Group #26130963 — subgroups

Subgroups — #26130963

Order 12032.

306 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 95 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	#20245611	C₂	1	4	yes	maximal, complemented
2	#20243006	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈×C₄₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₇×#1712	C₂	1	1	yes	maximal, characteristic
5	#15689528	—	2	1	no	—
6	#15689528	C₂²	1	2	yes	—
7	#15689503	—	2	1	no	—
8	#15688924	—	2	8	no	—
9	C₄×C₄₇⋊C₁₆	C₂²	1	2	yes	—
10	C₄₇×#493	—	2	1	no	—
11	C₄₇×#493	—	2	1	no	—
12	C₄₇×#493	C₂²	1	2	yes	—
13	C₈²×C₄₇	C₂²	1	1	yes	characteristic
14	C₂×Dic₁₈₈	—	4	2	no	—
15	#13094024	—	2	2	no	—
16	C₂×C₄₇⋊C₁₆	D₄	1	4	yes	—
17	#13093958	—	2	4	no	—
18	#13093958	—	4	2	no	—
19	C₂×D₈×C₄₇	—	2	4	no	—
20	C₂×D₈×C₄₇	—	4	2	no	—
21	C₂×D₈×C₄₇	—	4	2	no	—
22	#127×C₄₇	—	2	2	no	—
23	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
24	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
25	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
26	C₂×D₄×C₄₇	—	4	2	no	—
27	C₂×D₄×C₄₇	—	4	2	no	—
28	C₂×D₄×C₄₇	—	4	2	no	—
29	C₂×Dic₉₄	—	4	2	no	—
30	D₈×C₄₇	—	4	8	no	—
31	D₈×C₄₇	—	8	4	no	—
32	D₈×C₄₇	—	8	4	no	—
33	C₂×C₈×C₄₇	D₈	1	2	yes	—
34	C₂×C₈×C₄₇	D₈	1	2	yes	—
35	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
36	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
37	C₄⋊Dic₄₇	—	4	2	no	—
38	C₄⋊Dic₄₇	—	4	2	no	—
39	Dic₁₈₈	—	8	4	no	—
40	C₄₇⋊C₁₆	—	2	4	no	—
41	C₂³×C₄₇	—	8	2	no	—
42	D₄×C₄₇	—	8	4	no	—
43	D₄×C₄₇	—	8	4	no	—
44	D₄×C₄₇	—	8	4	no	—
45	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
46	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
47	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
48	C₂×Dic₄₇	—	8	2	no	—
49	Dic₉₄	—	8	4	no	—
50	C₈×C₄₇	D₁₆	1	4	yes	—
51	C₈×C₄₇	SD₃₂	1	4	yes	—
52	C₈×C₄₇	#135	1	4	yes	—
53	#12509	—	47	1	no	maximal, Sylow
54	C₂²×C₄₇	—	16	2	no	—
55	C₂²×C₄₇	—	16	2	no	—
56	C₂²×C₄₇	—	16	2	no	—
57	C₂²×C₄₇	#493	1	1	yes	characteristic
58	C₄×C₄₇	#495	1	2	yes	—
59	C₄×C₄₇	C₂×D₁₆	1	2	yes	—
60	C₄×C₄₇	C₂×SD₃₂	1	2	yes	—
61	Dic₄₇	—	16	2	no	—
62	#2240	—	47	4	no	—
63	#1713	—	47	1	no	—
64	#1712	D₄₇	1	1	yes	characteristic
65	C₈×C₁₆	—	47	1	no	—
66	C₂×C₄₇	—	32	2	no	—
67	C₂×C₄₇	#1709	1	1	yes	characteristic
68	C₂×C₄₇	#2246	1	1	yes	characteristic
69	C₂×C₄₇	#2248	1	1	yes	characteristic
70	#500	—	47	2	no	—
71	#500	—	94	1	no	—
72	#494	—	94	1	no	—
73	#493	—	2	1	no	—
74	#493	—	2	1	no	—
75	#493	C₂×D₄₇	1	2	yes	—
76	#357	—	94	8	no	—
77	C₄×C₁₆	—	47	2	no	—
78	C₈²	C₂×D₄₇	1	1	yes	characteristic
79	C₄₇	#12509	1	1	yes	characteristic, complemented, Sylow
80	C₂×Q₁₆	—	188	2	no	—
81	C₂×D₈	—	2	4	no	—
82	C₂×D₈	—	4	2	no	—
83	C₂×D₈	—	4	2	no	—
84	C₄⋊Q₈	—	94	2	no	—
85	#127	—	2	2	no	—
86	C₂×C₁₆	—	47	4	no	—
87	#107	—	94	4	no	—
88	#107	—	188	2	no	—
89	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
90	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
91	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
92	C₂×Q₈	—	188	2	no	—
93	C₂×D₄	—	4	2	no	—
94	C₂×D₄	—	4	2	no	—
95	C₂×D₄	—	4	2	no	—
96	Q₁₆	—	376	4	no	—
97	D₈	—	4	8	no	—
98	D₈	—	8	4	no	—
99	D₈	—	8	4	no	—
100	C₂×C₈	#10625334	1	2	yes	—
101	C₂×C₈	#10625334	1	2	yes	—
102	C₂×C₈	D₄×D₄₇	1	2	yes	—
103	C₄⋊C₄	—	188	2	no	—
104	C₄⋊C₄	—	188	2	no	—
105	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
106	C₁₆	—	94	4	no	—
107	C₂³	—	8	2	no	—
108	Q₈	—	376	4	no	—
109	D₄	—	8	4	no	—
110	D₄	—	8	4	no	—
111	D₄	—	8	4	no	—
112	C₂×C₄	—	376	2	no	—
113	C₂×C₄	#13094020	1	1	yes	characteristic
114	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
115	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
116	C₈	#13093965	1	4	yes	—
117	C₈	#13093968	1	4	yes	—
118	C₈	#13094043	1	4	yes	—
119	C₂²	—	16	2	no	—
120	C₂²	—	16	2	no	—
121	C₂²	—	16	2	no	—
122	C₂²	#15689499	1	1	yes	characteristic
123	C₄	—	752	2	no	—
124	C₄	#15689506	1	2	yes	—
125	C₄	C₂×#13093965	1	2	yes	—
126	C₄	C₂×#13093968	1	2	yes	—
127	C₂	—	32	2	no	—
128	C₂	#20242993	1	1	yes	characteristic
129	C₂	#20245637	1	1	yes	characteristic
130	C₂	#20245643	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20245611

C₂

yes

maximal, complemented

#20243006

C₂

yes

maximal, characteristic, complemented

C₈×C₄₇⋊C₁₆

C₂

yes

maximal, characteristic, complemented

C₄₇×#1712

C₂

yes

maximal, characteristic

#15689528

—

#15689528

C₂²

yes

—

#15689503

—

#15688924

—

C₄×C₄₇⋊C₁₆

C₂²

yes

—

C₄₇×#493

—

C₄₇×#493

—

C₄₇×#493

C₂²

yes

—

C₈²×C₄₇

C₂²

yes

characteristic

C₂×Dic₁₈₈

—

#13094024

—

C₂×C₄₇⋊C₁₆

D₄

yes

—

#13093958

—

#13093958

—

C₂×D₈×C₄₇

—

C₂×D₈×C₄₇

—

C₂×D₈×C₄₇

—

#127×C₄₇

—

C₄×C₈×C₄₇

D₄

yes

characteristic

C₄×C₈×C₄₇

D₄

yes

characteristic

C₄×C₈×C₄₇

C₂³

yes

characteristic

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×Dic₉₄

—

D₈×C₄₇

—

D₈×C₄₇

—

D₈×C₄₇

—

C₂×C₈×C₄₇

D₈

yes

—

C₂×C₈×C₄₇

D₈

yes

—

C₂×C₈×C₄₇

C₂×D₄

yes

—

C₄²×C₄₇

C₂×D₄

yes

characteristic

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

Dic₁₈₈

—

C₄₇⋊C₁₆

—

C₂³×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

#127

yes

characteristic

C₂×C₄×C₄₇

C₂×D₈

yes

characteristic

C₂×C₄×C₄₇

C₂×D₈

yes

characteristic

C₂×Dic₄₇

—

Dic₉₄

—

C₈×C₄₇

D₁₆

yes

—

C₈×C₄₇

SD₃₂

yes

—

C₈×C₄₇

#135

yes

—

#12509

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#493

yes

characteristic

C₄×C₄₇

#495

yes

—

C₄×C₄₇

C₂×D₁₆

yes

—

C₄×C₄₇

C₂×SD₃₂

yes

—

Dic₄₇

—

#2240

—

#1713

—

#1712

D₄₇

yes

characteristic

C₈×C₁₆

—

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

—

—

—

—

C₂×D₄₇

yes

—

#357

—

C₄×C₁₆

—

C₈²

C₂×D₄₇

yes

characteristic

C₄₇

#12509

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

188

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

#127

—

C₂×C₁₆

—

#107

—

#107

—

188

—

C₄×C₈

C₂²⋊D₄₇

yes

characteristic

C₄×C₈

C₂²⋊D₄₇

yes

characteristic

C₄×C₈

C₂²×D₄₇

yes

characteristic

C₂×Q₈

—

188

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

Q₁₆

—

376

—

D₈

—

D₈

—

D₈

—

100

C₂×C₈

#10625334

yes

—

101

C₂×C₈

#10625334

yes

—

102

C₂×C₈

D₄×D₄₇

yes

—

103

C₄⋊C₄

—

188

—

104

C₄⋊C₄

—

188

—

105

C₄²

C₂×C₂²⋊D₄₇

yes

characteristic

106

C₁₆

—

107

C₂³

—

108

Q₈

—

376

—

109

D₄

—

110

D₄

—

111

D₄

—

112

C₂×C₄

—

376

—

113

C₂×C₄

#13094020

yes

characteristic

114

C₂×C₄

C₂×#10625334

yes

characteristic

115

C₂×C₄

C₂×#10625334

yes

characteristic

116

C₈

#13093965

yes

—

117

C₈

#13093968

yes

—

118

C₈

#13094043

yes

—

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

#15689499

yes

characteristic

123

C₄

—

752

—

124

C₄

#15689506

yes

—

125

C₄

C₂×#13093965

yes

—

126

C₄

C₂×#13093968

yes

—

127

C₂

—

128

C₂

#20242993

yes

characteristic

129

C₂

#20245637

yes

characteristic

130

C₂

#20245643

yes

characteristic

	subgroup	G/H
center	C₂²	#15689499
commutator, cosocle	C₄×C₈×C₄₇	C₂³
Frattini	C₄×C₈	C₂²×D₄₇
Fitting, prime radical	C₄₇×#1712	C₂
socle	C₂²×C₄₇	#493
hypercenter	#1712	D₄₇
hypocenter	C₄₇	#12509
LCS	G ▷ C₄×C₈×C₄₇ ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1712

subgroup

G/H

center

C₂²

#15689499

commutator, cosocle

C₄×C₈×C₄₇

C₂³

Frattini

C₄×C₈

C₂²×D₄₇

Fitting, prime radical

C₄₇×#1712

C₂

C₂²×C₄₇

D₄₇

C₄₇

G ▷ C₄×C₈×C₄₇ ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1712

N	H ≅ G/N	type
1. #20245611	C₂	semidirect
2. #20243006	C₂	semidirect
3. C₈×C₄₇⋊C₁₆	C₂	semidirect
79. C₄₇	#12509	semidirect

H ≅ G/N

type

1. #20245611

C₂

semidirect

2. #20243006

C₂

semidirect

3. C₈×C₄₇⋊C₁₆

C₂

semidirect

79. C₄₇

#12509

semidirect

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

subgroup	index
#20245611	2
#20243006	2
C₈×C₄₇⋊C₁₆	2
C₄₇×#1712	2
#12509	47

subgroup

index

#20245611

#20243006

C₈×C₄₇⋊C₁₆

C₄₇×#1712

#12509

Subgroups — #26130963

Conjugacy classes of subgroups

Complements

Maximal subgroups