Group #22550074 — subgroups

Subgroups — #22550074

Order 8128.

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

normal subgroups only

This group has 128 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×#14523235	C₂	1	1	yes	maximal, characteristic, complemented
2	#16861205	C₂	1	1	yes	maximal, characteristic
3	Q₈×Dic₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
4	Dic₁₂₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	#16861131	C₂	1	1	yes	maximal, characteristic, complemented
6	#16861130	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×SD₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₅₀₈	C₂²	1	1	yes	characteristic
9	C₂×C₂²⋊D₁₂₇	—	2	1	no	—
10	C₂×C₂²⋊D₁₂₇	—	2	1	no	—
11	#14523235	—	2	2	no	—
12	C₂×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
13	C₄⋊Dic₁₂₇	—	2	1	no	—
14	C₄⋊Dic₁₂₇	C₂²	1	1	yes	characteristic
15	C₄×Dic₁₂₇	—	2	1	no	—
16	C₄×Dic₁₂₇	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
18	C₂×D₄×C₁₂₇	C₂²	1	1	yes	characteristic
19	SD₁₆×C₁₂₇	—	2	2	no	—
20	C₂×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
21	C₂²×D₁₂₇	—	2	1	no	—
22	D₅₀₈	—	2	1	no	—
23	D₅₀₈	—	2	1	no	—
24	C₂²⋊D₁₂₇	—	4	2	no	—
25	C₂²⋊D₁₂₇	—	4	2	no	—
26	C₂×Dic₁₂₇	—	2	1	no	—
27	C₂×Dic₁₂₇	D₄	1	1	yes	characteristic
28	C₂×Dic₁₂₇	D₄	1	1	yes	characteristic
29	C₁₂₇⋊C₈	—	2	1	no	—
30	C₂³×C₁₂₇	—	2	1	no	—
31	Q₈×C₁₂₇	—	2	1	no	—
32	Q₈×C₁₂₇	D₄	1	2	yes	complemented
33	D₄×C₁₂₇	—	2	1	no	—
34	D₄×C₁₂₇	—	2	1	no	—
35	C₂×C₄×C₁₂₇	—	2	1	no	—
36	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
37	C₈×C₁₂₇	—	2	1	no	—
38	C₂²×C₁₂₇	—	4	1	no	—
39	C₂²×C₁₂₇	—	4	1	no	—
40	C₂²×C₁₂₇	—	4	1	no	—
41	C₂²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
42	C₂×D₁₂₇	—	4	1	no	—
43	C₂×D₁₂₇	—	4	1	no	—
44	C₂×D₁₂₇	—	4	1	no	—
45	C₄×C₁₂₇	—	2	2	no	—
46	C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
47	C₄×C₁₂₇	#41	1	1	yes	characteristic
48	Dic₁₂₇	—	2	1	no	—
49	Dic₁₂₇	—	4	1	no	—
50	Dic₁₂₇	SD₁₆	1	2	yes	complemented
51	C₂×C₁₂₇	—	8	1	no	—
52	C₂×C₁₂₇	#121	1	1	yes	characteristic
53	C₂×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
54	C₂×C₁₂₇	AGL(1,ℤ/8)	1	1	yes	characteristic
55	D₁₂₇	—	8	1	no	—
56	C₁₂₇	#460	1	1	yes	characteristic, complemented, Sylow
57	#460	—	127	1	no	maximal, Sylow
58	C₂×SD₁₆	—	127	1	no	—
59	C₂×SD₁₆	D₁₂₇	1	1	yes	characteristic, complemented
60	#127	—	127	1	no	—
61	C₄×Q₈	—	127	1	no	—
62	C₄⋊C₈	—	127	1	no	—
63	#102	—	127	1	no	—
64	#102	—	127	1	no	—
65	C₂×Q₈	C₂×D₁₂₇	1	1	yes	characteristic
66	C₂×D₄	—	127	1	no	—
67	C₂×D₄	—	254	1	no	—
68	C₂×D₄	—	254	1	no	—
69	C₂×D₄	C₂×D₁₂₇	1	1	yes	characteristic
70	SD₁₆	—	2	2	no	—
71	SD₁₆	—	254	2	no	—
72	C₂×C₈	—	127	1	no	—
73	C₂×C₈	C₂×D₁₂₇	1	1	yes	characteristic
74	C₄⋊C₄	—	127	1	no	—
75	C₄⋊C₄	—	254	1	no	—
76	C₄²	—	127	1	no	—
77	C₄²	—	254	1	no	—
78	C₂³	—	2	1	no	—
79	C₂³	—	254	1	no	—
80	Q₈	—	2	1	no	—
81	Q₈	C₂²⋊D₁₂₇	1	2	yes	complemented
82	D₄	—	2	1	no	—
83	D₄	—	2	1	no	—
84	D₄	—	508	2	no	—
85	D₄	—	508	2	no	—
86	D₄	—	254	1	no	—
87	D₄	—	254	1	no	—
88	C₂×C₄	—	2	1	no	—
89	C₂×C₄	—	127	1	no	—
90	C₂×C₄	—	127	1	no	—
91	C₂×C₄	—	254	1	no	—
92	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
93	C₈	—	2	1	no	—
94	C₈	—	254	1	no	—
95	C₂²	—	4	1	no	—
96	C₂²	—	4	1	no	—
97	C₂²	—	4	1	no	—
98	C₂²	—	508	1	no	—
99	C₂²	—	508	1	no	—
100	C₂²	—	508	1	no	—
101	C₂²	D₄×D₁₂₇	1	1	yes	characteristic
102	C₄	—	2	2	no	—
103	C₄	—	127	2	no	—
104	C₄	—	508	1	no	—
105	C₄	—	254	1	no	—
106	C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
107	C₄	#14523247	1	1	yes	characteristic
108	C₂	—	8	1	no	—
109	C₂	—	1016	1	no	—
110	C₂	#16861181	1	1	yes	characteristic
111	C₂	SD₁₆×D₁₂₇	1	1	yes	characteristic
112	C₂	#16861236	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14523235

C₂

yes

maximal, characteristic, complemented

#16861205

C₂

yes

maximal, characteristic

Q₈×Dic₁₂₇

C₂

yes

maximal, characteristic, complemented

Dic₁₂₇⋊C₈

C₂

yes

maximal, characteristic, complemented

#16861131

C₂

yes

maximal, characteristic, complemented

#16861130

C₂

yes

maximal, characteristic, complemented

C₂×SD₁₆×C₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂×D₅₀₈

C₂²

yes

characteristic

C₂×C₂²⋊D₁₂₇

—

C₂×C₂²⋊D₁₂₇

—

#14523235

—

C₂×C₁₂₇⋊C₈

C₂²

yes

characteristic

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

C₂²

yes

characteristic

C₄×Dic₁₂₇

—

C₄×Dic₁₂₇

C₂²

yes

characteristic

C₂×Q₈×C₁₂₇

C₂²

yes

characteristic

C₂×D₄×C₁₂₇

C₂²

yes

characteristic

SD₁₆×C₁₂₇

—

C₂×C₈×C₁₂₇

C₂²

yes

characteristic

C₂²×D₁₂₇

—

D₅₀₈

—

D₅₀₈

—

C₂²⋊D₁₂₇

—

C₂²⋊D₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

D₄

yes

characteristic

C₂×Dic₁₂₇

D₄

yes

characteristic

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

—

Q₈×C₁₂₇

—

Q₈×C₁₂₇

D₄

yes

complemented

D₄×C₁₂₇

—

D₄×C₁₂₇

—

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

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₄×C₁₂₇

#41

yes

characteristic

Dic₁₂₇

—

Dic₁₂₇

—

Dic₁₂₇

SD₁₆

yes

complemented

C₂×C₁₂₇

—

C₂×C₁₂₇

#121

yes

characteristic

C₂×C₁₂₇

C₂×SD₁₆

yes

characteristic

C₂×C₁₂₇

AGL(1,ℤ/8)

yes

characteristic

D₁₂₇

—

C₁₂₇

#460

yes

characteristic, complemented, Sylow

#460

—

127

maximal, Sylow

C₂×SD₁₆

—

127

—

C₂×SD₁₆

D₁₂₇

yes

characteristic, complemented

#127

—

127

—

C₄×Q₈

—

127

—

C₄⋊C₈

—

127

—

#102

—

127

—

#102

—

127

—

C₂×Q₈

C₂×D₁₂₇

yes

characteristic

C₂×D₄

—

127

—

C₂×D₄

—

254

—

C₂×D₄

—

254

—

C₂×D₄

C₂×D₁₂₇

yes

characteristic

SD₁₆

—

SD₁₆

—

254

—

C₂×C₈

—

127

—

C₂×C₈

C₂×D₁₂₇

yes

characteristic

C₄⋊C₄

—

127

—

C₄⋊C₄

—

254

—

C₄²

—

127

—

C₄²

—

254

—

C₂³

—

C₂³

—

254

—

Q₈

—

Q₈

C₂²⋊D₁₂₇

yes

complemented

D₄

—

D₄

—

D₄

—

508

—

D₄

—

508

—

D₄

—

254

—

D₄

—

254

—

C₂×C₄

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

254

—

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

C₈

—

C₈

—

254

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

508

—

C₂²

—

508

—

100

C₂²

—

508

—

101

C₂²

D₄×D₁₂₇

yes

characteristic

102

C₄

—

103

C₄

—

127

—

104

C₄

—

508

—

105

C₄

—

254

—

106

C₄

C₂×C₂²⋊D₁₂₇

yes

characteristic

107

C₄

#14523247

yes

characteristic

108

C₂

—

109

C₂

—

1016

—

110

C₂

#16861181

yes

characteristic

111

C₂

SD₁₆×D₁₂₇

yes

characteristic

112

C₂

#16861236

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×D₁₂₇
commutator	C₂×C₄×C₁₂₇	C₂³
Frattini	C₂×C₄	C₂²×D₁₂₇
Fitting	C₂×SD₁₆×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂×D₄
hypercenter	C₂×SD₁₆	D₁₂₇
hypocenter	C₁₂₇	#460
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×SD₁₆

subgroup

G/H

center

C₂²

D₄×D₁₂₇

commutator

C₂×C₄×C₁₂₇

C₂³

Frattini

C₂×C₄

C₂²×D₁₂₇

Fitting

C₂×SD₁₆×C₁₂₇

C₂

socle

C₂²×C₁₂₇

C₂×D₄

hypercenter

C₂×SD₁₆

D₁₂₇

hypocenter

C₁₂₇

#460

LCS

G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×SD₁₆

N	H ≅ G/N	type
1. C₂×#14523235	C₂	semidirect
3. Q₈×Dic₁₂₇	C₂	semidirect
4. Dic₁₂₇⋊C₈	C₂	semidirect
5. #16861131	C₂	semidirect
6. #16861130	C₂	semidirect
7. C₂×SD₁₆×C₁₂₇	C₂	semidirect
32. Q₈×C₁₂₇	D₄	semidirect
50. Dic₁₂₇	SD₁₆	semidirect
56. C₁₂₇	#460	semidirect
59. C₂×SD₁₆	D₁₂₇	semidirect
81. Q₈	C₂²⋊D₁₂₇	semidirect

H ≅ G/N

type

1. C₂×#14523235

C₂

semidirect

3. Q₈×Dic₁₂₇

C₂

semidirect

4. Dic₁₂₇⋊C₈

C₂

semidirect

5. #16861131

C₂

semidirect

6. #16861130

C₂

semidirect

7. C₂×SD₁₆×C₁₂₇

C₂

semidirect

32. Q₈×C₁₂₇

D₄

semidirect

50. Dic₁₂₇

SD₁₆

semidirect

56. C₁₂₇

#460

semidirect

59. C₂×SD₁₆

D₁₂₇

semidirect

81. Q₈

C₂²⋊D₁₂₇

semidirect

Subgroups — #22550074

Conjugacy classes of subgroups

Complements