Group #22550197 — subgroups

Subgroups — #22550197

Order 8128.

106 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 106 conjugacy classes of subgroups — too many to draw. Only its 39 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	#16861168	C₂	1	1	yes	maximal, characteristic, complemented
2	#16861142	C₂	1	1	yes	maximal, characteristic, complemented
3	#16861140	C₂	1	1	yes	maximal, characteristic, complemented
4	#16861130	C₂	1	1	yes	maximal, characteristic, complemented
5	#16861130	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²⋊(C₁₂₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
7	#121×C₁₂₇	C₂	1	1	yes	maximal, characteristic
8	C₂×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₁₂₇	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₁₂₇	C₂²	1	1	yes	characteristic
12	C₂²⋊Dic₁₂₇	—	2	1	no	—
13	C₄×Dic₁₂₇	—	2	1	no	—
14	C₂×D₄×C₁₂₇	—	2	1	no	—
15	C₂×D₄×C₁₂₇	C₂²	1	1	yes	characteristic
16	C₂²×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
17	C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
18	C₂²⋊C₄×C₁₂₇	—	2	1	no	—
19	C₂×Dic₁₂₇	—	2	1	no	—
20	C₂×Dic₁₂₇	—	2	1	no	—
21	C₁₂₇⋊C₈	—	2	1	no	—
22	C₁₂₇⋊C₈	—	2	1	no	—
23	C₂³×C₁₂₇	—	2	1	no	—
24	C₂³×C₁₂₇	D₄	1	1	yes	characteristic
25	D₄×C₁₂₇	—	2	1	no	—
26	D₄×C₁₂₇	—	2	1	no	—
27	D₄×C₁₂₇	—	4	2	no	—
28	C₂×C₄×C₁₂₇	—	2	1	no	—
29	C₂×C₄×C₁₂₇	—	2	1	no	—
30	C₂×C₄×C₁₂₇	—	2	1	no	—
31	C₂×C₄×C₁₂₇	D₄	1	1	yes	characteristic
32	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
33	C₂²×C₁₂₇	—	2	1	no	—
34	C₂²×C₁₂₇	—	2	1	no	—
35	C₂²×C₁₂₇	—	2	1	no	—
36	C₂²×C₁₂₇	—	4	1	no	—
37	C₂²×C₁₂₇	—	4	1	no	—
38	C₂²×C₁₂₇	—	4	1	no	—
39	C₂²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
40	C₄×C₁₂₇	—	2	1	no	—
41	C₄×C₁₂₇	—	4	1	no	—
42	C₄×C₁₂₇	#41	1	1	yes	characteristic
43	C₄×C₁₂₇	#41	1	1	yes	characteristic
44	Dic₁₂₇	—	4	1	no	—
45	Dic₁₂₇	—	4	1	no	—
46	C₂×C₁₂₇	—	4	1	no	—
47	C₂×C₁₂₇	—	8	1	no	—
48	C₂×C₁₂₇	#123	1	1	yes	characteristic
49	C₂×C₁₂₇	#135	1	1	yes	characteristic
50	C₂×C₁₂₇	AGL(1,ℤ/8)	1	1	yes	characteristic
51	C₁₂₇	#482	1	1	yes	characteristic, complemented, Sylow
52	#482	—	127	1	no	maximal, Sylow
53	#121	D₁₂₇	1	1	yes	characteristic
54	#117	—	127	1	no	—
55	#107	—	127	1	no	—
56	#106	—	127	1	no	—
57	#102	—	127	1	no	—
58	#102	—	127	1	no	—
59	C₂²⋊C₈	—	127	1	no	—
60	C₂×D₄	—	2	1	no	—
61	C₂×D₄	C₂×D₁₂₇	1	1	yes	characteristic
62	C₂²×C₄	C₂×D₁₂₇	1	1	yes	characteristic
63	C₂×C₈	—	127	1	no	—
64	C₂×C₈	—	127	1	no	—
65	C₄⋊C₄	—	127	1	no	—
66	C₄⋊C₄	—	127	1	no	—
67	C₄⋊C₄	C₂×D₁₂₇	1	1	yes	characteristic
68	C₂²⋊C₄	—	2	1	no	—
69	C₂²⋊C₄	—	254	1	no	—
70	C₄²	—	254	1	no	—
71	C₂³	—	2	1	no	—
72	C₂³	C₂²⋊D₁₂₇	1	1	yes	characteristic
73	D₄	—	2	1	no	—
74	D₄	—	2	1	no	—
75	D₄	—	4	2	no	—
76	C₂×C₄	—	2	1	no	—
77	C₂×C₄	—	2	1	no	—
78	C₂×C₄	—	2	1	no	—
79	C₂×C₄	—	254	1	no	—
80	C₂×C₄	—	254	1	no	—
81	C₂×C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
82	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
83	C₈	—	254	1	no	—
84	C₈	—	254	1	no	—
85	C₂²	—	2	1	no	—
86	C₂²	—	2	1	no	—
87	C₂²	—	2	1	no	—
88	C₂²	—	4	1	no	—
89	C₂²	—	4	1	no	—
90	C₂²	—	4	1	no	—
91	C₂²	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
92	C₄	—	2	1	no	—
93	C₄	—	4	1	no	—
94	C₄	—	508	1	no	—
95	C₄	—	508	1	no	—
96	C₄	#14523247	1	1	yes	characteristic
97	C₄	#14523247	1	1	yes	characteristic
98	C₂	—	4	1	no	—
99	C₂	—	8	1	no	—
100	C₂	#16861190	1	1	yes	characteristic
101	C₂	#16861229	1	1	yes	characteristic
102	C₂	#16861235	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16861168

C₂

yes

maximal, characteristic, complemented

#16861142

C₂

yes

maximal, characteristic, complemented

#16861140

C₂

yes

maximal, characteristic, complemented

#16861130

C₂

yes

maximal, characteristic, complemented

#16861130

C₂

yes

maximal, characteristic, complemented

C₂²⋊(C₁₂₇⋊C₈)

C₂

yes

maximal, characteristic, complemented

#121×C₁₂₇

C₂

yes

maximal, characteristic

C₂×C₁₂₇⋊C₈

C₂²

yes

characteristic

C₂×C₁₂₇⋊C₈

C₂²

yes

characteristic

C₄⋊Dic₁₂₇

C₂²

yes

characteristic

C₄⋊Dic₁₂₇

C₂²

yes

characteristic

C₂²⋊Dic₁₂₇

—

C₄×Dic₁₂₇

—

C₂×D₄×C₁₂₇

—

C₂×D₄×C₁₂₇

C₂²

yes

characteristic

C₂²×C₄×C₁₂₇

C₂²

yes

characteristic

C₄⋊C₄×C₁₂₇

C₂²

yes

characteristic

C₂²⋊C₄×C₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₁₂₇⋊C₈

—

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

—

C₂³×C₁₂₇

D₄

yes

characteristic

D₄×C₁₂₇

—

D₄×C₁₂₇

—

D₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

D₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂³

yes

characteristic

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₂×D₄

yes

characteristic

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

yes

C₄×C₁₂₇

yes

Dic₁₂₇

—

Dic₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

C₁₂₇

yes

characteristic, complemented, Sylow

#482

—

127

maximal, Sylow

#121

D₁₂₇

yes

characteristic

#117

—

127

—

#107

—

127

—

#106

—

127

—

#102

—

127

—

#102

—

127

—

C₂²⋊C₈

—

127

—

C₂×D₄

—

C₂×D₄

C₂×D₁₂₇

yes

characteristic

C₂²×C₄

C₂×D₁₂₇

yes

characteristic

C₂×C₈

—

127

—

C₂×C₈

—

127

—

C₄⋊C₄

—

127

—

C₄⋊C₄

—

127

—

C₄⋊C₄

C₂×D₁₂₇

yes

characteristic

C₂²⋊C₄

—

C₂²⋊C₄

—

254

—

C₄²

—

254

—

C₂³

—

C₂³

C₂²⋊D₁₂₇

yes

characteristic

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

254

—

C₂×C₄

—

254

—

C₂×C₄

C₂²⋊D₁₂₇

yes

characteristic

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

C₈

—

254

—

C₈

—

254

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

C₂×C₂²⋊D₁₂₇

yes

characteristic

C₄

—

C₄

—

C₄

—

508

—

C₄

—

508

—

C₄

yes

C₄

yes

C₂

—

C₂

—

100

C₂

#16861190

yes

characteristic

101

C₂

#16861229

yes

characteristic

102

C₂

#16861235

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₂₇
commutator	C₂×C₄×C₁₂₇	C₂³
Frattini	C₂×C₄	C₂²×D₁₂₇
Fitting	#121×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂×D₄
hypercenter	#121	D₁₂₇
hypocenter	C₁₂₇	#482
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #121

subgroup

G/H

center

C₂²

C₂×C₂²⋊D₁₂₇

commutator

C₂×C₄×C₁₂₇

C₂³

Frattini

C₂×C₄

C₂²×D₁₂₇

Fitting

#121×C₁₂₇

C₂

socle

C₂²×C₁₂₇

C₂×D₄

D₁₂₇

C₁₂₇

G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #121

N	H ≅ G/N	type
1. #16861168	C₂	semidirect
2. #16861142	C₂	semidirect
3. #16861140	C₂	semidirect
4. #16861130	C₂	semidirect
5. #16861130	C₂	semidirect
6. C₂²⋊(C₁₂₇⋊C₈)	C₂	semidirect
51. C₁₂₇	#482	semidirect

H ≅ G/N

type

1. #16861168

C₂

semidirect

2. #16861142

C₂

semidirect

3. #16861140

C₂

semidirect

4. #16861130

C₂

semidirect

5. #16861130

C₂

semidirect

6. C₂²⋊(C₁₂₇⋊C₈)

C₂

semidirect

51. C₁₂₇

#482

semidirect

Subgroups — #22550197

Conjugacy classes of subgroups

Complements