Group #26106705 — subgroups

Subgroups — #26106705

Order 12032.

478 conjugacy classes of subgroups; only the 235 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 235 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	C₄×#13093935	C₂	1	1	yes	maximal, characteristic
2	C₄²×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₄₇⋊#1273	C₂	1	4	yes	maximal
4	C₄₇×#1744	C₂	1	1	yes	maximal, characteristic
5	C₂×#13093935	C₄	1	2	yes	—
6	C₂×#13093935	C₄	1	2	yes	—
7	C₂×C₄×C₄₇⋊C₈	C₄	1	4	yes	—
8	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
9	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
11	C₄³×C₄₇	C₂²	1	1	yes	characteristic
12	C₄₇×#336	C₄	1	4	yes	—
13	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
14	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
15	#13093935	—	2	4	no	—
16	#13093935	C₈	1	8	yes	complemented
17	C₄×C₄₇⋊C₈	—	2	4	no	—
18	C₄×C₄₇⋊C₈	—	2	4	no	—
19	C₄×C₄₇⋊C₈	—	2	8	no	—
20	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
21	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
22	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
26	C₄×C₈×C₄₇	—	2	4	no	—
27	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
28	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
29	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₈×C₄₇	—	2	4	no	—
33	C₂×C₈×C₄₇	—	2	8	no	—
34	C₄²×C₄₇	—	2	4	no	—
35	C₄²×C₄₇	—	2	4	no	—
36	C₄²×C₄₇	C₂×C₈	1	4	yes	—
37	C₄²×C₄₇	M₁₆	1	4	yes	—
38	C₄²×C₄₇	#41	1	4	yes	—
39	C₂×C₄₇⋊C₈	—	2	4	no	—
40	C₂×C₄₇⋊C₈	—	2	4	no	—
41	C₂×C₄₇⋊C₈	—	2	4	no	—
42	C₂×C₄₇⋊C₈	—	2	4	no	—
43	C₂×C₄₇⋊C₈	C₂×C₈	1	8	yes	—
44	C₂×C₄₇⋊C₈	M₁₆	1	8	yes	—
45	C₂³×C₄₇	C₂×C₄²	1	1	yes	characteristic
46	C₂×C₄×C₄₇	—	2	2	no	—
47	C₂×C₄×C₄₇	—	2	2	no	—
48	C₂×C₄×C₄₇	—	2	2	no	—
49	C₂×C₄×C₄₇	—	2	2	no	—
50	C₂×C₄×C₄₇	C₄×C₈	1	4	yes	—
51	C₂×C₄×C₄₇	#97	1	4	yes	—
52	C₂×C₄×C₄₇	#117	1	2	yes	—
53	C₂×C₄×C₄₇	#117	1	2	yes	—
54	C₂×C₄×C₄₇	#117	1	2	yes	—
55	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
56	C₂×C₄×C₄₇	C₂×M₁₆	1	4	yes	—
57	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
58	C₂×C₄×C₄₇	#131	1	4	yes	—
59	C₈×C₄₇	—	4	4	no	—
60	C₄₇⋊C₈	—	2	8	no	—
61	C₄₇⋊C₈	—	2	8	no	—
62	#8132	—	47	1	no	maximal, Sylow
63	C₂²×C₄₇	#376	1	1	yes	characteristic
64	C₂²×C₄₇	C₂×C₄×C₈	1	1	yes	characteristic
65	C₂²×C₄₇	C₂×#97	1	1	yes	characteristic
66	C₂²×C₄₇	C₄×M₁₆	1	2	yes	—
67	C₂²×C₄₇	#405	1	2	yes	—
68	C₄×C₄₇	—	2	4	no	—
69	C₄×C₄₇	—	2	4	no	—
70	C₄×C₄₇	#431	1	4	yes	—
71	C₄×C₄₇	#432	1	4	yes	—
72	C₄×C₄₇	#433	1	4	yes	—
73	#1744	D₄₇	1	1	yes	characteristic
74	C₄×#97	—	47	1	no	—
75	C₄²×C₈	—	47	1	no	—
76	#1273	—	47	4	no	—
77	C₂×C₄₇	C₈×M₁₆	1	2	yes	—
78	C₂×C₄₇	#1451	1	2	yes	—
79	C₂×C₄₇	#1744	1	1	yes	characteristic
80	C₂×C₄₇	#1745	1	1	yes	characteristic
81	C₂×C₄₇	#1746	1	1	yes	characteristic
82	C₂×#97	—	47	2	no	—
83	C₂×#97	—	47	2	no	—
84	C₂×C₄×C₈	—	47	4	no	—
85	C₂×C₄×C₈	—	47	2	no	—
86	C₂×C₄×C₈	—	47	2	no	—
87	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
88	C₄³	C₂×D₄₇	1	1	yes	characteristic
89	#336	Dic₄₇	1	4	yes	—
90	C₄₇	#8132	1	1	yes	characteristic, complemented, Sylow
91	C₂²×C₈	—	47	4	no	—
92	C₂²×C₈	—	47	4	no	—
93	C₂²×C₈	C₂×Dic₄₇	1	4	yes	—
94	C₂×C₄²	C₄×D₄₇	1	2	yes	—
95	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
96	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
97	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
98	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
99	#97	—	47	8	no	—
100	#97	—	94	4	no	—
101	C₄×C₈	—	2	4	no	—
102	C₄×C₈	—	94	8	no	—
103	C₄×C₈	—	94	4	no	—
104	C₄×C₈	—	94	4	no	—
105	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
106	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
107	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
108	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
109	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
110	C₂×C₈	—	2	4	no	—
111	C₂×C₈	—	2	8	no	—
112	C₂×C₈	—	47	8	no	—
113	C₂×C₈	—	47	8	no	—
114	C₂×C₈	—	94	4	no	—
115	C₂×C₈	—	94	4	no	—
116	C₂×C₈	—	94	4	no	—
117	C₂×C₈	—	94	4	no	—
118	C₄²	—	2	4	no	—
119	C₄²	—	2	4	no	—
120	C₄²	C₈×D₄₇	1	4	yes	—
121	C₄²	#10625324	1	4	yes	—
122	C₄²	#10625350	1	4	yes	—
123	C₂³	C₂×C₄×Dic₄₇	1	1	yes	characteristic
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	C₈×Dic₄₇	1	4	yes	—
129	C₂×C₄	#13093934	1	4	yes	—
130	C₂×C₄	#13093981	1	2	yes	—
131	C₂×C₄	#13093981	1	2	yes	—
132	C₂×C₄	#13093983	1	2	yes	—
133	C₂×C₄	C₂×C₈×D₄₇	1	2	yes	—
134	C₂×C₄	M₁₆×D₄₇	1	4	yes	—
135	C₂×C₄	C₂×#10625324	1	2	yes	—
136	C₂×C₄	#13094031	1	4	yes	—
137	C₈	—	4	4	no	—
138	C₈	—	94	8	no	—
139	C₈	—	94	8	no	—
140	C₂²	#15688969	1	1	yes	characteristic
141	C₂²	C₂×C₈×Dic₄₇	1	1	yes	characteristic
142	C₂²	C₂×#13093934	1	1	yes	characteristic
143	C₂²	M₁₆×Dic₄₇	1	2	yes	—
144	C₂²	#15689079	1	2	yes	—
145	C₄	—	2	4	no	—
146	C₄	—	2	4	no	—
147	C₄	#15689174	1	4	yes	—
148	C₄	#15689178	1	4	yes	—
149	C₄	#15689184	1	4	yes	—
150	C₂	#20241669	1	2	yes	—
151	C₂	#20241678	1	2	yes	—
152	C₂	#20243133	1	1	yes	characteristic
153	C₂	#20243136	1	1	yes	characteristic
154	C₂	#20243140	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#13093935

C₂

yes

maximal, characteristic

C₄²×C₄₇⋊C₈

C₂

yes

maximal, characteristic

C₄₇⋊#1273

C₂

yes

maximal

C₄₇×#1744

C₂

yes

maximal, characteristic

C₂×#13093935

C₄

yes

—

C₂×#13093935

C₄

yes

—

C₂×C₄×C₄₇⋊C₈

C₄

yes

—

C₂×C₄×C₄₇⋊C₈

C₂²

yes

—

C₂×C₄×C₄₇⋊C₈

C₂²

yes

—

C₂×C₄×C₈×C₄₇

C₂²

yes

—

C₄³×C₄₇

C₂²

yes

characteristic

C₄₇×#336

C₄

yes

—

C₂²×C₄₇⋊C₈

C₂×C₄

yes

—

C₂²×C₄₇⋊C₈

C₂×C₄

yes

—

#13093935

—

#13093935

C₈

yes

complemented

C₄×C₄₇⋊C₈

—

C₄×C₄₇⋊C₈

—

C₄×C₄₇⋊C₈

—

C₂²×C₈×C₄₇

C₂×C₄

yes

—

C₂×C₄²×C₄₇

C₂×C₄

yes

—

C₂×C₄²×C₄₇

C₂×C₄

yes

—

C₂×C₄²×C₄₇

C₂×C₄

yes

characteristic

C₂×C₄²×C₄₇

C₂×C₄

yes

characteristic

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄×C₈×C₄₇

—

C₂²×C₄×C₄₇

C₄²

yes

—

C₂²×C₄×C₄₇

C₄²

yes

—

C₂²×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

C₂×C₈

yes

—

C₄²×C₄₇

M₁₆

yes

—

C₄²×C₄₇

#41

yes

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

C₂×C₈

yes

—

C₂×C₄₇⋊C₈

M₁₆

yes

—

C₂³×C₄₇

C₂×C₄²

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₄×C₈

yes

—

C₂×C₄×C₄₇

#97

yes

—

C₂×C₄×C₄₇

#117

yes

—

C₂×C₄×C₄₇

#117

yes

—

C₂×C₄×C₄₇

#117

yes

—

C₂×C₄×C₄₇

C₂²×C₈

yes

—

C₂×C₄×C₄₇

C₂×M₁₆

yes

—

C₂×C₄×C₄₇

C₂×M₁₆

yes

—

C₂×C₄×C₄₇

#131

yes

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

—

C₂²×C₄₇

yes

C₂²×C₄₇

C₂×C₄×C₈

yes

characteristic

C₂²×C₄₇

C₂×#97

yes

characteristic

C₂²×C₄₇

C₄×M₁₆

yes

—

C₂²×C₄₇

#405

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#431

yes

—

C₄×C₄₇

#432

yes

—

C₄×C₄₇

#433

yes

—

#1744

D₄₇

yes

characteristic

C₄×#97

—

C₄²×C₈

—

#1273

—

C₂×C₄₇

C₈×M₁₆

yes

—

C₂×C₄₇

#1451

yes

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×#97

—

C₂×#97

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₄₇

yes

—

C₄³

C₂×D₄₇

yes

characteristic

#336

Dic₄₇

yes

—

C₄₇

#8132

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×Dic₄₇

yes

—

C₂×C₄²

C₄×D₄₇

yes

—

C₂×C₄²

C₄×D₄₇

yes

characteristic

C₂×C₄²

C₄×D₄₇

yes

characteristic

C₂×C₄²

C₂×Dic₄₇

yes

—

C₂×C₄²

C₂²×D₄₇

yes

characteristic

#97

—

100

#97

—

101

C₄×C₈

—

102

C₄×C₈

—

103

C₄×C₈

—

104

C₄×C₈

—

105

C₂²×C₄

C₄×Dic₄₇

yes

—

106

C₂²×C₄

C₄×Dic₄₇

yes

—

107

C₂²×C₄

C₂×C₄×D₄₇

yes

characteristic

108

C₂²×C₄

C₂×C₄×D₄₇

yes

characteristic

109

C₂²×C₄

C₂²×Dic₄₇

yes

characteristic

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₄²

—

119

C₄²

—

120

C₄²

C₈×D₄₇

yes

—

121

C₄²

#10625324

yes

—

122

C₄²

#10625350

yes

—

123

C₂³

C₂×C₄×Dic₄₇

yes

characteristic

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

C₈×Dic₄₇

yes

—

129

C₂×C₄

#13093934

yes

—

130

C₂×C₄

#13093981

yes

—

131

C₂×C₄

#13093981

yes

—

132

C₂×C₄

#13093983

yes

—

133

C₂×C₄

C₂×C₈×D₄₇

yes

—

134

C₂×C₄

M₁₆×D₄₇

yes

—

135

C₂×C₄

C₂×#10625324

yes

—

136

C₂×C₄

#13094031

yes

—

137

C₈

—

138

C₈

—

139

C₈

—

140

C₂²

#15688969

yes

characteristic

141

C₂²

C₂×C₈×Dic₄₇

yes

characteristic

142

C₂²

C₂×#13093934

yes

characteristic

143

C₂²

M₁₆×Dic₄₇

yes

—

144

C₂²

#15689079

yes

—

145

C₄

—

146

C₄

—

147

C₄

#15689174

yes

—

148

C₄

#15689178

yes

—

149

C₄

#15689184

yes

—

150

C₂

#20241669

yes

—

151

C₂

#20241678

yes

—

152

C₂

#20243133

yes

characteristic

153

C₂

#20243136

yes

characteristic

154

C₂

#20243140

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₄₇
commutator	C₂²×C₄₇	C₂×C₄×C₈
Fitting	C₄₇×#1744	C₂
socle, prime radical	C₂³×C₄₇	C₂×C₄²
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1744	D₄₇
hypocenter	C₄₇	#8132
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄² ◁ #1744

subgroup

G/H

center, Frattini

C₂×C₄²

C₂²×D₄₇

commutator

C₂²×C₄₇

C₂×C₄×C₈

Fitting

C₄₇×#1744

C₂

socle, prime radical

C₂³×C₄₇

C₂×C₄²

cosocle

C₂×C₄²×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₂²×C₄₇ ▷ C₄₇

C₁ ◁ C₂×C₄² ◁ #1744

N	H ≅ G/N	type
16. #13093935	C₈	semidirect
90. C₄₇	#8132	semidirect

H ≅ G/N

type

16. #13093935

C₈

semidirect

90. C₄₇

#8132

semidirect

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

subgroup	index
C₄×#13093935	2
C₄²×C₄₇⋊C₈	2
C₄₇⋊#1273	2
C₄₇×#1744	2
#8132	47

subgroup

index

C₄×#13093935

C₄²×C₄₇⋊C₈

C₄₇⋊#1273

C₄₇×#1744

#8132

Subgroups — #26106705

Conjugacy classes of subgroups

Complements

Maximal subgroups