Group #20249496 — subgroups

Subgroups — #20249496

Order 6016.

844 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 844 conjugacy classes of subgroups — too many to draw. Only its 165 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×#13094021	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#13094020	C₂	1	2	yes	maximal, complemented
3	C₂×#13094013	C₂	1	2	yes	maximal, complemented
4	#15689032	C₂	1	8	yes	maximal, complemented
5	#15688969	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#124×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₁₈₈	C₂²	1	2	yes	—
8	C₂²×D₁₈₈	C₂²	1	4	yes	—
9	C₂²×C₂²⋊D₄₇	C₂²	1	8	yes	—
10	#13094021	—	2	4	no	—
11	#13094020	—	2	8	no	—
12	#13094013	—	2	8	no	—
13	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	8	yes	—
14	C₂×C₄×Dic₄₇	C₂²	1	2	yes	—
15	C₄₇⋊#95	C₂²	1	4	yes	—
16	C₂²×Q₈×C₄₇	C₂²	1	1	yes	characteristic
17	C₂²×D₄×C₄₇	C₂²	1	1	yes	characteristic
18	#124×C₄₇	—	2	4	no	—
19	C₂×C₂²⋊C₄×C₄₇	C₂²	1	4	yes	—
20	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
21	C₂⁴×C₄₇	C₂³	1	2	yes	—
22	C₂³×D₄₇	C₂³	1	4	yes	—
23	C₂×Q₈×C₄₇	—	2	2	no	—
24	C₂×Q₈×C₄₇	—	2	4	no	—
25	C₂×D₄×C₄₇	—	2	2	no	—
26	C₂×D₄×C₄₇	—	2	4	no	—
27	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
28	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
29	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
30	C₂×C₂²⋊D₄₇	—	2	16	no	—
31	C₂×C₂²⋊D₄₇	—	4	16	no	—
32	C₂²×Dic₄₇	C₂³	1	4	yes	—
33	C₂×D₁₈₈	—	2	4	no	—
34	C₂×D₁₈₈	—	2	4	no	—
35	C₂×D₁₈₈	—	2	8	no	—
36	C₂×D₁₈₈	—	2	4	no	—
37	C₂×D₁₈₈	—	4	8	no	—
38	C₂²⋊C₄×C₄₇	—	2	8	no	—
39	C₄²×C₄₇	D₄	1	4	yes	complemented
40	(C₂×D₄₇)⋊C₄	—	2	16	no	—
41	C₄×Dic₄₇	D₄	1	8	yes	complemented
42	C₂³×C₄₇	—	2	2	no	—
43	C₂³×C₄₇	—	2	4	no	—
44	C₂³×C₄₇	—	2	8	no	—
45	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
46	C₂²×D₄₇	—	2	4	no	—
47	C₂²×D₄₇	—	2	4	no	—
48	C₂²×D₄₇	—	2	4	no	—
49	C₂²×D₄₇	—	2	16	no	—
50	Q₈×C₄₇	—	4	4	no	—
51	D₄×C₄₇	—	4	4	no	—
52	C₂×C₄×C₄₇	—	2	4	no	—
53	C₂×C₄×C₄₇	—	2	4	no	—
54	C₂×C₄×C₄₇	—	2	2	no	—
55	C₂×C₄×C₄₇	C₂×D₄	1	4	yes	—
56	C₂×C₄×C₄₇	C₂×D₄	1	4	yes	—
57	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
58	C₂²⋊D₄₇	—	4	32	no	—
59	C₂×Dic₄₇	—	2	8	no	—
60	C₂×Dic₄₇	C₂×D₄	1	8	yes	—
61	D₁₈₈	—	4	16	no	—
62	D₁₈₈	—	4	8	no	—
63	C₂²×C₄₇	—	4	2	no	—
64	C₂²×C₄₇	—	4	8	no	—
65	C₂²×C₄₇	—	4	4	no	—
66	C₂²×C₄₇	C₂²×D₄	1	2	yes	—
67	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
68	C₂²×C₄₇	H(5,𝔽₂)	1	4	yes	—
69	C₂×D₄₇	—	4	4	no	—
70	C₂×D₄₇	—	4	8	no	—
71	C₂×D₄₇	—	4	8	no	—
72	C₂×D₄₇	—	4	8	no	—
73	C₄×C₄₇	—	2	4	no	—
74	C₄×C₄₇	—	4	2	no	—
75	C₄×C₄₇	#127	1	4	yes	—
76	Dic₄₇	—	2	8	no	—
77	#2657	—	47	1	no	maximal, Sylow
78	C₂×C₄₇	—	8	2	no	—
79	C₂×C₄₇	C₂×#127	1	1	yes	characteristic
80	C₂×C₄₇	#535	1	2	yes	—
81	C₂×C₄₇	#535	1	4	yes	—
82	D₄₇	—	8	4	no	—
83	C₂×#127	—	47	2	no	—
84	C₂×#127	—	47	1	no	—
85	C₂×#124	—	47	2	no	—
86	C₂×#124	D₄₇	1	1	yes	characteristic, complemented
87	#392	—	47	8	no	—
88	#376	—	47	1	no	—
89	C₄₇	#2657	1	1	yes	characteristic, complemented, Sylow
90	C₂²×Q₈	C₂×D₄₇	1	1	yes	characteristic
91	C₂²×D₄	—	47	8	no	—
92	C₂²×D₄	—	47	2	no	—
93	C₂²×D₄	—	47	4	no	—
94	C₂²×D₄	C₂×D₄₇	1	1	yes	characteristic
95	#127	—	94	8	no	—
96	#127	—	94	4	no	—
97	#124	—	2	4	no	—
98	#124	—	94	8	no	—
99	C₂×C₂²⋊C₄	—	47	8	no	—
100	C₂×C₂²⋊C₄	C₂×D₄₇	1	4	yes	—
101	C₂×C₄²	—	47	2	no	—
102	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
103	#95	—	47	4	no	—
104	C₂⁴	—	47	4	no	—
105	C₂⁴	C₂²×D₄₇	1	2	yes	—
106	C₂×Q₈	—	2	4	no	—
107	C₂×Q₈	—	2	2	no	—
108	C₂×D₄	—	2	4	no	—
109	C₂×D₄	—	2	2	no	—
110	C₂×D₄	—	94	16	no	—
111	C₂×D₄	—	94	4	no	—
112	C₂×D₄	—	94	4	no	—
113	C₂×D₄	—	94	8	no	—
114	C₂×D₄	—	94	4	no	—
115	C₂×D₄	—	188	8	no	—
116	C₂×D₄	—	188	16	no	—
117	C₂²×C₄	—	47	4	no	—
118	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
119	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
120	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
121	C₂²⋊C₄	—	2	8	no	—
122	C₂²⋊C₄	—	94	16	no	—
123	C₄²	—	47	8	no	—
124	C₄²	C₂²⋊D₄₇	1	4	yes	complemented
125	C₂³	—	2	4	no	—
126	C₂³	—	2	2	no	—
127	C₂³	—	2	8	no	—
128	C₂³	—	94	4	no	—
129	C₂³	—	94	4	no	—
130	C₂³	—	94	4	no	—
131	C₂³	—	94	16	no	—
132	C₂³	C₂³×D₄₇	1	1	yes	characteristic
133	Q₈	—	4	4	no	—
134	D₄	—	4	4	no	—
135	D₄	—	188	16	no	—
136	D₄	—	188	32	no	—
137	D₄	—	188	8	no	—
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	—	2	4	no	—
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	47	8	no	—
142	C₂×C₄	—	94	8	no	—
143	C₂×C₄	D₄×D₄₇	1	4	yes	—
144	C₂×C₄	C₂×C₂²⋊D₄₇	1	4	yes	—
145	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
146	C₂²	—	4	2	no	—
147	C₂²	—	4	4	no	—
148	C₂²	—	4	8	no	—
149	C₂²	—	188	4	no	—
150	C₂²	—	188	8	no	—
151	C₂²	—	188	8	no	—
152	C₂²	—	188	8	no	—
153	C₂²	C₂×D₄×D₄₇	1	2	yes	—
154	C₂²	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
155	C₂²	#13094070	1	4	yes	—
156	C₄	—	2	4	no	—
157	C₄	—	4	2	no	—
158	C₄	—	94	8	no	—
159	C₄	#13094020	1	4	yes	—
160	C₂	—	8	2	no	—
161	C₂	—	376	4	no	—
162	C₂	C₂×#13094020	1	1	yes	characteristic
163	C₂	#15689687	1	2	yes	—
164	C₂	#15689691	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13094021

C₂

yes

maximal, characteristic, complemented

C₂×#13094020

C₂

yes

maximal, complemented

C₂×#13094013

C₂

yes

maximal, complemented

#15689032

C₂

yes

maximal, complemented

#15688969

C₂

yes

maximal, characteristic, complemented

C₂×#124×C₄₇

C₂

yes

maximal, characteristic, complemented

C₂²×D₁₈₈

C₂²

yes

—

C₂²×D₁₈₈

C₂²

yes

—

C₂²×C₂²⋊D₄₇

C₂²

yes

—

#13094021

—

#13094020

—

#13094013

—

C₂×(C₂×D₄₇)⋊C₄

C₂²

yes

—

C₂×C₄×Dic₄₇

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₂²×Q₈×C₄₇

C₂²

yes

characteristic

C₂²×D₄×C₄₇

C₂²

yes

characteristic

#124×C₄₇

—

C₂×C₂²⋊C₄×C₄₇

C₂²

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₂⁴×C₄₇

C₂³

yes

—

C₂³×D₄₇

C₂³

yes

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂×C₂²⋊D₄₇

—

C₂×C₂²⋊D₄₇

—

C₂²×Dic₄₇

C₂³

yes

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

D₄

yes

complemented

(C₂×D₄₇)⋊C₄

—

C₄×Dic₄₇

D₄

yes

complemented

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

C₂⁴

yes

characteristic

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

Q₈×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂²⋊D₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

C₂×D₄

yes

—

D₁₈₈

—

D₁₈₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂²×D₄

yes

—

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₂²×C₄₇

H(5,𝔽₂)

yes

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#127

yes

—

Dic₄₇

—

#2657

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

C₂×#127

yes

characteristic

C₂×C₄₇

#535

yes

—

C₂×C₄₇

#535

yes

—

D₄₇

—

—

—

—

D₄₇

yes

characteristic, complemented

#392

—

#376

—

C₄₇

#2657

yes

characteristic, complemented, Sylow

C₂²×Q₈

C₂×D₄₇

yes

characteristic

C₂²×D₄

—

C₂²×D₄

—

C₂²×D₄

—

C₂²×D₄

C₂×D₄₇

yes

—

—

—

—

C₂×C₂²⋊C₄

—

100

C₂×C₂²⋊C₄

C₂×D₄₇

yes

—

101

C₂×C₄²

—

102

C₂×C₄²

C₂×D₄₇

yes

characteristic

103

#95

—

104

C₂⁴

—

105

C₂⁴

C₂²×D₄₇

yes

—

106

C₂×Q₈

—

107

C₂×Q₈

—

108

C₂×D₄

—

109

C₂×D₄

—

110

C₂×D₄

—

111

C₂×D₄

—

112

C₂×D₄

—

113

C₂×D₄

—

114

C₂×D₄

—

115

C₂×D₄

—

188

—

116

C₂×D₄

—

188

—

117

C₂²×C₄

—

118

C₂²×C₄

C₂²×D₄₇

yes

—

119

C₂²×C₄

C₂²×D₄₇

yes

—

120

C₂²×C₄

C₂²×D₄₇

yes

characteristic

121

C₂²⋊C₄

—

122

C₂²⋊C₄

—

123

C₄²

—

124

C₄²

C₂²⋊D₄₇

yes

complemented

125

C₂³

—

126

C₂³

—

127

C₂³

—

128

C₂³

—

129

C₂³

—

130

C₂³

—

131

C₂³

—

132

C₂³

C₂³×D₄₇

yes

characteristic

133

Q₈

—

134

D₄

—

135

D₄

—

188

—

136

D₄

—

188

—

137

D₄

—

188

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

D₄×D₄₇

yes

—

144

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

145

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

—

188

—

150

C₂²

—

188

—

151

C₂²

—

188

—

152

C₂²

—

188

—

153

C₂²

C₂×D₄×D₄₇

yes

—

154

C₂²

C₂²×C₂²⋊D₄₇

yes

characteristic

155

C₂²

#13094070

yes

—

156

C₄

—

157

C₄

—

158

C₄

—

159

C₄

#13094020

yes

—

160

C₂

—

161

C₂

—

376

—

162

C₂

C₂×#13094020

yes

characteristic

163

C₂

#15689687

yes

—

164

C₂

#15689691

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂⁴
Fitting	C₂×#124×C₄₇	C₂
hypercenter	C₂×#124	D₄₇
hypocenter	C₄₇	#2657
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×#124

subgroup

G/H

center, Frattini

C₂³

C₂³×D₄₇

commutator, socle

C₂³×C₄₇

C₂⁴

Fitting

C₂×#124×C₄₇

C₂

D₄₇

C₄₇

G ▷ C₂³×C₄₇ ▷ C₄₇

C₁ ◁ C₂³ ◁ C₂×#124

N	H ≅ G/N	type
1. C₂×#13094021	C₂	semidirect
2. C₂×#13094020	C₂	semidirect
3. C₂×#13094013	C₂	semidirect
4. #15689032	C₂	semidirect
5. #15688969	C₂	semidirect
6. C₂×#124×C₄₇	C₂	semidirect
39. C₄²×C₄₇	D₄	semidirect
41. C₄×Dic₄₇	D₄	semidirect
86. C₂×#124	D₄₇	semidirect
89. C₄₇	#2657	semidirect
124. C₄²	C₂²⋊D₄₇	semidirect

H ≅ G/N

type

1. C₂×#13094021

C₂

semidirect

2. C₂×#13094020

C₂

semidirect

3. C₂×#13094013

C₂

semidirect

4. #15689032

C₂

semidirect

5. #15688969

C₂

semidirect

6. C₂×#124×C₄₇

C₂

semidirect

39. C₄²×C₄₇

D₄

semidirect

41. C₄×Dic₄₇

D₄

semidirect

86. C₂×#124

D₄₇

semidirect

89. C₄₇

#2657

semidirect

124. C₄²

C₂²⋊D₄₇

semidirect

Subgroups — #20249496

Conjugacy classes of subgroups

Complements