Group #22550618 — subgroups

Subgroups — #22550618

Order 8128.

260 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 91 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	#16861198	C₂	1	2	yes	maximal, complemented
2	#16861197	C₂	1	2	yes	maximal, complemented
3	#16861195	C₂	1	4	yes	maximal, complemented
4	#16861179	C₂	1	4	yes	maximal, complemented
5	#16861177	C₂	1	2	yes	maximal, complemented
6	#124×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₁₂₇	C₂²	1	2	yes	complemented
8	C₂×Dic₂₅₄	C₂²	1	2	yes	complemented
9	C₂×D₅₀₈	C₂²	1	2	yes	—
10	C₂×D₅₀₈	C₂²	1	2	yes	—
11	C₂×C₂²⋊D₁₂₇	C₂²	1	4	yes	—
12	C₂²⋊Dic₁₂₇	C₂²	1	2	yes	complemented
13	(C₂×D₁₂₇)⋊C₄	C₂²	1	4	yes	complemented
14	(C₂×D₁₂₇)⋊C₄	C₂²	1	4	yes	complemented
15	(C₂×D₁₂₇)⋊C₄	C₂²	1	4	yes	complemented
16	C₄×Dic₁₂₇	C₂²	1	2	yes	complemented
17	C₂×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic, complemented
18	C₂×D₄×C₁₂₇	C₂²	1	1	yes	characteristic
19	C₂²⋊C₄×C₁₂₇	C₂²	1	4	yes	complemented
20	C₄²×C₁₂₇	C₂²	1	1	yes	characteristic, complemented
21	C₂²×D₁₂₇	—	4	4	no	—
22	C₂²×D₁₂₇	—	4	2	no	—
23	C₂²×D₁₂₇	C₂³	1	4	yes	—
24	Dic₂₅₄	—	4	2	no	—
25	D₅₀₈	—	4	2	no	—
26	D₅₀₈	—	4	2	no	—
27	C₂²⋊D₁₂₇	—	4	4	no	—
28	C₂×Dic₁₂₇	C₂³	1	4	yes	—
29	C₂³×C₁₂₇	C₂³	1	2	yes	—
30	Q₈×C₁₂₇	—	4	1	no	—
31	D₄×C₁₂₇	—	4	1	no	—
32	C₂×C₄×C₁₂₇	C₂³	1	2	yes	—
33	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
34	C₂×C₄×C₁₂₇	C₂³	1	2	yes	—
35	C₂²×C₁₂₇	—	2	2	no	—
36	C₂²×C₁₂₇	—	2	4	no	—
37	C₂²×C₁₂₇	C₂⁴	1	1	yes	characteristic
38	C₂×D₁₂₇	—	2	4	no	—
39	C₂×D₁₂₇	—	2	4	no	—
40	C₂×D₁₂₇	—	2	4	no	—
41	C₂×D₁₂₇	—	4	8	no	—
42	C₄×C₁₂₇	—	2	1	no	—
43	C₄×C₁₂₇	—	2	2	no	—
44	C₄×C₁₂₇	—	2	2	no	—
45	Dic₁₂₇	—	2	4	no	—
46	C₂×C₁₂₇	—	4	2	no	—
47	C₂×C₁₂₇	H(5,𝔽₂)	1	1	yes	characteristic
48	C₂×C₁₂₇	H(5,𝔽₂)	1	2	yes	—
49	D₁₂₇	—	4	4	no	—
50	C₁₂₇	H(3,𝔽₄)	1	1	yes	characteristic, complemented, Sylow
51	H(3,𝔽₄)	—	127	1	no	maximal, Sylow
52	#124	—	127	2	no	—
53	#124	—	127	4	no	—
54	#124	—	127	2	no	—
55	#124	D₁₂₇	1	1	yes	characteristic, complemented
56	#120	—	127	2	no	—
57	#120	—	127	4	no	—
58	C₂⁴	—	127	2	no	—
59	C₂×Q₈	—	127	2	no	—
60	C₂×Q₈	C₂×D₁₂₇	1	1	yes	characteristic, complemented
61	C₂×D₄	—	127	4	no	—
62	C₂×D₄	—	127	2	no	—
63	C₂×D₄	—	127	2	no	—
64	C₂×D₄	C₂×D₁₂₇	1	1	yes	characteristic
65	C₂²⋊C₄	—	127	2	no	—
66	C₂²⋊C₄	—	127	4	no	—
67	C₂²⋊C₄	—	127	4	no	—
68	C₂²⋊C₄	—	127	4	no	—
69	C₂²⋊C₄	C₂×D₁₂₇	1	4	yes	complemented
70	C₄²	—	127	2	no	—
71	C₄²	C₂×D₁₂₇	1	1	yes	characteristic, complemented
72	C₂³	—	127	4	no	—
73	C₂³	—	508	2	no	—
74	C₂³	—	508	4	no	—
75	C₂³	C₂²×D₁₂₇	1	2	yes	—
76	Q₈	—	4	1	no	—
77	Q₈	—	508	2	no	—
78	D₄	—	4	1	no	—
79	D₄	—	508	2	no	—
80	D₄	—	508	2	no	—
81	D₄	—	508	4	no	—
82	C₂×C₄	—	127	4	no	—
83	C₂×C₄	C₂²×D₁₂₇	1	2	yes	—
84	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
85	C₂×C₄	C₂²×D₁₂₇	1	2	yes	—
86	C₂²	—	2	2	no	—
87	C₂²	—	2	4	no	—
88	C₂²	—	508	8	no	—
89	C₂²	—	254	4	no	—
90	C₂²	—	254	4	no	—
91	C₂²	—	254	4	no	—
92	C₂²	C₂³×D₁₂₇	1	1	yes	characteristic
93	C₄	—	2	1	no	—
94	C₄	—	2	2	no	—
95	C₄	—	2	2	no	—
96	C₄	—	254	4	no	—
97	C₂	—	4	2	no	—
98	C₂	—	508	4	no	—
99	C₂	#16861254	1	1	yes	characteristic
100	C₂	#16861255	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16861198

C₂

yes

maximal, complemented

#16861197

C₂

yes

maximal, complemented

#16861195

C₂

yes

maximal, complemented

#16861179

C₂

yes

maximal, complemented

#16861177

C₂

yes

maximal, complemented

#124×C₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂³×D₁₂₇

C₂²

yes

complemented

C₂×Dic₂₅₄

C₂²

yes

complemented

C₂×D₅₀₈

C₂²

yes

—

C₂×D₅₀₈

C₂²

yes

—

C₂×C₂²⋊D₁₂₇

C₂²

yes

—

C₂²⋊Dic₁₂₇

C₂²

yes

complemented

(C₂×D₁₂₇)⋊C₄

C₂²

yes

complemented

(C₂×D₁₂₇)⋊C₄

C₂²

yes

complemented

(C₂×D₁₂₇)⋊C₄

C₂²

yes

complemented

C₄×Dic₁₂₇

C₂²

yes

complemented

C₂×Q₈×C₁₂₇

C₂²

yes

characteristic, complemented

C₂×D₄×C₁₂₇

C₂²

yes

characteristic

C₂²⋊C₄×C₁₂₇

C₂²

yes

complemented

C₄²×C₁₂₇

C₂²

yes

characteristic, complemented

C₂²×D₁₂₇

—

C₂²×D₁₂₇

—

C₂²×D₁₂₇

C₂³

yes

—

Dic₂₅₄

—

D₅₀₈

—

D₅₀₈

—

C₂²⋊D₁₂₇

—

C₂×Dic₁₂₇

C₂³

yes

—

C₂³×C₁₂₇

C₂³

yes

—

Q₈×C₁₂₇

—

D₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂³

yes

—

C₂×C₄×C₁₂₇

C₂³

yes

characteristic

C₂×C₄×C₁₂₇

C₂³

yes

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₂⁴

yes

characteristic

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

Dic₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

H(5,𝔽₂)

yes

characteristic

C₂×C₁₂₇

H(5,𝔽₂)

yes

—

D₁₂₇

—

C₁₂₇

H(3,𝔽₄)

yes

characteristic, complemented, Sylow

H(3,𝔽₄)

—

127

maximal, Sylow

#124

—

127

—

#124

—

127

—

#124

—

127

—

#124

D₁₂₇

yes

characteristic, complemented

#120

—

127

—

#120

—

127

—

C₂⁴

—

127

—

C₂×Q₈

—

127

—

C₂×Q₈

C₂×D₁₂₇

yes

characteristic, complemented

C₂×D₄

—

127

—

C₂×D₄

—

127

—

C₂×D₄

—

127

—

C₂×D₄

C₂×D₁₂₇

yes

characteristic

C₂²⋊C₄

—

127

—

C₂²⋊C₄

—

127

—

C₂²⋊C₄

—

127

—

C₂²⋊C₄

—

127

—

C₂²⋊C₄

C₂×D₁₂₇

yes

complemented

C₄²

—

127

—

C₄²

C₂×D₁₂₇

yes

characteristic, complemented

C₂³

—

127

—

C₂³

—

508

—

C₂³

—

508

—

C₂³

C₂²×D₁₂₇

yes

—

Q₈

—

Q₈

—

508

—

D₄

—

D₄

—

508

—

D₄

—

508

—

D₄

—

508

—

C₂×C₄

—

127

—

C₂×C₄

C₂²×D₁₂₇

yes

—

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

C₂×C₄

C₂²×D₁₂₇

yes

—

C₂²

—

C₂²

—

C₂²

—

508

—

C₂²

—

254

—

C₂²

—

254

—

C₂²

—

254

—

C₂²

C₂³×D₁₂₇

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

—

254

—

C₂

—

C₂

—

508

—

C₂

#16861254

yes

characteristic

100

C₂

#16861255

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₂₇
commutator, socle	C₂²×C₁₂₇	C₂⁴
Fitting	#124×C₁₂₇	C₂
hypercenter	#124	D₁₂₇
hypocenter	C₁₂₇	H(3,𝔽₄)
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ #124

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₂₇

commutator, socle

C₂²×C₁₂₇

C₂⁴

Fitting

#124×C₁₂₇

C₂

D₁₂₇

C₁₂₇

G ▷ C₂²×C₁₂₇ ▷ C₁₂₇

C₁ ◁ C₂² ◁ #124

N	H ≅ G/N	type
1. #16861198	C₂	semidirect
2. #16861197	C₂	semidirect
3. #16861195	C₂	semidirect
4. #16861179	C₂	semidirect
5. #16861177	C₂	semidirect
6. #124×C₁₂₇	C₂	semidirect
7. C₂³×D₁₂₇	C₂²	semidirect
8. C₂×Dic₂₅₄	C₂²	semidirect
12. C₂²⋊Dic₁₂₇	C₂²	semidirect
13. (C₂×D₁₂₇)⋊C₄	C₂²	semidirect
14. (C₂×D₁₂₇)⋊C₄	C₂²	semidirect
15. (C₂×D₁₂₇)⋊C₄	C₂²	semidirect
16. C₄×Dic₁₂₇	C₂²	semidirect
17. C₂×Q₈×C₁₂₇	C₂²	semidirect
19. C₂²⋊C₄×C₁₂₇	C₂²	semidirect
20. C₄²×C₁₂₇	C₂²	semidirect
50. C₁₂₇	H(3,𝔽₄)	semidirect
55. #124	D₁₂₇	semidirect
60. C₂×Q₈	C₂×D₁₂₇	semidirect
69. C₂²⋊C₄	C₂×D₁₂₇	semidirect
71. C₄²	C₂×D₁₂₇	semidirect

H ≅ G/N

type

1. #16861198

C₂

semidirect

2. #16861197

C₂

semidirect

3. #16861195

C₂

semidirect

4. #16861179

C₂

semidirect

5. #16861177

C₂

semidirect

6. #124×C₁₂₇

C₂

semidirect

7. C₂³×D₁₂₇

C₂²

semidirect

8. C₂×Dic₂₅₄

C₂²

semidirect

12. C₂²⋊Dic₁₂₇

C₂²

semidirect

13. (C₂×D₁₂₇)⋊C₄

C₂²

semidirect

14. (C₂×D₁₂₇)⋊C₄

C₂²

semidirect

15. (C₂×D₁₂₇)⋊C₄

C₂²

semidirect

16. C₄×Dic₁₂₇

C₂²

semidirect

17. C₂×Q₈×C₁₂₇

C₂²

semidirect

19. C₂²⋊C₄×C₁₂₇

C₂²

semidirect

20. C₄²×C₁₂₇

C₂²

semidirect

50. C₁₂₇

H(3,𝔽₄)

semidirect

55. #124

D₁₂₇

semidirect

60. C₂×Q₈

C₂×D₁₂₇

semidirect

69. C₂²⋊C₄

C₂×D₁₂₇

semidirect

71. C₄²

C₂×D₁₂₇

semidirect

Subgroups — #22550618

Conjugacy classes of subgroups

Complements