Group #26121328 — subgroups

Subgroups — #26121328

Order 12032.

526 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 165 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	#20244194	C₂	1	3	yes	maximal
2	#20243548	C₂	1	3	yes	maximal, complemented
3	C₄₇×#1273	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#13094021	C₂²	1	1	yes	characteristic
5	C₂×#13093958	—	2	6	no	—
6	C₂×#13093946	—	2	6	no	—
7	#15688993	C₂²	1	3	yes	—
8	C₂×C₄×C₈×C₄₇	C₂²	1	3	yes	—
9	C₂²×D₁₈₈	—	2	3	no	—
10	#13094021	—	2	3	no	—
11	#13094021	Q₈	1	2	yes	—
12	C₂×C₄⋊Dic₄₇	—	2	3	no	—
13	C₂×C₄⋊Dic₄₇	—	2	6	no	—
14	#13093958	—	4	12	no	—
15	#13093946	—	4	12	no	—
16	C₂²×C₈×C₄₇	D₄	1	6	yes	—
17	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
18	C₄×C₈×C₄₇	—	2	6	no	—
19	C₂³×D₄₇	—	4	1	no	—
20	C₂²×C₄×C₄₇	C₂×D₄	1	3	yes	—
21	C₂²×Dic₄₇	—	4	3	no	—
22	C₂×D₁₈₈	—	2	6	no	—
23	C₂×D₁₈₈	—	4	6	no	—
24	C₂×D₁₈₈	—	4	6	no	—
25	C₂×D₁₈₈	—	4	3	no	—
26	C₂×C₈×C₄₇	—	2	6	no	—
27	C₂×C₈×C₄₇	—	2	6	no	—
28	C₂×C₈×C₄₇	D₈	1	12	yes	—
29	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
30	C₄²×C₄₇	#41	1	3	yes	—
31	C₄⋊Dic₄₇	—	4	6	no	—
32	C₄⋊Dic₄₇	—	4	12	no	—
33	C₂³×C₄₇	#127	1	1	yes	characteristic
34	C₂²×D₄₇	—	4	2	no	—
35	C₂²×D₄₇	—	8	3	no	—
36	C₂²×D₄₇	—	8	3	no	—
37	C₂×C₄×C₄₇	C₂²⋊Q₈	1	3	yes	—
38	C₂×C₄×C₄₇	#123	1	3	yes	—
39	C₂×C₄×C₄₇	C₂×D₈	1	6	yes	—
40	C₂×C₄×C₄₇	#137	1	6	yes	—
41	C₂×Dic₄₇	—	8	3	no	—
42	C₂×Dic₄₇	—	8	6	no	—
43	D₁₈₈	—	4	12	no	—
44	D₁₈₈	—	8	6	no	—
45	C₈×C₄₇	—	2	12	no	—
46	#10673	—	47	1	no	maximal, Sylow
47	C₂²×C₄₇	#399	1	1	yes	characteristic
48	C₂²×C₄₇	#493	1	3	yes	—
49	C₂²×C₄₇	#497	1	3	yes	—
50	C₂×D₄₇	—	8	6	no	—
51	C₂×D₄₇	—	16	1	no	—
52	C₂×D₄₇	—	16	3	no	—
53	C₄×C₄₇	#474	1	6	yes	—
54	C₄×C₄₇	#480	1	6	yes	—
55	Dic₄₇	—	16	3	no	—
56	#1957	—	47	3	no	—
57	#1845	—	47	3	no	—
58	#1273	D₄₇	1	1	yes	characteristic, complemented
59	C₂×C₄₇	#1712	1	1	yes	characteristic
60	C₂×C₄₇	#1722	1	3	yes	—
61	C₂×C₄₇	#2093	1	3	yes	—
62	D₄₇	—	16	2	no	—
63	C₂×#127	—	47	1	no	—
64	C₂×#107	—	94	6	no	—
65	C₂×#102	—	94	6	no	—
66	C₂×C₄×C₈	C₂×D₄₇	1	3	yes	—
67	#384	—	47	3	no	—
68	C₄₇	#10673	1	1	yes	characteristic, complemented, Sylow
69	C₂²×D₄	—	94	3	no	—
70	C₂²×C₈	D₁₈₈	1	6	yes	—
71	#127	—	47	2	no	—
72	#127	—	94	3	no	—
73	C₂×C₄⋊C₄	—	94	3	no	—
74	C₂×C₄⋊C₄	—	94	6	no	—
75	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
76	#107	—	188	12	no	—
77	#102	—	188	12	no	—
78	C₄×C₈	—	2	6	no	—
79	C₂⁴	—	188	1	no	—
80	C₂×D₄	—	94	6	no	—
81	C₂×D₄	—	188	3	no	—
82	C₂×D₄	—	188	6	no	—
83	C₂×D₄	—	188	6	no	—
84	C₂²×C₄	—	188	3	no	—
85	C₂²×C₄	C₂×D₁₈₈	1	3	yes	—
86	C₂×C₈	—	2	6	no	—
87	C₂×C₈	—	2	6	no	—
88	C₂×C₈	D₃₇₆	1	12	yes	—
89	C₄⋊C₄	—	188	6	no	—
90	C₄⋊C₄	—	188	12	no	—
91	C₄²	#10625352	1	3	yes	—
92	C₄²	Q₈×D₄₇	1	1	yes	characteristic
93	C₂³	—	376	3	no	—
94	C₂³	—	376	3	no	—
95	C₂³	—	188	2	no	—
96	C₂³	#13094021	1	1	yes	characteristic
97	D₄	—	376	6	no	—
98	D₄	—	188	12	no	—
99	C₂×C₄	—	376	3	no	—
100	C₂×C₄	—	376	6	no	—
101	C₂×C₄	#13094004	1	3	yes	—
102	C₂×C₄	#13094008	1	3	yes	—
103	C₂×C₄	C₂×D₃₇₆	1	6	yes	—
104	C₂×C₄	#13094057	1	6	yes	—
105	C₈	—	2	12	no	—
106	C₂²	—	376	6	no	—
107	C₂²	—	752	1	no	—
108	C₂²	—	752	3	no	—
109	C₂²	#15689060	1	1	yes	characteristic
110	C₂²	#15689500	1	3	yes	—
111	C₂²	#15689517	1	3	yes	—
112	C₄	—	752	3	no	—
113	C₄	#15689404	1	6	yes	—
114	C₄	#15689437	1	6	yes	—
115	C₂	—	752	2	no	—
116	C₂	#20243003	1	1	yes	characteristic
117	C₂	#20243059	1	3	yes	—
118	C₂	#20244915	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20244194

C₂

yes

maximal

#20243548

C₂

yes

maximal, complemented

C₄₇×#1273

C₂

yes

maximal, characteristic, complemented

C₂×#13094021

C₂²

yes

—

—

C₂²

yes

—

C₂×C₄×C₈×C₄₇

C₂²

yes

—

C₂²×D₁₈₈

—

#13094021

—

#13094021

Q₈

yes

—

C₂×C₄⋊Dic₄₇

—

C₂×C₄⋊Dic₄₇

—

#13093958

—

#13093946

—

C₂²×C₈×C₄₇

D₄

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄×C₈×C₄₇

—

C₂³×D₄₇

—

C₂²×C₄×C₄₇

C₂×D₄

yes

—

C₂²×Dic₄₇

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

D₈

yes

—

C₄²×C₄₇

C₂×Q₈

yes

characteristic

C₄²×C₄₇

#41

yes

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₂³×C₄₇

#127

yes

characteristic

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂×C₄×C₄₇

C₂²⋊Q₈

yes

—

C₂×C₄×C₄₇

#123

yes

—

C₂×C₄×C₄₇

C₂×D₈

yes

—

C₂×C₄×C₄₇

#137

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

D₁₈₈

—

D₁₈₈

—

C₈×C₄₇

—

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

—

C₂²×C₄₇

#497

yes

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

#474

yes

—

C₄×C₄₇

#480

yes

—

Dic₄₇

—

#1957

—

#1845

—

#1273

D₄₇

yes

characteristic, complemented

C₂×C₄₇

#1712

yes

characteristic

C₂×C₄₇

#1722

yes

—

C₂×C₄₇

#2093

yes

—

D₄₇

—

C₂×#127

—

C₂×#107

—

C₂×#102

—

C₂×C₄×C₈

C₂×D₄₇

yes

—

#384

—

C₄₇

#10673

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂²×C₈

D₁₈₈

yes

—

#127

—

#127

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂²×D₄₇

yes

characteristic

#107

—

188

—

#102

—

188

—

C₄×C₈

—

C₂⁴

—

188

—

C₂×D₄

—

C₂×D₄

—

188

—

C₂×D₄

—

188

—

C₂×D₄

—

188

—

C₂²×C₄

—

188

—

C₂²×C₄

C₂×D₁₈₈

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₃₇₆

yes

—

C₄⋊C₄

—

188

—

C₄⋊C₄

—

188

—

C₄²

#10625352

yes

—

C₄²

Q₈×D₄₇

yes

characteristic

C₂³

—

376

—

C₂³

—

376

—

C₂³

—

188

—

C₂³

#13094021

yes

characteristic

D₄

—

376

—

D₄

—

188

—

C₂×C₄

—

376

—

100

C₂×C₄

—

376

—

101

C₂×C₄

#13094004

yes

—

102

C₂×C₄

#13094008

yes

—

103

C₂×C₄

C₂×D₃₇₆

yes

—

104

C₂×C₄

#13094057

yes

—

105

C₈

—

106

C₂²

—

376

—

107

C₂²

—

752

—

108

C₂²

—

752

—

109

C₂²

#15689060

yes

characteristic

110

C₂²

#15689500

yes

—

111

C₂²

#15689517

yes

—

112

C₄

—

752

—

113

C₄

#15689404

yes

—

114

C₄

#15689437

yes

—

115

C₂

—

752

—

116

C₂

#20243003

yes

characteristic

117

C₂

#20243059

yes

—

118

C₂

#20244915

yes

—

	subgroup	G/H
center	C₂³	#13094021
commutator, cosocle	C₂×C₄²×C₄₇	C₂³
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1273	C₂
socle	C₂³×C₄₇	#127
prime radical	C₂×#13094021	C₂²
hypercenter	#1273	D₄₇
hypocenter	C₄₇	#10673
LCS	G ▷ C₂×C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1273

subgroup

G/H

center

C₂³

#13094021

commutator, cosocle

C₂×C₄²×C₄₇

C₂³

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#1273

C₂

C₂³×C₄₇

C₂²

D₄₇

C₄₇

G ▷ C₂×C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1273

N	H ≅ G/N	type
2. #20243548	C₂	semidirect
3. C₄₇×#1273	C₂	semidirect
58. #1273	D₄₇	semidirect
68. C₄₇	#10673	semidirect

H ≅ G/N

type

2. #20243548

C₂

semidirect

3. C₄₇×#1273

C₂

semidirect

58. #1273

D₄₇

semidirect

68. C₄₇

#10673

semidirect

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

subgroup	index
#20244194	2
#20243548	2
C₄₇×#1273	2
#10673	47

subgroup

index

#20244194

#20243548

C₄₇×#1273

#10673

Subgroups — #26121328

Conjugacy classes of subgroups

Complements

Maximal subgroups