Group #26019573 — subgroups

Subgroups — #26019573

Order 11392.

196 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 45 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	#19152346	C₂	1	1	yes	maximal, characteristic
2	#19151960	C₂	1	2	yes	maximal, complemented
3	#19151340	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151337	C₂	1	2	yes	maximal, complemented
5	#491×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#15668000	C₂²	1	2	yes	—
7	#15667997	—	2	2	no	—
8	#15667990	—	2	2	no	—
9	#15667987	—	2	2	no	—
10	#15667935	—	2	2	no	—
11	#15667932	—	2	2	no	—
12	#15667923	C₂²	1	2	yes	—
13	C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
14	#125×C₈₉	C₂²	1	1	yes	characteristic
15	#103×C₈₉	—	2	2	no	—
16	#97×C₈₉	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₇₈	D₄	1	2	yes	—
18	C₂×D₃₅₆	—	2	2	no	—
19	C₂×D₃₅₆	D₄	1	2	yes	—
20	C₂×C₄×D₈₉	—	2	2	no	—
21	C₂×#10623494	—	2	2	no	—
22	#13083553	—	2	2	no	—
23	#13083551	—	2	2	no	—
24	#13083551	—	2	2	no	—
25	#13083551	—	2	2	no	—
26	C₂×Q₈×C₈₉	—	2	1	no	—
27	C₂×C₈×C₈₉	—	2	1	no	—
28	C₄⋊C₄×C₈₉	—	2	2	no	—
29	C₄⋊C₄×C₈₉	—	2	2	no	—
30	C₄⋊C₄×C₈₉	D₄	1	2	yes	—
31	C₄²×C₈₉	C₂³	1	1	yes	characteristic
32	C₂²×D₈₉	—	2	2	no	—
33	Q₈×C₈₉	—	4	2	no	—
34	C₂×C₄×C₈₉	—	2	2	no	—
35	C₂×C₄×C₈₉	—	4	1	no	—
36	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
37	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
38	C₂×Dic₈₉	—	2	2	no	—
39	D₃₅₆	—	4	2	no	—
40	D₃₅₆	—	8	2	no	—
41	C₄×D₈₉	—	8	2	no	—
42	Dic₁₇₈	—	4	2	no	—
43	C₈×C₈₉	—	4	1	no	—
44	#10623494	—	4	2	no	—
45	C₂²×C₈₉	#120	1	1	yes	characteristic
46	C₂×D₈₉	—	4	2	no	—
47	C₂×D₈₉	—	4	2	no	—
48	C₂×D₈₉	—	4	2	no	—
49	C₄×C₈₉	—	2	1	no	—
50	C₄×C₈₉	—	2	2	no	—
51	C₄×C₈₉	—	4	2	no	—
52	C₄×C₈₉	—	8	1	no	—
53	Dic₈₉	—	4	2	no	—
54	C₂×C₈₉	#454	1	1	yes	characteristic
55	C₂×C₈₉	#455	1	2	yes	—
56	D₈₉	—	8	2	no	—
57	#1662	—	89	1	no	maximal, Sylow
58	C₈₉	#1662	1	1	yes	characteristic, complemented, Sylow
59	#562	—	89	1	no	—
60	#491	D₈₉	1	1	yes	characteristic, complemented
61	#489	—	89	2	no	—
62	#330	—	89	1	no	—
63	#329	—	89	2	no	—
64	C₄⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
65	#125	C₂×D₈₉	1	1	yes	characteristic
66	#124	—	89	2	no	—
67	#123	—	178	2	no	—
68	C₂²⋊Q₈	—	178	2	no	—
69	#121	—	178	2	no	—
70	#103	—	2	2	no	—
71	#103	—	178	2	no	—
72	#102	—	178	2	no	—
73	#97	—	89	2	no	—
74	#97	C₂×D₈₉	1	1	yes	characteristic
75	C₂×Q₈	—	2	1	no	—
76	C₂×Q₈	—	89	2	no	—
77	C₂×D₄	—	89	2	no	—
78	C₂×D₄	—	178	2	no	—
79	C₂²×C₄	—	178	2	no	—
80	C₂×C₈	—	2	1	no	—
81	C₂×C₈	—	178	2	no	—
82	C₄⋊C₄	—	2	2	no	—
83	C₄⋊C₄	—	2	2	no	—
84	C₄⋊C₄	—	178	2	no	—
85	C₄⋊C₄	C₂²⋊D₈₉	1	2	yes	—
86	C₂²⋊C₄	—	178	2	no	—
87	C₂²⋊C₄	—	178	2	no	—
88	C₂²⋊C₄	—	178	2	no	—
89	C₄²	C₂²×D₈₉	1	1	yes	characteristic
90	C₂³	—	178	2	no	—
91	Q₈	—	4	2	no	—
92	Q₈	—	356	2	no	—
93	D₄	—	356	2	no	—
94	D₄	—	712	2	no	—
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	4	1	no	—
97	C₂×C₄	—	178	2	no	—
98	C₂×C₄	—	712	2	no	—
99	C₂×C₄	D₄×D₈₉	1	2	yes	—
100	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
101	C₈	—	4	1	no	—
102	C₈	—	356	2	no	—
103	C₂²	—	356	2	no	—
104	C₂²	—	356	2	no	—
105	C₂²	—	356	2	no	—
106	C₂²	#15667980	1	1	yes	characteristic
107	C₄	—	2	1	no	—
108	C₄	—	2	2	no	—
109	C₄	—	4	2	no	—
110	C₄	—	8	1	no	—
111	C₄	—	356	2	no	—
112	C₂	—	712	2	no	—
113	C₂	#19151770	1	1	yes	characteristic
114	C₂	#19151774	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19152346

C₂

yes

maximal, characteristic

#19151960

C₂

yes

maximal, complemented

#19151340

C₂

yes

maximal, characteristic, complemented

#19151337

C₂

yes

maximal, complemented

#491×C₈₉

C₂

yes

maximal, characteristic, complemented

#15668000

C₂²

yes

—

—

—

—

—

—

C₂²

yes

—

C₄⋊Q₈×C₈₉

C₂²

yes

characteristic

#125×C₈₉

C₂²

yes

characteristic

#103×C₈₉

—

#97×C₈₉

C₂²

yes

characteristic

C₂×Dic₁₇₈

D₄

yes

—

C₂×D₃₅₆

—

C₂×D₃₅₆

D₄

yes

—

C₂×C₄×D₈₉

—

—

—

—

—

—

C₂×Q₈×C₈₉

—

C₂×C₈×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

D₄

yes

—

C₄²×C₈₉

C₂³

yes

characteristic

C₂²×D₈₉

—

Q₈×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×Dic₈₉

—

D₃₅₆

—

D₃₅₆

—

C₄×D₈₉

—

Dic₁₇₈

—

C₈×C₈₉

—

#10623494

—

C₂²×C₈₉

#120

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

Dic₈₉

—

C₂×C₈₉

#454

yes

characteristic

C₂×C₈₉

#455

yes

—

D₈₉

—

#1662

—

maximal, Sylow

C₈₉

#1662

yes

characteristic, complemented, Sylow

#562

—

#491

D₈₉

yes

characteristic, complemented

#489

—

#330

—

#329

—

C₄⋊Q₈

C₂×D₈₉

yes

characteristic

#125

C₂×D₈₉

yes

characteristic

#124

—

#123

—

178

—

C₂²⋊Q₈

—

178

—

#121

—

178

—

#103

—

#103

—

178

—

#102

—

178

—

#97

—

#97

C₂×D₈₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

178

—

C₂²×C₄

—

178

—

C₂×C₈

—

C₂×C₈

—

178

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

178

—

C₄⋊C₄

C₂²⋊D₈₉

yes

—

C₂²⋊C₄

—

178

—

C₂²⋊C₄

—

178

—

C₂²⋊C₄

—

178

—

C₄²

C₂²×D₈₉

yes

characteristic

C₂³

—

178

—

Q₈

—

Q₈

—

356

—

D₄

—

356

—

D₄

—

712

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

178

—

C₂×C₄

—

712

—

C₂×C₄

D₄×D₈₉

yes

—

100

C₂×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

101

C₈

—

102

C₈

—

356

—

103

C₂²

—

356

—

104

C₂²

—

356

—

105

C₂²

—

356

—

106

C₂²

#15667980

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

—

356

—

112

C₂

—

712

—

113

C₂

#19151770

yes

characteristic

114

C₂

#19151774

yes

—

	subgroup	G/H
center	C₂²	#15667980
commutator, cosocle	C₄²×C₈₉	C₂³
Frattini	C₄²	C₂²×D₈₉
Fitting	#491×C₈₉	C₂
socle	C₂²×C₈₉	#120
prime radical	#19152346	C₂
hypercenter	#491	D₈₉
hypocenter	C₈₉	#1662
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #491

subgroup

G/H

center

C₂²

#15667980

commutator, cosocle

C₄²×C₈₉

C₂³

Frattini

C₄²

C₂²×D₈₉

Fitting

#491×C₈₉

C₂

C₂²×C₈₉

C₂

D₈₉

C₈₉

G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #491

N	H ≅ G/N	type
2. #19151960	C₂	semidirect
3. #19151340	C₂	semidirect
4. #19151337	C₂	semidirect
5. #491×C₈₉	C₂	semidirect
58. C₈₉	#1662	semidirect
60. #491	D₈₉	semidirect

H ≅ G/N

type

2. #19151960

C₂

semidirect

3. #19151340

C₂

semidirect

4. #19151337

C₂

semidirect

5. #491×C₈₉

C₂

semidirect

58. C₈₉

#1662

semidirect

60. #491

D₈₉

semidirect

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

subgroup	index
#19152346	2
#19151960	2
#19151340	2
#19151337	2
#491×C₈₉	2
#1662	89

subgroup

index

#491×C₈₉

Subgroups — #26019573

Conjugacy classes of subgroups

Complements

Maximal subgroups