Group #22581504 — subgroups

Subgroups — #22581504

Order 8896.

310 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 111 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×#14555786	C₂	1	2	yes	maximal, complemented
2	#17984839	C₂	1	1	yes	maximal, characteristic, complemented
3	#17984831	C₂	1	2	yes	maximal, complemented
4	#17984817	C₂	1	4	yes	maximal, complemented
5	#17984804	C₂	1	4	yes	maximal, complemented
6	C₄²×D₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄⋊Q₈×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#14555786	—	2	8	no	—
9	C₂×D₅₅₆	C₂²	1	4	yes	—
10	C₂×D₅₅₆	C₂²	1	4	yes	—
11	C₂×D₅₅₆	C₂²	1	2	yes	—
12	C₂×C₄×D₁₃₉	C₂²	1	2	yes	—
13	C₂×C₄×D₁₃₉	C₂²	1	4	yes	—
14	C₂×C₄×D₁₃₉	C₂²	1	1	yes	characteristic
15	(C₂×D₁₃₉)⋊C₄	C₂²	1	8	yes	—
16	C₄×Dic₁₃₉	C₂²	1	2	yes	—
17	C₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
18	C₂×Q₈×C₁₃₉	C₂²	1	2	yes	—
19	C₄⋊C₄×C₁₃₉	C₂²	1	4	yes	—
20	C₄²×C₁₃₉	C₂²	1	1	yes	characteristic
21	C₂²×D₁₃₉	C₂³	1	4	yes	—
22	C₂²×D₁₃₉	C₂³	1	1	yes	characteristic
23	D₅₅₆	—	2	8	no	—
24	D₅₅₆	—	2	4	no	—
25	D₅₅₆	—	2	8	no	—
26	C₂×Dic₁₃₉	C₂³	1	2	yes	—
27	C₂×Dic₁₃₉	C₂³	1	1	yes	characteristic
28	C₄×D₁₃₉	—	2	4	no	—
29	C₄×D₁₃₉	—	2	8	no	—
30	C₄×D₁₃₉	D₄	1	4	yes	complemented
31	Q₈×C₁₃₉	—	2	4	no	—
32	C₂×C₄×C₁₃₉	C₂³	1	4	yes	—
33	C₂×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
34	C₂×C₄×C₁₃₉	C₂³	1	2	yes	—
35	C₂²×C₁₃₉	C₂⁴	1	1	yes	characteristic
36	C₂×D₁₃₉	—	2	2	no	—
37	C₂×D₁₃₉	—	2	4	no	—
38	C₂×D₁₃₉	—	2	8	no	—
39	C₂×D₁₃₉	C₂×D₄	1	2	yes	—
40	C₄×C₁₃₉	—	2	4	no	—
41	C₄×C₁₃₉	C₂×D₄	1	2	yes	—
42	C₄×C₁₃₉	#41	1	4	yes	complemented
43	Dic₁₃₉	—	2	2	no	—
44	Dic₁₃₉	C₂×D₄	1	2	yes	—
45	C₂×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
46	C₂×C₁₃₉	C₂×#41	1	2	yes	—
47	D₁₃₉	—	2	2	no	—
48	D₁₃₉	—	4	4	no	—
49	C₁₃₉	#532	1	1	yes	characteristic, complemented, Sylow
50	#532	—	139	1	no	maximal, Sylow
51	C₂×#41	—	139	2	no	—
52	C₄⋊Q₈	D₁₃₉	1	1	yes	characteristic, complemented
53	#127	—	139	1	no	—
54	#124	—	139	2	no	—
55	#121	—	139	4	no	—
56	C₄×D₄	—	139	4	no	—
57	C₂×C₄²	—	139	1	no	—
58	#41	—	278	8	no	—
59	C₂×Q₈	C₂×D₁₃₉	1	2	yes	—
60	C₂×D₄	—	139	4	no	—
61	C₂×D₄	—	139	4	no	—
62	C₂×D₄	—	139	2	no	—
63	C₂²×C₄	—	139	2	no	—
64	C₂²×C₄	—	139	4	no	—
65	C₂²×C₄	—	139	1	no	—
66	C₄⋊C₄	C₂×D₁₃₉	1	4	yes	—
67	C₂²⋊C₄	—	139	8	no	—
68	C₄²	—	139	2	no	—
69	C₄²	—	139	1	no	—
70	C₄²	C₂×D₁₃₉	1	1	yes	characteristic
71	C₂³	—	139	4	no	—
72	C₂³	—	139	1	no	—
73	Q₈	—	2	4	no	—
74	D₄	—	278	8	no	—
75	D₄	—	278	4	no	—
76	D₄	—	278	8	no	—
77	C₂×C₄	—	278	8	no	—
78	C₂×C₄	—	278	4	no	—
79	C₂×C₄	—	139	2	no	—
80	C₂×C₄	—	139	1	no	—
81	C₂×C₄	—	139	4	no	—
82	C₂×C₄	C₂²×D₁₃₉	1	4	yes	—
83	C₂×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
84	C₂×C₄	C₂²×D₁₃₉	1	2	yes	—
85	C₂²	—	278	2	no	—
86	C₂²	—	278	4	no	—
87	C₂²	—	278	8	no	—
88	C₂²	—	139	2	no	—
89	C₂²	C₂³×D₁₃₉	1	1	yes	characteristic
90	C₄	—	2	4	no	—
91	C₄	—	278	2	no	—
92	C₄	—	139	2	no	—
93	C₄	D₄×D₁₃₉	1	2	yes	—
94	C₄	#14555786	1	4	yes	complemented
95	C₂	—	278	2	no	—
96	C₂	—	556	4	no	—
97	C₂	C₂×D₄×D₁₃₉	1	1	yes	characteristic
98	C₂	C₂×#14555786	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14555786

C₂

yes

maximal, complemented

#17984839

C₂

yes

maximal, characteristic, complemented

#17984831

C₂

yes

maximal, complemented

#17984817

C₂

yes

maximal, complemented

#17984804

C₂

yes

maximal, complemented

C₄²×D₁₃₉

C₂

yes

maximal, characteristic, complemented

C₄⋊Q₈×C₁₃₉

C₂

yes

maximal, characteristic, complemented

#14555786

—

C₂×D₅₅₆

C₂²

yes

—

C₂×D₅₅₆

C₂²

yes

—

C₂×D₅₅₆

C₂²

yes

—

C₂×C₄×D₁₃₉

C₂²

yes

—

C₂×C₄×D₁₃₉

C₂²

yes

—

C₂×C₄×D₁₃₉

C₂²

yes

characteristic

(C₂×D₁₃₉)⋊C₄

C₂²

yes

—

C₄×Dic₁₃₉

C₂²

yes

—

C₄×Dic₁₃₉

C₂²

yes

characteristic

C₂×Q₈×C₁₃₉

C₂²

yes

—

C₄⋊C₄×C₁₃₉

C₂²

yes

—

C₄²×C₁₃₉

C₂²

yes

characteristic

C₂²×D₁₃₉

C₂³

yes

—

C₂²×D₁₃₉

C₂³

yes

characteristic

D₅₅₆

—

D₅₅₆

—

D₅₅₆

—

C₂×Dic₁₃₉

C₂³

yes

—

C₂×Dic₁₃₉

C₂³

yes

characteristic

C₄×D₁₃₉

—

C₄×D₁₃₉

—

C₄×D₁₃₉

D₄

yes

complemented

Q₈×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂³

yes

—

C₂×C₄×C₁₃₉

C₂³

yes

characteristic

C₂×C₄×C₁₃₉

C₂³

yes

—

C₂²×C₁₃₉

C₂⁴

yes

characteristic

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₂×D₁₃₉

C₂×D₄

yes

—

C₄×C₁₃₉

—

C₄×C₁₃₉

C₂×D₄

yes

—

C₄×C₁₃₉

#41

yes

complemented

Dic₁₃₉

—

Dic₁₃₉

C₂×D₄

yes

—

C₂×C₁₃₉

C₂²×D₄

yes

characteristic

C₂×C₁₃₉

C₂×#41

yes

—

D₁₃₉

—

D₁₃₉

—

C₁₃₉

#532

yes

characteristic, complemented, Sylow

#532

—

139

maximal, Sylow

C₂×#41

—

139

—

C₄⋊Q₈

D₁₃₉

yes

characteristic, complemented

#127

—

139

—

#124

—

139

—

#121

—

139

—

C₄×D₄

—

139

—

C₂×C₄²

—

139

—

#41

—

278

—

C₂×Q₈

C₂×D₁₃₉

yes

—

C₂×D₄

—

139

—

C₂×D₄

—

139

—

C₂×D₄

—

139

—

C₂²×C₄

—

139

—

C₂²×C₄

—

139

—

C₂²×C₄

—

139

—

C₄⋊C₄

C₂×D₁₃₉

yes

—

C₂²⋊C₄

—

139

—

C₄²

—

139

—

C₄²

—

139

—

C₄²

C₂×D₁₃₉

yes

characteristic

C₂³

—

139

—

C₂³

—

139

—

Q₈

—

D₄

—

278

—

D₄

—

278

—

D₄

—

278

—

C₂×C₄

—

278

—

C₂×C₄

—

278

—

C₂×C₄

—

139

—

C₂×C₄

—

139

—

C₂×C₄

—

139

—

C₂×C₄

C₂²×D₁₃₉

yes

—

C₂×C₄

C₂²×D₁₃₉

yes

characteristic

C₂×C₄

C₂²×D₁₃₉

yes

—

C₂²

—

278

—

C₂²

—

278

—

C₂²

—

278

—

C₂²

—

139

—

C₂²

C₂³×D₁₃₉

yes

characteristic

C₄

—

C₄

—

278

—

C₄

—

139

—

C₄

D₄×D₁₃₉

yes

—

C₄

#14555786

yes

complemented

C₂

—

278

—

C₂

—

556

—

C₂

C₂×D₄×D₁₃₉

yes

characteristic

C₂

C₂×#14555786

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₃₉
commutator, socle	C₂²×C₁₃₉	C₂⁴
Fitting	C₄⋊Q₈×C₁₃₉	C₂
hypercenter	C₄⋊Q₈	D₁₃₉
hypocenter	C₁₃₉	#532
LCS	G ▷ C₂²×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₃₉

commutator, socle

C₂²×C₁₃₉

C₂⁴

Fitting

C₄⋊Q₈×C₁₃₉

C₂

hypercenter

C₄⋊Q₈

D₁₃₉

C₁₃₉

G ▷ C₂²×C₁₃₉ ▷ C₁₃₉

C₁ ◁ C₂² ◁ C₄⋊Q₈

N	H ≅ G/N	type
1. C₂×#14555786	C₂	semidirect
2. #17984839	C₂	semidirect
3. #17984831	C₂	semidirect
4. #17984817	C₂	semidirect
5. #17984804	C₂	semidirect
6. C₄²×D₁₃₉	C₂	semidirect
7. C₄⋊Q₈×C₁₃₉	C₂	semidirect
30. C₄×D₁₃₉	D₄	semidirect
42. C₄×C₁₃₉	#41	semidirect
49. C₁₃₉	#532	semidirect
52. C₄⋊Q₈	D₁₃₉	semidirect
94. C₄	#14555786	semidirect

H ≅ G/N

type

1. C₂×#14555786

C₂

semidirect

2. #17984839

C₂

semidirect

3. #17984831

C₂

semidirect

4. #17984817

C₂

semidirect

5. #17984804

C₂

semidirect

6. C₄²×D₁₃₉

C₂

semidirect

7. C₄⋊Q₈×C₁₃₉

C₂

semidirect

30. C₄×D₁₃₉

D₄

semidirect

42. C₄×C₁₃₉

#41

semidirect

49. C₁₃₉

#532

semidirect

52. C₄⋊Q₈

D₁₃₉

semidirect

94. C₄

#14555786

semidirect

Subgroups — #22581504

Conjugacy classes of subgroups

Complements