Group #22581063 — subgroups

Subgroups — #22581063

Order 8896.

132 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 132 conjugacy classes of subgroups — too many to draw. Only its 51 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×#14555770	C₂	1	1	yes	maximal, characteristic
2	C₄×D₅₅₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#17984773	C₂	1	1	yes	maximal, characteristic, complemented
4	#17984763	C₂	1	1	yes	maximal, characteristic, complemented
5	#17984762	C₂	1	1	yes	maximal, characteristic, complemented
6	#17984753	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×D₄×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₅₅₆	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₁₃₉	—	2	1	no	—
10	#14555770	C₄	1	4	yes	complemented
11	C₂×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
13	C₄⋊Dic₁₃₉	C₂²	1	1	yes	characteristic
14	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
15	C₂×D₄×C₁₃₉	C₂²	1	1	yes	characteristic
16	C₂²×C₄×C₁₃₉	—	2	1	no	—
17	C₄⋊C₄×C₁₃₉	C₂²	1	1	yes	characteristic
18	C₂²⋊C₄×C₁₃₉	—	2	1	no	—
19	C₄²×C₁₃₉	C₂²	1	1	yes	characteristic
20	C₂²×D₁₃₉	—	2	1	no	—
21	D₅₅₆	—	2	1	no	—
22	D₅₅₆	C₂×C₄	1	2	yes	complemented
23	C₂×Dic₁₃₉	—	2	1	no	—
24	C₄×D₁₃₉	—	4	2	no	—
25	C₁₃₉⋊C₈	—	2	1	no	—
26	C₁₃₉⋊C₈	C₂×C₄	1	2	yes	complemented
27	C₂³×C₁₃₉	—	2	1	no	—
28	D₄×C₁₃₉	—	2	1	no	—
29	D₄×C₁₃₉	C₂×C₄	1	2	yes	complemented
30	C₂×C₄×C₁₃₉	—	2	1	no	—
31	C₂×C₄×C₁₃₉	—	4	2	no	—
32	C₂×C₄×C₁₃₉	D₄	1	1	yes	characteristic
33	C₂×C₄×C₁₃₉	D₄	1	1	yes	characteristic
34	C₂×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
35	C₂²×C₁₃₉	—	2	2	no	—
36	C₂²×C₁₃₉	—	4	1	no	—
37	C₂²×C₁₃₉	—	4	1	no	—
38	C₂²×C₁₃₉	C₂×D₄	1	1	yes	characteristic
39	C₂×D₁₃₉	—	2	2	no	—
40	C₂×D₁₃₉	—	4	1	no	—
41	C₂×D₁₃₉	—	4	1	no	—
42	C₄×C₁₃₉	—	2	1	no	—
43	C₄×C₁₃₉	—	2	1	no	—
44	C₄×C₁₃₉	—	4	1	no	—
45	C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
46	C₄×C₁₃₉	#41	1	1	yes	characteristic
47	Dic₁₃₉	—	4	1	no	—
48	C₂×C₁₃₉	—	4	2	no	—
49	C₂×C₁₃₉	C₄×D₄	1	1	yes	characteristic
50	C₂×C₁₃₉	AGL(1,ℤ/8)	1	1	yes	characteristic
51	C₂×C₁₃₉	AGL(1,ℤ/8)	1	1	yes	characteristic
52	D₁₃₉	—	4	2	no	—
53	C₁₃₉	#442	1	1	yes	characteristic, complemented, Sylow
54	#442	—	139	1	no	maximal, Sylow
55	C₂×D₈	—	139	1	no	—
56	C₄×D₄	—	139	1	no	—
57	C₄×D₄	D₁₃₉	1	1	yes	characteristic, complemented
58	#106	—	139	1	no	—
59	#102	—	139	1	no	—
60	#102	—	139	1	no	—
61	#97	—	139	1	no	—
62	C₂×D₄	—	139	1	no	—
63	C₂×D₄	C₂×D₁₃₉	1	1	yes	characteristic
64	C₂²×C₄	—	2	1	no	—
65	C₂²×C₄	—	278	1	no	—
66	D₈	—	139	4	no	—
67	C₂×C₈	—	139	1	no	—
68	C₂×C₈	—	139	1	no	—
69	C₄⋊C₄	—	139	1	no	—
70	C₄⋊C₄	C₂×D₁₃₉	1	1	yes	characteristic
71	C₂²⋊C₄	—	2	1	no	—
72	C₂²⋊C₄	—	278	1	no	—
73	C₄²	C₂×D₁₃₉	1	1	yes	characteristic
74	C₂³	—	2	1	no	—
75	C₂³	—	278	1	no	—
76	D₄	—	2	1	no	—
77	D₄	—	278	1	no	—
78	D₄	—	139	2	no	—
79	D₄	C₄×D₁₃₉	1	2	yes	complemented
80	C₂×C₄	—	2	1	no	—
81	C₂×C₄	—	4	2	no	—
82	C₂×C₄	—	278	1	no	—
83	C₂×C₄	—	556	2	no	—
84	C₂×C₄	C₂²⋊D₁₃₉	1	1	yes	characteristic
85	C₂×C₄	C₂²⋊D₁₃₉	1	1	yes	characteristic
86	C₂×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
87	C₈	—	278	1	no	—
88	C₈	—	139	2	no	—
89	C₂²	—	2	2	no	—
90	C₂²	—	4	1	no	—
91	C₂²	—	4	1	no	—
92	C₂²	—	278	2	no	—
93	C₂²	—	556	1	no	—
94	C₂²	—	556	1	no	—
95	C₂²	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
96	C₄	—	2	1	no	—
97	C₄	—	2	1	no	—
98	C₄	—	4	1	no	—
99	C₄	—	556	1	no	—
100	C₄	C₂×C₄×D₁₃₉	1	1	yes	characteristic
101	C₄	#14555785	1	1	yes	characteristic
102	C₂	—	4	2	no	—
103	C₂	—	556	2	no	—
104	C₂	C₄×C₂²⋊D₁₃₉	1	1	yes	characteristic
105	C₂	#17984867	1	1	yes	characteristic
106	C₂	#17984868	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14555770

C₂

yes

maximal, characteristic

C₄×D₅₅₆

C₂

yes

maximal, characteristic, complemented

#17984773

C₂

yes

maximal, characteristic, complemented

#17984763

C₂

yes

maximal, characteristic, complemented

#17984762

C₂

yes

maximal, characteristic, complemented

#17984753

C₂

yes

maximal, characteristic, complemented

C₄×D₄×C₁₃₉

C₂

yes

maximal, characteristic, complemented

C₂×D₅₅₆

C₂²

yes

characteristic

C₂×C₄×D₁₃₉

—

#14555770

C₄

yes

complemented

C₂×C₁₃₉⋊C₈

C₂²

yes

characteristic

C₂×C₁₃₉⋊C₈

C₂²

yes

characteristic

C₄⋊Dic₁₃₉

C₂²

yes

characteristic

(C₂×D₁₃₉)⋊C₄

—

C₂×D₄×C₁₃₉

C₂²

yes

characteristic

C₂²×C₄×C₁₃₉

—

C₄⋊C₄×C₁₃₉

C₂²

yes

characteristic

C₂²⋊C₄×C₁₃₉

—

C₄²×C₁₃₉

C₂²

yes

characteristic

C₂²×D₁₃₉

—

D₅₅₆

—

D₅₅₆

C₂×C₄

yes

complemented

C₂×Dic₁₃₉

—

C₄×D₁₃₉

—

C₁₃₉⋊C₈

—

C₁₃₉⋊C₈

C₂×C₄

yes

complemented

C₂³×C₁₃₉

—

D₄×C₁₃₉

—

D₄×C₁₃₉

C₂×C₄

yes

complemented

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

D₄

yes

characteristic

C₂×C₄×C₁₃₉

D₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂³

yes

characteristic

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

C₂×D₄

yes

characteristic

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

C₂²×C₄

yes

characteristic

C₄×C₁₃₉

#41

yes

characteristic

Dic₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

C₄×D₄

yes

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

D₁₃₉

—

C₁₃₉

#442

yes

characteristic, complemented, Sylow

#442

—

139

maximal, Sylow

C₂×D₈

—

139

—

C₄×D₄

—

139

—

C₄×D₄

D₁₃₉

yes

characteristic, complemented

#106

—

139

—

#102

—

139

—

#102

—

139

—

#97

—

139

—

C₂×D₄

—

139

—

C₂×D₄

C₂×D₁₃₉

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

278

—

D₈

—

139

—

C₂×C₈

—

139

—

C₂×C₈

—

139

—

C₄⋊C₄

—

139

—

C₄⋊C₄

C₂×D₁₃₉

yes

characteristic

C₂²⋊C₄

—

C₂²⋊C₄

—

278

—

C₄²

C₂×D₁₃₉

yes

characteristic

C₂³

—

C₂³

—

278

—

D₄

—

D₄

—

278

—

D₄

—

139

—

D₄

C₄×D₁₃₉

yes

complemented

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

278

—

C₂×C₄

—

556

—

C₂×C₄

C₂²⋊D₁₃₉

yes

characteristic

C₂×C₄

C₂²⋊D₁₃₉

yes

characteristic

C₂×C₄

C₂²×D₁₃₉

yes

characteristic

C₈

—

278

—

C₈

—

139

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

278

—

C₂²

—

556

—

C₂²

—

556

—

C₂²

C₂×C₂²⋊D₁₃₉

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

—

556

—

100

C₄

C₂×C₄×D₁₃₉

yes

characteristic

101

C₄

#14555785

yes

characteristic

102

C₂

—

103

C₂

—

556

—

104

C₂

C₄×C₂²⋊D₁₃₉

yes

characteristic

105

C₂

#17984867

yes

characteristic

106

C₂

#17984868

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₃₉
commutator	C₄×C₁₃₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₁₃₉
Fitting	C₄×D₄×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂×D₄
hypercenter	C₄×D₄	D₁₃₉
hypocenter	C₁₃₉	#442
LCS	G ▷ C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×D₄

subgroup

G/H

center

C₂²

C₂×C₂²⋊D₁₃₉

commutator

C₄×C₁₃₉

C₂²×C₄

Frattini

C₂×C₄

C₂²×D₁₃₉

Fitting

C₄×D₄×C₁₃₉

C₂

socle

C₂²×C₁₃₉

C₂×D₄

hypercenter

C₄×D₄

D₁₃₉

hypocenter

C₁₃₉

#442

LCS

G ▷ C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×D₄

N	H ≅ G/N	type
2. C₄×D₅₅₆	C₂	semidirect
3. #17984773	C₂	semidirect
4. #17984763	C₂	semidirect
5. #17984762	C₂	semidirect
6. #17984753	C₂	semidirect
7. C₄×D₄×C₁₃₉	C₂	semidirect
10. #14555770	C₄	semidirect
22. D₅₅₆	C₂×C₄	semidirect
26. C₁₃₉⋊C₈	C₂×C₄	semidirect
29. D₄×C₁₃₉	C₂×C₄	semidirect
53. C₁₃₉	#442	semidirect
57. C₄×D₄	D₁₃₉	semidirect
79. D₄	C₄×D₁₃₉	semidirect

H ≅ G/N

type

2. C₄×D₅₅₆

C₂

semidirect

3. #17984773

C₂

semidirect

4. #17984763

C₂

semidirect

5. #17984762

C₂

semidirect

6. #17984753

C₂

semidirect

7. C₄×D₄×C₁₃₉

C₂

semidirect

10. #14555770

C₄

semidirect

22. D₅₅₆

C₂×C₄

semidirect

26. C₁₃₉⋊C₈

C₂×C₄

semidirect

29. D₄×C₁₃₉

C₂×C₄

semidirect

53. C₁₃₉

#442

semidirect

57. C₄×D₄

D₁₃₉

semidirect

79. D₄

C₄×D₁₃₉

semidirect

Subgroups — #22581063

Conjugacy classes of subgroups

Complements