Group #26125185 — subgroups

Subgroups — #26125185

Order 12032.

1246 conjugacy classes of subgroups; only the 387 normal subgroups are shown.

normal subgroups only

This group has 1246 conjugacy classes of subgroups — too many to draw. Only its 387 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	C₂×#15688878	C₂	1	6	yes	maximal
2	C₂²×C₄×Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
3	C₂³×C₄₇⋊C₈	C₂²	1	4	yes	—
4	#15688878	—	2	12	no	—
5	C₂³×Q₈×C₄₇	C₄	1	1	yes	characteristic
6	C₂×C₄×Q₈×C₄₇	—	2	12	no	—
7	C₂²×C₄⋊C₄×C₄₇	C₄	1	3	yes	—
8	C₂²×C₄²×C₄₇	C₂²	1	3	yes	—
9	C₂²×C₄₇⋊C₈	—	2	12	no	—
10	C₂²×C₄₇⋊C₈	—	2	12	no	—
11	C₂²×C₄₇⋊C₈	Q₈	1	8	yes	complemented
12	C₂²×Q₈×C₄₇	—	2	6	no	—
13	C₂²×Q₈×C₄₇	—	2	6	no	—
14	C₂²×Q₈×C₄₇	C₈	1	4	yes	complemented
15	C₂³×C₄×C₄₇	C₂×C₄	1	3	yes	—
16	C₂³×C₄×C₄₇	C₂×C₄	1	3	yes	—
17	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
18	C₄×Q₈×C₄₇	—	2	32	no	—
19	C₂×C₄⋊C₄×C₄₇	—	2	12	no	—
20	C₂×C₄⋊C₄×C₄₇	—	2	6	no	—
21	C₂×C₄²×C₄₇	—	2	12	no	—
22	C₂×C₄²×C₄₇	D₄	1	12	yes	—
23	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
24	C₂×Q₈×C₄₇	—	2	8	no	—
25	C₂×Q₈×C₄₇	—	2	24	no	—
26	C₂×Q₈×C₄₇	—	2	24	no	—
27	C₂²×C₄×C₄₇	—	2	6	no	—
28	C₂²×C₄×C₄₇	—	2	6	no	—
29	C₂²×C₄×C₄₇	—	2	6	no	—
30	C₂²×C₄×C₄₇	—	2	12	no	—
31	C₂²×C₄×C₄₇	C₂²⋊C₄	1	6	yes	—
32	C₂²×C₄×C₄₇	C₂²⋊C₄	1	6	yes	—
33	C₂²×C₄×C₄₇	C₂×C₈	1	6	yes	—
34	C₂²×C₄×C₄₇	M₁₆	1	6	yes	—
35	C₂²×C₄×C₄₇	C₂×D₄	1	6	yes	—
36	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
37	C₂²×C₄×C₄₇	#41	1	4	yes	—
38	C₄⋊C₄×C₄₇	—	2	24	no	—
39	C₄²×C₄₇	—	2	24	no	—
40	C₂×C₄₇⋊C₈	—	2	24	no	—
41	C₂×C₄₇⋊C₈	—	4	4	no	—
42	C₂×C₄₇⋊C₈	—	4	12	no	—
43	C₂³×C₄₇	C₂×C₂²⋊C₄	1	3	yes	—
44	C₂³×C₄₇	C₄×Q₈	1	4	yes	—
45	C₂³×C₄₇	C₂²×C₈	1	1	yes	characteristic
46	C₂³×C₄₇	C₂×M₁₆	1	3	yes	—
47	C₂³×C₄₇	#131	1	4	yes	—
48	Q₈×C₄₇	—	2	32	no	—
49	C₂×C₄×C₄₇	—	2	6	no	—
50	C₂×C₄×C₄₇	—	2	12	no	—
51	C₂×C₄×C₄₇	—	2	6	no	—
52	C₂×C₄×C₄₇	—	2	12	no	—
53	C₂×C₄×C₄₇	—	2	6	no	—
54	C₂×C₄×C₄₇	—	2	12	no	—
55	C₂×C₄×C₄₇	—	4	12	no	—
56	C₂×C₄×C₄₇	C₂²⋊C₈	1	12	yes	—
57	C₂×C₄×C₄₇	#120	1	4	yes	—
58	C₂×C₄×C₄₇	C₂²⋊Q₈	1	12	yes	—
59	C₂×C₄×C₄₇	#124	1	6	yes	—
60	C₂×C₄×C₄₇	C₄⋊Q₈	1	6	yes	—
61	C₄₇⋊C₈	—	4	8	no	—
62	#11380	—	47	1	no	maximal, Sylow
63	C₂²×C₄₇	#379	1	1	yes	characteristic
64	C₂²×C₄₇	#391	1	6	yes	—
65	C₂²×C₄₇	C₂×C₂²⋊C₈	1	3	yes	—
66	C₂²×C₄₇	#407	1	3	yes	—
67	C₂²×C₄₇	#408	1	6	yes	—
68	C₂²×C₄₇	C₈×Q₈	1	4	yes	—
69	C₂²×C₄₇	#446	1	12	yes	—
70	C₄×C₄₇	—	2	12	no	—
71	C₄×C₄₇	—	4	6	no	—
72	C₄×C₄₇	#393	1	4	yes	—
73	C₄×C₄₇	#395	1	4	yes	—
74	C₂²×C₄×Q₈	D₄₇	1	1	yes	characteristic
75	C₂×#336	—	47	6	no	—
76	C₂×C₄₇	#1446	1	1	yes	characteristic
77	C₂×C₄₇	#1779	1	1	yes	characteristic
78	C₂×C₄₇	#1780	1	1	yes	characteristic
79	C₂×C₄₇	#1984	1	6	yes	—
80	C₂×C₄₇	#1985	1	6	yes	—
81	C₂³×Q₈	Dic₄₇	1	1	yes	characteristic
82	C₂³×C₈	—	47	4	no	—
83	C₂×C₄×Q₈	—	2	12	no	—
84	C₂²×C₄⋊C₄	Dic₄₇	1	3	yes	—
85	C₂²×C₄²	C₂×D₄₇	1	3	yes	—
86	#336	—	94	12	no	—
87	C₄₇	#11380	1	1	yes	characteristic, complemented, Sylow
88	C₂²×Q₈	—	2	6	no	—
89	C₂²×Q₈	—	2	6	no	—
90	C₂²×Q₈	C₄₇⋊C₈	1	4	yes	complemented
91	C₂³×C₄	C₂×Dic₄₇	1	3	yes	—
92	C₂³×C₄	C₂×Dic₄₇	1	3	yes	—
93	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
94	C₂²×C₈	—	47	8	no	—
95	C₂²×C₈	—	94	12	no	—
96	C₂²×C₈	—	94	12	no	—
97	C₄×Q₈	—	2	32	no	—
98	C₂×C₄⋊C₄	—	2	12	no	—
99	C₂×C₄⋊C₄	—	2	6	no	—
100	C₂×C₄²	—	2	12	no	—
101	C₂×C₄²	C₂²⋊D₄₇	1	12	yes	—
102	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
103	C₂×Q₈	—	2	8	no	—
104	C₂×Q₈	—	2	24	no	—
105	C₂×Q₈	—	2	24	no	—
106	C₂²×C₄	—	2	6	no	—
107	C₂²×C₄	—	2	6	no	—
108	C₂²×C₄	—	2	6	no	—
109	C₂²×C₄	—	2	12	no	—
110	C₂²×C₄	C₂×C₄₇⋊C₈	1	6	yes	—
111	C₂²×C₄	#10625329	1	6	yes	—
112	C₂²×C₄	C₂²⋊Dic₄₇	1	6	yes	—
113	C₂²×C₄	C₂²⋊Dic₄₇	1	6	yes	—
114	C₂²×C₄	Q₈×D₄₇	1	4	yes	—
115	C₂²×C₄	#10625354	1	4	yes	—
116	C₂²×C₄	C₂×C₂²⋊D₄₇	1	6	yes	—
117	C₂×C₈	—	94	24	no	—
118	C₂×C₈	—	188	4	no	—
119	C₂×C₈	—	188	12	no	—
120	C₄⋊C₄	—	2	24	no	—
121	C₄²	—	2	24	no	—
122	C₂³	C₂×C₂²⋊Dic₄₇	1	3	yes	—
123	C₂³	Q₈×Dic₄₇	1	4	yes	—
124	C₂³	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
125	C₂³	C₂×#10625329	1	3	yes	—
126	C₂³	#13094031	1	4	yes	—
127	Q₈	—	2	32	no	—
128	C₂×C₄	—	2	6	no	—
129	C₂×C₄	—	2	12	no	—
130	C₂×C₄	—	2	12	no	—
131	C₂×C₄	—	2	6	no	—
132	C₂×C₄	—	2	6	no	—
133	C₂×C₄	—	2	12	no	—
134	C₂×C₄	—	4	12	no	—
135	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	12	yes	—
136	C₂×C₄	#13093993	1	4	yes	—
137	C₂×C₄	#13094000	1	12	yes	—
138	C₂×C₄	#13094013	1	6	yes	—
139	C₂×C₄	#13094022	1	6	yes	—
140	C₈	—	188	8	no	—
141	C₂²	#15688979	1	1	yes	characteristic
142	C₂²	#15689030	1	6	yes	—
143	C₂²	C₂×C₂²⋊(C₄₇⋊C₈)	1	3	yes	—
144	C₂²	#15689087	1	3	yes	—
145	C₂²	#15689090	1	6	yes	—
146	C₂²	Q₈×C₄₇⋊C₈	1	4	yes	—
147	C₂²	#15689254	1	12	yes	—
148	C₄	—	2	12	no	—
149	C₄	—	4	6	no	—
150	C₄	#15689034	1	4	yes	—
151	C₄	#15689042	1	4	yes	—
152	C₂	#20241659	1	1	yes	characteristic
153	C₂	#20243298	1	1	yes	characteristic
154	C₂	#20243300	1	1	yes	characteristic
155	C₂	#20244325	1	6	yes	—
156	C₂	#20244329	1	6	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15688878

C₂

yes

maximal

C₂²×C₄×Q₈×C₄₇

C₂

yes

maximal, characteristic

C₂³×C₄₇⋊C₈

C₂²

yes

—

#15688878

—

C₂³×Q₈×C₄₇

C₄

yes

characteristic

C₂×C₄×Q₈×C₄₇

—

C₂²×C₄⋊C₄×C₄₇

C₄

yes

—

C₂²×C₄²×C₄₇

C₂²

yes

—

C₂²×C₄₇⋊C₈

—

C₂²×C₄₇⋊C₈

—

C₂²×C₄₇⋊C₈

Q₈

yes

complemented

C₂²×Q₈×C₄₇

—

C₂²×Q₈×C₄₇

—

C₂²×Q₈×C₄₇

C₈

yes

complemented

C₂³×C₄×C₄₇

C₂×C₄

yes

—

C₂³×C₄×C₄₇

C₂×C₄

yes

—

C₂³×C₄×C₄₇

C₂³

yes

characteristic

C₄×Q₈×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄²×C₄₇

—

C₂×C₄²×C₄₇

D₄

yes

—

C₂⁴×C₄₇

C₂²×C₄

yes

characteristic

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₄₇

C₂×C₈

yes

—

C₂²×C₄×C₄₇

M₁₆

yes

—

C₂²×C₄×C₄₇

C₂×D₄

yes

—

C₂²×C₄×C₄₇

C₂×Q₈

yes

—

C₂²×C₄×C₄₇

#41

yes

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

C₂×C₂²⋊C₄

yes

—

C₂³×C₄₇

C₄×Q₈

yes

—

C₂³×C₄₇

C₂²×C₈

yes

characteristic

C₂³×C₄₇

C₂×M₁₆

yes

—

C₂³×C₄₇

#131

yes

—

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²⋊C₈

yes

—

C₂×C₄×C₄₇

#120

yes

—

C₂×C₄×C₄₇

C₂²⋊Q₈

yes

—

C₂×C₄×C₄₇

#124

yes

—

C₂×C₄×C₄₇

C₄⋊Q₈

yes

—

C₄₇⋊C₈

—

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

—

C₂²×C₄₇

C₂×C₂²⋊C₈

yes

—

C₂²×C₄₇

#407

yes

—

C₂²×C₄₇

#408

yes

—

C₂²×C₄₇

C₈×Q₈

yes

—

C₂²×C₄₇

#446

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#393

yes

—

C₄×C₄₇

#395

yes

—

C₂²×C₄×Q₈

D₄₇

yes

characteristic

C₂×#336

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

C₂×C₄₇

#1985

yes

—

C₂³×Q₈

Dic₄₇

yes

characteristic

C₂³×C₈

—

C₂×C₄×Q₈

—

C₂²×C₄⋊C₄

Dic₄₇

yes

—

C₂²×C₄²

C₂×D₄₇

yes

—

#336

—

C₄₇

#11380

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×Q₈

—

C₂²×Q₈

C₄₇⋊C₈

yes

complemented

C₂³×C₄

C₂×Dic₄₇

yes

—

C₂³×C₄

C₂×Dic₄₇

yes

—

C₂³×C₄

C₂²×D₄₇

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

100

C₂×C₄²

—

101

C₂×C₄²

C₂²⋊D₄₇

yes

—

102

C₂⁴

C₂²×Dic₄₇

yes

characteristic

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

C₂×C₄₇⋊C₈

yes

—

111

C₂²×C₄

#10625329

yes

—

112

C₂²×C₄

C₂²⋊Dic₄₇

yes

—

113

C₂²×C₄

C₂²⋊Dic₄₇

yes

—

114

C₂²×C₄

Q₈×D₄₇

yes

—

115

C₂²×C₄

#10625354

yes

—

116

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

—

117

C₂×C₈

—

118

C₂×C₈

—

188

—

119

C₂×C₈

—

188

—

120

C₄⋊C₄

—

121

C₄²

—

122

C₂³

C₂×C₂²⋊Dic₄₇

yes

—

123

C₂³

Q₈×Dic₄₇

yes

—

124

C₂³

C₂²×C₄₇⋊C₈

yes

characteristic

125

C₂³

C₂×#10625329

yes

—

126

C₂³

#13094031

yes

—

127

Q₈

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

C₂²⋊(C₄₇⋊C₈)

yes

—

136

C₂×C₄

#13093993

yes

—

137

C₂×C₄

#13094000

yes

—

138

C₂×C₄

#13094013

yes

—

139

C₂×C₄

#13094022

yes

—

140

C₈

—

188

—

141

C₂²

#15688979

yes

characteristic

142

C₂²

#15689030

yes

—

143

C₂²

C₂×C₂²⋊(C₄₇⋊C₈)

yes

—

144

C₂²

#15689087

yes

—

145

C₂²

#15689090

yes

—

146

C₂²

Q₈×C₄₇⋊C₈

yes

—

147

C₂²

#15689254

yes

—

148

C₄

—

149

C₄

—

150

C₄

#15689034

yes

—

151

C₄

#15689042

yes

—

152

C₂

#20241659

yes

characteristic

153

C₂

#20243298

yes

characteristic

154

C₂

#20243300

yes

characteristic

155

C₂

#20244325

yes

—

156

C₂

#20244329

yes

—

	subgroup	G/H
center, Frattini	C₂³×C₄	C₂²×D₄₇
commutator	C₂³×C₄₇	C₂²×C₈
Fitting	C₂²×C₄×Q₈×C₄₇	C₂
socle, prime radical	C₂⁴×C₄₇	C₂²×C₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂²×C₄×Q₈	D₄₇
hypocenter	C₄₇	#11380
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₄×Q₈

subgroup

G/H

center, Frattini

C₂³×C₄

C₂²×D₄₇

commutator

C₂³×C₄₇

C₂²×C₈

Fitting

C₂²×C₄×Q₈×C₄₇

C₂

socle, prime radical

C₂⁴×C₄₇

C₂²×C₄

cosocle

C₂³×C₄×C₄₇

C₂³

hypercenter

C₂²×C₄×Q₈

D₄₇

C₄₇

G ▷ C₂³×C₄₇ ▷ C₄₇

C₁ ◁ C₂³×C₄ ◁ C₂²×C₄×Q₈

N	H ≅ G/N	type
11. C₂²×C₄₇⋊C₈	Q₈	semidirect
14. C₂²×Q₈×C₄₇	C₈	semidirect
87. C₄₇	#11380	semidirect
90. C₂²×Q₈	C₄₇⋊C₈	semidirect

H ≅ G/N

type

11. C₂²×C₄₇⋊C₈

Q₈

semidirect

14. C₂²×Q₈×C₄₇

C₈

semidirect

87. C₄₇

#11380

semidirect

90. C₂²×Q₈

C₄₇⋊C₈

semidirect

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

subgroup	index
C₂×#15688878	2
C₂²×C₄×Q₈×C₄₇	2
#11380	47

subgroup

index

C₂×#15688878

C₂²×C₄×Q₈×C₄₇

#11380

Subgroups — #26125185

Conjugacy classes of subgroups

Complements

Maximal subgroups