Group #22550028 — subgroups

Subgroups — #22550028

Order 8128.

188 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 45 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×D₄×D₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#14523237	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#14523236	C₂	1	1	yes	maximal, characteristic, complemented
4	#16861189	C₂	1	1	yes	maximal, characteristic, complemented
5	#16861129	C₂	1	1	yes	maximal, characteristic, complemented
6	(C₂×D₁₂₇)⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	#102×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×D₁₂₇	—	2	1	no	—
9	C₂×Dic₂₅₄	C₂²	1	1	yes	characteristic, complemented
10	C₂×D₅₀₈	C₂²	1	1	yes	characteristic
11	C₂×C₂²⋊D₁₂₇	—	2	1	no	—
12	D₄×D₁₂₇	—	2	2	no	—
13	D₄×D₁₂₇	—	2	2	no	—
14	C₂×C₄×D₁₂₇	C₂²	1	1	yes	characteristic
15	#14523237	—	2	2	no	—
16	#14523236	—	2	2	no	—
17	C₂×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic, complemented
18	C₄⋊Dic₁₂₇	—	2	1	no	—
19	(C₂×D₁₂₇)⋊C₄	—	2	1	no	—
20	C₂×D₄×C₁₂₇	C₂²	1	1	yes	characteristic
21	C₂×C₈×C₁₂₇	C₂²	1	1	yes	characteristic, complemented
22	C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic, complemented
23	C₂²×D₁₂₇	—	2	4	no	—
24	C₂²×D₁₂₇	—	2	1	no	—
25	C₂²×D₁₂₇	—	4	2	no	—
26	C₂²×D₁₂₇	—	4	2	no	—
27	C₂²×D₁₂₇	D₄	1	1	yes	characteristic
28	Dic₂₅₄	—	2	1	no	—
29	Dic₂₅₄	—	2	1	no	—
30	D₅₀₈	—	2	1	no	—
31	D₅₀₈	D₄	1	2	yes	—
32	C₂²⋊D₁₂₇	—	2	4	no	—
33	C₂×Dic₁₂₇	—	2	1	no	—
34	C₂×Dic₁₂₇	D₄	1	1	yes	characteristic
35	C₄×D₁₂₇	—	2	1	no	—
36	C₄×D₁₂₇	—	2	1	no	—
37	C₁₂₇⋊C₈	—	2	1	no	—
38	C₂³×C₁₂₇	—	2	1	no	—
39	D₄×C₁₂₇	—	2	1	no	—
40	D₄×C₁₂₇	D₄	1	2	yes	—
41	C₂×C₄×C₁₂₇	—	2	1	no	—
42	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
43	C₈×C₁₂₇	—	2	1	no	—
44	C₂²×C₁₂₇	—	2	2	no	—
45	C₂²×C₁₂₇	—	4	1	no	—
46	C₂²×C₁₂₇	—	4	1	no	—
47	C₂²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
48	C₂×D₁₂₇	—	2	1	no	—
49	C₂×D₁₂₇	—	2	1	no	—
50	C₂×D₁₂₇	—	2	2	no	—
51	C₂×D₁₂₇	—	4	8	no	—
52	C₂×D₁₂₇	—	4	1	no	—
53	C₂×D₁₂₇	—	4	1	no	—
54	C₂×D₁₂₇	SD₁₆	1	2	yes	complemented
55	C₄×C₁₂₇	—	4	1	no	—
56	C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
57	C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
58	Dic₁₂₇	—	2	1	no	—
59	Dic₁₂₇	—	4	1	no	—
60	C₂×C₁₂₇	—	4	2	no	—
61	C₂×C₁₂₇	#120	1	1	yes	characteristic
62	C₂×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
63	C₂×C₁₂₇	AGL(1,ℤ/8)	1	1	yes	characteristic
64	D₁₂₇	—	2	2	no	—
65	D₁₂₇	—	4	2	no	—
66	C₁₂₇	#450	1	1	yes	characteristic, complemented, Sylow
67	#450	—	127	1	no	maximal, Sylow
68	C₂²×D₄	—	127	1	no	—
69	C₂×SD₁₆	—	127	1	no	—
70	C₂×SD₁₆	—	127	1	no	—
71	C₂²⋊Q₈	—	127	1	no	—
72	#102	—	127	1	no	—
73	#102	D₁₂₇	1	1	yes	characteristic, complemented
74	C₂²⋊C₈	—	127	1	no	—
75	C₂⁴	—	254	1	no	—
76	C₂×Q₈	—	127	1	no	—
77	C₂×D₄	—	127	1	no	—
78	C₂×D₄	—	254	1	no	—
79	C₂×D₄	—	254	2	no	—
80	C₂×D₄	—	254	2	no	—
81	C₂×D₄	C₂×D₁₂₇	1	1	yes	characteristic
82	C₂²×C₄	—	127	1	no	—
83	SD₁₆	—	254	2	no	—
84	SD₁₆	—	254	2	no	—
85	C₂×C₈	—	127	1	no	—
86	C₂×C₈	C₂×D₁₂₇	1	1	yes	characteristic, complemented
87	C₄⋊C₄	—	254	1	no	—
88	C₄⋊C₄	C₂×D₁₂₇	1	1	yes	characteristic, complemented
89	C₂²⋊C₄	—	254	1	no	—
90	C₂³	—	2	1	no	—
91	C₂³	—	127	1	no	—
92	C₂³	—	508	2	no	—
93	C₂³	—	508	2	no	—
94	C₂³	—	254	4	no	—
95	C₂³	—	254	1	no	—
96	Q₈	—	254	1	no	—
97	Q₈	—	254	1	no	—
98	D₄	—	2	1	no	—
99	D₄	—	127	2	no	—
100	D₄	—	254	4	no	—
101	D₄	—	254	1	no	—
102	D₄	D₅₀₈	1	2	yes	—
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	—	127	1	no	—
105	C₂×C₄	—	254	1	no	—
106	C₂×C₄	—	254	1	no	—
107	C₂×C₄	—	254	1	no	—
108	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
109	C₈	—	2	1	no	—
110	C₈	—	254	1	no	—
111	C₂²	—	2	2	no	—
112	C₂²	—	4	1	no	—
113	C₂²	—	4	1	no	—
114	C₂²	—	127	2	no	—
115	C₂²	—	508	8	no	—
116	C₂²	—	508	1	no	—
117	C₂²	—	508	1	no	—
118	C₂²	—	254	2	no	—
119	C₂²	—	254	1	no	—
120	C₂²	—	254	1	no	—
121	C₂²	D₄×D₁₂₇	1	1	yes	characteristic
122	C₄	—	4	1	no	—
123	C₄	—	508	1	no	—
124	C₄	—	254	1	no	—
125	C₄	D₄×D₁₂₇	1	1	yes	characteristic
126	C₄	C₂×D₅₀₈	1	1	yes	characteristic
127	C₂	—	4	2	no	—
128	C₂	—	508	2	no	—
129	C₂	—	254	2	no	—
130	C₂	#16861179	1	1	yes	characteristic
131	C₂	SD₁₆×D₁₂₇	1	1	yes	characteristic
132	C₂	#16861233	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×D₄×D₁₂₇

C₂

yes

maximal, characteristic

C₂×#14523237

C₂

yes

maximal, characteristic, complemented

C₂×#14523236

C₂

yes

maximal, characteristic, complemented

#16861189

C₂

yes

maximal, characteristic, complemented

#16861129

C₂

yes

maximal, characteristic, complemented

(C₂×D₁₂₇)⋊C₈

C₂

yes

maximal, characteristic, complemented

#102×C₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂³×D₁₂₇

—

C₂×Dic₂₅₄

C₂²

yes

characteristic, complemented

C₂×D₅₀₈

C₂²

yes

characteristic

C₂×C₂²⋊D₁₂₇

—

D₄×D₁₂₇

—

D₄×D₁₂₇

—

C₂×C₄×D₁₂₇

C₂²

yes

characteristic

#14523237

—

#14523236

—

C₂×C₁₂₇⋊C₈

C₂²

yes

characteristic, complemented

C₄⋊Dic₁₂₇

—

(C₂×D₁₂₇)⋊C₄

—

C₂×D₄×C₁₂₇

C₂²

yes

characteristic

C₂×C₈×C₁₂₇

C₂²

yes

characteristic, complemented

C₄⋊C₄×C₁₂₇

C₂²

yes

characteristic, complemented

C₂²×D₁₂₇

—

C₂²×D₁₂₇

—

C₂²×D₁₂₇

—

C₂²×D₁₂₇

—

C₂²×D₁₂₇

D₄

yes

characteristic

Dic₂₅₄

—

Dic₂₅₄

—

D₅₀₈

—

D₅₀₈

D₄

yes

—

C₂²⋊D₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

D₄

yes

characteristic

C₄×D₁₂₇

—

C₄×D₁₂₇

—

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

—

D₄×C₁₂₇

—

D₄×C₁₂₇

D₄

yes

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂³

yes

characteristic

C₈×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₂×D₄

yes

characteristic

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

SD₁₆

yes

complemented

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₄×C₁₂₇

C₂×D₄

yes

characteristic

Dic₁₂₇

—

Dic₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

#120

yes

characteristic

C₂×C₁₂₇

C₂×SD₁₆

yes

characteristic

C₂×C₁₂₇

AGL(1,ℤ/8)

yes

characteristic

D₁₂₇

—

D₁₂₇

—

C₁₂₇

#450

yes

characteristic, complemented, Sylow

#450

—

127

maximal, Sylow

C₂²×D₄

—

127

—

C₂×SD₁₆

—

127

—

C₂×SD₁₆

—

127

—

C₂²⋊Q₈

—

127

—

#102

—

127

—

#102

D₁₂₇

yes

characteristic, complemented

C₂²⋊C₈

—

127

—

C₂⁴

—

254

—

C₂×Q₈

—

127

—

C₂×D₄

—

127

—

C₂×D₄

—

254

—

C₂×D₄

—

254

—

C₂×D₄

—

254

—

C₂×D₄

C₂×D₁₂₇

yes

characteristic

C₂²×C₄

—

127

—

SD₁₆

—

254

—

SD₁₆

—

254

—

C₂×C₈

—

127

—

C₂×C₈

C₂×D₁₂₇

yes

characteristic, complemented

C₄⋊C₄

—

254

—

C₄⋊C₄

C₂×D₁₂₇

yes

characteristic, complemented

C₂²⋊C₄

—

254

—

C₂³

—

C₂³

—

127

—

C₂³

—

508

—

C₂³

—

508

—

C₂³

—

254

—

C₂³

—

254

—

Q₈

—

254

—

Q₈

—

254

—

D₄

—

D₄

—

127

—

100

D₄

—

254

—

101

D₄

—

254

—

102

D₄

D₅₀₈

yes

—

103

C₂×C₄

—

104

C₂×C₄

—

127

—

105

C₂×C₄

—

254

—

106

C₂×C₄

—

254

—

107

C₂×C₄

—

254

—

108

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

109

C₈

—

110

C₈

—

254

—

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

—

127

—

115

C₂²

—

508

—

116

C₂²

—

508

—

117

C₂²

—

508

—

118

C₂²

—

254

—

119

C₂²

—

254

—

120

C₂²

—

254

—

121

C₂²

D₄×D₁₂₇

yes

characteristic

122

C₄

—

123

C₄

—

508

—

124

C₄

—

254

—

125

C₄

D₄×D₁₂₇

yes

characteristic

126

C₄

C₂×D₅₀₈

yes

characteristic

127

C₂

—

128

C₂

—

508

—

129

C₂

—

254

—

130

C₂

#16861179

yes

characteristic

131

C₂

SD₁₆×D₁₂₇

yes

characteristic

132

C₂

#16861233

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×D₁₂₇
commutator	C₂×C₄×C₁₂₇	C₂³
Frattini	C₂×C₄	C₂²×D₁₂₇
Fitting	#102×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂×D₄
hypercenter	#102	D₁₂₇
hypocenter	C₁₂₇	#450
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #102

subgroup

G/H

center

C₂²

D₄×D₁₂₇

commutator

C₂×C₄×C₁₂₇

C₂³

Frattini

C₂×C₄

C₂²×D₁₂₇

Fitting

#102×C₁₂₇

C₂

socle

C₂²×C₁₂₇

C₂×D₄

D₁₂₇

C₁₂₇

G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ #102

N	H ≅ G/N	type
2. C₂×#14523237	C₂	semidirect
3. C₂×#14523236	C₂	semidirect
4. #16861189	C₂	semidirect
5. #16861129	C₂	semidirect
6. (C₂×D₁₂₇)⋊C₈	C₂	semidirect
7. #102×C₁₂₇	C₂	semidirect
9. C₂×Dic₂₅₄	C₂²	semidirect
17. C₂×C₁₂₇⋊C₈	C₂²	semidirect
21. C₂×C₈×C₁₂₇	C₂²	semidirect
22. C₄⋊C₄×C₁₂₇	C₂²	semidirect
54. C₂×D₁₂₇	SD₁₆	semidirect
66. C₁₂₇	#450	semidirect
73. #102	D₁₂₇	semidirect
86. C₂×C₈	C₂×D₁₂₇	semidirect
88. C₄⋊C₄	C₂×D₁₂₇	semidirect

H ≅ G/N

type

2. C₂×#14523237

C₂

semidirect

3. C₂×#14523236

C₂

semidirect

4. #16861189

C₂

semidirect

5. #16861129

C₂

semidirect

6. (C₂×D₁₂₇)⋊C₈

C₂

semidirect

7. #102×C₁₂₇

C₂

semidirect

9. C₂×Dic₂₅₄

C₂²

semidirect

17. C₂×C₁₂₇⋊C₈

C₂²

semidirect

21. C₂×C₈×C₁₂₇

C₂²

semidirect

22. C₄⋊C₄×C₁₂₇

C₂²

semidirect

54. C₂×D₁₂₇

SD₁₆

semidirect

66. C₁₂₇

#450

semidirect

73. #102

D₁₂₇

semidirect

86. C₂×C₈

C₂×D₁₂₇

semidirect

88. C₄⋊C₄

C₂×D₁₂₇

semidirect

Subgroups — #22550028

Conjugacy classes of subgroups

Complements