Group #22581038 — subgroups

Subgroups — #22581038

Order 8896.

122 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 122 conjugacy classes of subgroups — too many to draw. Only its 51 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×#14555772	C₂	1	1	yes	maximal, characteristic
2	Q₈×Dic₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#17984804	C₂	1	1	yes	maximal, characteristic
4	#17984773	C₂	1	1	yes	maximal, characteristic, complemented
5	#17984764	C₂	1	1	yes	maximal, characteristic
6	C₈×Dic₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
7	#103×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₅₅₆	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₁₃₉	—	2	1	no	—
10	#14555772	C₄	1	4	yes	complemented
11	C₂×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₁₃₉	—	2	1	no	—
13	C₄⋊Dic₁₃₉	C₂²	1	1	yes	characteristic
14	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
15	C₄×Dic₁₃₉	—	2	1	no	—
16	C₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
18	C₂×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
19	C₄⋊C₄×C₁₃₉	C₂²	1	1	yes	characteristic
20	C₂²×D₁₃₉	—	2	1	no	—
21	D₅₅₆	—	2	1	no	—
22	D₅₅₆	C₂×C₄	1	2	yes	—
23	C₂×Dic₁₃₉	—	2	1	no	—
24	C₂×Dic₁₃₉	D₄	1	1	yes	characteristic
25	C₂×Dic₁₃₉	D₄	1	1	yes	characteristic
26	C₄×D₁₃₉	—	4	2	no	—
27	C₁₃₉⋊C₈	C₂×C₄	1	2	yes	complemented
28	Q₈×C₁₃₉	—	2	1	no	—
29	Q₈×C₁₃₉	C₂×C₄	1	2	yes	complemented
30	C₂×C₄×C₁₃₉	—	2	1	no	—
31	C₂×C₄×C₁₃₉	—	2	1	no	—
32	C₂×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
33	C₈×C₁₃₉	—	2	1	no	—
34	C₂²×C₁₃₉	C₂×D₄	1	1	yes	characteristic
35	C₂×D₁₃₉	—	2	2	no	—
36	C₂×D₁₃₉	—	4	1	no	—
37	C₂×D₁₃₉	—	4	1	no	—
38	C₄×C₁₃₉	—	2	2	no	—
39	C₄×C₁₃₉	—	4	1	no	—
40	C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
41	C₄×C₁₃₉	#41	1	1	yes	characteristic
42	Dic₁₃₉	—	2	1	no	—
43	Dic₁₃₉	—	4	1	no	—
44	Dic₁₃₉	SD₁₆	1	2	yes	complemented
45	C₂×C₁₃₉	C₄×D₄	1	1	yes	characteristic
46	C₂×C₁₃₉	C₂×SD₁₆	1	1	yes	characteristic
47	C₂×C₁₃₉	#135	1	1	yes	characteristic
48	D₁₃₉	—	4	2	no	—
49	C₁₃₉	C₄×SD₁₆	1	1	yes	characteristic, complemented, Sylow
50	C₄×SD₁₆	—	139	1	no	maximal, Sylow
51	C₂×SD₁₆	—	139	1	no	—
52	C₄×Q₈	—	139	1	no	—
53	C₄×D₄	—	139	1	no	—
54	#106	—	139	1	no	—
55	#103	D₁₃₉	1	1	yes	characteristic, complemented
56	#102	—	139	1	no	—
57	C₄×C₈	—	139	1	no	—
58	C₂×Q₈	C₂×D₁₃₉	1	1	yes	characteristic
59	C₂×D₄	—	139	1	no	—
60	C₂²×C₄	—	278	1	no	—
61	SD₁₆	—	139	4	no	—
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	C₂×D₁₃₉	1	1	yes	characteristic
67	C₂²⋊C₄	—	278	1	no	—
68	C₄²	—	278	1	no	—
69	C₄²	—	139	1	no	—
70	C₂³	—	278	1	no	—
71	Q₈	—	2	1	no	—
72	Q₈	C₄×D₁₃₉	1	2	yes	complemented
73	D₄	—	278	1	no	—
74	D₄	—	139	2	no	—
75	C₂×C₄	—	2	1	no	—
76	C₂×C₄	—	2	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	—	2	1	no	—
83	C₈	—	139	2	no	—
84	C₂²	—	278	2	no	—
85	C₂²	—	556	1	no	—
86	C₂²	—	556	1	no	—
87	C₂²	D₄×D₁₃₉	1	1	yes	characteristic
88	C₄	—	2	2	no	—
89	C₄	—	4	1	no	—
90	C₄	—	278	1	no	—
91	C₄	—	556	1	no	—
92	C₄	—	139	2	no	—
93	C₄	C₂×C₄×D₁₃₉	1	1	yes	characteristic
94	C₄	#14555784	1	1	yes	characteristic
95	C₂	—	556	2	no	—
96	C₂	(C₂²⋊D₁₃₉)⋊C₄	1	1	yes	characteristic
97	C₂	SD₁₆×D₁₃₉	1	1	yes	characteristic
98	C₂	#17984865	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14555772

C₂

yes

maximal, characteristic

Q₈×Dic₁₃₉

C₂

yes

maximal, characteristic, complemented

#17984804

C₂

yes

maximal, characteristic

#17984773

C₂

yes

maximal, characteristic, complemented

#17984764

C₂

yes

maximal, characteristic

C₈×Dic₁₃₉

C₂

yes

maximal, characteristic, complemented

#103×C₁₃₉

C₂

yes

maximal, characteristic, complemented

C₂×D₅₅₆

C₂²

yes

characteristic

C₂×C₄×D₁₃₉

—

#14555772

C₄

yes

complemented

C₂×C₁₃₉⋊C₈

C₂²

yes

characteristic

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

C₂²

yes

characteristic

(C₂×D₁₃₉)⋊C₄

—

C₄×Dic₁₃₉

—

C₄×Dic₁₃₉

C₂²

yes

characteristic

C₂×Q₈×C₁₃₉

C₂²

yes

characteristic

C₂×C₈×C₁₃₉

C₂²

yes

characteristic

C₄⋊C₄×C₁₃₉

C₂²

yes

characteristic

C₂²×D₁₃₉

—

D₅₅₆

—

D₅₅₆

C₂×C₄

yes

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

D₄

yes

characteristic

C₂×Dic₁₃₉

D₄

yes

characteristic

C₄×D₁₃₉

—

C₁₃₉⋊C₈

C₂×C₄

yes

complemented

Q₈×C₁₃₉

—

Q₈×C₁₃₉

C₂×C₄

yes

complemented

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂³

yes

characteristic

C₈×C₁₃₉

—

C₂²×C₁₃₉

C₂×D₄

yes

characteristic

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

C₂²×C₄

yes

characteristic

C₄×C₁₃₉

#41

yes

characteristic

Dic₁₃₉

—

Dic₁₃₉

—

Dic₁₃₉

SD₁₆

yes

complemented

C₂×C₁₃₉

C₄×D₄

yes

characteristic

C₂×C₁₃₉

C₂×SD₁₆

yes

characteristic

C₂×C₁₃₉

#135

yes

characteristic

D₁₃₉

—

C₁₃₉

C₄×SD₁₆

yes

characteristic, complemented, Sylow

C₄×SD₁₆

—

139

maximal, Sylow

C₂×SD₁₆

—

139

—

C₄×Q₈

—

139

—

C₄×D₄

—

139

—

#106

—

139

—

#103

D₁₃₉

yes

characteristic, complemented

#102

—

139

—

C₄×C₈

—

139

—

C₂×Q₈

C₂×D₁₃₉

yes

characteristic

C₂×D₄

—

139

—

C₂²×C₄

—

278

—

SD₁₆

—

139

—

C₂×C₈

—

139

—

C₂×C₈

C₂×D₁₃₉

yes

characteristic

C₄⋊C₄

—

278

—

C₄⋊C₄

—

139

—

C₄⋊C₄

C₂×D₁₃₉

yes

characteristic

C₂²⋊C₄

—

278

—

C₄²

—

278

—

C₄²

—

139

—

C₂³

—

278

—

Q₈

—

Q₈

C₄×D₁₃₉

yes

complemented

D₄

—

278

—

D₄

—

139

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

278

—

C₂×C₄

—

556

—

C₂×C₄

—

139

—

C₂×C₄

—

139

—

C₂×C₄

C₂²×D₁₃₉

yes

characteristic

C₈

—

C₈

—

139

—

C₂²

—

278

—

C₂²

—

556

—

C₂²

—

556

—

C₂²

D₄×D₁₃₉

yes

characteristic

C₄

—

C₄

—

C₄

—

278

—

C₄

—

556

—

C₄

—

139

—

C₄

C₂×C₄×D₁₃₉

yes

characteristic

C₄

#14555784

yes

characteristic

C₂

—

556

—

C₂

(C₂²⋊D₁₃₉)⋊C₄

yes

characteristic

C₂

SD₁₆×D₁₃₉

yes

characteristic

C₂

#17984865

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×D₁₃₉
commutator	C₄×C₁₃₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₁₃₉
Fitting	#103×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂×D₄
hypercenter	#103	D₁₃₉
hypocenter	C₁₃₉	C₄×SD₁₆
LCS	G ▷ C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #103

subgroup

G/H

center

C₂²

D₄×D₁₃₉

commutator

C₄×C₁₃₉

C₂²×C₄

Frattini

C₂×C₄

C₂²×D₁₃₉

Fitting

#103×C₁₃₉

C₂

socle

C₂²×C₁₃₉

C₂×D₄

hypercenter

#103

D₁₃₉

hypocenter

C₁₃₉

C₄×SD₁₆

LCS

G ▷ C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ #103

N	H ≅ G/N	type
2. Q₈×Dic₁₃₉	C₂	semidirect
4. #17984773	C₂	semidirect
6. C₈×Dic₁₃₉	C₂	semidirect
7. #103×C₁₃₉	C₂	semidirect
10. #14555772	C₄	semidirect
27. C₁₃₉⋊C₈	C₂×C₄	semidirect
29. Q₈×C₁₃₉	C₂×C₄	semidirect
44. Dic₁₃₉	SD₁₆	semidirect
49. C₁₃₉	C₄×SD₁₆	semidirect
55. #103	D₁₃₉	semidirect
72. Q₈	C₄×D₁₃₉	semidirect

H ≅ G/N

type

2. Q₈×Dic₁₃₉

C₂

semidirect

4. #17984773

C₂

semidirect

6. C₈×Dic₁₃₉

C₂

semidirect

7. #103×C₁₃₉

C₂

semidirect

10. #14555772

C₄

semidirect

27. C₁₃₉⋊C₈

C₂×C₄

semidirect

29. Q₈×C₁₃₉

C₂×C₄

semidirect

44. Dic₁₃₉

SD₁₆

semidirect

49. C₁₃₉

C₄×SD₁₆

semidirect

55. #103

D₁₃₉

semidirect

72. Q₈

C₄×D₁₃₉

semidirect

Subgroups — #22581038

Conjugacy classes of subgroups

Complements