Group #36448099 — subgroups

Subgroups — #36448099

Order 22784.

362 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 179 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	#26020404	C₂	1	2	yes	maximal
2	#26018087	C₂	1	4	yes	maximal
3	C₈₉×#1744	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊#10623494	—	2	2	no	—
5	C₂×C₄⋊#10623494	—	2	2	no	—
6	C₂×C₄⋊#10623494	C₂²	1	4	yes	—
7	C₂×C₄×C₈×C₈₉	C₂²	1	2	yes	—
8	C₄³×C₈₉	C₂²	1	1	yes	characteristic
9	#336×C₈₉	C₄	1	4	yes	—
10	C₂²×#10623494	—	2	4	no	—
11	C₄⋊#10623494	—	2	8	no	—
12	C₄⋊#10623494	—	4	4	no	—
13	C₄⋊#10623494	—	4	4	no	—
14	C₂²×C₈×C₈₉	C₂×C₄	1	4	yes	—
15	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
16	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
17	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
18	C₂×C₄²×C₈₉	Q₈	1	2	yes	—
19	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
20	C₄×C₈×C₈₉	—	2	4	no	—
21	C₂×#10623494	—	4	4	no	—
22	C₂×#10623494	—	4	4	no	—
23	C₂×#10623494	—	4	4	no	—
24	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
25	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
26	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
28	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
29	C₂×C₈×C₈₉	—	2	4	no	—
30	C₂×C₈×C₈₉	—	2	8	no	—
31	C₄²×C₈₉	—	2	4	no	—
32	C₄²×C₈₉	—	4	2	no	—
33	C₄²×C₈₉	D₈	1	2	yes	—
34	C₄²×C₈₉	SD₁₆	1	4	yes	—
35	C₄²×C₈₉	Q₁₆	1	2	yes	—
36	C₄²×C₈₉	#41	1	4	yes	—
37	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₄×C₈₉	—	2	2	no	—
39	C₂×C₄×C₈₉	—	2	2	no	—
40	C₂×C₄×C₈₉	—	2	2	no	—
41	C₂×C₄×C₈₉	—	2	2	no	—
42	C₂×C₄×C₈₉	—	2	2	no	—
43	C₂×C₄×C₈₉	#106	1	4	yes	—
44	C₂×C₄×C₈₉	#107	1	4	yes	—
45	C₂×C₄×C₈₉	#117	1	2	yes	—
46	C₂×C₄×C₈₉	#124	1	2	yes	—
47	C₂×C₄×C₈₉	#125	1	2	yes	—
48	C₂×C₄×C₈₉	#131	1	4	yes	—
49	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
50	C₂×C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
51	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
52	C₈×C₈₉	—	4	4	no	—
53	#10623494	—	8	4	no	—
54	C₂²×C₈₉	#382	1	1	yes	characteristic
55	C₂²×C₈₉	#424	1	2	yes	—
56	C₂²×C₈₉	C₂×#106	1	1	yes	characteristic
57	C₂²×C₈₉	C₂×#107	1	1	yes	characteristic
58	C₂²×C₈₉	#430	1	2	yes	—
59	C₄×C₈₉	—	2	4	no	—
60	C₄×C₈₉	—	2	4	no	—
61	C₄×C₈₉	—	2	2	no	—
62	C₄×C₈₉	#433	1	4	yes	—
63	C₄×C₈₉	#486	1	2	yes	—
64	C₄×C₈₉	#487	1	2	yes	—
65	#8787	—	89	1	no	maximal, Sylow
66	C₂×C₈₉	#1570	1	2	yes	—
67	C₂×C₈₉	#1571	1	2	yes	—
68	C₂×C₈₉	#1832	1	1	yes	characteristic
69	C₂×C₈₉	#1834	1	1	yes	characteristic
70	C₂×C₈₉	#1838	1	1	yes	characteristic
71	#1842	—	89	2	no	—
72	#1744	D₈₉	1	1	yes	characteristic
73	#1277	—	89	4	no	—
74	C₈₉	#8787	1	1	yes	characteristic, complemented, Sylow
75	C₂×C₄⋊C₈	—	89	4	no	—
76	C₂×C₄⋊C₈	—	178	2	no	—
77	C₂×C₄⋊C₈	—	178	2	no	—
78	C₂×C₄×C₈	C₂×D₈₉	1	2	yes	—
79	C₄³	C₂×D₈₉	1	1	yes	characteristic
80	#336	Dic₈₉	1	4	yes	—
81	C₂²×C₈	—	178	4	no	—
82	C₂²×C₈	C₂×Dic₈₉	1	4	yes	—
83	C₂×C₄²	Dic₁₇₈	1	2	yes	—
84	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
85	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
86	C₂×C₄²	C₂×Dic₈₉	1	2	yes	—
87	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
88	C₄⋊C₈	—	356	4	no	—
89	C₄⋊C₈	—	356	4	no	—
90	C₄⋊C₈	—	178	8	no	—
91	C₄×C₈	—	2	4	no	—
92	C₂²×C₄	C₄⋊Dic₈₉	1	2	yes	—
93	C₂²×C₄	C₄⋊Dic₈₉	1	2	yes	—
94	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
95	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
96	C₂²×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
97	C₂×C₈	—	2	4	no	—
98	C₂×C₈	—	2	8	no	—
99	C₂×C₈	—	356	4	no	—
100	C₂×C₈	—	356	4	no	—
101	C₂×C₈	—	356	4	no	—
102	C₄²	—	2	4	no	—
103	C₄²	—	4	2	no	—
104	C₄²	D₇₁₂	1	2	yes	—
105	C₄²	#13083563	1	4	yes	—
106	C₄²	Dic₃₅₆	1	2	yes	—
107	C₄²	#13083574	1	4	yes	—
108	C₂³	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	#15667944	1	4	yes	—
115	C₂×C₄	#15667946	1	4	yes	—
116	C₂×C₄	#15667971	1	2	yes	—
117	C₂×C₄	#15668000	1	2	yes	—
118	C₂×C₄	#15668004	1	2	yes	—
119	C₂×C₄	#15668019	1	4	yes	—
120	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
121	C₂×C₄	C₂×#13083563	1	2	yes	—
122	C₂×C₄	C₂×Dic₃₅₆	1	1	yes	characteristic
123	C₈	—	4	4	no	—
124	C₈	—	712	4	no	—
125	C₂²	#19151464	1	1	yes	characteristic
126	C₂²	#19151625	1	2	yes	—
127	C₂²	C₂×#15667944	1	1	yes	characteristic
128	C₂²	C₂×#15667946	1	1	yes	characteristic
129	C₂²	#19151647	1	2	yes	—
130	C₄	—	2	2	no	—
131	C₄	—	2	4	no	—
132	C₄	—	2	4	no	—
133	C₄	#19151659	1	4	yes	—
134	C₄	#19151945	1	2	yes	—
135	C₄	#19151949	1	2	yes	—
136	C₂	#26019023	1	2	yes	—
137	C₂	#26019027	1	2	yes	—
138	C₂	#26020372	1	1	yes	characteristic
139	C₂	#26020380	1	1	yes	characteristic
140	C₂	#26020393	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26020404

C₂

yes

maximal

#26018087

C₂

yes

maximal

C₈₉×#1744

C₂

yes

maximal, characteristic

C₂×C₄⋊#10623494

—

C₂×C₄⋊#10623494

—

C₂×C₄⋊#10623494

C₂²

yes

—

C₂×C₄×C₈×C₈₉

C₂²

yes

—

C₄³×C₈₉

C₂²

yes

characteristic

#336×C₈₉

C₄

yes

—

—

—

—

—

C₂²×C₈×C₈₉

C₂×C₄

yes

—

C₂×C₄²×C₈₉

C₂×C₄

yes

—

C₂×C₄²×C₈₉

D₄

yes

characteristic

C₂×C₄²×C₈₉

D₄

yes

characteristic

C₂×C₄²×C₈₉

Q₈

yes

—

C₂×C₄²×C₈₉

C₂³

yes

characteristic

C₄×C₈×C₈₉

—

C₂×#10623494

—

C₂×#10623494

—

C₂×#10623494

—

C₂²×C₄×C₈₉

C₄⋊C₄

yes

—

C₂²×C₄×C₈₉

C₄⋊C₄

yes

—

C₂²×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₉

C₂×Q₈

yes

characteristic

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

D₈

yes

—

C₄²×C₈₉

SD₁₆

yes

—

C₄²×C₈₉

Q₁₆

yes

—

C₄²×C₈₉

#41

yes

—

C₂³×C₈₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

#106

yes

—

C₂×C₄×C₈₉

#107

yes

—

C₂×C₄×C₈₉

#117

yes

—

C₂×C₄×C₈₉

#124

yes

—

C₂×C₄×C₈₉

#125

yes

—

C₂×C₄×C₈₉

#131

yes

—

C₂×C₄×C₈₉

C₂×D₈

yes

characteristic

C₂×C₄×C₈₉

C₂×SD₁₆

yes

—

C₂×C₄×C₈₉

C₂×Q₁₆

yes

characteristic

C₈×C₈₉

—

#10623494

—

C₂²×C₈₉

#382

yes

characteristic

C₂²×C₈₉

#424

yes

—

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#433

yes

—

C₄×C₈₉

#486

yes

—

C₄×C₈₉

#487

yes

—

#8787

—

maximal, Sylow

C₂×C₈₉

#1570

yes

—

C₂×C₈₉

#1571

yes

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

D₈₉

yes

characteristic

#1277

—

C₈₉

#8787

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

178

—

C₂×C₄⋊C₈

—

178

—

C₂×C₄×C₈

C₂×D₈₉

yes

—

C₄³

C₂×D₈₉

yes

characteristic

#336

Dic₈₉

yes

—

C₂²×C₈

—

178

—

C₂²×C₈

C₂×Dic₈₉

yes

—

C₂×C₄²

Dic₁₇₈

yes

—

C₂×C₄²

D₃₅₆

yes

characteristic

C₂×C₄²

D₃₅₆

yes

characteristic

C₂×C₄²

C₂×Dic₈₉

yes

—

C₂×C₄²

C₂²×D₈₉

yes

characteristic

C₄⋊C₈

—

356

—

C₄⋊C₈

—

356

—

C₄⋊C₈

—

178

—

C₄×C₈

—

C₂²×C₄

C₄⋊Dic₈₉

yes

—

C₂²×C₄

C₄⋊Dic₈₉

yes

—

C₂²×C₄

C₂²×Dic₈₉

yes

characteristic

C₂²×C₄

C₂×D₃₅₆

yes

characteristic

C₂²×C₄

C₂×Dic₁₇₈

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

356

—

100

C₂×C₈

—

356

—

101

C₂×C₈

—

356

—

102

C₄²

—

103

C₄²

—

104

C₄²

D₇₁₂

yes

—

105

C₄²

#13083563

yes

—

106

C₄²

Dic₃₅₆

yes

—

107

C₄²

#13083574

yes

—

108

C₂³

C₂×C₄⋊Dic₈₉

yes

characteristic

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

#15667944

yes

—

115

C₂×C₄

#15667946

yes

—

116

C₂×C₄

#15667971

yes

—

117

C₂×C₄

#15668000

yes

—

118

C₂×C₄

#15668004

yes

—

119

C₂×C₄

#15668019

yes

—

120

C₂×C₄

C₂×D₇₁₂

yes

characteristic

121

C₂×C₄

C₂×#13083563

yes

—

122

C₂×C₄

C₂×Dic₃₅₆

yes

characteristic

123

C₈

—

124

C₈

—

712

—

125

C₂²

#19151464

yes

characteristic

126

C₂²

#19151625

yes

—

127

C₂²

C₂×#15667944

yes

characteristic

128

C₂²

C₂×#15667946

yes

characteristic

129

C₂²

#19151647

yes

—

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

#19151659

yes

—

134

C₄

#19151945

yes

—

135

C₄

#19151949

yes

—

136

C₂

#26019023

yes

—

137

C₂

#26019027

yes

—

138

C₂

#26020372

yes

characteristic

139

C₂

#26020380

yes

characteristic

140

C₂

#26020393

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₃₅₆
commutator	C₂²×C₄×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1744	C₂
socle, prime radical	C₂³×C₈₉	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1744	D₈₉
hypocenter	C₈₉	#8787
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1744

subgroup

G/H

center

C₂²×C₄

C₂×D₃₅₆

commutator

C₂²×C₄×C₈₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₉

Fitting

C₈₉×#1744

C₂

socle, prime radical

C₂³×C₈₉

C₂×C₄⋊C₄

cosocle

C₂×C₄²×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1744

N	H ≅ G/N	type
74. C₈₉	#8787	semidirect

H ≅ G/N

type

74. C₈₉

#8787

semidirect

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

subgroup	index
#26020404	2
#26018087	2
C₈₉×#1744	2
#8787	89

subgroup

index

#26020404

#26018087

C₈₉×#1744

#8787

Subgroups — #36448099

Conjugacy classes of subgroups

Complements

Maximal subgroups