Group #22550431 — subgroups

Subgroups — #22550431

Order 8128.

222 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 111 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	#16861209	C₂	1	1	yes	maximal, characteristic, complemented
2	#16861200	C₂	1	2	yes	maximal, complemented
3	#16861189	C₂	1	2	yes	maximal
4	#16861185	C₂	1	2	yes	maximal
5	Q₈×Dic₁₂₇	C₂	1	2	yes	maximal, complemented
6	#16861174	C₂	1	2	yes	maximal, complemented
7	#16861167	C₂	1	2	yes	maximal
8	C₄²×D₁₂₇	C₂	1	1	yes	maximal, characteristic
9	C₄⋊Q₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Dic₂₅₄	C₂²	1	2	yes	—
11	C₂×C₄×D₁₂₇	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₁₂₇	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₁₂₇	C₂²	1	1	yes	characteristic
14	C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
15	C₄⋊Dic₁₂₇	C₂²	1	4	yes	—
16	C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
17	Dic₁₂₇⋊C₄	C₂²	1	4	yes	—
18	(C₂×D₁₂₇)⋊C₄	C₂²	1	4	yes	—
19	C₄×Dic₁₂₇	C₂²	1	1	yes	characteristic
20	C₄×Dic₁₂₇	C₂²	1	1	yes	characteristic
21	C₄×Dic₁₂₇	C₂²	1	1	yes	characteristic
22	C₄×Dic₁₂₇	C₂²	1	4	yes	—
23	C₂×Q₈×C₁₂₇	C₂²	1	2	yes	—
24	C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
25	C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
26	C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
27	C₂²×D₁₂₇	C₂³	1	1	yes	characteristic
28	Dic₂₅₄	—	2	4	no	—
29	C₂×Dic₁₂₇	C₂³	1	1	yes	characteristic
30	C₂×Dic₁₂₇	C₂³	1	1	yes	characteristic
31	C₂×Dic₁₂₇	C₂³	1	4	yes	—
32	C₂×Dic₁₂₇	C₂³	1	1	yes	characteristic
33	C₄×D₁₂₇	—	2	2	no	—
34	C₄×D₁₂₇	—	2	2	no	—
35	C₄×D₁₂₇	Q₈	1	4	yes	complemented
36	Q₈×C₁₂₇	—	2	4	no	—
37	C₂×C₄×C₁₂₇	C₂³	1	4	yes	—
38	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
39	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
40	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
41	C₂²×C₁₂₇	C₂⁴	1	1	yes	characteristic
42	C₂×D₁₂₇	—	2	1	no	—
43	C₂×D₁₂₇	—	2	1	no	—
44	C₂×D₁₂₇	C₂×Q₈	1	2	yes	—
45	C₄×C₁₂₇	—	2	4	no	—
46	C₄×C₁₂₇	C₂×Q₈	1	2	yes	—
47	C₄×C₁₂₇	#41	1	2	yes	—
48	C₄×C₁₂₇	#41	1	2	yes	—
49	Dic₁₂₇	—	2	1	no	—
50	Dic₁₂₇	—	2	1	no	—
51	Dic₁₂₇	—	2	4	no	—
52	Dic₁₂₇	C₂×Q₈	1	2	yes	—
53	C₂×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
54	C₂×C₁₂₇	C₂×#41	1	1	yes	characteristic
55	C₂×C₁₂₇	C₂×#41	1	1	yes	characteristic
56	D₁₂₇	—	2	2	no	—
57	C₁₂₇	#533	1	1	yes	characteristic, complemented, Sylow
58	#533	—	127	1	no	maximal, Sylow
59	C₄⋊Q₈	—	127	1	no	—
60	C₄⋊Q₈	D₁₂₇	1	1	yes	characteristic, complemented
61	#125	—	127	2	no	—
62	C₂²⋊Q₈	—	127	2	no	—
63	C₂²⋊Q₈	—	127	2	no	—
64	C₄×Q₈	—	127	2	no	—
65	C₄×Q₈	—	127	2	no	—
66	#117	—	127	2	no	—
67	C₂×C₄²	—	127	1	no	—
68	C₂×Q₈	—	127	2	no	—
69	C₂×Q₈	C₂×D₁₂₇	1	2	yes	—
70	C₂²×C₄	—	127	1	no	—
71	C₂²×C₄	—	127	1	no	—
72	C₂²×C₄	—	127	1	no	—
73	C₄⋊C₄	—	127	4	no	—
74	C₄⋊C₄	—	127	2	no	—
75	C₄⋊C₄	—	127	4	no	—
76	C₄⋊C₄	—	127	2	no	—
77	C₄⋊C₄	C₂×D₁₂₇	1	2	yes	—
78	C₄⋊C₄	C₂×D₁₂₇	1	2	yes	—
79	C₂²⋊C₄	—	127	4	no	—
80	C₄²	—	127	1	no	—
81	C₄²	—	127	1	no	—
82	C₄²	—	127	1	no	—
83	C₄²	—	127	4	no	—
84	C₄²	C₂×D₁₂₇	1	1	yes	characteristic
85	C₂³	—	127	1	no	—
86	Q₈	—	2	4	no	—
87	Q₈	—	254	4	no	—
88	C₂×C₄	—	127	1	no	—
89	C₂×C₄	—	127	4	no	—
90	C₂×C₄	—	127	1	no	—
91	C₂×C₄	—	127	1	no	—
92	C₂×C₄	—	127	4	no	—
93	C₂×C₄	—	254	2	no	—
94	C₂×C₄	—	254	2	no	—
95	C₂×C₄	C₂²×D₁₂₇	1	4	yes	—
96	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
97	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
98	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
99	C₂²	—	127	2	no	—
100	C₂²	—	254	1	no	—
101	C₂²	—	254	1	no	—
102	C₂²	C₂³×D₁₂₇	1	1	yes	characteristic
103	C₄	—	2	4	no	—
104	C₄	—	127	2	no	—
105	C₄	—	254	1	no	—
106	C₄	—	254	1	no	—
107	C₄	—	254	4	no	—
108	C₄	Q₈×D₁₂₇	1	2	yes	—
109	C₄	#14523247	1	2	yes	—
110	C₄	#14523249	1	2	yes	—
111	C₂	—	254	2	no	—
112	C₂	C₂×Q₈×D₁₂₇	1	1	yes	characteristic
113	C₂	C₂×#14523247	1	1	yes	characteristic
114	C₂	C₂×#14523249	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16861209

C₂

yes

maximal, characteristic, complemented

#16861200

C₂

yes

maximal, complemented

#16861189

C₂

yes

maximal

#16861185

C₂

yes

maximal

Q₈×Dic₁₂₇

C₂

yes

maximal, complemented

#16861174

C₂

yes

maximal, complemented

#16861167

C₂

yes

maximal

C₄²×D₁₂₇

C₂

yes

maximal, characteristic

C₄⋊Q₈×C₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₅₄

C₂²

yes

—

C₂×C₄×D₁₂₇

C₂²

yes

characteristic

C₂×C₄×D₁₂₇

C₂²

yes

characteristic

C₂×C₄×D₁₂₇

C₂²

yes

characteristic

C₄⋊Dic₁₂₇

C₂²

yes

—

C₄⋊Dic₁₂₇

C₂²

yes

—

C₄⋊Dic₁₂₇

C₂²

yes

—

Dic₁₂₇⋊C₄

C₂²

yes

—

(C₂×D₁₂₇)⋊C₄

C₂²

yes

—

C₄×Dic₁₂₇

C₂²

yes

characteristic

C₄×Dic₁₂₇

C₂²

yes

characteristic

C₄×Dic₁₂₇

C₂²

yes

characteristic

C₄×Dic₁₂₇

C₂²

yes

—

C₂×Q₈×C₁₂₇

C₂²

yes

—

C₄⋊C₄×C₁₂₇

C₂²

yes

—

C₄⋊C₄×C₁₂₇

C₂²

yes

—

C₄²×C₁₂₇

C₂²

yes

characteristic

C₂²×D₁₂₇

C₂³

yes

characteristic

Dic₂₅₄

—

C₂×Dic₁₂₇

C₂³

yes

characteristic

C₂×Dic₁₂₇

C₂³

yes

characteristic

C₂×Dic₁₂₇

C₂³

yes

—

C₂×Dic₁₂₇

C₂³

yes

characteristic

C₄×D₁₂₇

—

C₄×D₁₂₇

—

C₄×D₁₂₇

Q₈

yes

complemented

Q₈×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂³

yes

—

C₂×C₄×C₁₂₇

C₂³

yes

characteristic

C₂×C₄×C₁₂₇

C₂³

yes

characteristic

C₂×C₄×C₁₂₇

C₂³

yes

characteristic

C₂²×C₁₂₇

C₂⁴

yes

characteristic

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

C₂×Q₈

yes

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂×Q₈

yes

—

C₄×C₁₂₇

#41

yes

—

C₄×C₁₂₇

#41

yes

—

Dic₁₂₇

—

Dic₁₂₇

—

Dic₁₂₇

—

Dic₁₂₇

C₂×Q₈

yes

—

C₂×C₁₂₇

C₂²×Q₈

yes

characteristic

C₂×C₁₂₇

C₂×#41

yes

characteristic

C₂×C₁₂₇

C₂×#41

yes

characteristic

D₁₂₇

—

C₁₂₇

#533

yes

characteristic, complemented, Sylow

#533

—

127

maximal, Sylow

C₄⋊Q₈

—

127

—

C₄⋊Q₈

D₁₂₇

yes

characteristic, complemented

#125

—

127

—

C₂²⋊Q₈

—

127

—

C₂²⋊Q₈

—

127

—

C₄×Q₈

—

127

—

C₄×Q₈

—

127

—

#117

—

127

—

C₂×C₄²

—

127

—

C₂×Q₈

—

127

—

C₂×Q₈

C₂×D₁₂₇

yes

—

C₂²×C₄

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

—

127

—

C₄⋊C₄

—

127

—

C₄⋊C₄

—

127

—

C₄⋊C₄

—

127

—

C₄⋊C₄

—

127

—

C₄⋊C₄

C₂×D₁₂₇

yes

—

C₄⋊C₄

C₂×D₁₂₇

yes

—

C₂²⋊C₄

—

127

—

C₄²

—

127

—

C₄²

—

127

—

C₄²

—

127

—

C₄²

—

127

—

C₄²

C₂×D₁₂₇

yes

characteristic

C₂³

—

127

—

Q₈

—

Q₈

—

254

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

254

—

C₂×C₄

—

254

—

C₂×C₄

C₂²×D₁₂₇

yes

—

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

C₂²

—

127

—

100

C₂²

—

254

—

101

C₂²

—

254

—

102

C₂²

C₂³×D₁₂₇

yes

characteristic

103

C₄

—

104

C₄

—

127

—

105

C₄

—

254

—

106

C₄

—

254

—

107

C₄

—

254

—

108

C₄

Q₈×D₁₂₇

yes

—

109

C₄

#14523247

yes

—

110

C₄

#14523249

yes

—

111

C₂

—

254

—

112

C₂

C₂×Q₈×D₁₂₇

yes

characteristic

113

C₂

C₂×#14523247

yes

characteristic

114

C₂

C₂×#14523249

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₂₇
commutator, socle	C₂²×C₁₂₇	C₂⁴
Fitting	C₄⋊Q₈×C₁₂₇	C₂
hypercenter	C₄⋊Q₈	D₁₂₇
hypocenter	C₁₂₇	#533
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₂₇

commutator, socle

C₂²×C₁₂₇

C₂⁴

Fitting

C₄⋊Q₈×C₁₂₇

C₂

hypercenter

C₄⋊Q₈

D₁₂₇

C₁₂₇

G ▷ C₂²×C₁₂₇ ▷ C₁₂₇

C₁ ◁ C₂² ◁ C₄⋊Q₈

N	H ≅ G/N	type
1. #16861209	C₂	semidirect
2. #16861200	C₂	semidirect
5. Q₈×Dic₁₂₇	C₂	semidirect
6. #16861174	C₂	semidirect
9. C₄⋊Q₈×C₁₂₇	C₂	semidirect
35. C₄×D₁₂₇	Q₈	semidirect
57. C₁₂₇	#533	semidirect
60. C₄⋊Q₈	D₁₂₇	semidirect

H ≅ G/N

type

1. #16861209

C₂

semidirect

2. #16861200

C₂

semidirect

5. Q₈×Dic₁₂₇

C₂

semidirect

6. #16861174

C₂

semidirect

9. C₄⋊Q₈×C₁₂₇

C₂

semidirect

35. C₄×D₁₂₇

Q₈

semidirect

57. C₁₂₇

#533

semidirect

60. C₄⋊Q₈

D₁₂₇

semidirect

Subgroups — #22550431

Conjugacy classes of subgroups

Complements