Group #22581152 — subgroups

Subgroups — #22581152

Order 8896.

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

normal subgroups only

This group has 128 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×#14555773	C₂	1	1	yes	maximal, characteristic, complemented
3	#17984838	C₂	1	1	yes	maximal, characteristic
4	#17984807	C₂	1	1	yes	maximal, characteristic, complemented
5	Dic₁₃₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	#17984762	C₂	1	1	yes	maximal, characteristic, complemented
7	#102×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₂₇₈	C₂²	1	1	yes	characteristic
9	C₂×D₅₅₆	C₂²	1	1	yes	characteristic
10	C₂×C₂²⋊D₁₃₉	—	2	1	no	—
11	C₂×C₂²⋊D₁₃₉	—	2	1	no	—
12	#14555774	—	2	2	no	—
13	#14555773	—	2	2	no	—
14	C₂×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
15	Dic₁₃₉⋊C₄	—	2	1	no	—
16	C₄×Dic₁₃₉	—	2	1	no	—
17	C₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
18	C₂×D₄×C₁₃₉	C₂²	1	1	yes	characteristic
19	C₂×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
20	C₄⋊C₄×C₁₃₉	C₂²	1	1	yes	characteristic
21	C₂²×D₁₃₉	—	2	1	no	—
22	Dic₂₇₈	—	2	1	no	—
23	Dic₂₇₈	D₄	1	2	yes	complemented
24	D₅₅₆	—	2	1	no	—
25	D₅₅₆	—	2	1	no	—
26	C₂²⋊D₁₃₉	—	4	2	no	—
27	C₂²⋊D₁₃₉	—	4	2	no	—
28	C₂×Dic₁₃₉	—	2	1	no	—
29	C₂×Dic₁₃₉	D₄	1	1	yes	characteristic
30	C₂×Dic₁₃₉	D₄	1	1	yes	characteristic
31	C₁₃₉⋊C₈	—	2	1	no	—
32	C₂³×C₁₃₉	—	2	1	no	—
33	D₄×C₁₃₉	—	2	1	no	—
34	D₄×C₁₃₉	—	2	1	no	—
35	C₂×C₄×C₁₃₉	—	2	1	no	—
36	C₂×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
37	C₈×C₁₃₉	—	2	1	no	—
38	C₂²×C₁₃₉	—	4	1	no	—
39	C₂²×C₁₃₉	—	4	1	no	—
40	C₂²×C₁₃₉	—	4	1	no	—
41	C₂²×C₁₃₉	C₂×D₄	1	1	yes	characteristic
42	C₂×D₁₃₉	—	4	1	no	—
43	C₂×D₁₃₉	—	4	1	no	—
44	C₂×D₁₃₉	—	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	2	no	—
49	Dic₁₃₉	—	2	1	no	—
50	Dic₁₃₉	SD₁₆	1	2	yes	complemented
51	C₂×C₁₃₉	—	8	1	no	—
52	C₂×C₁₃₉	#121	1	1	yes	characteristic
53	C₂×C₁₃₉	C₂×SD₁₆	1	1	yes	characteristic
54	C₂×C₁₃₉	AGL(1,ℤ/8)	1	1	yes	characteristic
55	D₁₃₉	—	8	1	no	—
56	C₁₃₉	#460	1	1	yes	characteristic, complemented, Sylow
57	#460	—	139	1	no	maximal, Sylow
58	C₂×SD₁₆	—	139	1	no	—
59	C₂×SD₁₆	—	139	1	no	—
60	#127	—	139	1	no	—
61	C₄×Q₈	—	139	1	no	—
62	C₄⋊C₈	—	139	1	no	—
63	#102	—	139	1	no	—
64	#102	D₁₃₉	1	1	yes	characteristic, complemented
65	C₂×Q₈	—	139	1	no	—
66	C₂×D₄	—	278	1	no	—
67	C₂×D₄	—	278	1	no	—
68	C₂×D₄	—	139	1	no	—
69	C₂×D₄	C₂×D₁₃₉	1	1	yes	characteristic
70	SD₁₆	—	278	2	no	—
71	SD₁₆	—	278	2	no	—
72	C₂×C₈	—	139	1	no	—
73	C₂×C₈	C₂×D₁₃₉	1	1	yes	characteristic
74	C₄⋊C₄	—	278	1	no	—
75	C₄⋊C₄	C₂×D₁₃₉	1	1	yes	characteristic
76	C₄²	—	278	1	no	—
77	C₄²	—	139	1	no	—
78	C₂³	—	2	1	no	—
79	C₂³	—	278	1	no	—
80	Q₈	—	278	1	no	—
81	Q₈	—	139	2	no	—
82	D₄	—	2	1	no	—
83	D₄	—	2	1	no	—
84	D₄	—	278	1	no	—
85	D₄	—	278	1	no	—
86	D₄	—	556	2	no	—
87	D₄	—	556	2	no	—
88	C₂×C₄	—	2	1	no	—
89	C₂×C₄	—	278	1	no	—
90	C₂×C₄	—	139	1	no	—
91	C₂×C₄	—	139	1	no	—
92	C₂×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
93	C₈	—	2	1	no	—
94	C₈	—	278	1	no	—
95	C₂²	—	4	1	no	—
96	C₂²	—	4	1	no	—
97	C₂²	—	4	1	no	—
98	C₂²	—	556	1	no	—
99	C₂²	—	556	1	no	—
100	C₂²	—	556	1	no	—
101	C₂²	D₄×D₁₃₉	1	1	yes	characteristic
102	C₄	—	4	1	no	—
103	C₄	—	278	2	no	—
104	C₄	—	278	1	no	—
105	C₄	—	139	2	no	—
106	C₄	D₄×D₁₃₉	1	1	yes	characteristic
107	C₄	#14555785	1	1	yes	characteristic
108	C₂	—	8	1	no	—
109	C₂	—	1112	1	no	—
110	C₂	#17984815	1	1	yes	characteristic
111	C₂	SD₁₆×D₁₃₉	1	1	yes	characteristic
112	C₂	#17984866	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14555774

C₂

yes

maximal, characteristic, complemented

C₂×#14555773

C₂

yes

maximal, characteristic, complemented

#17984838

C₂

yes

maximal, characteristic

#17984807

C₂

yes

maximal, characteristic, complemented

Dic₁₃₉⋊C₈

C₂

yes

maximal, characteristic, complemented

#17984762

C₂

yes

maximal, characteristic, complemented

#102×C₁₃₉

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₇₈

C₂²

yes

characteristic

C₂×D₅₅₆

C₂²

yes

characteristic

C₂×C₂²⋊D₁₃₉

—

C₂×C₂²⋊D₁₃₉

—

#14555774

—

#14555773

—

C₂×C₁₃₉⋊C₈

C₂²

yes

characteristic

Dic₁₃₉⋊C₄

—

C₄×Dic₁₃₉

—

C₄×Dic₁₃₉

C₂²

yes

characteristic

C₂×D₄×C₁₃₉

C₂²

yes

characteristic

C₂×C₈×C₁₃₉

C₂²

yes

characteristic

C₄⋊C₄×C₁₃₉

C₂²

yes

characteristic

C₂²×D₁₃₉

—

Dic₂₇₈

—

Dic₂₇₈

D₄

yes

complemented

D₅₅₆

—

D₅₅₆

—

C₂²⋊D₁₃₉

—

C₂²⋊D₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

D₄

yes

characteristic

C₂×Dic₁₃₉

D₄

yes

characteristic

C₁₃₉⋊C₈

—

C₂³×C₁₃₉

—

D₄×C₁₃₉

—

D₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂³

yes

characteristic

C₈×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

C₂×D₄

yes

characteristic

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₄×C₁₃₉

#41

yes

characteristic

Dic₁₃₉

—

Dic₁₃₉

—

Dic₁₃₉

SD₁₆

yes

complemented

C₂×C₁₃₉

—

C₂×C₁₃₉

#121

yes

characteristic

C₂×C₁₃₉

C₂×SD₁₆

yes

characteristic

C₂×C₁₃₉

AGL(1,ℤ/8)

yes

characteristic

D₁₃₉

—

C₁₃₉

#460

yes

characteristic, complemented, Sylow

#460

—

139

maximal, Sylow

C₂×SD₁₆

—

139

—

C₂×SD₁₆

—

139

—

#127

—

139

—

C₄×Q₈

—

139

—

C₄⋊C₈

—

139

—

#102

—

139

—

#102

D₁₃₉

yes

characteristic, complemented

C₂×Q₈

—

139

—

C₂×D₄

—

278

—

C₂×D₄

—

278

—

C₂×D₄

—

139

—

C₂×D₄

C₂×D₁₃₉

yes

characteristic

SD₁₆

—

278

—

SD₁₆

—

278

—

C₂×C₈

—

139

—

C₂×C₈

C₂×D₁₃₉

yes

characteristic

C₄⋊C₄

—

278

—

C₄⋊C₄

C₂×D₁₃₉

yes

characteristic

C₄²

—

278

—

C₄²

—

139

—

C₂³

—

C₂³

—

278

—

Q₈

—

278

—

Q₈

—

139

—

D₄

—

D₄

—

D₄

—

278

—

D₄

—

278

—

D₄

—

556

—

D₄

—

556

—

C₂×C₄

—

C₂×C₄

—

278

—

C₂×C₄

—

139

—

C₂×C₄

—

139

—

C₂×C₄

C₂²×D₁₃₉

yes

characteristic

C₈

—

C₈

—

278

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

556

—

C₂²

—

556

—

100

C₂²

—

556

—

101

C₂²

D₄×D₁₃₉

yes

characteristic

102

C₄

—

103

C₄

—

278

—

104

C₄

—

278

—

105

C₄

—

139

—

106

C₄

D₄×D₁₃₉

yes

characteristic

107

C₄

#14555785

yes

characteristic

108

C₂

—

109

C₂

—

1112

—

110

C₂

#17984815

yes

characteristic

111

C₂

SD₁₆×D₁₃₉

yes

characteristic

112

C₂

#17984866

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×D₁₃₉
commutator	C₂×C₄×C₁₃₉	C₂³
Frattini	C₂×C₄	C₂²×D₁₃₉
Fitting	#102×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂×D₄
hypercenter	#102	D₁₃₉
hypocenter	C₁₃₉	#460
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #102

subgroup

G/H

center

C₂²

D₄×D₁₃₉

commutator

C₂×C₄×C₁₃₉

C₂³

Frattini

C₂×C₄

C₂²×D₁₃₉

Fitting

#102×C₁₃₉

C₂

socle

C₂²×C₁₃₉

C₂×D₄

D₁₃₉

C₁₃₉

G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ #102

N	H ≅ G/N	type
1. C₂×#14555774	C₂	semidirect
2. C₂×#14555773	C₂	semidirect
4. #17984807	C₂	semidirect
5. Dic₁₃₉⋊C₈	C₂	semidirect
6. #17984762	C₂	semidirect
7. #102×C₁₃₉	C₂	semidirect
23. Dic₂₇₈	D₄	semidirect
50. Dic₁₃₉	SD₁₆	semidirect
56. C₁₃₉	#460	semidirect
64. #102	D₁₃₉	semidirect

H ≅ G/N

type

1. C₂×#14555774

C₂

semidirect

2. C₂×#14555773

C₂

semidirect

4. #17984807

C₂

semidirect

5. Dic₁₃₉⋊C₈

C₂

semidirect

6. #17984762

C₂

semidirect

7. #102×C₁₃₉

C₂

semidirect

23. Dic₂₇₈

D₄

semidirect

50. Dic₁₃₉

SD₁₆

semidirect

56. C₁₃₉

#460

semidirect

64. #102

D₁₃₉

semidirect

Subgroups — #22581152

Conjugacy classes of subgroups

Complements