Group #22581694 — subgroups

Subgroups — #22581694

Order 8896.

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

normal subgroups only

This group has 252 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	#17984839	C₂	1	1	yes	maximal, characteristic, complemented
2	#17984836	C₂	1	1	yes	maximal, characteristic, complemented
3	#17984827	C₂	1	3	yes	maximal, complemented
4	#17984817	C₂	1	3	yes	maximal, complemented
5	#17984815	C₂	1	3	yes	maximal, complemented
6	#17984812	C₂	1	3	yes	maximal, complemented
7	#126×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×D₁₃₉	C₂²	1	1	yes	characteristic
9	C₂×D₅₅₆	C₂²	1	3	yes	—
10	C₂×D₅₅₆	C₂²	1	3	yes	—
11	C₂×D₅₅₆	C₂²	1	3	yes	complemented
12	C₂×C₂²⋊D₁₃₉	C₂²	1	3	yes	—
13	C₂×C₄×D₁₃₉	C₂²	1	3	yes	complemented
14	Dic₁₃₉⋊C₄	C₂²	1	3	yes	complemented
15	(C₂×D₁₃₉)⋊C₄	C₂²	1	3	yes	—
16	(C₂×D₁₃₉)⋊C₄	C₂²	1	3	yes	—
17	(C₂×D₁₃₉)⋊C₄	C₂²	1	3	yes	complemented
18	C₄⋊C₄×C₁₃₉	C₂²	1	3	yes	complemented
19	C₂²⋊C₄×C₁₃₉	C₂²	1	3	yes	—
20	C₄²×C₁₃₉	C₂²	1	1	yes	characteristic, complemented
21	C₂²×D₁₃₉	—	4	3	no	—
22	C₂²×D₁₃₉	C₂³	1	1	yes	characteristic
23	C₂²×D₁₃₉	C₂³	1	1	yes	characteristic
24	C₂²×D₁₃₉	C₂³	1	3	yes	—
25	D₅₅₆	—	4	3	no	—
26	D₅₅₆	—	4	3	no	—
27	D₅₅₆	—	4	3	no	—
28	C₂²⋊D₁₃₉	—	4	3	no	—
29	C₂×Dic₁₃₉	C₂³	1	3	yes	—
30	C₄×D₁₃₉	—	4	3	no	—
31	C₂³×C₁₃₉	C₂³	1	1	yes	characteristic
32	C₂×C₄×C₁₃₉	C₂³	1	3	yes	—
33	C₂×C₄×C₁₃₉	C₂³	1	3	yes	—
34	C₂²×C₁₃₉	—	2	3	no	—
35	C₂²×C₁₃₉	C₂⁴	1	1	yes	characteristic
36	C₂×D₁₃₉	—	2	3	no	—
37	C₂×D₁₃₉	—	2	3	no	—
38	C₂×D₁₃₉	—	2	3	no	—
39	C₂×D₁₃₉	—	2	3	no	—
40	C₂×D₁₃₉	—	2	3	no	—
41	C₂×D₁₃₉	—	4	4	no	—
42	C₄×C₁₃₉	—	2	3	no	—
43	C₄×C₁₃₉	—	2	3	no	—
44	Dic₁₃₉	—	2	3	no	—
45	C₂×C₁₃₉	—	4	1	no	—
46	C₂×C₁₃₉	H(5,𝔽₂)	1	3	yes	—
47	D₁₃₉	—	4	3	no	—
48	D₁₃₉	—	4	1	no	—
49	D₁₃₉	—	4	1	no	—
50	C₁₃₉	#560	1	1	yes	characteristic, complemented, Sylow
51	#560	—	139	1	no	maximal, Sylow
52	#127	—	139	1	no	—
53	#126	—	139	1	no	—
54	#126	D₁₃₉	1	1	yes	characteristic, complemented
55	#123	—	139	3	no	—
56	#121	—	139	3	no	—
57	#121	—	139	3	no	—
58	#120	—	139	3	no	—
59	C₂⁴	—	139	1	no	—
60	C₂×D₄	—	139	3	no	—
61	C₂×D₄	—	139	3	no	—
62	C₂×D₄	—	139	3	no	—
63	C₂×D₄	—	139	3	no	—
64	C₂²×C₄	—	139	3	no	—
65	C₄⋊C₄	—	139	3	no	—
66	C₄⋊C₄	C₂×D₁₃₉	1	3	yes	complemented
67	C₂²⋊C₄	—	139	3	no	—
68	C₂²⋊C₄	—	139	3	no	—
69	C₂²⋊C₄	—	139	3	no	—
70	C₂²⋊C₄	C₂×D₁₃₉	1	3	yes	—
71	C₄²	C₂×D₁₃₉	1	1	yes	characteristic, complemented
72	C₂³	—	556	3	no	—
73	C₂³	—	139	1	no	—
74	C₂³	—	139	1	no	—
75	C₂³	—	139	3	no	—
76	C₂³	C₂²×D₁₃₉	1	1	yes	characteristic
77	D₄	—	556	3	no	—
78	D₄	—	556	3	no	—
79	D₄	—	556	3	no	—
80	D₄	—	556	3	no	—
81	C₂×C₄	—	556	3	no	—
82	C₂×C₄	—	139	3	no	—
83	C₂×C₄	C₂²×D₁₃₉	1	3	yes	—
84	C₂×C₄	C₂²×D₁₃₉	1	3	yes	—
85	C₂²	—	2	3	no	—
86	C₂²	—	278	3	no	—
87	C₂²	—	278	3	no	—
88	C₂²	—	278	3	no	—
89	C₂²	—	278	3	no	—
90	C₂²	—	278	3	no	—
91	C₂²	—	556	4	no	—
92	C₂²	C₂³×D₁₃₉	1	1	yes	characteristic
93	C₄	—	2	3	no	—
94	C₄	—	2	3	no	—
95	C₄	—	278	3	no	—
96	C₂	—	4	1	no	—
97	C₂	—	556	1	no	—
98	C₂	—	556	1	no	—
99	C₂	—	556	3	no	—
100	C₂	#17984888	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#17984839

C₂

yes

maximal, characteristic, complemented

#17984836

C₂

yes

maximal, characteristic, complemented

#17984827

C₂

yes

maximal, complemented

#17984817

C₂

yes

maximal, complemented

#17984815

C₂

yes

maximal, complemented

#17984812

C₂

yes

maximal, complemented

#126×C₁₃₉

C₂

yes

maximal, characteristic, complemented

C₂³×D₁₃₉

C₂²

yes

characteristic

C₂×D₅₅₆

C₂²

yes

—

C₂×D₅₅₆

C₂²

yes

—

C₂×D₅₅₆

C₂²

yes

complemented

C₂×C₂²⋊D₁₃₉

C₂²

yes

—

C₂×C₄×D₁₃₉

C₂²

yes

complemented

Dic₁₃₉⋊C₄

C₂²

yes

complemented

(C₂×D₁₃₉)⋊C₄

C₂²

yes

—

(C₂×D₁₃₉)⋊C₄

C₂²

yes

—

(C₂×D₁₃₉)⋊C₄

C₂²

yes

complemented

C₄⋊C₄×C₁₃₉

C₂²

yes

complemented

C₂²⋊C₄×C₁₃₉

C₂²

yes

—

C₄²×C₁₃₉

C₂²

yes

characteristic, complemented

C₂²×D₁₃₉

—

C₂²×D₁₃₉

C₂³

yes

characteristic

C₂²×D₁₃₉

C₂³

yes

characteristic

C₂²×D₁₃₉

C₂³

yes

—

D₅₅₆

—

D₅₅₆

—

D₅₅₆

—

C₂²⋊D₁₃₉

—

C₂×Dic₁₃₉

C₂³

yes

—

C₄×D₁₃₉

—

C₂³×C₁₃₉

C₂³

yes

characteristic

C₂×C₄×C₁₃₉

C₂³

yes

—

C₂×C₄×C₁₃₉

C₂³

yes

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

C₂⁴

yes

characteristic

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

—

Dic₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

H(5,𝔽₂)

yes

—

D₁₃₉

—

D₁₃₉

—

D₁₃₉

—

C₁₃₉

#560

yes

characteristic, complemented, Sylow

#560

—

139

maximal, Sylow

#127

—

139

—

#126

—

139

—

#126

D₁₃₉

yes

characteristic, complemented

#123

—

139

—

#121

—

139

—

#121

—

139

—

#120

—

139

—

C₂⁴

—

139

—

C₂×D₄

—

139

—

C₂×D₄

—

139

—

C₂×D₄

—

139

—

C₂×D₄

—

139

—

C₂²×C₄

—

139

—

C₄⋊C₄

—

139

—

C₄⋊C₄

C₂×D₁₃₉

yes

complemented

C₂²⋊C₄

—

139

—

C₂²⋊C₄

—

139

—

C₂²⋊C₄

—

139

—

C₂²⋊C₄

C₂×D₁₃₉

yes

—

C₄²

C₂×D₁₃₉

yes

characteristic, complemented

C₂³

—

556

—

C₂³

—

139

—

C₂³

—

139

—

C₂³

—

139

—

C₂³

C₂²×D₁₃₉

yes

characteristic

D₄

—

556

—

D₄

—

556

—

D₄

—

556

—

D₄

—

556

—

C₂×C₄

—

556

—

C₂×C₄

—

139

—

C₂×C₄

C₂²×D₁₃₉

yes

—

C₂×C₄

C₂²×D₁₃₉

yes

—

C₂²

—

C₂²

—

278

—

C₂²

—

278

—

C₂²

—

278

—

C₂²

—

278

—

C₂²

—

278

—

C₂²

—

556

—

C₂²

C₂³×D₁₃₉

yes

characteristic

C₄

—

C₄

—

C₄

—

278

—

C₂

—

C₂

—

556

—

C₂

—

556

—

C₂

—

556

—

100

C₂

#17984888

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₃₉
commutator, socle	C₂²×C₁₃₉	C₂⁴
Fitting	#126×C₁₃₉	C₂
hypercenter	#126	D₁₃₉
hypocenter	C₁₃₉	#560
LCS	G ▷ C₂²×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ #126

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₃₉

commutator, socle

C₂²×C₁₃₉

C₂⁴

Fitting

#126×C₁₃₉

C₂

D₁₃₉

C₁₃₉

G ▷ C₂²×C₁₃₉ ▷ C₁₃₉

C₁ ◁ C₂² ◁ #126

N	H ≅ G/N	type
1. #17984839	C₂	semidirect
2. #17984836	C₂	semidirect
3. #17984827	C₂	semidirect
4. #17984817	C₂	semidirect
5. #17984815	C₂	semidirect
6. #17984812	C₂	semidirect
7. #126×C₁₃₉	C₂	semidirect
11. C₂×D₅₅₆	C₂²	semidirect
13. C₂×C₄×D₁₃₉	C₂²	semidirect
14. Dic₁₃₉⋊C₄	C₂²	semidirect
17. (C₂×D₁₃₉)⋊C₄	C₂²	semidirect
18. C₄⋊C₄×C₁₃₉	C₂²	semidirect
20. C₄²×C₁₃₉	C₂²	semidirect
50. C₁₃₉	#560	semidirect
54. #126	D₁₃₉	semidirect
66. C₄⋊C₄	C₂×D₁₃₉	semidirect
71. C₄²	C₂×D₁₃₉	semidirect

H ≅ G/N

type

1. #17984839

C₂

semidirect

2. #17984836

C₂

semidirect

3. #17984827

C₂

semidirect

4. #17984817

C₂

semidirect

5. #17984815

C₂

semidirect

6. #17984812

C₂

semidirect

7. #126×C₁₃₉

C₂

semidirect

11. C₂×D₅₅₆

C₂²

semidirect

13. C₂×C₄×D₁₃₉

C₂²

semidirect

14. Dic₁₃₉⋊C₄

C₂²

semidirect

17. (C₂×D₁₃₉)⋊C₄

C₂²

semidirect

18. C₄⋊C₄×C₁₃₉

C₂²

semidirect

20. C₄²×C₁₃₉

C₂²

semidirect

50. C₁₃₉

#560

semidirect

54. #126

D₁₃₉

semidirect

66. C₄⋊C₄

C₂×D₁₃₉

semidirect

71. C₄²

C₂×D₁₃₉

semidirect

Subgroups — #22581694

Conjugacy classes of subgroups

Complements