Group #22581156 — subgroups

Subgroups — #22581156

Order 8896.

120 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 120 conjugacy classes of subgroups — too many to draw. Only its 43 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×#14555773	C₂	1	1	yes	maximal, characteristic
2	#17984840	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄×Dic₁₃₉	C₂	1	1	yes	maximal, characteristic
4	Dic₁₃₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	#17984767	C₂	1	1	yes	maximal, characteristic, complemented
6	#17984766	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×SD₁₆×C₁₃₉	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₂₇₈	C₂²	1	1	yes	characteristic
9	C₂²×Dic₁₃₉	—	2	1	no	—
10	#14555773	—	2	2	no	—
11	C₂×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₁₃₉	C₂²	1	1	yes	characteristic
13	Dic₁₃₉⋊C₄	—	2	1	no	—
14	Dic₁₃₉⋊C₄	—	2	1	no	—
15	C₂²⋊Dic₁₃₉	—	2	1	no	—
16	C₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
18	C₂×D₄×C₁₃₉	C₂²	1	1	yes	characteristic
19	SD₁₆×C₁₃₉	—	2	2	no	—
20	C₂×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
21	Dic₂₇₈	—	2	1	no	—
22	Dic₂₇₈	—	2	1	no	—
23	C₂×Dic₁₃₉	—	2	1	no	—
24	C₂×Dic₁₃₉	—	2	1	no	—
25	C₂×Dic₁₃₉	—	4	2	no	—
26	C₂×Dic₁₃₉	D₄	1	1	yes	characteristic
27	C₂×Dic₁₃₉	D₄	1	1	yes	characteristic
28	C₁₃₉⋊C₈	—	2	1	no	—
29	C₂³×C₁₃₉	—	2	1	no	—
30	Q₈×C₁₃₉	—	2	1	no	—
31	Q₈×C₁₃₉	—	2	1	no	—
32	D₄×C₁₃₉	—	2	1	no	—
33	D₄×C₁₃₉	D₄	1	2	yes	—
34	C₂×C₄×C₁₃₉	—	2	1	no	—
35	C₂×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
36	C₈×C₁₃₉	—	2	1	no	—
37	C₂²×C₁₃₉	—	2	2	no	—
38	C₂²×C₁₃₉	—	4	1	no	—
39	C₂²×C₁₃₉	—	4	1	no	—
40	C₂²×C₁₃₉	C₂×D₄	1	1	yes	characteristic
41	C₄×C₁₃₉	—	4	1	no	—
42	C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
43	C₄×C₁₃₉	#41	1	1	yes	characteristic
44	Dic₁₃₉	—	2	1	no	—
45	Dic₁₃₉	—	4	1	no	—
46	Dic₁₃₉	—	4	1	no	—
47	Dic₁₃₉	SD₁₆	1	2	yes	complemented
48	C₂×C₁₃₉	—	4	2	no	—
49	C₂×C₁₃₉	#121	1	1	yes	characteristic
50	C₂×C₁₃₉	C₂×SD₁₆	1	1	yes	characteristic
51	C₂×C₁₃₉	#137	1	1	yes	characteristic
52	C₁₃₉	#461	1	1	yes	characteristic, complemented, Sylow
53	#461	—	139	1	no	maximal, Sylow
54	C₂×SD₁₆	—	139	1	no	—
55	C₂×SD₁₆	D₁₃₉	1	1	yes	characteristic
56	C₄⋊Q₈	—	139	1	no	—
57	C₄×D₄	—	139	1	no	—
58	C₄⋊C₈	—	139	1	no	—
59	#103	—	139	1	no	—
60	#103	—	139	1	no	—
61	C₂×Q₈	—	139	1	no	—
62	C₂×Q₈	C₂×D₁₃₉	1	1	yes	characteristic
63	C₂×D₄	C₂×D₁₃₉	1	1	yes	characteristic
64	C₂²×C₄	—	278	1	no	—
65	SD₁₆	—	2	2	no	—
66	SD₁₆	—	278	2	no	—
67	C₂×C₈	—	139	1	no	—
68	C₂×C₈	C₂×D₁₃₉	1	1	yes	characteristic
69	C₄⋊C₄	—	278	1	no	—
70	C₄⋊C₄	—	278	1	no	—
71	C₄⋊C₄	—	139	1	no	—
72	C₂²⋊C₄	—	278	1	no	—
73	C₄²	—	139	1	no	—
74	C₂³	—	2	1	no	—
75	Q₈	—	2	1	no	—
76	Q₈	—	2	1	no	—
77	Q₈	—	278	1	no	—
78	Q₈	—	278	1	no	—
79	D₄	—	2	1	no	—
80	D₄	C₂²⋊D₁₃₉	1	2	yes	—
81	C₂×C₄	—	2	1	no	—
82	C₂×C₄	—	278	1	no	—
83	C₂×C₄	—	278	1	no	—
84	C₂×C₄	—	556	2	no	—
85	C₂×C₄	—	139	1	no	—
86	C₂×C₄	—	139	1	no	—
87	C₂×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
88	C₈	—	2	1	no	—
89	C₈	—	278	1	no	—
90	C₂²	—	2	2	no	—
91	C₂²	—	4	1	no	—
92	C₂²	—	4	1	no	—
93	C₂²	D₄×D₁₃₉	1	1	yes	characteristic
94	C₄	—	4	1	no	—
95	C₄	—	278	1	no	—
96	C₄	—	556	1	no	—
97	C₄	—	556	1	no	—
98	C₄	—	139	2	no	—
99	C₄	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
100	C₄	#14555784	1	1	yes	characteristic
101	C₂	—	4	2	no	—
102	C₂	#17984814	1	1	yes	characteristic
103	C₂	SD₁₆×D₁₃₉	1	1	yes	characteristic
104	C₂	#17984871	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14555773

C₂

yes

maximal, characteristic

#17984840

C₂

yes

maximal, characteristic, complemented

D₄×Dic₁₃₉

C₂

yes

maximal, characteristic

Dic₁₃₉⋊C₈

C₂

yes

maximal, characteristic, complemented

#17984767

C₂

yes

maximal, characteristic, complemented

#17984766

C₂

yes

maximal, characteristic, complemented

C₂×SD₁₆×C₁₃₉

C₂

yes

maximal, characteristic

C₂×Dic₂₇₈

C₂²

yes

characteristic

C₂²×Dic₁₃₉

—

#14555773

—

C₂×C₁₃₉⋊C₈

C₂²

yes

characteristic

C₄⋊Dic₁₃₉

C₂²

yes

characteristic

Dic₁₃₉⋊C₄

—

Dic₁₃₉⋊C₄

—

C₂²⋊Dic₁₃₉

—

C₄×Dic₁₃₉

C₂²

yes

characteristic

C₂×Q₈×C₁₃₉

C₂²

yes

characteristic

C₂×D₄×C₁₃₉

C₂²

yes

characteristic

SD₁₆×C₁₃₉

—

C₂×C₈×C₁₃₉

C₂²

yes

characteristic

Dic₂₇₈

—

Dic₂₇₈

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

D₄

yes

characteristic

C₂×Dic₁₃₉

D₄

yes

characteristic

C₁₃₉⋊C₈

—

C₂³×C₁₃₉

—

Q₈×C₁₃₉

—

Q₈×C₁₃₉

—

D₄×C₁₃₉

—

D₄×C₁₃₉

D₄

yes

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂³

yes

characteristic

C₈×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

C₂×D₄

yes

characteristic

C₄×C₁₃₉

—

C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₄×C₁₃₉

#41

yes

characteristic

Dic₁₃₉

—

Dic₁₃₉

—

Dic₁₃₉

—

Dic₁₃₉

SD₁₆

yes

complemented

C₂×C₁₃₉

—

C₂×C₁₃₉

#121

yes

characteristic

C₂×C₁₃₉

C₂×SD₁₆

yes

C₂×C₁₃₉

yes

C₁₃₉

yes

characteristic, complemented, Sylow

#461

—

139

maximal, Sylow

C₂×SD₁₆

—

139

—

C₂×SD₁₆

D₁₃₉

yes

characteristic

C₄⋊Q₈

—

139

—

C₄×D₄

—

139

—

C₄⋊C₈

—

139

—

#103

—

139

—

#103

—

139

—

C₂×Q₈

—

139

—

C₂×Q₈

C₂×D₁₃₉

yes

characteristic

C₂×D₄

C₂×D₁₃₉

yes

characteristic

C₂²×C₄

—

278

—

SD₁₆

—

SD₁₆

—

278

—

C₂×C₈

—

139

—

C₂×C₈

C₂×D₁₃₉

yes

characteristic

C₄⋊C₄

—

278

—

C₄⋊C₄

—

278

—

C₄⋊C₄

—

139

—

C₂²⋊C₄

—

278

—

C₄²

—

139

—

C₂³

—

Q₈

—

Q₈

—

Q₈

—

278

—

Q₈

—

278

—

D₄

—

D₄

C₂²⋊D₁₃₉

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

—

C₈

—

278

—

C₂²

—

C₂²

—

C₂²

—

C₂²

D₄×D₁₃₉

yes

characteristic

C₄

—

C₄

—

278

—

C₄

—

556

—

C₄

—

556

—

C₄

—

139

—

C₄

C₂×C₂²⋊D₁₃₉

yes

characteristic

100

C₄

#14555784

yes

characteristic

101

C₂

—

102

C₂

#17984814

yes

characteristic

103

C₂

SD₁₆×D₁₃₉

yes

characteristic

104

C₂

#17984871

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×D₁₃₉
commutator	C₂×C₄×C₁₃₉	C₂³
Frattini	C₂×C₄	C₂²×D₁₃₉
Fitting	C₂×SD₁₆×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂×D₄
hypercenter	C₂×SD₁₆	D₁₃₉
hypocenter	C₁₃₉	#461
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×SD₁₆

subgroup

G/H

center

C₂²

D₄×D₁₃₉

commutator

C₂×C₄×C₁₃₉

C₂³

Frattini

C₂×C₄

C₂²×D₁₃₉

Fitting

C₂×SD₁₆×C₁₃₉

C₂

socle

C₂²×C₁₃₉

C₂×D₄

hypercenter

C₂×SD₁₆

D₁₃₉

hypocenter

C₁₃₉

#461

LCS

G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×SD₁₆

N	H ≅ G/N	type
2. #17984840	C₂	semidirect
4. Dic₁₃₉⋊C₈	C₂	semidirect
5. #17984767	C₂	semidirect
6. #17984766	C₂	semidirect
47. Dic₁₃₉	SD₁₆	semidirect
52. C₁₃₉	#461	semidirect

H ≅ G/N

type

2. #17984840

C₂

semidirect

4. Dic₁₃₉⋊C₈

C₂

semidirect

5. #17984767

C₂

semidirect

6. #17984766

C₂

semidirect

47. Dic₁₃₉

SD₁₆

semidirect

52. C₁₃₉

#461

semidirect

Subgroups — #22581156

Conjugacy classes of subgroups

Complements