Group #22581054 — subgroups

Subgroups — #22581054

Order 8896.

120 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 120 conjugacy classes of subgroups — too many to draw. Only its 57 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	Q₈×Dic₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×Dic₁₃₉	C₂	1	1	yes	maximal, characteristic
3	#17984776	C₂	1	1	yes	maximal, characteristic, complemented
4	#17984766	C₂	1	1	yes	maximal, characteristic, complemented
5	#17984763	C₂	1	1	yes	maximal, characteristic
6	#17984752	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×SD₁₆×C₁₃₉	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₃₉	—	2	1	no	—
9	C₂×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₁₃₉	—	2	1	no	—
11	C₄⋊Dic₁₃₉	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₁₃₉	C₂²	1	1	yes	characteristic
13	C₂²⋊Dic₁₃₉	—	2	1	no	—
14	C₄×Dic₁₃₉	—	2	1	no	—
15	C₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
16	C₂×Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
17	C₂×D₄×C₁₃₉	C₂²	1	1	yes	characteristic
18	SD₁₆×C₁₃₉	C₄	1	4	yes	complemented
19	C₂×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
20	C₂×Dic₁₃₉	—	2	1	no	—
21	C₂×Dic₁₃₉	—	2	1	no	—
22	C₂×Dic₁₃₉	—	4	2	no	—
23	C₂×Dic₁₃₉	D₄	1	1	yes	characteristic
24	C₂×Dic₁₃₉	D₄	1	1	yes	characteristic
25	C₁₃₉⋊C₈	—	2	1	no	—
26	C₂³×C₁₃₉	—	2	1	no	—
27	Q₈×C₁₃₉	—	2	1	no	—
28	Q₈×C₁₃₉	C₂×C₄	1	2	yes	complemented
29	D₄×C₁₃₉	—	2	1	no	—
30	D₄×C₁₃₉	C₂×C₄	1	2	yes	—
31	C₂×C₄×C₁₃₉	—	2	1	no	—
32	C₂×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
33	C₈×C₁₃₉	C₂×C₄	1	2	yes	complemented
34	C₂²×C₁₃₉	—	2	2	no	—
35	C₂²×C₁₃₉	—	4	1	no	—
36	C₂²×C₁₃₉	—	4	1	no	—
37	C₂²×C₁₃₉	C₂×D₄	1	1	yes	characteristic
38	C₄×C₁₃₉	—	2	2	no	—
39	C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
40	C₄×C₁₃₉	#41	1	1	yes	characteristic
41	Dic₁₃₉	—	2	1	no	—
42	Dic₁₃₉	—	2	1	no	—
43	Dic₁₃₉	—	4	1	no	—
44	Dic₁₃₉	—	4	1	no	—
45	C₂×C₁₃₉	—	4	2	no	—
46	C₂×C₁₃₉	C₄×D₄	1	1	yes	characteristic
47	C₂×C₁₃₉	AGL(1,ℤ/8)	1	1	yes	characteristic
48	C₂×C₁₃₉	#137	1	1	yes	characteristic
49	C₁₃₉	#440	1	1	yes	characteristic, complemented, Sylow
50	#440	—	139	1	no	maximal, Sylow
51	C₂×SD₁₆	D₁₃₉	1	1	yes	characteristic
52	C₄×Q₈	—	139	1	no	—
53	C₄×D₄	—	139	1	no	—
54	#107	—	139	1	no	—
55	#103	—	139	1	no	—
56	#102	—	139	1	no	—
57	#97	—	139	1	no	—
58	C₂×Q₈	C₂×D₁₃₉	1	1	yes	characteristic
59	C₂×D₄	C₂×D₁₃₉	1	1	yes	characteristic
60	C₂²×C₄	—	278	1	no	—
61	SD₁₆	Dic₁₃₉	1	4	yes	complemented
62	C₂×C₈	—	139	1	no	—
63	C₂×C₈	C₂×D₁₃₉	1	1	yes	characteristic
64	C₄⋊C₄	—	278	1	no	—
65	C₄⋊C₄	—	139	1	no	—
66	C₄⋊C₄	—	139	1	no	—
67	C₂²⋊C₄	—	278	1	no	—
68	C₄²	—	278	1	no	—
69	C₄²	—	139	1	no	—
70	C₂³	—	2	1	no	—
71	Q₈	—	2	1	no	—
72	Q₈	C₂×Dic₁₃₉	1	2	yes	complemented
73	D₄	—	2	1	no	—
74	D₄	C₂×Dic₁₃₉	1	2	yes	—
75	C₂×C₄	—	2	1	no	—
76	C₂×C₄	—	278	1	no	—
77	C₂×C₄	—	278	1	no	—
78	C₂×C₄	—	556	2	no	—
79	C₂×C₄	—	139	1	no	—
80	C₂×C₄	—	139	1	no	—
81	C₂×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
82	C₈	—	278	1	no	—
83	C₈	C₂×Dic₁₃₉	1	2	yes	complemented
84	C₂²	—	2	2	no	—
85	C₂²	—	4	1	no	—
86	C₂²	—	4	1	no	—
87	C₂²	D₄×D₁₃₉	1	1	yes	characteristic
88	C₄	—	2	2	no	—
89	C₄	—	278	1	no	—
90	C₄	—	278	1	no	—
91	C₄	—	556	1	no	—
92	C₄	—	556	1	no	—
93	C₄	C₂²×Dic₁₃₉	1	1	yes	characteristic
94	C₄	#14555784	1	1	yes	characteristic
95	C₂	—	4	2	no	—
96	C₂	D₄×Dic₁₃₉	1	1	yes	characteristic
97	C₂	#17984869	1	1	yes	characteristic
98	C₂	#17984871	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×Dic₁₃₉

C₂

yes

maximal, characteristic, complemented

D₄×Dic₁₃₉

C₂

yes

maximal, characteristic

#17984776

C₂

yes

maximal, characteristic, complemented

#17984766

C₂

yes

maximal, characteristic, complemented

#17984763

C₂

yes

maximal, characteristic

#17984752

C₂

yes

maximal, characteristic, complemented

C₂×SD₁₆×C₁₃₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₃₉

—

C₂×C₁₃₉⋊C₈

C₂²

yes

characteristic

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

C₂²

yes

characteristic

C₄⋊Dic₁₃₉

C₂²

yes

characteristic

C₂²⋊Dic₁₃₉

—

C₄×Dic₁₃₉

—

C₄×Dic₁₃₉

C₂²

yes

characteristic

C₂×Q₈×C₁₃₉

C₂²

yes

characteristic

C₂×D₄×C₁₃₉

C₂²

yes

characteristic

SD₁₆×C₁₃₉

C₄

yes

complemented

C₂×C₈×C₁₃₉

C₂²

yes

characteristic

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

D₄

yes

characteristic

C₂×Dic₁₃₉

D₄

yes

characteristic

C₁₃₉⋊C₈

—

C₂³×C₁₃₉

—

Q₈×C₁₃₉

—

Q₈×C₁₃₉

C₂×C₄

yes

complemented

D₄×C₁₃₉

—

D₄×C₁₃₉

C₂×C₄

yes

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂³

yes

characteristic

C₈×C₁₃₉

C₂×C₄

yes

complemented

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

C₂×D₄

yes

characteristic

C₄×C₁₃₉

—

C₄×C₁₃₉

C₂²×C₄

yes

characteristic

C₄×C₁₃₉

#41

yes

characteristic

Dic₁₃₉

—

Dic₁₃₉

—

Dic₁₃₉

—

Dic₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

C₄×D₄

yes

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

C₁₃₉

yes

characteristic, complemented, Sylow

#440

—

139

maximal, Sylow

C₂×SD₁₆

D₁₃₉

yes

characteristic

C₄×Q₈

—

139

—

C₄×D₄

—

139

—

#107

—

139

—

#103

—

139

—

#102

—

139

—

#97

—

139

—

C₂×Q₈

C₂×D₁₃₉

yes

characteristic

C₂×D₄

C₂×D₁₃₉

yes

characteristic

C₂²×C₄

—

278

—

SD₁₆

Dic₁₃₉

yes

complemented

C₂×C₈

—

139

—

C₂×C₈

C₂×D₁₃₉

yes

characteristic

C₄⋊C₄

—

278

—

C₄⋊C₄

—

139

—

C₄⋊C₄

—

139

—

C₂²⋊C₄

—

278

—

C₄²

—

278

—

C₄²

—

139

—

C₂³

—

Q₈

—

Q₈

C₂×Dic₁₃₉

yes

complemented

D₄

—

D₄

C₂×Dic₁₃₉

yes

—

C₂×C₄

—

C₂×C₄

—

278

—

C₂×C₄

—

278

—

C₂×C₄

—

556

—

C₂×C₄

—

139

—

C₂×C₄

—

139

—

C₂×C₄

C₂²×D₁₃₉

yes

characteristic

C₈

—

278

—

C₈

C₂×Dic₁₃₉

yes

complemented

C₂²

—

C₂²

—

C₂²

—

C₂²

D₄×D₁₃₉

yes

characteristic

C₄

—

C₄

—

278

—

C₄

—

278

—

C₄

—

556

—

C₄

—

556

—

C₄

C₂²×Dic₁₃₉

yes

characteristic

C₄

#14555784

yes

characteristic

C₂

—

C₂

D₄×Dic₁₃₉

yes

C₂

yes

C₂

yes

	subgroup	G/H
center	C₂²	D₄×D₁₃₉
commutator	C₄×C₁₃₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₁₃₉
Fitting	C₂×SD₁₆×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂×D₄
hypercenter	C₂×SD₁₆	D₁₃₉
hypocenter	C₁₃₉	#440
LCS	G ▷ C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×SD₁₆

subgroup

G/H

center

C₂²

D₄×D₁₃₉

commutator

C₄×C₁₃₉

C₂²×C₄

Frattini

C₂×C₄

C₂²×D₁₃₉

Fitting

C₂×SD₁₆×C₁₃₉

C₂

socle

C₂²×C₁₃₉

C₂×D₄

hypercenter

C₂×SD₁₆

D₁₃₉

hypocenter

C₁₃₉

#440

LCS

G ▷ C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×SD₁₆

N	H ≅ G/N	type
1. Q₈×Dic₁₃₉	C₂	semidirect
3. #17984776	C₂	semidirect
4. #17984766	C₂	semidirect
6. #17984752	C₂	semidirect
18. SD₁₆×C₁₃₉	C₄	semidirect
28. Q₈×C₁₃₉	C₂×C₄	semidirect
33. C₈×C₁₃₉	C₂×C₄	semidirect
49. C₁₃₉	#440	semidirect
61. SD₁₆	Dic₁₃₉	semidirect
72. Q₈	C₂×Dic₁₃₉	semidirect
83. C₈	C₂×Dic₁₃₉	semidirect

H ≅ G/N

type

1. Q₈×Dic₁₃₉

C₂

semidirect

3. #17984776

C₂

semidirect

4. #17984766

C₂

semidirect

6. #17984752

C₂

semidirect

18. SD₁₆×C₁₃₉

C₄

semidirect

28. Q₈×C₁₃₉

C₂×C₄

semidirect

33. C₈×C₁₃₉

C₂×C₄

semidirect

49. C₁₃₉

#440

semidirect

61. SD₁₆

Dic₁₃₉

semidirect

72. Q₈

C₂×Dic₁₃₉

semidirect

83. C₈

C₂×Dic₁₃₉

semidirect

Subgroups — #22581054

Conjugacy classes of subgroups

Complements