Group #22581200 — subgroups

Subgroups — #22581200

Order 8896.

120 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 120 conjugacy classes of subgroups — too many to draw. Only its 41 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×#14555774	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14555768	C₂	1	1	yes	maximal, characteristic, complemented
3	#17984822	C₂	1	1	yes	maximal, characteristic, complemented
4	#17984817	C₂	1	1	yes	maximal, characteristic
5	#17984765	C₂	1	1	yes	maximal, characteristic, complemented
6	#17984762	C₂	1	1	yes	maximal, characteristic, complemented
7	#107×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₂₇₈	C₂²	1	1	yes	characteristic
9	C₂×D₅₅₆	—	2	1	no	—
10	C₂×D₅₅₆	C₂²	1	1	yes	characteristic
11	C₂×C₄×D₁₃₉	C₂²	1	1	yes	characteristic
12	#14555774	—	2	2	no	—
13	#14555768	—	2	2	no	—
14	C₂×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
15	C₄⋊Dic₁₃₉	—	2	1	no	—
16	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
17	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
18	C₂×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
19	C₄⋊C₄×C₁₃₉	C₂²	1	1	yes	characteristic
20	C₄⋊C₄×C₁₃₉	C₂²	1	1	yes	characteristic
21	C₂²×D₁₃₉	—	2	1	no	—
22	C₂²×D₁₃₉	D₄	1	1	yes	characteristic
23	Dic₂₇₈	—	2	1	no	—
24	Dic₂₇₈	—	2	1	no	—
25	D₅₅₆	—	2	1	no	—
26	D₅₅₆	—	2	1	no	—
27	D₅₅₆	—	4	2	no	—
28	C₂×Dic₁₃₉	—	2	1	no	—
29	C₂×Dic₁₃₉	D₄	1	1	yes	characteristic
30	C₄×D₁₃₉	—	2	1	no	—
31	C₄×D₁₃₉	—	2	1	no	—
32	C₁₃₉⋊C₈	—	2	1	no	—
33	C₂×C₄×C₁₃₉	—	2	1	no	—
34	C₂×C₄×C₁₃₉	—	2	1	no	—
35	C₂×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
36	C₈×C₁₃₉	D₄	1	2	yes	complemented
37	C₂²×C₁₃₉	C₂×D₄	1	1	yes	characteristic
38	C₂×D₁₃₉	—	2	1	no	—
39	C₂×D₁₃₉	—	2	1	no	—
40	C₂×D₁₃₉	—	2	1	no	—
41	C₂×D₁₃₉	—	4	1	no	—
42	C₂×D₁₃₉	—	4	1	no	—
43	C₂×D₁₃₉	—	4	1	no	—
44	C₄×C₁₃₉	—	4	1	no	—
45	C₄×C₁₃₉	—	4	1	no	—
46	C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
47	C₄×C₁₃₉	#41	1	1	yes	characteristic
48	Dic₁₃₉	—	2	1	no	—
49	Dic₁₃₉	—	4	1	no	—
50	C₂×C₁₃₉	#121	1	1	yes	characteristic
51	C₂×C₁₃₉	AGL(1,ℤ/8)	1	1	yes	characteristic
52	C₂×C₁₃₉	#137	1	1	yes	characteristic
53	D₁₃₉	—	4	1	no	—
54	D₁₃₉	—	8	1	no	—
55	C₁₃₉	#468	1	1	yes	characteristic, complemented, Sylow
56	#468	—	139	1	no	maximal, Sylow
57	C₂×SD₁₆	—	139	1	no	—
58	C₂×M₁₆	—	139	1	no	—
59	C₂²⋊Q₈	—	139	1	no	—
60	#121	—	139	1	no	—
61	#107	D₁₃₉	1	1	yes	characteristic, complemented
62	#103	—	139	1	no	—
63	#102	—	139	1	no	—
64	C₂×Q₈	—	139	1	no	—
65	C₂×D₄	—	278	1	no	—
66	C₂×D₄	—	139	1	no	—
67	C₂²×C₄	—	139	1	no	—
68	SD₁₆	—	278	2	no	—
69	M₁₆	—	278	2	no	—
70	C₂×C₈	—	139	1	no	—
71	C₂×C₈	C₂×D₁₃₉	1	1	yes	characteristic
72	C₄⋊C₄	—	278	1	no	—
73	C₄⋊C₄	C₂×D₁₃₉	1	1	yes	characteristic
74	C₄⋊C₄	C₂×D₁₃₉	1	1	yes	characteristic
75	C₂²⋊C₄	—	278	1	no	—
76	C₂²⋊C₄	—	278	1	no	—
77	C₂³	—	278	1	no	—
78	C₂³	—	139	1	no	—
79	Q₈	—	278	1	no	—
80	Q₈	—	278	1	no	—
81	D₄	—	278	1	no	—
82	D₄	—	278	1	no	—
83	D₄	—	556	2	no	—
84	C₂×C₄	—	2	1	no	—
85	C₂×C₄	—	2	1	no	—
86	C₂×C₄	—	278	1	no	—
87	C₂×C₄	—	278	1	no	—
88	C₂×C₄	—	278	1	no	—
89	C₂×C₄	—	139	1	no	—
90	C₂×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
91	C₈	—	278	1	no	—
92	C₈	D₅₅₆	1	2	yes	complemented
93	C₂²	—	278	1	no	—
94	C₂²	—	278	1	no	—
95	C₂²	—	278	1	no	—
96	C₂²	—	556	1	no	—
97	C₂²	—	556	1	no	—
98	C₂²	—	556	1	no	—
99	C₂²	D₄×D₁₃₉	1	1	yes	characteristic
100	C₄	—	4	1	no	—
101	C₄	—	4	1	no	—
102	C₄	—	278	1	no	—
103	C₄	—	556	1	no	—
104	C₄	C₂×D₅₅₆	1	1	yes	characteristic
105	C₄	#14555786	1	1	yes	characteristic
106	C₂	—	556	1	no	—
107	C₂	—	1112	1	no	—
108	C₂	#17984817	1	1	yes	characteristic
109	C₂	#17984866	1	1	yes	characteristic
110	C₂	#17984874	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14555774

C₂

yes

maximal, characteristic, complemented

C₂×#14555768

C₂

yes

maximal, characteristic, complemented

#17984822

C₂

yes

maximal, characteristic, complemented

#17984817

C₂

yes

maximal, characteristic

#17984765

C₂

yes

maximal, characteristic, complemented

#17984762

C₂

yes

maximal, characteristic, complemented

#107×C₁₃₉

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₇₈

C₂²

yes

characteristic

C₂×D₅₅₆

—

C₂×D₅₅₆

C₂²

yes

characteristic

C₂×C₄×D₁₃₉

C₂²

yes

characteristic

#14555774

—

#14555768

—

C₂×C₁₃₉⋊C₈

C₂²

yes

characteristic

C₄⋊Dic₁₃₉

—

(C₂×D₁₃₉)⋊C₄

—

(C₂×D₁₃₉)⋊C₄

—

C₂×C₈×C₁₃₉

C₂²

yes

characteristic

C₄⋊C₄×C₁₃₉

C₂²

yes

characteristic

C₄⋊C₄×C₁₃₉

C₂²

yes

characteristic

C₂²×D₁₃₉

—

C₂²×D₁₃₉

D₄

yes

characteristic

Dic₂₇₈

—

Dic₂₇₈

—

D₅₅₆

—

D₅₅₆

—

D₅₅₆

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

D₄

yes

characteristic

C₄×D₁₃₉

—

C₄×D₁₃₉

—

C₁₃₉⋊C₈

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂³

yes

characteristic

C₈×C₁₃₉

D₄

yes

complemented

C₂²×C₁₃₉

C₂×D₄

yes

characteristic

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₄×C₁₃₉

#41

yes

characteristic

Dic₁₃₉

—

Dic₁₃₉

—

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

D₁₃₉

—

D₁₃₉

—

C₁₃₉

#468

yes

characteristic, complemented, Sylow

#468

—

139

maximal, Sylow

C₂×SD₁₆

—

139

—

C₂×M₁₆

—

139

—

C₂²⋊Q₈

—

139

—

#121

—

139

—

#107

D₁₃₉

yes

characteristic, complemented

#103

—

139

—

#102

—

139

—

C₂×Q₈

—

139

—

C₂×D₄

—

278

—

C₂×D₄

—

139

—

C₂²×C₄

—

139

—

SD₁₆

—

278

—

M₁₆

—

278

—

C₂×C₈

—

139

—

C₂×C₈

C₂×D₁₃₉

yes

characteristic

C₄⋊C₄

—

278

—

C₄⋊C₄

C₂×D₁₃₉

yes

characteristic

C₄⋊C₄

C₂×D₁₃₉

yes

characteristic

C₂²⋊C₄

—

278

—

C₂²⋊C₄

—

278

—

C₂³

—

278

—

C₂³

—

139

—

Q₈

—

278

—

Q₈

—

278

—

D₄

—

278

—

D₄

—

278

—

D₄

—

556

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

278

—

C₂×C₄

—

278

—

C₂×C₄

—

278

—

C₂×C₄

—

139

—

C₂×C₄

C₂²×D₁₃₉

yes

characteristic

C₈

—

278

—

C₈

D₅₅₆

yes

complemented

C₂²

—

278

—

C₂²

—

278

—

C₂²

—

278

—

C₂²

—

556

—

C₂²

—

556

—

C₂²

—

556

—

C₂²

D₄×D₁₃₉

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

—

278

—

103

C₄

—

556

—

104

C₄

C₂×D₅₅₆

yes

characteristic

105

C₄

#14555786

yes

characteristic

106

C₂

—

556

—

107

C₂

—

1112

—

108

C₂

#17984817

yes

characteristic

109

C₂

#17984866

yes

characteristic

110

C₂

#17984874

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×D₁₃₉
commutator	C₂×C₄×C₁₃₉	C₂³
Frattini	C₂×C₄	C₂²×D₁₃₉
Fitting	#107×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂×D₄
hypercenter	#107	D₁₃₉
hypocenter	C₁₃₉	#468
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

subgroup

G/H

center

C₂²

D₄×D₁₃₉

commutator

C₂×C₄×C₁₃₉

C₂³

Frattini

C₂×C₄

C₂²×D₁₃₉

Fitting

#107×C₁₃₉

C₂

socle

C₂²×C₁₃₉

C₂×D₄

D₁₃₉

C₁₃₉

G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

N	H ≅ G/N	type
1. C₂×#14555774	C₂	semidirect
2. C₂×#14555768	C₂	semidirect
3. #17984822	C₂	semidirect
5. #17984765	C₂	semidirect
6. #17984762	C₂	semidirect
7. #107×C₁₃₉	C₂	semidirect
36. C₈×C₁₃₉	D₄	semidirect
55. C₁₃₉	#468	semidirect
61. #107	D₁₃₉	semidirect
92. C₈	D₅₅₆	semidirect

H ≅ G/N

type

1. C₂×#14555774

C₂

semidirect

2. C₂×#14555768

C₂

semidirect

3. #17984822

C₂

semidirect

5. #17984765

C₂

semidirect

6. #17984762

C₂

semidirect

7. #107×C₁₃₉

C₂

semidirect

36. C₈×C₁₃₉

D₄

semidirect

55. C₁₃₉

#468

semidirect

61. #107

D₁₃₉

semidirect

92. C₈

D₅₅₆

semidirect

Subgroups — #22581200

Conjugacy classes of subgroups

Complements