Group #22581021 — subgroups

Subgroups — #22581021

Order 8896.

124 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 124 conjugacy classes of subgroups — too many to draw. Only its 53 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×#14555768	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₁₃₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₂²⋊D₁₃₉	C₂	1	1	yes	maximal, characteristic
4	Dic₁₃₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	(C₂×D₁₃₉)⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	#17984752	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²⋊C₈×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₂²⋊D₁₃₉	C₄	1	1	yes	characteristic
9	C₂×C₄×D₁₃₉	C₂²	1	1	yes	characteristic
10	#14555768	—	2	2	no	—
11	C₂×C₁₃₉⋊C₈	—	2	2	no	—
12	C₂×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
14	Dic₁₃₉⋊C₄	C₄	1	1	yes	characteristic
15	C₂²⋊Dic₁₃₉	C₄	1	1	yes	characteristic
16	(C₂×D₁₃₉)⋊C₄	C₄	1	1	yes	characteristic
17	C₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
18	C₂²×C₄×C₁₃₉	C₂²	1	1	yes	characteristic
19	C₂×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
20	C₂×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
21	C₂²×D₁₃₉	C₂×C₄	1	1	yes	characteristic
22	C₂²⋊D₁₃₉	—	2	2	no	—
23	C₂×Dic₁₃₉	C₂×C₄	1	1	yes	characteristic
24	C₂×Dic₁₃₉	C₂×C₄	1	1	yes	characteristic
25	C₂×Dic₁₃₉	C₂×C₄	1	1	yes	characteristic
26	C₄×D₁₃₉	—	2	1	no	—
27	C₄×D₁₃₉	—	2	1	no	—
28	C₁₃₉⋊C₈	—	2	1	no	—
29	C₁₃₉⋊C₈	D₄	1	2	yes	complemented
30	C₂³×C₁₃₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×C₁₃₉	—	2	1	no	—
32	C₂×C₄×C₁₃₉	—	2	1	no	—
33	C₂×C₄×C₁₃₉	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
35	C₈×C₁₃₉	—	2	1	no	—
36	C₈×C₁₃₉	—	2	1	no	—
37	C₂²×C₁₃₉	—	2	1	no	—
38	C₂²×C₁₃₉	—	2	1	no	—
39	C₂²×C₁₃₉	M₁₆	1	2	yes	complemented
40	C₂²×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
41	C₂×D₁₃₉	—	2	1	no	—
42	C₂×D₁₃₉	—	2	1	no	—
43	C₂×D₁₃₉	—	2	1	no	—
44	C₄×C₁₃₉	—	2	1	no	—
45	C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
46	C₄×C₁₃₉	#41	1	1	yes	characteristic
47	Dic₁₃₉	—	2	1	no	—
48	Dic₁₃₉	—	2	1	no	—
49	Dic₁₃₉	—	2	1	no	—
50	C₂×C₁₃₉	—	2	2	no	—
51	C₂×C₁₃₉	C₄×D₄	1	1	yes	characteristic
52	C₂×C₁₃₉	C₂×M₁₆	1	1	yes	characteristic
53	C₂×C₁₃₉	#131	1	1	yes	characteristic
54	D₁₃₉	—	4	1	no	—
55	C₁₃₉	#435	1	1	yes	characteristic, complemented, Sylow
56	#435	—	139	1	no	maximal, Sylow
57	C₂×M₁₆	—	139	1	no	—
58	C₂²×C₈	—	139	1	no	—
59	C₄×D₄	—	139	1	no	—
60	C₄⋊C₈	—	139	1	no	—
61	C₂²⋊C₈	—	139	1	no	—
62	C₂²⋊C₈	D₁₃₉	1	1	yes	characteristic, complemented
63	#97	—	139	1	no	—
64	C₂×D₄	—	139	1	no	—
65	C₂²×C₄	—	139	1	no	—
66	C₂²×C₄	C₂×D₁₃₉	1	1	yes	characteristic
67	M₁₆	—	278	2	no	—
68	C₂×C₈	—	278	2	no	—
69	C₂×C₈	—	139	1	no	—
70	C₂×C₈	—	139	1	no	—
71	C₂×C₈	C₂×D₁₃₉	1	1	yes	characteristic
72	C₂×C₈	C₂×D₁₃₉	1	1	yes	characteristic
73	C₄⋊C₄	—	139	1	no	—
74	C₂²⋊C₄	—	139	1	no	—
75	C₂²⋊C₄	—	139	1	no	—
76	C₄²	—	139	1	no	—
77	C₂³	—	139	1	no	—
78	C₂³	C₄×D₁₃₉	1	1	yes	characteristic
79	D₄	—	278	2	no	—
80	C₂×C₄	—	2	1	no	—
81	C₂×C₄	—	2	1	no	—
82	C₂×C₄	—	278	1	no	—
83	C₂×C₄	—	278	1	no	—
84	C₂×C₄	—	139	1	no	—
85	C₂×C₄	—	139	1	no	—
86	C₂×C₄	—	139	1	no	—
87	C₂×C₄	C₄×D₁₃₉	1	1	yes	characteristic
88	C₂×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
89	C₈	—	2	1	no	—
90	C₈	—	2	1	no	—
91	C₈	—	278	1	no	—
92	C₈	—	139	2	no	—
93	C₂²	—	2	1	no	—
94	C₂²	—	2	1	no	—
95	C₂²	—	278	1	no	—
96	C₂²	—	278	1	no	—
97	C₂²	—	278	1	no	—
98	C₂²	#14555768	1	2	yes	complemented
99	C₂²	C₂×C₄×D₁₃₉	1	1	yes	characteristic
100	C₄	—	2	1	no	—
101	C₄	—	278	1	no	—
102	C₄	—	278	1	no	—
103	C₄	—	278	1	no	—
104	C₄	D₄×D₁₃₉	1	1	yes	characteristic
105	C₄	#14555784	1	1	yes	characteristic
106	C₂	—	2	2	no	—
107	C₂	—	556	1	no	—
108	C₂	(C₂²⋊D₁₃₉)⋊C₄	1	1	yes	characteristic
109	C₂	C₂×#14555768	1	1	yes	characteristic
110	C₂	#17984850	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14555768

C₂

yes

maximal, characteristic, complemented

C₂²×C₁₃₉⋊C₈

C₂

yes

maximal, characteristic, complemented

C₄×C₂²⋊D₁₃₉

C₂

yes

maximal, characteristic

Dic₁₃₉⋊C₈

C₂

yes

maximal, characteristic, complemented

(C₂×D₁₃₉)⋊C₈

C₂

yes

maximal, characteristic, complemented

#17984752

C₂

yes

maximal, characteristic, complemented

C₂²⋊C₈×C₁₃₉

C₂

yes

maximal, characteristic, complemented

C₂×C₂²⋊D₁₃₉

C₄

yes

characteristic

C₂×C₄×D₁₃₉

C₂²

yes

characteristic

#14555768

—

C₂×C₁₃₉⋊C₈

—

C₂×C₁₃₉⋊C₈

C₂²

yes

characteristic

C₂×C₁₃₉⋊C₈

C₂²

yes

characteristic

Dic₁₃₉⋊C₄

C₄

yes

characteristic

C₂²⋊Dic₁₃₉

C₄

yes

characteristic

(C₂×D₁₃₉)⋊C₄

C₄

yes

characteristic

C₄×Dic₁₃₉

C₂²

yes

characteristic

C₂²×C₄×C₁₃₉

C₂²

yes

characteristic

C₂×C₈×C₁₃₉

C₂²

yes

characteristic

C₂×C₈×C₁₃₉

C₂²

yes

characteristic

C₂²×D₁₃₉

C₂×C₄

yes

characteristic

C₂²⋊D₁₃₉

—

C₂×Dic₁₃₉

C₂×C₄

yes

characteristic

C₂×Dic₁₃₉

C₂×C₄

yes

characteristic

C₂×Dic₁₃₉

C₂×C₄

yes

characteristic

C₄×D₁₃₉

—

C₄×D₁₃₉

—

C₁₃₉⋊C₈

—

C₁₃₉⋊C₈

D₄

yes

complemented

C₂³×C₁₃₉

C₂×C₄

yes

characteristic

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂×C₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂³

yes

characteristic

C₈×C₁₃₉

—

C₈×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

M₁₆

yes

complemented

C₂²×C₁₃₉

C₂²×C₄

yes

characteristic

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₄×C₁₃₉

#41

yes

characteristic

Dic₁₃₉

—

Dic₁₃₉

—

Dic₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

C₄×D₄

yes

characteristic

C₂×C₁₃₉

C₂×M₁₆

yes

characteristic

C₂×C₁₃₉

#131

yes

characteristic

D₁₃₉

—

C₁₃₉

#435

yes

characteristic, complemented, Sylow

#435

—

139

maximal, Sylow

C₂×M₁₆

—

139

—

C₂²×C₈

—

139

—

C₄×D₄

—

139

—

C₄⋊C₈

—

139

—

C₂²⋊C₈

—

139

—

C₂²⋊C₈

D₁₃₉

yes

characteristic, complemented

#97

—

139

—

C₂×D₄

—

139

—

C₂²×C₄

—

139

—

C₂²×C₄

C₂×D₁₃₉

yes

characteristic

M₁₆

—

278

—

C₂×C₈

—

278

—

C₂×C₈

—

139

—

C₂×C₈

—

139

—

C₂×C₈

C₂×D₁₃₉

yes

characteristic

C₂×C₈

C₂×D₁₃₉

yes

characteristic

C₄⋊C₄

—

139

—

C₂²⋊C₄

—

139

—

C₂²⋊C₄

—

139

—

C₄²

—

139

—

C₂³

—

139

—

C₂³

C₄×D₁₃₉

yes

characteristic

D₄

—

278

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

278

—

C₂×C₄

—

278

—

C₂×C₄

—

139

—

C₂×C₄

—

139

—

C₂×C₄

—

139

—

C₂×C₄

C₄×D₁₃₉

yes

characteristic

C₂×C₄

C₂²×D₁₃₉

yes

characteristic

C₈

—

C₈

—

C₈

—

278

—

C₈

—

139

—

C₂²

—

C₂²

—

C₂²

—

278

—

C₂²

—

278

—

C₂²

—

278

—

C₂²

#14555768

yes

complemented

C₂²

C₂×C₄×D₁₃₉

yes

characteristic

100

C₄

—

101

C₄

—

278

—

102

C₄

—

278

—

103

C₄

—

278

—

104

C₄

D₄×D₁₃₉

yes

characteristic

105

C₄

#14555784

yes

characteristic

106

C₂

—

107

C₂

—

556

—

108

C₂

(C₂²⋊D₁₃₉)⋊C₄

yes

characteristic

109

C₂

C₂×#14555768

yes

characteristic

110

C₂

#17984850

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₁₃₉
commutator, socle	C₂²×C₁₃₉	C₂²×C₄
Fitting	C₂²⋊C₈×C₁₃₉	C₂
hypercenter	C₂²⋊C₈	D₁₃₉
hypocenter	C₁₃₉	#435
LCS	G ▷ C₂²×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²⋊C₈

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×D₁₃₉

commutator, socle

C₂²×C₁₃₉

C₂²×C₄

Fitting

C₂²⋊C₈×C₁₃₉

C₂

hypercenter

C₂²⋊C₈

D₁₃₉

C₁₃₉

G ▷ C₂²×C₁₃₉ ▷ C₁₃₉

C₁ ◁ C₂×C₄ ◁ C₂²⋊C₈

N	H ≅ G/N	type
1. C₂×#14555768	C₂	semidirect
2. C₂²×C₁₃₉⋊C₈	C₂	semidirect
4. Dic₁₃₉⋊C₈	C₂	semidirect
5. (C₂×D₁₃₉)⋊C₈	C₂	semidirect
6. #17984752	C₂	semidirect
7. C₂²⋊C₈×C₁₃₉	C₂	semidirect
29. C₁₃₉⋊C₈	D₄	semidirect
39. C₂²×C₁₃₉	M₁₆	semidirect
55. C₁₃₉	#435	semidirect
62. C₂²⋊C₈	D₁₃₉	semidirect
98. C₂²	#14555768	semidirect

H ≅ G/N

type

1. C₂×#14555768

C₂

semidirect

2. C₂²×C₁₃₉⋊C₈

C₂

semidirect

4. Dic₁₃₉⋊C₈

C₂

semidirect

5. (C₂×D₁₃₉)⋊C₈

C₂

semidirect

6. #17984752

C₂

semidirect

7. C₂²⋊C₈×C₁₃₉

C₂

semidirect

29. C₁₃₉⋊C₈

D₄

semidirect

39. C₂²×C₁₃₉

M₁₆

semidirect

55. C₁₃₉

#435

semidirect

62. C₂²⋊C₈

D₁₃₉

semidirect

98. C₂²

#14555768

semidirect

Subgroups — #22581021

Conjugacy classes of subgroups

Complements