Group #22550625 — subgroups

Subgroups — #22550625

Order 8128.

196 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 91 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	#16861207	C₂	1	1	yes	maximal, characteristic, complemented
2	#16861204	C₂	1	2	yes	maximal, complemented
3	#16861203	C₂	1	2	yes	maximal, complemented
4	#16861199	C₂	1	2	yes	maximal, complemented
5	#16861198	C₂	1	1	yes	maximal, characteristic
6	#16861194	C₂	1	2	yes	maximal
7	#16861187	C₂	1	2	yes	maximal, complemented
8	#16861185	C₂	1	2	yes	maximal, complemented
9	C₄⋊Q₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Dic₂₅₄	C₂²	1	1	yes	characteristic
11	C₂×D₅₀₈	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₁₂₇	C₂²	1	2	yes	—
13	C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
14	Dic₁₂₇⋊C₄	C₂²	1	2	yes	—
15	Dic₁₂₇⋊C₄	C₂²	1	2	yes	—
16	Dic₁₂₇⋊C₄	C₂²	1	2	yes	—
17	Dic₁₂₇⋊C₄	C₂²	1	2	yes	—
18	Dic₁₂₇⋊C₄	C₂²	1	2	yes	—
19	(C₂×D₁₂₇)⋊C₄	C₂²	1	2	yes	—
20	(C₂×D₁₂₇)⋊C₄	C₂²	1	2	yes	—
21	(C₂×D₁₂₇)⋊C₄	C₂²	1	2	yes	—
22	(C₂×D₁₂₇)⋊C₄	C₂²	1	2	yes	—
23	(C₂×D₁₂₇)⋊C₄	C₂²	1	2	yes	—
24	C₄×Dic₁₂₇	C₂²	1	1	yes	characteristic
25	C₄×Dic₁₂₇	C₂²	1	1	yes	characteristic
26	C₂×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
27	C₂×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
28	C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
29	C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
30	C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
31	C₂²×D₁₂₇	C₂³	1	2	yes	—
32	Dic₂₅₄	—	4	1	no	—
33	D₅₀₈	—	4	1	no	—
34	C₂×Dic₁₂₇	C₂³	1	2	yes	—
35	C₂×Dic₁₂₇	C₂³	1	2	yes	—
36	C₂×Dic₁₂₇	C₂³	1	2	yes	—
37	C₄×D₁₂₇	—	4	2	no	—
38	Q₈×C₁₂₇	—	4	1	no	—
39	Q₈×C₁₂₇	—	4	1	no	—
40	C₂×C₄×C₁₂₇	C₂³	1	2	yes	—
41	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
42	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
43	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
44	C₂×C₄×C₁₂₇	C₂³	1	2	yes	—
45	C₂²×C₁₂₇	C₂⁴	1	1	yes	characteristic
46	C₂×D₁₂₇	—	2	2	no	—
47	C₂×D₁₂₇	—	2	4	no	—
48	C₄×C₁₂₇	—	2	1	no	—
49	C₄×C₁₂₇	—	2	2	no	—
50	C₄×C₁₂₇	—	2	1	no	—
51	C₄×C₁₂₇	—	2	1	no	—
52	C₄×C₁₂₇	—	2	2	no	—
53	Dic₁₂₇	—	2	2	no	—
54	Dic₁₂₇	—	2	2	no	—
55	Dic₁₂₇	—	2	2	no	—
56	C₂×C₁₂₇	H(5,𝔽₂)	1	1	yes	characteristic
57	C₂×C₁₂₇	2₋¹⁺⁴	1	2	yes	—
58	D₁₂₇	—	4	2	no	—
59	C₁₂₇	#563	1	1	yes	characteristic, complemented, Sylow
60	#563	—	127	1	no	maximal, Sylow
61	C₄⋊Q₈	—	127	1	no	—
62	C₄⋊Q₈	D₁₂₇	1	1	yes	characteristic, complemented
63	#126	—	127	2	no	—
64	#126	—	127	2	no	—
65	#125	—	127	2	no	—
66	#124	—	127	1	no	—
67	#123	—	127	2	no	—
68	C₂²⋊Q₈	—	127	2	no	—
69	C₂²⋊Q₈	—	127	2	no	—
70	C₂×Q₈	—	127	1	no	—
71	C₂×Q₈	C₂×D₁₂₇	1	1	yes	characteristic
72	C₂×Q₈	C₂×D₁₂₇	1	1	yes	characteristic
73	C₂×D₄	—	127	1	no	—
74	C₂²×C₄	—	127	2	no	—
75	C₄⋊C₄	—	127	2	no	—
76	C₄⋊C₄	—	127	2	no	—
77	C₄⋊C₄	—	127	2	no	—
78	C₄⋊C₄	—	127	2	no	—
79	C₄⋊C₄	—	127	2	no	—
80	C₄⋊C₄	—	127	2	no	—
81	C₄⋊C₄	C₂×D₁₂₇	1	2	yes	—
82	C₄⋊C₄	C₂×D₁₂₇	1	2	yes	—
83	C₂²⋊C₄	—	127	2	no	—
84	C₂²⋊C₄	—	127	2	no	—
85	C₂²⋊C₄	—	127	2	no	—
86	C₂²⋊C₄	—	127	2	no	—
87	C₂²⋊C₄	—	127	2	no	—
88	C₄²	—	127	1	no	—
89	C₄²	—	127	1	no	—
90	C₄²	C₂×D₁₂₇	1	1	yes	characteristic
91	C₂³	—	127	2	no	—
92	Q₈	—	4	1	no	—
93	Q₈	—	4	1	no	—
94	Q₈	—	508	1	no	—
95	D₄	—	508	1	no	—
96	C₂×C₄	—	127	2	no	—
97	C₂×C₄	—	127	2	no	—
98	C₂×C₄	—	127	2	no	—
99	C₂×C₄	—	508	2	no	—
100	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
101	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
102	C₂×C₄	C₂²×D₁₂₇	1	2	yes	—
103	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
104	C₂×C₄	C₂²×D₁₂₇	1	2	yes	—
105	C₂²	—	254	2	no	—
106	C₂²	—	254	4	no	—
107	C₂²	C₂³×D₁₂₇	1	1	yes	characteristic
108	C₄	—	2	2	no	—
109	C₄	—	2	1	no	—
110	C₄	—	2	1	no	—
111	C₄	—	2	1	no	—
112	C₄	—	2	2	no	—
113	C₄	—	254	2	no	—
114	C₄	—	254	2	no	—
115	C₄	—	254	2	no	—
116	C₂	—	508	2	no	—
117	C₂	#16861254	1	1	yes	characteristic
118	C₂	#16861257	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16861207

C₂

yes

maximal, characteristic, complemented

#16861204

C₂

yes

maximal, complemented

#16861203

C₂

yes

maximal, complemented

#16861199

C₂

yes

maximal, complemented

#16861198

C₂

yes

maximal, characteristic

#16861194

C₂

yes

maximal

#16861187

C₂

yes

maximal, complemented

#16861185

C₂

yes

maximal, complemented

C₄⋊Q₈×C₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₅₄

C₂²

yes

characteristic

C₂×D₅₀₈

C₂²

yes

characteristic

C₂×C₄×D₁₂₇

C₂²

yes

—

C₄⋊Dic₁₂₇

C₂²

yes

—

Dic₁₂₇⋊C₄

C₂²

yes

—

Dic₁₂₇⋊C₄

C₂²

yes

—

Dic₁₂₇⋊C₄

C₂²

yes

—

Dic₁₂₇⋊C₄

C₂²

yes

—

Dic₁₂₇⋊C₄

C₂²

yes

—

(C₂×D₁₂₇)⋊C₄

C₂²

yes

—

(C₂×D₁₂₇)⋊C₄

C₂²

yes

—

(C₂×D₁₂₇)⋊C₄

C₂²

yes

—

(C₂×D₁₂₇)⋊C₄

C₂²

yes

—

(C₂×D₁₂₇)⋊C₄

C₂²

yes

—

C₄×Dic₁₂₇

C₂²

yes

characteristic

C₄×Dic₁₂₇

C₂²

yes

characteristic

C₂×Q₈×C₁₂₇

C₂²

yes

characteristic

C₂×Q₈×C₁₂₇

C₂²

yes

characteristic

C₄⋊C₄×C₁₂₇

C₂²

yes

—

C₄⋊C₄×C₁₂₇

C₂²

yes

—

C₄²×C₁₂₇

C₂²

yes

characteristic

C₂²×D₁₂₇

C₂³

yes

—

Dic₂₅₄

—

D₅₀₈

—

C₂×Dic₁₂₇

C₂³

yes

—

C₂×Dic₁₂₇

C₂³

yes

—

C₂×Dic₁₂₇

C₂³

yes

—

C₄×D₁₂₇

—

Q₈×C₁₂₇

—

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

C₂×C₄×C₁₂₇

C₂³

yes

—

C₂²×C₁₂₇

C₂⁴

yes

characteristic

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

Dic₁₂₇

—

Dic₁₂₇

—

Dic₁₂₇

—

C₂×C₁₂₇

H(5,𝔽₂)

yes

characteristic

C₂×C₁₂₇

2₋¹⁺⁴

yes

—

D₁₂₇

—

C₁₂₇

#563

yes

characteristic, complemented, Sylow

#563

—

127

maximal, Sylow

C₄⋊Q₈

—

127

—

C₄⋊Q₈

D₁₂₇

yes

characteristic, complemented

#126

—

127

—

#126

—

127

—

#125

—

127

—

#124

—

127

—

#123

—

127

—

C₂²⋊Q₈

—

127

—

C₂²⋊Q₈

—

127

—

C₂×Q₈

—

127

—

C₂×Q₈

C₂×D₁₂₇

yes

characteristic

C₂×Q₈

C₂×D₁₂₇

yes

characteristic

C₂×D₄

—

127

—

C₂²×C₄

—

127

—

C₄⋊C₄

—

127

—

C₄⋊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

—

C₂²⋊C₄

—

127

—

C₂²⋊C₄

—

127

—

C₂²⋊C₄

—

127

—

C₂²⋊C₄

—

127

—

C₄²

—

127

—

C₄²

—

127

—

C₄²

C₂×D₁₂₇

yes

characteristic

C₂³

—

127

—

Q₈

—

Q₈

—

Q₈

—

508

—

D₄

—

508

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

508

—

100

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

101

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

102

C₂×C₄

C₂²×D₁₂₇

yes

—

103

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

104

C₂×C₄

C₂²×D₁₂₇

yes

—

105

C₂²

—

254

—

106

C₂²

—

254

—

107

C₂²

C₂³×D₁₂₇

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

—

254

—

114

C₄

—

254

—

115

C₄

—

254

—

116

C₂

—

508

—

117

C₂

#16861254

yes

characteristic

118

C₂

#16861257

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₂₇
commutator, socle	C₂²×C₁₂₇	C₂⁴
Fitting	C₄⋊Q₈×C₁₂₇	C₂
hypercenter	C₄⋊Q₈	D₁₂₇
hypocenter	C₁₂₇	#563
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. #16861207	C₂	semidirect
2. #16861204	C₂	semidirect
3. #16861203	C₂	semidirect
4. #16861199	C₂	semidirect
7. #16861187	C₂	semidirect
8. #16861185	C₂	semidirect
9. C₄⋊Q₈×C₁₂₇	C₂	semidirect
59. C₁₂₇	#563	semidirect
62. C₄⋊Q₈	D₁₂₇	semidirect

H ≅ G/N

type

1. #16861207

C₂

semidirect

2. #16861204

C₂

semidirect

3. #16861203

C₂

semidirect

4. #16861199

C₂

semidirect

7. #16861187

C₂

semidirect

8. #16861185

C₂

semidirect

9. C₄⋊Q₈×C₁₂₇

C₂

semidirect

59. C₁₂₇

#563

semidirect

62. C₄⋊Q₈

D₁₂₇

semidirect

Subgroups — #22550625

Conjugacy classes of subgroups

Complements