Group #26020104 — subgroups

Subgroups — #26020104

Order 11392.

292 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 137 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×#15667941	C₂	1	2	yes	maximal
2	C₂×#15667922	C₂	1	2	yes	maximal
3	C₄²×Dic₈₉	C₂	1	1	yes	maximal, characteristic
4	#19151353	C₂	1	1	yes	maximal, characteristic
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	#15667941	C₄	1	8	yes	complemented
12	#15667922	—	2	4	no	—
13	C₂²×C₈×C₈₉	C₂²	1	2	yes	—
14	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
15	C₂²×Dic₈₉	C₂×C₄	1	1	yes	characteristic
16	C₂²×Dic₈₉	C₂×C₄	1	2	yes	—
17	C₂²×Dic₈₉	C₂×C₄	1	1	yes	characteristic
18	C₂×#10623494	—	2	2	no	—
19	C₂×#10623494	—	2	2	no	—
20	C₂×#10623494	C₂×C₄	1	4	yes	—
21	C₄×Dic₈₉	—	2	4	no	—
22	C₄×Dic₈₉	—	2	4	no	—
23	C₄×Dic₈₉	C₂×C₄	1	4	yes	—
24	C₄×Dic₈₉	D₄	1	2	yes	—
25	C₄×Dic₈₉	Q₈	1	2	yes	—
26	C₂²×C₄×C₈₉	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
28	C₂×C₈×C₈₉	—	2	2	no	—
29	C₂×C₈×C₈₉	—	2	2	no	—
30	C₂×C₈×C₈₉	C₂×C₄	1	4	yes	—
31	C₄²×C₈₉	—	2	2	no	—
32	C₂³×C₈₉	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₄×C₈₉	—	2	2	no	—
34	C₂×C₄×C₈₉	—	2	2	no	—
35	C₂×C₄×C₈₉	M₁₆	1	4	yes	—
36	C₂×C₄×C₈₉	C₂²×C₄	1	2	yes	—
37	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₄×C₈₉	#41	1	2	yes	—
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	C₄⋊C₄	1	4	yes	—
46	C₂×Dic₈₉	M₁₆	1	4	yes	—
47	C₈×C₈₉	—	2	4	no	—
48	#10623494	—	2	4	no	—
49	C₂²×C₈₉	C₂×C₄²	1	1	yes	characteristic
50	C₂²×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
51	C₂²×C₈₉	#117	1	1	yes	characteristic
52	C₂²×C₈₉	C₂×M₁₆	1	2	yes	—
53	C₂²×C₈₉	C₂×M₁₆	1	2	yes	—
54	C₄×C₈₉	—	2	4	no	—
55	C₄×C₈₉	C₄×D₄	1	2	yes	—
56	C₄×C₈₉	C₄×Q₈	1	2	yes	—
57	Dic₈₉	—	2	2	no	—
58	Dic₈₉	—	2	4	no	—
59	Dic₈₉	#97	1	4	yes	complemented
60	C₂×C₈₉	C₄×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₈₉	C₂×#97	1	1	yes	characteristic
62	C₂×C₈₉	C₄×M₁₆	1	1	yes	characteristic
63	C₂×C₈₉	#423	1	2	yes	—
64	C₂×C₈₉	#432	1	2	yes	—
65	#1767	—	89	1	no	maximal, Sylow
66	C₈₉	#1767	1	1	yes	characteristic, complemented, Sylow
67	C₂×C₄⋊C₈	—	89	2	no	—
68	C₂×#97	—	89	2	no	—
69	C₄³	—	89	1	no	—
70	#336	—	89	1	no	—
71	#336	D₈₉	1	1	yes	characteristic
72	C₂²×C₈	—	89	2	no	—
73	C₂²×C₈	C₂×D₈₉	1	2	yes	—
74	C₂×C₄²	—	89	1	no	—
75	C₂×C₄²	—	89	1	no	—
76	C₂×C₄²	—	89	2	no	—
77	C₂×C₄²	—	89	2	no	—
78	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
79	C₄⋊C₈	—	89	8	no	—
80	#97	—	178	4	no	—
81	C₂²×C₄	—	89	1	no	—
82	C₂²×C₄	—	89	1	no	—
83	C₂²×C₄	—	89	2	no	—
84	C₂²×C₄	C₄×D₈₉	1	2	yes	—
85	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	—	89	4	no	—
89	C₂×C₈	—	178	2	no	—
90	C₂×C₈	—	178	2	no	—
91	C₂×C₈	C₄×D₈₉	1	4	yes	—
92	C₄²	—	2	2	no	—
93	C₄²	—	89	4	no	—
94	C₄²	—	89	2	no	—
95	C₄²	—	89	2	no	—
96	C₄²	—	178	4	no	—
97	C₄²	—	178	4	no	—
98	C₂³	C₂×C₄×D₈₉	1	1	yes	characteristic
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	89	4	no	—
102	C₂×C₄	—	89	2	no	—
103	C₂×C₄	—	89	2	no	—
104	C₂×C₄	—	89	4	no	—
105	C₂×C₄	—	178	2	no	—
106	C₂×C₄	—	178	2	no	—
107	C₂×C₄	—	178	2	no	—
108	C₂×C₄	#13083557	1	4	yes	—
109	C₂×C₄	C₂×C₄×D₈₉	1	2	yes	—
110	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
111	C₂×C₄	Q₈×D₈₉	1	1	yes	characteristic
112	C₂×C₄	#13083573	1	2	yes	—
113	C₈	—	2	4	no	—
114	C₈	—	178	4	no	—
115	C₂²	C₄²×D₈₉	1	1	yes	characteristic
116	C₂²	C₄⋊C₄×D₈₉	1	1	yes	characteristic
117	C₂²	#15667968	1	1	yes	characteristic
118	C₂²	M₁₆×D₈₉	1	2	yes	—
119	C₂²	C₂×#13083557	1	2	yes	—
120	C₄	—	2	4	no	—
121	C₄	—	89	4	no	—
122	C₄	—	178	4	no	—
123	C₄	—	178	2	no	—
124	C₄	#15667972	1	2	yes	—
125	C₄	#15667977	1	2	yes	—
126	C₂	#19151449	1	1	yes	characteristic
127	C₂	#97×D₈₉	1	1	yes	characteristic
128	C₂	C₄×#13083557	1	1	yes	characteristic
129	C₂	#19151620	1	2	yes	—
130	C₂	#19151652	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15667941

C₂

yes

maximal

C₂×#15667922

C₂

yes

maximal

C₄²×Dic₈₉

C₂

yes

maximal, characteristic

#19151353

C₂

yes

maximal, characteristic

#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

#15667941

C₄

yes

complemented

#15667922

—

C₂²×C₈×C₈₉

C₂²

yes

—

C₂×C₄²×C₈₉

C₂²

yes

characteristic

C₂²×Dic₈₉

C₂×C₄

yes

characteristic

C₂²×Dic₈₉

C₂×C₄

yes

—

C₂²×Dic₈₉

C₂×C₄

yes

—

—

C₂×C₄

yes

—

C₄×Dic₈₉

—

C₄×Dic₈₉

—

C₄×Dic₈₉

C₂×C₄

yes

—

C₄×Dic₈₉

D₄

yes

—

C₄×Dic₈₉

Q₈

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

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₈₉

#41

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

C₄²

yes

—

C₂×Dic₈₉

C₄²

yes

—

C₂×Dic₈₉

C₄⋊C₄

yes

—

C₂×Dic₈₉

M₁₆

yes

—

C₈×C₈₉

—

#10623494

—

C₂²×C₈₉

C₂×C₄²

yes

characteristic

C₂²×C₈₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₈₉

#117

yes

characteristic

C₂²×C₈₉

C₂×M₁₆

yes

—

C₂²×C₈₉

C₂×M₁₆

yes

—

C₄×C₈₉

—

C₄×C₈₉

C₄×D₄

yes

—

C₄×C₈₉

C₄×Q₈

yes

—

Dic₈₉

—

Dic₈₉

—

Dic₈₉

#97

yes

complemented

C₂×C₈₉

C₄×C₄⋊C₄

yes

characteristic

C₂×C₈₉

C₂×#97

yes

characteristic

C₂×C₈₉

C₄×M₁₆

yes

characteristic

C₂×C₈₉

#423

yes

—

C₂×C₈₉

#432

yes

—

#1767

—

maximal, Sylow

C₈₉

#1767

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

C₂×#97

—

C₄³

—

#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

C₄⋊C₈

—

#97

—

178

—

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

—

C₄²

—

C₄²

—

178

—

C₄²

—

178

—

C₂³

C₂×C₄×D₈₉

yes

characteristic

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

178

—

106

C₂×C₄

—

178

—

107

C₂×C₄

—

178

—

108

C₂×C₄

#13083557

yes

—

109

C₂×C₄

C₂×C₄×D₈₉

yes

—

110

C₂×C₄

D₄×D₈₉

yes

characteristic

111

C₂×C₄

Q₈×D₈₉

yes

characteristic

112

C₂×C₄

#13083573

yes

—

113

C₈

—

114

C₈

—

178

—

115

C₂²

C₄²×D₈₉

yes

characteristic

116

C₂²

C₄⋊C₄×D₈₉

yes

characteristic

117

C₂²

#15667968

yes

characteristic

118

C₂²

M₁₆×D₈₉

yes

—

119

C₂²

C₂×#13083557

yes

—

120

C₄

—

121

C₄

—

122

C₄

—

178

—

123

C₄

—

178

—

124

C₄

#15667972

yes

—

125

C₄

#15667977

yes

—

126

C₂

#19151449

yes

characteristic

127

C₂

#97×D₈₉

yes

characteristic

128

C₂

C₄×#13083557

yes

characteristic

129

C₂

#19151620

yes

—

130

C₂

#19151652

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	#1767
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. #15667941	C₄	semidirect
59. Dic₈₉	#97	semidirect
66. C₈₉	#1767	semidirect

H ≅ G/N

type

11. #15667941

C₄

semidirect

59. Dic₈₉

#97

semidirect

66. C₈₉

#1767

semidirect

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

subgroup	index
C₂×#15667941	2
C₂×#15667922	2
C₄²×Dic₈₉	2
#19151353	2
#336×C₈₉	2
#1767	89

subgroup

index

C₄²×Dic₈₉

#336×C₈₉

Subgroups — #26020104

Conjugacy classes of subgroups

Complements

Maximal subgroups