Group #26022023 — subgroups

Subgroups — #26022023

Order 11392.

144 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 65 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	M₃₂×D₈₉	C₂	1	1	yes	maximal, characteristic
2	#108×D₈₉	C₂	1	1	yes	maximal, characteristic
3	#19151633	C₂	1	1	yes	maximal, characteristic
4	#19151422	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151421	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151420	C₂	1	1	yes	maximal, characteristic, complemented
7	#365×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	M₁₆×D₈₉	—	2	1	no	—
9	C₂×C₈×D₈₉	C₂²	1	1	yes	characteristic
10	#15667970	—	2	1	no	—
11	#15667952	C₂²	1	1	yes	characteristic
12	#15667951	C₄	1	2	yes	complemented
13	C₁₆×D₈₉	C₄	1	2	yes	complemented
14	#15667948	C₂²	1	1	yes	characteristic
15	#15667947	—	2	1	no	—
16	#15667945	—	2	1	no	—
17	#15667944	C₂²	1	1	yes	characteristic
18	M₃₂×C₈₉	C₂²	1	1	yes	characteristic
19	#108×C₈₉	C₂²	1	1	yes	characteristic
20	#106×C₈₉	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₈₉	D₄	1	1	yes	characteristic
22	#13083558	—	2	1	no	—
23	#13083557	—	4	1	no	—
24	C₂×#10623494	D₄	1	1	yes	characteristic
25	C₈×D₈₉	—	4	1	no	—
26	C₈×D₈₉	C₂×C₄	1	1	yes	characteristic
27	C₈×D₈₉	C₂×C₄	1	1	yes	characteristic
28	C₈×D₈₉	Q₈	1	1	yes	characteristic
29	C₈×D₈₉	Q₈	1	1	yes	characteristic
30	C₄⋊Dic₈₉	—	2	1	no	—
31	#13083551	—	4	1	no	—
32	C₄×Dic₈₉	—	4	1	no	—
33	#13083549	C₂×C₄	1	2	yes	—
34	M₁₆×C₈₉	—	2	1	no	—
35	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
36	C₄⋊C₄×C₈₉	—	2	1	no	—
37	C₁₆×C₈₉	C₂×C₄	1	2	yes	—
38	C₂²×D₈₉	SD₁₆	1	1	yes	characteristic
39	C₂×C₄×C₈₉	—	4	1	no	—
40	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
41	C₂×Dic₈₉	—	4	1	no	—
42	C₂×Dic₈₉	SD₁₆	1	1	yes	characteristic
43	C₄×D₈₉	C₄⋊C₄	1	1	yes	characteristic
44	C₄×D₈₉	C₄⋊C₄	1	1	yes	characteristic
45	C₄×D₈₉	SD₁₆	1	1	yes	characteristic
46	C₄×D₈₉	SD₁₆	1	1	yes	characteristic
47	C₈×C₈₉	—	4	1	no	—
48	C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
49	C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
50	#10623494	—	4	1	no	—
51	#10623494	C₄⋊C₄	1	1	yes	characteristic
52	#10623494	C₄⋊C₄	1	1	yes	characteristic
53	C₂²×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
54	C₂×D₈₉	—	4	1	no	—
55	C₂×D₈₉	#106	1	1	yes	characteristic
56	C₂×D₈₉	#106	1	1	yes	characteristic
57	C₄×C₈₉	—	8	1	no	—
58	C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
60	Dic₈₉	—	8	1	no	—
61	Dic₈₉	#106	1	1	yes	characteristic
62	Dic₈₉	#106	1	1	yes	characteristic
63	C₂×C₈₉	—	2	1	no	—
64	C₂×C₈₉	C₂×#106	1	1	yes	characteristic
65	D₈₉	—	2	1	no	—
66	D₈₉	—	2	1	no	—
67	#2155	—	89	1	no	maximal, Sylow
68	C₈₉	#2155	1	1	yes	characteristic, complemented, Sylow
69	C₂×M₃₂	—	89	1	no	—
70	C₂×#108	—	89	1	no	—
71	#427	—	89	1	no	—
72	#365	—	89	1	no	—
73	#365	—	89	1	no	—
74	#365	—	89	1	no	—
75	#365	D₈₉	1	1	yes	characteristic, complemented
76	C₂×M₁₆	—	178	1	no	—
77	C₂²×C₈	—	89	1	no	—
78	#117	—	178	1	no	—
79	M₃₂	—	89	1	no	—
80	M₃₂	—	89	2	no	—
81	M₃₂	C₂×D₈₉	1	1	yes	characteristic
82	C₂×C₁₆	—	89	2	no	—
83	#108	—	89	1	no	—
84	#108	—	178	1	no	—
85	#108	C₂×D₈₉	1	1	yes	characteristic
86	#107	—	178	1	no	—
87	#106	—	89	1	no	—
88	#106	C₂×D₈₉	1	1	yes	characteristic
89	C₂²×C₄	—	89	1	no	—
90	M₁₆	—	2	1	no	—
91	M₁₆	—	356	1	no	—
92	M₁₆	—	178	1	no	—
93	C₂×C₈	—	89	1	no	—
94	C₂×C₈	—	89	1	no	—
95	C₂×C₈	—	89	1	no	—
96	C₂×C₈	—	89	1	no	—
97	C₂×C₈	—	89	1	no	—
98	C₂×C₈	—	356	1	no	—
99	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
100	C₄⋊C₄	—	2	1	no	—
101	C₄⋊C₄	—	178	1	no	—
102	C₂²⋊C₄	—	356	1	no	—
103	C₄²	—	356	1	no	—
104	C₁₆	—	89	2	no	—
105	C₁₆	C₄×D₈₉	1	2	yes	—
106	C₂³	—	89	1	no	—
107	C₂×C₄	—	4	1	no	—
108	C₂×C₄	—	89	1	no	—
109	C₂×C₄	—	89	1	no	—
110	C₂×C₄	—	89	1	no	—
111	C₂×C₄	—	89	1	no	—
112	C₂×C₄	—	89	1	no	—
113	C₂×C₄	—	356	1	no	—
114	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
115	C₈	—	4	1	no	—
116	C₈	—	89	1	no	—
117	C₈	—	89	1	no	—
118	C₈	—	356	1	no	—
119	C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
120	C₈	Q₈×D₈₉	1	1	yes	characteristic
121	C₂²	—	89	1	no	—
122	C₂²	—	89	1	no	—
123	C₂²	—	356	1	no	—
124	C₂²	SD₁₆×D₈₉	1	1	yes	characteristic
125	C₄	—	8	1	no	—
126	C₄	—	89	1	no	—
127	C₄	—	89	1	no	—
128	C₄	—	712	1	no	—
129	C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
130	C₄	SD₁₆×D₈₉	1	1	yes	characteristic
131	C₂	—	2	1	no	—
132	C₂	—	178	1	no	—
133	C₂	—	178	1	no	—
134	C₂	#106×D₈₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

M₃₂×D₈₉

C₂

yes

maximal, characteristic

#108×D₈₉

C₂

yes

maximal, characteristic

#19151633

C₂

yes

maximal, characteristic

#19151422

C₂

yes

maximal, characteristic, complemented

#19151421

C₂

yes

maximal, characteristic, complemented

#19151420

C₂

yes

maximal, characteristic, complemented

#365×C₈₉

C₂

yes

maximal, characteristic, complemented

M₁₆×D₈₉

—

C₂×C₈×D₈₉

C₂²

yes

characteristic

#15667970

—

#15667952

C₂²

yes

characteristic

#15667951

C₄

yes

complemented

C₁₆×D₈₉

C₄

yes

complemented

#15667948

C₂²

yes

—

—

C₂²

yes

characteristic

M₃₂×C₈₉

C₂²

yes

characteristic

#108×C₈₉

C₂²

yes

characteristic

#106×C₈₉

C₂²

yes

characteristic

C₂×C₄×D₈₉

D₄

yes

—

—

D₄

yes

characteristic

C₈×D₈₉

—

C₈×D₈₉

C₂×C₄

yes

characteristic

C₈×D₈₉

C₂×C₄

yes

characteristic

C₈×D₈₉

Q₈

yes

characteristic

C₈×D₈₉

Q₈

yes

characteristic

C₄⋊Dic₈₉

—

#13083551

—

C₄×Dic₈₉

—

#13083549

C₂×C₄

yes

—

M₁₆×C₈₉

—

C₂×C₈×C₈₉

C₂³

yes

characteristic

C₄⋊C₄×C₈₉

—

C₁₆×C₈₉

C₂×C₄

yes

—

C₂²×D₈₉

SD₁₆

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

SD₁₆

yes

characteristic

C₄×D₈₉

C₄⋊C₄

yes

characteristic

C₄×D₈₉

C₄⋊C₄

yes

characteristic

C₄×D₈₉

SD₁₆

yes

characteristic

C₄×D₈₉

SD₁₆

yes

characteristic

C₈×C₈₉

—

C₈×C₈₉

C₂²×C₄

yes

characteristic

C₈×C₈₉

C₂×Q₈

yes

characteristic

#10623494

—

#10623494

C₄⋊C₄

yes

characteristic

#10623494

C₄⋊C₄

yes

characteristic

C₂²×C₈₉

C₂×SD₁₆

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

yes

C₂×D₈₉

yes

C₄×C₈₉

—

C₄×C₈₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₈₉

C₂×SD₁₆

yes

characteristic

Dic₈₉

—

Dic₈₉

yes

Dic₈₉

yes

C₂×C₈₉

—

C₂×C₈₉

C₂×#106

yes

characteristic

D₈₉

—

D₈₉

—

#2155

—

maximal, Sylow

C₈₉

#2155

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

—

—

—

—

—

D₈₉

yes

characteristic, complemented

C₂×M₁₆

—

178

—

C₂²×C₈

—

#117

—

178

—

M₃₂

—

M₃₂

—

M₃₂

C₂×D₈₉

yes

characteristic

C₂×C₁₆

—

#108

—

#108

—

178

—

#108

C₂×D₈₉

yes

characteristic

#107

—

178

—

#106

—

#106

C₂×D₈₉

yes

characteristic

C₂²×C₄

—

M₁₆

—

M₁₆

—

356

—

M₁₆

—

178

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

356

—

C₂×C₈

C₂²×D₈₉

yes

characteristic

100

C₄⋊C₄

—

101

C₄⋊C₄

—

178

—

102

C₂²⋊C₄

—

356

—

103

C₄²

—

356

—

104

C₁₆

—

105

C₁₆

C₄×D₈₉

yes

—

106

C₂³

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

356

—

114

C₂×C₄

D₄×D₈₉

yes

characteristic

115

C₈

—

116

C₈

—

117

C₈

—

118

C₈

—

356

—

119

C₈

C₂×C₄×D₈₉

yes

characteristic

120

C₈

Q₈×D₈₉

yes

characteristic

121

C₂²

—

122

C₂²

—

123

C₂²

—

356

—

124

C₂²

SD₁₆×D₈₉

yes

characteristic

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

—

712

—

129

C₄

C₄⋊C₄×D₈₉

yes

characteristic

130

C₄

SD₁₆×D₈₉

yes

characteristic

131

C₂

—

132

C₂

—

178

—

133

C₂

—

178

—

134

C₂

#106×D₈₉

yes

characteristic

	subgroup	G/H
center	C₂	#106×D₈₉
commutator	C₈×C₈₉	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₈₉
Fitting	#365×C₈₉	C₂
socle	C₂×C₈₉	C₂×#106
prime radical	C₂²×D₈₉	SD₁₆
cosocle	C₂×C₈×C₈₉	C₂³
hypercenter	#365	D₈₉
hypocenter	C₈₉	#2155
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #365

subgroup

G/H

center

C₂

#106×D₈₉

commutator

C₈×C₈₉

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₈₉

Fitting

#365×C₈₉

C₂

socle

C₂×C₈₉

C₂×#106

prime radical

C₂²×D₈₉

SD₁₆

cosocle

C₂×C₈×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #365

N	H ≅ G/N	type
4. #19151422	C₂	semidirect
5. #19151421	C₂	semidirect
6. #19151420	C₂	semidirect
7. #365×C₈₉	C₂	semidirect
12. #15667951	C₄	semidirect
13. C₁₆×D₈₉	C₄	semidirect
68. C₈₉	#2155	semidirect
75. #365	D₈₉	semidirect

H ≅ G/N

type

4. #19151422

C₂

semidirect

5. #19151421

C₂

semidirect

6. #19151420

C₂

semidirect

7. #365×C₈₉

C₂

semidirect

12. #15667951

C₄

semidirect

13. C₁₆×D₈₉

C₄

semidirect

68. C₈₉

#2155

semidirect

75. #365

D₈₉

semidirect

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

subgroup	index
M₃₂×D₈₉	2
#108×D₈₉	2
#19151633	2
#19151422	2
#19151421	2
#19151420	2
#365×C₈₉	2
#2155	89

subgroup

index

M₃₂×D₈₉

#108×D₈₉

#365×C₈₉

Subgroups — #26022023

Conjugacy classes of subgroups

Complements

Maximal subgroups