Group #20242546 — subgroups

Subgroups — #20242546

Order 6016.

188 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 61 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	#15689475	C₂	1	1	yes	maximal, characteristic
2	#15689418	C₂	1	1	yes	maximal, characteristic
3	#15689418	C₂	1	1	yes	maximal, characteristic
4	#15688869	C₂	1	1	yes	maximal, characteristic
5	Q₈⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
6	Q₈⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
7	C₄₇×#557	C₂	1	1	yes	maximal, characteristic
8	#13094024	C₂²	1	1	yes	characteristic
9	#13093957	—	2	1	no	—
10	#13093957	—	2	1	no	—
11	C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
12	C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
13	#13093948	—	2	1	no	—
14	#13093948	—	2	1	no	—
15	#13093947	—	2	2	no	—
16	C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
17	C₄⋊Q₈×C₄₇	—	2	1	no	—
18	C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
19	#125×C₄₇	—	2	2	no	—
20	#125×C₄₇	—	2	1	no	—
21	C₄×Q₈×C₄₇	—	2	1	no	—
22	C₄×Q₈×C₄₇	—	2	1	no	—
23	C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
24	C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
25	C₂×Q₈×C₄₇	—	2	1	no	—
26	C₂×Q₈×C₄₇	D₄	1	1	yes	characteristic
27	C₂×Q₈×C₄₇	D₄	1	1	yes	characteristic
28	C₂×Dic₉₄	—	2	1	no	—
29	C₄⋊C₄×C₄₇	—	2	2	no	—
30	C₄⋊C₄×C₄₇	—	2	2	no	—
31	C₄⋊C₄×C₄₇	—	2	2	no	—
32	C₄⋊C₄×C₄₇	—	2	1	no	—
33	C₄⋊C₄×C₄₇	—	2	1	no	—
34	C₄⋊C₄×C₄₇	—	2	1	no	—
35	C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
36	C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
37	C₄⋊C₄×C₄₇	D₄	1	2	yes	—
38	C₄²×C₄₇	—	2	2	no	—
39	C₄²×C₄₇	—	2	1	no	—
40	C₄²×C₄₇	—	2	1	no	—
41	C₄²×C₄₇	C₂³	1	1	yes	characteristic
42	C₄⋊Dic₄₇	—	2	1	no	—
43	C₄⋊Dic₄₇	—	2	1	no	—
44	C₂×C₄₇⋊C₈	—	2	1	no	—
45	C₂×C₄₇⋊C₈	—	2	1	no	—
46	C₂×C₄₇⋊C₈	—	2	1	no	—
47	Q₈×C₄₇	—	2	1	no	—
48	Q₈×C₄₇	—	2	1	no	—
49	Q₈×C₄₇	—	2	1	no	—
50	Q₈×C₄₇	—	4	1	no	—
51	Q₈×C₄₇	—	4	1	no	—
52	Q₈×C₄₇	Q₁₆	1	2	yes	—
53	C₂×C₄×C₄₇	—	2	1	no	—
54	C₂×C₄×C₄₇	—	2	1	no	—
55	C₂×C₄×C₄₇	—	2	1	no	—
56	C₂×C₄×C₄₇	—	2	1	no	—
57	C₂×C₄×C₄₇	—	2	2	no	—
58	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
61	C₂×Dic₄₇	—	4	1	no	—
62	Dic₉₄	—	4	2	no	—
63	C₄₇⋊C₈	—	4	1	no	—
64	C₄₇⋊C₈	—	4	1	no	—
65	C₄₇⋊C₈	—	4	1	no	—
66	C₂²×C₄₇	#120	1	1	yes	characteristic
67	C₄×C₄₇	—	2	2	no	—
68	C₄×C₄₇	—	2	1	no	—
69	C₄×C₄₇	—	4	1	no	—
70	C₄×C₄₇	—	4	2	no	—
71	C₄×C₄₇	—	4	1	no	—
72	C₄×C₄₇	—	4	1	no	—
73	C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
74	C₄×C₄₇	#135	1	1	yes	characteristic
75	C₄×C₄₇	#137	1	1	yes	characteristic
76	C₄×C₄₇	#137	1	1	yes	characteristic
77	Dic₄₇	—	8	1	no	—
78	#1636	—	47	1	no	maximal, Sylow
79	C₂×C₄₇	#451	1	1	yes	characteristic
80	C₂×C₄₇	#452	1	1	yes	characteristic
81	C₂×C₄₇	#454	1	1	yes	characteristic
82	#557	D₄₇	1	1	yes	characteristic
83	#487	—	47	1	no	—
84	#477	—	47	1	no	—
85	#477	—	47	1	no	—
86	#332	—	47	1	no	—
87	Q₈⋊C₈	—	47	1	no	—
88	Q₈⋊C₈	—	47	1	no	—
89	C₄₇	#1636	1	1	yes	characteristic, complemented, Sylow
90	C₄⋊Q₈	—	2	1	no	—
91	C₄⋊Q₈	—	47	1	no	—
92	C₄⋊Q₈	C₂×D₄₇	1	1	yes	characteristic
93	#125	—	2	2	no	—
94	#125	—	2	1	no	—
95	C₄×Q₈	—	2	1	no	—
96	C₄×Q₈	—	2	1	no	—
97	C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
98	C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
99	#107	—	94	1	no	—
100	#107	—	94	1	no	—
101	C₄⋊C₈	—	47	1	no	—
102	C₄⋊C₈	—	47	1	no	—
103	#103	—	94	1	no	—
104	#103	—	94	1	no	—
105	#103	—	94	2	no	—
106	C₄×C₈	—	47	1	no	—
107	C₂×Q₈	—	2	1	no	—
108	C₂×Q₈	—	94	1	no	—
109	C₂×Q₈	C₂²⋊D₄₇	1	1	yes	characteristic
110	C₂×Q₈	C₂²⋊D₄₇	1	1	yes	characteristic
111	C₂×C₈	—	94	1	no	—
112	C₂×C₈	—	94	1	no	—
113	C₂×C₈	—	94	1	no	—
114	C₄⋊C₄	—	2	2	no	—
115	C₄⋊C₄	—	2	2	no	—
116	C₄⋊C₄	—	2	2	no	—
117	C₄⋊C₄	—	2	1	no	—
118	C₄⋊C₄	—	2	1	no	—
119	C₄⋊C₄	—	2	1	no	—
120	C₄⋊C₄	—	94	1	no	—
121	C₄⋊C₄	—	94	1	no	—
122	C₄⋊C₄	C₂²⋊D₄₇	1	2	yes	—
123	C₄⋊C₄	C₂²⋊D₄₇	1	1	yes	characteristic
124	C₄⋊C₄	C₂²⋊D₄₇	1	1	yes	characteristic
125	C₄²	—	2	2	no	—
126	C₄²	—	2	1	no	—
127	C₄²	—	2	1	no	—
128	C₄²	C₂²×D₄₇	1	1	yes	characteristic
129	Q₈	—	2	1	no	—
130	Q₈	—	2	1	no	—
131	Q₈	—	2	1	no	—
132	Q₈	—	4	1	no	—
133	Q₈	—	4	1	no	—
134	Q₈	—	188	2	no	—
135	Q₈	#10625337	1	2	yes	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	—	2	1	no	—
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	188	1	no	—
142	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
143	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
144	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
145	C₈	—	188	1	no	—
146	C₈	—	188	1	no	—
147	C₈	—	188	1	no	—
148	C₂²	#13093993	1	1	yes	characteristic
149	C₄	—	2	2	no	—
150	C₄	—	2	1	no	—
151	C₄	—	4	1	no	—
152	C₄	—	4	2	no	—
153	C₄	—	4	1	no	—
154	C₄	—	4	1	no	—
155	C₄	—	376	1	no	—
156	C₄	C₂×#10625337	1	1	yes	characteristic
157	C₄	#13094044	1	1	yes	characteristic
158	C₄	#13094054	1	1	yes	characteristic
159	C₄	#13094055	1	1	yes	characteristic
160	C₂	#15689279	1	1	yes	characteristic
161	C₂	#15689285	1	1	yes	characteristic
162	C₂	#15689296	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689475

C₂

yes

maximal, characteristic

#15689418

C₂

yes

maximal, characteristic

#15689418

C₂

yes

maximal, characteristic

#15688869

C₂

yes

maximal, characteristic

Q₈⋊(C₄₇⋊C₈)

C₂

yes

maximal, characteristic

Q₈⋊(C₄₇⋊C₈)

C₂

yes

maximal, characteristic

C₄₇×#557

C₂

yes

maximal, characteristic

#13094024

C₂²

yes

characteristic

#13093957

—

#13093957

—

C₄⋊(C₄₇⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₄₇⋊C₈)

C₂²

yes

—

—

—

C₄×C₄₇⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₄₇

—

C₄⋊Q₈×C₄₇

C₂²

yes

characteristic

#125×C₄₇

—

#125×C₄₇

—

C₄×Q₈×C₄₇

—

C₄×Q₈×C₄₇

—

C₄×Q₈×C₄₇

C₂²

yes

characteristic

C₄×Q₈×C₄₇

C₂²

yes

characteristic

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

D₄

yes

characteristic

C₂×Q₈×C₄₇

D₄

yes

characteristic

C₂×Dic₉₄

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

D₄

yes

characteristic

C₄⋊C₄×C₄₇

D₄

yes

characteristic

C₄⋊C₄×C₄₇

D₄

yes

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

C₂³

yes

characteristic

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

Q₈×C₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

Q₁₆

yes

—

C₂×C₄×C₄₇

—

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×D₄

yes

characteristic

C₂×Dic₄₇

—

Dic₉₄

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₂²×C₄₇

#120

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂×Q₁₆

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

Dic₄₇

—

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

D₄₇

yes

—

—

—

—

Q₈⋊C₈

—

Q₈⋊C₈

—

C₄₇

#1636

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₄₇

yes

characteristic

#125

—

#125

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

C₂×D₄₇

yes

characteristic

C₄×Q₈

C₂×D₄₇

yes

characteristic

#107

—

100

#107

—

101

C₄⋊C₈

—

102

C₄⋊C₈

—

103

#103

—

104

#103

—

105

#103

—

106

C₄×C₈

—

107

C₂×Q₈

—

108

C₂×Q₈

—

109

C₂×Q₈

C₂²⋊D₄₇

yes

characteristic

110

C₂×Q₈

C₂²⋊D₄₇

yes

characteristic

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

C₂²⋊D₄₇

yes

—

123

C₄⋊C₄

C₂²⋊D₄₇

yes

characteristic

124

C₄⋊C₄

C₂²⋊D₄₇

yes

characteristic

125

C₄²

—

126

C₄²

—

127

C₄²

—

128

C₄²

C₂²×D₄₇

yes

characteristic

129

Q₈

—

130

Q₈

—

131

Q₈

—

132

Q₈

—

133

Q₈

—

134

Q₈

—

188

—

135

Q₈

#10625337

yes

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

188

—

142

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

143

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

144

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

145

C₈

—

188

—

146

C₈

—

188

—

147

C₈

—

188

—

148

C₂²

#13093993

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

—

376

—

156

C₄

C₂×#10625337

yes

characteristic

157

C₄

#13094044

yes

characteristic

158

C₄

#13094054

yes

characteristic

159

C₄

#13094055

yes

characteristic

160

C₂

#15689279

yes

characteristic

161

C₂

#15689285

yes

characteristic

162

C₂

#15689296

yes

characteristic

	subgroup	G/H
center	C₂²	#13093993
commutator	C₄²×C₄₇	C₂³
Frattini	C₄²	C₂²×D₄₇
Fitting	C₄₇×#557	C₂
socle	C₂²×C₄₇	#120
hypercenter	#557	D₄₇
hypocenter	C₄₇	#1636
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #557

subgroup

G/H

center

C₂²

#13093993

commutator

C₄²×C₄₇

C₂³

Frattini

C₄²

C₂²×D₄₇

Fitting

C₄₇×#557

C₂

C₂²×C₄₇

D₄₇

C₄₇

G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #557

N	H ≅ G/N	type
89. C₄₇	#1636	semidirect

H ≅ G/N

type

89. C₄₇

#1636

semidirect

Subgroups — #20242546

Conjugacy classes of subgroups

Complements