Group #26019999 — subgroups

Subgroups — #26019999

Order 11392.

272 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 125 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×#15667922	C₂	1	2	yes	maximal
2	C₄²×Dic₈₉	C₂	1	1	yes	maximal, characteristic
3	#19151353	C₂	1	1	yes	maximal, characteristic
4	#19151352	C₂	1	2	yes	maximal
5	#336×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10623494	C₂²	1	2	yes	—
7	C₂×C₄×Dic₈₉	C₄	1	2	yes	—
8	C₂×C₄×Dic₈₉	C₄	1	2	yes	—
9	C₂×C₄×Dic₈₉	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₈₉	C₂²	1	1	yes	characteristic
11	#15667922	C₄	1	8	yes	complemented
12	C₂²×C₈×C₈₉	C₂²	1	2	yes	—
13	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
14	C₂²×Dic₈₉	C₂×C₄	1	1	yes	characteristic
15	C₂²×Dic₈₉	C₂×C₄	1	1	yes	characteristic
16	C₂²×Dic₈₉	C₂×C₄	1	2	yes	—
17	C₂×#10623494	—	2	2	no	—
18	C₂×#10623494	—	2	2	no	—
19	C₂×#10623494	C₂×C₄	1	4	yes	—
20	C₄×Dic₈₉	—	2	4	no	—
21	C₄×Dic₈₉	—	2	1	no	—
22	C₄×Dic₈₉	—	2	1	no	—
23	C₄×Dic₈₉	—	2	4	no	—
24	C₄×Dic₈₉	C₂×C₄	1	4	yes	—
25	C₂²×C₄×C₈₉	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
27	C₂×C₈×C₈₉	—	2	2	no	—
28	C₂×C₈×C₈₉	—	2	2	no	—
29	C₂×C₈×C₈₉	C₂×C₄	1	4	yes	—
30	C₄²×C₈₉	—	2	2	no	—
31	C₂³×C₈₉	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₄×C₈₉	—	2	2	no	—
33	C₂×C₄×C₈₉	—	2	2	no	—
34	C₂×C₄×C₈₉	M₁₆	1	4	yes	—
35	C₂×C₄×C₈₉	C₂²×C₄	1	2	yes	—
36	C₂×C₄×C₈₉	#41	1	2	yes	—
37	C₂×C₄×C₈₉	#41	1	1	yes	characteristic
38	C₂×C₄×C₈₉	#41	1	1	yes	characteristic
39	C₂×Dic₈₉	—	2	2	no	—
40	C₂×Dic₈₉	—	2	2	no	—
41	C₂×Dic₈₉	—	2	2	no	—
42	C₂×Dic₈₉	—	2	2	no	—
43	C₂×Dic₈₉	C₄²	1	2	yes	—
44	C₂×Dic₈₉	C₄²	1	2	yes	—
45	C₂×Dic₈₉	M₁₆	1	4	yes	—
46	C₈×C₈₉	—	2	4	no	—
47	#10623494	—	2	4	no	—
48	C₂²×C₈₉	C₂×C₄²	1	1	yes	characteristic
49	C₂²×C₈₉	#117	1	1	yes	characteristic
50	C₂²×C₈₉	#117	1	1	yes	characteristic
51	C₂²×C₈₉	C₂×M₁₆	1	2	yes	—
52	C₂²×C₈₉	C₂×M₁₆	1	2	yes	—
53	C₄×C₈₉	—	2	4	no	—
54	C₄×C₈₉	#117	1	2	yes	—
55	C₄×C₈₉	#117	1	2	yes	—
56	Dic₈₉	—	2	2	no	—
57	Dic₈₉	—	2	2	no	—
58	Dic₈₉	—	2	4	no	—
59	C₂×C₈₉	#376	1	1	yes	characteristic
60	C₂×C₈₉	C₄×M₁₆	1	1	yes	characteristic
61	C₂×C₈₉	C₄×M₁₆	1	1	yes	characteristic
62	C₂×C₈₉	#432	1	2	yes	—
63	C₂×C₈₉	#432	1	2	yes	—
64	#1745	—	89	1	no	maximal, Sylow
65	C₈₉	#1745	1	1	yes	characteristic, complemented, Sylow
66	C₂×#97	—	89	2	no	—
67	C₄³	—	89	1	no	—
68	#336	—	89	2	no	—
69	#336	—	89	1	no	—
70	#336	D₈₉	1	1	yes	characteristic
71	C₂²×C₈	—	89	2	no	—
72	C₂²×C₈	C₂×D₈₉	1	2	yes	—
73	C₂×C₄²	—	89	1	no	—
74	C₂×C₄²	—	89	2	no	—
75	C₂×C₄²	—	89	2	no	—
76	C₂×C₄²	—	89	1	no	—
77	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
78	#97	—	89	8	no	—
79	C₂²×C₄	—	89	2	no	—
80	C₂²×C₄	—	89	1	no	—
81	C₂²×C₄	—	89	1	no	—
82	C₂²×C₄	C₄×D₈₉	1	2	yes	—
83	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	—	89	4	no	—
87	C₂×C₈	—	178	2	no	—
88	C₂×C₈	—	178	2	no	—
89	C₂×C₈	C₄×D₈₉	1	4	yes	—
90	C₄²	—	2	2	no	—
91	C₄²	—	89	4	no	—
92	C₄²	—	178	4	no	—
93	C₄²	—	178	4	no	—
94	C₄²	—	178	1	no	—
95	C₄²	—	178	1	no	—
96	C₂³	C₂×C₄×D₈₉	1	1	yes	characteristic
97	C₂×C₄	—	2	2	no	—
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	—	89	4	no	—
100	C₂×C₄	—	89	2	no	—
101	C₂×C₄	—	89	2	no	—
102	C₂×C₄	—	178	2	no	—
103	C₂×C₄	—	178	2	no	—
104	C₂×C₄	—	178	2	no	—
105	C₂×C₄	—	178	2	no	—
106	C₂×C₄	#13083557	1	4	yes	—
107	C₂×C₄	C₂×C₄×D₈₉	1	2	yes	—
108	C₂×C₄	#13083573	1	1	yes	characteristic
109	C₂×C₄	#13083573	1	2	yes	—
110	C₂×C₄	#13083575	1	1	yes	characteristic
111	C₈	—	2	4	no	—
112	C₈	—	178	4	no	—
113	C₂²	C₄²×D₈₉	1	1	yes	characteristic
114	C₂²	#15667968	1	1	yes	characteristic
115	C₂²	#15667970	1	1	yes	characteristic
116	C₂²	M₁₆×D₈₉	1	2	yes	—
117	C₂²	C₂×#13083557	1	2	yes	—
118	C₄	—	2	4	no	—
119	C₄	—	178	4	no	—
120	C₄	—	178	2	no	—
121	C₄	—	178	2	no	—
122	C₄	#15667968	1	2	yes	—
123	C₄	#15667970	1	2	yes	—
124	C₂	#19151443	1	1	yes	characteristic
125	C₂	C₄×#13083557	1	1	yes	characteristic
126	C₂	#19151550	1	1	yes	characteristic
127	C₂	#19151652	1	2	yes	—
128	C₂	#19151654	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15667922

C₂

yes

maximal

C₄²×Dic₈₉

C₂

yes

maximal, characteristic

#19151353

C₂

yes

maximal, characteristic

#19151352

C₂

yes

maximal

#336×C₈₉

C₂

yes

maximal, characteristic

C₂²×#10623494

C₂²

yes

—

C₂×C₄×Dic₈₉

C₄

yes

—

C₂×C₄×Dic₈₉

C₄

yes

—

C₂×C₄×Dic₈₉

C₂²

yes

characteristic

C₂×C₄×Dic₈₉

C₂²

yes

characteristic

#15667922

C₄

yes

complemented

C₂²×C₈×C₈₉

C₂²

yes

—

C₂×C₄²×C₈₉

C₂²

yes

characteristic

C₂²×Dic₈₉

C₂×C₄

yes

characteristic

C₂²×Dic₈₉

C₂×C₄

yes

characteristic

C₂²×Dic₈₉

C₂×C₄

yes

—

C₂×#10623494

—

C₂×#10623494

—

C₂×#10623494

C₂×C₄

yes

—

C₄×Dic₈₉

—

C₄×Dic₈₉

—

C₄×Dic₈₉

—

C₄×Dic₈₉

—

C₄×Dic₈₉

C₂×C₄

yes

—

C₂²×C₄×C₈₉

C₂×C₄

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂×C₄

yes

—

C₄²×C₈₉

—

C₂³×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

M₁₆

yes

—

C₂×C₄×C₈₉

C₂²×C₄

yes

—

C₂×C₄×C₈₉

#41

yes

—

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

C₄²

yes

—

C₂×Dic₈₉

C₄²

yes

—

C₂×Dic₈₉

M₁₆

yes

—

C₈×C₈₉

—

#10623494

—

C₂²×C₈₉

C₂×C₄²

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

C₂×M₁₆

yes

—

C₂²×C₈₉

C₂×M₁₆

yes

—

C₄×C₈₉

—

C₄×C₈₉

#117

yes

—

C₄×C₈₉

#117

yes

—

Dic₈₉

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

#376

yes

characteristic

C₂×C₈₉

C₄×M₁₆

yes

characteristic

C₂×C₈₉

C₄×M₁₆

yes

characteristic

C₂×C₈₉

#432

yes

—

C₂×C₈₉

#432

yes

—

#1745

—

maximal, Sylow

C₈₉

#1745

yes

characteristic, complemented, Sylow

C₂×#97

—

C₄³

—

#336

—

#336

—

#336

D₈₉

yes

characteristic

C₂²×C₈

—

C₂²×C₈

C₂×D₈₉

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₈₉

yes

characteristic

#97

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₈₉

yes

—

C₂²×C₄

C₂²×D₈₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

178

—

C₂×C₈

—

178

—

C₂×C₈

C₄×D₈₉

yes

—

C₄²

—

C₄²

—

C₄²

—

178

—

C₄²

—

178

—

C₄²

—

178

—

C₄²

—

178

—

C₂³

C₂×C₄×D₈₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

178

—

103

C₂×C₄

—

178

—

104

C₂×C₄

—

178

—

105

C₂×C₄

—

178

—

106

C₂×C₄

#13083557

yes

—

107

C₂×C₄

C₂×C₄×D₈₉

yes

—

108

C₂×C₄

#13083573

yes

characteristic

109

C₂×C₄

#13083573

yes

—

110

C₂×C₄

#13083575

yes

characteristic

111

C₈

—

112

C₈

—

178

—

113

C₂²

C₄²×D₈₉

yes

characteristic

114

C₂²

#15667968

yes

characteristic

115

C₂²

#15667970

yes

characteristic

116

C₂²

M₁₆×D₈₉

yes

—

117

C₂²

C₂×#13083557

yes

—

118

C₄

—

119

C₄

—

178

—

120

C₄

—

178

—

121

C₄

—

178

—

122

C₄

#15667968

yes

—

123

C₄

#15667970

yes

—

124

C₂

#19151443

yes

characteristic

125

C₂

C₄×#13083557

yes

characteristic

126

C₂

#19151550

yes

characteristic

127

C₂

#19151652

yes

—

128

C₂

#19151654

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₈₉
commutator	C₂²×C₈₉	C₂×C₄²
Fitting	#336×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂²×C₄
cosocle	C₂²×C₄×C₈₉	C₂³
hypercenter	#336	D₈₉
hypocenter	C₈₉	#1745
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ #336

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₈₉

commutator

C₂²×C₈₉

C₂×C₄²

Fitting

#336×C₈₉

C₂

socle, prime radical

C₂³×C₈₉

C₂²×C₄

cosocle

C₂²×C₄×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂²×C₈₉ ▷ C₈₉

C₁ ◁ C₂²×C₄ ◁ #336

N	H ≅ G/N	type
11. #15667922	C₄	semidirect
65. C₈₉	#1745	semidirect

H ≅ G/N

type

11. #15667922

C₄

semidirect

65. C₈₉

#1745

semidirect

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

subgroup	index
C₂×#15667922	2
C₄²×Dic₈₉	2
#19151353	2
#19151352	2
#336×C₈₉	2
#1745	89

subgroup

index

C₄²×Dic₈₉

#336×C₈₉

Subgroups — #26019999

Conjugacy classes of subgroups

Complements

Maximal subgroups