Group #22549883 — subgroups

Subgroups — #22549883

Order 8128.

150 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 55 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	1	yes	maximal, characteristic
2	C₂×#14523231	C₂	1	1	yes	maximal, characteristic
3	#16861167	C₂	1	1	yes	maximal, characteristic
4	#16861134	C₂	1	1	yes	maximal, characteristic, complemented
5	#16861133	C₂	1	1	yes	maximal, characteristic, complemented
6	#16861132	C₂	1	1	yes	maximal, characteristic, complemented
7	#103×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₂₅₄	—	2	1	no	—
9	C₂×Dic₂₅₄	C₂²	1	1	yes	characteristic
10	Q₈×D₁₂₇	—	2	2	no	—
11	Q₈×D₁₂₇	C₄	1	4	yes	complemented
12	C₂×C₄×D₁₂₇	—	2	1	no	—
13	C₂×C₄×D₁₂₇	C₂²	1	1	yes	characteristic
14	#14523231	—	2	2	no	—
15	C₂×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
16	C₄⋊Dic₁₂₇	C₂²	1	1	yes	characteristic
17	(C₂×D₁₂₇)⋊C₄	—	2	1	no	—
18	C₄×Dic₁₂₇	—	2	1	no	—
19	C₂×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
20	C₂×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
21	C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
22	C₂²×D₁₂₇	D₄	1	1	yes	characteristic
23	Dic₂₅₄	—	2	1	no	—
24	Dic₂₅₄	—	2	4	no	—
25	Dic₂₅₄	C₂×C₄	1	2	yes	—
26	C₂×Dic₁₂₇	—	2	1	no	—
27	C₂×Dic₁₂₇	—	2	1	no	—
28	C₂×Dic₁₂₇	D₄	1	1	yes	characteristic
29	C₄×D₁₂₇	—	2	4	no	—
30	C₄×D₁₂₇	C₂×C₄	1	2	yes	—
31	C₄×D₁₂₇	D₄	1	2	yes	—
32	C₁₂₇⋊C₈	—	2	1	no	—
33	Q₈×C₁₂₇	—	2	1	no	—
34	Q₈×C₁₂₇	C₂×C₄	1	2	yes	—
35	C₂×C₄×C₁₂₇	—	2	1	no	—
36	C₂×C₄×C₁₂₇	—	2	1	no	—
37	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
38	C₈×C₁₂₇	—	2	1	no	—
39	C₂²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
40	C₂×D₁₂₇	—	2	1	no	—
41	C₂×D₁₂₇	—	2	1	no	—
42	C₂×D₁₂₇	C₂²⋊C₄	1	2	yes	—
43	C₄×C₁₂₇	—	2	2	no	—
44	C₄×C₁₂₇	—	4	1	no	—
45	C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
46	C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
47	Dic₁₂₇	—	2	2	no	—
48	Dic₁₂₇	—	4	1	no	—
49	Dic₁₂₇	C₂²⋊C₄	1	2	yes	—
50	C₂×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₁₂₇	#137	1	1	yes	characteristic
52	C₂×C₁₂₇	#137	1	1	yes	characteristic
53	D₁₂₇	—	2	2	no	—
54	C₁₂₇	#419	1	1	yes	characteristic, complemented, Sylow
55	#419	—	127	1	no	maximal, Sylow
56	C₂²×Q₈	—	127	1	no	—
57	C₂×M₁₆	—	127	1	no	—
58	#117	—	127	1	no	—
59	#103	—	127	1	no	—
60	#103	—	127	1	no	—
61	#103	—	127	1	no	—
62	#103	D₁₂₇	1	1	yes	characteristic, complemented
63	C₂×Q₈	—	127	4	no	—
64	C₂×Q₈	—	127	1	no	—
65	C₂×Q₈	—	254	1	no	—
66	C₂×Q₈	—	254	2	no	—
67	C₂×Q₈	C₂×D₁₂₇	1	1	yes	characteristic
68	C₂²×C₄	—	127	1	no	—
69	C₂²×C₄	—	254	1	no	—
70	M₁₆	—	254	2	no	—
71	C₂×C₈	—	127	1	no	—
72	C₂×C₈	C₂×D₁₂₇	1	1	yes	characteristic
73	C₄⋊C₄	—	127	1	no	—
74	C₄⋊C₄	C₂×D₁₂₇	1	1	yes	characteristic
75	C₂²⋊C₄	—	254	1	no	—
76	C₄²	—	254	1	no	—
77	C₂³	—	127	1	no	—
78	Q₈	—	2	1	no	—
79	Q₈	—	127	2	no	—
80	Q₈	—	254	4	no	—
81	Q₈	—	254	1	no	—
82	Q₈	C₄×D₁₂₇	1	2	yes	—
83	C₂×C₄	—	2	1	no	—
84	C₂×C₄	—	2	1	no	—
85	C₂×C₄	—	127	1	no	—
86	C₂×C₄	—	127	2	no	—
87	C₂×C₄	—	127	2	no	—
88	C₂×C₄	—	254	4	no	—
89	C₂×C₄	—	254	1	no	—
90	C₂×C₄	—	254	1	no	—
91	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
92	C₈	—	2	1	no	—
93	C₈	—	254	1	no	—
94	C₂²	—	127	2	no	—
95	C₂²	—	254	1	no	—
96	C₂²	—	254	1	no	—
97	C₂²	D₄×D₁₂₇	1	1	yes	characteristic
98	C₄	—	2	2	no	—
99	C₄	—	4	1	no	—
100	C₄	—	127	2	no	—
101	C₄	—	508	1	no	—
102	C₄	—	254	2	no	—
103	C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
104	C₄	D₄×D₁₂₇	1	1	yes	characteristic
105	C₂	—	254	2	no	—
106	C₂	C₂²⋊C₄×D₁₂₇	1	1	yes	characteristic
107	C₂	#16861238	1	1	yes	characteristic
108	C₂	#16861241	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×D₁₂₇

C₂

yes

maximal, characteristic

C₂×#14523231

C₂

yes

maximal, characteristic

#16861167

C₂

yes

maximal, characteristic

#16861134

C₂

yes

maximal, characteristic, complemented

#16861133

C₂

yes

maximal, characteristic, complemented

#16861132

C₂

yes

maximal, characteristic, complemented

#103×C₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₅₄

—

C₂×Dic₂₅₄

C₂²

yes

characteristic

Q₈×D₁₂₇

—

Q₈×D₁₂₇

C₄

yes

complemented

C₂×C₄×D₁₂₇

—

C₂×C₄×D₁₂₇

C₂²

yes

characteristic

#14523231

—

C₂×C₁₂₇⋊C₈

C₂²

yes

characteristic

C₄⋊Dic₁₂₇

C₂²

yes

characteristic

(C₂×D₁₂₇)⋊C₄

—

C₄×Dic₁₂₇

—

C₂×Q₈×C₁₂₇

C₂²

yes

characteristic

C₂×C₈×C₁₂₇

C₂²

yes

characteristic

C₄⋊C₄×C₁₂₇

C₂²

yes

characteristic

C₂²×D₁₂₇

D₄

yes

characteristic

Dic₂₅₄

—

Dic₂₅₄

—

Dic₂₅₄

C₂×C₄

yes

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

D₄

yes

characteristic

C₄×D₁₂₇

—

C₄×D₁₂₇

C₂×C₄

yes

—

C₄×D₁₂₇

D₄

yes

—

C₁₂₇⋊C₈

—

Q₈×C₁₂₇

—

Q₈×C₁₂₇

C₂×C₄

yes

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂³

yes

characteristic

C₈×C₁₂₇

—

C₂²×C₁₂₇

C₂×D₄

yes

characteristic

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

C₂²⋊C₄

yes

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂²×C₄

yes

characteristic

C₄×C₁₂₇

C₂×D₄

yes

characteristic

Dic₁₂₇

—

Dic₁₂₇

—

Dic₁₂₇

C₂²⋊C₄

yes

—

C₂×C₁₂₇

C₂×C₂²⋊C₄

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

D₁₂₇

—

C₁₂₇

#419

yes

characteristic, complemented, Sylow

#419

—

127

maximal, Sylow

C₂²×Q₈

—

127

—

C₂×M₁₆

—

127

—

#117

—

127

—

#103

—

127

—

#103

—

127

—

#103

—

127

—

#103

D₁₂₇

yes

characteristic, complemented

C₂×Q₈

—

127

—

C₂×Q₈

—

127

—

C₂×Q₈

—

254

—

C₂×Q₈

—

254

—

C₂×Q₈

C₂×D₁₂₇

yes

characteristic

C₂²×C₄

—

127

—

C₂²×C₄

—

254

—

M₁₆

—

254

—

C₂×C₈

—

127

—

C₂×C₈

C₂×D₁₂₇

yes

characteristic

C₄⋊C₄

—

127

—

C₄⋊C₄

C₂×D₁₂₇

yes

characteristic

C₂²⋊C₄

—

254

—

C₄²

—

254

—

C₂³

—

127

—

Q₈

—

Q₈

—

127

—

Q₈

—

254

—

Q₈

—

254

—

Q₈

C₄×D₁₂₇

yes

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

254

—

C₂×C₄

—

254

—

C₂×C₄

—

254

—

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

C₈

—

C₈

—

254

—

C₂²

—

127

—

C₂²

—

254

—

C₂²

—

254

—

C₂²

D₄×D₁₂₇

yes

characteristic

C₄

—

C₄

—

100

C₄

—

127

—

101

C₄

—

508

—

102

C₄

—

254

—

103

C₄

C₂×C₄×D₁₂₇

yes

characteristic

104

C₄

D₄×D₁₂₇

yes

characteristic

105

C₂

—

254

—

106

C₂

C₂²⋊C₄×D₁₂₇

yes

characteristic

107

C₂

#16861238

yes

characteristic

108

C₂

#16861241

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×D₁₂₇
commutator	C₄×C₁₂₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₁₂₇
Fitting	#103×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂×D₄
hypercenter	#103	D₁₂₇
hypocenter	C₁₂₇	#419
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #103

subgroup

G/H

center

C₂²

D₄×D₁₂₇

commutator

C₄×C₁₂₇

C₂²×C₄

Frattini

C₂×C₄

C₂²×D₁₂₇

Fitting

#103×C₁₂₇

C₂

socle

C₂²×C₁₂₇

C₂×D₄

D₁₂₇

C₁₂₇

G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ #103

N	H ≅ G/N	type
4. #16861134	C₂	semidirect
5. #16861133	C₂	semidirect
6. #16861132	C₂	semidirect
7. #103×C₁₂₇	C₂	semidirect
11. Q₈×D₁₂₇	C₄	semidirect
54. C₁₂₇	#419	semidirect
62. #103	D₁₂₇	semidirect

H ≅ G/N

type

4. #16861134

C₂

semidirect

5. #16861133

C₂

semidirect

6. #16861132

C₂

semidirect

7. #103×C₁₂₇

C₂

semidirect

11. Q₈×D₁₂₇

C₄

semidirect

54. C₁₂₇

#419

semidirect

62. #103

D₁₂₇

semidirect

Subgroups — #22549883

Conjugacy classes of subgroups

Complements