Group #26110055 — subgroups

Subgroups — #26110055

Order 12032.

422 conjugacy classes of subgroups; only the 227 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 227 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	#20243548	C₂	1	2	yes	maximal
2	#20243516	C₂	1	4	yes	maximal
3	C₂×C₈²×C₄₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#13093958	—	2	4	no	—
5	#15688993	C₂²	1	4	yes	—
6	#15688886	—	2	8	no	—
7	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
9	C₈²×C₄₇	C₄	1	4	yes	complemented
10	C₂×C₄⋊Dic₄₇	—	2	8	no	—
11	C₂×C₄⋊Dic₄₇	—	2	4	no	—
12	#13093958	—	4	8	no	—
13	C₂²×C₈×C₄₇	D₄	1	4	yes	—
14	C₂²×C₈×C₄₇	D₄	1	2	yes	—
15	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
16	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
17	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
18	C₄×C₈×C₄₇	Q₈	1	4	yes	—
19	C₄×C₈×C₄₇	Q₈	1	2	yes	—
20	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
21	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
22	C₂²×Dic₄₇	—	4	4	no	—
23	C₂×C₈×C₄₇	—	2	8	no	—
24	C₂×C₈×C₄₇	C₄⋊C₄	1	8	yes	—
25	C₂×C₈×C₄₇	C₄⋊C₄	1	4	yes	—
26	C₂×C₈×C₄₇	D₈	1	4	yes	—
27	C₂×C₈×C₄₇	Q₁₆	1	4	yes	—
28	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
29	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
30	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
31	C₄⋊Dic₄₇	—	4	8	no	—
32	C₄⋊Dic₄₇	—	4	16	no	—
33	C₂³×C₄₇	#127	1	1	yes	characteristic
34	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
36	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
37	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
38	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
39	C₂×C₄×C₄₇	C₂×Q₁₆	1	2	yes	—
40	C₂×C₄×C₄₇	#135	1	8	yes	—
41	C₂×Dic₄₇	—	8	4	no	—
42	C₂×Dic₄₇	—	8	8	no	—
43	C₈×C₄₇	—	2	8	no	—
44	C₈×C₄₇	#107	1	8	yes	—
45	#8816	—	47	1	no	maximal, Sylow
46	C₂²×C₄₇	#384	1	1	yes	characteristic
47	C₂²×C₄₇	#493	1	1	yes	characteristic
48	C₂²×C₄₇	#494	1	1	yes	characteristic
49	C₂²×C₄₇	#495	1	4	yes	—
50	C₄×C₄₇	C₂×#107	1	2	yes	—
51	C₄×C₄₇	#427	1	4	yes	—
52	C₄×C₄₇	#499	1	4	yes	—
53	C₄×C₄₇	#500	1	2	yes	—
54	Dic₄₇	—	16	4	no	—
55	#1845	—	47	2	no	—
56	#1834	—	47	4	no	—
57	C₂×C₈²	D₄₇	1	1	yes	characteristic
58	C₂×C₄₇	#1709	1	2	yes	—
59	C₂×C₄₇	#1710	1	2	yes	—
60	C₂×C₄₇	#1845	1	1	yes	characteristic
61	C₂×C₄₇	#1846	1	2	yes	—
62	C₂×#107	—	94	4	no	—
63	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
64	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
65	#384	—	47	4	no	—
66	#340	—	94	8	no	—
67	C₈²	Dic₄₇	1	4	yes	complemented
68	C₄₇	#8816	1	1	yes	characteristic, complemented, Sylow
69	C₂²×C₈	D₁₈₈	1	4	yes	—
70	C₂²×C₈	D₁₈₈	1	2	yes	—
71	C₂×C₄⋊C₄	—	94	8	no	—
72	C₂×C₄⋊C₄	—	94	4	no	—
73	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
74	#107	—	188	8	no	—
75	C₄×C₈	Dic₉₄	1	4	yes	—
76	C₄×C₈	Dic₉₄	1	2	yes	—
77	C₄×C₈	C₂×Dic₄₇	1	4	yes	—
78	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
79	C₂²×C₄	—	188	4	no	—
80	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
81	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
82	C₂×C₈	—	2	8	no	—
83	C₂×C₈	D₃₇₆	1	4	yes	—
84	C₂×C₈	Dic₁₈₈	1	4	yes	—
85	C₂×C₈	C₄⋊Dic₄₇	1	8	yes	—
86	C₂×C₈	C₄⋊Dic₄₇	1	4	yes	—
87	C₄⋊C₄	—	188	8	no	—
88	C₄⋊C₄	—	188	16	no	—
89	C₄²	C₂×Dic₉₄	1	1	yes	characteristic
90	C₄²	C₂×Dic₉₄	1	2	yes	—
91	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
92	C₂³	#13094021	1	1	yes	characteristic
93	C₂×C₄	—	376	4	no	—
94	C₂×C₄	—	376	8	no	—
95	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
96	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
97	C₂×C₄	#13094024	1	2	yes	—
98	C₂×C₄	#13094024	1	1	yes	characteristic
99	C₂×C₄	C₂×D₃₇₆	1	2	yes	—
100	C₂×C₄	C₂×Dic₁₈₈	1	2	yes	—
101	C₂×C₄	#13094046	1	8	yes	—
102	C₈	—	2	8	no	—
103	C₈	#13093958	1	8	yes	—
104	C₂²	#15688993	1	1	yes	characteristic
105	C₂²	#15689500	1	1	yes	characteristic
106	C₂²	#15689503	1	1	yes	characteristic
107	C₂²	#15689507	1	4	yes	—
108	C₄	—	752	4	no	—
109	C₄	C₂×#13093958	1	2	yes	—
110	C₄	#15689161	1	4	yes	—
111	C₄	#15689525	1	4	yes	—
112	C₄	#15689528	1	2	yes	—
113	C₂	#20242994	1	2	yes	—
114	C₂	#20242998	1	2	yes	—
115	C₂	#20243548	1	1	yes	characteristic
116	C₂	#20243552	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20243548

C₂

yes

maximal

#20243516

C₂

yes

maximal

C₂×C₈²×C₄₇

C₂

yes

maximal, characteristic

C₂×#13093958

—

#15688993

C₂²

yes

—

#15688886

—

C₂×C₄×C₈×C₄₇

C₂²

yes

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₈²×C₄₇

C₄

yes

complemented

C₂×C₄⋊Dic₄₇

—

C₂×C₄⋊Dic₄₇

—

#13093958

—

C₂²×C₈×C₄₇

D₄

yes

—

C₂²×C₈×C₄₇

D₄

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄×C₈×C₄₇

C₂×C₄

yes

—

C₄×C₈×C₄₇

C₂×C₄

yes

—

C₄×C₈×C₄₇

Q₈

yes

—

C₄×C₈×C₄₇

Q₈

yes

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×D₄

yes

—

C₂²×Dic₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₄⋊C₄

yes

—

C₂×C₈×C₄₇

C₄⋊C₄

yes

—

C₂×C₈×C₄₇

D₈

yes

—

C₂×C₈×C₄₇

Q₁₆

yes

—

C₄²×C₄₇

C₂²×C₄

yes

characteristic

C₄²×C₄₇

C₂×Q₈

yes

characteristic

C₄²×C₄₇

C₂×Q₈

yes

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₂³×C₄₇

#127

yes

characteristic

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₄₇

C₄⋊Q₈

yes

—

C₂×C₄×C₄₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₄₇

C₂×D₈

yes

—

C₂×C₄×C₄₇

C₂×Q₁₆

yes

—

C₂×C₄×C₄₇

#135

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₈×C₄₇

—

C₈×C₄₇

#107

yes

—

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

—

C₄×C₄₇

C₂×#107

yes

—

C₄×C₄₇

#427

yes

—

C₄×C₄₇

#499

yes

—

C₄×C₄₇

#500

yes

—

Dic₄₇

—

#1845

—

#1834

—

C₂×C₈²

D₄₇

yes

characteristic

C₂×C₄₇

#1709

yes

—

C₂×C₄₇

#1710

yes

—

C₂×C₄₇

#1845

yes

characteristic

C₂×C₄₇

#1846

yes

—

C₂×#107

—

C₂×C₄×C₈

C₂×D₄₇

yes

—

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

#384

—

#340

—

C₈²

Dic₄₇

yes

complemented

C₄₇

#8816

yes

characteristic, complemented, Sylow

C₂²×C₈

D₁₈₈

yes

—

C₂²×C₈

D₁₈₈

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂²×D₄₇

yes

characteristic

#107

—

188

—

C₄×C₈

Dic₉₄

yes

—

C₄×C₈

Dic₉₄

yes

—

C₄×C₈

C₂×Dic₄₇

yes

—

C₄×C₈

C₂×Dic₄₇

yes

—

C₂²×C₄

—

188

—

C₂²×C₄

C₂×D₁₈₈

yes

characteristic

C₂²×C₄

C₂×D₁₈₈

yes

—

C₂×C₈

—

C₂×C₈

D₃₇₆

yes

—

C₂×C₈

Dic₁₈₈

yes

—

C₂×C₈

C₄⋊Dic₄₇

yes

—

C₂×C₈

C₄⋊Dic₄₇

yes

—

C₄⋊C₄

—

188

—

C₄⋊C₄

—

188

—

C₄²

C₂×Dic₉₄

yes

characteristic

C₄²

C₂×Dic₉₄

yes

—

C₄²

C₂²×Dic₄₇

yes

characteristic

C₂³

#13094021

yes

characteristic

C₂×C₄

—

376

—

C₂×C₄

—

376

—

C₂×C₄

C₂×C₄⋊Dic₄₇

yes

characteristic

C₂×C₄

C₂×C₄⋊Dic₄₇

yes

—

C₂×C₄

#13094024

yes

—

C₂×C₄

#13094024

yes

characteristic

C₂×C₄

C₂×D₃₇₆

yes

—

100

C₂×C₄

C₂×Dic₁₈₈

yes

—

101

C₂×C₄

#13094046

yes

—

102

C₈

—

103

C₈

#13093958

yes

—

104

C₂²

#15688993

yes

characteristic

105

C₂²

#15689500

yes

characteristic

106

C₂²

#15689503

yes

characteristic

107

C₂²

#15689507

yes

—

108

C₄

—

752

—

109

C₄

C₂×#13093958

yes

—

110

C₄

#15689161

yes

—

111

C₄

#15689525

yes

—

112

C₄

#15689528

yes

—

113

C₂

#20242994

yes

—

114

C₂

#20242998

yes

—

115

C₂

#20243548

yes

characteristic

116

C₂

#20243552

yes

—

	subgroup	G/H
center	C₂³	#13094021
commutator	C₄²×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₂×C₈²×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	#127
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₂×C₈²	D₄₇
hypocenter	C₄₇	#8816
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈²

subgroup

G/H

center

C₂³

#13094021

commutator

C₄²×C₄₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₂×C₈²×C₄₇

C₂

socle, prime radical

C₂³×C₄₇

#127

cosocle

C₂×C₄²×C₄₇

C₂³

hypercenter

C₂×C₈²

D₄₇

hypocenter

C₄₇

#8816

LCS

G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈²

N	H ≅ G/N	type
9. C₈²×C₄₇	C₄	semidirect
67. C₈²	Dic₄₇	semidirect
68. C₄₇	#8816	semidirect

H ≅ G/N

type

9. C₈²×C₄₇

C₄

semidirect

67. C₈²

Dic₄₇

semidirect

68. C₄₇

#8816

semidirect

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

subgroup	index
#20243548	2
#20243516	2
C₂×C₈²×C₄₇	2
#8816	47

subgroup

index

#20243548

#20243516

C₂×C₈²×C₄₇

#8816

Subgroups — #26110055

Conjugacy classes of subgroups

Complements

Maximal subgroups