Group #20244901 — subgroups

Subgroups — #20244901

Order 6016.

176 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 69 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	#15689166	C₂	1	1	yes	maximal, characteristic
2	C₂×#13093955	C₂	1	1	yes	maximal, characteristic
3	#15689150	C₂	1	1	yes	maximal, characteristic
4	#15688892	C₂	1	1	yes	maximal, characteristic
5	#15688891	C₂	1	1	yes	maximal, characteristic
6	#15688887	C₂	1	1	yes	maximal, characteristic
7	C₄₇×#428	C₂	1	1	yes	maximal, characteristic
8	C₂×#10625329	C₂²	1	1	yes	characteristic
9	C₂×#10625329	C₂²	1	1	yes	characteristic
10	C₂²×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
12	#13093957	—	2	1	no	—
13	#13093955	—	2	1	no	—
14	#13093955	—	2	2	no	—
15	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
16	C₂×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
17	#117×C₄₇	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
19	#107×C₄₇	—	2	1	no	—
20	#106×C₄₇	—	2	1	no	—
21	C₂²×C₄×C₄₇	—	2	1	no	—
22	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
23	C₂²×Dic₄₇	—	2	1	no	—
24	M₁₆×C₄₇	—	2	1	no	—
25	M₁₆×C₄₇	Q₈	1	2	yes	—
26	C₂×C₈×C₄₇	—	2	1	no	—
27	C₄⋊C₄×C₄₇	—	2	1	no	—
28	C₄⋊C₄×C₄₇	—	2	1	no	—
29	C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
30	C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
31	C₂²⋊C₄×C₄₇	—	2	1	no	—
32	C₄²×C₄₇	—	2	1	no	—
33	C₄⋊Dic₄₇	—	2	1	no	—
34	C₄⋊Dic₄₇	—	2	1	no	—
35	#10625329	—	2	1	no	—
36	#10625329	—	2	1	no	—
37	#10625329	—	2	1	no	—
38	#10625329	Q₈	1	2	yes	—
39	C₂×C₄₇⋊C₈	—	2	1	no	—
40	C₂×C₄₇⋊C₈	—	2	1	no	—
41	C₂×C₄₇⋊C₈	—	2	1	no	—
42	C₂×C₄₇⋊C₈	—	2	1	no	—
43	C₂×C₄₇⋊C₈	D₄	1	2	yes	—
44	C₂³×C₄₇	#41	1	1	yes	characteristic
45	C₂×C₄×C₄₇	—	2	1	no	—
46	C₂×C₄×C₄₇	—	2	1	no	—
47	C₂×C₄×C₄₇	—	4	1	no	—
48	C₂×C₄×C₄₇	—	8	1	no	—
49	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
54	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
55	C₂×Dic₄₇	—	4	1	no	—
56	C₂×Dic₄₇	—	8	1	no	—
57	C₈×C₄₇	—	2	2	no	—
58	C₄₇⋊C₈	—	2	2	no	—
59	C₄₇⋊C₈	—	2	2	no	—
60	C₄₇⋊C₈	—	4	1	no	—
61	C₂²×C₄₇	—	2	1	no	—
62	C₂²×C₄₇	—	2	1	no	—
63	C₂²×C₄₇	#121	1	1	yes	characteristic
64	C₂²×C₄₇	#125	1	1	yes	characteristic
65	C₂²×C₄₇	#125	1	1	yes	characteristic
66	C₄×C₄₇	—	4	1	no	—
67	C₄×C₄₇	—	4	1	no	—
68	C₄×C₄₇	—	8	1	no	—
69	C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
70	C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
71	C₄×C₄₇	#123	1	1	yes	characteristic
72	C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
73	Dic₄₇	—	8	1	no	—
74	#2090	—	47	1	no	maximal, Sylow
75	C₂×C₄₇	—	2	1	no	—
76	C₂×C₄₇	—	2	1	no	—
77	C₂×C₄₇	#398	1	1	yes	characteristic
78	C₂×C₄₇	#471	1	1	yes	characteristic
79	C₂×C₄₇	#471	1	1	yes	characteristic
80	#428	—	47	1	no	—
81	#428	D₄₇	1	1	yes	characteristic
82	C₂×#106	—	47	1	no	—
83	#424	—	47	1	no	—
84	#343	—	47	1	no	—
85	#343	—	47	1	no	—
86	#341	—	47	1	no	—
87	C₄₇	#2090	1	1	yes	characteristic, complemented, Sylow
88	C₂×M₁₆	—	47	1	no	—
89	C₂×M₁₆	—	47	1	no	—
90	C₂×M₁₆	C₂×D₄₇	1	1	yes	characteristic
91	C₂²×C₈	—	47	1	no	—
92	#117	C₂×D₄₇	1	1	yes	characteristic
93	C₂×C₄⋊C₄	—	47	1	no	—
94	C₂×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
95	#107	—	2	1	no	—
96	#107	—	94	1	no	—
97	#106	—	2	1	no	—
98	#106	—	94	2	no	—
99	#106	—	94	1	no	—
100	C₄⋊C₈	—	94	2	no	—
101	C₂²×C₄	—	2	1	no	—
102	C₂²×C₄	—	94	1	no	—
103	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
104	M₁₆	—	2	1	no	—
105	M₁₆	—	47	2	no	—
106	M₁₆	—	94	1	no	—
107	M₁₆	—	94	1	no	—
108	M₁₆	—	94	1	no	—
109	M₁₆	Dic₉₄	1	2	yes	—
110	C₂×C₈	—	2	1	no	—
111	C₂×C₈	—	47	2	no	—
112	C₂×C₈	—	94	1	no	—
113	C₂×C₈	—	94	1	no	—
114	C₂×C₈	—	94	1	no	—
115	C₂×C₈	—	94	1	no	—
116	C₄⋊C₄	—	2	1	no	—
117	C₄⋊C₄	—	2	1	no	—
118	C₄⋊C₄	—	94	1	no	—
119	C₄⋊C₄	—	94	1	no	—
120	C₄⋊C₄	C₂²⋊D₄₇	1	1	yes	characteristic
121	C₄⋊C₄	C₂²⋊D₄₇	1	1	yes	characteristic
122	C₂²⋊C₄	—	2	1	no	—
123	C₄²	—	2	1	no	—
124	C₂³	#10625352	1	1	yes	characteristic
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	—	8	1	no	—
129	C₂×C₄	—	376	1	no	—
130	C₂×C₄	—	188	1	no	—
131	C₂×C₄	C₂×Dic₉₄	1	1	yes	characteristic
132	C₂×C₄	#10625350	1	1	yes	characteristic
133	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
134	C₂×C₄	Q₈×D₄₇	1	1	yes	characteristic
135	C₂×C₄	#10625354	1	1	yes	characteristic
136	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
137	C₈	—	2	2	no	—
138	C₈	—	94	2	no	—
139	C₈	—	94	2	no	—
140	C₈	—	188	1	no	—
141	C₂²	—	2	1	no	—
142	C₂²	—	2	1	no	—
143	C₂²	#13093996	1	1	yes	characteristic
144	C₂²	#13094014	1	1	yes	characteristic
145	C₂²	#13094015	1	1	yes	characteristic
146	C₄	—	4	1	no	—
147	C₄	—	4	1	no	—
148	C₄	—	8	1	no	—
149	C₄	—	376	1	no	—
150	C₄	#13094000	1	1	yes	characteristic
151	C₄	#13094003	1	1	yes	characteristic
152	C₄	#13094009	1	1	yes	characteristic
153	C₄	#13094023	1	1	yes	characteristic
154	C₂	—	2	1	no	—
155	C₂	—	2	1	no	—
156	C₂	#15689056	1	1	yes	characteristic
157	C₂	#15689384	1	1	yes	characteristic
158	C₂	#15689388	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689166

C₂

yes

maximal, characteristic

C₂×#13093955

C₂

yes

maximal, characteristic

#15689150

C₂

yes

maximal, characteristic

#15688892

C₂

yes

maximal, characteristic

#15688891

C₂

yes

maximal, characteristic

#15688887

C₂

yes

maximal, characteristic

C₄₇×#428

C₂

yes

maximal, characteristic

C₂×#10625329

C₂²

yes

characteristic

C₂×#10625329

C₂²

yes

characteristic

C₂²×C₄₇⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₄₇

C₂²

yes

—

—

—

C₄⋊(C₄₇⋊C₈)

—

C₂×M₁₆×C₄₇

C₂²

yes

characteristic

#117×C₄₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₄₇

C₂²

yes

characteristic

#107×C₄₇

—

#106×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

—

M₁₆×C₄₇

—

M₁₆×C₄₇

Q₈

yes

—

C₂×C₈×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

D₄

yes

characteristic

C₄⋊C₄×C₄₇

D₄

yes

characteristic

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

—

—

—

Q₈

yes

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

D₄

yes

—

C₂³×C₄₇

#41

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₄₇

C₂×Q₈

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂²⋊Q₈

yes

characteristic

C₄×C₄₇

C₂²⋊Q₈

yes

characteristic

C₄×C₄₇

#123

yes

characteristic

C₄×C₄₇

C₄⋊Q₈

yes

characteristic

Dic₄₇

—

#2090

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

D₄₇

yes

—

—

—

—

—

C₄₇

#2090

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₄₇

yes

characteristic

C₂²×C₈

—

#117

C₂×D₄₇

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₄₇

yes

—

—

—

—

—

100

C₄⋊C₈

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

C₂²×D₄₇

yes

characteristic

104

M₁₆

—

105

M₁₆

—

106

M₁₆

—

107

M₁₆

—

108

M₁₆

—

109

M₁₆

Dic₉₄

yes

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

C₂²⋊D₄₇

yes

characteristic

121

C₄⋊C₄

C₂²⋊D₄₇

yes

characteristic

122

C₂²⋊C₄

—

123

C₄²

—

124

C₂³

#10625352

yes

characteristic

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

376

—

130

C₂×C₄

—

188

—

131

C₂×C₄

C₂×Dic₉₄

yes

characteristic

132

C₂×C₄

#10625350

yes

characteristic

133

C₂×C₄

D₄×D₄₇

yes

characteristic

134

C₂×C₄

Q₈×D₄₇

yes

characteristic

135

C₂×C₄

#10625354

yes

characteristic

136

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

137

C₈

—

138

C₈

—

139

C₈

—

140

C₈

—

188

—

141

C₂²

—

142

C₂²

—

143

C₂²

#13093996

yes

characteristic

144

C₂²

#13094014

yes

characteristic

145

C₂²

#13094015

yes

characteristic

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

376

—

150

C₄

#13094000

yes

characteristic

151

C₄

#13094003

yes

characteristic

152

C₄

#13094009

yes

characteristic

153

C₄

#13094023

yes

characteristic

154

C₂

—

155

C₂

—

156

C₂

#15689056

yes

characteristic

157

C₂

#15689384

yes

characteristic

158

C₂

#15689388

yes

characteristic

	subgroup	G/H
center	C₂²	#13093996
commutator	C₂²×C₄×C₄₇	C₂³
Frattini	C₂²×C₄	C₂²×D₄₇
Fitting	C₄₇×#428	C₂
socle	C₂²×C₄₇	#121
hypercenter	#428	D₄₇
hypocenter	C₄₇	#2090
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #428

subgroup

G/H

center

C₂²

#13093996

commutator

C₂²×C₄×C₄₇

C₂³

Frattini

C₂²×C₄

C₂²×D₄₇

Fitting

C₄₇×#428

C₂

C₂²×C₄₇

D₄₇

C₄₇

G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #428

N	H ≅ G/N	type
87. C₄₇	#2090	semidirect

H ≅ G/N

type

87. C₄₇

#2090

semidirect

Subgroups — #20244901

Conjugacy classes of subgroups

Complements