Group #26030756 — subgroups

Subgroups — C₂×#19152000 #26030756

Order 11392.

360 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 181 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×#15668004	C₂	1	2	yes	maximal
2	#19152000	C₂	1	8	yes	maximal, direct product factor
3	C₂×#15667946	C₂	1	2	yes	maximal
4	C₂×#15667944	C₂	1	2	yes	maximal
5	C₂×C₄×C₈×C₈₉	C₂	1	1	yes	maximal, characteristic
6	#15668004	—	2	4	no	—
7	#15668004	C₂²	1	8	yes	—
8	C₂×C₄⋊Dic₈₉	C₂²	1	4	yes	—
9	C₂×#13083553	—	2	4	no	—
10	#15667946	C₂²	1	8	yes	—
11	#15667944	C₂²	1	8	yes	—
12	C₂²×C₈×C₈₉	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₈₉	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₈₉	C₂²	1	4	yes	—
16	C₂²×Dic₈₉	—	2	4	no	—
17	C₄⋊Dic₈₉	—	2	4	no	—
18	C₄⋊Dic₈₉	C₂³	1	8	yes	—
19	#13083553	—	2	16	no	—
20	C₂²×C₄×C₈₉	D₄	1	2	yes	—
21	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
22	C₂×C₈×C₈₉	Q₈	1	4	yes	—
23	C₂×C₈×C₈₉	Q₈	1	4	yes	—
24	C₂×C₈×C₈₉	C₂³	1	2	yes	—
25	C₂×C₈×C₈₉	C₂³	1	2	yes	—
26	C₄²×C₈₉	D₄	1	2	yes	—
27	C₄²×C₈₉	C₂³	1	2	yes	—
28	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
29	C₂×C₄×C₈₉	—	2	4	no	—
30	C₂×C₄×C₈₉	C₂×D₄	1	4	yes	—
31	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₈₉	C₂×Q₈	1	2	yes	—
33	C₂×C₄×C₈₉	C₂×Q₈	1	2	yes	—
34	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
35	C₂×Dic₈₉	—	2	8	no	—
36	C₂×Dic₈₉	—	4	8	no	—
37	C₈×C₈₉	C₂×Q₈	1	4	yes	—
38	C₈×C₈₉	C₂×Q₈	1	4	yes	—
39	C₂²×C₈₉	C₄⋊Q₈	1	2	yes	—
40	C₂²×C₈₉	#135	1	4	yes	—
41	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
42	C₄×C₈₉	—	2	4	no	—
43	C₄×C₈₉	C₄⋊Q₈	1	2	yes	—
44	C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
45	C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
46	Dic₈₉	—	4	8	no	—
47	C₂×C₈₉	#499	1	4	yes	complemented
48	C₂×C₈₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
49	C₂×C₈₉	C₂×#135	1	2	yes	—
50	C₂×#499	—	89	1	no	maximal, Sylow
51	C₈₉	C₂×#499	1	1	yes	characteristic, complemented, Sylow
52	C₂×#125	—	89	2	no	—
53	#499	—	89	8	no	—
54	C₂×#107	—	89	2	no	—
55	C₂×#106	—	89	2	no	—
56	C₂×C₄×C₈	D₈₉	1	1	yes	characteristic
57	C₂²×C₈	C₂×D₈₉	1	1	yes	characteristic
58	C₂²×C₈	C₂×D₈₉	1	1	yes	characteristic
59	#125	—	89	8	no	—
60	#125	—	178	4	no	—
61	C₂×C₄⋊C₄	—	89	4	no	—
62	C₂×C₄⋊C₄	—	178	4	no	—
63	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
64	#107	—	89	8	no	—
65	#106	—	89	8	no	—
66	C₄×C₈	C₂×D₈₉	1	4	yes	—
67	C₂²×C₄	—	178	4	no	—
68	C₂²×C₄	D₃₅₆	1	2	yes	—
69	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
70	C₂×C₈	Dic₁₇₈	1	4	yes	—
71	C₂×C₈	Dic₁₇₈	1	4	yes	—
72	C₂×C₈	C₂²×D₈₉	1	2	yes	—
73	C₂×C₈	C₂²×D₈₉	1	2	yes	—
74	C₄⋊C₄	—	89	8	no	—
75	C₄⋊C₄	—	178	4	no	—
76	C₄⋊C₄	—	178	16	no	—
77	C₄²	D₃₅₆	1	2	yes	—
78	C₄²	C₂²×D₈₉	1	2	yes	—
79	C₂³	C₂×D₃₅₆	1	1	yes	characteristic
80	C₂×C₄	—	2	4	no	—
81	C₂×C₄	—	356	8	no	—
82	C₂×C₄	—	178	8	no	—
83	C₂×C₄	C₂×D₃₅₆	1	4	yes	—
84	C₂×C₄	C₂×D₃₅₆	1	1	yes	characteristic
85	C₂×C₄	C₂×Dic₁₇₈	1	2	yes	—
86	C₂×C₄	C₂×Dic₁₇₈	1	2	yes	—
87	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
88	C₈	C₂×Dic₁₇₈	1	4	yes	—
89	C₈	C₂×Dic₁₇₈	1	4	yes	—
90	C₂²	#15668012	1	2	yes	—
91	C₂²	#15668034	1	4	yes	—
92	C₂²	C₂²×D₃₅₆	1	1	yes	characteristic
93	C₄	—	2	4	no	—
94	C₄	—	356	8	no	—
95	C₄	#15668012	1	2	yes	—
96	C₄	C₂²×Dic₁₇₈	1	1	yes	characteristic
97	C₄	C₂²×Dic₁₇₈	1	1	yes	characteristic
98	C₂	#19152000	1	4	yes	direct product factor
99	C₂	C₂×#15668012	1	1	yes	characteristic
100	C₂	C₂×#15668034	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15668004

C₂

yes

maximal

#19152000

C₂

yes

maximal, direct product factor

C₂×#15667946

C₂

yes

maximal

C₂×#15667944

C₂

yes

maximal

C₂×C₄×C₈×C₈₉

C₂

yes

maximal, characteristic

#15668004

—

#15668004

C₂²

yes

—

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×#13083553

—

#15667946

C₂²

yes

—

#15667944

C₂²

yes

—

C₂²×C₈×C₈₉

C₂²

yes

characteristic

C₂²×C₈×C₈₉

C₂²

yes

characteristic

C₂×C₄²×C₈₉

C₂²

yes

characteristic

C₄×C₈×C₈₉

C₂²

yes

—

C₂²×Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

C₂³

yes

—

#13083553

—

C₂²×C₄×C₈₉

D₄

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₂×C₈×C₈₉

Q₈

yes

—

C₂×C₈×C₈₉

Q₈

yes

—

C₂×C₈×C₈₉

C₂³

yes

—

C₂×C₈×C₈₉

C₂³

yes

—

C₄²×C₈₉

D₄

yes

—

C₄²×C₈₉

C₂³

yes

—

C₂³×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×Q₈

yes

—

C₂×C₄×C₈₉

C₂×Q₈

yes

—

C₂×C₄×C₈₉

C₂⁴

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₈×C₈₉

C₂×Q₈

yes

—

C₈×C₈₉

C₂×Q₈

yes

—

C₂²×C₈₉

C₄⋊Q₈

yes

—

C₂²×C₈₉

#135

yes

—

C₂²×C₈₉

C₂²×D₄

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

C₄⋊Q₈

yes

—

C₄×C₈₉

C₂²×Q₈

yes

characteristic

C₄×C₈₉

C₂²×Q₈

yes

characteristic

Dic₈₉

—

C₂×C₈₉

#499

yes

complemented

C₂×C₈₉

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₈₉

C₂×#135

yes

—

C₂×#499

—

maximal, Sylow

C₈₉

C₂×#499

yes

characteristic, complemented, Sylow

—

—

—

—

C₂×C₄×C₈

D₈₉

yes

characteristic

C₂²×C₈

C₂×D₈₉

yes

characteristic

C₂²×C₈

C₂×D₈₉

yes

characteristic

#125

—

#125

—

178

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

178

—

C₂×C₄²

C₂×D₈₉

yes

characteristic

#107

—

#106

—

C₄×C₈

C₂×D₈₉

yes

—

C₂²×C₄

—

178

—

C₂²×C₄

D₃₅₆

yes

—

C₂²×C₄

C₂²×D₈₉

yes

characteristic

C₂×C₈

Dic₁₇₈

yes

—

C₂×C₈

Dic₁₇₈

yes

—

C₂×C₈

C₂²×D₈₉

yes

—

C₂×C₈

C₂²×D₈₉

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

178

—

C₄⋊C₄

—

178

—

C₄²

D₃₅₆

yes

—

C₄²

C₂²×D₈₉

yes

—

C₂³

C₂×D₃₅₆

yes

characteristic

C₂×C₄

—

C₂×C₄

—

356

—

C₂×C₄

—

178

—

C₂×C₄

C₂×D₃₅₆

yes

—

C₂×C₄

C₂×D₃₅₆

yes

characteristic

C₂×C₄

C₂×Dic₁₇₈

yes

—

C₂×C₄

C₂×Dic₁₇₈

yes

—

C₂×C₄

C₂³×D₈₉

yes

characteristic

C₈

C₂×Dic₁₇₈

yes

—

C₈

C₂×Dic₁₇₈

yes

—

C₂²

#15668012

yes

—

C₂²

#15668034

yes

—

C₂²

C₂²×D₃₅₆

yes

characteristic

C₄

—

C₄

—

356

—

C₄

#15668012

yes

—

C₄

C₂²×Dic₁₇₈

yes

characteristic

C₄

C₂²×Dic₁₇₈

yes

characteristic

C₂

#19152000

yes

direct product factor

C₂

C₂×#15668012

yes

characteristic

100

C₂

C₂×#15668034

yes

—

	subgroup	G/H
center	C₂³	C₂×D₃₅₆
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	C₂×C₄×C₈×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂×D₄
hypercenter	C₂×C₄×C₈	D₈₉
hypocenter	C₈₉	C₂×#499
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂³

C₂×D₃₅₆

commutator, cosocle

C₂×C₄×C₈₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₈₉

Fitting

C₂×C₄×C₈×C₈₉

C₂

socle, prime radical

C₂³×C₈₉

C₂×D₄

hypercenter

C₂×C₄×C₈

D₈₉

hypocenter

C₈₉

C₂×#499

LCS

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
2. #19152000	C₂	direct (N × H)
47. C₂×C₈₉	#499	semidirect
51. C₈₉	C₂×#499	semidirect
98. C₂	#19152000	direct (N × H)

H ≅ G/N

type

2. #19152000

C₂

direct (N × H)

47. C₂×C₈₉

#499

semidirect

51. C₈₉

C₂×#499

semidirect

98. C₂

#19152000

direct (N × H)

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#15668004	2
#19152000	2
C₂×#15667946	2
C₂×#15667944	2
C₂×C₄×C₈×C₈₉	2
C₂×#499	89

subgroup

index

C₂×C₄×C₈×C₈₉

Subgroups — C₂×#19152000 #26030756

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#19152000 #26030756

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#19152000 #26030756