Group #22580854 — subgroups

Subgroups — #22580854

Order 8896.

258 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 83 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	C₂²×D₅₅₆	C₂	1	1	yes	maximal, characteristic
2	C₂×(C₂×D₁₃₉)⋊C₄	C₂	1	4	yes	maximal, complemented
3	C₂×C₄×Dic₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄⋊C₄×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×D₁₃₉	C₂²	1	2	yes	—
6	C₂×D₅₅₆	—	2	2	no	—
7	C₂×D₅₅₆	—	2	2	no	—
8	C₂×D₅₅₆	C₄	1	4	yes	complemented
9	C₂²×Dic₁₃₉	C₂²	1	2	yes	—
10	(C₂×D₁₃₉)⋊C₄	—	2	8	no	—
11	C₄×Dic₁₃₉	—	2	2	no	—
12	C₂²×C₄×C₁₃₉	C₂²	1	2	yes	—
13	C₂²×C₄×C₁₃₉	C₂²	1	1	yes	characteristic
14	C₄⋊C₄×C₁₃₉	—	2	2	no	—
15	C₂²×D₁₃₉	—	2	4	no	—
16	C₂²×D₁₃₉	—	2	2	no	—
17	C₂²×D₁₃₉	—	2	2	no	—
18	C₂²×D₁₃₉	—	2	4	no	—
19	C₂²×D₁₃₉	C₂×C₄	1	4	yes	—
20	D₅₅₆	—	2	8	no	—
21	C₂×Dic₁₃₉	—	2	2	no	—
22	C₂×Dic₁₃₉	—	2	2	no	—
23	C₂×Dic₁₃₉	D₄	1	4	yes	complemented
24	C₂³×C₁₃₉	C₂³	1	1	yes	characteristic
25	C₂×C₄×C₁₃₉	—	2	2	no	—
26	C₂×C₄×C₁₃₉	—	2	4	no	—
27	C₂×C₄×C₁₃₉	C₂×C₄	1	2	yes	—
28	C₂×C₄×C₁₃₉	D₄	1	2	yes	—
29	C₂×C₄×C₁₃₉	D₄	1	2	yes	—
30	C₂²×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
31	C₂²×C₁₃₉	C₂×D₄	1	2	yes	—
32	C₂²×C₁₃₉	C₂×D₄	1	1	yes	characteristic
33	C₂²×C₁₃₉	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₁₃₉	#41	1	2	yes	—
35	C₂×D₁₃₉	—	2	4	no	—
36	C₂×D₁₃₉	—	2	8	no	—
37	C₂×D₁₃₉	—	4	4	no	—
38	C₂×D₁₃₉	—	4	2	no	—
39	C₂×D₁₃₉	—	4	2	no	—
40	C₄×C₁₃₉	—	4	2	no	—
41	C₄×C₁₃₉	C₂²⋊C₄	1	4	yes	complemented
42	Dic₁₃₉	—	2	4	no	—
43	C₂×C₁₃₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₁₃₉	C₄×D₄	1	2	yes	—
45	C₂×C₁₃₉	#121	1	2	yes	—
46	C₂×C₁₃₉	#124	1	1	yes	characteristic
47	C₂×C₁₃₉	#127	1	1	yes	characteristic
48	D₁₃₉	—	4	4	no	—
49	C₁₃₉	#390	1	1	yes	characteristic, complemented, Sylow
50	#390	—	139	1	no	maximal, Sylow
51	C₂²×D₄	—	139	1	no	—
52	C₂×C₄⋊C₄	D₁₃₉	1	1	yes	characteristic, complemented
53	C₂×C₂²⋊C₄	—	139	4	no	—
54	C₂×C₄²	—	139	1	no	—
55	C₂⁴	—	139	2	no	—
56	C₂×D₄	—	278	2	no	—
57	C₂×D₄	—	278	2	no	—
58	C₂×D₄	—	139	4	no	—
59	C₂²×C₄	—	139	2	no	—
60	C₂²×C₄	C₂×D₁₃₉	1	2	yes	—
61	C₂²×C₄	C₂×D₁₃₉	1	1	yes	characteristic
62	C₄⋊C₄	—	2	2	no	—
63	C₂²⋊C₄	—	278	8	no	—
64	C₄²	—	278	2	no	—
65	C₂³	—	278	4	no	—
66	C₂³	—	278	2	no	—
67	C₂³	—	278	2	no	—
68	C₂³	—	278	4	no	—
69	C₂³	—	139	4	no	—
70	C₂³	C₂²×D₁₃₉	1	1	yes	characteristic
71	D₄	—	278	8	no	—
72	C₂×C₄	—	2	4	no	—
73	C₂×C₄	—	2	2	no	—
74	C₂×C₄	—	278	2	no	—
75	C₂×C₄	—	278	2	no	—
76	C₂×C₄	—	139	4	no	—
77	C₂×C₄	C₄×D₁₃₉	1	2	yes	—
78	C₂×C₄	C₂²⋊D₁₃₉	1	2	yes	—
79	C₂×C₄	D₅₅₆	1	2	yes	—
80	C₂²	—	278	4	no	—
81	C₂²	—	278	8	no	—
82	C₂²	—	556	4	no	—
83	C₂²	—	556	2	no	—
84	C₂²	—	556	2	no	—
85	C₂²	C₂×C₄×D₁₃₉	1	1	yes	characteristic
86	C₂²	D₄×D₁₃₉	1	2	yes	—
87	C₂²	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
88	C₂²	C₂×D₅₅₆	1	1	yes	characteristic
89	C₂²	#14555786	1	2	yes	—
90	C₄	—	4	2	no	—
91	C₄	—	278	4	no	—
92	C₄	(C₂×D₁₃₉)⋊C₄	1	4	yes	complemented
93	C₂	—	556	4	no	—
94	C₂	C₂×(C₂×D₁₃₉)⋊C₄	1	1	yes	characteristic
95	C₂	#17984804	1	2	yes	—
96	C₂	#17984817	1	2	yes	—
97	C₂	#17984831	1	1	yes	characteristic
98	C₂	#17984838	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×D₅₅₆

C₂

yes

maximal, characteristic

C₂×(C₂×D₁₃₉)⋊C₄

C₂

yes

maximal, complemented

C₂×C₄×Dic₁₃₉

C₂

yes

maximal, characteristic, complemented

C₂×C₄⋊C₄×C₁₃₉

C₂

yes

maximal, characteristic, complemented

C₂³×D₁₃₉

C₂²

yes

—

C₂×D₅₅₆

—

C₂×D₅₅₆

—

C₂×D₅₅₆

C₄

yes

complemented

C₂²×Dic₁₃₉

C₂²

yes

—

(C₂×D₁₃₉)⋊C₄

—

C₄×Dic₁₃₉

—

C₂²×C₄×C₁₃₉

C₂²

yes

—

C₂²×C₄×C₁₃₉

C₂²

yes

characteristic

C₄⋊C₄×C₁₃₉

—

C₂²×D₁₃₉

—

C₂²×D₁₃₉

—

C₂²×D₁₃₉

—

C₂²×D₁₃₉

—

C₂²×D₁₃₉

C₂×C₄

yes

—

D₅₅₆

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

D₄

yes

complemented

C₂³×C₁₃₉

C₂³

yes

characteristic

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂×C₄

yes

—

C₂×C₄×C₁₃₉

D₄

yes

—

C₂×C₄×C₁₃₉

D₄

yes

—

C₂²×C₁₃₉

C₂²×C₄

yes

characteristic

C₂²×C₁₃₉

C₂×D₄

yes

—

C₂²×C₁₃₉

C₂×D₄

yes

characteristic

C₂²×C₁₃₉

C₂×D₄

yes

characteristic

C₂²×C₁₃₉

#41

yes

—

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

C₂²⋊C₄

yes

complemented

Dic₁₃₉

—

C₂×C₁₃₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₁₃₉

C₄×D₄

yes

—

C₂×C₁₃₉

#121

yes

—

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

D₁₃₉

—

C₁₃₉

#390

yes

characteristic, complemented, Sylow

#390

—

139

maximal, Sylow

C₂²×D₄

—

139

—

C₂×C₄⋊C₄

D₁₃₉

yes

characteristic, complemented

C₂×C₂²⋊C₄

—

139

—

C₂×C₄²

—

139

—

C₂⁴

—

139

—

C₂×D₄

—

278

—

C₂×D₄

—

278

—

C₂×D₄

—

139

—

C₂²×C₄

—

139

—

C₂²×C₄

C₂×D₁₃₉

yes

—

C₂²×C₄

C₂×D₁₃₉

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

278

—

C₄²

—

278

—

C₂³

—

278

—

C₂³

—

278

—

C₂³

—

278

—

C₂³

—

278

—

C₂³

—

139

—

C₂³

C₂²×D₁₃₉

yes

characteristic

D₄

—

278

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

278

—

C₂×C₄

—

278

—

C₂×C₄

—

139

—

C₂×C₄

C₄×D₁₃₉

yes

—

C₂×C₄

C₂²⋊D₁₃₉

yes

—

C₂×C₄

D₅₅₆

yes

—

C₂²

—

278

—

C₂²

—

278

—

C₂²

—

556

—

C₂²

—

556

—

C₂²

—

556

—

C₂²

C₂×C₄×D₁₃₉

yes

characteristic

C₂²

D₄×D₁₃₉

yes

—

C₂²

C₂×C₂²⋊D₁₃₉

yes

characteristic

C₂²

C₂×D₅₅₆

yes

characteristic

C₂²

#14555786

yes

—

C₄

—

C₄

—

278

—

C₄

(C₂×D₁₃₉)⋊C₄

yes

complemented

C₂

—

556

—

C₂

C₂×(C₂×D₁₃₉)⋊C₄

yes

characteristic

C₂

#17984804

yes

—

C₂

#17984817

yes

—

C₂

yes

C₂

yes

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₁₃₉
commutator	C₂²×C₁₃₉	C₂²×C₄
Fitting	C₂×C₄⋊C₄×C₁₃₉	C₂
socle	C₂³×C₁₃₉	C₂³
hypercenter	C₂×C₄⋊C₄	D₁₃₉
hypocenter	C₁₃₉	#390
LCS	G ▷ C₂²×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

subgroup

G/H

center, Frattini

C₂³

C₂²×D₁₃₉

commutator

C₂²×C₁₃₉

C₂²×C₄

Fitting

C₂×C₄⋊C₄×C₁₃₉

C₂

socle

C₂³×C₁₃₉

C₂³

hypercenter

C₂×C₄⋊C₄

D₁₃₉

C₁₃₉

G ▷ C₂²×C₁₃₉ ▷ C₁₃₉

C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

N	H ≅ G/N	type
2. C₂×(C₂×D₁₃₉)⋊C₄	C₂	semidirect
3. C₂×C₄×Dic₁₃₉	C₂	semidirect
4. C₂×C₄⋊C₄×C₁₃₉	C₂	semidirect
8. C₂×D₅₅₆	C₄	semidirect
23. C₂×Dic₁₃₉	D₄	semidirect
41. C₄×C₁₃₉	C₂²⋊C₄	semidirect
49. C₁₃₉	#390	semidirect
52. C₂×C₄⋊C₄	D₁₃₉	semidirect
92. C₄	(C₂×D₁₃₉)⋊C₄	semidirect

H ≅ G/N

type

2. C₂×(C₂×D₁₃₉)⋊C₄

C₂

semidirect

3. C₂×C₄×Dic₁₃₉

C₂

semidirect

4. C₂×C₄⋊C₄×C₁₃₉

C₂

semidirect

8. C₂×D₅₅₆

C₄

semidirect

23. C₂×Dic₁₃₉

D₄

semidirect

41. C₄×C₁₃₉

C₂²⋊C₄

semidirect

49. C₁₃₉

#390

semidirect

52. C₂×C₄⋊C₄

D₁₃₉

semidirect

92. C₄

(C₂×D₁₃₉)⋊C₄

semidirect

Subgroups — #22580854

Conjugacy classes of subgroups

Complements