Group #22550097 — subgroups

Subgroups — #22550097

Order 8128.

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

normal subgroups only

This group has 112 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	C₂×Dic₅₀₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14523236	C₂	1	1	yes	maximal, characteristic
3	#16861196	C₂	1	1	yes	maximal, characteristic
4	#16861174	C₂	1	1	yes	maximal, characteristic, complemented
5	Dic₁₂₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	#16861132	C₂	1	1	yes	maximal, characteristic, complemented
7	#102×C₁₂₇	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₂₅₄	C₂²	1	1	yes	characteristic
9	C₂×Dic₂₅₄	C₂²	1	1	yes	characteristic
10	Dic₅₀₈	—	2	2	no	—
11	#14523236	—	2	2	no	—
12	C₂×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
13	Dic₁₂₇⋊C₄	—	2	1	no	—
14	C₂²⋊Dic₁₂₇	—	2	1	no	—
15	C₂²⋊Dic₁₂₇	—	2	1	no	—
16	C₄×Dic₁₂₇	—	2	1	no	—
17	C₄×Dic₁₂₇	C₂²	1	1	yes	characteristic
18	C₂×D₄×C₁₂₇	C₂²	1	1	yes	characteristic
19	C₂×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
20	C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
21	Dic₂₅₄	—	2	1	no	—
22	Dic₂₅₄	—	2	1	no	—
23	Dic₂₅₄	—	2	1	no	—
24	Dic₂₅₄	D₄	1	2	yes	—
25	C₂×Dic₁₂₇	—	2	1	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	D₄×C₁₂₇	—	2	1	no	—
32	D₄×C₁₂₇	—	2	1	no	—
33	C₂×C₄×C₁₂₇	—	2	1	no	—
34	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
35	C₈×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	—	4	1	no	—
41	C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
42	C₄×C₁₂₇	#41	1	1	yes	characteristic
43	Dic₁₂₇	—	2	2	no	—
44	Dic₁₂₇	—	2	1	no	—
45	Dic₁₂₇	—	2	1	no	—
46	Dic₁₂₇	—	4	1	no	—
47	C₂×C₁₂₇	—	8	1	no	—
48	C₂×C₁₂₇	#121	1	1	yes	characteristic
49	C₂×C₁₂₇	#135	1	1	yes	characteristic
50	C₂×C₁₂₇	#137	1	1	yes	characteristic
51	C₁₂₇	#464	1	1	yes	characteristic, complemented, Sylow
52	#464	—	127	1	no	maximal, Sylow
53	C₂×Q₁₆	—	127	1	no	—
54	C₂×SD₁₆	—	127	1	no	—
55	#124	—	127	1	no	—
56	C₄×Q₈	—	127	1	no	—
57	C₄⋊C₈	—	127	1	no	—
58	#103	—	127	1	no	—
59	#102	D₁₂₇	1	1	yes	characteristic
60	C₂×Q₈	—	127	1	no	—
61	C₂×Q₈	—	127	1	no	—
62	C₂×D₄	C₂×D₁₂₇	1	1	yes	characteristic
63	Q₁₆	—	254	2	no	—
64	SD₁₆	—	254	2	no	—
65	C₂×C₈	—	127	1	no	—
66	C₂×C₈	C₂×D₁₂₇	1	1	yes	characteristic
67	C₄⋊C₄	—	254	1	no	—
68	C₄⋊C₄	C₂×D₁₂₇	1	1	yes	characteristic
69	C₂²⋊C₄	—	254	1	no	—
70	C₂²⋊C₄	—	254	1	no	—
71	C₄²	—	127	1	no	—
72	C₄²	—	254	1	no	—
73	C₂³	—	2	1	no	—
74	Q₈	—	127	2	no	—
75	Q₈	—	254	1	no	—
76	Q₈	—	254	1	no	—
77	Q₈	—	254	1	no	—
78	D₄	—	2	1	no	—
79	D₄	—	2	1	no	—
80	C₂×C₄	—	2	1	no	—
81	C₂×C₄	—	127	1	no	—
82	C₂×C₄	—	127	1	no	—
83	C₂×C₄	—	254	1	no	—
84	C₂×C₄	—	254	1	no	—
85	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
86	C₈	—	2	1	no	—
87	C₈	—	254	1	no	—
88	C₂²	—	4	1	no	—
89	C₂²	—	4	1	no	—
90	C₂²	—	4	1	no	—
91	C₂²	D₄×D₁₂₇	1	1	yes	characteristic
92	C₄	—	4	1	no	—
93	C₄	—	508	1	no	—
94	C₄	—	254	1	no	—
95	C₄	—	254	1	no	—
96	C₄	—	254	2	no	—
97	C₄	D₄×D₁₂₇	1	1	yes	characteristic
98	C₄	#14523248	1	1	yes	characteristic
99	C₂	—	8	1	no	—
100	C₂	#16861182	1	1	yes	characteristic
101	C₂	#16861228	1	1	yes	characteristic
102	C₂	#16861238	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₅₀₈

C₂

yes

maximal, characteristic, complemented

C₂×#14523236

C₂

yes

maximal, characteristic

#16861196

C₂

yes

maximal, characteristic

#16861174

C₂

yes

maximal, characteristic, complemented

Dic₁₂₇⋊C₈

C₂

yes

maximal, characteristic, complemented

#16861132

C₂

yes

maximal, characteristic, complemented

#102×C₁₂₇

C₂

yes

maximal, characteristic

C₂×Dic₂₅₄

C₂²

yes

characteristic

C₂×Dic₂₅₄

C₂²

yes

characteristic

Dic₅₀₈

—

#14523236

—

C₂×C₁₂₇⋊C₈

C₂²

yes

characteristic

Dic₁₂₇⋊C₄

—

C₂²⋊Dic₁₂₇

—

C₂²⋊Dic₁₂₇

—

C₄×Dic₁₂₇

—

C₄×Dic₁₂₇

C₂²

yes

characteristic

C₂×D₄×C₁₂₇

C₂²

yes

characteristic

C₂×C₈×C₁₂₇

C₂²

yes

characteristic

C₄⋊C₄×C₁₂₇

C₂²

yes

characteristic

Dic₂₅₄

—

Dic₂₅₄

—

Dic₂₅₄

—

Dic₂₅₄

D₄

yes

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

D₄

yes

characteristic

C₂×Dic₁₂₇

D₄

yes

characteristic

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

—

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

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₄×C₁₂₇

#41

yes

characteristic

Dic₁₂₇

—

Dic₁₂₇

—

Dic₁₂₇

—

Dic₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

C₁₂₇

yes

characteristic, complemented, Sylow

#464

—

127

maximal, Sylow

C₂×Q₁₆

—

127

—

C₂×SD₁₆

—

127

—

#124

—

127

—

C₄×Q₈

—

127

—

C₄⋊C₈

—

127

—

#103

—

127

—

#102

D₁₂₇

yes

characteristic

C₂×Q₈

—

127

—

C₂×Q₈

—

127

—

C₂×D₄

C₂×D₁₂₇

yes

characteristic

Q₁₆

—

254

—

SD₁₆

—

254

—

C₂×C₈

—

127

—

C₂×C₈

C₂×D₁₂₇

yes

characteristic

C₄⋊C₄

—

254

—

C₄⋊C₄

C₂×D₁₂₇

yes

characteristic

C₂²⋊C₄

—

254

—

C₂²⋊C₄

—

254

—

C₄²

—

127

—

C₄²

—

254

—

C₂³

—

Q₈

—

127

—

Q₈

—

254

—

Q₈

—

254

—

Q₈

—

254

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

254

—

C₂×C₄

—

254

—

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

C₈

—

C₈

—

254

—

C₂²

—

C₂²

—

C₂²

—

C₂²

D₄×D₁₂₇

yes

characteristic

C₄

—

C₄

—

508

—

C₄

—

254

—

C₄

—

254

—

C₄

—

254

—

C₄

D₄×D₁₂₇

yes

characteristic

C₄

#14523248

yes

characteristic

C₂

—

100

C₂

#16861182

yes

characteristic

101

C₂

#16861228

yes

characteristic

102

C₂

#16861238

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×D₁₂₇
commutator	C₂×C₄×C₁₂₇	C₂³
Frattini	C₂×C₄	C₂²×D₁₂₇
Fitting	#102×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂×D₄
hypercenter	#102	D₁₂₇
hypocenter	C₁₂₇	#464
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #102

subgroup

G/H

center

C₂²

D₄×D₁₂₇

commutator

C₂×C₄×C₁₂₇

C₂³

Frattini

C₂×C₄

C₂²×D₁₂₇

Fitting

#102×C₁₂₇

C₂

socle

C₂²×C₁₂₇

C₂×D₄

D₁₂₇

C₁₂₇

G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ #102

N	H ≅ G/N	type
1. C₂×Dic₅₀₈	C₂	semidirect
4. #16861174	C₂	semidirect
5. Dic₁₂₇⋊C₈	C₂	semidirect
6. #16861132	C₂	semidirect
51. C₁₂₇	#464	semidirect

H ≅ G/N

type

1. C₂×Dic₅₀₈

C₂

semidirect

4. #16861174

C₂

semidirect

5. Dic₁₂₇⋊C₈

C₂

semidirect

6. #16861132

C₂

semidirect

51. C₁₂₇

#464

semidirect

Subgroups — #22550097

Conjugacy classes of subgroups

Complements