Group #22550541 — subgroups

Subgroups — #22550541

Order 8128.

280 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 115 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×Q₈×D₁₂₇	C₂	1	2	yes	maximal, complemented
2	#16861208	C₂	1	2	yes	maximal, complemented
3	#16861187	C₂	1	4	yes	maximal, complemented
4	#16861185	C₂	1	2	yes	maximal, complemented
5	C₄×Dic₂₅₄	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×D₅₀₈	C₂	1	1	yes	maximal, characteristic
7	#16861171	C₂	1	2	yes	maximal
8	C₄⋊Q₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×Dic₂₅₄	C₂²	1	4	yes	—
10	C₂×Dic₂₅₄	C₂²	1	1	yes	characteristic
11	Q₈×D₁₂₇	—	2	8	no	—
12	C₂×D₅₀₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₁₂₇	C₂²	1	2	yes	—
14	C₂×C₄×D₁₂₇	C₂²	1	4	yes	—
15	C₄⋊Dic₁₂₇	C₂²	1	1	yes	characteristic
16	C₄⋊Dic₁₂₇	C₂²	1	1	yes	characteristic
17	Dic₁₂₇⋊C₄	C₂²	1	4	yes	—
18	Dic₁₂₇⋊C₄	C₂²	1	2	yes	—
19	(C₂×D₁₂₇)⋊C₄	C₂²	1	4	yes	—
20	(C₂×D₁₂₇)⋊C₄	C₂²	1	2	yes	—
21	C₄×Dic₁₂₇	C₂²	1	2	yes	—
22	C₂×Q₈×C₁₂₇	C₂²	1	2	yes	—
23	C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
24	C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
25	C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
26	C₂²×D₁₂₇	C₂³	1	2	yes	—
27	Dic₂₅₄	—	2	8	no	—
28	Dic₂₅₄	D₄	1	4	yes	complemented
29	D₅₀₈	Q₈	1	4	yes	complemented
30	C₂×Dic₁₂₇	C₂³	1	2	yes	—
31	C₂×Dic₁₂₇	C₂³	1	2	yes	—
32	C₂×Dic₁₂₇	C₂³	1	2	yes	—
33	C₄×D₁₂₇	—	2	4	no	—
34	C₄×D₁₂₇	—	2	8	no	—
35	Q₈×C₁₂₇	—	2	4	no	—
36	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
37	C₂×C₄×C₁₂₇	C₂³	1	4	yes	—
38	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
39	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
40	C₂²×C₁₂₇	C₂⁴	1	1	yes	characteristic
41	C₂×D₁₂₇	—	2	2	no	—
42	C₂×D₁₂₇	—	2	2	no	—
43	C₂×D₁₂₇	C₂×Q₈	1	4	yes	complemented
44	C₄×C₁₂₇	—	2	4	no	—
45	C₄×C₁₂₇	—	2	1	no	—
46	C₄×C₁₂₇	C₂×D₄	1	2	yes	—
47	C₄×C₁₂₇	C₂×Q₈	1	2	yes	complemented
48	Dic₁₂₇	—	2	2	no	—
49	Dic₁₂₇	—	2	2	no	—
50	Dic₁₂₇	C₂×D₄	1	4	yes	—
51	C₂×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
52	C₂×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
53	C₂×C₁₂₇	2₋¹⁺⁴	1	1	yes	characteristic
54	D₁₂₇	—	2	4	no	—
55	C₁₂₇	D₄×Q₈	1	1	yes	characteristic, complemented, Sylow
56	D₄×Q₈	—	127	1	no	maximal, Sylow
57	C₂²×Q₈	—	127	2	no	—
58	C₄⋊Q₈	—	127	2	no	—
59	C₄⋊Q₈	D₁₂₇	1	1	yes	characteristic, complemented
60	C₂²⋊Q₈	—	127	2	no	—
61	C₂²⋊Q₈	—	127	4	no	—
62	C₄×Q₈	—	127	1	no	—
63	C₄×D₄	—	127	2	no	—
64	C₄×D₄	—	127	1	no	—
65	C₂×Q₈	—	127	1	no	—
66	C₂×Q₈	—	127	4	no	—
67	C₂×Q₈	—	254	8	no	—
68	C₂×Q₈	C₂×D₁₂₇	1	2	yes	—
69	C₂×D₄	—	127	1	no	—
70	C₂²×C₄	—	127	4	no	—
71	C₂²×C₄	—	127	2	no	—
72	C₄⋊C₄	—	127	2	no	—
73	C₄⋊C₄	—	127	4	no	—
74	C₄⋊C₄	—	127	1	no	—
75	C₄⋊C₄	—	127	1	no	—
76	C₄⋊C₄	C₂×D₁₂₇	1	2	yes	—
77	C₄⋊C₄	C₂×D₁₂₇	1	2	yes	—
78	C₂²⋊C₄	—	127	4	no	—
79	C₂²⋊C₄	—	127	2	no	—
80	C₄²	—	127	2	no	—
81	C₄²	C₂×D₁₂₇	1	1	yes	characteristic
82	C₂³	—	127	2	no	—
83	Q₈	—	2	4	no	—
84	Q₈	—	127	4	no	—
85	Q₈	—	254	8	no	—
86	D₄	—	127	4	no	—
87	C₂×C₄	—	127	2	no	—
88	C₂×C₄	—	127	2	no	—
89	C₂×C₄	—	127	2	no	—
90	C₂×C₄	—	254	8	no	—
91	C₂×C₄	—	254	4	no	—
92	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
93	C₂×C₄	C₂²×D₁₂₇	1	4	yes	—
94	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
95	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
96	C₂²	—	127	4	no	—
97	C₂²	—	254	2	no	—
98	C₂²	—	254	2	no	—
99	C₂²	C₂³×D₁₂₇	1	1	yes	characteristic
100	C₄	—	2	4	no	—
101	C₄	—	2	1	no	—
102	C₄	—	127	4	no	—
103	C₄	—	254	2	no	—
104	C₄	—	254	2	no	—
105	C₄	D₄×D₁₂₇	1	2	yes	—
106	C₄	Q₈×D₁₂₇	1	2	yes	complemented
107	C₂	—	254	4	no	—
108	C₂	C₂×D₄×D₁₂₇	1	1	yes	characteristic
109	C₂	C₂×Q₈×D₁₂₇	1	1	yes	characteristic
110	C₂	#16861257	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×D₁₂₇

C₂

yes

maximal, complemented

#16861208

C₂

yes

maximal, complemented

#16861187

C₂

yes

maximal, complemented

#16861185

C₂

yes

maximal, complemented

C₄×Dic₂₅₄

C₂

yes

maximal, characteristic, complemented

C₄×D₅₀₈

C₂

yes

maximal, characteristic

#16861171

C₂

yes

maximal

C₄⋊Q₈×C₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₅₄

C₂²

yes

—

C₂×Dic₂₅₄

C₂²

yes

characteristic

Q₈×D₁₂₇

—

C₂×D₅₀₈

C₂²

yes

characteristic

C₂×C₄×D₁₂₇

C₂²

yes

—

C₂×C₄×D₁₂₇

C₂²

yes

—

C₄⋊Dic₁₂₇

C₂²

yes

characteristic

C₄⋊Dic₁₂₇

C₂²

yes

characteristic

Dic₁₂₇⋊C₄

C₂²

yes

—

Dic₁₂₇⋊C₄

C₂²

yes

—

(C₂×D₁₂₇)⋊C₄

C₂²

yes

—

(C₂×D₁₂₇)⋊C₄

C₂²

yes

—

C₄×Dic₁₂₇

C₂²

yes

—

C₂×Q₈×C₁₂₇

C₂²

yes

—

C₄⋊C₄×C₁₂₇

C₂²

yes

—

C₄⋊C₄×C₁₂₇

C₂²

yes

—

C₄²×C₁₂₇

C₂²

yes

characteristic

C₂²×D₁₂₇

C₂³

yes

—

Dic₂₅₄

—

Dic₂₅₄

D₄

yes

complemented

D₅₀₈

Q₈

yes

complemented

C₂×Dic₁₂₇

C₂³

yes

—

C₂×Dic₁₂₇

C₂³

yes

—

C₂×Dic₁₂₇

C₂³

yes

—

C₄×D₁₂₇

—

C₄×D₁₂₇

—

Q₈×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

C₂⁴

yes

characteristic

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

C₂×Q₈

yes

complemented

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂×D₄

yes

—

C₄×C₁₂₇

C₂×Q₈

yes

complemented

Dic₁₂₇

—

Dic₁₂₇

—

Dic₁₂₇

C₂×D₄

yes

—

C₂×C₁₂₇

C₂²×D₄

yes

characteristic

C₂×C₁₂₇

C₂²×Q₈

yes

characteristic

C₂×C₁₂₇

2₋¹⁺⁴

yes

characteristic

D₁₂₇

—

C₁₂₇

D₄×Q₈

yes

characteristic, complemented, Sylow

D₄×Q₈

—

127

maximal, Sylow

C₂²×Q₈

—

127

—

C₄⋊Q₈

—

127

—

C₄⋊Q₈

D₁₂₇

yes

characteristic, complemented

C₂²⋊Q₈

—

127

—

C₂²⋊Q₈

—

127

—

C₄×Q₈

—

127

—

C₄×D₄

—

127

—

C₄×D₄

—

127

—

C₂×Q₈

—

127

—

C₂×Q₈

—

127

—

C₂×Q₈

—

254

—

C₂×Q₈

C₂×D₁₂₇

yes

—

C₂×D₄

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

—

127

—

C₄⋊C₄

—

127

—

C₄⋊C₄

—

127

—

C₄⋊C₄

—

127

—

C₄⋊C₄

—

127

—

C₄⋊C₄

C₂×D₁₂₇

yes

—

C₄⋊C₄

C₂×D₁₂₇

yes

—

C₂²⋊C₄

—

127

—

C₂²⋊C₄

—

127

—

C₄²

—

127

—

C₄²

C₂×D₁₂₇

yes

characteristic

C₂³

—

127

—

Q₈

—

Q₈

—

127

—

Q₈

—

254

—

D₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

254

—

C₂×C₄

—

254

—

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

C₂×C₄

C₂²×D₁₂₇

yes

—

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

C₂²

—

127

—

C₂²

—

254

—

C₂²

—

254

—

C₂²

C₂³×D₁₂₇

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

—

127

—

103

C₄

—

254

—

104

C₄

—

254

—

105

C₄

D₄×D₁₂₇

yes

—

106

C₄

Q₈×D₁₂₇

yes

complemented

107

C₂

—

254

—

108

C₂

C₂×D₄×D₁₂₇

yes

characteristic

109

C₂

C₂×Q₈×D₁₂₇

yes

characteristic

110

C₂

#16861257

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₂₇
commutator, socle	C₂²×C₁₂₇	C₂⁴
Fitting	C₄⋊Q₈×C₁₂₇	C₂
hypercenter	C₄⋊Q₈	D₁₂₇
hypocenter	C₁₂₇	D₄×Q₈
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₂₇

commutator, socle

C₂²×C₁₂₇

C₂⁴

Fitting

C₄⋊Q₈×C₁₂₇

C₂

hypercenter

C₄⋊Q₈

D₁₂₇

hypocenter

C₁₂₇

D₄×Q₈

LCS

G ▷ C₂²×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂² ◁ C₄⋊Q₈

N	H ≅ G/N	type
1. C₂×Q₈×D₁₂₇	C₂	semidirect
2. #16861208	C₂	semidirect
3. #16861187	C₂	semidirect
4. #16861185	C₂	semidirect
5. C₄×Dic₂₅₄	C₂	semidirect
8. C₄⋊Q₈×C₁₂₇	C₂	semidirect
28. Dic₂₅₄	D₄	semidirect
29. D₅₀₈	Q₈	semidirect
43. C₂×D₁₂₇	C₂×Q₈	semidirect
47. C₄×C₁₂₇	C₂×Q₈	semidirect
55. C₁₂₇	D₄×Q₈	semidirect
59. C₄⋊Q₈	D₁₂₇	semidirect
106. C₄	Q₈×D₁₂₇	semidirect

H ≅ G/N

type

1. C₂×Q₈×D₁₂₇

C₂

semidirect

2. #16861208

C₂

semidirect

3. #16861187

C₂

semidirect

4. #16861185

C₂

semidirect

5. C₄×Dic₂₅₄

C₂

semidirect

8. C₄⋊Q₈×C₁₂₇

C₂

semidirect

28. Dic₂₅₄

D₄

semidirect

29. D₅₀₈

Q₈

semidirect

43. C₂×D₁₂₇

C₂×Q₈

semidirect

47. C₄×C₁₂₇

C₂×Q₈

semidirect

55. C₁₂₇

D₄×Q₈

semidirect

59. C₄⋊Q₈

D₁₂₇

semidirect

106. C₄

Q₈×D₁₂₇

semidirect

Subgroups — #22550541

Conjugacy classes of subgroups

Complements