Group #22549874 — subgroups

Subgroups — #22549874

Order 8128.

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

normal subgroups only

This group has 162 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×#14523247	C₂	1	1	yes	maximal, characteristic
2	C₂×#14523231	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄⋊C₄×D₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#16861134	C₂	1	1	yes	maximal, characteristic, complemented
5	#16861132	C₂	1	1	yes	maximal, characteristic, complemented
6	#16861130	C₂	1	1	yes	maximal, characteristic, complemented
7	#102×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#14523247	—	2	2	no	—
9	#14523247	C₄	1	4	yes	complemented
10	C₂×Dic₂₅₄	C₂²	1	1	yes	characteristic
11	C₂×C₂²⋊D₁₂₇	—	2	1	no	—
12	C₂²×Dic₁₂₇	—	2	1	no	—
13	C₂×C₄×D₁₂₇	—	2	1	no	—
14	C₂×C₄×D₁₂₇	C₂²	1	1	yes	characteristic
15	#14523231	—	2	2	no	—
16	C₂×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
17	C₄⋊Dic₁₂₇	C₂²	1	1	yes	characteristic
18	Dic₁₂₇⋊C₄	—	2	1	no	—
19	C₂×D₄×C₁₂₇	C₂²	1	1	yes	characteristic
20	C₂×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
21	C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
22	C₂²×D₁₂₇	D₄	1	1	yes	characteristic
23	Dic₂₅₄	—	2	1	no	—
24	Dic₂₅₄	C₂×C₄	1	2	yes	complemented
25	C₂²⋊D₁₂₇	—	2	4	no	—
26	C₂×Dic₁₂₇	—	2	1	no	—
27	C₂×Dic₁₂₇	—	2	4	no	—
28	C₂×Dic₁₂₇	—	2	1	no	—
29	C₂×Dic₁₂₇	D₄	1	1	yes	characteristic
30	C₄×D₁₂₇	—	4	2	no	—
31	C₄×D₁₂₇	C₂×C₄	1	2	yes	—
32	C₄×D₁₂₇	D₄	1	2	yes	—
33	C₁₂₇⋊C₈	—	2	1	no	—
34	C₂³×C₁₂₇	—	2	1	no	—
35	D₄×C₁₂₇	—	2	1	no	—
36	D₄×C₁₂₇	C₂×C₄	1	2	yes	complemented
37	C₂×C₄×C₁₂₇	—	2	1	no	—
38	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
39	C₈×C₁₂₇	—	2	1	no	—
40	C₂²×C₁₂₇	—	2	2	no	—
41	C₂²×C₁₂₇	—	4	1	no	—
42	C₂²×C₁₂₇	—	4	1	no	—
43	C₂²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
44	C₂×D₁₂₇	—	2	1	no	—
45	C₂×D₁₂₇	—	2	1	no	—
46	C₂×D₁₂₇	C₂²⋊C₄	1	2	yes	—
47	C₄×C₁₂₇	—	4	1	no	—
48	C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
49	C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
50	Dic₁₂₇	—	2	2	no	—
51	Dic₁₂₇	—	4	1	no	—
52	Dic₁₂₇	C₂²⋊C₄	1	2	yes	—
53	C₂×C₁₂₇	—	4	2	no	—
54	C₂×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₁₂₇	AGL(1,ℤ/8)	1	1	yes	characteristic
56	C₂×C₁₂₇	#137	1	1	yes	characteristic
57	D₁₂₇	—	2	2	no	—
58	C₁₂₇	#417	1	1	yes	characteristic, complemented, Sylow
59	#417	—	127	1	no	maximal, Sylow
60	C₂×#41	—	127	1	no	—
61	C₂×M₁₆	—	127	1	no	—
62	C₂×C₄⋊C₄	—	127	1	no	—
63	#103	—	127	1	no	—
64	#103	—	127	1	no	—
65	#102	—	127	1	no	—
66	#102	D₁₂₇	1	1	yes	characteristic, complemented
67	#41	—	127	4	no	—
68	#41	—	254	2	no	—
69	C₂×Q₈	—	127	1	no	—
70	C₂×D₄	—	254	1	no	—
71	C₂×D₄	C₂×D₁₂₇	1	1	yes	characteristic
72	C₂²×C₄	—	127	1	no	—
73	C₂²×C₄	—	254	1	no	—
74	C₂²×C₄	—	254	1	no	—
75	M₁₆	—	254	2	no	—
76	C₂×C₈	—	127	1	no	—
77	C₂×C₈	C₂×D₁₂₇	1	1	yes	characteristic
78	C₄⋊C₄	—	127	1	no	—
79	C₄⋊C₄	—	254	1	no	—
80	C₄⋊C₄	C₂×D₁₂₇	1	1	yes	characteristic
81	C₂³	—	2	1	no	—
82	C₂³	—	127	1	no	—
83	Q₈	—	127	2	no	—
84	Q₈	—	254	1	no	—
85	D₄	—	2	1	no	—
86	D₄	—	254	4	no	—
87	D₄	C₄×D₁₂₇	1	2	yes	complemented
88	C₂×C₄	—	2	1	no	—
89	C₂×C₄	—	127	1	no	—
90	C₂×C₄	—	127	2	no	—
91	C₂×C₄	—	127	2	no	—
92	C₂×C₄	—	508	2	no	—
93	C₂×C₄	—	254	4	no	—
94	C₂×C₄	—	254	1	no	—
95	C₂×C₄	—	254	1	no	—
96	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
97	C₈	—	2	1	no	—
98	C₈	—	254	1	no	—
99	C₂²	—	2	2	no	—
100	C₂²	—	4	1	no	—
101	C₂²	—	4	1	no	—
102	C₂²	—	127	2	no	—
103	C₂²	—	254	1	no	—
104	C₂²	—	254	1	no	—
105	C₂²	D₄×D₁₂₇	1	1	yes	characteristic
106	C₄	—	4	1	no	—
107	C₄	—	127	2	no	—
108	C₄	—	508	1	no	—
109	C₄	—	254	2	no	—
110	C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
111	C₄	D₄×D₁₂₇	1	1	yes	characteristic
112	C₂	—	4	2	no	—
113	C₂	—	254	2	no	—
114	C₂	C₂²⋊C₄×D₁₂₇	1	1	yes	characteristic
115	C₂	#16861233	1	1	yes	characteristic
116	C₂	#16861238	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14523247

C₂

yes

maximal, characteristic

C₂×#14523231

C₂

yes

maximal, characteristic, complemented

C₄⋊C₄×D₁₂₇

C₂

yes

maximal, characteristic, complemented

#16861134

C₂

yes

maximal, characteristic, complemented

#16861132

C₂

yes

maximal, characteristic, complemented

#16861130

C₂

yes

maximal, characteristic, complemented

#102×C₁₂₇

C₂

yes

maximal, characteristic, complemented

#14523247

—

#14523247

C₄

yes

complemented

C₂×Dic₂₅₄

C₂²

yes

characteristic

C₂×C₂²⋊D₁₂₇

—

C₂²×Dic₁₂₇

—

C₂×C₄×D₁₂₇

—

C₂×C₄×D₁₂₇

C₂²

yes

characteristic

#14523231

—

C₂×C₁₂₇⋊C₈

C₂²

yes

characteristic

C₄⋊Dic₁₂₇

C₂²

yes

characteristic

Dic₁₂₇⋊C₄

—

C₂×D₄×C₁₂₇

C₂²

yes

characteristic

C₂×C₈×C₁₂₇

C₂²

yes

characteristic

C₄⋊C₄×C₁₂₇

C₂²

yes

characteristic

C₂²×D₁₂₇

D₄

yes

characteristic

Dic₂₅₄

—

Dic₂₅₄

C₂×C₄

yes

complemented

C₂²⋊D₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

D₄

yes

characteristic

C₄×D₁₂₇

—

C₄×D₁₂₇

C₂×C₄

yes

—

C₄×D₁₂₇

D₄

yes

—

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

—

D₄×C₁₂₇

—

D₄×C₁₂₇

C₂×C₄

yes

complemented

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

C₂²⋊C₄

yes

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂²×C₄

yes

characteristic

C₄×C₁₂₇

C₂×D₄

yes

characteristic

Dic₁₂₇

—

Dic₁₂₇

—

Dic₁₂₇

C₂²⋊C₄

yes

—

C₂×C₁₂₇

—

C₂×C₁₂₇

C₂×C₂²⋊C₄

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

D₁₂₇

—

C₁₂₇

#417

yes

characteristic, complemented, Sylow

#417

—

127

maximal, Sylow

C₂×#41

—

127

—

C₂×M₁₆

—

127

—

C₂×C₄⋊C₄

—

127

—

#103

—

127

—

#103

—

127

—

#102

—

127

—

#102

D₁₂₇

yes

characteristic, complemented

#41

—

127

—

#41

—

254

—

C₂×Q₈

—

127

—

C₂×D₄

—

254

—

C₂×D₄

C₂×D₁₂₇

yes

characteristic

C₂²×C₄

—

127

—

C₂²×C₄

—

254

—

C₂²×C₄

—

254

—

M₁₆

—

254

—

C₂×C₈

—

127

—

C₂×C₈

C₂×D₁₂₇

yes

characteristic

C₄⋊C₄

—

127

—

C₄⋊C₄

—

254

—

C₄⋊C₄

C₂×D₁₂₇

yes

characteristic

C₂³

—

C₂³

—

127

—

Q₈

—

127

—

Q₈

—

254

—

D₄

—

D₄

—

254

—

D₄

C₄×D₁₂₇

yes

complemented

C₂×C₄

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

508

—

C₂×C₄

—

254

—

C₂×C₄

—

254

—

C₂×C₄

—

254

—

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

C₈

—

C₈

—

254

—

C₂²

—

100

C₂²

—

101

C₂²

—

102

C₂²

—

127

—

103

C₂²

—

254

—

104

C₂²

—

254

—

105

C₂²

D₄×D₁₂₇

yes

characteristic

106

C₄

—

107

C₄

—

127

—

108

C₄

—

508

—

109

C₄

—

254

—

110

C₄

C₂×C₄×D₁₂₇

yes

characteristic

111

C₄

D₄×D₁₂₇

yes

characteristic

112

C₂

—

113

C₂

—

254

—

114

C₂

C₂²⋊C₄×D₁₂₇

yes

characteristic

115

C₂

#16861233

yes

characteristic

116

C₂

#16861238

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	#417
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. C₄⋊C₄×D₁₂₇	C₂	semidirect
4. #16861134	C₂	semidirect
5. #16861132	C₂	semidirect
6. #16861130	C₂	semidirect
7. #102×C₁₂₇	C₂	semidirect
9. #14523247	C₄	semidirect
24. Dic₂₅₄	C₂×C₄	semidirect
36. D₄×C₁₂₇	C₂×C₄	semidirect
58. C₁₂₇	#417	semidirect
66. #102	D₁₂₇	semidirect
87. D₄	C₄×D₁₂₇	semidirect

H ≅ G/N

type

2. C₂×#14523231

C₂

semidirect

3. C₄⋊C₄×D₁₂₇

C₂

semidirect

4. #16861134

C₂

semidirect

5. #16861132

C₂

semidirect

6. #16861130

C₂

semidirect

7. #102×C₁₂₇

C₂

semidirect

9. #14523247

C₄

semidirect

24. Dic₂₅₄

C₂×C₄

semidirect

36. D₄×C₁₂₇

C₂×C₄

semidirect

58. C₁₂₇

#417

semidirect

66. #102

D₁₂₇

semidirect

87. D₄

C₄×D₁₂₇

semidirect

Subgroups — #22549874

Conjugacy classes of subgroups

Complements