Group #22550504 — subgroups

Subgroups — #22550504

Order 8128.

206 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 111 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	2	yes	maximal, complemented
2	#16861208	C₂	1	2	yes	maximal, complemented
3	#16861201	C₂	1	2	yes	maximal, complemented
4	#16861200	C₂	1	2	yes	maximal, complemented
5	#16861186	C₂	1	4	yes	maximal
6	#16861168	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄⋊Dic₁₂₇	C₂	1	1	yes	maximal, characteristic
8	#117×C₁₂₇	C₂	1	1	yes	maximal, characteristic
9	C₂×Dic₂₅₄	C₂²	1	4	yes	—
10	C₂²×Dic₁₂₇	C₂²	1	2	yes	—
11	C₄⋊Dic₁₂₇	C₂²	1	4	yes	—
12	C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
13	C₄⋊Dic₁₂₇	C₂²	1	1	yes	characteristic
14	C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
15	C₄⋊Dic₁₂₇	C₂²	1	1	yes	characteristic
16	Dic₁₂₇⋊C₄	C₂²	1	4	yes	—
17	Dic₁₂₇⋊C₄	C₂²	1	4	yes	—
18	C₂²⋊Dic₁₂₇	C₂²	1	2	yes	—
19	C₄×Dic₁₂₇	C₂²	1	2	yes	—
20	C₂²×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
21	C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
22	C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
23	C₂²⋊C₄×C₁₂₇	C₂²	1	2	yes	—
24	C₄²×C₁₂₇	C₂²	1	2	yes	—
25	Dic₂₅₄	—	4	4	no	—
26	C₂×Dic₁₂₇	—	4	2	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	1	yes	characteristic
31	C₂×C₄×C₁₂₇	Q₈	1	4	yes	complemented
32	C₂×C₄×C₁₂₇	C₂³	1	2	yes	—
33	C₂×C₄×C₁₂₇	C₂³	1	2	yes	—
34	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
35	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
36	C₂²×C₁₂₇	—	2	1	no	—
37	C₂²×C₁₂₇	—	2	1	no	—
38	C₂²×C₁₂₇	C₂×Q₈	1	2	yes	—
39	C₂²×C₁₂₇	C₂⁴	1	1	yes	characteristic
40	C₄×C₁₂₇	—	2	2	no	—
41	C₄×C₁₂₇	—	2	2	no	—
42	C₄×C₁₂₇	C₂×Q₈	1	2	yes	—
43	C₄×C₁₂₇	C₂×Q₈	1	2	yes	—
44	Dic₁₂₇	—	2	4	no	—
45	Dic₁₂₇	—	2	2	no	—
46	Dic₁₂₇	—	2	2	no	—
47	C₂×C₁₂₇	—	2	2	no	—
48	C₂×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
49	C₂×C₁₂₇	H(5,𝔽₂)	1	1	yes	characteristic
50	C₂×C₁₂₇	2₋¹⁺⁴	1	1	yes	characteristic
51	C₁₂₇	#544	1	1	yes	characteristic, complemented, Sylow
52	#544	—	127	1	no	maximal, Sylow
53	C₄⋊Q₈	—	127	2	no	—
54	C₄⋊Q₈	—	127	2	no	—
55	#125	—	127	2	no	—
56	#125	—	127	2	no	—
57	C₂²⋊Q₈	—	127	4	no	—
58	#117	—	127	1	no	—
59	#117	D₁₂₇	1	1	yes	characteristic
60	C₂×C₄⋊C₄	—	127	1	no	—
61	C₂×Q₈	—	127	4	no	—
62	C₂²×C₄	—	127	2	no	—
63	C₂²×C₄	C₂×D₁₂₇	1	1	yes	characteristic
64	C₄⋊C₄	—	127	4	no	—
65	C₄⋊C₄	—	127	4	no	—
66	C₄⋊C₄	—	127	2	no	—
67	C₄⋊C₄	—	127	2	no	—
68	C₄⋊C₄	—	127	1	no	—
69	C₄⋊C₄	—	127	1	no	—
70	C₄⋊C₄	—	127	4	no	—
71	C₄⋊C₄	C₂×D₁₂₇	1	1	yes	characteristic
72	C₄⋊C₄	C₂×D₁₂₇	1	1	yes	characteristic
73	C₂²⋊C₄	—	127	2	no	—
74	C₂²⋊C₄	C₂×D₁₂₇	1	2	yes	—
75	C₄²	—	127	2	no	—
76	C₄²	C₂×D₁₂₇	1	2	yes	—
77	C₂³	C₂²×D₁₂₇	1	1	yes	characteristic
78	Q₈	—	508	4	no	—
79	C₂×C₄	—	127	2	no	—
80	C₂×C₄	—	127	2	no	—
81	C₂×C₄	—	127	4	no	—
82	C₂×C₄	—	508	2	no	—
83	C₂×C₄	Dic₂₅₄	1	4	yes	complemented
84	C₂×C₄	C₂²×D₁₂₇	1	2	yes	—
85	C₂×C₄	C₂²×D₁₂₇	1	2	yes	—
86	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
87	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
88	C₂²	—	2	1	no	—
89	C₂²	—	2	1	no	—
90	C₂²	C₂×Dic₂₅₄	1	2	yes	—
91	C₂²	C₂³×D₁₂₇	1	1	yes	characteristic
92	C₄	—	2	2	no	—
93	C₄	—	2	2	no	—
94	C₄	—	254	4	no	—
95	C₄	—	254	2	no	—
96	C₄	—	254	2	no	—
97	C₄	C₂×Dic₂₅₄	1	2	yes	—
98	C₄	C₂×Dic₂₅₄	1	2	yes	—
99	C₂	—	2	2	no	—
100	C₂	C₂²×Dic₂₅₄	1	1	yes	characteristic
101	C₂	#16861255	1	1	yes	characteristic
102	C₂	#16861256	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16861209

C₂

yes

maximal, complemented

#16861208

C₂

yes

maximal, complemented

#16861201

C₂

yes

maximal, complemented

#16861200

C₂

yes

maximal, complemented

#16861186

C₂

yes

maximal

#16861168

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₁₂₇

C₂

yes

maximal, characteristic

#117×C₁₂₇

C₂

yes

maximal, characteristic

C₂×Dic₂₅₄

C₂²

yes

—

C₂²×Dic₁₂₇

C₂²

yes

—

C₄⋊Dic₁₂₇

C₂²

yes

—

C₄⋊Dic₁₂₇

C₂²

yes

—

C₄⋊Dic₁₂₇

C₂²

yes

characteristic

C₄⋊Dic₁₂₇

C₂²

yes

—

C₄⋊Dic₁₂₇

C₂²

yes

characteristic

Dic₁₂₇⋊C₄

C₂²

yes

—

Dic₁₂₇⋊C₄

C₂²

yes

—

C₂²⋊Dic₁₂₇

C₂²

yes

—

C₄×Dic₁₂₇

C₂²

yes

—

C₂²×C₄×C₁₂₇

C₂²

yes

characteristic

C₄⋊C₄×C₁₂₇

C₂²

yes

characteristic

C₄⋊C₄×C₁₂₇

C₂²

yes

characteristic

C₂²⋊C₄×C₁₂₇

C₂²

yes

—

C₄²×C₁₂₇

C₂²

yes

—

Dic₂₅₄

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

C₂³

yes

—

C₂×Dic₁₂₇

C₂³

yes

—

C₂×Dic₁₂₇

C₂³

yes

—

C₂³×C₁₂₇

C₂³

yes

characteristic

C₂×C₄×C₁₂₇

Q₈

yes

complemented

C₂×C₄×C₁₂₇

C₂³

yes

—

C₂×C₄×C₁₂₇

C₂³

yes

—

C₂×C₄×C₁₂₇

C₂³

yes

characteristic

C₂×C₄×C₁₂₇

C₂³

yes

characteristic

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₂×Q₈

yes

—

C₂²×C₁₂₇

C₂⁴

yes

characteristic

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂×Q₈

yes

—

C₄×C₁₂₇

C₂×Q₈

yes

—

Dic₁₂₇

—

Dic₁₂₇

—

Dic₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

C₂²×Q₈

yes

characteristic

C₂×C₁₂₇

H(5,𝔽₂)

yes

characteristic

C₂×C₁₂₇

2₋¹⁺⁴

yes

characteristic

C₁₂₇

#544

yes

characteristic, complemented, Sylow

#544

—

127

maximal, Sylow

C₄⋊Q₈

—

127

—

C₄⋊Q₈

—

127

—

#125

—

127

—

#125

—

127

—

C₂²⋊Q₈

—

127

—

#117

—

127

—

#117

D₁₂₇

yes

characteristic

C₂×C₄⋊C₄

—

127

—

C₂×Q₈

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

C₂×D₁₂₇

yes

characteristic

C₄⋊C₄

—

127

—

C₄⋊C₄

—

127

—

C₄⋊C₄

—

127

—

C₄⋊C₄

—

127

—

C₄⋊C₄

—

127

—

C₄⋊C₄

—

127

—

C₄⋊C₄

—

127

—

C₄⋊C₄

C₂×D₁₂₇

yes

characteristic

C₄⋊C₄

C₂×D₁₂₇

yes

characteristic

C₂²⋊C₄

—

127

—

C₂²⋊C₄

C₂×D₁₂₇

yes

—

C₄²

—

127

—

C₄²

C₂×D₁₂₇

yes

—

C₂³

C₂²×D₁₂₇

yes

characteristic

Q₈

—

508

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

508

—

C₂×C₄

Dic₂₅₄

yes

complemented

C₂×C₄

C₂²×D₁₂₇

yes

—

C₂×C₄

C₂²×D₁₂₇

yes

—

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

C₂²

—

C₂²

—

C₂²

C₂×Dic₂₅₄

yes

—

C₂²

C₂³×D₁₂₇

yes

characteristic

C₄

—

C₄

—

C₄

—

254

—

C₄

—

254

—

C₄

—

254

—

C₄

C₂×Dic₂₅₄

yes

—

C₄

C₂×Dic₂₅₄

yes

—

C₂

—

100

C₂

C₂²×Dic₂₅₄

yes

characteristic

101

C₂

#16861255

yes

characteristic

102

C₂

#16861256

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₂₇
commutator, socle	C₂²×C₁₂₇	C₂⁴
Fitting	#117×C₁₂₇	C₂
hypercenter	#117	D₁₂₇
hypocenter	C₁₂₇	#544
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ #117

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₂₇

commutator, socle

C₂²×C₁₂₇

C₂⁴

Fitting

#117×C₁₂₇

C₂

D₁₂₇

C₁₂₇

G ▷ C₂²×C₁₂₇ ▷ C₁₂₇

C₁ ◁ C₂² ◁ #117

N	H ≅ G/N	type
1. #16861209	C₂	semidirect
2. #16861208	C₂	semidirect
3. #16861201	C₂	semidirect
4. #16861200	C₂	semidirect
31. C₂×C₄×C₁₂₇	Q₈	semidirect
51. C₁₂₇	#544	semidirect
83. C₂×C₄	Dic₂₅₄	semidirect

H ≅ G/N

type

1. #16861209

C₂

semidirect

2. #16861208

C₂

semidirect

3. #16861201

C₂

semidirect

4. #16861200

C₂

semidirect

31. C₂×C₄×C₁₂₇

Q₈

semidirect

51. C₁₂₇

#544

semidirect

83. C₂×C₄

Dic₂₅₄

semidirect

Subgroups — #22550504

Conjugacy classes of subgroups

Complements