Group #22581274 — subgroups

Subgroups — #22581274

Order 8896.

106 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 106 conjugacy classes of subgroups — too many to draw. Only its 39 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	#17984801	C₂	1	1	yes	maximal, characteristic, complemented
2	#17984775	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	#17984763	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²⋊(C₁₃₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
7	#121×C₁₃₉	C₂	1	1	yes	maximal, characteristic
8	C₂×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₁₃₉	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₁₃₉	C₂²	1	1	yes	characteristic
12	C₂²⋊Dic₁₃₉	—	2	1	no	—
13	C₄×Dic₁₃₉	—	2	1	no	—
14	C₂×D₄×C₁₃₉	—	2	1	no	—
15	C₂×D₄×C₁₃₉	C₂²	1	1	yes	characteristic
16	C₂²×C₄×C₁₃₉	C₂²	1	1	yes	characteristic
17	C₄⋊C₄×C₁₃₉	C₂²	1	1	yes	characteristic
18	C₂²⋊C₄×C₁₃₉	—	2	1	no	—
19	C₂×Dic₁₃₉	—	2	1	no	—
20	C₂×Dic₁₃₉	—	2	1	no	—
21	C₁₃₉⋊C₈	—	2	1	no	—
22	C₁₃₉⋊C₈	—	2	1	no	—
23	C₂³×C₁₃₉	—	2	1	no	—
24	C₂³×C₁₃₉	D₄	1	1	yes	characteristic
25	D₄×C₁₃₉	—	2	1	no	—
26	D₄×C₁₃₉	—	2	1	no	—
27	D₄×C₁₃₉	—	4	2	no	—
28	C₂×C₄×C₁₃₉	—	2	1	no	—
29	C₂×C₄×C₁₃₉	—	2	1	no	—
30	C₂×C₄×C₁₃₉	—	2	1	no	—
31	C₂×C₄×C₁₃₉	D₄	1	1	yes	characteristic
32	C₂×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
33	C₂²×C₁₃₉	—	2	1	no	—
34	C₂²×C₁₃₉	—	2	1	no	—
35	C₂²×C₁₃₉	—	2	1	no	—
36	C₂²×C₁₃₉	—	4	1	no	—
37	C₂²×C₁₃₉	—	4	1	no	—
38	C₂²×C₁₃₉	—	4	1	no	—
39	C₂²×C₁₃₉	C₂×D₄	1	1	yes	characteristic
40	C₄×C₁₃₉	—	2	1	no	—
41	C₄×C₁₃₉	—	4	1	no	—
42	C₄×C₁₃₉	#41	1	1	yes	characteristic
43	C₄×C₁₃₉	#41	1	1	yes	characteristic
44	Dic₁₃₉	—	4	1	no	—
45	Dic₁₃₉	—	4	1	no	—
46	C₂×C₁₃₉	—	4	1	no	—
47	C₂×C₁₃₉	—	8	1	no	—
48	C₂×C₁₃₉	#123	1	1	yes	characteristic
49	C₂×C₁₃₉	#135	1	1	yes	characteristic
50	C₂×C₁₃₉	AGL(1,ℤ/8)	1	1	yes	characteristic
51	C₁₃₉	#482	1	1	yes	characteristic, complemented, Sylow
52	#482	—	139	1	no	maximal, Sylow
53	#121	D₁₃₉	1	1	yes	characteristic
54	#117	—	139	1	no	—
55	#107	—	139	1	no	—
56	#106	—	139	1	no	—
57	#102	—	139	1	no	—
58	#102	—	139	1	no	—
59	C₂²⋊C₈	—	139	1	no	—
60	C₂×D₄	—	2	1	no	—
61	C₂×D₄	C₂×D₁₃₉	1	1	yes	characteristic
62	C₂²×C₄	C₂×D₁₃₉	1	1	yes	characteristic
63	C₂×C₈	—	139	1	no	—
64	C₂×C₈	—	139	1	no	—
65	C₄⋊C₄	—	139	1	no	—
66	C₄⋊C₄	—	139	1	no	—
67	C₄⋊C₄	C₂×D₁₃₉	1	1	yes	characteristic
68	C₂²⋊C₄	—	2	1	no	—
69	C₂²⋊C₄	—	278	1	no	—
70	C₄²	—	278	1	no	—
71	C₂³	—	2	1	no	—
72	C₂³	C₂²⋊D₁₃₉	1	1	yes	characteristic
73	D₄	—	2	1	no	—
74	D₄	—	2	1	no	—
75	D₄	—	4	2	no	—
76	C₂×C₄	—	2	1	no	—
77	C₂×C₄	—	2	1	no	—
78	C₂×C₄	—	2	1	no	—
79	C₂×C₄	—	278	1	no	—
80	C₂×C₄	—	278	1	no	—
81	C₂×C₄	C₂²⋊D₁₃₉	1	1	yes	characteristic
82	C₂×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
83	C₈	—	278	1	no	—
84	C₈	—	278	1	no	—
85	C₂²	—	2	1	no	—
86	C₂²	—	2	1	no	—
87	C₂²	—	2	1	no	—
88	C₂²	—	4	1	no	—
89	C₂²	—	4	1	no	—
90	C₂²	—	4	1	no	—
91	C₂²	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
92	C₄	—	2	1	no	—
93	C₄	—	4	1	no	—
94	C₄	—	556	1	no	—
95	C₄	—	556	1	no	—
96	C₄	#14555784	1	1	yes	characteristic
97	C₄	#14555784	1	1	yes	characteristic
98	C₂	—	4	1	no	—
99	C₂	—	8	1	no	—
100	C₂	#17984823	1	1	yes	characteristic
101	C₂	#17984862	1	1	yes	characteristic
102	C₂	#17984868	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17984801

C₂

yes

maximal, characteristic, complemented

#17984775

C₂

yes

maximal, characteristic, complemented

#17984773

C₂

yes

maximal, characteristic, complemented

#17984763

C₂

yes

maximal, characteristic, complemented

#17984763

C₂

yes

maximal, characteristic, complemented

C₂²⋊(C₁₃₉⋊C₈)

C₂

yes

maximal, characteristic, complemented

#121×C₁₃₉

C₂

yes

maximal, characteristic

C₂×C₁₃₉⋊C₈

C₂²

yes

characteristic

C₂×C₁₃₉⋊C₈

C₂²

yes

characteristic

C₄⋊Dic₁₃₉

C₂²

yes

characteristic

C₄⋊Dic₁₃₉

C₂²

yes

characteristic

C₂²⋊Dic₁₃₉

—

C₄×Dic₁₃₉

—

C₂×D₄×C₁₃₉

—

C₂×D₄×C₁₃₉

C₂²

yes

characteristic

C₂²×C₄×C₁₃₉

C₂²

yes

characteristic

C₄⋊C₄×C₁₃₉

C₂²

yes

characteristic

C₂²⋊C₄×C₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₁₃₉⋊C₈

—

C₁₃₉⋊C₈

—

C₂³×C₁₃₉

—

C₂³×C₁₃₉

D₄

yes

characteristic

D₄×C₁₃₉

—

D₄×C₁₃₉

—

D₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

D₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂³

yes

characteristic

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

C₂×D₄

yes

characteristic

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

yes

C₄×C₁₃₉

yes

Dic₁₃₉

—

Dic₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

C₁₃₉

yes

characteristic, complemented, Sylow

#482

—

139

maximal, Sylow

#121

D₁₃₉

yes

characteristic

#117

—

139

—

#107

—

139

—

#106

—

139

—

#102

—

139

—

#102

—

139

—

C₂²⋊C₈

—

139

—

C₂×D₄

—

C₂×D₄

C₂×D₁₃₉

yes

characteristic

C₂²×C₄

C₂×D₁₃₉

yes

characteristic

C₂×C₈

—

139

—

C₂×C₈

—

139

—

C₄⋊C₄

—

139

—

C₄⋊C₄

—

139

—

C₄⋊C₄

C₂×D₁₃₉

yes

characteristic

C₂²⋊C₄

—

C₂²⋊C₄

—

278

—

C₄²

—

278

—

C₂³

—

C₂³

C₂²⋊D₁₃₉

yes

characteristic

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

278

—

C₂×C₄

—

278

—

C₂×C₄

C₂²⋊D₁₃₉

yes

characteristic

C₂×C₄

C₂²×D₁₃₉

yes

characteristic

C₈

—

278

—

C₈

—

278

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

C₂×C₂²⋊D₁₃₉

yes

characteristic

C₄

—

C₄

—

C₄

—

556

—

C₄

—

556

—

C₄

yes

C₄

yes

C₂

—

C₂

—

100

C₂

#17984823

yes

characteristic

101

C₂

#17984862

yes

characteristic

102

C₂

#17984868

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₃₉
commutator	C₂×C₄×C₁₃₉	C₂³
Frattini	C₂×C₄	C₂²×D₁₃₉
Fitting	#121×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂×D₄
hypercenter	#121	D₁₃₉
hypocenter	C₁₃₉	#482
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #121

subgroup

G/H

center

C₂²

C₂×C₂²⋊D₁₃₉

commutator

C₂×C₄×C₁₃₉

C₂³

Frattini

C₂×C₄

C₂²×D₁₃₉

Fitting

#121×C₁₃₉

C₂

socle

C₂²×C₁₃₉

C₂×D₄

D₁₃₉

C₁₃₉

G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #121

N	H ≅ G/N	type
1. #17984801	C₂	semidirect
2. #17984775	C₂	semidirect
3. #17984773	C₂	semidirect
4. #17984763	C₂	semidirect
5. #17984763	C₂	semidirect
6. C₂²⋊(C₁₃₉⋊C₈)	C₂	semidirect
51. C₁₃₉	#482	semidirect

H ≅ G/N

type

1. #17984801

C₂

semidirect

2. #17984775

C₂

semidirect

3. #17984773

C₂

semidirect

4. #17984763

C₂

semidirect

5. #17984763

C₂

semidirect

6. C₂²⋊(C₁₃₉⋊C₈)

C₂

semidirect

51. C₁₃₉

#482

semidirect

Subgroups — #22581274

Conjugacy classes of subgroups

Complements