Group #22550106 — subgroups

Subgroups — #22550106

Order 8128.

124 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 124 conjugacy classes of subgroups — too many to draw. Only its 43 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×#14523237	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₈×D₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#16861189	C₂	1	1	yes	maximal, characteristic, complemented
4	#16861184	C₂	1	1	yes	maximal, characteristic
5	#16861132	C₂	1	1	yes	maximal, characteristic, complemented
6	#16861129	C₂	1	1	yes	maximal, characteristic, complemented
7	#106×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₂₅₄	C₂²	1	1	yes	characteristic
9	C₂×D₅₀₈	—	2	1	no	—
10	C₂×D₅₀₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×D₁₂₇	C₂²	1	1	yes	characteristic
12	#14523237	—	2	2	no	—
13	C₂×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
14	C₈×D₁₂₇	—	2	2	no	—
15	C₄⋊Dic₁₂₇	—	2	1	no	—
16	(C₂×D₁₂₇)⋊C₄	—	2	1	no	—
17	(C₂×D₁₂₇)⋊C₄	—	2	1	no	—
18	C₂×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
19	C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
20	C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
21	C₂²×D₁₂₇	—	2	1	no	—
22	C₂²×D₁₂₇	D₄	1	1	yes	characteristic
23	Dic₂₅₄	—	2	1	no	—
24	Dic₂₅₄	—	2	1	no	—
25	D₅₀₈	—	2	1	no	—
26	D₅₀₈	—	2	1	no	—
27	D₅₀₈	—	4	2	no	—
28	C₂×Dic₁₂₇	—	2	1	no	—
29	C₂×Dic₁₂₇	D₄	1	1	yes	characteristic
30	C₄×D₁₂₇	—	2	1	no	—
31	C₄×D₁₂₇	—	2	1	no	—
32	C₁₂₇⋊C₈	—	2	1	no	—
33	C₂×C₄×C₁₂₇	—	2	1	no	—
34	C₂×C₄×C₁₂₇	—	2	1	no	—
35	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
36	C₈×C₁₂₇	D₄	1	2	yes	complemented
37	C₂²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
38	C₂×D₁₂₇	—	2	1	no	—
39	C₂×D₁₂₇	—	2	1	no	—
40	C₂×D₁₂₇	—	4	1	no	—
41	C₂×D₁₂₇	—	4	1	no	—
42	C₂×D₁₂₇	—	4	1	no	—
43	C₂×D₁₂₇	SD₁₆	1	2	yes	complemented
44	C₄×C₁₂₇	—	4	1	no	—
45	C₄×C₁₂₇	—	4	1	no	—
46	C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
47	C₄×C₁₂₇	#41	1	1	yes	characteristic
48	Dic₁₂₇	—	2	1	no	—
49	Dic₁₂₇	—	4	1	no	—
50	C₂×C₁₂₇	#121	1	1	yes	characteristic
51	C₂×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
52	C₂×C₁₂₇	#135	1	1	yes	characteristic
53	D₁₂₇	—	2	2	no	—
54	D₁₂₇	—	8	1	no	—
55	C₁₂₇	#465	1	1	yes	characteristic, complemented, Sylow
56	#465	—	127	1	no	maximal, Sylow
57	C₂×SD₁₆	—	127	1	no	—
58	C₂²×C₈	—	127	1	no	—
59	C₂²⋊Q₈	—	127	1	no	—
60	#121	—	127	1	no	—
61	#106	D₁₂₇	1	1	yes	characteristic, complemented
62	#103	—	127	1	no	—
63	#102	—	127	1	no	—
64	C₂×Q₈	—	127	1	no	—
65	C₂×D₄	—	127	1	no	—
66	C₂×D₄	—	254	1	no	—
67	C₂²×C₄	—	127	1	no	—
68	SD₁₆	—	254	2	no	—
69	C₂×C₈	—	127	1	no	—
70	C₂×C₈	—	254	2	no	—
71	C₂×C₈	C₂×D₁₂₇	1	1	yes	characteristic
72	C₄⋊C₄	—	254	1	no	—
73	C₄⋊C₄	C₂×D₁₂₇	1	1	yes	characteristic
74	C₄⋊C₄	C₂×D₁₂₇	1	1	yes	characteristic
75	C₂²⋊C₄	—	254	1	no	—
76	C₂²⋊C₄	—	254	1	no	—
77	C₂³	—	127	1	no	—
78	C₂³	—	254	1	no	—
79	Q₈	—	254	1	no	—
80	Q₈	—	254	1	no	—
81	D₄	—	508	2	no	—
82	D₄	—	254	1	no	—
83	D₄	—	254	1	no	—
84	C₂×C₄	—	2	1	no	—
85	C₂×C₄	—	2	1	no	—
86	C₂×C₄	—	127	1	no	—
87	C₂×C₄	—	254	1	no	—
88	C₂×C₄	—	254	1	no	—
89	C₂×C₄	—	254	1	no	—
90	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
91	C₈	—	254	1	no	—
92	C₈	D₅₀₈	1	2	yes	complemented
93	C₂²	—	127	2	no	—
94	C₂²	—	508	1	no	—
95	C₂²	—	508	1	no	—
96	C₂²	—	508	1	no	—
97	C₂²	—	254	1	no	—
98	C₂²	—	254	1	no	—
99	C₂²	D₄×D₁₂₇	1	1	yes	characteristic
100	C₄	—	4	1	no	—
101	C₄	—	4	1	no	—
102	C₄	—	508	1	no	—
103	C₄	—	254	1	no	—
104	C₄	C₂×D₅₀₈	1	1	yes	characteristic
105	C₄	#14523249	1	1	yes	characteristic
106	C₂	—	1016	1	no	—
107	C₂	—	254	2	no	—
108	C₂	#16861184	1	1	yes	characteristic
109	C₂	SD₁₆×D₁₂₇	1	1	yes	characteristic
110	C₂	#16861230	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14523237

C₂

yes

maximal, characteristic, complemented

C₂×C₈×D₁₂₇

C₂

yes

maximal, characteristic, complemented

#16861189

C₂

yes

maximal, characteristic, complemented

#16861184

C₂

yes

maximal, characteristic

#16861132

C₂

yes

maximal, characteristic, complemented

#16861129

C₂

yes

maximal, characteristic, complemented

#106×C₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₅₄

C₂²

yes

characteristic

C₂×D₅₀₈

—

C₂×D₅₀₈

C₂²

yes

characteristic

C₂×C₄×D₁₂₇

C₂²

yes

characteristic

#14523237

—

C₂×C₁₂₇⋊C₈

C₂²

yes

characteristic

C₈×D₁₂₇

—

C₄⋊Dic₁₂₇

—

(C₂×D₁₂₇)⋊C₄

—

(C₂×D₁₂₇)⋊C₄

—

C₂×C₈×C₁₂₇

C₂²

yes

characteristic

C₄⋊C₄×C₁₂₇

C₂²

yes

characteristic

C₄⋊C₄×C₁₂₇

C₂²

yes

characteristic

C₂²×D₁₂₇

—

C₂²×D₁₂₇

D₄

yes

characteristic

Dic₂₅₄

—

Dic₂₅₄

—

D₅₀₈

—

D₅₀₈

—

D₅₀₈

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

D₄

yes

characteristic

C₄×D₁₂₇

—

C₄×D₁₂₇

—

C₁₂₇⋊C₈

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂³

yes

characteristic

C₈×C₁₂₇

D₄

yes

complemented

C₂²×C₁₂₇

C₂×D₄

yes

characteristic

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

SD₁₆

yes

complemented

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₄×C₁₂₇

#41

yes

characteristic

Dic₁₂₇

—

Dic₁₂₇

—

C₂×C₁₂₇

#121

yes

characteristic

C₂×C₁₂₇

C₂×SD₁₆

yes

characteristic

C₂×C₁₂₇

#135

yes

characteristic

D₁₂₇

—

D₁₂₇

—

C₁₂₇

#465

yes

characteristic, complemented, Sylow

#465

—

127

maximal, Sylow

C₂×SD₁₆

—

127

—

C₂²×C₈

—

127

—

C₂²⋊Q₈

—

127

—

#121

—

127

—

#106

D₁₂₇

yes

characteristic, complemented

#103

—

127

—

#102

—

127

—

C₂×Q₈

—

127

—

C₂×D₄

—

127

—

C₂×D₄

—

254

—

C₂²×C₄

—

127

—

SD₁₆

—

254

—

C₂×C₈

—

127

—

C₂×C₈

—

254

—

C₂×C₈

C₂×D₁₂₇

yes

characteristic

C₄⋊C₄

—

254

—

C₄⋊C₄

C₂×D₁₂₇

yes

characteristic

C₄⋊C₄

C₂×D₁₂₇

yes

characteristic

C₂²⋊C₄

—

254

—

C₂²⋊C₄

—

254

—

C₂³

—

127

—

C₂³

—

254

—

Q₈

—

254

—

Q₈

—

254

—

D₄

—

508

—

D₄

—

254

—

D₄

—

254

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

127

—

C₂×C₄

—

254

—

C₂×C₄

—

254

—

C₂×C₄

—

254

—

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

C₈

—

254

—

C₈

D₅₀₈

yes

complemented

C₂²

—

127

—

C₂²

—

508

—

C₂²

—

508

—

C₂²

—

508

—

C₂²

—

254

—

C₂²

—

254

—

C₂²

D₄×D₁₂₇

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

—

508

—

103

C₄

—

254

—

104

C₄

C₂×D₅₀₈

yes

characteristic

105

C₄

#14523249

yes

characteristic

106

C₂

—

1016

—

107

C₂

—

254

—

108

C₂

#16861184

yes

characteristic

109

C₂

SD₁₆×D₁₂₇

yes

characteristic

110

C₂

#16861230

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×D₁₂₇
commutator	C₂×C₄×C₁₂₇	C₂³
Frattini	C₂×C₄	C₂²×D₁₂₇
Fitting	#106×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂×D₄
hypercenter	#106	D₁₂₇
hypocenter	C₁₂₇	#465
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #106

subgroup

G/H

center

C₂²

D₄×D₁₂₇

commutator

C₂×C₄×C₁₂₇

C₂³

Frattini

C₂×C₄

C₂²×D₁₂₇

Fitting

#106×C₁₂₇

C₂

socle

C₂²×C₁₂₇

C₂×D₄

D₁₂₇

C₁₂₇

G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ #106

N	H ≅ G/N	type
1. C₂×#14523237	C₂	semidirect
2. C₂×C₈×D₁₂₇	C₂	semidirect
3. #16861189	C₂	semidirect
5. #16861132	C₂	semidirect
6. #16861129	C₂	semidirect
7. #106×C₁₂₇	C₂	semidirect
36. C₈×C₁₂₇	D₄	semidirect
43. C₂×D₁₂₇	SD₁₆	semidirect
55. C₁₂₇	#465	semidirect
61. #106	D₁₂₇	semidirect
92. C₈	D₅₀₈	semidirect

H ≅ G/N

type

1. C₂×#14523237

C₂

semidirect

2. C₂×C₈×D₁₂₇

C₂

semidirect

3. #16861189

C₂

semidirect

5. #16861132

C₂

semidirect

6. #16861129

C₂

semidirect

7. #106×C₁₂₇

C₂

semidirect

36. C₈×C₁₂₇

D₄

semidirect

43. C₂×D₁₂₇

SD₁₆

semidirect

55. C₁₂₇

#465

semidirect

61. #106

D₁₂₇

semidirect

92. C₈

D₅₀₈

semidirect

Subgroups — #22550106

Conjugacy classes of subgroups

Complements