Group #22550019 — subgroups

Subgroups — #22550019

Order 8128.

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

normal subgroups only

This group has 162 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×#14523248	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14523236	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#14523233	C₂	1	1	yes	maximal, characteristic, complemented
4	#16861132	C₂	1	1	yes	maximal, characteristic, complemented
5	#16861129	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²⋊(C₁₂₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
7	#121×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#14523248	—	2	2	no	—
9	#14523248	—	2	2	no	—
10	C₂×Dic₂₅₄	C₂²	1	1	yes	characteristic
11	C₂×D₅₀₈	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊D₁₂₇	—	2	1	no	—
13	C₂×C₄×D₁₂₇	—	2	1	no	—
14	#14523236	—	2	2	no	—
15	#14523233	—	2	2	no	—
16	C₂×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×D₄×C₁₂₇	—	2	1	no	—
19	C₂×D₄×C₁₂₇	C₂²	1	1	yes	characteristic
20	C₂²×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
21	C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
22	C₂²⋊C₄×C₁₂₇	—	2	1	no	—
23	C₂²×D₁₂₇	—	2	1	no	—
24	Dic₂₅₄	—	2	1	no	—
25	Dic₂₅₄	D₄	1	2	yes	complemented
26	D₅₀₈	—	2	1	no	—
27	D₅₀₈	D₄	1	2	yes	complemented
28	C₂²⋊D₁₂₇	—	2	4	no	—
29	C₂×Dic₁₂₇	—	2	1	no	—
30	C₄×D₁₂₇	—	2	4	no	—
31	C₁₂₇⋊C₈	—	2	1	no	—
32	C₁₂₇⋊C₈	—	2	1	no	—
33	C₂³×C₁₂₇	—	2	1	no	—
34	C₂³×C₁₂₇	D₄	1	1	yes	characteristic
35	D₄×C₁₂₇	—	2	1	no	—
36	D₄×C₁₂₇	—	2	1	no	—
37	D₄×C₁₂₇	—	4	2	no	—
38	C₂×C₄×C₁₂₇	—	2	1	no	—
39	C₂×C₄×C₁₂₇	—	2	1	no	—
40	C₂×C₄×C₁₂₇	—	2	1	no	—
41	C₂×C₄×C₁₂₇	D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
43	C₂²×C₁₂₇	—	2	1	no	—
44	C₂²×C₁₂₇	—	2	1	no	—
45	C₂²×C₁₂₇	—	2	1	no	—
46	C₂²×C₁₂₇	—	4	1	no	—
47	C₂²×C₁₂₇	—	4	1	no	—
48	C₂²×C₁₂₇	—	4	1	no	—
49	C₂²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
50	C₂×D₁₂₇	—	2	2	no	—
51	C₂×D₁₂₇	—	4	1	no	—
52	C₂×D₁₂₇	—	4	1	no	—
53	C₄×C₁₂₇	—	2	1	no	—
54	C₄×C₁₂₇	—	4	1	no	—
55	C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
56	C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
57	Dic₁₂₇	—	2	2	no	—
58	C₂×C₁₂₇	—	4	1	no	—
59	C₂×C₁₂₇	—	8	1	no	—
60	C₂×C₁₂₇	#120	1	1	yes	characteristic
61	C₂×C₁₂₇	#135	1	1	yes	characteristic
62	C₂×C₁₂₇	AGL(1,ℤ/8)	1	1	yes	characteristic
63	D₁₂₇	—	4	2	no	—
64	C₁₂₇	#449	1	1	yes	characteristic, complemented, Sylow
65	#449	—	127	1	no	maximal, Sylow
66	C₂×#41	—	127	1	no	—
67	C₂×SD₁₆	—	127	1	no	—
68	C₂×D₈	—	127	1	no	—
69	#121	D₁₂₇	1	1	yes	characteristic, complemented
70	#103	—	127	1	no	—
71	#102	—	127	1	no	—
72	C₂²⋊C₈	—	127	1	no	—
73	#41	—	254	2	no	—
74	#41	—	254	2	no	—
75	C₂×Q₈	—	127	1	no	—
76	C₂×D₄	—	2	1	no	—
77	C₂×D₄	—	127	1	no	—
78	C₂×D₄	—	254	1	no	—
79	C₂×D₄	C₂×D₁₂₇	1	1	yes	characteristic
80	C₂²×C₄	—	254	1	no	—
81	C₂²×C₄	C₂×D₁₂₇	1	1	yes	characteristic
82	SD₁₆	—	254	2	no	—
83	D₈	—	254	2	no	—
84	C₂×C₈	—	127	1	no	—
85	C₂×C₈	—	127	1	no	—
86	C₄⋊C₄	C₂×D₁₂₇	1	1	yes	characteristic
87	C₂²⋊C₄	—	2	1	no	—
88	C₂³	—	2	1	no	—
89	C₂³	—	254	1	no	—
90	C₂³	C₂²⋊D₁₂₇	1	1	yes	characteristic
91	Q₈	—	127	2	no	—
92	Q₈	—	254	1	no	—
93	D₄	—	2	1	no	—
94	D₄	—	2	1	no	—
95	D₄	—	4	2	no	—
96	D₄	—	127	2	no	—
97	D₄	—	254	4	no	—
98	D₄	—	254	1	no	—
99	C₂×C₄	—	2	1	no	—
100	C₂×C₄	—	2	1	no	—
101	C₂×C₄	—	2	1	no	—
102	C₂×C₄	—	254	4	no	—
103	C₂×C₄	—	254	1	no	—
104	C₂×C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
105	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
106	C₈	—	254	1	no	—
107	C₈	—	254	1	no	—
108	C₂²	—	2	1	no	—
109	C₂²	—	2	1	no	—
110	C₂²	—	2	1	no	—
111	C₂²	—	4	1	no	—
112	C₂²	—	4	1	no	—
113	C₂²	—	4	1	no	—
114	C₂²	—	508	1	no	—
115	C₂²	—	508	1	no	—
116	C₂²	—	254	2	no	—
117	C₂²	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
118	C₄	—	2	1	no	—
119	C₄	—	4	1	no	—
120	C₄	—	254	2	no	—
121	C₄	D₄×D₁₂₇	1	1	yes	characteristic
122	C₄	D₄×D₁₂₇	1	1	yes	characteristic
123	C₂	—	4	1	no	—
124	C₂	—	8	1	no	—
125	C₂	—	508	2	no	—
126	C₂	#16861177	1	1	yes	characteristic
127	C₂	#16861229	1	1	yes	characteristic
128	C₂	#16861235	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14523248

C₂

yes

maximal, characteristic, complemented

C₂×#14523236

C₂

yes

maximal, characteristic, complemented

C₂×#14523233

C₂

yes

maximal, characteristic, complemented

#16861132

C₂

yes

maximal, characteristic, complemented

#16861129

C₂

yes

maximal, characteristic, complemented

C₂²⋊(C₁₂₇⋊C₈)

C₂

yes

maximal, characteristic, complemented

#121×C₁₂₇

C₂

yes

maximal, characteristic, complemented

#14523248

—

#14523248

—

C₂×Dic₂₅₄

C₂²

yes

characteristic

C₂×D₅₀₈

C₂²

yes

characteristic

C₂×C₂²⋊D₁₂₇

—

C₂×C₄×D₁₂₇

—

#14523236

—

#14523233

—

C₂×C₁₂₇⋊C₈

C₂²

yes

characteristic

C₂×C₁₂₇⋊C₈

C₂²

yes

characteristic

C₂×D₄×C₁₂₇

—

C₂×D₄×C₁₂₇

C₂²

yes

characteristic

C₂²×C₄×C₁₂₇

C₂²

yes

characteristic

C₄⋊C₄×C₁₂₇

C₂²

yes

characteristic

C₂²⋊C₄×C₁₂₇

—

C₂²×D₁₂₇

—

Dic₂₅₄

—

Dic₂₅₄

D₄

yes

complemented

D₅₀₈

—

D₅₀₈

D₄

yes

complemented

C₂²⋊D₁₂₇

—

C₂×Dic₁₂₇

—

C₄×D₁₂₇

—

C₁₂₇⋊C₈

—

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

—

C₂³×C₁₂₇

D₄

yes

characteristic

D₄×C₁₂₇

—

D₄×C₁₂₇

—

D₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

D₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂³

yes

characteristic

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₂×D₄

yes

characteristic

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₄×C₁₂₇

C₂×D₄

yes

characteristic

Dic₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

D₁₂₇

—

C₁₂₇

#449

yes

characteristic, complemented, Sylow

#449

—

127

maximal, Sylow

C₂×#41

—

127

—

C₂×SD₁₆

—

127

—

C₂×D₈

—

127

—

#121

D₁₂₇

yes

characteristic, complemented

#103

—

127

—

#102

—

127

—

C₂²⋊C₈

—

127

—

#41

—

254

—

#41

—

254

—

C₂×Q₈

—

127

—

C₂×D₄

—

C₂×D₄

—

127

—

C₂×D₄

—

254

—

C₂×D₄

C₂×D₁₂₇

yes

characteristic

C₂²×C₄

—

254

—

C₂²×C₄

C₂×D₁₂₇

yes

characteristic

SD₁₆

—

254

—

D₈

—

254

—

C₂×C₈

—

127

—

C₂×C₈

—

127

—

C₄⋊C₄

C₂×D₁₂₇

yes

characteristic

C₂²⋊C₄

—

C₂³

—

C₂³

—

254

—

C₂³

C₂²⋊D₁₂₇

yes

characteristic

Q₈

—

127

—

Q₈

—

254

—

D₄

—

D₄

—

D₄

—

D₄

—

127

—

D₄

—

254

—

D₄

—

254

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

254

—

103

C₂×C₄

—

254

—

104

C₂×C₄

C₂²⋊D₁₂₇

yes

characteristic

105

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

106

C₈

—

254

—

107

C₈

—

254

—

108

C₂²

—

109

C₂²

—

110

C₂²

—

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

—

508

—

115

C₂²

—

508

—

116

C₂²

—

254

—

117

C₂²

C₂×C₂²⋊D₁₂₇

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

254

—

121

C₄

D₄×D₁₂₇

yes

characteristic

122

C₄

D₄×D₁₂₇

yes

characteristic

123

C₂

—

124

C₂

—

125

C₂

—

508

—

126

C₂

#16861177

yes

characteristic

127

C₂

#16861229

yes

characteristic

128

C₂

#16861235

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₂₇
commutator	C₂×C₄×C₁₂₇	C₂³
Frattini	C₂×C₄	C₂²×D₁₂₇
Fitting	#121×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂×D₄
hypercenter	#121	D₁₂₇
hypocenter	C₁₂₇	#449
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #121

subgroup

G/H

center

C₂²

C₂×C₂²⋊D₁₂₇

commutator

C₂×C₄×C₁₂₇

C₂³

Frattini

C₂×C₄

C₂²×D₁₂₇

Fitting

#121×C₁₂₇

C₂

socle

C₂²×C₁₂₇

C₂×D₄

D₁₂₇

C₁₂₇

G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #121

N	H ≅ G/N	type
1. C₂×#14523248	C₂	semidirect
2. C₂×#14523236	C₂	semidirect
3. C₂×#14523233	C₂	semidirect
4. #16861132	C₂	semidirect
5. #16861129	C₂	semidirect
6. C₂²⋊(C₁₂₇⋊C₈)	C₂	semidirect
7. #121×C₁₂₇	C₂	semidirect
25. Dic₂₅₄	D₄	semidirect
27. D₅₀₈	D₄	semidirect
64. C₁₂₇	#449	semidirect
69. #121	D₁₂₇	semidirect

H ≅ G/N

type

1. C₂×#14523248

C₂

semidirect

2. C₂×#14523236

C₂

semidirect

3. C₂×#14523233

C₂

semidirect

4. #16861132

C₂

semidirect

5. #16861129

C₂

semidirect

6. C₂²⋊(C₁₂₇⋊C₈)

C₂

semidirect

7. #121×C₁₂₇

C₂

semidirect

25. Dic₂₅₄

D₄

semidirect

27. D₅₀₈

D₄

semidirect

64. C₁₂₇

#449

semidirect

69. #121

D₁₂₇

semidirect

Subgroups — #22550019

Conjugacy classes of subgroups

Complements