Group #22550687 — subgroups

Subgroups — #22550687

Order 8128.

258 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 107 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	#16861256	C₂	1	1	yes	maximal, characteristic, complemented
2	#16861239	C₂	1	3	yes	maximal, complemented
3	#16861235	C₂	1	3	yes	maximal, complemented
4	#16861229	C₂	1	3	yes	maximal, complemented
5	C₂×#14523236	C₂	1	3	yes	maximal, complemented
6	#16861216	C₂	1	1	yes	maximal, characteristic, complemented
7	H(5,𝔽₂)×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#14523248	C₂²	1	3	yes	complemented
9	#14523247	—	2	3	no	—
10	C₂×Dic₂₅₄	C₂²	1	3	yes	complemented
11	Q₈×D₁₂₇	—	2	1	no	—
12	#14523238	C₂²	1	3	yes	complemented
13	#14523236	C₂²	1	3	yes	complemented
14	#14523236	C₂²	1	6	yes	complemented
15	#14523235	C₂²	1	1	yes	characteristic, complemented
16	#14523233	C₂²	1	3	yes	complemented
17	#14523232	C₂²	1	3	yes	complemented
18	C₂×C₁₂₇⋊C₈	C₂²	1	3	yes	complemented
19	#41×C₁₂₇	—	2	1	no	—
20	#41×C₁₂₇	C₂²	1	3	yes	—
21	#41×C₁₂₇	C₂²	1	1	yes	characteristic
22	C₂×D₄×C₁₂₇	—	2	3	no	—
23	C₂×D₄×C₁₂₇	C₂²	1	3	yes	—
24	Dic₂₅₄	—	2	3	no	—
25	Dic₂₅₄	C₂³	1	3	yes	—
26	D₅₀₈	C₂³	1	1	yes	characteristic
27	C₂²⋊D₁₂₇	—	2	3	no	—
28	C₂×Dic₁₂₇	—	2	3	no	—
29	C₄×D₁₂₇	—	2	3	no	—
30	C₁₂₇⋊C₈	C₂³	1	1	yes	characteristic
31	C₁₂₇⋊C₈	C₂³	1	3	yes	—
32	C₂³×C₁₂₇	—	2	3	no	—
33	Q₈×C₁₂₇	D₄	1	1	yes	characteristic
34	Q₈×C₁₂₇	C₂³	1	1	yes	characteristic
35	D₄×C₁₂₇	—	2	3	no	—
36	D₄×C₁₂₇	—	2	3	no	—
37	D₄×C₁₂₇	D₄	1	3	yes	—
38	D₄×C₁₂₇	C₂³	1	3	yes	—
39	C₂×C₄×C₁₂₇	—	2	3	no	—
40	C₂×C₄×C₁₂₇	C₂³	1	3	yes	—
41	C₂²×C₁₂₇	—	2	3	no	—
42	C₂²×C₁₂₇	—	8	3	no	—
43	C₂²×C₁₂₇	C₂×D₄	1	3	yes	—
44	C₂×D₁₂₇	—	2	1	no	—
45	C₄×C₁₂₇	—	2	1	no	—
46	C₄×C₁₂₇	C₂×D₄	1	3	yes	—
47	C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
48	Dic₁₂₇	—	2	3	no	—
49	C₂×C₁₂₇	—	2	3	no	—
50	C₂×C₁₂₇	—	4	3	no	—
51	C₂×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
52	D₁₂₇	—	4	1	no	—
53	C₁₂₇	#577	1	1	yes	characteristic, complemented, Sylow
54	#577	—	127	1	no	maximal, Sylow
55	2₋¹⁺⁴	—	127	1	no	—
56	H(5,𝔽₂)	D₁₂₇	1	1	yes	characteristic, complemented
57	#137	—	127	3	no	—
58	AGL(1,ℤ/8)	—	127	3	no	—
59	#135	—	127	3	no	—
60	C₂×SD₁₆	—	127	3	no	—
61	#131	—	127	1	no	—
62	#41	—	2	1	no	—
63	#41	—	127	3	no	—
64	#41	—	254	3	no	—
65	#41	C₂×D₁₂₇	1	3	yes	—
66	#41	C₂×D₁₂₇	1	1	yes	characteristic
67	C₂×Q₈	—	127	3	no	—
68	C₂×Q₈	—	254	1	no	—
69	C₂×D₄	—	2	3	no	—
70	C₂×D₄	C₂×D₁₂₇	1	3	yes	—
71	Q₁₆	—	127	3	no	—
72	SD₁₆	—	127	1	no	—
73	SD₁₆	—	127	3	no	—
74	SD₁₆	—	127	6	no	—
75	D₈	—	127	3	no	—
76	M₁₆	—	127	3	no	—
77	C₂×C₈	—	127	3	no	—
78	C₂³	—	2	3	no	—
79	Q₈	—	127	3	no	—
80	Q₈	—	254	3	no	—
81	Q₈	C₂²⋊D₁₂₇	1	1	yes	characteristic
82	Q₈	C₂²×D₁₂₇	1	1	yes	characteristic
83	D₄	—	2	3	no	—
84	D₄	—	2	3	no	—
85	D₄	—	127	1	no	—
86	D₄	—	254	3	no	—
87	D₄	C₂²⋊D₁₂₇	1	3	yes	—
88	D₄	C₂²×D₁₂₇	1	3	yes	—
89	C₂×C₄	—	2	3	no	—
90	C₂×C₄	—	254	3	no	—
91	C₂×C₄	—	254	3	no	—
92	C₂×C₄	C₂²×D₁₂₇	1	3	yes	—
93	C₈	—	127	1	no	—
94	C₈	—	127	3	no	—
95	C₂²	—	2	3	no	—
96	C₂²	—	8	3	no	—
97	C₂²	—	254	1	no	—
98	C₂²	C₂×C₂²⋊D₁₂₇	1	3	yes	—
99	C₄	—	2	1	no	—
100	C₄	—	254	3	no	—
101	C₄	C₂×C₂²⋊D₁₂₇	1	3	yes	—
102	C₄	C₂³×D₁₂₇	1	1	yes	characteristic
103	C₂	—	2	3	no	—
104	C₂	—	4	3	no	—
105	C₂	—	508	1	no	—
106	C₂	C₂²×C₂²⋊D₁₂₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16861256

C₂

yes

maximal, characteristic, complemented

#16861239

C₂

yes

maximal, complemented

#16861235

C₂

yes

maximal, complemented

#16861229

C₂

yes

maximal, complemented

C₂×#14523236

C₂

yes

maximal, complemented

#16861216

C₂

yes

maximal, characteristic, complemented

H(5,𝔽₂)×C₁₂₇

C₂

yes

maximal, characteristic, complemented

#14523248

C₂²

yes

complemented

#14523247

—

C₂×Dic₂₅₄

C₂²

yes

complemented

Q₈×D₁₂₇

—

#14523238

C₂²

yes

complemented

#14523236

C₂²

yes

complemented

#14523236

C₂²

yes

complemented

#14523235

C₂²

yes

characteristic, complemented

#14523233

C₂²

yes

complemented

#14523232

C₂²

yes

complemented

C₂×C₁₂₇⋊C₈

C₂²

yes

complemented

#41×C₁₂₇

—

#41×C₁₂₇

C₂²

yes

—

#41×C₁₂₇

C₂²

yes

characteristic

C₂×D₄×C₁₂₇

—

C₂×D₄×C₁₂₇

C₂²

yes

—

Dic₂₅₄

—

Dic₂₅₄

C₂³

yes

—

D₅₀₈

C₂³

yes

characteristic

C₂²⋊D₁₂₇

—

C₂×Dic₁₂₇

—

C₄×D₁₂₇

—

C₁₂₇⋊C₈

C₂³

yes

characteristic

C₁₂₇⋊C₈

C₂³

yes

—

C₂³×C₁₂₇

—

Q₈×C₁₂₇

D₄

yes

characteristic

Q₈×C₁₂₇

C₂³

yes

characteristic

D₄×C₁₂₇

—

D₄×C₁₂₇

—

D₄×C₁₂₇

D₄

yes

—

D₄×C₁₂₇

C₂³

yes

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂³

yes

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₂×D₄

yes

—

C₂×D₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂×D₄

yes

—

C₄×C₁₂₇

C₂⁴

yes

characteristic

Dic₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

C₂²×D₄

yes

characteristic

D₁₂₇

—

C₁₂₇

#577

yes

characteristic, complemented, Sylow

#577

—

127

maximal, Sylow

2₋¹⁺⁴

—

127

—

H(5,𝔽₂)

D₁₂₇

yes

characteristic, complemented

#137

—

127

—

AGL(1,ℤ/8)

—

127

—

#135

—

127

—

C₂×SD₁₆

—

127

—

#131

—

127

—

#41

—

#41

—

127

—

#41

—

254

—

#41

C₂×D₁₂₇

yes

—

#41

C₂×D₁₂₇

yes

characteristic

C₂×Q₈

—

127

—

C₂×Q₈

—

254

—

C₂×D₄

—

C₂×D₄

C₂×D₁₂₇

yes

—

Q₁₆

—

127

—

SD₁₆

—

127

—

SD₁₆

—

127

—

SD₁₆

—

127

—

D₈

—

127

—

M₁₆

—

127

—

C₂×C₈

—

127

—

C₂³

—

Q₈

—

127

—

Q₈

—

254

—

Q₈

C₂²⋊D₁₂₇

yes

characteristic

Q₈

C₂²×D₁₂₇

yes

characteristic

D₄

—

D₄

—

D₄

—

127

—

D₄

—

254

—

D₄

C₂²⋊D₁₂₇

yes

—

D₄

C₂²×D₁₂₇

yes

—

C₂×C₄

—

C₂×C₄

—

254

—

C₂×C₄

—

254

—

C₂×C₄

C₂²×D₁₂₇

yes

—

C₈

—

127

—

C₈

—

127

—

C₂²

—

C₂²

—

C₂²

—

254

—

C₂²

C₂×C₂²⋊D₁₂₇

yes

—

C₄

—

100

C₄

—

254

—

101

C₄

C₂×C₂²⋊D₁₂₇

yes

—

102

C₄

C₂³×D₁₂₇

yes

characteristic

103

C₂

—

104

C₂

—

105

C₂

—

508

—

106

C₂

C₂²×C₂²⋊D₁₂₇

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×C₂²⋊D₁₂₇
commutator	C₄×C₁₂₇	C₂⁴
Frattini	C₄	C₂³×D₁₂₇
Fitting	H(5,𝔽₂)×C₁₂₇	C₂
socle	C₂×C₁₂₇	C₂²×D₄
hypercenter	H(5,𝔽₂)	D₁₂₇
hypocenter	C₁₂₇	#577
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂ ◁ #41 ◁ H(5,𝔽₂)

subgroup

G/H

center

C₂

C₂²×C₂²⋊D₁₂₇

commutator

C₄×C₁₂₇

C₂⁴

Frattini

C₄

C₂³×D₁₂₇

Fitting

H(5,𝔽₂)×C₁₂₇

C₂

socle

C₂×C₁₂₇

C₂²×D₄

D₁₂₇

C₁₂₇

G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂ ◁ #41 ◁ H(5,𝔽₂)

N	H ≅ G/N	type
1. #16861256	C₂	semidirect
2. #16861239	C₂	semidirect
3. #16861235	C₂	semidirect
4. #16861229	C₂	semidirect
5. C₂×#14523236	C₂	semidirect
6. #16861216	C₂	semidirect
7. H(5,𝔽₂)×C₁₂₇	C₂	semidirect
8. #14523248	C₂²	semidirect
10. C₂×Dic₂₅₄	C₂²	semidirect
12. #14523238	C₂²	semidirect
13. #14523236	C₂²	semidirect
14. #14523236	C₂²	semidirect
15. #14523235	C₂²	semidirect
16. #14523233	C₂²	semidirect
17. #14523232	C₂²	semidirect
18. C₂×C₁₂₇⋊C₈	C₂²	semidirect
53. C₁₂₇	#577	semidirect
56. H(5,𝔽₂)	D₁₂₇	semidirect

H ≅ G/N

type

1. #16861256

C₂

semidirect

2. #16861239

C₂

semidirect

3. #16861235

C₂

semidirect

4. #16861229

C₂

semidirect

5. C₂×#14523236

C₂

semidirect

6. #16861216

C₂

semidirect

7. H(5,𝔽₂)×C₁₂₇

C₂

semidirect

8. #14523248

C₂²

semidirect

10. C₂×Dic₂₅₄

C₂²

semidirect

12. #14523238

C₂²

semidirect

13. #14523236

C₂²

semidirect

14. #14523236

C₂²

semidirect

15. #14523235

C₂²

semidirect

16. #14523233

C₂²

semidirect

17. #14523232

C₂²

semidirect

18. C₂×C₁₂₇⋊C₈

C₂²

semidirect

53. C₁₂₇

#577

semidirect

56. H(5,𝔽₂)

D₁₂₇

semidirect

Subgroups — #22550687

Conjugacy classes of subgroups

Complements