Group #22549869 — subgroups

Subgroups — #22549869

Order 8128.

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

normal subgroups only

This group has 158 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×C₈×D₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#16861167	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	C₂²	1	1	yes	characteristic
14	C₂×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
15	C₈×D₁₂₇	—	2	2	no	—
16	C₄⋊Dic₁₂₇	C₂²	1	1	yes	characteristic
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
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	—
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	C₂×C₄	1	2	yes	—
31	C₄×D₁₂₇	D₄	1	2	yes	—
32	C₁₂₇⋊C₈	—	2	1	no	—
33	C₂³×C₁₂₇	—	2	1	no	—
34	D₄×C₁₂₇	—	2	1	no	—
35	D₄×C₁₂₇	C₂×C₄	1	2	yes	—
36	C₂×C₄×C₁₂₇	—	2	1	no	—
37	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
38	C₈×C₁₂₇	—	2	1	no	—
39	C₂²×C₁₂₇	—	2	2	no	—
40	C₂²×C₁₂₇	—	4	1	no	—
41	C₂²×C₁₂₇	—	4	1	no	—
42	C₂²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
43	C₂×D₁₂₇	—	2	1	no	—
44	C₂×D₁₂₇	—	2	1	no	—
45	C₂×D₁₂₇	C₂²⋊C₄	1	2	yes	—
46	C₄×C₁₂₇	—	4	1	no	—
47	C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
48	C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
49	Dic₁₂₇	—	2	2	no	—
50	Dic₁₂₇	—	4	1	no	—
51	Dic₁₂₇	C₂²⋊C₄	1	2	yes	—
52	C₂×C₁₂₇	—	4	2	no	—
53	C₂×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₁₂₇	#135	1	1	yes	characteristic
55	C₂×C₁₂₇	#135	1	1	yes	characteristic
56	D₁₂₇	—	2	2	no	—
57	C₁₂₇	#416	1	1	yes	characteristic, complemented, Sylow
58	#416	—	127	1	no	maximal, Sylow
59	C₂×#41	—	127	1	no	—
60	C₂²×C₈	—	127	1	no	—
61	#117	—	127	1	no	—
62	#103	—	127	1	no	—
63	#103	—	127	1	no	—
64	#102	—	127	1	no	—
65	#102	D₁₂₇	1	1	yes	characteristic, complemented
66	#41	—	127	4	no	—
67	#41	—	254	2	no	—
68	C₂×Q₈	—	127	1	no	—
69	C₂×D₄	—	254	1	no	—
70	C₂×D₄	C₂×D₁₂₇	1	1	yes	characteristic
71	C₂²×C₄	—	127	1	no	—
72	C₂²×C₄	—	254	1	no	—
73	C₂×C₈	—	127	1	no	—
74	C₂×C₈	—	254	2	no	—
75	C₂×C₈	C₂×D₁₂₇	1	1	yes	characteristic
76	C₄⋊C₄	—	127	1	no	—
77	C₄⋊C₄	C₂×D₁₂₇	1	1	yes	characteristic
78	C₂²⋊C₄	—	254	1	no	—
79	C₄²	—	254	1	no	—
80	C₂³	—	2	1	no	—
81	C₂³	—	127	1	no	—
82	Q₈	—	127	2	no	—
83	Q₈	—	254	1	no	—
84	D₄	—	2	1	no	—
85	D₄	—	254	4	no	—
86	D₄	C₄×D₁₂₇	1	2	yes	—
87	C₂×C₄	—	2	1	no	—
88	C₂×C₄	—	127	1	no	—
89	C₂×C₄	—	127	2	no	—
90	C₂×C₄	—	127	2	no	—
91	C₂×C₄	—	254	4	no	—
92	C₂×C₄	—	254	1	no	—
93	C₂×C₄	—	254	1	no	—
94	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
95	C₈	—	2	1	no	—
96	C₈	—	254	1	no	—
97	C₂²	—	2	2	no	—
98	C₂²	—	4	1	no	—
99	C₂²	—	4	1	no	—
100	C₂²	—	127	2	no	—
101	C₂²	—	254	1	no	—
102	C₂²	—	254	1	no	—
103	C₂²	D₄×D₁₂₇	1	1	yes	characteristic
104	C₄	—	4	1	no	—
105	C₄	—	127	2	no	—
106	C₄	—	508	1	no	—
107	C₄	—	254	2	no	—
108	C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
109	C₄	D₄×D₁₂₇	1	1	yes	characteristic
110	C₂	—	4	2	no	—
111	C₂	—	254	2	no	—
112	C₂	C₂²⋊C₄×D₁₂₇	1	1	yes	characteristic
113	C₂	#16861228	1	1	yes	characteristic
114	C₂	#16861230	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14523247

C₂

yes

maximal, characteristic

C₂×C₈×D₁₂₇

C₂

yes

maximal, characteristic, complemented

#16861167

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

C₂²

yes

characteristic

C₂×C₁₂₇⋊C₈

C₂²

yes

characteristic

C₈×D₁₂₇

—

C₄⋊Dic₁₂₇

C₂²

yes

characteristic

(C₂×D₁₂₇)⋊C₄

—

C₄×Dic₁₂₇

—

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

—

C₂²⋊D₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

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

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

#416

yes

characteristic, complemented, Sylow

#416

—

127

maximal, Sylow

C₂×#41

—

127

—

C₂²×C₈

—

127

—

#117

—

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

—

C₂×C₈

—

127

—

C₂×C₈

—

254

—

C₂×C₈

C₂×D₁₂₇

yes

characteristic

C₄⋊C₄

—

127

—

C₄⋊C₄

C₂×D₁₂₇

yes

characteristic

C₂²⋊C₄

—

254

—

C₄²

—

254

—

C₂³

—

C₂³

—

127

—

Q₈

—

127

—

Q₈

—

254

—

D₄

—

D₄

—

254

—

D₄

C₄×D₁₂₇

yes

—

C₂×C₄

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

254

—

C₂×C₄

—

254

—

C₂×C₄

—

254

—

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

C₈

—

C₈

—

254

—

C₂²

—

C₂²

—

C₂²

—

100

C₂²

—

127

—

101

C₂²

—

254

—

102

C₂²

—

254

—

103

C₂²

D₄×D₁₂₇

yes

characteristic

104

C₄

—

105

C₄

—

127

—

106

C₄

—

508

—

107

C₄

—

254

—

108

C₄

C₂×C₄×D₁₂₇

yes

characteristic

109

C₄

D₄×D₁₂₇

yes

characteristic

110

C₂

—

111

C₂

—

254

—

112

C₂

C₂²⋊C₄×D₁₂₇

yes

characteristic

113

C₂

#16861228

yes

characteristic

114

C₂

#16861230

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	#416
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×C₈×D₁₂₇	C₂	semidirect
3. #16861167	C₂	semidirect
4. #16861134	C₂	semidirect
5. #16861132	C₂	semidirect
6. #16861130	C₂	semidirect
7. #102×C₁₂₇	C₂	semidirect
9. #14523247	C₄	semidirect
57. C₁₂₇	#416	semidirect
65. #102	D₁₂₇	semidirect

H ≅ G/N

type

2. C₂×C₈×D₁₂₇

C₂

semidirect

3. #16861167

C₂

semidirect

4. #16861134

C₂

semidirect

5. #16861132

C₂

semidirect

6. #16861130

C₂

semidirect

7. #102×C₁₂₇

C₂

semidirect

9. #14523247

C₄

semidirect

57. C₁₂₇

#416

semidirect

65. #102

D₁₂₇

semidirect

Subgroups — #22549869

Conjugacy classes of subgroups

Complements