄	—	94	8	no	—
129	D₄	—	94	4	no	—
130	D₄	—	188	4	no	—
131	C₂×C₄	—	4	2	no	—
132	C₂×C₄	—	47	4	no	—
133	C₂×C₄	—	94	4	no	—
134	C₂×C₄	—	94	4	no	—
135	C₂×C₄	C₂×C₄×D₄₇	1	2	yes	—
136	C₂×C₄	C₂×C₄×D₄₇	1	2	yes	—
137	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
138	C₂×C₄	D₄×D₄₇	1	2	yes	—
139	C₂×C₄	#10625354	1	2	yes	—
140	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
141	C₈	—	2	4	no	—
142	C₈	—	47	4	no	—
143	C₈	—	94	2	no	—
144	C₂²	—	2	2	no	—
145	C₂²	—	94	4	no	—
146	C₂²	—	94	8	no	—
147	C₂²	#13093986	1	2	yes	—
148	C₂²	C₂²×C₄×D₄₇	1	1	yes	characteristic
149	C₄	—	2	2	no	—
150	C₄	—	2	2	no	—
151	C₄	—	94	4	no	—
152	C₄	#13093986	1	2	yes	—
153	C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
154	C₄	C₂×#10625354	1	1	yes	characteristic
155	C₂	—	2	2	no	—
156	C₂	—	188	4	no	—
157	C₂	C₂×#13093986	1	1	yes	characteristic
158	C₂	#15689889	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13094032

C₂

yes

maximal, complemented

#15689673

C₂

yes

maximal, characteristic

#15689193

C₂

yes

maximal, complemented

#15689091

C₂

yes

maximal, complemented

C₂×#13093935

C₂

yes

maximal, characteristic, complemented

C₄₇×#423

C₂

yes

maximal, characteristic, complemented

C₂×#10625350

C₂²

yes

—

#13094032

—

C₂×#10625324

C₂²

yes

—

C₂²×C₄₇⋊C₈

C₂²

yes

characteristic

C₂²×C₄₇⋊C₈

C₂²

yes

characteristic

C₂×C₈×D₄₇

C₂²

yes

—

#13094024

C₄

yes

characteristic

#13094021

C₄

yes

characteristic

#13094012

C₄

yes

—

#13093998

C₄

yes

—

C₄×D₁₈₈

C₂²

yes

—

(C₂×D₄₇)⋊C₈

C₂²

yes

—

#13093935

C₂²

yes

—

C₂×M₁₆×C₄₇

C₂²

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₄⋊C₈×C₄₇

C₂²

yes

—

C₂²×C₄×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂×C₂²⋊D₄₇

C₂×C₄

yes

—

#10625350

—

#10625350

—

C₂×D₁₈₈

C₂×C₄

yes

—

C₂×D₁₈₈

C₂×C₄

yes

—

C₂×C₄×D₄₇

C₂³

yes

—

C₂×Dic₉₄

C₂×C₄

yes

—

M₁₆×C₄₇

—

C₂×C₈×C₄₇

C₂³

yes

—

C₄²×C₄₇

C₂×C₄

yes

characteristic

C₄²×C₄₇

C₂×C₄

yes

characteristic

C₄²×C₄₇

C₂³

yes

—

(C₂×D₄₇)⋊C₄

C₂×C₄

yes

—

C₄⋊Dic₄₇

C₂×C₄

yes

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

D₄

yes

complemented

C₂×C₄₇⋊C₈

C₂³

yes

—

C₂×C₄₇⋊C₈

C₂³

yes

—

#10625324

—

C₈×D₄₇

—

C₂³×C₄₇

C₂²×C₄

yes

characteristic

C₂²×D₄₇

C₂²×C₄

yes

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²×C₄

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

#41

yes

—

C₂×C₄×C₄₇

C₂⁴

yes

characteristic

C₂²⋊D₄₇

—

C₂×Dic₄₇

C₂²×C₄

yes

—

D₁₈₈

—

D₁₈₈

—

C₄×D₄₇

—

C₄×D₄₇

—

Dic₉₄

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

C₂×D₄

yes

—

C₂²×C₄₇

—

C₂²×C₄₇

C₄×D₄

yes

—

C₂²×C₄₇

C₂³×C₄

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₄×D₄

yes

—

C₄×C₄₇

C₂²×D₄

yes

characteristic

C₄×C₄₇

C₂×#41

yes

characteristic

Dic₄₇

—

#2932

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

C₂×C₄×D₄

yes

characteristic

C₂×C₄₇

#568

yes

—

D₄₇

—

—

—

—

D₄₇

yes

characteristic, complemented

#408

—

C₂×#97

—

C₄₇

#2932

yes

characteristic, complemented, Sylow

C₂×#41

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₄₇

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₄⋊Q₈

—

#127

—

#124

—

#121

—

C₄×D₄

—

C₂×C₄²

C₂×D₄₇

yes

characteristic

100

C₄⋊C₈

C₂×D₄₇

yes

—

101

C₂²⋊C₈

—

102

#97

—

103

#41

—

104

#41

—

105

C₂×Q₈

—

106

C₂×D₄

—

107

C₂×D₄

—

108

C₂×D₄

—

109

C₂²×C₄

—

110

C₂²×C₄

C₄×D₄₇

yes

—

111

C₂²×C₄

C₂²×D₄₇

yes

characteristic

112

M₁₆

—

113

M₁₆

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

C₂²×D₄₇

yes

—

120

C₄⋊C₄

—

121

C₂²⋊C₄

—

122

C₄²

C₄×D₄₇

yes

characteristic

123

C₄²

C₄×D₄₇

yes

characteristic

124

C₄²

C₂²×D₄₇

yes

—

125

C₂³

—

126

C₂³

C₂×C₄×D₄₇

yes

characteristic

127

Q₈

—

128

D₄

—

129

D₄

—

130

D₄

—

188

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

C₂×C₄×D₄₇

yes

—

136

C₂×C₄

C₂×C₄×D₄₇

yes

—

137

C₂×C₄

C₂×C₄×D₄₇

yes

characteristic

138

C₂×C₄

D₄×D₄₇

yes

—

139

C₂×C₄

#10625354

yes

—

140

C₂×C₄

C₂³×D₄₇

yes

characteristic

141

C₈

—

142

C₈

—

143

C₈

—

144

C₂²

—

145

C₂²

—

146

C₂²

—

147

C₂²

#13093986

yes

—

148

C₂²

C₂²×C₄×D₄₇

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

#13093986

yes

—

153

C₄

C₂×D₄×D₄₇

yes

characteristic

154

C₄

C₂×#10625354

yes

characteristic

155

C₂

—

156

C₂

—

188

—

157

C₂

C₂×#13093986

yes

characteristic

158

C₂

#15689889

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₄₇
commutator, socle	C₂²×C₄₇	C₂³×C₄
Fitting	C₄₇×#423	C₂
hypercenter	#423	D₄₇
hypocenter	C₄₇	#2932
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ #423

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₄₇

commutator, socle

C₂²×C₄₇

C₂³×C₄

Fitting

C₄₇×#423

C₂

D₄₇

C₄₇

G ▷ C₂²×C₄₇ ▷ C₄₇

C₁ ◁ C₂×C₄ ◁ #423

N	H ≅ G/N	type
1. C₂×#13094032	C₂	semidirect
3. #15689193	C₂	semidirect
4. #15689091	C₂	semidirect
5. C₂×#13093935	C₂	semidirect
6. C₄₇×#423	C₂	semidirect
40. C₂×C₄₇⋊C₈	D₄	semidirect
83. #423	D₄₇	semidirect
86. C₄₇	#2932	semidirect

H ≅ G/N

type

1. C₂×#13094032

C₂

semidirect

3. #15689193

C₂

semidirect

4. #15689091

C₂

semidirect

5. C₂×#13093935

C₂

semidirect

6. C₄₇×#423

C₂

semidirect

40. C₂×C₄₇⋊C₈

D₄

semidirect

83. #423

D₄₇

semidirect

86. C₄₇

#2932

semidirect

Subgroups — #20252105

Conjugacy classes of subgroups

Complements