Group #22550596 — subgroups

Subgroups — #22550596

Order 8128.

212 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 99 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	#16861209	C₂	1	1	yes	maximal, characteristic, complemented
2	#16861202	C₂	1	4	yes	maximal, complemented
3	#16861196	C₂	1	2	yes	maximal
4	#16861188	C₂	1	2	yes	maximal, complemented
5	C₄×Dic₂₅₄	C₂	1	1	yes	maximal, characteristic, complemented
6	#16861174	C₂	1	2	yes	maximal, complemented
7	#16861168	C₂	1	2	yes	maximal, complemented
8	C₄×D₄×C₁₂₇	C₂	1	1	yes	maximal, characteristic
9	C₂×Dic₂₅₄	C₂²	1	1	yes	characteristic
10	C₂×Dic₂₅₄	C₂²	1	2	yes	—
11	C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
12	C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
13	C₄⋊Dic₁₂₇	C₂²	1	1	yes	characteristic
14	Dic₁₂₇⋊C₄	C₂²	1	4	yes	—
15	Dic₁₂₇⋊C₄	C₂²	1	2	yes	—
16	C₂²⋊Dic₁₂₇	C₂²	1	4	yes	—
17	C₂²⋊Dic₁₂₇	C₂²	1	4	yes	—
18	C₄×Dic₁₂₇	C₂²	1	2	yes	—
19	C₄×Dic₁₂₇	C₂²	1	4	yes	—
20	C₂×D₄×C₁₂₇	C₂²	1	1	yes	characteristic
21	C₂²×C₄×C₁₂₇	C₂²	1	2	yes	—
22	C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
23	C₂²⋊C₄×C₁₂₇	C₂²	1	2	yes	—
24	C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
25	Dic₂₅₄	—	2	2	no	—
26	Dic₂₅₄	—	2	4	no	—
27	C₂×Dic₁₂₇	C₂³	1	2	yes	—
28	C₂×Dic₁₂₇	C₂³	1	2	yes	—
29	C₂×Dic₁₂₇	C₂³	1	4	yes	—
30	C₂³×C₁₂₇	C₂³	1	2	yes	—
31	D₄×C₁₂₇	—	2	2	no	—
32	C₂×C₄×C₁₂₇	—	2	4	no	—
33	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
34	C₂×C₄×C₁₂₇	C₂³	1	2	yes	—
35	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
36	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
37	C₂²×C₁₂₇	—	2	2	no	—
38	C₂²×C₁₂₇	—	2	2	no	—
39	C₂²×C₁₂₇	—	2	2	no	—
40	C₂²×C₁₂₇	C₂⁴	1	1	yes	characteristic
41	C₄×C₁₂₇	—	2	2	no	—
42	C₄×C₁₂₇	—	2	1	no	—
43	C₄×C₁₂₇	#41	1	2	yes	—
44	C₄×C₁₂₇	#41	1	2	yes	—
45	Dic₁₂₇	—	2	2	no	—
46	Dic₁₂₇	—	2	4	no	—
47	Dic₁₂₇	—	2	2	no	—
48	C₂×C₁₂₇	—	4	2	no	—
49	C₂×C₁₂₇	C₂×#41	1	1	yes	characteristic
50	C₂×C₁₂₇	C₂×#41	1	1	yes	characteristic
51	C₂×C₁₂₇	2₋¹⁺⁴	1	1	yes	characteristic
52	C₁₂₇	#556	1	1	yes	characteristic, complemented, Sylow
53	#556	—	127	1	no	maximal, Sylow
54	C₄⋊Q₈	—	127	1	no	—
55	#126	—	127	4	no	—
56	#124	—	127	2	no	—
57	C₂²⋊Q₈	—	127	2	no	—
58	C₄×Q₈	—	127	2	no	—
59	C₄×Q₈	—	127	1	no	—
60	C₄×D₄	D₁₂₇	1	1	yes	characteristic
61	#117	—	127	2	no	—
62	C₂×Q₈	—	127	1	no	—
63	C₂×Q₈	—	127	2	no	—
64	C₂×D₄	C₂×D₁₂₇	1	1	yes	characteristic
65	C₂²×C₄	C₂×D₁₂₇	1	2	yes	—
66	C₄⋊C₄	—	127	4	no	—
67	C₄⋊C₄	—	127	2	no	—
68	C₄⋊C₄	—	127	1	no	—
69	C₄⋊C₄	—	127	2	no	—
70	C₄⋊C₄	—	127	2	no	—
71	C₄⋊C₄	C₂×D₁₂₇	1	1	yes	characteristic
72	C₂²⋊C₄	—	127	4	no	—
73	C₂²⋊C₄	—	127	4	no	—
74	C₂²⋊C₄	C₂×D₁₂₇	1	2	yes	—
75	C₄²	—	127	4	no	—
76	C₄²	—	127	2	no	—
77	C₄²	C₂×D₁₂₇	1	1	yes	characteristic
78	C₂³	C₂²×D₁₂₇	1	2	yes	—
79	Q₈	—	254	2	no	—
80	Q₈	—	254	4	no	—
81	D₄	—	2	2	no	—
82	C₂×C₄	—	2	4	no	—
83	C₂×C₄	—	127	2	no	—
84	C₂×C₄	—	127	2	no	—
85	C₂×C₄	—	127	4	no	—
86	C₂×C₄	C₂²×D₁₂₇	1	2	yes	—
87	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
88	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
89	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
90	C₂²	—	2	2	no	—
91	C₂²	—	2	2	no	—
92	C₂²	—	2	2	no	—
93	C₂²	C₂³×D₁₂₇	1	1	yes	characteristic
94	C₄	—	2	2	no	—
95	C₄	—	2	1	no	—
96	C₄	—	254	2	no	—
97	C₄	—	254	4	no	—
98	C₄	—	254	2	no	—
99	C₄	#14523247	1	2	yes	—
100	C₄	#14523248	1	2	yes	—
101	C₂	—	4	2	no	—
102	C₂	C₂×#14523247	1	1	yes	characteristic
103	C₂	C₂×#14523248	1	1	yes	characteristic
104	C₂	#16861256	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16861209

C₂

yes

maximal, characteristic, complemented

#16861202

C₂

yes

maximal, complemented

#16861196

C₂

yes

maximal

#16861188

C₂

yes

maximal, complemented

C₄×Dic₂₅₄

C₂

yes

maximal, characteristic, complemented

#16861174

C₂

yes

maximal, complemented

#16861168

C₂

yes

maximal, complemented

C₄×D₄×C₁₂₇

C₂

yes

maximal, characteristic

C₂×Dic₂₅₄

C₂²

yes

characteristic

C₂×Dic₂₅₄

C₂²

yes

—

C₄⋊Dic₁₂₇

C₂²

yes

—

C₄⋊Dic₁₂₇

C₂²

yes

—

C₄⋊Dic₁₂₇

C₂²

yes

characteristic

Dic₁₂₇⋊C₄

C₂²

yes

—

Dic₁₂₇⋊C₄

C₂²

yes

—

C₂²⋊Dic₁₂₇

C₂²

yes

—

C₂²⋊Dic₁₂₇

C₂²

yes

—

C₄×Dic₁₂₇

C₂²

yes

—

C₄×Dic₁₂₇

C₂²

yes

—

C₂×D₄×C₁₂₇

C₂²

yes

characteristic

C₂²×C₄×C₁₂₇

C₂²

yes

—

C₄⋊C₄×C₁₂₇

C₂²

yes

characteristic

C₂²⋊C₄×C₁₂₇

C₂²

yes

—

C₄²×C₁₂₇

C₂²

yes

characteristic

Dic₂₅₄

—

Dic₂₅₄

—

C₂×Dic₁₂₇

C₂³

yes

—

C₂×Dic₁₂₇

C₂³

yes

—

C₂×Dic₁₂₇

C₂³

yes

—

C₂³×C₁₂₇

C₂³

yes

—

D₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂³

yes

characteristic

C₂×C₄×C₁₂₇

C₂³

yes

—

C₂×C₄×C₁₂₇

C₂³

yes

characteristic

C₂×C₄×C₁₂₇

C₂³

yes

characteristic

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₂⁴

yes

characteristic

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

#41

yes

—

C₄×C₁₂₇

#41

yes

—

Dic₁₂₇

—

Dic₁₂₇

—

Dic₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

C₂×#41

yes

characteristic

C₂×C₁₂₇

C₂×#41

yes

characteristic

C₂×C₁₂₇

2₋¹⁺⁴

yes

characteristic

C₁₂₇

#556

yes

characteristic, complemented, Sylow

#556

—

127

maximal, Sylow

C₄⋊Q₈

—

127

—

#126

—

127

—

#124

—

127

—

C₂²⋊Q₈

—

127

—

C₄×Q₈

—

127

—

C₄×Q₈

—

127

—

C₄×D₄

D₁₂₇

yes

characteristic

#117

—

127

—

C₂×Q₈

—

127

—

C₂×Q₈

—

127

—

C₂×D₄

C₂×D₁₂₇

yes

characteristic

C₂²×C₄

C₂×D₁₂₇

yes

—

C₄⋊C₄

—

127

—

C₄⋊C₄

—

127

—

C₄⋊C₄

—

127

—

C₄⋊C₄

—

127

—

C₄⋊C₄

—

127

—

C₄⋊C₄

C₂×D₁₂₇

yes

characteristic

C₂²⋊C₄

—

127

—

C₂²⋊C₄

—

127

—

C₂²⋊C₄

C₂×D₁₂₇

yes

—

C₄²

—

127

—

C₄²

—

127

—

C₄²

C₂×D₁₂₇

yes

characteristic

C₂³

C₂²×D₁₂₇

yes

—

Q₈

—

254

—

Q₈

—

254

—

D₄

—

C₂×C₄

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

C₂²×D₁₂₇

yes

—

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

C₂²

—

C₂²

—

C₂²

—

C₂²

C₂³×D₁₂₇

yes

characteristic

C₄

—

C₄

—

C₄

—

254

—

C₄

—

254

—

C₄

—

254

—

C₄

#14523247

yes

—

100

C₄

#14523248

yes

—

101

C₂

—

102

C₂

C₂×#14523247

yes

characteristic

103

C₂

C₂×#14523248

yes

characteristic

104

C₂

#16861256

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₂₇
commutator, socle	C₂²×C₁₂₇	C₂⁴
Fitting	C₄×D₄×C₁₂₇	C₂
hypercenter	C₄×D₄	D₁₂₇
hypocenter	C₁₂₇	#556
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₄×D₄

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₂₇

commutator, socle

C₂²×C₁₂₇

C₂⁴

Fitting

C₄×D₄×C₁₂₇

C₂

hypercenter

C₄×D₄

D₁₂₇

C₁₂₇

G ▷ C₂²×C₁₂₇ ▷ C₁₂₇

C₁ ◁ C₂² ◁ C₄×D₄

N	H ≅ G/N	type
1. #16861209	C₂	semidirect
2. #16861202	C₂	semidirect
4. #16861188	C₂	semidirect
5. C₄×Dic₂₅₄	C₂	semidirect
6. #16861174	C₂	semidirect
7. #16861168	C₂	semidirect
52. C₁₂₇	#556	semidirect

H ≅ G/N

type

1. #16861209

C₂

semidirect

2. #16861202

C₂

semidirect

4. #16861188

C₂

semidirect

5. C₄×Dic₂₅₄

C₂

semidirect

6. #16861174

C₂

semidirect

7. #16861168

C₂

semidirect

52. C₁₂₇

#556

semidirect

Subgroups — #22550596

Conjugacy classes of subgroups

Complements