Group #28335175 — subgroups

Subgroups — #28335175

Order 13696.

332 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 85 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	#20292684	C₂	1	2	yes	maximal, complemented
2	#20292371	C₂	1	4	yes	maximal, complemented
3	C₂×#127×C₁₀₇	C₂	1	1	yes	maximal, characteristic
4	C₄⋊(C₁₀₇⋊C₈)	C₂²	1	4	yes	complemented
5	C₂²⋊(C₁₀₇⋊C₈)	—	2	2	no	—
6	#16812020	—	2	2	no	—
7	C₂²×D₄×C₁₀₇	—	2	2	no	—
8	C₂²×D₄×C₁₀₇	C₄	1	2	yes	—
9	#127×C₁₀₇	—	2	2	no	—
10	#127×C₁₀₇	C₄	1	2	yes	—
11	#127×C₁₀₇	C₂²	1	2	yes	—
12	C₂×C₄²×C₁₀₇	C₂²	1	1	yes	characteristic
13	C₂×C₁₀₇⋊C₈	—	2	4	no	—
14	C₂⁴×C₁₀₇	—	2	2	no	—
15	C₂×D₄×C₁₀₇	—	2	2	no	—
16	C₂×D₄×C₁₀₇	—	2	4	no	—
17	C₂×D₄×C₁₀₇	—	2	4	no	—
18	C₂×D₄×C₁₀₇	—	4	4	no	—
19	C₂×D₄×C₁₀₇	—	4	8	no	—
20	C₂×D₄×C₁₀₇	C₂×C₄	1	4	yes	—
21	C₂²×C₄×C₁₀₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₁₀₇	D₄	1	2	yes	—
23	C₄²×C₁₀₇	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₁₀₇	D₄	1	2	yes	—
25	C₄²×C₁₀₇	C₂³	1	1	yes	characteristic
26	C₂³×C₁₀₇	—	2	4	no	—
27	C₂³×C₁₀₇	—	4	8	no	—
28	C₂³×C₁₀₇	—	4	4	no	—
29	C₂³×C₁₀₇	C₂²⋊C₄	1	1	yes	characteristic
30	D₄×C₁₀₇	—	4	4	no	—
31	D₄×C₁₀₇	—	4	4	no	—
32	D₄×C₁₀₇	—	4	8	no	—
33	D₄×C₁₀₇	—	4	8	no	—
34	C₂×C₄×C₁₀₇	—	2	2	no	—
35	C₂×C₄×C₁₀₇	—	2	4	no	—
36	C₂×C₄×C₁₀₇	C₂²⋊C₄	1	2	yes	—
37	C₂×C₄×C₁₀₇	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₄×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₁₀₇	C₂×D₄	1	2	yes	—
40	C₁₀₇⋊C₈	—	4	4	no	—
41	C₂²×C₁₀₇	—	2	2	no	—
42	C₂²×C₁₀₇	—	4	4	no	—
43	C₂²×C₁₀₇	—	4	8	no	—
44	C₂²×C₁₀₇	—	8	8	no	—
45	C₂²×C₁₀₇	#100	1	2	yes	—
46	C₂²×C₁₀₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
47	C₄×C₁₀₇	—	2	1	no	—
48	C₄×C₁₀₇	—	2	1	no	—
49	C₄×C₁₀₇	—	2	2	no	—
50	C₄×C₁₀₇	AGL(1,ℤ/8)	1	4	yes	—
51	C₂×C₁₀₇	—	2	2	no	—
52	C₂×C₁₀₇	—	8	4	no	—
53	C₂×C₁₀₇	C₂×#100	1	1	yes	characteristic
54	C₂×C₁₀₇	#418	1	2	yes	—
55	#1555	—	107	1	no	maximal, Sylow
56	C₁₀₇	#1555	1	1	yes	characteristic, complemented, Sylow
57	C₂×#127	D₁₀₇	1	1	yes	characteristic
58	#432	—	107	2	no	—
59	#331	—	107	4	no	—
60	C₂²×D₄	—	2	2	no	—
61	C₂²×D₄	Dic₁₀₇	1	2	yes	—
62	#127	—	2	2	no	—
63	#127	Dic₁₀₇	1	2	yes	—
64	#127	C₂×D₁₀₇	1	2	yes	—
65	C₂×C₄²	C₂×D₁₀₇	1	1	yes	characteristic
66	C₄⋊C₈	—	107	4	no	—
67	C₂²⋊C₈	—	214	2	no	—
68	#97	—	214	2	no	—
69	C₂⁴	—	2	2	no	—
70	C₂×D₄	—	2	2	no	—
71	C₂×D₄	—	2	4	no	—
72	C₂×D₄	—	2	4	no	—
73	C₂×D₄	—	4	4	no	—
74	C₂×D₄	—	4	8	no	—
75	C₂×D₄	C₂×Dic₁₀₇	1	4	yes	—
76	C₂²×C₄	C₂×Dic₁₀₇	1	1	yes	characteristic
77	C₂²×C₄	C₂²⋊D₁₀₇	1	2	yes	—
78	C₂×C₈	—	214	4	no	—
79	C₄²	C₂×Dic₁₀₇	1	1	yes	characteristic
80	C₄²	C₂²⋊D₁₀₇	1	2	yes	—
81	C₄²	C₂²×D₁₀₇	1	1	yes	characteristic
82	C₂³	—	2	4	no	—
83	C₂³	—	4	8	no	—
84	C₂³	—	4	4	no	—
85	C₂³	C₂²⋊Dic₁₀₇	1	1	yes	characteristic
86	D₄	—	4	4	no	—
87	D₄	—	4	8	no	—
88	D₄	—	4	4	no	—
89	D₄	—	4	8	no	—
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	—	2	4	no	—
92	C₂×C₄	C₂²⋊Dic₁₀₇	1	2	yes	—
93	C₂×C₄	C₂²⋊Dic₁₀₇	1	1	yes	characteristic
94	C₂×C₄	C₂²×Dic₁₀₇	1	1	yes	characteristic
95	C₂×C₄	C₂×C₂²⋊D₁₀₇	1	2	yes	—
96	C₈	—	428	4	no	—
97	C₂²	—	2	2	no	—
98	C₂²	—	4	8	no	—
99	C₂²	—	4	4	no	—
100	C₂²	—	8	8	no	—
101	C₂²	#16812026	1	2	yes	—
102	C₂²	C₂×C₂²⋊Dic₁₀₇	1	1	yes	characteristic
103	C₄	—	2	2	no	—
104	C₄	—	2	1	no	—
105	C₄	—	2	1	no	—
106	C₄	#16812135	1	4	yes	—
107	C₂	—	2	2	no	—
108	C₂	—	8	4	no	—
109	C₂	C₂×#16812026	1	1	yes	characteristic
110	C₂	#20292632	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20292684

C₂

yes

maximal, complemented

#20292371

C₂

yes

maximal, complemented

C₂×#127×C₁₀₇

C₂

yes

maximal, characteristic

C₄⋊(C₁₀₇⋊C₈)

C₂²

yes

complemented

C₂²⋊(C₁₀₇⋊C₈)

—

#16812020

—

C₂²×D₄×C₁₀₇

—

C₂²×D₄×C₁₀₇

C₄

yes

—

#127×C₁₀₇

—

#127×C₁₀₇

C₄

yes

—

#127×C₁₀₇

C₂²

yes

—

C₂×C₄²×C₁₀₇

C₂²

yes

characteristic

C₂×C₁₀₇⋊C₈

—

C₂⁴×C₁₀₇

—

C₂×D₄×C₁₀₇

—

C₂×D₄×C₁₀₇

—

C₂×D₄×C₁₀₇

—

C₂×D₄×C₁₀₇

—

C₂×D₄×C₁₀₇

—

C₂×D₄×C₁₀₇

C₂×C₄

yes

—

C₂²×C₄×C₁₀₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₇

D₄

yes

—

C₄²×C₁₀₇

C₂×C₄

yes

characteristic

C₄²×C₁₀₇

D₄

yes

—

C₄²×C₁₀₇

C₂³

yes

characteristic

C₂³×C₁₀₇

—

C₂³×C₁₀₇

—

C₂³×C₁₀₇

—

C₂³×C₁₀₇

C₂²⋊C₄

yes

characteristic

D₄×C₁₀₇

—

D₄×C₁₀₇

—

D₄×C₁₀₇

—

D₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₀₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₇

C₂×D₄

yes

—

C₁₀₇⋊C₈

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

#100

yes

—

C₂²×C₁₀₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

AGL(1,ℤ/8)

yes

—

C₂×C₁₀₇

—

C₂×C₁₀₇

—

C₂×C₁₀₇

C₂×#100

yes

characteristic

C₂×C₁₀₇

#418

yes

—

#1555

—

107

maximal, Sylow

C₁₀₇

#1555

yes

characteristic, complemented, Sylow

C₂×#127

D₁₀₇

yes

characteristic

#432

—

107

—

#331

—

107

—

C₂²×D₄

—

C₂²×D₄

Dic₁₀₇

yes

—

#127

—

#127

Dic₁₀₇

yes

—

#127

C₂×D₁₀₇

yes

—

C₂×C₄²

C₂×D₁₀₇

yes

characteristic

C₄⋊C₈

—

107

—

C₂²⋊C₈

—

214

—

#97

—

214

—

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×Dic₁₀₇

yes

—

C₂²×C₄

C₂×Dic₁₀₇

yes

characteristic

C₂²×C₄

C₂²⋊D₁₀₇

yes

—

C₂×C₈

—

214

—

C₄²

C₂×Dic₁₀₇

yes

characteristic

C₄²

C₂²⋊D₁₀₇

yes

—

C₄²

C₂²×D₁₀₇

yes

characteristic

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂²⋊Dic₁₀₇

yes

characteristic

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊Dic₁₀₇

yes

—

C₂×C₄

C₂²⋊Dic₁₀₇

yes

characteristic

C₂×C₄

C₂²×Dic₁₀₇

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₁₀₇

yes

—

C₈

—

428

—

C₂²

—

C₂²

—

C₂²

—

100

C₂²

—

101

C₂²

#16812026

yes

—

102

C₂²

C₂×C₂²⋊Dic₁₀₇

yes

characteristic

103

C₄

—

104

C₄

—

105

C₄

—

106

C₄

#16812135

yes

—

107

C₂

—

108

C₂

—

109

C₂

C₂×#16812026

yes

characteristic

110

C₂

#20292632

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₁₀₇
commutator	C₂×C₄×C₁₀₇	C₂²×C₄
Frattini	C₄²	C₂²×D₁₀₇
Fitting, prime radical	C₂×#127×C₁₀₇	C₂
socle	C₂²×C₁₀₇	C₂×C₂²⋊C₄
cosocle	C₄²×C₁₀₇	C₂³
hypercenter	C₂×#127	D₁₀₇
hypocenter	C₁₀₇	#1555
LCS	G ▷ C₂×C₄×C₁₀₇ ▷ C₂²×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#127

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₁₀₇

commutator

C₂×C₄×C₁₀₇

C₂²×C₄

Frattini

C₄²

C₂²×D₁₀₇

Fitting, prime radical

C₂×#127×C₁₀₇

C₂

socle

C₂²×C₁₀₇

C₂×C₂²⋊C₄

cosocle

C₄²×C₁₀₇

C₂³

D₁₀₇

C₁₀₇

G ▷ C₂×C₄×C₁₀₇ ▷ C₂²×C₁₀₇ ▷ C₁₀₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#127

N	H ≅ G/N	type
1. #20292684	C₂	semidirect
2. #20292371	C₂	semidirect
4. C₄⋊(C₁₀₇⋊C₈)	C₂²	semidirect
56. C₁₀₇	#1555	semidirect

H ≅ G/N

type

1. #20292684

C₂

semidirect

2. #20292371

C₂

semidirect

4. C₄⋊(C₁₀₇⋊C₈)

C₂²

semidirect

56. C₁₀₇

#1555

semidirect

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

subgroup	index
#20292684	2
#20292371	2
C₂×#127×C₁₀₇	2
#1555	107

subgroup

index

#20292684

#20292371

C₂×#127×C₁₀₇

#1555

107

Subgroups — #28335175

Conjugacy classes of subgroups

Complements

Maximal subgroups