Group #26130971 — subgroups

Subgroups — #26130971

Order 12032.

260 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 79 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	#20245615	C₂	1	2	yes	maximal, complemented
2	#20245611	C₂	1	2	yes	maximal
3	#20242998	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈×C₄₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₇×#1709	C₂	1	1	yes	maximal, characteristic
6	#15689528	C₂²	1	2	yes	—
7	#15689474	—	2	2	no	—
8	#15688927	—	2	4	no	—
9	#15688924	—	2	4	no	—
10	C₄×C₄₇⋊C₁₆	C₂²	1	2	yes	—
11	C₄₇×#494	C₂²	1	1	yes	characteristic
12	C₄₇×#493	C₂²	1	1	yes	characteristic
13	C₄₇×#486	—	2	2	no	—
14	C₈²×C₄₇	C₂²	1	1	yes	characteristic
15	#13094024	—	2	2	no	—
16	C₂×C₄₇⋊C₁₆	D₄	1	4	yes	—
17	#13093958	—	2	2	no	—
18	#13093958	—	2	2	no	—
19	#13093949	—	4	4	no	—
20	C₂×Q₁₆×C₄₇	—	2	2	no	—
21	C₂×D₈×C₄₇	—	2	2	no	—
22	C₄⋊Q₈×C₄₇	—	2	1	no	—
23	#127×C₄₇	—	2	1	no	—
24	#102×C₄₇	—	4	4	no	—
25	C₄×C₈×C₄₇	D₄	1	2	yes	—
26	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
27	C₂×Q₈×C₄₇	—	4	1	no	—
28	C₂×D₄×C₄₇	—	4	1	no	—
29	C₂×D₄×C₄₇	—	4	2	no	—
30	C₂×Dic₉₄	—	4	2	no	—
31	Q₁₆×C₄₇	—	4	4	no	—
32	D₈×C₄₇	—	4	4	no	—
33	C₂×C₈×C₄₇	D₈	1	4	yes	—
34	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
35	C₄⋊C₄×C₄₇	—	4	2	no	—
36	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
37	C₄⋊Dic₄₇	—	4	2	no	—
38	C₄⋊Dic₄₇	—	4	2	no	—
39	C₄₇⋊C₁₆	—	2	4	no	—
40	C₂³×C₄₇	—	8	1	no	—
41	Q₈×C₄₇	—	8	2	no	—
42	D₄×C₄₇	—	8	2	no	—
43	D₄×C₄₇	—	8	4	no	—
44	C₂×C₄×C₄₇	—	8	1	no	—
45	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
46	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
47	C₂×Dic₄₇	—	8	2	no	—
48	Dic₉₄	—	8	4	no	—
49	C₈×C₄₇	—	2	4	no	—
50	C₈×C₄₇	#135	1	4	yes	—
51	#12511	—	47	1	no	maximal, Sylow
52	C₂²×C₄₇	—	16	1	no	—
53	C₂²×C₄₇	—	16	2	no	—
54	C₂²×C₄₇	#493	1	1	yes	characteristic
55	C₄×C₄₇	—	16	1	no	—
56	C₄×C₄₇	#495	1	2	yes	—
57	C₄×C₄₇	#508	1	4	yes	—
58	Dic₄₇	—	16	2	no	—
59	#2241	—	47	2	no	—
60	#2240	—	47	2	no	—
61	#1710	—	47	1	no	—
62	#1709	D₄₇	1	1	yes	characteristic
63	C₈×C₁₆	—	47	1	no	—
64	C₂×C₄₇	—	32	1	no	—
65	C₂×C₄₇	#1709	1	1	yes	characteristic
66	C₂×C₄₇	#2249	1	2	yes	—
67	#500	—	47	2	no	—
68	#494	C₂×D₄₇	1	1	yes	characteristic
69	#493	C₂×D₄₇	1	1	yes	characteristic
70	#487	—	94	2	no	—
71	#486	—	2	2	no	—
72	#358	—	94	4	no	—
73	#357	—	94	4	no	—
74	C₄×C₁₆	—	47	2	no	—
75	C₈²	C₂×D₄₇	1	1	yes	characteristic
76	C₄₇	#12511	1	1	yes	characteristic, complemented, Sylow
77	C₂×Q₁₆	—	2	2	no	—
78	C₂×D₈	—	2	2	no	—
79	C₄⋊Q₈	—	2	1	no	—
80	C₄⋊Q₈	—	94	2	no	—
81	#127	—	2	1	no	—
82	C₂×C₁₆	—	47	4	no	—
83	#107	—	94	2	no	—
84	#107	—	94	2	no	—
85	#103	—	188	4	no	—
86	#102	—	4	4	no	—
87	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
88	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
89	C₂×Q₈	—	4	1	no	—
90	C₂×Q₈	—	188	2	no	—
91	C₂×D₄	—	4	1	no	—
92	C₂×D₄	—	4	2	no	—
93	Q₁₆	—	4	4	no	—
94	D₈	—	4	4	no	—
95	C₂×C₈	#10625334	1	4	yes	—
96	C₂×C₈	D₄×D₄₇	1	2	yes	—
97	C₄⋊C₄	—	4	2	no	—
98	C₄⋊C₄	—	188	2	no	—
99	C₄⋊C₄	—	188	2	no	—
100	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
101	C₁₆	—	94	4	no	—
102	C₂³	—	8	1	no	—
103	Q₈	—	8	2	no	—
104	Q₈	—	376	4	no	—
105	D₄	—	8	2	no	—
106	D₄	—	8	4	no	—
107	C₂×C₄	—	8	1	no	—
108	C₂×C₄	—	376	2	no	—
109	C₂×C₄	#13094020	1	1	yes	characteristic
110	C₂×C₄	C₂×#10625334	1	2	yes	—
111	C₈	—	2	4	no	—
112	C₈	#13094043	1	4	yes	—
113	C₂²	—	16	1	no	—
114	C₂²	—	16	2	no	—
115	C₂²	#15689499	1	1	yes	characteristic
116	C₄	—	16	1	no	—
117	C₄	—	752	2	no	—
118	C₄	#15689506	1	2	yes	—
119	C₄	#15689553	1	4	yes	—
120	C₂	—	32	1	no	—
121	C₂	#20242993	1	1	yes	characteristic
122	C₂	#20245647	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20245615

C₂

yes

maximal, complemented

#20245611

C₂

yes

maximal

#20242998

C₂

yes

maximal, characteristic, complemented

C₈×C₄₇⋊C₁₆

C₂

yes

maximal, characteristic, complemented

C₄₇×#1709

C₂

yes

maximal, characteristic

#15689528

C₂²

yes

—

#15689474

—

#15688927

—

#15688924

—

C₄×C₄₇⋊C₁₆

C₂²

yes

—

C₄₇×#494

C₂²

yes

characteristic

C₄₇×#493

C₂²

yes

characteristic

C₄₇×#486

—

C₈²×C₄₇

C₂²

yes

characteristic

#13094024

—

C₂×C₄₇⋊C₁₆

D₄

yes

—

#13093958

—

#13093958

—

#13093949

—

C₂×Q₁₆×C₄₇

—

C₂×D₈×C₄₇

—

C₄⋊Q₈×C₄₇

—

#127×C₄₇

—

#102×C₄₇

—

C₄×C₈×C₄₇

D₄

yes

—

C₄×C₈×C₄₇

C₂³

yes

characteristic

C₂×Q₈×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×Dic₉₄

—

Q₁₆×C₄₇

—

D₈×C₄₇

—

C₂×C₈×C₄₇

D₈

yes

—

C₂×C₈×C₄₇

C₂×D₄

yes

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

C₂×D₄

yes

characteristic

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄₇⋊C₁₆

—

C₂³×C₄₇

—

Q₈×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#127

yes

characteristic

C₂×C₄×C₄₇

C₂×D₈

yes

—

C₂×Dic₄₇

—

Dic₉₄

—

C₈×C₄₇

—

C₈×C₄₇

#135

yes

—

#12511

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#493

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

#495

yes

—

C₄×C₄₇

#508

yes

—

Dic₄₇

—

—

—

—

D₄₇

yes

characteristic

C₈×C₁₆

—

C₂×C₄₇

—

C₂×C₄₇

#1709

yes

characteristic

C₂×C₄₇

#2249

yes

—

#500

—

#494

C₂×D₄₇

yes

characteristic

#493

C₂×D₄₇

yes

—

—

—

—

C₄×C₁₆

—

C₈²

C₂×D₄₇

yes

characteristic

C₄₇

#12511

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×D₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#127

—

C₂×C₁₆

—

#107

—

#107

—

#103

—

188

—

#102

—

C₄×C₈

C₂²⋊D₄₇

yes

—

C₄×C₈

C₂²×D₄₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

188

—

C₂×D₄

—

C₂×D₄

—

Q₁₆

—

D₈

—

C₂×C₈

#10625334

yes

—

C₂×C₈

D₄×D₄₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

188

—

C₄⋊C₄

—

188

—

100

C₄²

C₂×C₂²⋊D₄₇

yes

characteristic

101

C₁₆

—

102

C₂³

—

103

Q₈

—

104

Q₈

—

376

—

105

D₄

—

106

D₄

—

107

C₂×C₄

—

108

C₂×C₄

—

376

—

109

C₂×C₄

#13094020

yes

characteristic

110

C₂×C₄

C₂×#10625334

yes

—

111

C₈

—

112

C₈

#13094043

yes

—

113

C₂²

—

114

C₂²

—

115

C₂²

#15689499

yes

characteristic

116

C₄

—

117

C₄

—

752

—

118

C₄

#15689506

yes

—

119

C₄

#15689553

yes

—

120

C₂

—

121

C₂

#20242993

yes

characteristic

122

C₂

#20245647

yes

—

	subgroup	G/H
center	C₂²	#15689499
commutator, cosocle	C₄×C₈×C₄₇	C₂³
Frattini	C₄×C₈	C₂²×D₄₇
Fitting	C₄₇×#1709	C₂
socle	C₂²×C₄₇	#493
prime radical	C₄₇×#493	C₂²
hypercenter	#1709	D₄₇
hypocenter	C₄₇	#12511
LCS	G ▷ C₄×C₈×C₄₇ ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1709

subgroup

G/H

center

C₂²

#15689499

commutator, cosocle

C₄×C₈×C₄₇

C₂³

Frattini

C₄×C₈

C₂²×D₄₇

Fitting

C₄₇×#1709

C₂

socle

C₂²×C₄₇

#493

prime radical

C₄₇×#493

C₂²

D₄₇

C₄₇

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1709

N	H ≅ G/N	type
1. #20245615	C₂	semidirect
3. #20242998	C₂	semidirect
4. C₈×C₄₇⋊C₁₆	C₂	semidirect
76. C₄₇	#12511	semidirect

H ≅ G/N

type

1. #20245615

C₂

semidirect

3. #20242998

C₂

semidirect

4. C₈×C₄₇⋊C₁₆

C₂

semidirect

76. C₄₇

#12511

semidirect

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

subgroup	index
#20245615	2
#20245611	2
#20242998	2
C₈×C₄₇⋊C₁₆	2
C₄₇×#1709	2
#12511	47

subgroup

index

#20245615

#20245611

#20242998

C₈×C₄₇⋊C₁₆

C₄₇×#1709

#12511

Subgroups — #26130971

Conjugacy classes of subgroups

Complements

Maximal subgroups