Group #36448101 — subgroups

Subgroups — #36448101

Order 22784.

322 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 139 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	#26020369	C₂	1	1	yes	maximal, characteristic
2	#26019998	C₂	1	1	yes	maximal, characteristic
3	#26018086	C₂	1	4	yes	maximal
4	C₈₉×#1831	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
6	C₂×C₄×#10623494	C₂²	1	2	yes	—
7	#19151353	—	2	2	no	—
8	#19151353	C₄	1	2	yes	—
9	#19151353	C₄	1	2	yes	—
10	C₂×C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
11	C₄³×C₈₉	C₂²	1	1	yes	characteristic
12	#336×C₈₉	—	2	2	no	—
13	C₂²×#10623494	—	2	2	no	—
14	C₂²×#10623494	C₂×C₄	1	4	yes	—
15	C₄⋊#10623494	—	2	4	no	—
16	C₄×#10623494	—	2	2	no	—
17	C₄×#10623494	—	2	2	no	—
18	C₂²×C₈×C₈₉	—	2	2	no	—
19	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄²×C₈₉	D₄	1	2	yes	—
22	C₂×C₄²×C₈₉	Q₈	1	2	yes	—
23	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
24	C₄⋊C₈×C₈₉	—	2	4	no	—
25	C₂×#10623494	—	2	4	no	—
26	C₂×#10623494	—	2	4	no	—
27	C₂×#10623494	—	2	4	no	—
28	C₂×#10623494	—	4	2	no	—
29	C₂×#10623494	—	4	4	no	—
30	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
31	C₂²×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
32	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
35	C₂×C₈×C₈₉	—	4	2	no	—
36	C₂×C₈×C₈₉	—	4	4	no	—
37	C₄²×C₈₉	—	2	2	no	—
38	C₄²×C₈₉	—	2	2	no	—
39	C₄²×C₈₉	—	2	4	no	—
40	C₄²×C₈₉	—	4	2	no	—
41	C₄²×C₈₉	#41	1	2	yes	—
42	C₄²×C₈₉	#41	1	2	yes	—
43	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₈₉	—	2	2	no	—
45	C₂×C₄×C₈₉	—	2	2	no	—
46	C₂×C₄×C₈₉	—	2	2	no	—
47	C₂×C₄×C₈₉	—	2	2	no	—
48	C₂×C₄×C₈₉	—	2	2	no	—
49	C₂×C₄×C₈₉	—	2	2	no	—
50	C₂×C₄×C₈₉	—	2	2	no	—
51	C₂×C₄×C₈₉	#117	1	2	yes	—
52	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
53	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
54	C₂×C₄×C₈₉	#125	1	2	yes	—
55	C₂×C₄×C₈₉	#131	1	2	yes	—
56	C₂×C₄×C₈₉	#131	1	2	yes	—
57	C₂×C₄×C₈₉	#135	1	4	yes	—
58	C₈×C₈₉	—	8	2	no	—
59	#10623494	—	4	4	no	—
60	#10623494	—	8	2	no	—
61	C₂²×C₈₉	#382	1	1	yes	characteristic
62	C₂²×C₈₉	#424	1	1	yes	characteristic
63	C₂²×C₈₉	#424	1	1	yes	characteristic
64	C₂²×C₈₉	#427	1	2	yes	—
65	C₂²×C₈₉	#430	1	2	yes	—
66	C₄×C₈₉	—	2	2	no	—
67	C₄×C₈₉	—	4	2	no	—
68	C₄×C₈₉	—	4	2	no	—
69	C₄×C₈₉	#433	1	4	yes	—
70	C₄×C₈₉	#488	1	2	yes	—
71	C₄×C₈₉	#488	1	2	yes	—
72	#8788	—	89	1	no	maximal, Sylow
73	C₂×C₈₉	#1572	1	2	yes	—
74	C₂×C₈₉	#1572	1	2	yes	—
75	C₂×C₈₉	#1832	1	1	yes	characteristic
76	C₂×C₈₉	#1835	1	1	yes	characteristic
77	C₂×C₈₉	#1838	1	1	yes	characteristic
78	#1831	—	89	1	no	—
79	#1831	D₈₉	1	1	yes	characteristic
80	#1744	—	89	1	no	—
81	#1277	—	89	4	no	—
82	C₈₉	#8788	1	1	yes	characteristic, complemented, Sylow
83	C₂×C₄⋊C₈	—	89	2	no	—
84	C₂×C₄⋊C₈	C₂×D₈₉	1	2	yes	—
85	C₂×C₄×C₈	—	89	2	no	—
86	C₄³	C₂×D₈₉	1	1	yes	characteristic
87	#336	—	2	2	no	—
88	#336	—	89	2	no	—
89	#336	—	89	2	no	—
90	#336	—	178	2	no	—
91	C₂²×C₈	—	2	2	no	—
92	C₂²×C₈	—	89	4	no	—
93	C₂²×C₈	—	178	2	no	—
94	C₂×C₄²	Dic₁₇₈	1	2	yes	—
95	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
96	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
97	C₂×C₄²	C₂²⋊D₈₉	1	2	yes	—
98	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
99	C₄⋊C₈	—	2	4	no	—
100	C₄⋊C₈	—	178	4	no	—
101	C₄×C₈	—	178	2	no	—
102	C₄×C₈	—	178	2	no	—
103	C₂²×C₄	#13083553	1	2	yes	—
104	C₂²×C₄	#13083553	1	2	yes	—
105	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
106	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
107	C₂²×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
108	C₂×C₈	—	4	2	no	—
109	C₂×C₈	—	4	4	no	—
110	C₂×C₈	—	356	2	no	—
111	C₂×C₈	—	356	4	no	—
112	C₂×C₈	—	178	4	no	—
113	C₂×C₈	—	178	4	no	—
114	C₂×C₈	—	178	4	no	—
115	C₄²	—	2	2	no	—
116	C₄²	—	2	2	no	—
117	C₄²	—	2	4	no	—
118	C₄²	—	4	2	no	—
119	C₄²	#13083573	1	2	yes	—
120	C₄²	#13083575	1	2	yes	—
121	C₂³	C₂×#13083553	1	1	yes	characteristic
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	#15667970	1	2	yes	—
130	C₂×C₄	#15667999	1	1	yes	characteristic
131	C₂×C₄	#15668001	1	1	yes	characteristic
132	C₂×C₄	#15668003	1	2	yes	—
133	C₂×C₄	#15668018	1	2	yes	—
134	C₂×C₄	#15668020	1	2	yes	—
135	C₂×C₄	#15668032	1	4	yes	—
136	C₈	—	8	2	no	—
137	C₈	—	356	4	no	—
138	C₈	—	712	2	no	—
139	C₂²	#19151463	1	1	yes	characteristic
140	C₂²	#19151624	1	1	yes	characteristic
141	C₂²	#19151626	1	1	yes	characteristic
142	C₂²	#19151634	1	2	yes	—
143	C₂²	#19151646	1	2	yes	—
144	C₄	—	2	2	no	—
145	C₄	—	4	2	no	—
146	C₄	—	4	2	no	—
147	C₄	#19151658	1	4	yes	—
148	C₄	#19151952	1	2	yes	—
149	C₄	#19151955	1	2	yes	—
150	C₂	#26019030	1	2	yes	—
151	C₂	#26019033	1	2	yes	—
152	C₂	#26020371	1	1	yes	characteristic
153	C₂	#26020382	1	1	yes	characteristic
154	C₂	#26020392	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26020369

C₂

yes

maximal, characteristic

#26019998

C₂

yes

maximal, characteristic

#26018086

C₂

yes

maximal

C₈₉×#1831

C₂

yes

maximal, characteristic

C₂×C₄⋊#10623494

C₂²

yes

—

C₂×C₄×#10623494

C₂²

yes

—

#19151353

—

#19151353

C₄

yes

—

#19151353

C₄

yes

—

C₂×C₄⋊C₈×C₈₉

C₂²

yes

—

C₄³×C₈₉

C₂²

yes

characteristic

#336×C₈₉

—

C₂²×#10623494

—

C₂²×#10623494

C₂×C₄

yes

—

C₄⋊#10623494

—

C₄×#10623494

—

C₄×#10623494

—

C₂²×C₈×C₈₉

—

C₂×C₄²×C₈₉

C₂×C₄

yes

characteristic

C₂×C₄²×C₈₉

C₂×C₄

yes

characteristic

C₂×C₄²×C₈₉

D₄

yes

—

C₂×C₄²×C₈₉

Q₈

yes

—

C₂×C₄²×C₈₉

C₂³

yes

characteristic

C₄⋊C₈×C₈₉

—

—

—

—

—

—

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

—

C₄²×C₈₉

—

C₄²×C₈₉

#41

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

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

#117

yes

—

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

—

C₂×C₄×C₈₉

#131

yes

—

C₂×C₄×C₈₉

#131

yes

—

C₂×C₄×C₈₉

#135

yes

—

C₈×C₈₉

—

#10623494

—

#10623494

—

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

—

C₂²×C₈₉

#430

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#433

yes

—

C₄×C₈₉

#488

yes

—

C₄×C₈₉

#488

yes

—

#8788

—

maximal, Sylow

C₂×C₈₉

#1572

yes

—

C₂×C₈₉

#1572

yes

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

D₈₉

yes

characteristic

#1744

—

#1277

—

C₈₉

#8788

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

C₂×D₈₉

yes

—

C₂×C₄×C₈

—

C₄³

C₂×D₈₉

yes

—

—

—

—

178

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

178

—

C₂×C₄²

Dic₁₇₈

yes

—

C₂×C₄²

C₄×D₈₉

yes

characteristic

C₂×C₄²

C₄×D₈₉

yes

characteristic

C₂×C₄²

C₂²⋊D₈₉

yes

—

C₂×C₄²

C₂²×D₈₉

yes

characteristic

C₄⋊C₈

—

100

C₄⋊C₈

—

178

—

101

C₄×C₈

—

178

—

102

C₄×C₈

—

178

—

103

C₂²×C₄

#13083553

yes

—

104

C₂²×C₄

#13083553

yes

—

105

C₂²×C₄

C₂×C₄×D₈₉

yes

characteristic

106

C₂²×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

107

C₂²×C₄

C₂×Dic₁₇₈

yes

characteristic

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

356

—

111

C₂×C₈

—

356

—

112

C₂×C₈

—

178

—

113

C₂×C₈

—

178

—

114

C₂×C₈

—

178

—

115

C₄²

—

116

C₄²

—

117

C₄²

—

118

C₄²

—

119

C₄²

#13083573

yes

—

120

C₄²

#13083575

yes

—

121

C₂³

C₂×#13083553

yes

characteristic

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

#15667970

yes

—

130

C₂×C₄

#15667999

yes

characteristic

131

C₂×C₄

#15668001

yes

characteristic

132

C₂×C₄

#15668003

yes

—

133

C₂×C₄

#15668018

yes

—

134

C₂×C₄

#15668020

yes

—

135

C₂×C₄

#15668032

yes

—

136

C₈

—

137

C₈

—

356

—

138

C₈

—

712

—

139

C₂²

#19151463

yes

characteristic

140

C₂²

#19151624

yes

characteristic

141

C₂²

#19151626

yes

characteristic

142

C₂²

#19151634

yes

—

143

C₂²

#19151646

yes

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

#19151658

yes

—

148

C₄

#19151952

yes

—

149

C₄

#19151955

yes

—

150

C₂

#26019030

yes

—

151

C₂

#26019033

yes

—

152

C₂

#26020371

yes

characteristic

153

C₂

#26020382

yes

characteristic

154

C₂

#26020392

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₈₉
commutator	C₂²×C₄×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1831	C₂
socle, prime radical	C₂³×C₈₉	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1831	D₈₉
hypocenter	C₈₉	#8788
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1831

subgroup

G/H

center

C₂²×C₄

C₂×C₂²⋊D₈₉

commutator

C₂²×C₄×C₈₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₉

Fitting

C₈₉×#1831

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 ◁ #1831

N	H ≅ G/N	type
82. C₈₉	#8788	semidirect

H ≅ G/N

type

82. C₈₉

#8788

semidirect

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

subgroup	index
#26020369	2
#26019998	2
#26018086	2
C₈₉×#1831	2
#8788	89

subgroup

index

C₈₉×#1831

Subgroups — #36448101

Conjugacy classes of subgroups

Complements

Maximal subgroups