Group #26131714 — subgroups

Subgroups — #26131714

Order 12032.

958 conjugacy classes of subgroups; only the 219 normal subgroups are shown.

normal subgroups only

This group has 958 conjugacy classes of subgroups — too many to draw. Only its 219 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×C₂²⋊(C₄₇⋊C₁₆)	C₂	1	6	yes	maximal, complemented
2	C₂²×#131×C₄₇	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄₇⋊C₁₆	C₂²	1	4	yes	—
4	C₂²⋊(C₄₇⋊C₁₆)	—	2	12	no	—
5	C₂²×#41×C₄₇	C₄	1	1	yes	characteristic
6	C₂×#131×C₄₇	—	2	12	no	—
7	C₂²×M₁₆×C₄₇	C₄	1	3	yes	—
8	C₂³×C₈×C₄₇	C₂²	1	3	yes	—
9	C₂×C₄₇⋊C₁₆	—	2	12	no	—
10	C₂×#41×C₄₇	—	2	6	no	—
11	C₂×#41×C₄₇	—	2	6	no	—
12	C₂²×Q₈×C₄₇	C₈	1	1	yes	characteristic
13	C₂²×D₄×C₄₇	C₈	1	3	yes	—
14	C₂³×C₄×C₄₇	C₂×C₄	1	3	yes	—
15	#131×C₄₇	—	2	32	no	—
16	C₂×M₁₆×C₄₇	—	2	6	no	—
17	C₂×M₁₆×C₄₇	—	2	12	no	—
18	C₂²×C₈×C₄₇	—	2	12	no	—
19	C₂²×C₈×C₄₇	C₂×C₄	1	3	yes	—
20	C₂²×C₈×C₄₇	D₄	1	12	yes	—
21	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
22	C₂⁴×C₄₇	C₂×C₈	1	3	yes	—
23	#41×C₄₇	—	2	8	no	—
24	#41×C₄₇	—	2	24	no	—
25	C₂×Q₈×C₄₇	—	2	6	no	—
26	C₂×D₄×C₄₇	—	2	12	no	—
27	C₂×D₄×C₄₇	—	2	6	no	—
28	C₂²×C₄×C₄₇	—	2	6	no	—
29	C₂²×C₄×C₄₇	—	2	6	no	—
30	C₂²×C₄×C₄₇	C₂²⋊C₄	1	6	yes	—
31	C₂²×C₄×C₄₇	C₂×C₈	1	3	yes	—
32	C₂²×C₄×C₄₇	M₁₆	1	6	yes	—
33	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
34	M₁₆×C₄₇	—	2	24	no	—
35	C₂×C₈×C₄₇	—	2	6	no	—
36	C₂×C₈×C₄₇	—	2	24	no	—
37	C₂×C₈×C₄₇	C₂²⋊C₄	1	6	yes	—
38	C₂×C₈×C₄₇	C₂×D₄	1	6	yes	—
39	C₄₇⋊C₁₆	—	4	4	no	—
40	C₂³×C₄₇	—	2	6	no	—
41	C₂³×C₄₇	—	2	12	no	—
42	C₂³×C₄₇	C₂²⋊C₈	1	6	yes	—
43	C₂³×C₄₇	C₂²×C₈	1	1	yes	characteristic
44	Q₈×C₄₇	—	2	8	no	—
45	D₄×C₄₇	—	2	24	no	—
46	C₂×C₄×C₄₇	—	2	6	no	—
47	C₂×C₄×C₄₇	—	2	6	no	—
48	C₂×C₄×C₄₇	—	2	6	no	—
49	C₂×C₄×C₄₇	—	2	12	no	—
50	C₂×C₄×C₄₇	C₂²⋊C₈	1	6	yes	—
51	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	3	yes	—
52	C₂×C₄×C₄₇	C₂×M₁₆	1	3	yes	—
53	C₈×C₄₇	—	2	6	no	—
54	C₈×C₄₇	#120	1	4	yes	—
55	#12680	—	47	1	no	maximal, Sylow
56	C₂²×C₄₇	—	2	12	no	—
57	C₂²×C₄₇	—	2	6	no	—
58	C₂²×C₄₇	—	4	12	no	—
59	C₂²×C₄₇	C₂×C₂²⋊C₈	1	3	yes	—
60	C₂²×C₄₇	#504	1	4	yes	—
61	C₄×C₄₇	—	2	6	no	—
62	C₄×C₄₇	#379	1	1	yes	characteristic
63	C₄×C₄₇	#407	1	3	yes	—
64	C₂²×#131	D₄₇	1	1	yes	characteristic
65	C₂×C₂²⋊C₁₆	—	47	6	no	—
66	C₂×C₄₇	—	4	6	no	—
67	C₂×C₄₇	#1779	1	1	yes	characteristic
68	C₂×C₄₇	#2113	1	6	yes	—
69	C₂²×#41	Dic₄₇	1	1	yes	characteristic
70	C₂×#131	—	2	12	no	—
71	C₂²×M₁₆	Dic₄₇	1	3	yes	—
72	C₂³×C₈	C₂×D₄₇	1	3	yes	—
73	C₂²×C₁₆	—	47	4	no	—
74	C₂²⋊C₁₆	—	94	12	no	—
75	C₄₇	#12680	1	1	yes	characteristic, complemented, Sylow
76	C₂×#41	—	2	6	no	—
77	C₂×#41	—	2	6	no	—
78	C₂²×Q₈	C₄₇⋊C₈	1	1	yes	characteristic
79	C₂²×D₄	C₄₇⋊C₈	1	3	yes	—
80	C₂³×C₄	C₂×Dic₄₇	1	3	yes	—
81	#131	—	2	32	no	—
82	C₂×M₁₆	—	2	6	no	—
83	C₂×M₁₆	—	2	12	no	—
84	C₂²×C₈	—	2	12	no	—
85	C₂²×C₈	C₂×Dic₄₇	1	3	yes	—
86	C₂²×C₈	C₂²⋊D₄₇	1	12	yes	—
87	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
88	C₂×C₁₆	—	94	12	no	—
89	C₂⁴	C₂×C₄₇⋊C₈	1	3	yes	—
90	#41	—	2	8	no	—
91	#41	—	2	24	no	—
92	C₂×Q₈	—	2	6	no	—
93	C₂×D₄	—	2	12	no	—
94	C₂×D₄	—	2	6	no	—
95	C₂²×C₄	—	2	6	no	—
96	C₂²×C₄	—	2	6	no	—
97	C₂²×C₄	C₂×C₄₇⋊C₈	1	3	yes	—
98	C₂²×C₄	#10625329	1	6	yes	—
99	C₂²×C₄	C₂²⋊Dic₄₇	1	6	yes	—
100	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
101	M₁₆	—	2	24	no	—
102	C₂×C₈	—	2	6	no	—
103	C₂×C₈	—	2	24	no	—
104	C₂×C₈	C₂²⋊Dic₄₇	1	6	yes	—
105	C₂×C₈	C₂×C₂²⋊D₄₇	1	6	yes	—
106	C₁₆	—	188	4	no	—
107	C₂³	—	2	12	no	—
108	C₂³	—	2	6	no	—
109	C₂³	C₂²⋊(C₄₇⋊C₈)	1	6	yes	—
110	C₂³	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
111	Q₈	—	2	8	no	—
112	D₄	—	2	24	no	—
113	C₂×C₄	—	2	6	no	—
114	C₂×C₄	—	2	6	no	—
115	C₂×C₄	—	2	12	no	—
116	C₂×C₄	—	2	6	no	—
117	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	6	yes	—
118	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	3	yes	—
119	C₂×C₄	C₂×#10625329	1	3	yes	—
120	C₈	—	2	6	no	—
121	C₈	#13093993	1	4	yes	—
122	C₂²	—	2	12	no	—
123	C₂²	—	2	6	no	—
124	C₂²	—	4	12	no	—
125	C₂²	C₂×C₂²⋊(C₄₇⋊C₈)	1	3	yes	—
126	C₂²	#15689540	1	4	yes	—
127	C₄	—	2	6	no	—
128	C₄	#15688979	1	1	yes	characteristic
129	C₄	#15689087	1	3	yes	—
130	C₂	—	4	6	no	—
131	C₂	#20243298	1	1	yes	characteristic
132	C₂	#20245002	1	6	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₂²⋊(C₄₇⋊C₁₆)

C₂

yes

maximal, complemented

C₂²×#131×C₄₇

C₂

yes

maximal, characteristic

C₂²×C₄₇⋊C₁₆

C₂²

yes

—

C₂²⋊(C₄₇⋊C₁₆)

—

C₂²×#41×C₄₇

C₄

yes

characteristic

C₂×#131×C₄₇

—

C₂²×M₁₆×C₄₇

C₄

yes

—

C₂³×C₈×C₄₇

C₂²

yes

—

C₂×C₄₇⋊C₁₆

—

C₂×#41×C₄₇

—

C₂×#41×C₄₇

—

C₂²×Q₈×C₄₇

C₈

yes

characteristic

C₂²×D₄×C₄₇

C₈

yes

—

C₂³×C₄×C₄₇

C₂×C₄

yes

—

#131×C₄₇

—

C₂×M₁₆×C₄₇

—

C₂×M₁₆×C₄₇

—

C₂²×C₈×C₄₇

—

C₂²×C₈×C₄₇

C₂×C₄

yes

—

C₂²×C₈×C₄₇

D₄

yes

—

C₂²×C₈×C₄₇

C₂³

yes

characteristic

C₂⁴×C₄₇

C₂×C₈

yes

—

#41×C₄₇

—

#41×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₄₇

C₂×C₈

yes

—

C₂²×C₄×C₄₇

M₁₆

yes

—

C₂²×C₄×C₄₇

C₂²×C₄

yes

characteristic

M₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₂²⋊C₄

yes

—

C₂×C₈×C₄₇

C₂×D₄

yes

—

C₄₇⋊C₁₆

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

C₂²⋊C₈

yes

—

C₂³×C₄₇

C₂²×C₈

yes

characteristic

Q₈×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²⋊C₈

yes

—

C₂×C₄×C₄₇

C₂×C₂²⋊C₄

yes

—

C₂×C₄×C₄₇

C₂×M₁₆

yes

—

C₈×C₄₇

—

C₈×C₄₇

#120

yes

—

#12680

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×C₂²⋊C₈

yes

—

C₂²×C₄₇

#504

yes

—

C₄×C₄₇

—

C₄×C₄₇

#379

yes

characteristic

C₄×C₄₇

#407

yes

—

C₂²×#131

D₄₇

yes

characteristic

C₂×C₂²⋊C₁₆

—

C₂×C₄₇

—

C₂×C₄₇

#1779

yes

characteristic

C₂×C₄₇

#2113

yes

—

C₂²×#41

Dic₄₇

yes

characteristic

C₂×#131

—

C₂²×M₁₆

Dic₄₇

yes

—

C₂³×C₈

C₂×D₄₇

yes

—

C₂²×C₁₆

—

C₂²⋊C₁₆

—

C₄₇

#12680

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂×#41

—

C₂²×Q₈

C₄₇⋊C₈

yes

characteristic

C₂²×D₄

C₄₇⋊C₈

yes

—

C₂³×C₄

C₂×Dic₄₇

yes

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂×Dic₄₇

yes

—

C₂²×C₈

C₂²⋊D₄₇

yes

—

C₂²×C₈

C₂²×D₄₇

yes

characteristic

C₂×C₁₆

—

C₂⁴

C₂×C₄₇⋊C₈

yes

—

#41

—

#41

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×C₄₇⋊C₈

yes

—

C₂²×C₄

#10625329

yes

—

C₂²×C₄

C₂²⋊Dic₄₇

yes

—

100

C₂²×C₄

C₂²×Dic₄₇

yes

characteristic

101

M₁₆

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

C₂²⋊Dic₄₇

yes

—

105

C₂×C₈

C₂×C₂²⋊D₄₇

yes

—

106

C₁₆

—

188

—

107

C₂³

—

108

C₂³

—

109

C₂³

C₂²⋊(C₄₇⋊C₈)

yes

—

110

C₂³

C₂²×C₄₇⋊C₈

yes

characteristic

111

Q₈

—

112

D₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

C₂²⋊(C₄₇⋊C₈)

yes

—

118

C₂×C₄

C₂×C₂²⋊Dic₄₇

yes

—

119

C₂×C₄

C₂×#10625329

yes

—

120

C₈

—

121

C₈

#13093993

yes

—

122

C₂²

—

123

C₂²

—

124

C₂²

—

125

C₂²

C₂×C₂²⋊(C₄₇⋊C₈)

yes

—

126

C₂²

#15689540

yes

—

127

C₄

—

128

C₄

#15688979

yes

characteristic

129

C₄

#15689087

yes

—

130

C₂

—

131

C₂

#20243298

yes

characteristic

132

C₂

#20245002

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₄₇
commutator, socle	C₂³×C₄₇	C₂²×C₈
Fitting	C₂²×#131×C₄₇	C₂
prime radical	C₂²×#41×C₄₇	C₄
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	C₂²×#131	D₄₇
hypocenter	C₄₇	#12680
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂²×#131

subgroup

G/H

center, Frattini

C₂²×C₈

C₂²×D₄₇

commutator, socle

C₂³×C₄₇

C₂²×C₈

Fitting

C₂²×#131×C₄₇

C₂

prime radical

C₂²×#41×C₄₇

C₄

cosocle

C₂²×C₈×C₄₇

C₂³

hypercenter

C₂²×#131

D₄₇

C₄₇

G ▷ C₂³×C₄₇ ▷ C₄₇

C₁ ◁ C₂²×C₈ ◁ C₂²×#131

N	H ≅ G/N	type
1. C₂×C₂²⋊(C₄₇⋊C₁₆)	C₂	semidirect
75. C₄₇	#12680	semidirect

H ≅ G/N

type

1. C₂×C₂²⋊(C₄₇⋊C₁₆)

C₂

semidirect

75. C₄₇

#12680

semidirect

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

subgroup	index
C₂×C₂²⋊(C₄₇⋊C₁₆)	2
C₂²×#131×C₄₇	2
#12680	47

subgroup

index

C₂×C₂²⋊(C₄₇⋊C₁₆)

C₂²×#131×C₄₇

#12680

Subgroups — #26131714

Conjugacy classes of subgroups

Complements

Maximal subgroups