Group #22550070 — subgroups

Subgroups — #22550070

Order 8128.

140 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 140 conjugacy classes of subgroups — too many to draw. Only its 41 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	C₂×D₁₀₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14523233	C₂	1	1	yes	maximal, characteristic, complemented
3	#16861205	C₂	1	1	yes	maximal, characteristic, complemented
4	#16861171	C₂	1	1	yes	maximal, characteristic, complemented
5	Dic₁₂₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	#16861129	C₂	1	1	yes	maximal, characteristic, complemented
7	#102×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₅₀₈	C₂²	1	1	yes	characteristic
9	C₂×D₅₀₈	C₂²	1	1	yes	characteristic
10	C₂×C₂²⋊D₁₂₇	—	2	1	no	—
11	C₂×C₂²⋊D₁₂₇	—	2	1	no	—
12	C₂×C₄×D₁₂₇	—	2	1	no	—
13	D₁₀₁₆	—	2	2	no	—
14	#14523233	—	2	2	no	—
15	C₂×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
16	(C₂×D₁₂₇)⋊C₄	—	2	1	no	—
17	C₄×Dic₁₂₇	C₂²	1	1	yes	characteristic
18	C₂×D₄×C₁₂₇	C₂²	1	1	yes	characteristic
19	C₂×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
20	C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
21	C₂²×D₁₂₇	—	2	1	no	—
22	C₂²×D₁₂₇	—	2	1	no	—
23	D₅₀₈	—	2	1	no	—
24	D₅₀₈	—	2	1	no	—
25	D₅₀₈	—	2	1	no	—
26	D₅₀₈	D₄	1	2	yes	complemented
27	C₂²⋊D₁₂₇	—	4	2	no	—
28	C₂²⋊D₁₂₇	—	4	2	no	—
29	C₂×Dic₁₂₇	D₄	1	1	yes	characteristic
30	C₂×Dic₁₂₇	D₄	1	1	yes	characteristic
31	C₄×D₁₂₇	—	4	2	no	—
32	C₁₂₇⋊C₈	—	2	1	no	—
33	C₂³×C₁₂₇	—	2	1	no	—
34	D₄×C₁₂₇	—	2	1	no	—
35	D₄×C₁₂₇	—	2	1	no	—
36	C₂×C₄×C₁₂₇	—	2	1	no	—
37	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
38	C₈×C₁₂₇	—	2	1	no	—
39	C₂²×C₁₂₇	—	4	1	no	—
40	C₂²×C₁₂₇	—	4	1	no	—
41	C₂²×C₁₂₇	—	4	1	no	—
42	C₂²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
43	C₂×D₁₂₇	—	2	2	no	—
44	C₂×D₁₂₇	—	4	1	no	—
45	C₂×D₁₂₇	—	4	1	no	—
46	C₂×D₁₂₇	—	4	1	no	—
47	C₂×D₁₂₇	—	4	1	no	—
48	C₂×D₁₂₇	—	4	1	no	—
49	C₄×C₁₂₇	—	4	1	no	—
50	C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
51	C₄×C₁₂₇	#41	1	1	yes	characteristic
52	Dic₁₂₇	—	2	1	no	—
53	Dic₁₂₇	D₈	1	2	yes	complemented
54	C₂×C₁₂₇	—	8	1	no	—
55	C₂×C₁₂₇	#121	1	1	yes	characteristic
56	C₂×C₁₂₇	C₂×D₈	1	1	yes	characteristic
57	C₂×C₁₂₇	AGL(1,ℤ/8)	1	1	yes	characteristic
58	D₁₂₇	—	4	2	no	—
59	D₁₂₇	—	8	1	no	—
60	C₁₂₇	#459	1	1	yes	characteristic, complemented, Sylow
61	#459	—	127	1	no	maximal, Sylow
62	C₂×D₈	—	127	1	no	—
63	C₂×D₈	—	127	1	no	—
64	#127	—	127	1	no	—
65	C₄×D₄	—	127	1	no	—
66	C₄⋊C₈	—	127	1	no	—
67	#102	—	127	1	no	—
68	#102	D₁₂₇	1	1	yes	characteristic, complemented
69	C₂×D₄	—	127	1	no	—
70	C₂×D₄	—	127	1	no	—
71	C₂×D₄	—	254	1	no	—
72	C₂×D₄	—	254	1	no	—
73	C₂×D₄	C₂×D₁₂₇	1	1	yes	characteristic
74	C₂²×C₄	—	254	1	no	—
75	D₈	—	254	2	no	—
76	D₈	—	254	2	no	—
77	C₂×C₈	—	127	1	no	—
78	C₂×C₈	C₂×D₁₂₇	1	1	yes	characteristic
79	C₄⋊C₄	C₂×D₁₂₇	1	1	yes	characteristic
80	C₂²⋊C₄	—	254	1	no	—
81	C₄²	—	127	1	no	—
82	C₂³	—	2	1	no	—
83	C₂³	—	254	1	no	—
84	C₂³	—	254	1	no	—
85	D₄	—	2	1	no	—
86	D₄	—	2	1	no	—
87	D₄	—	127	2	no	—
88	D₄	—	508	2	no	—
89	D₄	—	508	2	no	—
90	D₄	—	254	1	no	—
91	D₄	—	254	1	no	—
92	D₄	—	254	1	no	—
93	C₂×C₄	—	2	1	no	—
94	C₂×C₄	—	127	1	no	—
95	C₂×C₄	—	127	1	no	—
96	C₂×C₄	—	508	2	no	—
97	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
98	C₈	—	2	1	no	—
99	C₈	—	254	1	no	—
100	C₂²	—	4	1	no	—
101	C₂²	—	4	1	no	—
102	C₂²	—	4	1	no	—
103	C₂²	—	508	1	no	—
104	C₂²	—	508	1	no	—
105	C₂²	—	508	1	no	—
106	C₂²	—	508	1	no	—
107	C₂²	—	508	1	no	—
108	C₂²	—	254	2	no	—
109	C₂²	D₄×D₁₂₇	1	1	yes	characteristic
110	C₄	—	4	1	no	—
111	C₄	—	127	2	no	—
112	C₄	—	254	1	no	—
113	C₄	D₄×D₁₂₇	1	1	yes	characteristic
114	C₄	#14523248	1	1	yes	characteristic
115	C₂	—	8	1	no	—
116	C₂	—	1016	1	no	—
117	C₂	—	508	2	no	—
118	C₂	#16861182	1	1	yes	characteristic
119	C₂	D₈×D₁₂₇	1	1	yes	characteristic
120	C₂	#16861236	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×D₁₀₁₆

C₂

yes

maximal, characteristic, complemented

C₂×#14523233

C₂

yes

maximal, characteristic, complemented

#16861205

C₂

yes

maximal, characteristic, complemented

#16861171

C₂

yes

maximal, characteristic, complemented

Dic₁₂₇⋊C₈

C₂

yes

maximal, characteristic, complemented

#16861129

C₂

yes

maximal, characteristic, complemented

#102×C₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂×D₅₀₈

C₂²

yes

characteristic

C₂×D₅₀₈

C₂²

yes

characteristic

C₂×C₂²⋊D₁₂₇

—

C₂×C₂²⋊D₁₂₇

—

C₂×C₄×D₁₂₇

—

D₁₀₁₆

—

#14523233

—

C₂×C₁₂₇⋊C₈

C₂²

yes

characteristic

(C₂×D₁₂₇)⋊C₄

—

C₄×Dic₁₂₇

C₂²

yes

characteristic

C₂×D₄×C₁₂₇

C₂²

yes

characteristic

C₂×C₈×C₁₂₇

C₂²

yes

characteristic

C₄⋊C₄×C₁₂₇

C₂²

yes

characteristic

C₂²×D₁₂₇

—

C₂²×D₁₂₇

—

D₅₀₈

—

D₅₀₈

—

D₅₀₈

—

D₅₀₈

D₄

yes

complemented

C₂²⋊D₁₂₇

—

C₂²⋊D₁₂₇

—

C₂×Dic₁₂₇

D₄

yes

characteristic

C₂×Dic₁₂₇

D₄

yes

characteristic

C₄×D₁₂₇

—

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

—

D₄×C₁₂₇

—

D₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂³

yes

characteristic

C₈×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₂×D₄

yes

characteristic

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₄×C₁₂₇

#41

yes

characteristic

Dic₁₂₇

—

Dic₁₂₇

D₈

yes

complemented

C₂×C₁₂₇

—

C₂×C₁₂₇

#121

yes

characteristic

C₂×C₁₂₇

C₂×D₈

yes

characteristic

C₂×C₁₂₇

AGL(1,ℤ/8)

yes

characteristic

D₁₂₇

—

D₁₂₇

—

C₁₂₇

#459

yes

characteristic, complemented, Sylow

#459

—

127

maximal, Sylow

C₂×D₈

—

127

—

C₂×D₈

—

127

—

#127

—

127

—

C₄×D₄

—

127

—

C₄⋊C₈

—

127

—

#102

—

127

—

#102

D₁₂₇

yes

characteristic, complemented

C₂×D₄

—

127

—

C₂×D₄

—

127

—

C₂×D₄

—

254

—

C₂×D₄

—

254

—

C₂×D₄

C₂×D₁₂₇

yes

characteristic

C₂²×C₄

—

254

—

D₈

—

254

—

D₈

—

254

—

C₂×C₈

—

127

—

C₂×C₈

C₂×D₁₂₇

yes

characteristic

C₄⋊C₄

C₂×D₁₂₇

yes

characteristic

C₂²⋊C₄

—

254

—

C₄²

—

127

—

C₂³

—

C₂³

—

254

—

C₂³

—

254

—

D₄

—

D₄

—

D₄

—

127

—

D₄

—

508

—

D₄

—

508

—

D₄

—

254

—

D₄

—

254

—

D₄

—

254

—

C₂×C₄

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

508

—

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

C₈

—

C₈

—

254

—

100

C₂²

—

101

C₂²

—

102

C₂²

—

103

C₂²

—

508

—

104

C₂²

—

508

—

105

C₂²

—

508

—

106

C₂²

—

508

—

107

C₂²

—

508

—

108

C₂²

—

254

—

109

C₂²

D₄×D₁₂₇

yes

characteristic

110

C₄

—

111

C₄

—

127

—

112

C₄

—

254

—

113

C₄

D₄×D₁₂₇

yes

characteristic

114

C₄

#14523248

yes

characteristic

115

C₂

—

116

C₂

—

1016

—

117

C₂

—

508

—

118

C₂

#16861182

yes

characteristic

119

C₂

D₈×D₁₂₇

yes

characteristic

120

C₂

#16861236

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×D₁₂₇
commutator	C₂×C₄×C₁₂₇	C₂³
Frattini	C₂×C₄	C₂²×D₁₂₇
Fitting	#102×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂×D₄
hypercenter	#102	D₁₂₇
hypocenter	C₁₂₇	#459
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #102

subgroup

G/H

center

C₂²

D₄×D₁₂₇

commutator

C₂×C₄×C₁₂₇

C₂³

Frattini

C₂×C₄

C₂²×D₁₂₇

Fitting

#102×C₁₂₇

C₂

socle

C₂²×C₁₂₇

C₂×D₄

D₁₂₇

C₁₂₇

G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ #102

N	H ≅ G/N	type
1. C₂×D₁₀₁₆	C₂	semidirect
2. C₂×#14523233	C₂	semidirect
3. #16861205	C₂	semidirect
4. #16861171	C₂	semidirect
5. Dic₁₂₇⋊C₈	C₂	semidirect
6. #16861129	C₂	semidirect
7. #102×C₁₂₇	C₂	semidirect
26. D₅₀₈	D₄	semidirect
53. Dic₁₂₇	D₈	semidirect
60. C₁₂₇	#459	semidirect
68. #102	D₁₂₇	semidirect

H ≅ G/N

type

1. C₂×D₁₀₁₆

C₂

semidirect

2. C₂×#14523233

C₂

semidirect

3. #16861205

C₂

semidirect

4. #16861171

C₂

semidirect

5. Dic₁₂₇⋊C₈

C₂

semidirect

6. #16861129

C₂

semidirect

7. #102×C₁₂₇

C₂

semidirect

26. D₅₀₈

D₄

semidirect

53. Dic₁₂₇

D₈

semidirect

60. C₁₂₇

#459

semidirect

68. #102

D₁₂₇

semidirect

Subgroups — #22550070

Conjugacy classes of subgroups

Complements