Group #20245986 — subgroups

Subgroups — #20245986

Order 6016.

672 conjugacy classes of subgroups; only the 229 normal subgroups are shown.

normal subgroups only

This group has 672 conjugacy classes of subgroups — too many to draw. Only its 229 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	#15689016	C₂	1	8	yes	maximal, complemented
2	#15688983	C₂	1	2	yes	maximal, complemented
3	#15688979	C₂	1	2	yes	maximal
4	C₄×C₂²⋊Dic₄₇	C₂	1	2	yes	maximal, complemented
5	C₂²×C₂²⋊C₄×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Dic₄₇	C₂²	1	4	yes	complemented
7	C₂×C₂²⋊Dic₄₇	C₂²	1	4	yes	—
8	C₂×C₂²⋊Dic₄₇	C₂²	1	4	yes	—
9	C₂×C₂²⋊Dic₄₇	C₂²	1	4	yes	—
10	C₂×C₄×Dic₄₇	C₂²	1	4	yes	complemented
11	C₄₇⋊#95	C₂²	1	4	yes	complemented
12	C₄₇⋊#95	C₂²	1	4	yes	complemented
13	C₂⁵×C₄₇	C₂²	1	1	yes	characteristic
14	C₂³×C₄×C₄₇	C₂²	1	1	yes	characteristic
15	C₂³×C₄×C₄₇	C₂²	1	1	yes	characteristic
16	C₂×C₂²⋊C₄×C₄₇	C₄	1	8	yes	complemented
17	C₂×C₂²⋊C₄×C₄₇	C₂²	1	4	yes	—
18	C₂⁴×C₄₇	—	2	4	no	—
19	C₂⁴×C₄₇	—	4	2	no	—
20	C₂⁴×C₄₇	—	4	2	no	—
21	C₂⁴×C₄₇	C₂×C₄	1	4	yes	—
22	C₂⁴×C₄₇	C₂³	1	2	yes	—
23	C₂⁴×C₄₇	C₂³	1	1	yes	characteristic
24	C₂²×C₄×C₄₇	—	4	2	no	—
25	C₂²×C₄×C₄₇	—	4	2	no	—
26	C₂²×C₄×C₄₇	C₂×C₄	1	4	yes	—
27	C₂²×C₄×C₄₇	C₂×C₄	1	4	yes	—
28	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
29	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
30	C₂²×Dic₄₇	C₂³	1	8	yes	—
31	C₂²⋊C₄×C₄₇	C₂×C₄	1	16	yes	—
32	C₂²⋊Dic₄₇	—	2	8	no	—
33	C₂²⋊Dic₄₇	—	4	4	no	—
34	C₂²⋊Dic₄₇	—	4	4	no	—
35	C₄⋊Dic₄₇	—	4	4	no	—
36	C₄×Dic₄₇	—	4	4	no	—
37	C₂³×C₄₇	—	2	2	no	—
38	C₂³×C₄₇	—	2	2	no	—
39	C₂³×C₄₇	—	2	8	no	—
40	C₂³×C₄₇	—	2	4	no	—
41	C₂³×C₄₇	—	2	4	no	—
42	C₂³×C₄₇	—	2	4	no	—
43	C₂³×C₄₇	—	4	4	no	—
44	C₂³×C₄₇	—	4	4	no	—
45	C₂³×C₄₇	—	4	16	no	—
46	C₂³×C₄₇	C₂²×C₄	1	4	yes	—
47	C₂³×C₄₇	C₂²×C₄	1	2	yes	—
48	C₂³×C₄₇	#41	1	4	yes	—
49	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
50	C₂×C₄×C₄₇	—	2	4	no	—
51	C₂×C₄×C₄₇	—	2	4	no	—
52	C₂×C₄×C₄₇	—	4	4	no	—
53	C₂×C₄×C₄₇	—	4	4	no	—
54	C₂×C₄×C₄₇	C₂²×C₄	1	4	yes	—
55	C₂×C₄×C₄₇	C₂²×C₄	1	4	yes	—
56	C₂×Dic₄₇	—	2	8	no	—
57	C₂×Dic₄₇	—	2	8	no	—
58	C₂×Dic₄₇	—	2	8	no	—
59	C₂²×C₄₇	—	2	2	no	—
60	C₂²×C₄₇	—	2	2	no	—
61	C₂²×C₄₇	—	2	8	no	—
62	C₂²×C₄₇	—	2	4	no	—
63	C₂²×C₄₇	—	2	4	no	—
64	C₂²×C₄₇	—	2	4	no	—
65	C₂²×C₄₇	—	4	16	no	—
66	C₂²×C₄₇	—	4	8	no	—
67	C₂²×C₄₇	#117	1	4	yes	—
68	C₂²×C₄₇	C₂³×C₄	1	1	yes	characteristic
69	C₂²×C₄₇	C₂×#41	1	2	yes	—
70	C₂²×C₄₇	H(5,𝔽₂)	1	2	yes	—
71	C₂²×C₄₇	H(5,𝔽₂)	1	2	yes	—
72	C₄×C₄₇	—	2	4	no	—
73	C₄×C₄₇	—	2	4	no	—
74	Dic₄₇	—	4	8	no	—
75	#2312	—	47	1	no	maximal, Sylow
76	C₂×C₄₇	—	2	4	no	—
77	C₂×C₄₇	—	4	4	no	—
78	C₂×C₄₇	C₂×#117	1	1	yes	characteristic
79	C₂×C₄₇	#518	1	1	yes	characteristic
80	C₂×C₄₇	#518	1	1	yes	characteristic
81	C₂×C₄₇	#538	1	4	yes	—
82	C₂²×C₂²⋊C₄	D₄₇	1	1	yes	characteristic
83	#388	—	47	8	no	—
84	#380	—	47	2	no	—
85	#379	—	47	2	no	—
86	C₄×C₂²⋊C₄	—	47	2	no	—
87	C₄₇	#2312	1	1	yes	characteristic, complemented, Sylow
88	C₂⁵	C₂×D₄₇	1	1	yes	characteristic
89	C₂³×C₄	C₂×D₄₇	1	1	yes	characteristic
90	C₂³×C₄	C₂×D₄₇	1	1	yes	characteristic
91	C₂×C₄⋊C₄	—	47	4	no	—
92	C₂×C₂²⋊C₄	—	47	4	no	—
93	C₂×C₂²⋊C₄	—	47	4	no	—
94	C₂×C₂²⋊C₄	—	47	4	no	—
95	C₂×C₂²⋊C₄	Dic₄₇	1	8	yes	complemented
96	C₂×C₂²⋊C₄	C₂×D₄₇	1	4	yes	—
97	C₂×C₄²	—	47	4	no	—
98	#95	—	47	4	no	—
99	#95	—	47	4	no	—
100	C₂⁴	—	2	4	no	—
101	C₂⁴	—	4	2	no	—
102	C₂⁴	—	4	2	no	—
103	C₂⁴	C₂×Dic₄₇	1	4	yes	—
104	C₂⁴	C₂²×D₄₇	1	2	yes	—
105	C₂⁴	C₂²×D₄₇	1	1	yes	characteristic
106	C₂²×C₄	—	4	2	no	—
107	C₂²×C₄	—	4	2	no	—
108	C₂²×C₄	—	47	8	no	—
109	C₂²×C₄	C₂×Dic₄₇	1	4	yes	—
110	C₂²×C₄	C₂×Dic₄₇	1	4	yes	—
111	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
112	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
113	C₄⋊C₄	—	188	4	no	—
114	C₂²⋊C₄	—	94	8	no	—
115	C₂²⋊C₄	—	188	4	no	—
116	C₂²⋊C₄	—	188	4	no	—
117	C₂²⋊C₄	C₂×Dic₄₇	1	16	yes	—
118	C₄²	—	188	4	no	—
119	C₂³	—	2	2	no	—
120	C₂³	—	2	2	no	—
121	C₂³	—	2	8	no	—
122	C₂³	—	2	4	no	—
123	C₂³	—	2	4	no	—
124	C₂³	—	2	4	no	—
125	C₂³	—	4	4	no	—
126	C₂³	—	4	4	no	—
127	C₂³	—	4	16	no	—
128	C₂³	#10625350	1	4	yes	—
129	C₂³	C₂²×Dic₄₇	1	4	yes	—
130	C₂³	C₂²×Dic₄₇	1	2	yes	—
131	C₂³	C₂³×D₄₇	1	1	yes	characteristic
132	C₂×C₄	—	2	4	no	—
133	C₂×C₄	—	2	4	no	—
134	C₂×C₄	—	4	4	no	—
135	C₂×C₄	—	4	4	no	—
136	C₂×C₄	—	94	8	no	—
137	C₂×C₄	—	94	8	no	—
138	C₂×C₄	—	94	8	no	—
139	C₂×C₄	C₂²×Dic₄₇	1	4	yes	—
140	C₂×C₄	C₂²×Dic₄₇	1	4	yes	—
141	C₂²	—	2	2	no	—
142	C₂²	—	2	2	no	—
143	C₂²	—	2	8	no	—
144	C₂²	—	2	4	no	—
145	C₂²	—	2	4	no	—
146	C₂²	—	2	4	no	—
147	C₂²	—	4	16	no	—
148	C₂²	—	4	8	no	—
149	C₂²	#13093983	1	4	yes	—
150	C₂²	C₂³×Dic₄₇	1	1	yes	characteristic
151	C₂²	C₂×#10625350	1	2	yes	—
152	C₂²	#13094069	1	2	yes	—
153	C₂²	#13094069	1	2	yes	—
154	C₄	—	2	4	no	—
155	C₄	—	2	4	no	—
156	C₄	—	188	8	no	—
157	C₂	—	2	4	no	—
158	C₂	—	4	4	no	—
159	C₂	C₂×#13093983	1	1	yes	characteristic
160	C₂	#15689598	1	1	yes	characteristic
161	C₂	#15689598	1	1	yes	characteristic
162	C₂	#15689711	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#15689016

C₂

yes

maximal, complemented

#15688983

C₂

yes

maximal, complemented

#15688979

C₂

yes

maximal

C₄×C₂²⋊Dic₄₇

C₂

yes

maximal, complemented

C₂²×C₂²⋊C₄×C₄₇

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₄₇

C₂²

yes

complemented

C₂×C₂²⋊Dic₄₇

C₂²

yes

—

C₂×C₂²⋊Dic₄₇

C₂²

yes

—

C₂×C₂²⋊Dic₄₇

C₂²

yes

—

C₂×C₄×Dic₄₇

C₂²

yes

complemented

C₄₇⋊#95

C₂²

yes

complemented

C₄₇⋊#95

C₂²

yes

complemented

C₂⁵×C₄₇

C₂²

yes

characteristic

C₂³×C₄×C₄₇

C₂²

yes

characteristic

C₂³×C₄×C₄₇

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₄₇

C₄

yes

complemented

C₂×C₂²⋊C₄×C₄₇

C₂²

yes

—

C₂⁴×C₄₇

—

C₂⁴×C₄₇

—

C₂⁴×C₄₇

—

C₂⁴×C₄₇

C₂×C₄

yes

—

C₂⁴×C₄₇

C₂³

yes

—

C₂⁴×C₄₇

C₂³

yes

characteristic

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

—

C₂²⋊C₄×C₄₇

C₂×C₄

yes

—

C₂²⋊Dic₄₇

—

C₂²⋊Dic₄₇

—

C₂²⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄×Dic₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

C₂²×C₄

yes

—

C₂³×C₄₇

C₂²×C₄

yes

—

C₂³×C₄₇

#41

yes

—

C₂³×C₄₇

C₂⁴

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²×C₄

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#117

yes

—

C₂²×C₄₇

C₂³×C₄

yes

characteristic

C₂²×C₄₇

C₂×#41

yes

—

C₂²×C₄₇

H(5,𝔽₂)

yes

—

C₂²×C₄₇

H(5,𝔽₂)

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

Dic₄₇

—

#2312

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

C₂²×C₂²⋊C₄

D₄₇

yes

—

—

—

C₄×C₂²⋊C₄

—

C₄₇

#2312

yes

characteristic, complemented, Sylow

C₂⁵

C₂×D₄₇

yes

characteristic

C₂³×C₄

C₂×D₄₇

yes

characteristic

C₂³×C₄

C₂×D₄₇

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

Dic₄₇

yes

complemented

C₂×C₂²⋊C₄

C₂×D₄₇

yes

—

C₂×C₄²

—

#95

—

#95

—

100

C₂⁴

—

101

C₂⁴

—

102

C₂⁴

—

103

C₂⁴

C₂×Dic₄₇

yes

—

104

C₂⁴

C₂²×D₄₇

yes

—

105

C₂⁴

C₂²×D₄₇

yes

characteristic

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

—

109

C₂²×C₄

C₂×Dic₄₇

yes

—

110

C₂²×C₄

C₂×Dic₄₇

yes

—

111

C₂²×C₄

C₂²×D₄₇

yes

—

112

C₂²×C₄

C₂²×D₄₇

yes

—

113

C₄⋊C₄

—

188

—

114

C₂²⋊C₄

—

115

C₂²⋊C₄

—

188

—

116

C₂²⋊C₄

—

188

—

117

C₂²⋊C₄

C₂×Dic₄₇

yes

—

118

C₄²

—

188

—

119

C₂³

—

120

C₂³

—

121

C₂³

—

122

C₂³

—

123

C₂³

—

124

C₂³

—

125

C₂³

—

126

C₂³

—

127

C₂³

—

128

C₂³

#10625350

yes

—

129

C₂³

C₂²×Dic₄₇

yes

—

130

C₂³

C₂²×Dic₄₇

yes

—

131

C₂³

C₂³×D₄₇

yes

characteristic

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

C₂²×Dic₄₇

yes

—

140

C₂×C₄

C₂²×Dic₄₇

yes

—

141

C₂²

—

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

#13093983

yes

—

150

C₂²

C₂³×Dic₄₇

yes

characteristic

151

C₂²

C₂×#10625350

yes

—

152

C₂²

#13094069

yes

—

153

C₂²

#13094069

yes

—

154

C₄

—

155

C₄

—

156

C₄

—

188

—

157

C₂

—

158

C₂

—

159

C₂

C₂×#13093983

yes

characteristic

160

C₂

#15689598

yes

characteristic

161

C₂

#15689598

yes

characteristic

162

C₂

#15689711

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂³×C₄
Fitting	C₂²×C₂²⋊C₄×C₄₇	C₂
socle	C₂³×C₄₇	C₂⁴
hypercenter	C₂²×C₂²⋊C₄	D₄₇
hypocenter	C₄₇	#2312
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₂²⋊C₄

subgroup

G/H

center, Frattini

C₂³

C₂³×D₄₇

commutator

C₂²×C₄₇

C₂³×C₄

Fitting

C₂²×C₂²⋊C₄×C₄₇

C₂

socle

C₂³×C₄₇

C₂⁴

hypercenter

C₂²×C₂²⋊C₄

D₄₇

C₄₇

G ▷ C₂²×C₄₇ ▷ C₄₇

C₁ ◁ C₂³ ◁ C₂²×C₂²⋊C₄

N	H ≅ G/N	type
1. #15689016	C₂	semidirect
2. #15688983	C₂	semidirect
4. C₄×C₂²⋊Dic₄₇	C₂	semidirect
6. C₂×C₄⋊Dic₄₇	C₂²	semidirect
10. C₂×C₄×Dic₄₇	C₂²	semidirect
11. C₄₇⋊#95	C₂²	semidirect
12. C₄₇⋊#95	C₂²	semidirect
16. C₂×C₂²⋊C₄×C₄₇	C₄	semidirect
87. C₄₇	#2312	semidirect
95. C₂×C₂²⋊C₄	Dic₄₇	semidirect

H ≅ G/N

type

1. #15689016

C₂

semidirect

2. #15688983

C₂

semidirect

4. C₄×C₂²⋊Dic₄₇

C₂

semidirect

6. C₂×C₄⋊Dic₄₇

C₂²

semidirect

10. C₂×C₄×Dic₄₇

C₂²

semidirect

11. C₄₇⋊#95

C₂²

semidirect

12. C₄₇⋊#95

C₂²

semidirect

16. C₂×C₂²⋊C₄×C₄₇

C₄

semidirect

87. C₄₇

#2312

semidirect

95. C₂×C₂²⋊C₄

Dic₄₇

semidirect

Subgroups — #20245986

Conjugacy classes of subgroups

Complements