Group #36468809 — subgroups

Subgroups — #36468809

Order 22784.

188 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 71 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	#26022527	C₂	1	1	yes	maximal, characteristic
2	#26019656	C₂	1	2	yes	maximal
3	#26018326	C₂	1	1	yes	maximal, characteristic
4	#26018309	C₂	1	2	yes	maximal
5	C₈₉×#2183	C₂	1	1	yes	maximal, characteristic
6	#19152003	C₂²	1	2	yes	—
7	#19151893	—	2	2	no	—
8	#19151424	—	2	1	no	—
9	#19151424	—	2	1	no	—
10	#19151351	C₂²	1	2	yes	—
11	#19151346	—	2	2	no	—
12	C₈×Q₈×C₈₉	C₂²	1	1	yes	characteristic
13	C₄⋊C₁₆×C₈₉	—	2	1	no	—
14	C₄⋊C₁₆×C₈₉	C₂²	1	1	yes	characteristic
15	#346×C₈₉	—	2	1	no	—
16	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
17	#15668012	—	2	2	no	—
18	#15667946	—	2	2	no	—
19	#15667946	—	2	2	no	—
20	#15667945	—	4	2	no	—
21	C₄⋊#10623494	—	2	2	no	—
22	#15667936	—	4	2	no	—
23	C₄×Q₈×C₈₉	D₄	1	1	yes	characteristic
24	C₂×C₁₆×C₈₉	—	2	1	no	—
25	C₂×C₁₆×C₈₉	D₄	1	1	yes	characteristic
26	C₂×C₁₆×C₈₉	D₄	1	1	yes	characteristic
27	C₄⋊C₈×C₈₉	—	2	1	no	—
28	C₄⋊C₈×C₈₉	D₄	1	1	yes	characteristic
29	C₄×C₈×C₈₉	—	2	1	no	—
30	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
31	C₂×Dic₁₇₈	—	4	2	no	—
32	C₂×#10623494	—	4	2	no	—
33	C₄⋊Dic₈₉	—	4	2	no	—
34	C₄⋊Dic₈₉	—	4	2	no	—
35	C₂×Q₈×C₈₉	D₈	1	1	yes	characteristic
36	C₂×C₈×C₈₉	—	2	1	no	—
37	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
38	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
39	C₄⋊C₄×C₈₉	—	2	1	no	—
40	C₄⋊C₄×C₈₉	D₈	1	1	yes	characteristic
41	C₄²×C₈₉	—	2	1	no	—
42	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
43	C₁₆×C₈₉	—	2	1	no	—
44	C₁₆×C₈₉	—	4	1	no	—
45	C₁₆×C₈₉	Q₁₆	1	2	yes	—
46	Q₈×C₈₉	—	2	2	no	—
47	C₂×C₄×C₈₉	—	2	1	no	—
48	C₂×C₄×C₈₉	—	2	1	no	—
49	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
50	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
51	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
52	C₂×Dic₈₉	—	8	2	no	—
53	Dic₁₇₈	—	16	2	no	—
54	C₈×C₈₉	—	2	1	no	—
55	C₈×C₈₉	—	4	1	no	—
56	C₈×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
57	C₈×C₈₉	#137	1	1	yes	characteristic
58	#10623494	—	8	2	no	—
59	C₂²×C₈₉	#466	1	1	yes	characteristic
60	C₄×C₈₉	—	2	1	no	—
61	C₄×C₈₉	—	4	1	no	—
62	C₄×C₈₉	—	4	1	no	—
63	C₄×C₈₉	#462	1	1	yes	characteristic
64	C₄×C₈₉	#473	1	1	yes	characteristic
65	C₄×C₈₉	#510	1	2	yes	—
66	Dic₈₉	—	16	2	no	—
67	#12461	—	89	1	no	maximal, Sylow
68	C₂×C₈₉	#1674	1	1	yes	characteristic
69	C₂×C₈₉	#2219	1	1	yes	characteristic
70	C₂×C₈₉	#2221	1	1	yes	characteristic
71	#2252	—	89	1	no	—
72	#2183	D₈₉	1	1	yes	characteristic
73	#1676	—	89	2	no	—
74	#1372	—	89	1	no	—
75	#1365	—	89	2	no	—
76	C₈₉	#12461	1	1	yes	characteristic, complemented, Sylow
77	#500	—	89	2	no	—
78	#477	—	178	2	no	—
79	C₈×Q₈	C₂×D₈₉	1	1	yes	characteristic
80	#366	—	178	1	no	—
81	#366	—	178	1	no	—
82	C₄⋊C₁₆	—	2	1	no	—
83	C₄⋊C₁₆	C₂×D₈₉	1	1	yes	characteristic
84	#346	—	2	1	no	—
85	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
86	#335	—	89	2	no	—
87	#333	—	178	2	no	—
88	C₄⋊Q₈	—	178	2	no	—
89	C₄×Q₈	D₃₅₆	1	1	yes	characteristic
90	C₂×C₁₆	—	2	1	no	—
91	C₂×C₁₆	C₂²⋊D₈₉	1	1	yes	characteristic
92	C₂×C₁₆	C₂²⋊D₈₉	1	1	yes	characteristic
93	#107	—	356	2	no	—
94	#107	—	178	2	no	—
95	#107	—	178	2	no	—
96	C₄⋊C₈	—	2	1	no	—
97	C₄⋊C₈	—	178	2	no	—
98	C₄⋊C₈	D₃₅₆	1	1	yes	characteristic
99	#103	—	356	2	no	—
100	C₄×C₈	—	2	1	no	—
101	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
102	C₂×Q₈	—	356	2	no	—
103	C₂×Q₈	D₇₁₂	1	1	yes	characteristic
104	C₂×C₈	—	2	1	no	—
105	C₂×C₈	—	356	2	no	—
106	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
107	C₂×C₈	#13083574	1	1	yes	characteristic
108	C₄⋊C₄	—	2	1	no	—
109	C₄⋊C₄	—	356	2	no	—
110	C₄⋊C₄	—	356	2	no	—
111	C₄⋊C₄	D₇₁₂	1	1	yes	characteristic
112	C₄²	—	2	1	no	—
113	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
114	C₁₆	—	2	1	no	—
115	C₁₆	—	4	1	no	—
116	C₁₆	#13083564	1	2	yes	—
117	Q₈	—	2	2	no	—
118	Q₈	—	1424	2	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	712	2	no	—
122	C₂×C₄	#15667986	1	1	yes	characteristic
123	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
124	C₂×C₄	#15668034	1	1	yes	characteristic
125	C₈	—	2	1	no	—
126	C₈	—	4	1	no	—
127	C₈	—	712	2	no	—
128	C₈	C₂×#13083564	1	1	yes	characteristic
129	C₈	#15668042	1	1	yes	characteristic
130	C₂²	#19151834	1	1	yes	characteristic
131	C₄	—	2	1	no	—
132	C₄	—	4	1	no	—
133	C₄	—	4	1	no	—
134	C₄	—	1424	2	no	—
135	C₄	#19151808	1	1	yes	characteristic
136	C₄	#19151873	1	1	yes	characteristic
137	C₄	#19152041	1	2	yes	—
138	C₂	#26019646	1	1	yes	characteristic
139	C₂	#26022360	1	1	yes	characteristic
140	C₂	#26022372	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26022527

C₂

yes

maximal, characteristic

#26019656

C₂

yes

maximal

#26018326

C₂

yes

maximal, characteristic

#26018309

C₂

yes

maximal

C₈₉×#2183

C₂

yes

maximal, characteristic

#19152003

C₂²

yes

—

—

—

—

C₂²

yes

—

#19151346

—

C₈×Q₈×C₈₉

C₂²

yes

characteristic

C₄⋊C₁₆×C₈₉

—

C₄⋊C₁₆×C₈₉

C₂²

yes

characteristic

#346×C₈₉

—

C₄×C₁₆×C₈₉

C₂²

yes

—

—

—

—

—

—

C₄×Q₈×C₈₉

D₄

yes

characteristic

C₂×C₁₆×C₈₉

—

C₂×C₁₆×C₈₉

D₄

yes

characteristic

C₂×C₁₆×C₈₉

D₄

yes

characteristic

C₄⋊C₈×C₈₉

—

C₄⋊C₈×C₈₉

D₄

yes

characteristic

C₄×C₈×C₈₉

—

C₄×C₈×C₈₉

C₂³

yes

characteristic

C₂×Dic₁₇₈

—

C₂×#10623494

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₂×Q₈×C₈₉

D₈

yes

characteristic

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

#41

yes

characteristic

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

D₈

yes

characteristic

C₄²×C₈₉

—

C₄²×C₈₉

C₂×D₄

yes

characteristic

C₁₆×C₈₉

—

C₁₆×C₈₉

—

C₁₆×C₈₉

Q₁₆

yes

—

Q₈×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

#121

yes

characteristic

C₂×C₄×C₈₉

C₂×D₈

yes

characteristic

C₂×C₄×C₈₉

#135

yes

characteristic

C₂×Dic₈₉

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

—

C₈×C₈₉

C₂×Q₁₆

yes

C₈×C₈₉

yes

—

C₂²×C₈₉

#466

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

—

Dic₈₉

—

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

D₈₉

yes

—

—

—

C₈₉

#12461

yes

characteristic, complemented, Sylow

#500

—

#477

—

178

—

C₈×Q₈

C₂×D₈₉

yes

characteristic

#366

—

178

—

#366

—

178

—

C₄⋊C₁₆

—

C₄⋊C₁₆

C₂×D₈₉

yes

characteristic

#346

—

C₄×C₁₆

C₂×D₈₉

yes

characteristic

#335

—

#333

—

178

—

C₄⋊Q₈

—

178

—

C₄×Q₈

D₃₅₆

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₈₉

yes

characteristic

C₂×C₁₆

C₂²⋊D₈₉

yes

characteristic

#107

—

356

—

#107

—

178

—

#107

—

178

—

C₄⋊C₈

—

C₄⋊C₈

—

178

—

C₄⋊C₈

D₃₅₆

yes

characteristic

#103

—

356

—

100

C₄×C₈

—

101

C₄×C₈

C₂²×D₈₉

yes

characteristic

102

C₂×Q₈

—

356

—

103

C₂×Q₈

D₇₁₂

yes

characteristic

104

C₂×C₈

—

105

C₂×C₈

—

356

—

106

C₂×C₈

C₂×C₂²⋊D₈₉

yes

characteristic

107

C₂×C₈

#13083574

yes

characteristic

108

C₄⋊C₄

—

109

C₄⋊C₄

—

356

—

110

C₄⋊C₄

—

356

—

111

C₄⋊C₄

D₇₁₂

yes

characteristic

112

C₄²

—

113

C₄²

C₂×D₃₅₆

yes

characteristic

114

C₁₆

—

115

C₁₆

—

116

C₁₆

#13083564

yes

—

117

Q₈

—

118

Q₈

—

1424

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

712

—

122

C₂×C₄

#15667986

yes

characteristic

123

C₂×C₄

C₂×D₇₁₂

yes

characteristic

124

C₂×C₄

#15668034

yes

characteristic

125

C₈

—

126

C₈

—

127

C₈

—

712

—

128

C₈

C₂×#13083564

yes

characteristic

129

C₈

#15668042

yes

characteristic

130

C₂²

#19151834

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

1424

—

135

C₄

#19151808

yes

characteristic

136

C₄

#19151873

yes

characteristic

137

C₄

#19152041

yes

—

138

C₂

#26019646

yes

characteristic

139

C₂

#26022360

yes

characteristic

140

C₂

#26022372

yes

characteristic

	subgroup	G/H
center	C₂²	#19151834
commutator, cosocle	C₄×C₈×C₈₉	C₂³
Frattini	C₄×C₈	C₂²×D₈₉
Fitting	C₈₉×#2183	C₂
socle, prime radical	C₂²×C₈₉	#466
hypercenter	#2183	D₈₉
hypocenter	C₈₉	#12461
LCS	G ▷ C₄×C₈×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈ ◁ #2183

subgroup

G/H

center

C₂²

#19151834

commutator, cosocle

C₄×C₈×C₈₉

C₂³

Frattini

C₄×C₈

C₂²×D₈₉

Fitting

C₈₉×#2183

C₂

C₂²×C₈₉

D₈₉

C₈₉

G ▷ C₄×C₈×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈ ◁ #2183

N	H ≅ G/N	type
76. C₈₉	#12461	semidirect

H ≅ G/N

type

76. C₈₉

#12461

semidirect

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

subgroup	index
#26022527	2
#26019656	2
#26018326	2
#26018309	2
C₈₉×#2183	2
#12461	89

subgroup

index

C₈₉×#2183

Subgroups — #36468809

Conjugacy classes of subgroups

Complements

Maximal subgroups