Group #26020388 — subgroups

Subgroups — #26020388

Order 11392.

188 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 81 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	#19151621	C₂	1	2	yes	maximal
2	C₂×C₈×Dic₈₉	C₂	1	1	yes	maximal, characteristic
3	#19151363	C₂	1	1	yes	maximal, characteristic
4	#19151362	C₂	1	2	yes	maximal
5	#341×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#13083558	C₂²	1	2	yes	—
7	C₂²×#10623494	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₈₉	C₂²	1	1	yes	characteristic
9	#15667941	—	2	4	no	—
10	C₈×Dic₈₉	C₄	1	2	yes	—
11	C₈×Dic₈₉	C₄	1	2	yes	—
12	C₂×M₁₆×C₈₉	C₂²	1	2	yes	—
13	C₂²×C₈×C₈₉	C₂²	1	1	yes	characteristic
14	C₂²×Dic₈₉	Q₈	1	2	yes	—
15	#13083558	—	2	2	no	—
16	#13083558	—	2	2	no	—
17	C₂×#10623494	—	2	2	no	—
18	C₂×#10623494	—	2	2	no	—
19	C₂×#10623494	C₂×C₄	1	2	yes	—
20	C₄×Dic₈₉	C₂×C₄	1	1	yes	characteristic
21	C₄×Dic₈₉	C₂×C₄	1	1	yes	characteristic
22	C₄×Dic₈₉	D₄	1	2	yes	—
23	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
24	M₁₆×C₈₉	—	2	2	no	—
25	M₁₆×C₈₉	—	2	2	no	—
26	C₂×C₈×C₈₉	—	2	2	no	—
27	C₂×C₈×C₈₉	—	2	2	no	—
28	C₂×C₈×C₈₉	C₂×C₄	1	2	yes	—
29	C₂³×C₈₉	C₂×Q₈	1	1	yes	characteristic
30	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
31	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₈₉	#41	1	2	yes	—
33	C₂×C₄×C₈₉	#41	1	2	yes	—
34	C₂×Dic₈₉	—	2	4	no	—
35	C₂×Dic₈₉	C₄⋊C₄	1	2	yes	—
36	C₂×Dic₈₉	C₄⋊C₄	1	2	yes	—
37	C₈×C₈₉	—	2	2	no	—
38	C₈×C₈₉	—	4	2	no	—
39	#10623494	—	2	2	no	—
40	#10623494	—	4	2	no	—
41	C₂²×C₈₉	—	2	2	no	—
42	C₂²×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
43	C₂²×C₈₉	#125	1	2	yes	—
44	C₄×C₈₉	#117	1	1	yes	characteristic
45	C₄×C₈₉	#117	1	1	yes	characteristic
46	C₄×C₈₉	#124	1	2	yes	—
47	Dic₈₉	—	2	2	no	—
48	Dic₈₉	#108	1	4	yes	—
49	C₂×C₈₉	—	2	2	no	—
50	C₂×C₈₉	#382	1	1	yes	characteristic
51	C₂×C₈₉	C₂×#108	1	2	yes	—
52	#1837	—	89	1	no	maximal, Sylow
53	C₈₉	#1837	1	1	yes	characteristic, complemented, Sylow
54	#423	—	89	2	no	—
55	C₂×C₄×C₈	—	89	1	no	—
56	#341	—	89	2	no	—
57	#341	—	89	1	no	—
58	#341	D₈₉	1	1	yes	characteristic
59	C₂×M₁₆	—	89	2	no	—
60	C₂×M₁₆	C₂×D₈₉	1	2	yes	—
61	C₂²×C₈	—	89	1	no	—
62	C₂²×C₈	C₂×D₈₉	1	1	yes	characteristic
63	C₂×C₄²	—	89	1	no	—
64	C₄⋊C₈	—	178	4	no	—
65	C₄×C₈	—	89	2	no	—
66	C₄×C₈	—	89	2	no	—
67	C₂²×C₄	—	89	2	no	—
68	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
69	M₁₆	—	2	2	no	—
70	M₁₆	—	2	2	no	—
71	M₁₆	—	178	2	no	—
72	M₁₆	—	178	2	no	—
73	C₂×C₈	—	2	2	no	—
74	C₂×C₈	—	2	2	no	—
75	C₂×C₈	—	89	2	no	—
76	C₂×C₈	—	178	2	no	—
77	C₂×C₈	—	178	2	no	—
78	C₂×C₈	C₄×D₈₉	1	2	yes	—
79	C₄²	—	89	1	no	—
80	C₄²	—	89	1	no	—
81	C₄²	—	89	2	no	—
82	C₂³	Q₈×D₈₉	1	1	yes	characteristic
83	C₂×C₄	—	89	2	no	—
84	C₂×C₄	—	89	2	no	—
85	C₂×C₄	—	178	4	no	—
86	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
87	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
88	C₂×C₄	#13083573	1	2	yes	—
89	C₂×C₄	#13083574	1	2	yes	—
90	C₈	—	2	2	no	—
91	C₈	—	4	2	no	—
92	C₈	—	356	2	no	—
93	C₈	—	178	2	no	—
94	C₂²	—	2	2	no	—
95	C₂²	C₄⋊C₄×D₈₉	1	1	yes	characteristic
96	C₂²	#15668002	1	2	yes	—
97	C₄	—	89	4	no	—
98	C₄	—	178	2	no	—
99	C₄	#15667968	1	1	yes	characteristic
100	C₄	#15667969	1	1	yes	characteristic
101	C₄	#15667998	1	2	yes	—
102	C₂	—	2	2	no	—
103	C₂	#19151462	1	1	yes	characteristic
104	C₂	#108×D₈₉	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19151621

C₂

yes

maximal

C₂×C₈×Dic₈₉

C₂

yes

maximal, characteristic

#19151363

C₂

yes

maximal, characteristic

#19151362

C₂

yes

maximal

#341×C₈₉

C₂

yes

maximal, characteristic

C₂×#13083558

C₂²

yes

—

C₂²×#10623494

C₂²

yes

characteristic

C₂×C₄×Dic₈₉

C₂²

yes

characteristic

#15667941

—

C₈×Dic₈₉

C₄

yes

—

C₈×Dic₈₉

C₄

yes

—

C₂×M₁₆×C₈₉

C₂²

yes

—

C₂²×C₈×C₈₉

C₂²

yes

characteristic

C₂²×Dic₈₉

Q₈

yes

—

—

—

—

—

C₂×C₄

yes

—

C₄×Dic₈₉

C₂×C₄

yes

characteristic

C₄×Dic₈₉

C₂×C₄

yes

characteristic

C₄×Dic₈₉

D₄

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

M₁₆×C₈₉

—

M₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂×C₄

yes

—

C₂³×C₈₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

#41

yes

—

C₂×C₄×C₈₉

#41

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

C₄⋊C₄

yes

—

C₂×Dic₈₉

C₄⋊C₄

yes

—

C₈×C₈₉

—

C₈×C₈₉

—

#10623494

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₈₉

#125

yes

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

—

Dic₈₉

—

Dic₈₉

#108

yes

—

C₂×C₈₉

—

C₂×C₈₉

#382

yes

characteristic

C₂×C₈₉

C₂×#108

yes

—

#1837

—

maximal, Sylow

C₈₉

#1837

yes

characteristic, complemented, Sylow

#423

—

C₂×C₄×C₈

—

#341

—

#341

—

#341

D₈₉

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

C₂×D₈₉

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×D₈₉

yes

characteristic

C₂×C₄²

—

C₄⋊C₈

—

178

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₈₉

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

178

—

M₁₆

—

178

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

178

—

C₂×C₈

—

178

—

C₂×C₈

C₄×D₈₉

yes

—

C₄²

—

C₄²

—

C₄²

—

C₂³

Q₈×D₈₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

178

—

C₂×C₄

C₂×C₄×D₈₉

yes

characteristic

C₂×C₄

D₄×D₈₉

yes

characteristic

C₂×C₄

#13083573

yes

—

C₂×C₄

#13083574

yes

—

C₈

—

C₈

—

C₈

—

356

—

C₈

—

178

—

C₂²

—

C₂²

C₄⋊C₄×D₈₉

yes

characteristic

C₂²

#15668002

yes

—

C₄

—

C₄

—

178

—

C₄

#15667968

yes

characteristic

100

C₄

#15667969

yes

characteristic

101

C₄

#15667998

yes

—

102

C₂

—

103

C₂

#19151462

yes

characteristic

104

C₂

#108×D₈₉

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₈₉
commutator	C₂×C₄×C₈₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₈₉
Fitting	#341×C₈₉	C₂
socle	C₂²×C₈₉	C₂×C₄⋊C₄
prime radical	C₂³×C₈₉	C₂×Q₈
cosocle	C₂²×C₄×C₈₉	C₂³
hypercenter	#341	D₈₉
hypocenter	C₈₉	#1837
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂×C₄

D₄×D₈₉

commutator

C₂×C₄×C₈₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₈₉

Fitting

#341×C₈₉

C₂

socle

C₂²×C₈₉

C₂×C₄⋊C₄

prime radical

C₂³×C₈₉

C₂×Q₈

cosocle

C₂²×C₄×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
53. C₈₉	#1837	semidirect

H ≅ G/N

type

53. C₈₉

#1837

semidirect

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

subgroup	index
#19151621	2
C₂×C₈×Dic₈₉	2
#19151363	2
#19151362	2
#341×C₈₉	2
#1837	89

subgroup

index

C₂×C₈×Dic₈₉

#341×C₈₉

Subgroups — #26020388

Conjugacy classes of subgroups

Complements

Maximal subgroups