Group #26109993 — subgroups

Subgroups — #26109993

Order 12032.

220 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 91 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	#20242151	C₂	1	1	yes	maximal, characteristic
2	#20241680	C₂	1	1	yes	maximal, characteristic
3	#20241251	C₂	1	2	yes	maximal
4	#20241249	C₂	1	2	yes	maximal
5	C₄₇×#1568	C₂	1	1	yes	maximal, characteristic
6	#15689180	C₂²	1	2	yes	—
7	#15689176	C₂²	1	1	yes	characteristic
8	#15689176	C₂²	1	1	yes	characteristic
9	#15688876	—	2	1	no	—
10	#15688874	—	2	1	no	—
11	#15688845	C₄	1	4	yes	—
12	C₄₇×#432	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
14	C₄₇×#335	—	2	1	no	—
15	C₄₇×#334	—	2	1	no	—
16	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
17	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
18	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
19	C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
20	C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
21	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
22	#13093935	—	2	2	no	—
23	C₄×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
24	C₄×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₄₇	Q₈	1	2	yes	—
26	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
27	C₄⋊C₈×C₄₇	—	2	2	no	—
28	C₂²⋊C₈×C₄₇	—	2	2	no	—
29	#97×C₄₇	—	2	2	no	—
30	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
31	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
32	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
33	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
34	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
35	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
36	C₂×C₈×C₄₇	—	4	2	no	—
37	C₂×C₈×C₄₇	—	4	2	no	—
38	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
39	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
40	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
41	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
42	C₄²×C₄₇	#41	1	1	yes	characteristic
43	C₄²×C₄₇	#41	1	1	yes	characteristic
44	C₂×C₄₇⋊C₈	—	2	2	no	—
45	C₂×C₄₇⋊C₈	—	2	2	no	—
46	C₂×C₄₇⋊C₈	—	4	2	no	—
47	C₂³×C₄₇	#125	1	1	yes	characteristic
48	C₂×C₄×C₄₇	—	2	2	no	—
49	C₂×C₄×C₄₇	—	4	1	no	—
50	C₂×C₄×C₄₇	—	4	1	no	—
51	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
53	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
54	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
55	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
56	C₂×C₄×C₄₇	#125	1	1	yes	characteristic
57	C₈×C₄₇	—	2	2	no	—
58	C₈×C₄₇	—	2	2	no	—
59	C₈×C₄₇	—	8	2	no	—
60	C₄₇⋊C₈	—	4	2	no	—
61	C₄₇⋊C₈	—	4	2	no	—
62	C₄₇⋊C₈	—	8	2	no	—
63	#8799	—	47	1	no	maximal, Sylow
64	C₂²×C₄₇	—	2	1	no	—
65	C₂²×C₄₇	—	2	1	no	—
66	C₂²×C₄₇	—	2	1	no	—
67	C₂²×C₄₇	#382	1	1	yes	characteristic
68	C₄×C₄₇	—	2	1	no	—
69	C₄×C₄₇	—	2	1	no	—
70	C₄×C₄₇	—	2	1	no	—
71	C₄×C₄₇	—	2	1	no	—
72	C₄×C₄₇	—	2	1	no	—
73	C₄×C₄₇	#430	1	2	yes	—
74	#1568	—	47	1	no	—
75	#1568	D₄₇	1	1	yes	characteristic
76	#1452	—	47	1	no	—
77	#1287	—	47	2	no	—
78	#1287	—	47	2	no	—
79	C₂×C₄₇	—	4	1	no	—
80	C₂×C₄₇	#1833	1	1	yes	characteristic
81	C₂×C₄₇	#1833	1	1	yes	characteristic
82	C₂×C₄₇	#1838	1	1	yes	characteristic
83	#432	—	47	2	no	—
84	#432	C₂×D₄₇	1	2	yes	—
85	#431	—	47	1	no	—
86	#431	—	47	1	no	—
87	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
88	#335	—	2	1	no	—
89	#335	—	94	1	no	—
90	#334	—	2	1	no	—
91	#334	—	94	1	no	—
92	#322	—	47	4	no	—
93	C₄₇	#8799	1	1	yes	characteristic, complemented, Sylow
94	C₂²×C₈	Dic₉₄	1	2	yes	—
95	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
96	C₄⋊C₈	—	2	2	no	—
97	C₄⋊C₈	—	94	1	no	—
98	C₄⋊C₈	—	94	1	no	—
99	C₄⋊C₈	—	94	2	no	—
100	C₂²⋊C₈	—	2	2	no	—
101	C₂²⋊C₈	—	94	1	no	—
102	C₂²⋊C₈	—	94	1	no	—
103	C₂²⋊C₈	—	94	2	no	—
104	#97	—	2	2	no	—
105	#97	—	94	2	no	—
106	C₄×C₈	—	47	2	no	—
107	C₄×C₈	—	47	2	no	—
108	C₄×C₈	C₄×D₄₇	1	2	yes	—
109	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
110	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
111	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
112	C₂²×C₄	#10625352	1	1	yes	characteristic
113	C₂²×C₄	#10625354	1	1	yes	characteristic
114	C₂×C₈	—	4	2	no	—
115	C₂×C₈	—	4	2	no	—
116	C₂×C₈	—	94	2	no	—
117	C₂×C₈	—	94	2	no	—
118	C₂×C₈	—	188	2	no	—
119	C₂×C₈	Dic₄₇⋊C₄	1	2	yes	—
120	C₂×C₈	Dic₄₇⋊C₄	1	2	yes	—
121	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
122	C₄²	#10625352	1	1	yes	characteristic
123	C₄²	#10625354	1	1	yes	characteristic
124	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
125	C₂³	#13094015	1	1	yes	characteristic
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	—	4	1	no	—
129	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
130	C₂×C₄	#13093982	1	1	yes	characteristic
131	C₂×C₄	#13093982	1	1	yes	characteristic
132	C₂×C₄	#13094011	1	1	yes	characteristic
133	C₂×C₄	#13094013	1	1	yes	characteristic
134	C₂×C₄	#13094015	1	1	yes	characteristic
135	C₈	—	2	2	no	—
136	C₈	—	2	2	no	—
137	C₈	—	8	2	no	—
138	C₈	—	376	2	no	—
139	C₈	—	188	2	no	—
140	C₈	—	188	2	no	—
141	C₂²	—	2	1	no	—
142	C₂²	—	2	1	no	—
143	C₂²	—	2	1	no	—
144	C₂²	#15688988	1	1	yes	characteristic
145	C₄	—	2	1	no	—
146	C₄	—	2	1	no	—
147	C₄	—	2	1	no	—
148	C₄	—	2	1	no	—
149	C₄	—	2	1	no	—
150	C₄	#15689171	1	2	yes	—
151	C₂	—	4	1	no	—
152	C₂	#20243511	1	1	yes	characteristic
153	C₂	#20243513	1	1	yes	characteristic
154	C₂	#20243528	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20242151

C₂

yes

maximal, characteristic

#20241680

C₂

yes

maximal, characteristic

#20241251

C₂

yes

maximal

#20241249

C₂

yes

maximal

C₄₇×#1568

C₂

yes

maximal, characteristic

#15689180

C₂²

yes

—

#15689176

C₂²

yes

characteristic

#15689176

C₂²

yes

—

—

C₄

yes

—

C₄₇×#432

C₂²

yes

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₄₇×#335

—

C₄₇×#334

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

—

C₂²⋊(C₄₇⋊C₈)

—

C₂²⋊(C₄₇⋊C₈)

—

C₂²⋊(C₄₇⋊C₈)

—

#13093935

—

C₄×C₄₇⋊C₈

C₂×C₄

yes

—

C₄×C₄₇⋊C₈

C₂×C₄

yes

—

C₂²×C₈×C₄₇

Q₈

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄⋊C₈×C₄₇

—

C₂²⋊C₈×C₄₇

—

#97×C₄₇

—

C₄×C₈×C₄₇

C₂×C₄

yes

—

C₄×C₈×C₄₇

D₄

yes

characteristic

C₄×C₈×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×Q₈

yes

C₂²×C₄×C₄₇

yes

C₂²×C₄×C₄₇

yes

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₄⋊C₄

yes

—

C₂×C₈×C₄₇

C₄⋊C₄

yes

—

C₄²×C₄₇

C₂²×C₄

yes

characteristic

C₄²×C₄₇

C₂×D₄

yes

C₄²×C₄₇

yes

C₄²×C₄₇

yes

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

#125

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

#8799

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#382

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#430

yes

—

#1568

—

#1568

D₄₇

yes

—

—

—

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

C₂×D₄₇

yes

—

#431

—

#431

—

C₂×C₄×C₈

C₂×D₄₇

yes

—

—

—

—

—

C₄₇

#8799

yes

characteristic, complemented, Sylow

C₂²×C₈

Dic₉₄

yes

—

C₂×C₄²

C₂²×D₄₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

100

C₂²⋊C₈

—

101

C₂²⋊C₈

—

102

C₂²⋊C₈

—

103

C₂²⋊C₈

—

104

#97

—

105

#97

—

106

C₄×C₈

—

107

C₄×C₈

—

108

C₄×C₈

C₄×D₄₇

yes

—

109

C₄×C₈

C₂²⋊D₄₇

yes

characteristic

110

C₄×C₈

C₂²⋊D₄₇

yes

characteristic

111

C₂²×C₄

C₂×Dic₉₄

yes

characteristic

112

C₂²×C₄

#10625352

yes

characteristic

113

C₂²×C₄

#10625354

yes

characteristic

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

188

—

119

C₂×C₈

Dic₄₇⋊C₄

yes

—

120

C₂×C₈

Dic₄₇⋊C₄

yes

—

121

C₄²

C₂×C₄×D₄₇

yes

characteristic

122

C₄²

#10625352

yes

characteristic

123

C₄²

#10625354

yes

characteristic

124

C₄²

C₂×C₂²⋊D₄₇

yes

characteristic

125

C₂³

#13094015

yes

characteristic

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

C₂×Dic₄₇⋊C₄

yes

characteristic

130

C₂×C₄

#13093982

yes

characteristic

131

C₂×C₄

#13093982

yes

characteristic

132

C₂×C₄

#13094011

yes

characteristic

133

C₂×C₄

#13094013

yes

characteristic

134

C₂×C₄

#13094015

yes

characteristic

135

C₈

—

136

C₈

—

137

C₈

—

138

C₈

—

376

—

139

C₈

—

188

—

140

C₈

—

188

—

141

C₂²

—

142

C₂²

—

143

C₂²

—

144

C₂²

#15688988

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

#15689171

yes

—

151

C₂

—

152

C₂

#20243511

yes

characteristic

153

C₂

#20243513

yes

characteristic

154

C₂

#20243528

yes

characteristic

	subgroup	G/H
center	C₂²	#15688988
commutator	C₄²×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1568	C₂
socle	C₂²×C₄₇	#382
prime radical	C₂³×C₄₇	#125
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1568	D₄₇
hypocenter	C₄₇	#8799
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1568

subgroup

G/H

center

C₂²

#15688988

commutator

C₄²×C₄₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#1568

C₂

C₂²×C₄₇

C₂³×C₄₇

C₂×C₄²×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1568

N	H ≅ G/N	type
93. C₄₇	#8799	semidirect

H ≅ G/N

type

93. C₄₇

#8799

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20242151	2
#20241680	2
#20241251	2
#20241249	2
C₄₇×#1568	2
#8799	47

subgroup

index

C₄₇×#1568

Subgroups — #26109993

Conjugacy classes of subgroups

Complements

Maximal subgroups