Group #26022083 — subgroups

Subgroups — #26022083

Order 11392.

162 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 71 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×#15667951	C₂	1	2	yes	maximal, complemented
2	C₈×D₃₅₆	C₂	1	1	yes	maximal, characteristic
3	#19151377	C₂	1	2	yes	maximal, complemented
4	C₄×#13083549	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄⋊C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₈×D₈₉	C₂²	1	2	yes	—
7	C₄×D₃₅₆	C₄	1	1	yes	characteristic
8	#15667951	—	2	4	no	—
9	C₂×#13083549	C₂²	1	1	yes	characteristic
10	C₂×#13083549	C₂²	1	1	yes	characteristic
11	C₄⋊#10623494	C₄	1	1	yes	characteristic
12	#15667924	C₄	1	2	yes	—
13	C₂×C₁₆×C₈₉	C₂²	1	2	yes	—
14	C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
15	C₂×D₃₅₆	C₈	1	1	yes	characteristic
16	C₂×C₄×D₈₉	C₂×C₄	1	2	yes	—
17	C₂×#10623494	C₂×C₄	1	2	yes	—
18	C₈×D₈₉	—	2	2	no	—
19	C₈×D₈₉	—	2	2	no	—
20	C₄⋊Dic₈₉	C₈	1	1	yes	characteristic
21	#13083551	C₈	1	2	yes	—
22	#13083549	—	2	1	no	—
23	#13083549	D₄	1	2	yes	complemented
24	C₂×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
26	C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
27	C₁₆×C₈₉	—	2	2	no	—
28	C₂²×D₈₉	C₂×C₈	1	2	yes	—
29	C₂×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
30	C₂×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
31	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
32	C₂×Dic₈₉	C₂×C₈	1	2	yes	—
33	D₃₅₆	—	2	2	no	—
34	C₄×D₈₉	—	2	2	no	—
35	C₄×D₈₉	—	2	2	no	—
36	C₈×C₈₉	—	2	1	no	—
37	C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
38	C₈×C₈₉	#41	1	1	yes	characteristic
39	#10623494	—	2	2	no	—
40	C₂²×C₈₉	C₂²×C₈	1	1	yes	characteristic
41	C₂×D₈₉	—	2	2	no	—
42	C₂×D₈₉	—	2	2	no	—
43	C₂×D₈₉	—	2	2	no	—
44	C₄×C₈₉	—	2	1	no	—
45	C₄×C₈₉	M₃₂	1	2	yes	complemented
46	C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
47	C₄×C₈₉	#131	1	1	yes	characteristic
48	Dic₈₉	—	2	2	no	—
49	C₂×C₈₉	C₈×D₄	1	1	yes	characteristic
50	C₂×C₈₉	C₂×M₃₂	1	1	yes	characteristic
51	C₂×C₈₉	#504	1	1	yes	characteristic
52	D₈₉	—	4	2	no	—
53	#2169	—	89	1	no	maximal, Sylow
54	C₈₉	#2169	1	1	yes	characteristic, complemented, Sylow
55	C₂×M₃₂	—	89	2	no	—
56	C₈×D₄	—	89	1	no	—
57	C₄⋊C₁₆	D₈₉	1	1	yes	characteristic, complemented
58	C₂²⋊C₁₆	—	89	2	no	—
59	C₄×C₁₆	—	89	1	no	—
60	C₂²×C₈	—	89	2	no	—
61	C₄×D₄	—	89	1	no	—
62	M₃₂	—	178	4	no	—
63	C₂×C₁₆	—	89	1	no	—
64	C₂×C₁₆	—	89	1	no	—
65	C₂×C₁₆	C₂×D₈₉	1	2	yes	—
66	C₄⋊C₈	—	89	1	no	—
67	C₂²⋊C₈	—	89	2	no	—
68	C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
69	C₂×D₄	—	89	1	no	—
70	C₂²×C₄	—	89	2	no	—
71	C₂×C₈	—	89	2	no	—
72	C₂×C₈	—	178	2	no	—
73	C₂×C₈	—	178	2	no	—
74	C₂×C₈	C₄×D₈₉	1	1	yes	characteristic
75	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
76	C₄⋊C₄	—	89	1	no	—
77	C₂²⋊C₄	—	89	2	no	—
78	C₄²	C₄×D₈₉	1	1	yes	characteristic
79	C₁₆	—	2	2	no	—
80	C₁₆	—	89	2	no	—
81	C₁₆	—	178	1	no	—
82	C₂³	—	89	2	no	—
83	D₄	—	178	2	no	—
84	C₂×C₄	—	89	2	no	—
85	C₂×C₄	—	178	2	no	—
86	C₂×C₄	—	178	2	no	—
87	C₂×C₄	C₈×D₈₉	1	1	yes	characteristic
88	C₂×C₄	C₈×D₈₉	1	1	yes	characteristic
89	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
90	C₈	—	2	1	no	—
91	C₈	—	178	2	no	—
92	C₈	D₄×D₈₉	1	1	yes	characteristic
93	C₈	#13083575	1	1	yes	characteristic
94	C₂²	—	178	2	no	—
95	C₂²	—	178	2	no	—
96	C₂²	—	178	2	no	—
97	C₂²	C₂×C₈×D₈₉	1	1	yes	characteristic
98	C₄	—	2	1	no	—
99	C₄	—	178	2	no	—
100	C₄	#15667951	1	2	yes	complemented
101	C₄	#15667974	1	1	yes	characteristic
102	C₄	#15668020	1	1	yes	characteristic
103	C₂	—	356	2	no	—
104	C₂	#19151663	1	1	yes	characteristic
105	C₂	C₂×#15667951	1	1	yes	characteristic
106	C₂	#19152016	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15667951

C₂

yes

maximal, complemented

C₈×D₃₅₆

C₂

yes

maximal, characteristic

#19151377

C₂

yes

maximal, complemented

C₄×#13083549

C₂

yes

maximal, characteristic, complemented

C₄⋊C₁₆×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂×C₈×D₈₉

C₂²

yes

—

C₄×D₃₅₆

C₄

yes

characteristic

#15667951

—

C₂×#13083549

C₂²

yes

characteristic

C₂×#13083549

C₂²

yes

characteristic

C₄⋊#10623494

C₄

yes

characteristic

#15667924

C₄

yes

—

C₂×C₁₆×C₈₉

C₂²

yes

—

C₄×C₈×C₈₉

C₂²

yes

characteristic

C₂×D₃₅₆

C₈

yes

characteristic

C₂×C₄×D₈₉

C₂×C₄

yes

—

C₂×#10623494

C₂×C₄

yes

—

C₈×D₈₉

—

C₈×D₈₉

—

C₄⋊Dic₈₉

C₈

yes

characteristic

#13083551

C₈

yes

—

#13083549

—

#13083549

D₄

yes

complemented

C₂×C₈×C₈₉

C₂×C₄

yes

characteristic

C₂×C₈×C₈₉

C₂³

yes

characteristic

C₄²×C₈₉

C₂×C₄

yes

characteristic

C₁₆×C₈₉

—

C₂²×D₈₉

C₂×C₈

yes

—

C₂×C₄×C₈₉

C₂×C₈

yes

characteristic

C₂×C₄×C₈₉

C₂×C₈

yes

characteristic

C₂×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂×Dic₈₉

C₂×C₈

yes

—

D₃₅₆

—

C₄×D₈₉

—

C₄×D₈₉

—

C₈×C₈₉

—

C₈×C₈₉

C₂×D₄

yes

C₈×C₈₉

yes

—

C₂²×C₈₉

C₂²×C₈

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

M₃₂

yes

complemented

C₄×C₈₉

C₄×D₄

yes

characteristic

C₄×C₈₉

#131

yes

characteristic

Dic₈₉

—

C₂×C₈₉

C₈×D₄

yes

characteristic

C₂×C₈₉

C₂×M₃₂

yes

characteristic

C₂×C₈₉

#504

yes

characteristic

D₈₉

—

#2169

—

maximal, Sylow

C₈₉

#2169

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₈×D₄

—

C₄⋊C₁₆

D₈₉

yes

characteristic, complemented

C₂²⋊C₁₆

—

C₄×C₁₆

—

C₂²×C₈

—

C₄×D₄

—

M₃₂

—

178

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₈₉

yes

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

C₂×D₈₉

yes

characteristic

C₂×D₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

178

—

C₂×C₈

—

178

—

C₂×C₈

C₄×D₈₉

yes

characteristic

C₂×C₈

C₂²×D₈₉

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

C₄×D₈₉

yes

characteristic

C₁₆

—

C₁₆

—

C₁₆

—

178

—

C₂³

—

D₄

—

178

—

C₂×C₄

—

C₂×C₄

—

178

—

C₂×C₄

—

178

—

C₂×C₄

C₈×D₈₉

yes

characteristic

C₂×C₄

C₈×D₈₉

yes

characteristic

C₂×C₄

C₂×C₄×D₈₉

yes

characteristic

C₈

—

C₈

—

178

—

C₈

D₄×D₈₉

yes

characteristic

C₈

#13083575

yes

characteristic

C₂²

—

178

—

C₂²

—

178

—

C₂²

—

178

—

C₂²

C₂×C₈×D₈₉

yes

characteristic

C₄

—

C₄

—

178

—

100

C₄

#15667951

yes

complemented

101

C₄

#15667974

yes

characteristic

102

C₄

#15668020

yes

characteristic

103

C₂

—

356

—

104

C₂

#19151663

yes

characteristic

105

C₂

C₂×#15667951

yes

characteristic

106

C₂

#19152016

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₈₉
commutator, socle	C₂²×C₈₉	C₂²×C₈
Fitting	C₄⋊C₁₆×C₈₉	C₂
prime radical	C₂×D₃₅₆	C₈
cosocle	C₂×C₈×C₈₉	C₂³
hypercenter	C₄⋊C₁₆	D₈₉
hypocenter	C₈₉	#2169
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ C₄⋊C₁₆

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₈₉

commutator, socle

C₂²×C₈₉

C₂²×C₈

Fitting

C₄⋊C₁₆×C₈₉

C₂

prime radical

C₂×D₃₅₆

C₈

cosocle

C₂×C₈×C₈₉

C₂³

hypercenter

C₄⋊C₁₆

D₈₉

C₈₉

G ▷ C₂²×C₈₉ ▷ C₈₉

C₁ ◁ C₂×C₈ ◁ C₄⋊C₁₆

N	H ≅ G/N	type
1. C₂×#15667951	C₂	semidirect
3. #19151377	C₂	semidirect
4. C₄×#13083549	C₂	semidirect
5. C₄⋊C₁₆×C₈₉	C₂	semidirect
23. #13083549	D₄	semidirect
45. C₄×C₈₉	M₃₂	semidirect
54. C₈₉	#2169	semidirect
57. C₄⋊C₁₆	D₈₉	semidirect
100. C₄	#15667951	semidirect

H ≅ G/N

type

1. C₂×#15667951

C₂

semidirect

3. #19151377

C₂

semidirect

4. C₄×#13083549

C₂

semidirect

5. C₄⋊C₁₆×C₈₉

C₂

semidirect

23. #13083549

D₄

semidirect

45. C₄×C₈₉

M₃₂

semidirect

54. C₈₉

#2169

semidirect

57. C₄⋊C₁₆

D₈₉

semidirect

100. C₄

#15667951

semidirect

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

subgroup	index
C₂×#15667951	2
C₈×D₃₅₆	2
#19151377	2
C₄×#13083549	2
C₄⋊C₁₆×C₈₉	2
#2169	89

subgroup

index

C₈×D₃₅₆

C₄⋊C₁₆×C₈₉

Subgroups — #26022083

Conjugacy classes of subgroups

Complements

Maximal subgroups