Group #26136837 — subgroups

Subgroups — #26136837

Order 12032.

202 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 113 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×#15688957	C₂	1	2	yes	maximal, complemented
2	C₄⋊(C₄₇⋊C₃₂)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²⋊(C₄₇⋊C₃₂)	C₂	1	2	yes	maximal, complemented
4	C₄×C₄₇⋊C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
5	D₄×C₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
6	#15688957	—	2	4	no	—
7	C₂×C₄₇⋊C₃₂	C₂²	1	1	yes	characteristic
8	C₂×C₄₇⋊C₃₂	C₂²	1	2	yes	—
9	C₂×C₄₇⋊C₃₂	C₂²	1	1	yes	characteristic
10	C₂²×C₁₆×C₄₇	C₂²	1	2	yes	—
11	C₈×D₄×C₄₇	C₄	1	1	yes	characteristic
12	C₄₇×C₄⋊C₁₆	C₄	1	1	yes	characteristic
13	C₄₇×C₂²⋊C₁₆	C₄	1	2	yes	—
14	C₄×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
15	C₄₇⋊C₃₂	—	2	2	no	—
16	C₄₇⋊C₃₂	—	2	1	no	—
17	C₄₇⋊C₃₂	D₄	1	2	yes	complemented
18	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
19	C₄×D₄×C₄₇	C₈	1	1	yes	characteristic
20	C₂×C₁₆×C₄₇	—	2	2	no	—
21	C₂×C₁₆×C₄₇	—	2	2	no	—
22	C₂×C₁₆×C₄₇	C₂×C₄	1	2	yes	—
23	C₂×C₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
25	C₄⋊C₈×C₄₇	C₈	1	1	yes	characteristic
26	C₂²⋊C₈×C₄₇	C₈	1	2	yes	—
27	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
28	C₂×D₄×C₄₇	C₁₆	1	1	yes	characteristic
29	C₂²×C₄×C₄₇	C₂×C₈	1	2	yes	—
30	C₂×C₈×C₄₇	—	2	2	no	—
31	C₂×C₈×C₄₇	—	2	2	no	—
32	C₂×C₈×C₄₇	C₂×C₈	1	2	yes	—
33	C₂×C₈×C₄₇	C₂×C₈	1	1	yes	characteristic
34	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
35	C₄⋊C₄×C₄₇	C₁₆	1	1	yes	characteristic
36	C₂²⋊C₄×C₄₇	C₁₆	1	2	yes	—
37	C₄²×C₄₇	C₂×C₈	1	1	yes	characteristic
38	C₁₆×C₄₇	—	2	2	no	—
39	C₁₆×C₄₇	—	2	1	no	—
40	C₁₆×C₄₇	C₂×D₄	1	1	yes	characteristic
41	C₁₆×C₄₇	#41	1	1	yes	characteristic
42	C₂³×C₄₇	C₂×C₁₆	1	2	yes	—
43	D₄×C₄₇	—	2	2	no	—
44	C₂×C₄×C₄₇	—	2	2	no	—
45	C₂×C₄×C₄₇	—	2	2	no	—
46	C₂×C₄×C₄₇	C₂×C₁₆	1	1	yes	characteristic
47	C₂×C₄×C₄₇	C₂×C₁₆	1	2	yes	—
48	C₂×C₄×C₄₇	C₂×C₁₆	1	1	yes	characteristic
49	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
50	C₈×C₄₇	—	2	2	no	—
51	C₈×C₄₇	—	2	1	no	—
52	C₈×C₄₇	C₄×D₄	1	1	yes	characteristic
53	C₈×C₄₇	#131	1	1	yes	characteristic
54	#13646	—	47	1	no	maximal, Sylow
55	C₂²×C₄₇	—	2	2	no	—
56	C₂²×C₄₇	—	2	2	no	—
57	C₂²×C₄₇	—	2	2	no	—
58	C₂²×C₄₇	C₂²×C₁₆	1	1	yes	characteristic
59	C₄×C₄₇	—	2	2	no	—
60	C₄×C₄₇	—	2	1	no	—
61	C₄×C₄₇	M₆₄	1	2	yes	complemented
62	C₄×C₄₇	C₈×D₄	1	1	yes	characteristic
63	C₄×C₄₇	#504	1	1	yes	characteristic
64	C₂×M₆₄	—	47	2	no	—
65	D₄×C₁₆	D₄₇	1	1	yes	characteristic
66	C₄⋊C₃₂	—	47	1	no	—
67	C₂²⋊C₃₂	—	47	2	no	—
68	C₄×C₃₂	—	47	1	no	—
69	C₂×C₄₇	—	4	2	no	—
70	C₂×C₄₇	D₄×C₁₆	1	1	yes	characteristic
71	C₂×C₄₇	C₂×M₆₄	1	1	yes	characteristic
72	C₂×C₄₇	#2258	1	1	yes	characteristic
73	C₂²×C₁₆	C₂×D₄₇	1	2	yes	—
74	C₈×D₄	Dic₄₇	1	1	yes	characteristic
75	M₆₄	—	94	4	no	—
76	C₂×C₃₂	—	47	1	no	—
77	C₂×C₃₂	—	47	1	no	—
78	C₂×C₃₂	—	47	2	no	—
79	C₄⋊C₁₆	Dic₄₇	1	1	yes	characteristic
80	C₂²⋊C₁₆	Dic₄₇	1	2	yes	—
81	C₄×C₁₆	C₂×D₄₇	1	1	yes	characteristic
82	C₄₇	#13646	1	1	yes	characteristic, complemented, Sylow
83	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
84	C₄×D₄	C₄₇⋊C₈	1	1	yes	characteristic
85	C₂×C₁₆	—	2	2	no	—
86	C₂×C₁₆	—	2	2	no	—
87	C₂×C₁₆	C₂×Dic₄₇	1	1	yes	characteristic
88	C₂×C₁₆	C₂×Dic₄₇	1	2	yes	—
89	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
90	C₄⋊C₈	C₄₇⋊C₈	1	1	yes	characteristic
91	C₂²⋊C₈	C₄₇⋊C₈	1	2	yes	—
92	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
93	C₃₂	—	47	2	no	—
94	C₃₂	—	94	2	no	—
95	C₃₂	—	94	1	no	—
96	C₂×D₄	C₄₇⋊C₁₆	1	1	yes	characteristic
97	C₂²×C₄	C₂×C₄₇⋊C₈	1	2	yes	—
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	C₂×C₄₇⋊C₈	1	1	yes	characteristic
101	C₂×C₈	C₂×C₄₇⋊C₈	1	2	yes	—
102	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
103	C₄⋊C₄	C₄₇⋊C₁₆	1	1	yes	characteristic
104	C₂²⋊C₄	C₄₇⋊C₁₆	1	2	yes	—
105	C₄²	C₂×C₄₇⋊C₈	1	1	yes	characteristic
106	C₁₆	—	2	2	no	—
107	C₁₆	—	2	1	no	—
108	C₁₆	D₄×D₄₇	1	1	yes	characteristic
109	C₁₆	#10625352	1	1	yes	characteristic
110	C₂³	C₂×C₄₇⋊C₁₆	1	2	yes	—
111	D₄	—	2	2	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	C₂×C₄₇⋊C₁₆	1	1	yes	characteristic
115	C₂×C₄	C₂×C₄₇⋊C₁₆	1	2	yes	—
116	C₂×C₄	C₂×C₄₇⋊C₁₆	1	1	yes	characteristic
117	C₂×C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
118	C₈	—	2	1	no	—
119	C₈	—	2	2	no	—
120	C₈	D₄×Dic₄₇	1	1	yes	characteristic
121	C₈	#13094031	1	1	yes	characteristic
122	C₂²	—	2	2	no	—
123	C₂²	—	2	2	no	—
124	C₂²	—	2	2	no	—
125	C₂²	C₂²×C₄₇⋊C₁₆	1	1	yes	characteristic
126	C₄	—	2	1	no	—
127	C₄	—	2	2	no	—
128	C₄	#15688957	1	2	yes	complemented
129	C₄	D₄×C₄₇⋊C₈	1	1	yes	characteristic
130	C₄	#15689540	1	1	yes	characteristic
131	C₂	—	4	2	no	—
132	C₂	D₄×C₄₇⋊C₁₆	1	1	yes	characteristic
133	C₂	C₂×#15688957	1	1	yes	characteristic
134	C₂	#20245682	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15688957

C₂

yes

maximal, complemented

C₄⋊(C₄₇⋊C₃₂)

C₂

yes

maximal, characteristic, complemented

C₂²⋊(C₄₇⋊C₃₂)

C₂

yes

maximal, complemented

C₄×C₄₇⋊C₃₂

C₂

yes

maximal, characteristic, complemented

D₄×C₁₆×C₄₇

C₂

yes

maximal, characteristic

#15688957

—

C₂×C₄₇⋊C₃₂

C₂²

yes

characteristic

C₂×C₄₇⋊C₃₂

C₂²

yes

—

C₂×C₄₇⋊C₃₂

C₂²

yes

characteristic

C₂²×C₁₆×C₄₇

C₂²

yes

—

C₈×D₄×C₄₇

C₄

yes

characteristic

C₄₇×C₄⋊C₁₆

C₄

yes

characteristic

C₄₇×C₂²⋊C₁₆

C₄

yes

—

C₄×C₁₆×C₄₇

C₂²

yes

characteristic

C₄₇⋊C₃₂

—

C₄₇⋊C₃₂

—

C₄₇⋊C₃₂

D₄

yes

complemented

C₂²×C₈×C₄₇

C₂×C₄

yes

—

C₄×D₄×C₄₇

C₈

yes

characteristic

C₂×C₁₆×C₄₇

—

C₂×C₁₆×C₄₇

—

C₂×C₁₆×C₄₇

C₂×C₄

yes

—

C₂×C₁₆×C₄₇

C₂×C₄

yes

characteristic

C₂×C₁₆×C₄₇

C₂³

yes

characteristic

C₄⋊C₈×C₄₇

C₈

yes

characteristic

C₂²⋊C₈×C₄₇

C₈

yes

—

C₄×C₈×C₄₇

C₂×C₄

yes

characteristic

C₂×D₄×C₄₇

C₁₆

yes

characteristic

C₂²×C₄×C₄₇

C₂×C₈

yes

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₂×C₈

yes

—

C₂×C₈×C₄₇

C₂×C₈

yes

characteristic

C₂×C₈×C₄₇

C₂²×C₄

yes

characteristic

C₄⋊C₄×C₄₇

C₁₆

yes

characteristic

C₂²⋊C₄×C₄₇

C₁₆

yes

—

C₄²×C₄₇

C₂×C₈

yes

characteristic

C₁₆×C₄₇

—

C₁₆×C₄₇

—

C₁₆×C₄₇

C₂×D₄

yes

characteristic

C₁₆×C₄₇

#41

yes

characteristic

C₂³×C₄₇

C₂×C₁₆

yes

—

D₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×C₁₆

yes

characteristic

C₂×C₄×C₄₇

C₂×C₁₆

yes

—

C₂×C₄×C₄₇

C₂×C₁₆

yes

characteristic

C₂×C₄×C₄₇

C₂²×C₈

yes

characteristic

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

C₄×D₄

yes

C₈×C₄₇

yes

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂²×C₁₆

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

M₆₄

yes

complemented

C₄×C₄₇

C₈×D₄

yes

characteristic

C₄×C₄₇

#504

yes

characteristic

C₂×M₆₄

—

D₄×C₁₆

D₄₇

yes

characteristic

C₄⋊C₃₂

—

C₂²⋊C₃₂

—

C₄×C₃₂

—

C₂×C₄₇

—

C₂×C₄₇

D₄×C₁₆

yes

characteristic

C₂×C₄₇

C₂×M₆₄

yes

characteristic

C₂×C₄₇

#2258

yes

characteristic

C₂²×C₁₆

C₂×D₄₇

yes

—

C₈×D₄

Dic₄₇

yes

characteristic

M₆₄

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

C₄⋊C₁₆

Dic₄₇

yes

characteristic

C₂²⋊C₁₆

Dic₄₇

yes

—

C₄×C₁₆

C₂×D₄₇

yes

characteristic

C₄₇

#13646

yes

characteristic, complemented, Sylow

C₂²×C₈

C₂×Dic₄₇

yes

—

C₄×D₄

C₄₇⋊C₈

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×Dic₄₇

yes

characteristic

C₂×C₁₆

C₂×Dic₄₇

yes

—

C₂×C₁₆

C₂²×D₄₇

yes

characteristic

C₄⋊C₈

C₄₇⋊C₈

yes

characteristic

C₂²⋊C₈

C₄₇⋊C₈

yes

—

C₄×C₈

C₂×Dic₄₇

yes

characteristic

C₃₂

—

C₃₂

—

C₃₂

—

C₂×D₄

C₄₇⋊C₁₆

yes

characteristic

C₂²×C₄

C₂×C₄₇⋊C₈

yes

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

C₂×C₄₇⋊C₈

yes

characteristic

101

C₂×C₈

C₂×C₄₇⋊C₈

yes

—

102

C₂×C₈

C₂²×Dic₄₇

yes

characteristic

103

C₄⋊C₄

C₄₇⋊C₁₆

yes

characteristic

104

C₂²⋊C₄

C₄₇⋊C₁₆

yes

—

105

C₄²

C₂×C₄₇⋊C₈

yes

characteristic

106

C₁₆

—

107

C₁₆

—

108

C₁₆

D₄×D₄₇

yes

characteristic

109

C₁₆

#10625352

yes

characteristic

110

C₂³

C₂×C₄₇⋊C₁₆

yes

—

111

D₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

C₂×C₄₇⋊C₁₆

yes

characteristic

115

C₂×C₄

C₂×C₄₇⋊C₁₆

yes

—

116

C₂×C₄

C₂×C₄₇⋊C₁₆

yes

characteristic

117

C₂×C₄

C₂²×C₄₇⋊C₈

yes

characteristic

118

C₈

—

119

C₈

—

120

C₈

D₄×Dic₄₇

yes

characteristic

121

C₈

#13094031

yes

characteristic

122

C₂²

—

123

C₂²

—

124

C₂²

—

125

C₂²

C₂²×C₄₇⋊C₁₆

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

#15688957

yes

complemented

129

C₄

D₄×C₄₇⋊C₈

yes

characteristic

130

C₄

#15689540

yes

characteristic

131

C₂

—

132

C₂

D₄×C₄₇⋊C₁₆

yes

characteristic

133

C₂

C₂×#15688957

yes

characteristic

134

C₂

#20245682

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂²×D₄₇
commutator, socle	C₂²×C₄₇	C₂²×C₁₆
Fitting	D₄×C₁₆×C₄₇	C₂
prime radical	C₂×D₄×C₄₇	C₁₆
cosocle	C₂×C₁₆×C₄₇	C₂³
hypercenter	D₄×C₁₆	D₄₇
hypocenter	C₄₇	#13646
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₁₆ ◁ D₄×C₁₆

subgroup

G/H

center, Frattini

C₂×C₁₆

C₂²×D₄₇

commutator, socle

C₂²×C₄₇

C₂²×C₁₆

Fitting

D₄×C₁₆×C₄₇

C₂

prime radical

C₂×D₄×C₄₇

C₁₆

cosocle

C₂×C₁₆×C₄₇

C₂³

hypercenter

D₄×C₁₆

D₄₇

C₄₇

G ▷ C₂²×C₄₇ ▷ C₄₇

C₁ ◁ C₂×C₁₆ ◁ D₄×C₁₆

N	H ≅ G/N	type
1. C₂×#15688957	C₂	semidirect
2. C₄⋊(C₄₇⋊C₃₂)	C₂	semidirect
3. C₂²⋊(C₄₇⋊C₃₂)	C₂	semidirect
4. C₄×C₄₇⋊C₃₂	C₂	semidirect
17. C₄₇⋊C₃₂	D₄	semidirect
61. C₄×C₄₇	M₆₄	semidirect
82. C₄₇	#13646	semidirect
128. C₄	#15688957	semidirect

H ≅ G/N

type

1. C₂×#15688957

C₂

semidirect

2. C₄⋊(C₄₇⋊C₃₂)

C₂

semidirect

3. C₂²⋊(C₄₇⋊C₃₂)

C₂

semidirect

4. C₄×C₄₇⋊C₃₂

C₂

semidirect

17. C₄₇⋊C₃₂

D₄

semidirect

61. C₄×C₄₇

M₆₄

semidirect

82. C₄₇

#13646

semidirect

128. C₄

#15688957

semidirect

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

subgroup	index
C₂×#15688957	2
C₄⋊(C₄₇⋊C₃₂)	2
C₂²⋊(C₄₇⋊C₃₂)	2
C₄×C₄₇⋊C₃₂	2
D₄×C₁₆×C₄₇	2
#13646	47

subgroup

index

C₂×#15688957

C₄⋊(C₄₇⋊C₃₂)

C₂²⋊(C₄₇⋊C₃₂)

C₄×C₄₇⋊C₃₂

D₄×C₁₆×C₄₇

#13646

Subgroups — #26136837

Conjugacy classes of subgroups

Complements

Maximal subgroups