Group #26106728 — subgroups

Subgroups — #26106728

Order 12032.

330 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 131 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	#20243075	C₂	1	1	yes	maximal, characteristic
2	#20241681	C₂	1	2	yes	maximal
3	#20241218	C₂	1	2	yes	maximal
4	#20241217	C₂	1	1	yes	maximal, characteristic
5	C₄₇×#1274	C₂	1	1	yes	maximal, characteristic
6	#15689176	C₂²	1	2	yes	—
7	C₂×#13093935	C₄	1	2	yes	—
8	C₂×#13093935	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₄₇⋊C₈	C₄	1	2	yes	—
10	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
11	#15688845	C₄	1	4	yes	complemented
12	#15688844	C₄	1	4	yes	complemented
13	C₄₇×#431	C₂²	1	2	yes	—
14	C₄₇×#376	C₂²	1	1	yes	characteristic
15	C₂²×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
18	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
19	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
20	#13093935	—	2	4	no	—
21	#13093935	—	2	1	no	—
22	#13093935	—	2	1	no	—
23	C₄×C₄₇⋊C₈	—	2	4	no	—
24	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
25	C₄×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
26	C₄×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
29	C₄⋊C₈×C₄₇	—	2	2	no	—
30	C₂²⋊C₈×C₄₇	—	2	2	no	—
31	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
32	#95×C₄₇	—	2	2	no	—
33	C₂²×C₄×C₄₇	—	2	2	no	—
34	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₄₇	#41	1	2	yes	—
36	C₂×C₈×C₄₇	—	2	4	no	—
37	C₄²×C₄₇	—	2	2	no	—
38	C₄²×C₄₇	M₁₆	1	4	yes	—
39	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
40	C₄²×C₄₇	#41	1	1	yes	characteristic
41	C₄²×C₄₇	#41	1	1	yes	characteristic
42	C₂×C₄₇⋊C₈	—	2	2	no	—
43	C₂×C₄₇⋊C₈	—	2	2	no	—
44	C₂×C₄₇⋊C₈	—	2	2	no	—
45	C₂×C₄₇⋊C₈	—	2	2	no	—
46	C₂×C₄₇⋊C₈	—	2	4	no	—
47	C₂×C₄₇⋊C₈	C₄²	1	1	yes	characteristic
48	C₂×C₄₇⋊C₈	C₄²	1	2	yes	—
49	C₂×C₄₇⋊C₈	C₄²	1	1	yes	characteristic
50	C₂×C₄₇⋊C₈	M₁₆	1	4	yes	—
51	C₂³×C₄₇	#117	1	1	yes	characteristic
52	C₂×C₄×C₄₇	—	2	4	no	—
53	C₂×C₄×C₄₇	—	4	2	no	—
54	C₂×C₄×C₄₇	—	4	4	no	—
55	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
56	C₂×C₄×C₄₇	#117	1	2	yes	—
57	C₂×C₄×C₄₇	#117	1	2	yes	—
58	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
59	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
60	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
61	C₈×C₄₇	—	4	4	no	—
62	C₄₇⋊C₈	—	2	1	no	—
63	C₄₇⋊C₈	—	2	1	no	—
64	C₄₇⋊C₈	—	2	2	no	—
65	C₄₇⋊C₈	—	2	4	no	—
66	C₄₇⋊C₈	—	4	4	no	—
67	#8140	—	47	1	no	maximal, Sylow
68	C₂²×C₄₇	—	2	2	no	—
69	C₂²×C₄₇	#376	1	1	yes	characteristic
70	C₂²×C₄₇	#432	1	2	yes	—
71	C₄×C₄₇	—	2	2	no	—
72	C₄×C₄₇	—	2	2	no	—
73	C₄×C₄₇	—	4	4	no	—
74	C₄×C₄₇	C₄×M₁₆	1	1	yes	characteristic
75	C₄×C₄₇	C₄×M₁₆	1	1	yes	characteristic
76	C₄×C₄₇	#432	1	2	yes	—
77	#1726	—	47	1	no	—
78	#1452	—	47	2	no	—
79	#1275	—	47	2	no	—
80	#1274	—	47	1	no	—
81	#1274	D₄₇	1	1	yes	characteristic
82	C₂×C₄₇	—	2	2	no	—
83	C₂×C₄₇	#1745	1	1	yes	characteristic
84	C₂×C₄₇	#1747	1	2	yes	—
85	#431	—	47	2	no	—
86	#431	C₂×D₄₇	1	2	yes	—
87	C₂×#97	—	47	2	no	—
88	C₂×#97	—	47	1	no	—
89	C₂×C₄×C₈	—	47	1	no	—
90	C₂×C₄×C₈	—	47	2	no	—
91	#376	C₂×D₄₇	1	1	yes	characteristic
92	#322	—	47	4	no	—
93	#322	—	47	4	no	—
94	C₄₇	#8140	1	1	yes	characteristic, complemented, Sylow
95	C₂²×C₈	—	47	1	no	—
96	C₂²×C₈	—	47	1	no	—
97	C₂²×C₈	—	47	2	no	—
98	C₂×C₄²	C₄×D₄₇	1	2	yes	—
99	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
100	C₄⋊C₈	—	2	2	no	—
101	C₄⋊C₈	—	94	2	no	—
102	C₂²⋊C₈	—	2	2	no	—
103	C₂²⋊C₈	—	94	2	no	—
104	#97	—	94	4	no	—
105	#97	—	94	1	no	—
106	#97	—	94	1	no	—
107	C₄×C₈	—	47	2	no	—
108	C₄×C₈	—	47	2	no	—
109	C₄×C₈	—	47	4	no	—
110	C₄×C₈	—	94	4	no	—
111	C₄×C₈	C₄×D₄₇	1	4	yes	—
112	#95	—	2	2	no	—
113	C₂²×C₄	—	2	2	no	—
114	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
115	C₂²×C₄	#10625352	1	2	yes	—
116	C₂×C₈	—	2	4	no	—
117	C₂×C₈	—	47	1	no	—
118	C₂×C₈	—	47	1	no	—
119	C₂×C₈	—	47	2	no	—
120	C₂×C₈	—	47	4	no	—
121	C₂×C₈	—	94	2	no	—
122	C₂×C₈	—	94	2	no	—
123	C₂×C₈	—	94	2	no	—
124	C₂×C₈	—	94	2	no	—
125	C₂×C₈	—	94	4	no	—
126	C₄²	—	2	2	no	—
127	C₄²	#10625324	1	4	yes	—
128	C₄²	C₂×C₄×D₄₇	1	2	yes	—
129	C₄²	#10625352	1	1	yes	characteristic
130	C₄²	#10625354	1	1	yes	characteristic
131	C₂³	#13093980	1	1	yes	characteristic
132	C₂×C₄	—	2	4	no	—
133	C₂×C₄	—	4	2	no	—
134	C₂×C₄	—	4	4	no	—
135	C₂×C₄	C₄²×D₄₇	1	1	yes	characteristic
136	C₂×C₄	#13093980	1	2	yes	—
137	C₂×C₄	#13093982	1	2	yes	—
138	C₂×C₄	#13093982	1	1	yes	characteristic
139	C₂×C₄	M₁₆×D₄₇	1	2	yes	—
140	C₂×C₄	C₂×#10625324	1	2	yes	—
141	C₈	—	4	4	no	—
142	C₈	—	94	1	no	—
143	C₈	—	94	1	no	—
144	C₈	—	94	2	no	—
145	C₈	—	94	4	no	—
146	C₈	—	188	4	no	—
147	C₂²	—	2	2	no	—
148	C₂²	#15688968	1	1	yes	characteristic
149	C₂²	#15689177	1	2	yes	—
150	C₄	—	2	2	no	—
151	C₄	—	2	2	no	—
152	C₄	—	4	4	no	—
153	C₄	C₄×#10625324	1	1	yes	characteristic
154	C₄	#15689075	1	1	yes	characteristic
155	C₄	#15689179	1	2	yes	—
156	C₂	—	2	2	no	—
157	C₂	#20243135	1	1	yes	characteristic
158	C₂	#20243142	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20243075

C₂

yes

maximal, characteristic

#20241681

C₂

yes

maximal

#20241218

C₂

yes

maximal

#20241217

C₂

yes

maximal, characteristic

C₄₇×#1274

C₂

yes

maximal, characteristic

#15689176

C₂²

yes

—

C₂×#13093935

C₄

yes

—

C₂×#13093935

C₂²

yes

characteristic

C₂×C₄×C₄₇⋊C₈

C₄

yes

—

C₂×C₄×C₄₇⋊C₈

C₂²

yes

characteristic

#15688845

C₄

yes

complemented

#15688844

C₄

yes

complemented

C₄₇×#431

C₂²

yes

—

C₄₇×#376

C₂²

yes

characteristic

C₂²×C₄₇⋊C₈

C₂×C₄

yes

characteristic

C₂²×C₄₇⋊C₈

C₂×C₄

yes

characteristic

C₂²×C₄₇⋊C₈

C₂×C₄

yes

—

C₄⋊(C₄₇⋊C₈)

—

C₂²⋊(C₄₇⋊C₈)

—

#13093935

—

#13093935

—

#13093935

—

C₄×C₄₇⋊C₈

—

C₄×C₄₇⋊C₈

C₂×C₄

yes

—

C₄×C₄₇⋊C₈

C₂×C₄

yes

—

C₄×C₄₇⋊C₈

C₂×C₄

yes

—

C₂×C₄²×C₄₇

C₂×C₄

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄⋊C₈×C₄₇

—

C₂²⋊C₈×C₄₇

—

C₄×C₈×C₄₇

C₂×C₄

yes

—

#95×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₄₇

#41

yes

—

C₂×C₈×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

M₁₆

yes

—

C₄²×C₄₇

C₂²×C₄

yes

—

C₄²×C₄₇

yes

C₄²×C₄₇

yes

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

C₄²

yes

characteristic

C₂×C₄₇⋊C₈

C₄²

yes

—

C₂×C₄₇⋊C₈

C₄²

yes

characteristic

C₂×C₄₇⋊C₈

M₁₆

yes

—

C₂³×C₄₇

#117

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×C₄²

yes

characteristic

C₂×C₄×C₄₇

#117

yes

—

C₂×C₄×C₄₇

#117

yes

—

C₂×C₄×C₄₇

#117

yes

characteristic

C₂×C₄×C₄₇

C₂×M₁₆

yes

—

C₂×C₄×C₄₇

C₂×M₁₆

yes

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

#8140

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

#376

yes

characteristic

C₂²×C₄₇

#432

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₄×M₁₆

yes

characteristic

C₄×C₄₇

C₄×M₁₆

yes

characteristic

C₄×C₄₇

#432

yes

—

—

—

—

—

D₄₇

yes

characteristic

C₂×C₄₇

—

C₂×C₄₇

#1745

yes

characteristic

C₂×C₄₇

#1747

yes

—

#431

—

#431

C₂×D₄₇

yes

—

C₂×#97

—

C₂×#97

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

#376

C₂×D₄₇

yes

characteristic

#322

—

#322

—

C₄₇

#8140

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

C₄×D₄₇

yes

—

C₂×C₄²

C₂²×D₄₇

yes

characteristic

100

C₄⋊C₈

—

101

C₄⋊C₈

—

102

C₂²⋊C₈

—

103

C₂²⋊C₈

—

104

#97

—

105

#97

—

106

#97

—

107

C₄×C₈

—

108

C₄×C₈

—

109

C₄×C₈

—

110

C₄×C₈

—

111

C₄×C₈

C₄×D₄₇

yes

—

112

#95

—

113

C₂²×C₄

—

114

C₂²×C₄

C₂×C₄×D₄₇

yes

characteristic

115

C₂²×C₄

#10625352

yes

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₄²

—

127

C₄²

#10625324

yes

—

128

C₄²

C₂×C₄×D₄₇

yes

—

129

C₄²

#10625352

yes

characteristic

130

C₄²

#10625354

yes

characteristic

131

C₂³

#13093980

yes

characteristic

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

C₄²×D₄₇

yes

characteristic

136

C₂×C₄

#13093980

yes

—

137

C₂×C₄

#13093982

yes

—

138

C₂×C₄

#13093982

yes

characteristic

139

C₂×C₄

M₁₆×D₄₇

yes

—

140

C₂×C₄

C₂×#10625324

yes

—

141

C₈

—

142

C₈

—

143

C₈

—

144

C₈

—

145

C₈

—

146

C₈

—

188

—

147

C₂²

—

148

C₂²

#15688968

yes

characteristic

149

C₂²

#15689177

yes

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

C₄×#10625324

yes

characteristic

154

C₄

#15689075

yes

characteristic

155

C₄

#15689179

yes

—

156

C₂

—

157

C₂

#20243135

yes

characteristic

158

C₂

#20243142

yes

—

	subgroup	G/H
center	C₂×C₄	C₄²×D₄₇
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1274	C₂
socle	C₂²×C₄₇	#376
prime radical	C₂³×C₄₇	#117
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1274	D₄₇
hypocenter	C₄₇	#8140
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ #1274

subgroup

G/H

center

C₂×C₄

C₄²×D₄₇

commutator

C₂×C₄×C₄₇

C₂×C₄²

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#1274

C₂

C₂²×C₄₇

C₂³×C₄₇

C₂×C₄²×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₄₇

C₁ ◁ C₂×C₄ ◁ #1274

N	H ≅ G/N	type
11. #15688845	C₄	semidirect
12. #15688844	C₄	semidirect
94. C₄₇	#8140	semidirect

H ≅ G/N

type

11. #15688845

C₄

semidirect

12. #15688844

C₄

semidirect

94. C₄₇

#8140

semidirect

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

subgroup	index
#20243075	2
#20241681	2
#20241218	2
#20241217	2
C₄₇×#1274	2
#8140	47

subgroup

index

C₄₇×#1274

Subgroups — #26106728

Conjugacy classes of subgroups

Complements

Maximal subgroups