Group #22549879 — subgroups

Subgroups — #22549879

Order 8128.

190 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 55 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×#14523231	C₂	1	1	yes	maximal, characteristic, complemented
3	#16861167	C₂	1	1	yes	maximal, characteristic, complemented
4	#16861131	C₂	1	1	yes	maximal, characteristic, complemented
5	#16861130	C₂	1	1	yes	maximal, characteristic, complemented
6	#16861129	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×D₅₀₈	C₂²	1	1	yes	characteristic
10	C₂×C₂²⋊D₁₂₇	—	2	1	no	—
11	D₄×D₁₂₇	—	2	2	no	—
12	D₄×D₁₂₇	C₄	1	4	yes	complemented
13	C₂×C₄×D₁₂₇	C₂²	1	1	yes	characteristic
14	#14523231	—	2	2	no	—
15	C₂×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic, complemented
16	C₄⋊Dic₁₂₇	C₂²	1	1	yes	characteristic, complemented
17	(C₂×D₁₂₇)⋊C₄	—	2	1	no	—
18	C₄×Dic₁₂₇	—	2	1	no	—
19	C₂×D₄×C₁₂₇	C₂²	1	1	yes	characteristic
20	C₂×C₈×C₁₂₇	C₂²	1	1	yes	characteristic, complemented
21	C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic, complemented
22	C₂²×D₁₂₇	—	2	1	no	—
23	C₂²×D₁₂₇	—	2	4	no	—
24	C₂²×D₁₂₇	—	4	2	no	—
25	C₂²×D₁₂₇	—	4	2	no	—
26	C₂²×D₁₂₇	D₄	1	1	yes	characteristic
27	D₅₀₈	—	2	1	no	—
28	D₅₀₈	C₂×C₄	1	2	yes	—
29	C₂²⋊D₁₂₇	—	2	4	no	—
30	C₂×Dic₁₂₇	—	2	1	no	—
31	C₂×Dic₁₂₇	D₄	1	1	yes	characteristic
32	C₄×D₁₂₇	C₂×C₄	1	2	yes	—
33	C₄×D₁₂₇	D₄	1	2	yes	—
34	C₁₂₇⋊C₈	—	2	1	no	—
35	C₂³×C₁₂₇	—	2	1	no	—
36	D₄×C₁₂₇	—	2	1	no	—
37	D₄×C₁₂₇	C₂×C₄	1	2	yes	—
38	C₂×C₄×C₁₂₇	—	2	1	no	—
39	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
40	C₈×C₁₂₇	—	2	1	no	—
41	C₂²×C₁₂₇	—	2	2	no	—
42	C₂²×C₁₂₇	—	4	1	no	—
43	C₂²×C₁₂₇	—	4	1	no	—
44	C₂²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
45	C₂×D₁₂₇	—	2	1	no	—
46	C₂×D₁₂₇	—	2	1	no	—
47	C₂×D₁₂₇	—	2	2	no	—
48	C₂×D₁₂₇	—	4	8	no	—
49	C₂×D₁₂₇	—	4	1	no	—
50	C₂×D₁₂₇	—	4	1	no	—
51	C₂×D₁₂₇	C₂²⋊C₄	1	2	yes	—
52	C₄×C₁₂₇	—	4	1	no	—
53	C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
54	C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
55	Dic₁₂₇	—	4	1	no	—
56	Dic₁₂₇	C₂²⋊C₄	1	2	yes	—
57	C₂×C₁₂₇	—	4	2	no	—
58	C₂×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₁₂₇	AGL(1,ℤ/8)	1	1	yes	characteristic
60	C₂×C₁₂₇	AGL(1,ℤ/8)	1	1	yes	characteristic
61	D₁₂₇	—	2	2	no	—
62	D₁₂₇	—	4	2	no	—
63	C₁₂₇	#418	1	1	yes	characteristic, complemented, Sylow
64	#418	—	127	1	no	maximal, Sylow
65	C₂²×D₄	—	127	1	no	—
66	C₂×M₁₆	—	127	1	no	—
67	#117	—	127	1	no	—
68	#102	—	127	1	no	—
69	#102	—	127	1	no	—
70	#102	—	127	1	no	—
71	#102	D₁₂₇	1	1	yes	characteristic, complemented
72	C₂⁴	—	254	1	no	—
73	C₂×D₄	—	127	4	no	—
74	C₂×D₄	—	127	1	no	—
75	C₂×D₄	—	254	2	no	—
76	C₂×D₄	—	254	1	no	—
77	C₂×D₄	C₂×D₁₂₇	1	1	yes	characteristic
78	C₂²×C₄	—	127	1	no	—
79	M₁₆	—	254	2	no	—
80	C₂×C₈	—	127	1	no	—
81	C₂×C₈	C₂×D₁₂₇	1	1	yes	characteristic, complemented
82	C₄⋊C₄	—	127	1	no	—
83	C₄⋊C₄	C₂×D₁₂₇	1	1	yes	characteristic, complemented
84	C₂²⋊C₄	—	254	1	no	—
85	C₄²	—	254	1	no	—
86	C₂³	—	2	1	no	—
87	C₂³	—	127	1	no	—
88	C₂³	—	508	2	no	—
89	C₂³	—	508	2	no	—
90	C₂³	—	254	4	no	—
91	C₂³	—	254	1	no	—
92	D₄	—	2	1	no	—
93	D₄	—	127	2	no	—
94	D₄	—	254	4	no	—
95	D₄	—	254	1	no	—
96	D₄	C₄×D₁₂₇	1	2	yes	—
97	C₂×C₄	—	2	1	no	—
98	C₂×C₄	—	127	1	no	—
99	C₂×C₄	—	127	2	no	—
100	C₂×C₄	—	127	2	no	—
101	C₂×C₄	—	254	1	no	—
102	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
103	C₈	—	2	1	no	—
104	C₈	—	254	1	no	—
105	C₂²	—	2	2	no	—
106	C₂²	—	4	1	no	—
107	C₂²	—	4	1	no	—
108	C₂²	—	127	2	no	—
109	C₂²	—	508	8	no	—
110	C₂²	—	508	1	no	—
111	C₂²	—	508	1	no	—
112	C₂²	—	254	2	no	—
113	C₂²	—	254	1	no	—
114	C₂²	—	254	1	no	—
115	C₂²	D₄×D₁₂₇	1	1	yes	characteristic
116	C₄	—	4	1	no	—
117	C₄	—	127	2	no	—
118	C₄	—	508	1	no	—
119	C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
120	C₄	D₄×D₁₂₇	1	1	yes	characteristic
121	C₂	—	4	2	no	—
122	C₂	—	508	2	no	—
123	C₂	—	254	2	no	—
124	C₂	C₂²⋊C₄×D₁₂₇	1	1	yes	characteristic
125	C₂	#16861233	1	1	yes	characteristic
126	C₂	#16861236	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×D₄×D₁₂₇

C₂

yes

maximal, characteristic

C₂×#14523231

C₂

yes

maximal, characteristic, complemented

#16861167

C₂

yes

maximal, characteristic, complemented

#16861131

C₂

yes

maximal, characteristic, complemented

#16861130

C₂

yes

maximal, characteristic, complemented

#16861129

C₂

yes

maximal, characteristic, complemented

#102×C₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂³×D₁₂₇

—

C₂×D₅₀₈

C₂²

yes

characteristic

C₂×C₂²⋊D₁₂₇

—

D₄×D₁₂₇

—

D₄×D₁₂₇

C₄

yes

complemented

C₂×C₄×D₁₂₇

C₂²

yes

characteristic

#14523231

—

C₂×C₁₂₇⋊C₈

C₂²

yes

characteristic, complemented

C₄⋊Dic₁₂₇

C₂²

yes

characteristic, complemented

(C₂×D₁₂₇)⋊C₄

—

C₄×Dic₁₂₇

—

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

D₅₀₈

—

D₅₀₈

C₂×C₄

yes

—

C₂²⋊D₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

D₄

yes

characteristic

C₄×D₁₂₇

C₂×C₄

yes

—

C₄×D₁₂₇

D₄

yes

—

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

—

D₄×C₁₂₇

—

D₄×C₁₂₇

C₂×C₄

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

C₂²⋊C₄

yes

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂²×C₄

yes

characteristic

C₄×C₁₂₇

C₂×D₄

yes

characteristic

Dic₁₂₇

—

Dic₁₂₇

C₂²⋊C₄

yes

—

C₂×C₁₂₇

—

C₂×C₁₂₇

C₂×C₂²⋊C₄

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

D₁₂₇

—

D₁₂₇

—

C₁₂₇

#418

yes

characteristic, complemented, Sylow

#418

—

127

maximal, Sylow

C₂²×D₄

—

127

—

C₂×M₁₆

—

127

—

#117

—

127

—

#102

—

127

—

#102

—

127

—

#102

—

127

—

#102

D₁₂₇

yes

characteristic, complemented

C₂⁴

—

254

—

C₂×D₄

—

127

—

C₂×D₄

—

127

—

C₂×D₄

—

254

—

C₂×D₄

—

254

—

C₂×D₄

C₂×D₁₂₇

yes

characteristic

C₂²×C₄

—

127

—

M₁₆

—

254

—

C₂×C₈

—

127

—

C₂×C₈

C₂×D₁₂₇

yes

characteristic, complemented

C₄⋊C₄

—

127

—

C₄⋊C₄

C₂×D₁₂₇

yes

characteristic, complemented

C₂²⋊C₄

—

254

—

C₄²

—

254

—

C₂³

—

C₂³

—

127

—

C₂³

—

508

—

C₂³

—

508

—

C₂³

—

254

—

C₂³

—

254

—

D₄

—

D₄

—

127

—

D₄

—

254

—

D₄

—

254

—

D₄

C₄×D₁₂₇

yes

—

C₂×C₄

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

100

C₂×C₄

—

127

—

101

C₂×C₄

—

254

—

102

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

103

C₈

—

104

C₈

—

254

—

105

C₂²

—

106

C₂²

—

107

C₂²

—

108

C₂²

—

127

—

109

C₂²

—

508

—

110

C₂²

—

508

—

111

C₂²

—

508

—

112

C₂²

—

254

—

113

C₂²

—

254

—

114

C₂²

—

254

—

115

C₂²

D₄×D₁₂₇

yes

characteristic

116

C₄

—

117

C₄

—

127

—

118

C₄

—

508

—

119

C₄

C₂×C₄×D₁₂₇

yes

characteristic

120

C₄

D₄×D₁₂₇

yes

characteristic

121

C₂

—

122

C₂

—

508

—

123

C₂

—

254

—

124

C₂

C₂²⋊C₄×D₁₂₇

yes

characteristic

125

C₂

#16861233

yes

characteristic

126

C₂

#16861236

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×D₁₂₇
commutator	C₄×C₁₂₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₁₂₇
Fitting	#102×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂×D₄
hypercenter	#102	D₁₂₇
hypocenter	C₁₂₇	#418
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #102

subgroup

G/H

center

C₂²

D₄×D₁₂₇

commutator

C₄×C₁₂₇

C₂²×C₄

Frattini

C₂×C₄

C₂²×D₁₂₇

Fitting

#102×C₁₂₇

C₂

socle

C₂²×C₁₂₇

C₂×D₄

D₁₂₇

C₁₂₇

G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ #102

N	H ≅ G/N	type
2. C₂×#14523231	C₂	semidirect
3. #16861167	C₂	semidirect
4. #16861131	C₂	semidirect
5. #16861130	C₂	semidirect
6. #16861129	C₂	semidirect
7. #102×C₁₂₇	C₂	semidirect
12. D₄×D₁₂₇	C₄	semidirect
15. C₂×C₁₂₇⋊C₈	C₂²	semidirect
16. C₄⋊Dic₁₂₇	C₂²	semidirect
20. C₂×C₈×C₁₂₇	C₂²	semidirect
21. C₄⋊C₄×C₁₂₇	C₂²	semidirect
63. C₁₂₇	#418	semidirect
71. #102	D₁₂₇	semidirect
81. C₂×C₈	C₂×D₁₂₇	semidirect
83. C₄⋊C₄	C₂×D₁₂₇	semidirect

H ≅ G/N

type

2. C₂×#14523231

C₂

semidirect

3. #16861167

C₂

semidirect

4. #16861131

C₂

semidirect

5. #16861130

C₂

semidirect

6. #16861129

C₂

semidirect

7. #102×C₁₂₇

C₂

semidirect

12. D₄×D₁₂₇

C₄

semidirect

15. C₂×C₁₂₇⋊C₈

C₂²

semidirect

16. C₄⋊Dic₁₂₇

C₂²

semidirect

20. C₂×C₈×C₁₂₇

C₂²

semidirect

21. C₄⋊C₄×C₁₂₇

C₂²

semidirect

63. C₁₂₇

#418

semidirect

71. #102

D₁₂₇

semidirect

81. C₂×C₈

C₂×D₁₂₇

semidirect

83. C₄⋊C₄

C₂×D₁₂₇

semidirect

Subgroups — #22549879

Conjugacy classes of subgroups

Complements