Group #26020333 — subgroups

Subgroups — #26020333

Order 11392.

272 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 85 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×#15667971	C₂	1	1	yes	maximal, characteristic
2	#19151562	C₂	1	1	yes	maximal, characteristic
3	#19151366	C₂	1	4	yes	maximal, complemented
4	#407×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#13083558	C₂²	1	2	yes	—
6	#15667971	—	2	2	no	—
7	#15667971	—	2	2	no	—
8	C₂×C₄⋊Dic₈₉	C₂²	1	2	yes	—
9	C₂×C₂²⋊Dic₈₉	C₄	1	2	yes	—
10	C₂×C₄×Dic₈₉	C₄	1	2	yes	—
11	C₂²⋊#10623494	—	2	2	no	—
12	C₂³×C₄×C₈₉	C₂²	1	1	yes	characteristic
13	C₂×M₁₆×C₈₉	C₂²	1	2	yes	—
14	C₂²⋊C₈×C₈₉	—	2	2	no	—
15	C₂²×Dic₈₉	C₂×C₄	1	4	yes	—
16	#13083558	—	2	4	no	—
17	C₂×#10623494	—	2	2	no	—
18	C₄⋊Dic₈₉	—	2	2	no	—
19	C₄⋊Dic₈₉	—	2	2	no	—
20	C₂²⋊Dic₈₉	—	2	4	no	—
21	C₄×Dic₈₉	—	2	4	no	—
22	C₂⁴×C₈₉	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₈₉	—	2	4	no	—
24	C₂²×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₈₉	D₄	1	2	yes	—
26	C₂²×C₄×C₈₉	D₄	1	2	yes	—
27	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
28	M₁₆×C₈₉	—	2	4	no	—
29	C₂×C₈×C₈₉	—	2	2	no	—
30	C₂³×C₈₉	—	2	2	no	—
31	C₂³×C₈₉	—	4	2	no	—
32	C₂³×C₈₉	C₂²⋊C₄	1	2	yes	—
33	C₂³×C₈₉	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₄×C₈₉	—	2	2	no	—
35	C₂×C₄×C₈₉	—	2	4	no	—
36	C₂×C₄×C₈₉	—	2	4	no	—
37	C₂×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
38	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₈₉	#41	1	4	yes	—
41	C₂×Dic₈₉	—	2	4	no	—
42	C₂×Dic₈₉	—	4	4	no	—
43	C₈×C₈₉	—	4	2	no	—
44	#10623494	—	4	2	no	—
45	C₂²×C₈₉	—	2	2	no	—
46	C₂²×C₈₉	—	2	2	no	—
47	C₂²×C₈₉	—	2	4	no	—
48	C₂²×C₈₉	—	4	4	no	—
49	C₂²×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₈₉	#117	1	2	yes	—
51	C₄×C₈₉	—	2	2	no	—
52	C₄×C₈₉	#123	1	2	yes	—
53	C₄×C₈₉	#123	1	2	yes	—
54	Dic₈₉	—	4	4	no	—
55	C₂×C₈₉	—	2	2	no	—
56	C₂×C₈₉	—	4	2	no	—
57	C₂×C₈₉	#381	1	1	yes	characteristic
58	C₂×C₈₉	#413	1	2	yes	—
59	#1822	—	89	1	no	maximal, Sylow
60	C₈₉	#1822	1	1	yes	characteristic, complemented, Sylow
61	C₂×#117	—	89	1	no	—
62	#407	—	89	1	no	—
63	#407	D₈₉	1	1	yes	characteristic
64	#343	—	89	4	no	—
65	C₂³×C₄	C₂×D₈₉	1	1	yes	characteristic
66	C₂×M₁₆	—	89	2	no	—
67	C₂×M₁₆	C₂×D₈₉	1	2	yes	—
68	#117	—	178	2	no	—
69	#117	—	178	2	no	—
70	C₂×C₄⋊C₄	—	89	2	no	—
71	C₂×C₂²⋊C₄	—	89	2	no	—
72	C₂×C₄²	—	89	2	no	—
73	C₂²⋊C₈	—	2	2	no	—
74	C₂²⋊C₈	—	178	2	no	—
75	C₂⁴	C₄×D₈₉	1	1	yes	characteristic
76	C₂²×C₄	—	2	4	no	—
77	C₂²×C₄	—	89	4	no	—
78	C₂²×C₄	C₄×D₈₉	1	1	yes	characteristic
79	C₂²×C₄	D₃₅₆	1	2	yes	—
80	C₂²×C₄	C₂²⋊D₈₉	1	2	yes	—
81	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
82	M₁₆	—	2	4	no	—
83	M₁₆	—	178	4	no	—
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	178	2	no	—
86	C₄⋊C₄	—	178	2	no	—
87	C₄⋊C₄	—	178	2	no	—
88	C₂²⋊C₄	—	178	4	no	—
89	C₄²	—	178	4	no	—
90	C₂³	—	2	2	no	—
91	C₂³	—	4	2	no	—
92	C₂³	#13083551	1	2	yes	—
93	C₂³	C₂×C₄×D₈₉	1	1	yes	characteristic
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	2	4	no	—
96	C₂×C₄	—	2	4	no	—
97	C₂×C₄	—	356	4	no	—
98	C₂×C₄	—	178	4	no	—
99	C₂×C₄	#13083551	1	2	yes	—
100	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
101	C₂×C₄	C₂×D₃₅₆	1	1	yes	characteristic
102	C₂×C₄	#13083573	1	4	yes	—
103	C₈	—	4	2	no	—
104	C₈	—	356	2	no	—
105	C₂²	—	2	2	no	—
106	C₂²	—	2	2	no	—
107	C₂²	—	2	4	no	—
108	C₂²	—	4	4	no	—
109	C₂²	C₂×#13083551	1	1	yes	characteristic
110	C₂²	#15667968	1	2	yes	—
111	C₄	—	2	2	no	—
112	C₄	—	356	4	no	—
113	C₄	#15667993	1	2	yes	—
114	C₄	#15667996	1	2	yes	—
115	C₂	—	2	2	no	—
116	C₂	—	4	2	no	—
117	C₂	#19151459	1	1	yes	characteristic
118	C₂	#19151580	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15667971

C₂

yes

maximal, characteristic

#19151562

C₂

yes

maximal, characteristic

#19151366

C₂

yes

maximal, complemented

#407×C₈₉

C₂

yes

maximal, characteristic

C₂×#13083558

C₂²

yes

—

#15667971

—

#15667971

—

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×C₂²⋊Dic₈₉

C₄

yes

—

C₂×C₄×Dic₈₉

C₄

yes

—

C₂²⋊#10623494

—

C₂³×C₄×C₈₉

C₂²

yes

characteristic

C₂×M₁₆×C₈₉

C₂²

yes

—

C₂²⋊C₈×C₈₉

—

C₂²×Dic₈₉

C₂×C₄

yes

—

#13083558

—

C₂×#10623494

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₂²⋊Dic₈₉

—

C₄×Dic₈₉

—

C₂⁴×C₈₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₉

D₄

yes

—

C₂²×C₄×C₈₉

D₄

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

M₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

C₂²⋊C₄

yes

—

C₂³×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂²⋊C₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

#41

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₈×C₈₉

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₈₉

#117

yes

—

C₄×C₈₉

—

C₄×C₈₉

#123

yes

—

C₄×C₈₉

#123

yes

—

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#381

yes

characteristic

C₂×C₈₉

#413

yes

—

#1822

—

maximal, Sylow

C₈₉

#1822

yes

characteristic, complemented, Sylow

C₂×#117

—

#407

—

#407

D₈₉

yes

characteristic

#343

—

C₂³×C₄

C₂×D₈₉

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

C₂×D₈₉

yes

—

#117

—

178

—

#117

—

178

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂²⋊C₈

—

C₂²⋊C₈

—

178

—

C₂⁴

C₄×D₈₉

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₈₉

yes

characteristic

C₂²×C₄

D₃₅₆

yes

—

C₂²×C₄

C₂²⋊D₈₉

yes

—

C₂²×C₄

C₂²×D₈₉

yes

characteristic

M₁₆

—

M₁₆

—

178

—

C₂×C₈

—

C₂×C₈

—

178

—

C₄⋊C₄

—

178

—

C₄⋊C₄

—

178

—

C₂²⋊C₄

—

178

—

C₄²

—

178

—

C₂³

—

C₂³

—

C₂³

#13083551

yes

—

C₂³

C₂×C₄×D₈₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

356

—

C₂×C₄

—

178

—

C₂×C₄

#13083551

yes

—

100

C₂×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

101

C₂×C₄

C₂×D₃₅₆

yes

characteristic

102

C₂×C₄

#13083573

yes

—

103

C₈

—

104

C₈

—

356

—

105

C₂²

—

106

C₂²

—

107

C₂²

—

108

C₂²

—

109

C₂²

C₂×#13083551

yes

characteristic

110

C₂²

#15667968

yes

—

111

C₄

—

112

C₄

—

356

—

113

C₄

#15667993

yes

—

114

C₄

#15667996

yes

—

115

C₂

—

116

C₂

—

117

C₂

#19151459

yes

characteristic

118

C₂

#19151580

yes

—

	subgroup	G/H
center	C₂²	C₂×#13083551
commutator	C₂³×C₈₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₈₉
Fitting	#407×C₈₉	C₂
socle	C₂²×C₈₉	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₈₉	C₂×C₄
cosocle	C₂²×C₄×C₈₉	C₂³
hypercenter	#407	D₈₉
hypocenter	C₈₉	#1822
LCS	G ▷ C₂³×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #407

subgroup

G/H

center

C₂²

C₂×#13083551

commutator

C₂³×C₈₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₈₉

Fitting

#407×C₈₉

C₂

socle

C₂²×C₈₉

C₂×C₂²⋊C₄

prime radical

C₂⁴×C₈₉

C₂×C₄

cosocle

C₂²×C₄×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂³×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #407

N	H ≅ G/N	type
3. #19151366	C₂	semidirect
60. C₈₉	#1822	semidirect

H ≅ G/N

type

3. #19151366

C₂

semidirect

60. C₈₉

#1822

semidirect

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

subgroup	index
C₂×#15667971	2
#19151562	2
#19151366	2
#407×C₈₉	2
#1822	89

subgroup

index

#407×C₈₉

Subgroups — #26020333

Conjugacy classes of subgroups

Complements

Maximal subgroups