Group #26021762 — subgroups

Subgroups — #26021762

Order 11392.

176 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 69 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	#19151641	C₂	1	2	yes	maximal
2	C₂×#15667944	C₂	1	1	yes	maximal, characteristic
3	#19151366	C₂	1	2	yes	maximal
4	#19151363	C₂	1	1	yes	maximal, characteristic
5	#424×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#13083558	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₈₉	C₂²	1	2	yes	—
8	#15667945	—	2	2	no	—
9	#15667944	—	2	2	no	—
10	#15667943	—	2	2	no	—
11	C₂×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
12	C₂²×C₈×C₈₉	C₂²	1	1	yes	characteristic
13	#117×C₈₉	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₈₉	—	2	2	no	—
15	C₂²×Dic₈₉	—	2	2	no	—
16	#13083558	—	2	2	no	—
17	#13083558	Q₈	1	4	yes	—
18	C₂×#10623494	—	2	2	no	—
19	C₄⋊Dic₈₉	—	2	2	no	—
20	C₄⋊Dic₈₉	—	2	2	no	—
21	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
22	M₁₆×C₈₉	—	2	2	no	—
23	C₂×C₈×C₈₉	—	2	2	no	—
24	C₂×C₈×C₈₉	—	2	1	no	—
25	C₂×C₈×C₈₉	D₄	1	2	yes	—
26	C₄⋊C₄×C₈₉	D₄	1	2	yes	—
27	C₂²⋊C₄×C₈₉	—	2	1	no	—
28	C₄²×C₈₉	—	2	1	no	—
29	C₂³×C₈₉	#41	1	1	yes	characteristic
30	C₂×C₄×C₈₉	—	2	2	no	—
31	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₈₉	C₂×Q₈	1	2	yes	—
34	C₂×C₄×C₈₉	#41	1	2	yes	—
35	C₂×Dic₈₉	—	4	2	no	—
36	C₂×Dic₈₉	—	8	2	no	—
37	C₈×C₈₉	—	2	2	no	—
38	C₈×C₈₉	—	4	1	no	—
39	#10623494	—	2	4	no	—
40	C₂²×C₈₉	—	2	2	no	—
41	C₂²×C₈₉	#121	1	1	yes	characteristic
42	C₂²×C₈₉	#125	1	2	yes	—
43	C₄×C₈₉	—	4	2	no	—
44	C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
45	C₄×C₈₉	#123	1	1	yes	characteristic
46	C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
47	Dic₈₉	—	8	2	no	—
48	C₂×C₈₉	—	2	2	no	—
49	C₂×C₈₉	#398	1	1	yes	characteristic
50	C₂×C₈₉	#471	1	2	yes	—
51	#2090	—	89	1	no	maximal, Sylow
52	C₈₉	#2090	1	1	yes	characteristic, complemented, Sylow
53	#428	—	89	2	no	—
54	C₂×#106	—	89	1	no	—
55	#424	D₈₉	1	1	yes	characteristic
56	#343	—	89	2	no	—
57	#341	—	89	1	no	—
58	C₂×M₁₆	—	89	2	no	—
59	C₂×M₁₆	C₂×D₈₉	1	1	yes	characteristic
60	C₂²×C₈	C₂×D₈₉	1	1	yes	characteristic
61	#117	C₂×D₈₉	1	1	yes	characteristic
62	C₂×C₄⋊C₄	—	89	2	no	—
63	#107	—	178	2	no	—
64	#106	—	178	2	no	—
65	#106	—	178	2	no	—
66	C₄⋊C₈	—	2	2	no	—
67	C₂²×C₄	—	178	2	no	—
68	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
69	M₁₆	—	2	2	no	—
70	M₁₆	—	89	4	no	—
71	M₁₆	—	178	2	no	—
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	—	2	1	no	—
74	C₂×C₈	—	178	2	no	—
75	C₂×C₈	C₂²⋊D₈₉	1	2	yes	—
76	C₄⋊C₄	—	178	2	no	—
77	C₄⋊C₄	—	178	2	no	—
78	C₄⋊C₄	D₃₅₆	1	2	yes	—
79	C₂²⋊C₄	—	2	1	no	—
80	C₄²	—	2	1	no	—
81	C₂³	#13083574	1	1	yes	characteristic
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	—	356	2	no	—
84	C₂×C₄	—	712	2	no	—
85	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
86	C₂×C₄	C₂×D₃₅₆	1	1	yes	characteristic
87	C₂×C₄	Q₈×D₈₉	1	2	yes	—
88	C₂×C₄	#13083573	1	2	yes	—
89	C₈	—	2	2	no	—
90	C₈	—	4	1	no	—
91	C₈	—	178	4	no	—
92	C₂²	—	2	2	no	—
93	C₂²	#15667986	1	1	yes	characteristic
94	C₂²	#15668002	1	2	yes	—
95	C₄	—	4	2	no	—
96	C₄	—	712	2	no	—
97	C₄	#15667992	1	2	yes	—
98	C₄	#15667993	1	1	yes	characteristic
99	C₄	#15668010	1	1	yes	characteristic
100	C₂	—	2	2	no	—
101	C₂	#19151528	1	1	yes	characteristic
102	C₂	#19151861	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19151641

C₂

yes

maximal

C₂×#15667944

C₂

yes

maximal, characteristic

#19151366

C₂

yes

maximal

#19151363

C₂

yes

maximal, characteristic

#424×C₈₉

C₂

yes

maximal, characteristic

C₂×#13083558

C₂²

yes

—

C₂×C₄⋊Dic₈₉

C₂²

yes

—

#15667945

—

#15667944

—

#15667943

—

C₂×M₁₆×C₈₉

C₂²

yes

characteristic

C₂²×C₈×C₈₉

C₂²

yes

characteristic

#117×C₈₉

C₂²

yes

characteristic

C₄⋊C₈×C₈₉

—

C₂²×Dic₈₉

—

#13083558

—

#13083558

Q₈

yes

—

C₂×#10623494

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

M₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

D₄

yes

—

C₄⋊C₄×C₈₉

D₄

yes

—

C₂²⋊C₄×C₈₉

—

C₄²×C₈₉

—

C₂³×C₈₉

#41

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×Q₈

yes

—

C₂×C₄×C₈₉

#41

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₈×C₈₉

—

C₈×C₈₉

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

#121

yes

characteristic

C₂²×C₈₉

#125

yes

—

C₄×C₈₉

—

C₄×C₈₉

C₂²⋊Q₈

yes

—

C₄×C₈₉

#123

yes

characteristic

C₄×C₈₉

C₄⋊Q₈

yes

characteristic

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#398

yes

characteristic

C₂×C₈₉

#471

yes

—

#2090

—

maximal, Sylow

C₈₉

#2090

yes

characteristic, complemented, Sylow

#428

—

C₂×#106

—

#424

D₈₉

yes

characteristic

#343

—

#341

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₈₉

yes

characteristic

C₂²×C₈

C₂×D₈₉

yes

characteristic

#117

C₂×D₈₉

yes

characteristic

C₂×C₄⋊C₄

—

#107

—

178

—

#106

—

178

—

#106

—

178

—

C₄⋊C₈

—

C₂²×C₄

—

178

—

C₂²×C₄

C₂²×D₈₉

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

178

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

178

—

C₂×C₈

C₂²⋊D₈₉

yes

—

C₄⋊C₄

—

178

—

C₄⋊C₄

—

178

—

C₄⋊C₄

D₃₅₆

yes

—

C₂²⋊C₄

—

C₄²

—

C₂³

#13083574

yes

characteristic

C₂×C₄

—

C₂×C₄

—

356

—

C₂×C₄

—

712

—

C₂×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

C₂×C₄

C₂×D₃₅₆

yes

characteristic

C₂×C₄

Q₈×D₈₉

yes

—

C₂×C₄

#13083573

yes

—

C₈

—

C₈

—

C₈

—

178

—

C₂²

—

C₂²

#15667986

yes

characteristic

C₂²

#15668002

yes

—

C₄

—

C₄

—

712

—

C₄

#15667992

yes

—

C₄

yes

C₄

yes

100

C₂

—

101

C₂

#19151528

yes

characteristic

102

C₂

#19151861

yes

—

	subgroup	G/H
center	C₂²	#15667986
commutator, cosocle	C₂²×C₄×C₈₉	C₂³
Frattini	C₂²×C₄	C₂²×D₈₉
Fitting	#424×C₈₉	C₂
socle	C₂²×C₈₉	#121
prime radical	C₂³×C₈₉	#41
hypercenter	#424	D₈₉
hypocenter	C₈₉	#2090
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #424

subgroup

G/H

center

C₂²

#15667986

commutator, cosocle

C₂²×C₄×C₈₉

C₂³

Frattini

C₂²×C₄

C₂²×D₈₉

Fitting

#424×C₈₉

C₂

C₂²×C₈₉

C₂³×C₈₉

D₈₉

C₈₉

G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #424

N	H ≅ G/N	type
52. C₈₉	#2090	semidirect

H ≅ G/N

type

52. C₈₉

#2090

semidirect

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

subgroup	index
#19151641	2
C₂×#15667944	2
#19151366	2
#19151363	2
#424×C₈₉	2
#2090	89

subgroup

index

#424×C₈₉

Subgroups — #26021762

Conjugacy classes of subgroups

Complements

Maximal subgroups