Group #26130989 — subgroups

Subgroups — #26130989

Order 12032.

398 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 103 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	#20245637	C₂	1	4	yes	maximal, complemented
2	#20243003	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, complemented
5	C₂×#13093965	—	2	8	no	—
6	#15689500	—	2	2	no	—
7	#15689500	C₂²	1	2	yes	—
8	C₄×C₄₇⋊C₁₆	C₂²	1	2	yes	—
9	C₄₇×#493	—	2	2	no	—
10	C₄₇×#493	C₂²	1	2	yes	—
11	C₈²×C₄₇	C₂²	1	1	yes	characteristic
12	C₂×D₃₇₆	—	2	4	no	—
13	C₂×D₃₇₆	—	4	4	no	—
14	#13094021	—	2	2	no	—
15	#13093965	—	4	16	no	—
16	C₂×C₄₇⋊C₁₆	D₄	1	4	yes	—
17	C₂×D₈×C₄₇	—	2	4	no	—
18	C₂×D₈×C₄₇	—	4	4	no	—
19	#127×C₄₇	—	2	2	no	—
20	C₄×C₈×C₄₇	D₄	1	2	yes	—
21	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
22	C₂×D₄×C₄₇	—	4	2	no	—
23	C₂×D₄×C₄₇	—	4	4	no	—
24	C₂×D₁₈₈	—	4	2	no	—
25	C₂×D₁₈₈	—	4	4	no	—
26	D₈×C₄₇	—	4	8	no	—
27	D₈×C₄₇	—	8	8	no	—
28	C₂×C₈×C₄₇	D₈	1	4	yes	—
29	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
30	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
31	D₃₇₆	—	4	8	no	—
32	D₃₇₆	—	8	8	no	—
33	C₄₇⋊C₁₆	D₈	1	8	yes	complemented
34	C₂³×C₄₇	—	8	2	no	—
35	C₂²×D₄₇	—	8	2	no	—
36	D₄×C₄₇	—	8	4	no	—
37	D₄×C₄₇	—	8	8	no	—
38	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
39	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
40	D₁₈₈	—	8	4	no	—
41	D₁₈₈	—	8	8	no	—
42	C₈×C₄₇	D₁₆	1	8	yes	complemented
43	C₈×C₄₇	C₂×D₈	1	4	yes	—
44	#12515	—	47	1	no	maximal, Sylow
45	C₂²×C₄₇	—	16	2	no	—
46	C₂²×C₄₇	—	16	4	no	—
47	C₂²×C₄₇	#493	1	1	yes	characteristic
48	C₂×D₄₇	—	16	2	no	—
49	C₂×D₄₇	—	16	4	no	—
50	C₄×C₄₇	#493	1	2	yes	—
51	C₄×C₄₇	C₂×D₁₆	1	4	yes	—
52	#2246	—	47	4	no	—
53	#1712	—	47	1	no	—
54	#1712	D₄₇	1	1	yes	characteristic, complemented
55	C₈×C₁₆	—	47	1	no	—
56	C₂×C₄₇	—	32	2	no	—
57	C₂×C₄₇	#1712	1	1	yes	characteristic
58	C₂×C₄₇	#2246	1	2	yes	—
59	D₄₇	—	32	2	no	—
60	C₂×D₁₆	—	94	8	no	—
61	#493	—	2	2	no	—
62	#493	—	47	2	no	—
63	#493	—	94	2	no	—
64	#493	C₂×D₄₇	1	2	yes	—
65	C₄×C₁₆	—	47	2	no	—
66	C₈²	C₂×D₄₇	1	1	yes	characteristic
67	C₄₇	#12515	1	1	yes	characteristic, complemented, Sylow
68	C₂×D₈	—	2	4	no	—
69	C₂×D₈	—	4	4	no	—
70	C₂×D₈	—	94	4	no	—
71	C₂×D₈	—	188	4	no	—
72	#127	—	2	2	no	—
73	#127	—	94	2	no	—
74	D₁₆	—	188	16	no	—
75	C₂×C₁₆	—	47	4	no	—
76	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
77	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
78	C₂×D₄	—	4	2	no	—
79	C₂×D₄	—	4	4	no	—
80	C₂×D₄	—	188	2	no	—
81	C₂×D₄	—	188	4	no	—
82	D₈	—	4	8	no	—
83	D₈	—	8	8	no	—
84	D₈	—	376	8	no	—
85	D₈	—	188	8	no	—
86	C₂×C₈	#10625334	1	4	yes	—
87	C₂×C₈	D₄×D₄₇	1	2	yes	—
88	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
89	C₁₆	—	47	8	no	—
90	C₂³	—	8	2	no	—
91	C₂³	—	376	2	no	—
92	D₄	—	8	4	no	—
93	D₄	—	8	8	no	—
94	D₄	—	376	4	no	—
95	D₄	—	376	8	no	—
96	C₂×C₄	#13094020	1	1	yes	characteristic
97	C₂×C₄	C₂×#10625334	1	2	yes	—
98	C₈	#13093965	1	8	yes	complemented
99	C₈	D₈×D₄₇	1	4	yes	—
100	C₂²	—	16	2	no	—
101	C₂²	—	16	4	no	—
102	C₂²	—	752	2	no	—
103	C₂²	—	752	4	no	—
104	C₂²	#15689499	1	1	yes	characteristic
105	C₄	#15689498	1	2	yes	—
106	C₄	C₂×#13093965	1	4	yes	—
107	C₂	—	32	2	no	—
108	C₂	—	1504	2	no	—
109	C₂	#20243002	1	1	yes	characteristic
110	C₂	#20245637	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20245637

C₂

yes

maximal, complemented

#20243003

C₂

yes

maximal, characteristic, complemented

C₈×C₄₇⋊C₁₆

C₂

yes

maximal, characteristic, complemented

C₄₇×#1712

C₂

yes

maximal, characteristic, complemented

C₂×#13093965

—

#15689500

—

#15689500

C₂²

yes

—

C₄×C₄₇⋊C₁₆

C₂²

yes

—

C₄₇×#493

—

C₄₇×#493

C₂²

yes

—

C₈²×C₄₇

C₂²

yes

characteristic

C₂×D₃₇₆

—

C₂×D₃₇₆

—

#13094021

—

#13093965

—

C₂×C₄₇⋊C₁₆

D₄

yes

—

C₂×D₈×C₄₇

—

C₂×D₈×C₄₇

—

#127×C₄₇

—

C₄×C₈×C₄₇

D₄

yes

—

C₄×C₈×C₄₇

C₂³

yes

characteristic

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

D₈×C₄₇

—

D₈×C₄₇

—

C₂×C₈×C₄₇

D₈

yes

—

C₂×C₈×C₄₇

C₂×D₄

yes

—

C₄²×C₄₇

C₂×D₄

yes

characteristic

D₃₇₆

—

D₃₇₆

—

C₄₇⋊C₁₆

D₈

yes

complemented

C₂³×C₄₇

—

C₂²×D₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

#127

yes

characteristic

C₂×C₄×C₄₇

C₂×D₈

yes

—

D₁₈₈

—

D₁₈₈

—

C₈×C₄₇

D₁₆

yes

complemented

C₈×C₄₇

C₂×D₈

yes

—

#12515

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#493

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

#493

yes

—

C₄×C₄₇

C₂×D₁₆

yes

—

#2246

—

#1712

—

#1712

D₄₇

yes

characteristic, complemented

C₈×C₁₆

—

C₂×C₄₇

—

C₂×C₄₇

#1712

yes

characteristic

C₂×C₄₇

#2246

yes

—

D₄₇

—

C₂×D₁₆

—

—

—

—

C₂×D₄₇

yes

—

C₄×C₁₆

—

C₈²

C₂×D₄₇

yes

characteristic

C₄₇

#12515

yes

characteristic, complemented, Sylow

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

188

—

#127

—

#127

—

D₁₆

—

188

—

C₂×C₁₆

—

C₄×C₈

C₂²⋊D₄₇

yes

—

C₄×C₈

C₂²×D₄₇

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

188

—

C₂×D₄

—

188

—

D₈

—

D₈

—

D₈

—

376

—

D₈

—

188

—

C₂×C₈

#10625334

yes

—

C₂×C₈

D₄×D₄₇

yes

—

C₄²

C₂×C₂²⋊D₄₇

yes

characteristic

C₁₆

—

C₂³

—

C₂³

—

376

—

D₄

—

D₄

—

D₄

—

376

—

D₄

—

376

—

C₂×C₄

#13094020

yes

characteristic

C₂×C₄

C₂×#10625334

yes

—

C₈

#13093965

yes

complemented

C₈

D₈×D₄₇

yes

—

100

C₂²

—

101

C₂²

—

102

C₂²

—

752

—

103

C₂²

—

752

—

104

C₂²

#15689499

yes

characteristic

105

C₄

#15689498

yes

—

106

C₄

C₂×#13093965

yes

—

107

C₂

—

108

C₂

—

1504

—

109

C₂

#20243002

yes

characteristic

110

C₂

#20245637

yes

—

	subgroup	G/H
center	C₂²	#15689499
commutator, cosocle	C₄×C₈×C₄₇	C₂³
Frattini	C₄×C₈	C₂²×D₄₇
Fitting	C₄₇×#1712	C₂
socle	C₂²×C₄₇	#493
hypercenter	#1712	D₄₇
hypocenter	C₄₇	#12515
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

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. #20245637	C₂	semidirect
2. #20243003	C₂	semidirect
3. C₈×C₄₇⋊C₁₆	C₂	semidirect
4. C₄₇×#1712	C₂	semidirect
33. C₄₇⋊C₁₆	D₈	semidirect
42. C₈×C₄₇	D₁₆	semidirect
54. #1712	D₄₇	semidirect
67. C₄₇	#12515	semidirect
98. C₈	#13093965	semidirect

H ≅ G/N

type

1. #20245637

C₂

semidirect

2. #20243003

C₂

semidirect

3. C₈×C₄₇⋊C₁₆

C₂

semidirect

4. C₄₇×#1712

C₂

semidirect

33. C₄₇⋊C₁₆

D₈

semidirect

42. C₈×C₄₇

D₁₆

semidirect

54. #1712

D₄₇

semidirect

67. C₄₇

#12515

semidirect

98. C₈

#13093965

semidirect

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

subgroup	index
#20245637	2
#20243003	2
C₈×C₄₇⋊C₁₆	2
C₄₇×#1712	2
#12515	47

subgroup

index

#20245637

#20243003

C₈×C₄₇⋊C₁₆

C₄₇×#1712

#12515

Subgroups — #26130989

Conjugacy classes of subgroups

Complements

Maximal subgroups