Group #26019652 — subgroups

Subgroups — #26019652

Order 11392.

160 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 57 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	#19152003	C₂	1	1	yes	maximal, characteristic, complemented
2	#19151889	C₂	1	2	yes	maximal
3	C₈×D₃₅₆	C₂	1	1	yes	maximal, characteristic
4	#19151346	C₂	1	2	yes	maximal, complemented
5	#335×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₈×D₈₉	—	2	1	no	—
7	#15668012	C₂²	1	2	yes	—
8	C₄×D₃₅₆	C₂²	1	1	yes	characteristic
9	#15667946	—	2	1	no	—
10	#15667946	—	2	1	no	—
11	#15667944	—	2	2	no	—
12	C₄⋊#10623494	C₂²	1	1	yes	characteristic
13	#15667934	—	2	2	no	—
14	#15667924	—	2	1	no	—
15	C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
16	C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₇₈	—	2	2	no	—
18	C₂×D₃₅₆	D₄	1	1	yes	characteristic
19	C₂×C₄×D₈₉	—	2	1	no	—
20	C₂×#10623494	—	2	1	no	—
21	C₈×D₈₉	—	4	2	no	—
22	C₄⋊Dic₈₉	—	2	2	no	—
23	C₄⋊Dic₈₉	—	2	2	no	—
24	C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
25	#13083551	—	2	1	no	—
26	C₂×C₈×C₈₉	—	2	2	no	—
27	C₂×C₈×C₈₉	D₄	1	1	yes	characteristic
28	C₂×C₈×C₈₉	D₄	1	1	yes	characteristic
29	C₄²×C₈₉	C₂³	1	1	yes	characteristic
30	C₂²×D₈₉	—	2	1	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	#41	1	1	yes	characteristic
34	C₂×Dic₈₉	—	2	1	no	—
35	C₂×Dic₈₉	—	4	2	no	—
36	D₃₅₆	—	2	1	no	—
37	D₃₅₆	Q₁₆	1	2	yes	—
38	C₄×D₈₉	—	4	1	no	—
39	C₄×D₈₉	—	4	1	no	—
40	Dic₁₇₈	—	8	2	no	—
41	C₈×C₈₉	—	2	1	no	—
42	C₈×C₈₉	—	4	2	no	—
43	C₈×C₈₉	SD₁₆	1	2	yes	—
44	#10623494	—	4	1	no	—
45	C₂²×C₈₉	#121	1	1	yes	characteristic
46	C₂×D₈₉	—	2	2	no	—
47	C₂×D₈₉	—	4	1	no	—
48	C₂×D₈₉	—	4	1	no	—
49	C₄×C₈₉	—	2	1	no	—
50	C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
51	C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
52	C₄×C₈₉	#135	1	1	yes	characteristic
53	C₄×C₈₉	#137	1	1	yes	characteristic
54	Dic₈₉	—	4	1	no	—
55	Dic₈₉	—	8	2	no	—
56	C₂×C₈₉	#461	1	1	yes	characteristic
57	C₂×C₈₉	#467	1	1	yes	characteristic
58	C₂×C₈₉	#472	1	1	yes	characteristic
59	D₈₉	—	4	2	no	—
60	#1675	—	89	1	no	maximal, Sylow
61	C₈₉	#1675	1	1	yes	characteristic, complemented, Sylow
62	#500	—	89	1	no	—
63	#476	—	89	2	no	—
64	C₈×D₄	—	89	1	no	—
65	#335	D₈₉	1	1	yes	characteristic, complemented
66	#333	—	89	2	no	—
67	C₂²×C₈	—	178	1	no	—
68	C₄⋊Q₈	—	89	2	no	—
69	C₄×D₄	—	89	1	no	—
70	#107	—	178	1	no	—
71	#107	—	178	1	no	—
72	#106	—	178	2	no	—
73	C₄⋊C₈	—	89	1	no	—
74	C₄⋊C₈	C₂×D₈₉	1	2	yes	—
75	#102	—	178	2	no	—
76	C₂²⋊C₈	—	178	1	no	—
77	C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
78	C₂×Q₈	—	178	2	no	—
79	C₂×D₄	—	89	1	no	—
80	C₂²×C₄	—	178	1	no	—
81	C₂×C₈	—	2	2	no	—
82	C₂×C₈	—	356	2	no	—
83	C₂×C₈	—	178	1	no	—
84	C₂×C₈	D₃₅₆	1	1	yes	characteristic
85	C₂×C₈	D₃₅₆	1	1	yes	characteristic
86	C₄⋊C₄	—	89	1	no	—
87	C₄⋊C₄	—	178	2	no	—
88	C₄⋊C₄	—	178	2	no	—
89	C₂²⋊C₄	—	178	1	no	—
90	C₄²	C₂²×D₈₉	1	1	yes	characteristic
91	C₂³	—	178	1	no	—
92	Q₈	—	712	2	no	—
93	D₄	—	89	2	no	—
94	D₄	—	178	1	no	—
95	C₂×C₄	—	356	1	no	—
96	C₂×C₄	—	356	1	no	—
97	C₂×C₄	—	356	2	no	—
98	C₂×C₄	—	178	1	no	—
99	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
100	C₂×C₄	C₂×D₃₅₆	1	1	yes	characteristic
101	C₂×C₄	#13083575	1	1	yes	characteristic
102	C₈	—	2	1	no	—
103	C₈	—	4	2	no	—
104	C₈	—	356	1	no	—
105	C₈	#13083563	1	2	yes	—
106	C₂²	—	356	1	no	—
107	C₂²	—	356	1	no	—
108	C₂²	—	178	2	no	—
109	C₂²	#15667987	1	1	yes	characteristic
110	C₄	—	2	1	no	—
111	C₄	—	356	1	no	—
112	C₄	—	712	2	no	—
113	C₄	C₂×#13083563	1	1	yes	characteristic
114	C₄	Q₁₆×D₈₉	1	1	yes	characteristic
115	C₄	#15668031	1	1	yes	characteristic
116	C₄	#15668045	1	1	yes	characteristic
117	C₂	—	356	2	no	—
118	C₂	#19151804	1	1	yes	characteristic
119	C₂	#19151840	1	1	yes	characteristic
120	C₂	#19151869	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19152003

C₂

yes

maximal, characteristic, complemented

#19151889

C₂

yes

maximal

C₈×D₃₅₆

C₂

yes

maximal, characteristic

#19151346

C₂

yes

maximal, complemented

#335×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂×C₈×D₈₉

—

#15668012

C₂²

yes

—

C₄×D₃₅₆

C₂²

yes

—

—

—

C₂²

yes

characteristic

#15667934

—

#15667924

—

C₄⋊C₈×C₈₉

C₂²

yes

—

C₄×C₈×C₈₉

C₂²

yes

characteristic

C₂×Dic₁₇₈

—

C₂×D₃₅₆

D₄

yes

characteristic

C₂×C₄×D₈₉

—

C₂×#10623494

—

C₈×D₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

D₄

yes

characteristic

#13083551

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

D₄

yes

characteristic

C₂×C₈×C₈₉

D₄

yes

characteristic

C₄²×C₈₉

C₂³

yes

characteristic

C₂²×D₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

#41

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

D₃₅₆

—

D₃₅₆

Q₁₆

yes

—

C₄×D₈₉

—

C₄×D₈₉

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

—

C₈×C₈₉

SD₁₆

yes

—

#10623494

—

C₂²×C₈₉

#121

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₂×SD₁₆

yes

characteristic

C₄×C₈₉

C₂×Q₁₆

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

D₈₉

—

#1675

—

maximal, Sylow

C₈₉

#1675

yes

characteristic, complemented, Sylow

#500

—

#476

—

C₈×D₄

—

#335

D₈₉

yes

characteristic, complemented

#333

—

C₂²×C₈

—

178

—

C₄⋊Q₈

—

C₄×D₄

—

#107

—

178

—

#107

—

178

—

#106

—

178

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₈₉

yes

—

#102

—

178

—

C₂²⋊C₈

—

178

—

C₄×C₈

C₂×D₈₉

yes

characteristic

C₂×Q₈

—

178

—

C₂×D₄

—

C₂²×C₄

—

178

—

C₂×C₈

—

C₂×C₈

—

356

—

C₂×C₈

—

178

—

C₂×C₈

D₃₅₆

yes

characteristic

C₂×C₈

D₃₅₆

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

178

—

C₄⋊C₄

—

178

—

C₂²⋊C₄

—

178

—

C₄²

C₂²×D₈₉

yes

characteristic

C₂³

—

178

—

Q₈

—

712

—

D₄

—

D₄

—

178

—

C₂×C₄

—

356

—

C₂×C₄

—

356

—

C₂×C₄

—

356

—

C₂×C₄

—

178

—

C₂×C₄

D₄×D₈₉

yes

characteristic

100

C₂×C₄

C₂×D₃₅₆

yes

characteristic

101

C₂×C₄

#13083575

yes

characteristic

102

C₈

—

103

C₈

—

104

C₈

—

356

—

105

C₈

#13083563

yes

—

106

C₂²

—

356

—

107

C₂²

—

356

—

108

C₂²

—

178

—

109

C₂²

#15667987

yes

characteristic

110

C₄

—

111

C₄

—

356

—

112

C₄

—

712

—

113

C₄

C₂×#13083563

yes

characteristic

114

C₄

Q₁₆×D₈₉

yes

characteristic

115

C₄

#15668031

yes

characteristic

116

C₄

#15668045

yes

characteristic

117

C₂

—

356

—

118

C₂

#19151804

yes

characteristic

119

C₂

#19151840

yes

characteristic

120

C₂

#19151869

yes

characteristic

	subgroup	G/H
center	C₂²	#15667987
commutator, cosocle	C₄²×C₈₉	C₂³
Frattini	C₄²	C₂²×D₈₉
Fitting	#335×C₈₉	C₂
socle	C₂²×C₈₉	#121
prime radical	C₂×D₃₅₆	D₄
hypercenter	#335	D₈₉
hypocenter	C₈₉	#1675
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #335

subgroup

G/H

center

C₂²

#15667987

commutator, cosocle

C₄²×C₈₉

C₂³

Frattini

C₄²

C₂²×D₈₉

Fitting

#335×C₈₉

C₂

socle

C₂²×C₈₉

#121

prime radical

C₂×D₃₅₆

D₄

D₈₉

C₈₉

G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #335

N	H ≅ G/N	type
1. #19152003	C₂	semidirect
4. #19151346	C₂	semidirect
5. #335×C₈₉	C₂	semidirect
61. C₈₉	#1675	semidirect
65. #335	D₈₉	semidirect

H ≅ G/N

type

1. #19152003

C₂

semidirect

4. #19151346

C₂

semidirect

5. #335×C₈₉

C₂

semidirect

61. C₈₉

#1675

semidirect

65. #335

D₈₉

semidirect

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

subgroup	index
#19152003	2
#19151889	2
C₈×D₃₅₆	2
#19151346	2
#335×C₈₉	2
#1675	89

subgroup

index

C₈×D₃₅₆

#335×C₈₉

Subgroups — #26019652

Conjugacy classes of subgroups

Complements

Maximal subgroups