Group #22549949 — subgroups

Subgroups — #22549949

Order 8128.

124 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 124 conjugacy classes of subgroups — too many to draw. Only its 53 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×#14523231	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₈×D₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×D₅₀₈	C₂	1	1	yes	maximal, characteristic
4	C₄⋊(C₁₂₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
5	(C₂×D₁₂₇)⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	(C₂×D₁₂₇)⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	#97×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₅₀₈	C₄	1	1	yes	characteristic
9	C₂×C₄×D₁₂₇	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₁₂₇	C₂²	1	1	yes	characteristic
11	#14523231	—	2	2	no	—
12	C₂×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
14	C₈×D₁₂₇	—	2	2	no	—
15	C₄⋊Dic₁₂₇	C₄	1	1	yes	characteristic
16	(C₂×D₁₂₇)⋊C₄	C₄	1	1	yes	characteristic
17	(C₂×D₁₂₇)⋊C₄	C₄	1	1	yes	characteristic
18	C₂×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
19	C₂×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
20	C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
21	C₂²×D₁₂₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×D₁₂₇	C₂×C₄	1	1	yes	characteristic
23	D₅₀₈	—	2	2	no	—
24	C₂×Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
25	C₂×Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
26	C₄×D₁₂₇	—	2	1	no	—
27	C₄×D₁₂₇	—	2	1	no	—
28	C₄×D₁₂₇	—	2	1	no	—
29	C₄×D₁₂₇	—	2	1	no	—
30	C₁₂₇⋊C₈	—	2	1	no	—
31	C₁₂₇⋊C₈	—	2	1	no	—
32	C₂×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
35	C₈×C₁₂₇	—	2	1	no	—
36	C₈×C₁₂₇	D₄	1	2	yes	complemented
37	C₂²×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
38	C₂×D₁₂₇	—	2	1	no	—
39	C₂×D₁₂₇	—	2	1	no	—
40	C₂×D₁₂₇	—	2	1	no	—
41	C₂×D₁₂₇	—	2	1	no	—
42	C₂×D₁₂₇	—	2	1	no	—
43	C₂×D₁₂₇	M₁₆	1	2	yes	complemented
44	C₄×C₁₂₇	—	2	1	no	—
45	C₄×C₁₂₇	—	2	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	—	2	1	no	—
50	C₂×C₁₂₇	C₄×D₄	1	1	yes	characteristic
51	C₂×C₁₂₇	C₂×M₁₆	1	1	yes	characteristic
52	C₂×C₁₂₇	#131	1	1	yes	characteristic
53	D₁₂₇	—	2	2	no	—
54	D₁₂₇	—	4	1	no	—
55	C₁₂₇	#435	1	1	yes	characteristic, complemented, Sylow
56	#435	—	127	1	no	maximal, Sylow
57	C₂×M₁₆	—	127	1	no	—
58	C₂²×C₈	—	127	1	no	—
59	C₄×D₄	—	127	1	no	—
60	C₄⋊C₈	—	127	1	no	—
61	C₂²⋊C₈	—	127	1	no	—
62	C₂²⋊C₈	—	127	1	no	—
63	#97	D₁₂₇	1	1	yes	characteristic, complemented
64	C₂×D₄	—	127	1	no	—
65	C₂²×C₄	—	127	1	no	—
66	C₂²×C₄	—	127	1	no	—
67	M₁₆	—	254	2	no	—
68	C₂×C₈	—	127	1	no	—
69	C₂×C₈	—	127	1	no	—
70	C₂×C₈	—	254	2	no	—
71	C₂×C₈	C₂×D₁₂₇	1	1	yes	characteristic
72	C₂×C₈	C₂×D₁₂₇	1	1	yes	characteristic
73	C₄⋊C₄	—	127	1	no	—
74	C₂²⋊C₄	—	127	1	no	—
75	C₂²⋊C₄	—	127	1	no	—
76	C₄²	C₂×D₁₂₇	1	1	yes	characteristic
77	C₂³	—	127	1	no	—
78	C₂³	—	127	1	no	—
79	D₄	—	254	2	no	—
80	C₂×C₄	—	127	1	no	—
81	C₂×C₄	—	127	1	no	—
82	C₂×C₄	—	254	1	no	—
83	C₂×C₄	—	254	1	no	—
84	C₂×C₄	—	254	1	no	—
85	C₂×C₄	—	254	1	no	—
86	C₂×C₄	C₄×D₁₂₇	1	1	yes	characteristic
87	C₂×C₄	C₄×D₁₂₇	1	1	yes	characteristic
88	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
89	C₈	—	2	1	no	—
90	C₈	—	254	1	no	—
91	C₈	—	254	1	no	—
92	C₈	D₅₀₈	1	2	yes	complemented
93	C₂²	—	127	2	no	—
94	C₂²	—	254	1	no	—
95	C₂²	—	254	1	no	—
96	C₂²	—	254	1	no	—
97	C₂²	—	254	1	no	—
98	C₂²	—	254	1	no	—
99	C₂²	C₂×C₄×D₁₂₇	1	1	yes	characteristic
100	C₄	—	2	1	no	—
101	C₄	—	2	1	no	—
102	C₄	—	254	1	no	—
103	C₄	—	254	1	no	—
104	C₄	C₂×D₅₀₈	1	1	yes	characteristic
105	C₄	#14523248	1	1	yes	characteristic
106	C₂	—	508	1	no	—
107	C₂	—	254	2	no	—
108	C₂	C₄×D₅₀₈	1	1	yes	characteristic
109	C₂	M₁₆×D₁₂₇	1	1	yes	characteristic
110	C₂	#16861217	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14523231

C₂

yes

maximal, characteristic, complemented

C₂×C₈×D₁₂₇

C₂

yes

maximal, characteristic, complemented

C₄×D₅₀₈

C₂

yes

maximal, characteristic

C₄⋊(C₁₂₇⋊C₈)

C₂

yes

maximal, characteristic, complemented

(C₂×D₁₂₇)⋊C₈

C₂

yes

maximal, characteristic, complemented

(C₂×D₁₂₇)⋊C₈

C₂

yes

maximal, characteristic, complemented

#97×C₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂×D₅₀₈

C₄

yes

characteristic

C₂×C₄×D₁₂₇

C₂²

yes

characteristic

C₂×C₄×D₁₂₇

C₂²

yes

characteristic

#14523231

—

C₂×C₁₂₇⋊C₈

C₂²

yes

characteristic

C₂×C₁₂₇⋊C₈

C₂²

yes

characteristic

C₈×D₁₂₇

—

C₄⋊Dic₁₂₇

C₄

yes

characteristic

(C₂×D₁₂₇)⋊C₄

C₄

yes

characteristic

(C₂×D₁₂₇)⋊C₄

C₄

yes

characteristic

C₂×C₈×C₁₂₇

C₂²

yes

characteristic

C₂×C₈×C₁₂₇

C₂²

yes

characteristic

C₄²×C₁₂₇

C₂²

yes

characteristic

C₂²×D₁₂₇

C₂×C₄

yes

characteristic

C₂²×D₁₂₇

C₂×C₄

yes

characteristic

D₅₀₈

—

C₂×Dic₁₂₇

C₂×C₄

yes

characteristic

C₂×Dic₁₂₇

C₂×C₄

yes

characteristic

C₄×D₁₂₇

—

C₄×D₁₂₇

—

C₄×D₁₂₇

—

C₄×D₁₂₇

—

C₁₂₇⋊C₈

—

C₁₂₇⋊C₈

—

C₂×C₄×C₁₂₇

C₂×C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂×C₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂³

yes

characteristic

C₈×C₁₂₇

—

C₈×C₁₂₇

D₄

yes

complemented

C₂²×C₁₂₇

C₂²×C₄

yes

characteristic

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

M₁₆

yes

complemented

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₄×C₁₂₇

#41

yes

characteristic

Dic₁₂₇

—

Dic₁₂₇

—

C₂×C₁₂₇

C₄×D₄

yes

characteristic

C₂×C₁₂₇

C₂×M₁₆

yes

characteristic

C₂×C₁₂₇

#131

yes

characteristic

D₁₂₇

—

D₁₂₇

—

C₁₂₇

#435

yes

characteristic, complemented, Sylow

#435

—

127

maximal, Sylow

C₂×M₁₆

—

127

—

C₂²×C₈

—

127

—

C₄×D₄

—

127

—

C₄⋊C₈

—

127

—

C₂²⋊C₈

—

127

—

C₂²⋊C₈

—

127

—

#97

D₁₂₇

yes

characteristic, complemented

C₂×D₄

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

—

127

—

M₁₆

—

254

—

C₂×C₈

—

127

—

C₂×C₈

—

127

—

C₂×C₈

—

254

—

C₂×C₈

C₂×D₁₂₇

yes

characteristic

C₂×C₈

C₂×D₁₂₇

yes

characteristic

C₄⋊C₄

—

127

—

C₂²⋊C₄

—

127

—

C₂²⋊C₄

—

127

—

C₄²

C₂×D₁₂₇

yes

characteristic

C₂³

—

127

—

C₂³

—

127

—

D₄

—

254

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

254

—

C₂×C₄

—

254

—

C₂×C₄

—

254

—

C₂×C₄

—

254

—

C₂×C₄

C₄×D₁₂₇

yes

characteristic

C₂×C₄

C₄×D₁₂₇

yes

characteristic

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

C₈

—

C₈

—

254

—

C₈

—

254

—

C₈

D₅₀₈

yes

complemented

C₂²

—

127

—

C₂²

—

254

—

C₂²

—

254

—

C₂²

—

254

—

C₂²

—

254

—

C₂²

—

254

—

C₂²

C₂×C₄×D₁₂₇

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

—

254

—

103

C₄

—

254

—

104

C₄

C₂×D₅₀₈

yes

characteristic

105

C₄

#14523248

yes

characteristic

106

C₂

—

508

—

107

C₂

—

254

—

108

C₂

C₄×D₅₀₈

yes

characteristic

109

C₂

M₁₆×D₁₂₇

yes

characteristic

110

C₂

#16861217

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₁₂₇
commutator, socle	C₂²×C₁₂₇	C₂²×C₄
Fitting	#97×C₁₂₇	C₂
hypercenter	#97	D₁₂₇
hypocenter	C₁₂₇	#435
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ #97

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×D₁₂₇

commutator, socle

C₂²×C₁₂₇

C₂²×C₄

Fitting

#97×C₁₂₇

C₂

D₁₂₇

C₁₂₇

G ▷ C₂²×C₁₂₇ ▷ C₁₂₇

C₁ ◁ C₂×C₄ ◁ #97

N	H ≅ G/N	type
1. C₂×#14523231	C₂	semidirect
2. C₂×C₈×D₁₂₇	C₂	semidirect
4. C₄⋊(C₁₂₇⋊C₈)	C₂	semidirect
5. (C₂×D₁₂₇)⋊C₈	C₂	semidirect
6. (C₂×D₁₂₇)⋊C₈	C₂	semidirect
7. #97×C₁₂₇	C₂	semidirect
36. C₈×C₁₂₇	D₄	semidirect
43. C₂×D₁₂₇	M₁₆	semidirect
55. C₁₂₇	#435	semidirect
63. #97	D₁₂₇	semidirect
92. C₈	D₅₀₈	semidirect

H ≅ G/N

type

1. C₂×#14523231

C₂

semidirect

2. C₂×C₈×D₁₂₇

C₂

semidirect

4. C₄⋊(C₁₂₇⋊C₈)

C₂

semidirect

5. (C₂×D₁₂₇)⋊C₈

C₂

semidirect

6. (C₂×D₁₂₇)⋊C₈

C₂

semidirect

7. #97×C₁₂₇

C₂

semidirect

36. C₈×C₁₂₇

D₄

semidirect

43. C₂×D₁₂₇

M₁₆

semidirect

55. C₁₂₇

#435

semidirect

63. #97

D₁₂₇

semidirect

92. C₈

D₅₀₈

semidirect

Subgroups — #22549949

Conjugacy classes of subgroups

Complements