Group #22550426 — subgroups

Subgroups — #22550426

Order 8128.

310 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 119 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×#14523248	C₂	1	2	yes	maximal, complemented
2	#16861209	C₂	1	1	yes	maximal, characteristic, complemented
3	#16861206	C₂	1	1	yes	maximal, characteristic, complemented
4	#16861197	C₂	1	2	yes	maximal, complemented
5	#16861183	C₂	1	4	yes	maximal, complemented
6	C₄×D₅₀₈	C₂	1	4	yes	maximal, complemented
7	C₂×C₄²×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#14523248	—	2	8	no	—
9	C₂×Dic₂₅₄	C₂²	1	2	yes	—
10	C₂×D₅₀₈	C₂²	1	2	yes	—
11	C₂×D₅₀₈	C₂²	1	4	yes	—
12	C₂×C₂²⋊D₁₂₇	C₂²	1	4	yes	—
13	C₂×C₄×D₁₂₇	C₂²	1	4	yes	—
14	C₄⋊Dic₁₂₇	C₂²	1	4	yes	—
15	(C₂×D₁₂₇)⋊C₄	C₂²	1	8	yes	—
16	C₂²×C₄×C₁₂₇	C₂²	1	2	yes	—
17	C₂²×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
18	C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
19	C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
20	C₄²×C₁₂₇	C₂²	1	2	yes	—
21	C₂²×D₁₂₇	C₂³	1	4	yes	—
22	Dic₂₅₄	—	2	4	no	—
23	D₅₀₈	—	2	4	no	—
24	D₅₀₈	—	2	8	no	—
25	C₂²⋊D₁₂₇	—	2	8	no	—
26	C₂×Dic₁₂₇	C₂³	1	4	yes	—
27	C₄×D₁₂₇	—	2	8	no	—
28	C₂³×C₁₂₇	C₂³	1	1	yes	characteristic
29	C₂×C₄×C₁₂₇	—	2	4	no	—
30	C₂×C₄×C₁₂₇	D₄	1	4	yes	complemented
31	C₂×C₄×C₁₂₇	C₂³	1	2	yes	—
32	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
33	C₂×C₄×C₁₂₇	C₂³	1	2	yes	—
34	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
35	C₂²×C₁₂₇	—	2	2	no	—
36	C₂²×C₁₂₇	C₂×D₄	1	2	yes	—
37	C₂²×C₁₂₇	C₂⁴	1	1	yes	characteristic
38	C₂×D₁₂₇	—	2	4	no	—
39	C₂×D₁₂₇	—	2	8	no	—
40	C₄×C₁₂₇	—	2	2	no	—
41	C₄×C₁₂₇	C₂×D₄	1	2	yes	—
42	C₄×C₁₂₇	C₂×D₄	1	2	yes	—
43	C₄×C₁₂₇	#41	1	4	yes	complemented
44	Dic₁₂₇	—	2	4	no	—
45	C₂×C₁₂₇	—	2	2	no	—
46	C₂×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
47	C₂×C₁₂₇	C₂×#41	1	2	yes	—
48	D₁₂₇	—	4	4	no	—
49	C₁₂₇	#532	1	1	yes	characteristic, complemented, Sylow
50	#532	—	127	1	no	maximal, Sylow
51	C₂×#41	—	127	2	no	—
52	C₄⋊Q₈	—	127	1	no	—
53	#127	—	127	1	no	—
54	#124	—	127	2	no	—
55	#121	—	127	4	no	—
56	C₄×D₄	—	127	4	no	—
57	C₂×C₄²	D₁₂₇	1	1	yes	characteristic, complemented
58	#41	—	254	8	no	—
59	C₂×Q₈	—	127	2	no	—
60	C₂×D₄	—	127	4	no	—
61	C₂×D₄	—	127	4	no	—
62	C₂×D₄	—	127	2	no	—
63	C₂²×C₄	—	127	4	no	—
64	C₂²×C₄	C₂×D₁₂₇	1	2	yes	—
65	C₂²×C₄	C₂×D₁₂₇	1	1	yes	characteristic
66	C₄⋊C₄	—	127	4	no	—
67	C₂²⋊C₄	—	127	8	no	—
68	C₄²	C₂×D₁₂₇	1	1	yes	characteristic
69	C₄²	C₂×D₁₂₇	1	1	yes	characteristic
70	C₄²	C₂×D₁₂₇	1	2	yes	—
71	C₂³	—	127	4	no	—
72	C₂³	C₂²×D₁₂₇	1	1	yes	characteristic
73	Q₈	—	254	4	no	—
74	D₄	—	254	8	no	—
75	D₄	—	254	4	no	—
76	D₄	—	254	8	no	—
77	C₂×C₄	—	2	4	no	—
78	C₂×C₄	—	127	4	no	—
79	C₂×C₄	—	254	8	no	—
80	C₂×C₄	D₅₀₈	1	4	yes	complemented
81	C₂×C₄	C₂²×D₁₂₇	1	2	yes	—
82	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
83	C₂×C₄	C₂²×D₁₂₇	1	2	yes	—
84	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
85	C₂²	—	2	2	no	—
86	C₂²	—	254	4	no	—
87	C₂²	—	254	8	no	—
88	C₂²	C₂×D₅₀₈	1	2	yes	—
89	C₂²	C₂³×D₁₂₇	1	1	yes	characteristic
90	C₄	—	2	2	no	—
91	C₄	—	254	4	no	—
92	C₄	C₂×D₅₀₈	1	2	yes	—
93	C₄	C₂×D₅₀₈	1	2	yes	—
94	C₄	#14523248	1	4	yes	complemented
95	C₂	—	2	2	no	—
96	C₂	—	508	4	no	—
97	C₂	C₂²×D₅₀₈	1	1	yes	characteristic
98	C₂	C₂×#14523248	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14523248

C₂

yes

maximal, complemented

#16861209

C₂

yes

maximal, characteristic, complemented

#16861206

C₂

yes

maximal, characteristic, complemented

#16861197

C₂

yes

maximal, complemented

#16861183

C₂

yes

maximal, complemented

C₄×D₅₀₈

C₂

yes

maximal, complemented

C₂×C₄²×C₁₂₇

C₂

yes

maximal, characteristic, complemented

#14523248

—

C₂×Dic₂₅₄

C₂²

yes

—

C₂×D₅₀₈

C₂²

yes

—

C₂×D₅₀₈

C₂²

yes

—

C₂×C₂²⋊D₁₂₇

C₂²

yes

—

C₂×C₄×D₁₂₇

C₂²

yes

—

C₄⋊Dic₁₂₇

C₂²

yes

—

(C₂×D₁₂₇)⋊C₄

C₂²

yes

—

C₂²×C₄×C₁₂₇

C₂²

yes

—

C₂²×C₄×C₁₂₇

C₂²

yes

characteristic

C₄²×C₁₂₇

C₂²

yes

characteristic

C₄²×C₁₂₇

C₂²

yes

characteristic

C₄²×C₁₂₇

C₂²

yes

—

C₂²×D₁₂₇

C₂³

yes

—

Dic₂₅₄

—

D₅₀₈

—

D₅₀₈

—

C₂²⋊D₁₂₇

—

C₂×Dic₁₂₇

C₂³

yes

—

C₄×D₁₂₇

—

C₂³×C₁₂₇

C₂³

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

D₄

yes

complemented

C₂×C₄×C₁₂₇

C₂³

yes

—

C₂×C₄×C₁₂₇

C₂³

yes

characteristic

C₂×C₄×C₁₂₇

C₂³

yes

—

C₂×C₄×C₁₂₇

C₂³

yes

characteristic

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₂×D₄

yes

—

C₂²×C₁₂₇

C₂⁴

yes

characteristic

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂×D₄

yes

—

C₄×C₁₂₇

C₂×D₄

yes

—

C₄×C₁₂₇

#41

yes

complemented

Dic₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

C₂²×D₄

yes

characteristic

C₂×C₁₂₇

C₂×#41

yes

—

D₁₂₇

—

C₁₂₇

#532

yes

characteristic, complemented, Sylow

#532

—

127

maximal, Sylow

C₂×#41

—

127

—

C₄⋊Q₈

—

127

—

#127

—

127

—

#124

—

127

—

#121

—

127

—

C₄×D₄

—

127

—

C₂×C₄²

D₁₂₇

yes

characteristic, complemented

#41

—

254

—

C₂×Q₈

—

127

—

C₂×D₄

—

127

—

C₂×D₄

—

127

—

C₂×D₄

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

C₂×D₁₂₇

yes

—

C₂²×C₄

C₂×D₁₂₇

yes

characteristic

C₄⋊C₄

—

127

—

C₂²⋊C₄

—

127

—

C₄²

C₂×D₁₂₇

yes

characteristic

C₄²

C₂×D₁₂₇

yes

characteristic

C₄²

C₂×D₁₂₇

yes

—

C₂³

—

127

—

C₂³

C₂²×D₁₂₇

yes

characteristic

Q₈

—

254

—

D₄

—

254

—

D₄

—

254

—

D₄

—

254

—

C₂×C₄

—

C₂×C₄

—

127

—

C₂×C₄

—

254

—

C₂×C₄

D₅₀₈

yes

complemented

C₂×C₄

C₂²×D₁₂₇

yes

—

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

C₂×C₄

C₂²×D₁₂₇

yes

—

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

C₂²

—

C₂²

—

254

—

C₂²

—

254

—

C₂²

C₂×D₅₀₈

yes

—

C₂²

C₂³×D₁₂₇

yes

characteristic

C₄

—

C₄

—

254

—

C₄

C₂×D₅₀₈

yes

—

C₄

C₂×D₅₀₈

yes

—

C₄

#14523248

yes

complemented

C₂

—

C₂

—

508

—

C₂

C₂²×D₅₀₈

yes

characteristic

C₂

C₂×#14523248

yes

—

	subgroup	G/H
center	C₂×C₄	C₂²×D₁₂₇
commutator, socle	C₂²×C₁₂₇	C₂⁴
Frattini	C₂²	C₂³×D₁₂₇
Fitting	C₂×C₄²×C₁₂₇	C₂
hypercenter	C₂×C₄²	D₁₂₇
hypocenter	C₁₂₇	#532
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄²

subgroup

G/H

center

C₂×C₄

C₂²×D₁₂₇

commutator, socle

C₂²×C₁₂₇

C₂⁴

Frattini

C₂²

C₂³×D₁₂₇

Fitting

C₂×C₄²×C₁₂₇

C₂

hypercenter

C₂×C₄²

D₁₂₇

C₁₂₇

G ▷ C₂²×C₁₂₇ ▷ C₁₂₇

C₁ ◁ C₂×C₄ ◁ C₂×C₄²

N	H ≅ G/N	type
1. C₂×#14523248	C₂	semidirect
2. #16861209	C₂	semidirect
3. #16861206	C₂	semidirect
4. #16861197	C₂	semidirect
5. #16861183	C₂	semidirect
6. C₄×D₅₀₈	C₂	semidirect
7. C₂×C₄²×C₁₂₇	C₂	semidirect
30. C₂×C₄×C₁₂₇	D₄	semidirect
43. C₄×C₁₂₇	#41	semidirect
49. C₁₂₇	#532	semidirect
57. C₂×C₄²	D₁₂₇	semidirect
80. C₂×C₄	D₅₀₈	semidirect
94. C₄	#14523248	semidirect

H ≅ G/N

type

1. C₂×#14523248

C₂

semidirect

2. #16861209

C₂

semidirect

3. #16861206

C₂

semidirect

4. #16861197

C₂

semidirect

5. #16861183

C₂

semidirect

6. C₄×D₅₀₈

C₂

semidirect

7. C₂×C₄²×C₁₂₇

C₂

semidirect

30. C₂×C₄×C₁₂₇

D₄

semidirect

43. C₄×C₁₂₇

#41

semidirect

49. C₁₂₇

#532

semidirect

57. C₂×C₄²

D₁₂₇

semidirect

80. C₂×C₄

D₅₀₈

semidirect

94. C₄

#14523248

semidirect

Subgroups — #22550426

Conjugacy classes of subgroups

Complements