Group #22550505 — subgroups

Subgroups — #22550505

Order 8128.

206 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 103 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	#16861208	C₂	1	2	yes	maximal, complemented
2	#16861207	C₂	1	2	yes	maximal, complemented
3	#16861199	C₂	1	2	yes	maximal, complemented
4	#16861199	C₂	1	2	yes	maximal, complemented
5	#16861188	C₂	1	1	yes	maximal, characteristic
6	#16861186	C₂	1	2	yes	maximal
7	#16861165	C₂	1	2	yes	maximal
8	C₂×Dic₁₂₇⋊C₄	C₂	1	1	yes	maximal, characteristic
9	C₂²⋊Q₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
10	C₂×Dic₂₅₄	C₂²	1	2	yes	—
11	C₂×Dic₂₅₄	C₂²	1	1	yes	characteristic
12	C₂²×Dic₁₂₇	C₂²	1	1	yes	characteristic
13	C₂²×Dic₁₂₇	C₂²	1	1	yes	characteristic
14	C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
15	C₄⋊Dic₁₂₇	C₂²	1	1	yes	characteristic
16	Dic₁₂₇⋊C₄	C₂²	1	2	yes	—
17	Dic₁₂₇⋊C₄	C₂²	1	2	yes	—
18	Dic₁₂₇⋊C₄	C₂²	1	2	yes	—
19	Dic₁₂₇⋊C₄	C₂²	1	2	yes	—
20	Dic₁₂₇⋊C₄	C₂²	1	2	yes	—
21	Dic₁₂₇⋊C₄	C₂²	1	2	yes	—
22	Dic₁₂₇⋊C₄	C₂²	1	2	yes	—
23	C₂²⋊Dic₁₂₇	C₂²	1	2	yes	—
24	C₄×Dic₁₂₇	C₂²	1	2	yes	—
25	C₄×Dic₁₂₇	C₂²	1	2	yes	—
26	C₂×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
27	C₂²×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
28	C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
29	C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
30	C₂²⋊C₄×C₁₂₇	C₂²	1	2	yes	—
31	Dic₂₅₄	—	4	2	no	—
32	Dic₂₅₄	—	4	1	no	—
33	C₂×Dic₁₂₇	—	4	1	no	—
34	C₂×Dic₁₂₇	Q₈	1	4	yes	complemented
35	C₂×Dic₁₂₇	C₂³	1	2	yes	—
36	C₂×Dic₁₂₇	C₂³	1	2	yes	—
37	C₂×Dic₁₂₇	C₂³	1	2	yes	—
38	C₂×Dic₁₂₇	C₂³	1	2	yes	—
39	C₂³×C₁₂₇	C₂³	1	1	yes	characteristic
40	Q₈×C₁₂₇	—	4	1	no	—
41	C₂×C₄×C₁₂₇	—	4	1	no	—
42	C₂×C₄×C₁₂₇	C₂³	1	2	yes	—
43	C₂×C₄×C₁₂₇	C₂³	1	2	yes	—
44	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
45	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
46	C₂²×C₁₂₇	—	2	1	no	—
47	C₂²×C₁₂₇	—	2	1	no	—
48	C₂²×C₁₂₇	C₂×Q₈	1	2	yes	—
49	C₂²×C₁₂₇	C₂⁴	1	1	yes	characteristic
50	C₄×C₁₂₇	—	2	1	no	—
51	C₄×C₁₂₇	—	2	1	no	—
52	C₄×C₁₂₇	—	2	2	no	—
53	C₄×C₁₂₇	—	2	2	no	—
54	Dic₁₂₇	—	2	2	no	—
55	Dic₁₂₇	—	2	2	no	—
56	Dic₁₂₇	—	2	2	no	—
57	Dic₁₂₇	C₂×Q₈	1	4	yes	—
58	C₂×C₁₂₇	—	2	2	no	—
59	C₂×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
60	C₂×C₁₂₇	H(5,𝔽₂)	1	1	yes	characteristic
61	C₂×C₁₂₇	2₋¹⁺⁴	1	1	yes	characteristic
62	C₁₂₇	#544	1	1	yes	characteristic, complemented, Sylow
63	#544	—	127	1	no	maximal, Sylow
64	C₄⋊Q₈	—	127	2	no	—
65	C₄⋊Q₈	—	127	2	no	—
66	#125	—	127	2	no	—
67	#125	—	127	2	no	—
68	C₂²⋊Q₈	—	127	1	no	—
69	C₂²⋊Q₈	—	127	2	no	—
70	C₂²⋊Q₈	D₁₂₇	1	1	yes	characteristic
71	#117	—	127	2	no	—
72	C₂×C₄⋊C₄	—	127	1	no	—
73	C₂×Q₈	—	127	1	no	—
74	C₂×Q₈	—	127	2	no	—
75	C₂×Q₈	C₂×D₁₂₇	1	1	yes	characteristic
76	C₂²×C₄	—	127	1	no	—
77	C₂²×C₄	—	127	1	no	—
78	C₂²×C₄	C₂×D₁₂₇	1	1	yes	characteristic
79	C₄⋊C₄	—	127	2	no	—
80	C₄⋊C₄	—	127	2	no	—
81	C₄⋊C₄	—	127	2	no	—
82	C₄⋊C₄	—	127	2	no	—
83	C₄⋊C₄	—	127	2	no	—
84	C₄⋊C₄	—	127	2	no	—
85	C₄⋊C₄	—	127	2	no	—
86	C₄⋊C₄	—	127	1	no	—
87	C₄⋊C₄	—	127	2	no	—
88	C₄⋊C₄	C₂×D₁₂₇	1	1	yes	characteristic
89	C₄⋊C₄	C₂×D₁₂₇	1	2	yes	—
90	C₂²⋊C₄	—	127	2	no	—
91	C₂²⋊C₄	C₂×D₁₂₇	1	2	yes	—
92	C₄²	—	127	2	no	—
93	C₄²	—	127	2	no	—
94	C₂³	C₂²×D₁₂₇	1	1	yes	characteristic
95	Q₈	—	4	1	no	—
96	Q₈	—	508	1	no	—
97	Q₈	—	508	2	no	—
98	C₂×C₄	—	4	1	no	—
99	C₂×C₄	—	127	4	no	—
100	C₂×C₄	—	127	2	no	—
101	C₂×C₄	—	127	2	no	—
102	C₂×C₄	—	127	2	no	—
103	C₂×C₄	—	127	2	no	—
104	C₂×C₄	—	508	1	no	—
105	C₂×C₄	C₂²×D₁₂₇	1	2	yes	—
106	C₂×C₄	C₂²×D₁₂₇	1	2	yes	—
107	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
108	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
109	C₂²	—	2	1	no	—
110	C₂²	—	2	1	no	—
111	C₂²	Q₈×D₁₂₇	1	2	yes	—
112	C₂²	C₂³×D₁₂₇	1	1	yes	characteristic
113	C₄	—	2	1	no	—
114	C₄	—	2	1	no	—
115	C₄	—	2	2	no	—
116	C₄	—	2	2	no	—
117	C₄	—	127	4	no	—
118	C₄	—	254	2	no	—
119	C₄	—	254	2	no	—
120	C₄	—	254	2	no	—
121	C₂	—	2	2	no	—
122	C₂	C₂×Q₈×D₁₂₇	1	1	yes	characteristic
123	C₂	#16861254	1	1	yes	characteristic
124	C₂	#16861256	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16861208

C₂

yes

maximal, complemented

#16861207

C₂

yes

maximal, complemented

#16861199

C₂

yes

maximal, complemented

#16861199

C₂

yes

maximal, complemented

#16861188

C₂

yes

maximal, characteristic

#16861186

C₂

yes

maximal

#16861165

C₂

yes

maximal

C₂×Dic₁₂₇⋊C₄

C₂

yes

maximal, characteristic

C₂²⋊Q₈×C₁₂₇

C₂

yes

maximal, characteristic

C₂×Dic₂₅₄

C₂²

yes

—

C₂×Dic₂₅₄

C₂²

yes

characteristic

C₂²×Dic₁₂₇

C₂²

yes

characteristic

C₂²×Dic₁₂₇

C₂²

yes

characteristic

C₄⋊Dic₁₂₇

C₂²

yes

—

C₄⋊Dic₁₂₇

C₂²

yes

characteristic

Dic₁₂₇⋊C₄

C₂²

yes

—

Dic₁₂₇⋊C₄

C₂²

yes

—

Dic₁₂₇⋊C₄

C₂²

yes

—

Dic₁₂₇⋊C₄

C₂²

yes

—

Dic₁₂₇⋊C₄

C₂²

yes

—

Dic₁₂₇⋊C₄

C₂²

yes

—

Dic₁₂₇⋊C₄

C₂²

yes

—

C₂²⋊Dic₁₂₇

C₂²

yes

—

C₄×Dic₁₂₇

C₂²

yes

—

C₄×Dic₁₂₇

C₂²

yes

—

C₂×Q₈×C₁₂₇

C₂²

yes

characteristic

C₂²×C₄×C₁₂₇

C₂²

yes

characteristic

C₄⋊C₄×C₁₂₇

C₂²

yes

characteristic

C₄⋊C₄×C₁₂₇

C₂²

yes

—

C₂²⋊C₄×C₁₂₇

C₂²

yes

—

Dic₂₅₄

—

Dic₂₅₄

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

Q₈

yes

complemented

C₂×Dic₁₂₇

C₂³

yes

—

C₂×Dic₁₂₇

C₂³

yes

—

C₂×Dic₁₂₇

C₂³

yes

—

C₂×Dic₁₂₇

C₂³

yes

—

C₂³×C₁₂₇

C₂³

yes

characteristic

Q₈×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂³

yes

—

C₂×C₄×C₁₂₇

C₂³

yes

—

C₂×C₄×C₁₂₇

C₂³

yes

characteristic

C₂×C₄×C₁₂₇

C₂³

yes

characteristic

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₂×Q₈

yes

—

C₂²×C₁₂₇

C₂⁴

yes

characteristic

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

Dic₁₂₇

—

Dic₁₂₇

—

Dic₁₂₇

—

Dic₁₂₇

C₂×Q₈

yes

—

C₂×C₁₂₇

—

C₂×C₁₂₇

C₂²×Q₈

yes

characteristic

C₂×C₁₂₇

H(5,𝔽₂)

yes

characteristic

C₂×C₁₂₇

2₋¹⁺⁴

yes

characteristic

C₁₂₇

#544

yes

characteristic, complemented, Sylow

#544

—

127

maximal, Sylow

C₄⋊Q₈

—

127

—

C₄⋊Q₈

—

127

—

#125

—

127

—

#125

—

127

—

C₂²⋊Q₈

—

127

—

C₂²⋊Q₈

—

127

—

C₂²⋊Q₈

D₁₂₇

yes

characteristic

#117

—

127

—

C₂×C₄⋊C₄

—

127

—

C₂×Q₈

—

127

—

C₂×Q₈

—

127

—

C₂×Q₈

C₂×D₁₂₇

yes

characteristic

C₂²×C₄

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

C₂×D₁₂₇

yes

characteristic

C₄⋊C₄

—

127

—

C₄⋊C₄

—

127

—

C₄⋊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

characteristic

C₄⋊C₄

C₂×D₁₂₇

yes

—

C₂²⋊C₄

—

127

—

C₂²⋊C₄

C₂×D₁₂₇

yes

—

C₄²

—

127

—

C₄²

—

127

—

C₂³

C₂²×D₁₂₇

yes

characteristic

Q₈

—

Q₈

—

508

—

Q₈

—

508

—

C₂×C₄

—

C₂×C₄

—

127

—

100

C₂×C₄

—

127

—

101

C₂×C₄

—

127

—

102

C₂×C₄

—

127

—

103

C₂×C₄

—

127

—

104

C₂×C₄

—

508

—

105

C₂×C₄

C₂²×D₁₂₇

yes

—

106

C₂×C₄

C₂²×D₁₂₇

yes

—

107

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

108

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

109

C₂²

—

110

C₂²

—

111

C₂²

Q₈×D₁₂₇

yes

—

112

C₂²

C₂³×D₁₂₇

yes

characteristic

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

—

127

—

118

C₄

—

254

—

119

C₄

—

254

—

120

C₄

—

254

—

121

C₂

—

122

C₂

C₂×Q₈×D₁₂₇

yes

characteristic

123

C₂

#16861254

yes

characteristic

124

C₂

#16861256

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₂₇
commutator, socle	C₂²×C₁₂₇	C₂⁴
Fitting	C₂²⋊Q₈×C₁₂₇	C₂
hypercenter	C₂²⋊Q₈	D₁₂₇
hypocenter	C₁₂₇	#544
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂²⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₂₇

commutator, socle

C₂²×C₁₂₇

C₂⁴

Fitting

C₂²⋊Q₈×C₁₂₇

C₂

hypercenter

C₂²⋊Q₈

D₁₂₇

C₁₂₇

G ▷ C₂²×C₁₂₇ ▷ C₁₂₇

C₁ ◁ C₂² ◁ C₂²⋊Q₈

N	H ≅ G/N	type
1. #16861208	C₂	semidirect
2. #16861207	C₂	semidirect
3. #16861199	C₂	semidirect
4. #16861199	C₂	semidirect
34. C₂×Dic₁₂₇	Q₈	semidirect
62. C₁₂₇	#544	semidirect

H ≅ G/N

type

1. #16861208

C₂

semidirect

2. #16861207

C₂

semidirect

3. #16861199

C₂

semidirect

4. #16861199

C₂

semidirect

34. C₂×Dic₁₂₇

Q₈

semidirect

62. C₁₂₇

#544

semidirect

Subgroups — #22550505

Conjugacy classes of subgroups

Complements