Group #36466280 — subgroups

Subgroups — #36466280

Order 22784.

200 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 87 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	#26022160	C₂	1	2	yes	maximal
2	C₈×Dic₃₅₆	C₂	1	1	yes	maximal, characteristic
3	#26018316	C₂	1	1	yes	maximal, characteristic
4	#26018232	C₂	1	2	yes	maximal
5	C₈₉×#1312	C₂	1	1	yes	maximal, characteristic
6	C₈×Dic₁₇₈	C₂²	1	2	yes	—
7	C₄×Dic₃₅₆	C₄	1	1	yes	characteristic
8	C₄⋊#13083549	C₂²	1	2	yes	—
9	#19151416	—	2	2	no	—
10	#19151373	—	2	2	no	—
11	#19151351	C₄	1	1	yes	characteristic
12	#19151330	C₄	1	2	yes	—
13	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
14	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
15	C₈²×C₈₉	C₂²	1	1	yes	characteristic
16	C₂×Dic₃₅₆	C₈	1	1	yes	characteristic
17	C₄×Dic₁₇₈	C₂×C₄	1	2	yes	—
18	C₂×#13083549	—	2	2	no	—
19	#15667946	C₈	1	1	yes	characteristic
20	C₄⋊#10623494	C₂×C₄	1	2	yes	—
21	#15667941	—	2	2	no	—
22	#15667937	C₈	1	2	yes	—
23	C₈×Dic₈₉	—	2	2	no	—
24	C₂×C₁₆×C₈₉	—	2	1	no	—
25	C₂×C₁₆×C₈₉	D₄	1	2	yes	—
26	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
29	C₂×Dic₁₇₈	C₂×C₈	1	2	yes	—
30	Dic₃₅₆	—	2	2	no	—
31	C₂×#10623494	—	2	2	no	—
32	C₄⋊Dic₈₉	C₂×C₈	1	2	yes	—
33	#13083553	—	2	2	no	—
34	C₄×Dic₈₉	—	2	2	no	—
35	#13083549	—	4	2	no	—
36	C₂×C₈×C₈₉	—	2	1	no	—
37	C₂×C₈×C₈₉	C₂×C₈	1	1	yes	characteristic
38	C₂×C₈×C₈₉	C₂×C₈	1	1	yes	characteristic
39	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
40	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
41	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
42	C₁₆×C₈₉	—	2	2	no	—
43	C₁₆×C₈₉	—	4	1	no	—
44	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
45	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
46	C₂×C₄×C₈₉	#131	1	1	yes	characteristic
47	C₂×Dic₈₉	—	2	2	no	—
48	C₂×Dic₈₉	—	2	2	no	—
49	Dic₁₇₈	—	2	2	no	—
50	Dic₁₇₈	—	2	2	no	—
51	C₈×C₈₉	—	2	1	no	—
52	C₈×C₈₉	—	2	1	no	—
53	C₈×C₈₉	—	4	1	no	—
54	C₈×C₈₉	M₃₂	1	2	yes	—
55	C₈×C₈₉	#137	1	2	yes	—
56	#10623494	—	4	2	no	—
57	C₂²×C₈₉	C₈×D₄	1	1	yes	characteristic
58	C₄×C₈₉	—	2	1	no	—
59	C₄×C₈₉	#441	1	1	yes	characteristic
60	C₄×C₈₉	#444	1	1	yes	characteristic
61	C₄×C₈₉	C₂×M₃₂	1	1	yes	characteristic
62	C₄×C₈₉	#504	1	1	yes	characteristic
63	Dic₈₉	—	4	2	no	—
64	Dic₈₉	—	4	2	no	—
65	#12061	—	89	1	no	maximal, Sylow
66	C₂×C₈₉	#1579	1	1	yes	characteristic
67	C₂×C₈₉	#2169	1	1	yes	characteristic
68	C₂×C₈₉	#2171	1	1	yes	characteristic
69	#2183	—	89	2	no	—
70	C₈×Q₁₆	—	89	1	no	—
71	#1367	—	89	1	no	—
72	Q₈⋊C₁₆	—	89	2	no	—
73	#1312	D₈₉	1	1	yes	characteristic
74	C₈₉	#12061	1	1	yes	characteristic, complemented, Sylow
75	C₈×Q₈	—	89	2	no	—
76	C₄×Q₁₆	—	89	1	no	—
77	C₄⋊C₁₆	—	89	2	no	—
78	C₄⋊C₁₆	—	178	2	no	—
79	#346	—	178	2	no	—
80	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
81	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
82	#335	—	89	1	no	—
83	Q₈⋊C₈	—	89	2	no	—
84	C₈²	C₂×D₈₉	1	1	yes	characteristic
85	C₂×Q₁₆	—	89	1	no	—
86	C₄×Q₈	—	89	2	no	—
87	C₂×C₁₆	—	2	1	no	—
88	C₂×C₁₆	—	178	2	no	—
89	C₂×C₁₆	D₃₅₆	1	2	yes	—
90	#107	—	89	1	no	—
91	C₄⋊C₈	—	89	2	no	—
92	C₄⋊C₈	—	178	2	no	—
93	#103	—	89	2	no	—
94	C₄×C₈	—	178	2	no	—
95	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
96	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
97	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
98	C₂×Q₈	—	89	2	no	—
99	Q₁₆	—	178	2	no	—
100	C₂×C₈	—	2	1	no	—
101	C₂×C₈	—	178	2	no	—
102	C₂×C₈	C₈×D₈₉	1	1	yes	characteristic
103	C₂×C₈	C₈×D₈₉	1	1	yes	characteristic
104	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
105	C₂×C₈	#13083574	1	1	yes	characteristic
106	C₄⋊C₄	—	89	2	no	—
107	C₄⋊C₄	—	178	2	no	—
108	C₄²	—	178	2	no	—
109	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
110	C₁₆	—	2	2	no	—
111	C₁₆	—	4	1	no	—
112	C₁₆	—	356	2	no	—
113	Q₈	—	178	2	no	—
114	Q₈	—	178	2	no	—
115	C₂×C₄	—	178	2	no	—
116	C₂×C₄	—	178	2	no	—
117	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
118	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
119	C₂×C₄	#15668018	1	1	yes	characteristic
120	C₈	—	2	1	no	—
121	C₈	—	2	1	no	—
122	C₈	—	4	1	no	—
123	C₈	—	356	2	no	—
124	C₈	#15667951	1	2	yes	—
125	C₈	#15668045	1	2	yes	—
126	C₂²	C₈×D₃₅₆	1	1	yes	characteristic
127	C₄	—	2	1	no	—
128	C₄	—	356	2	no	—
129	C₄	—	356	2	no	—
130	C₄	#19151706	1	1	yes	characteristic
131	C₄	#19151721	1	1	yes	characteristic
132	C₄	C₂×#15667951	1	1	yes	characteristic
133	C₄	#19152014	1	1	yes	characteristic
134	C₂	#26019072	1	1	yes	characteristic
135	C₂	#26022085	1	1	yes	characteristic
136	C₂	#26022095	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26022160

C₂

yes

maximal

C₈×Dic₃₅₆

C₂

yes

maximal, characteristic

#26018316

C₂

yes

maximal, characteristic

#26018232

C₂

yes

maximal

C₈₉×#1312

C₂

yes

maximal, characteristic

C₈×Dic₁₇₈

C₂²

yes

—

C₄×Dic₃₅₆

C₄

yes

characteristic

C₄⋊#13083549

C₂²

yes

—

#19151416

—

#19151373

—

#19151351

C₄

yes

characteristic

#19151330

C₄

yes

—

C₄×C₁₆×C₈₉

C₂²

yes

characteristic

C₄×C₁₆×C₈₉

C₂²

yes

characteristic

C₈²×C₈₉

C₂²

yes

characteristic

C₂×Dic₃₅₆

C₈

yes

characteristic

C₄×Dic₁₇₈

C₂×C₄

yes

—

C₂×#13083549

—

#15667946

C₈

yes

characteristic

C₄⋊#10623494

C₂×C₄

yes

—

#15667941

—

#15667937

C₈

yes

—

C₈×Dic₈₉

—

C₂×C₁₆×C₈₉

—

C₂×C₁₆×C₈₉

D₄

yes

—

C₄×C₈×C₈₉

C₂×C₄

yes

characteristic

C₄×C₈×C₈₉

C₂×C₄

yes

characteristic

C₄×C₈×C₈₉

C₂³

yes

characteristic

C₂×Dic₁₇₈

C₂×C₈

yes

—

Dic₃₅₆

—

C₂×#10623494

—

C₄⋊Dic₈₉

C₂×C₈

yes

—

#13083553

—

C₄×Dic₈₉

—

#13083549

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂×C₈

yes

characteristic

C₂×C₈×C₈₉

C₂×C₈

yes

characteristic

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

#41

yes

characteristic

C₄²×C₈₉

C₂²×C₄

yes

characteristic

C₁₆×C₈₉

—

C₁₆×C₈₉

—

C₂×C₄×C₈₉

C₄×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂²×C₈

yes

characteristic

C₂×C₄×C₈₉

#131

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

—

C₈×C₈₉

—

C₈×C₈₉

M₃₂

yes

—

C₈×C₈₉

#137

yes

—

#10623494

—

C₂²×C₈₉

C₈×D₄

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

C₂×M₃₂

yes

characteristic

C₄×C₈₉

#504

yes

characteristic

Dic₈₉

—

Dic₈₉

—

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₈×Q₁₆

—

#1367

—

Q₈⋊C₁₆

—

#1312

D₈₉

yes

characteristic

C₈₉

#12061

yes

characteristic, complemented, Sylow

C₈×Q₈

—

C₄×Q₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

178

—

#346

—

178

—

C₄×C₁₆

C₂×D₈₉

yes

characteristic

C₄×C₁₆

C₂×D₈₉

yes

characteristic

#335

—

Q₈⋊C₈

—

C₈²

C₂×D₈₉

yes

characteristic

C₂×Q₁₆

—

C₄×Q₈

—

C₂×C₁₆

—

C₂×C₁₆

—

178

—

C₂×C₁₆

D₃₅₆

yes

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

178

—

#103

—

C₄×C₈

—

178

—

C₄×C₈

C₄×D₈₉

yes

characteristic

C₄×C₈

C₄×D₈₉

yes

characteristic

C₄×C₈

C₂²×D₈₉

yes

characteristic

C₂×Q₈

—

Q₁₆

—

178

—

100

C₂×C₈

—

101

C₂×C₈

—

178

—

102

C₂×C₈

C₈×D₈₉

yes

characteristic

103

C₂×C₈

C₈×D₈₉

yes

characteristic

104

C₂×C₈

C₂×D₃₅₆

yes

characteristic

105

C₂×C₈

#13083574

yes

characteristic

106

C₄⋊C₄

—

107

C₄⋊C₄

—

178

—

108

C₄²

—

178

—

109

C₄²

C₂×C₄×D₈₉

yes

characteristic

110

C₁₆

—

111

C₁₆

—

112

C₁₆

—

356

—

113

Q₈

—

178

—

114

Q₈

—

178

—

115

C₂×C₄

—

178

—

116

C₂×C₄

—

178

—

117

C₂×C₄

C₄×D₃₅₆

yes

characteristic

118

C₂×C₄

C₂×C₈×D₈₉

yes

characteristic

119

C₂×C₄

#15668018

yes

characteristic

120

C₈

—

121

C₈

—

122

C₈

—

123

C₈

—

356

—

124

C₈

#15667951

yes

—

125

C₈

#15668045

yes

—

126

C₂²

C₈×D₃₅₆

yes

characteristic

127

C₄

—

128

C₄

—

356

—

129

C₄

—

356

—

130

C₄

#19151706

yes

characteristic

131

C₄

#19151721

yes

characteristic

132

C₄

C₂×#15667951

yes

characteristic

133

C₄

#19152014

yes

characteristic

134

C₂

#26019072

yes

characteristic

135

C₂

#26022085

yes

characteristic

136

C₂

#26022095

yes

characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₃₅₆
commutator	C₂×C₄×C₈₉	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₈₉
Fitting	C₈₉×#1312	C₂
socle, prime radical	C₂²×C₈₉	C₈×D₄
cosocle	C₄×C₈×C₈₉	C₂³
hypercenter	#1312	D₈₉
hypocenter	C₈₉	#12061
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₁₆ ◁ #1312

subgroup

G/H

center

C₂×C₈

C₂×D₃₅₆

commutator

C₂×C₄×C₈₉

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₈₉

Fitting

C₈₉×#1312

C₂

socle, prime radical

C₂²×C₈₉

C₈×D₄

cosocle

C₄×C₈×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂×C₈ ◁ C₄×C₁₆ ◁ #1312

N	H ≅ G/N	type
74. C₈₉	#12061	semidirect

H ≅ G/N

type

74. C₈₉

#12061

semidirect

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

subgroup	index
#26022160	2
C₈×Dic₃₅₆	2
#26018316	2
#26018232	2
C₈₉×#1312	2
#12061	89

subgroup

index

C₈×Dic₃₅₆

C₈₉×#1312

Subgroups — #36466280

Conjugacy classes of subgroups

Complements

Maximal subgroups