Group #36466161 — subgroups

Subgroups — #36466161

Order 22784.

226 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 85 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	C₄×Dic₇₁₂	C₂	1	1	yes	maximal, characteristic
2	#26019058	C₂	1	2	yes	maximal
3	#26018215	C₂	1	2	yes	maximal
4	C₁₆×#10623494	C₂	1	1	yes	maximal, characteristic
5	C₈₉×#1371	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₇₁₂	C₄	1	1	yes	characteristic
7	#19151722	—	2	2	no	—
8	C₄×Dic₃₅₆	C₂²	1	2	yes	—
9	#19151424	C₄	1	1	yes	characteristic
10	#19151403	C₄	1	2	yes	—
11	C₄×#13083549	C₂²	1	1	yes	characteristic
12	#19151328	—	2	2	no	—
13	C₈×#10623494	C₂²	1	1	yes	characteristic
14	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
15	#335×C₈₉	C₂²	1	2	yes	—
16	C₂×Dic₃₅₆	C₂×C₄	1	2	yes	—
17	C₄×Dic₁₇₈	—	2	2	no	—
18	Dic₇₁₂	C₈	1	4	yes	complemented
19	C₂×#13083549	—	2	1	no	—
20	#15667946	C₂×C₄	1	2	yes	—
21	#15667941	—	4	2	no	—
22	#15667937	—	2	2	no	—
23	C₄×#10623494	D₄	1	1	yes	characteristic
24	C₄×#10623494	D₄	1	1	yes	characteristic
25	C₈×Dic₈₉	—	4	2	no	—
26	C₂×C₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
27	C₂×C₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
28	C₄⋊C₈×C₈₉	—	2	2	no	—
29	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
30	C₂×Dic₁₇₈	—	2	2	no	—
31	Dic₃₅₆	—	2	2	no	—
32	Dic₃₅₆	C₂×C₈	1	4	yes	—
33	C₂×#10623494	—	2	1	no	—
34	C₂×#10623494	D₈	1	1	yes	characteristic
35	C₂×#10623494	D₈	1	1	yes	characteristic
36	C₄⋊Dic₈₉	—	2	2	no	—
37	#13083553	—	4	2	no	—
38	C₄×Dic₈₉	—	4	2	no	—
39	#13083549	—	2	2	no	—
40	C₂×C₈×C₈₉	—	4	2	no	—
41	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
43	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
44	C₁₆×C₈₉	—	2	1	no	—
45	C₁₆×C₈₉	C₂×C₈	1	2	yes	—
46	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
47	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
48	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
49	C₂×Dic₈₉	—	4	2	no	—
50	C₂×Dic₈₉	—	4	2	no	—
51	Dic₁₇₈	—	2	4	no	—
52	Dic₁₇₈	—	4	2	no	—
53	C₈×C₈₉	—	2	1	no	—
54	C₈×C₈₉	—	8	2	no	—
55	C₈×C₈₉	C₂²×C₈	1	1	yes	characteristic
56	C₈×C₈₉	#131	1	1	yes	characteristic
57	#10623494	—	2	2	no	—
58	#10623494	—	2	1	no	—
59	#10623494	Q₃₂	1	2	yes	complemented
60	C₂²×C₈₉	C₄×D₈	1	1	yes	characteristic
61	C₄×C₈₉	—	2	1	no	—
62	C₄×C₈₉	C₈×D₄	1	1	yes	characteristic
63	C₄×C₈₉	#443	1	1	yes	characteristic
64	C₄×C₈₉	C₂×Q₃₂	1	1	yes	characteristic
65	C₄×C₈₉	#508	1	1	yes	characteristic
66	Dic₈₉	—	4	4	no	—
67	Dic₈₉	—	8	2	no	—
68	C₈×Q₃₂	—	89	1	no	maximal, Sylow
69	C₂×C₈₉	C₈×D₈	1	1	yes	characteristic
70	C₂×C₈₉	C₄×Q₃₂	1	1	yes	characteristic
71	C₂×C₈₉	#2178	1	1	yes	characteristic
72	C₄×Q₃₂	—	89	1	no	—
73	C₈×Q₁₆	—	89	2	no	—
74	#1371	D₈₉	1	1	yes	characteristic
75	#1332	—	89	2	no	—
76	C₈×C₁₆	—	89	1	no	—
77	C₈₉	C₈×Q₃₂	1	1	yes	characteristic, complemented, Sylow
78	C₂×Q₃₂	—	89	1	no	—
79	C₈×Q₈	—	178	2	no	—
80	C₄×Q₁₆	—	89	2	no	—
81	#366	—	89	1	no	—
82	#358	—	89	2	no	—
83	C₄×C₁₆	—	89	1	no	—
84	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
85	#335	C₂×D₈₉	1	2	yes	—
86	Q₈⋊C₈	—	178	2	no	—
87	C₈²	—	89	1	no	—
88	C₂×Q₁₆	—	89	2	no	—
89	C₄×Q₈	—	178	2	no	—
90	Q₃₂	—	89	4	no	—
91	C₂×C₁₆	—	178	1	no	—
92	C₂×C₁₆	C₄×D₈₉	1	1	yes	characteristic
93	C₂×C₁₆	C₄×D₈₉	1	1	yes	characteristic
94	#107	—	89	2	no	—
95	C₄⋊C₈	—	2	2	no	—
96	C₄⋊C₈	—	356	2	no	—
97	#103	—	178	2	no	—
98	C₄×C₈	—	89	1	no	—
99	C₄×C₈	—	89	1	no	—
100	C₄×C₈	—	356	2	no	—
101	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
102	C₂×Q₈	—	178	2	no	—
103	Q₁₆	—	89	4	no	—
104	Q₁₆	—	178	2	no	—
105	C₂×C₈	—	4	2	no	—
106	C₂×C₈	—	89	1	no	—
107	C₂×C₈	—	89	1	no	—
108	C₂×C₈	—	178	1	no	—
109	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
110	C₂×C₈	#13083575	1	1	yes	characteristic
111	C₄⋊C₄	—	356	2	no	—
112	C₄⋊C₄	—	178	2	no	—
113	C₄²	—	356	2	no	—
114	C₄²	D₄×D₈₉	1	1	yes	characteristic
115	C₁₆	—	2	1	no	—
116	C₁₆	—	178	2	no	—
117	C₁₆	C₈×D₈₉	1	2	yes	—
118	Q₈	—	356	2	no	—
119	Q₈	—	178	4	no	—
120	C₂×C₄	—	356	2	no	—
121	C₂×C₄	—	356	2	no	—
122	C₂×C₄	#15667974	1	1	yes	characteristic
123	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
124	C₂×C₄	#15668035	1	1	yes	characteristic
125	C₈	—	2	1	no	—
126	C₈	—	8	2	no	—
127	C₈	—	89	2	no	—
128	C₈	—	178	2	no	—
129	C₈	—	178	1	no	—
130	C₈	C₂×C₈×D₈₉	1	1	yes	characteristic
131	C₈	#15668020	1	1	yes	characteristic
132	C₂²	#19151680	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	356	4	no	—
135	C₄	—	712	2	no	—
136	C₄	#19151663	1	1	yes	characteristic
137	C₄	#19151714	1	1	yes	characteristic
138	C₄	Q₃₂×D₈₉	1	1	yes	characteristic
139	C₄	#19152027	1	1	yes	characteristic
140	C₂	#26019046	1	1	yes	characteristic
141	C₂	#26022110	1	1	yes	characteristic
142	C₂	#26022132	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×Dic₇₁₂

C₂

yes

maximal, characteristic

#26019058

C₂

yes

maximal

#26018215

C₂

yes

maximal

C₁₆×#10623494

C₂

yes

maximal, characteristic

C₈₉×#1371

C₂

yes

maximal, characteristic

C₂×Dic₇₁₂

C₄

yes

characteristic

#19151722

—

C₄×Dic₃₅₆

C₂²

yes

—

#19151424

C₄

yes

characteristic

#19151403

C₄

yes

—

C₄×#13083549

C₂²

yes

characteristic

#19151328

—

C₈×#10623494

C₂²

yes

characteristic

C₄×C₁₆×C₈₉

C₂²

yes

characteristic

#335×C₈₉

C₂²

yes

—

C₂×Dic₃₅₆

C₂×C₄

yes

—

C₄×Dic₁₇₈

—

Dic₇₁₂

C₈

yes

complemented

C₂×#13083549

—

#15667946

C₂×C₄

yes

—

#15667941

—

#15667937

—

C₄×#10623494

D₄

yes

characteristic

C₄×#10623494

D₄

yes

characteristic

C₈×Dic₈₉

—

C₂×C₁₆×C₈₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₈₉

C₂×C₄

yes

characteristic

C₄⋊C₈×C₈₉

—

C₄×C₈×C₈₉

C₂³

yes

characteristic

C₂×Dic₁₇₈

—

Dic₃₅₆

—

Dic₃₅₆

C₂×C₈

yes

—

C₂×#10623494

—

C₂×#10623494

D₈

yes

characteristic

C₂×#10623494

D₈

yes

characteristic

C₄⋊Dic₈₉

—

#13083553

—

C₄×Dic₈₉

—

#13083549

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₈₉

#41

yes

characteristic

C₄²×C₈₉

C₂×D₄

yes

characteristic

C₁₆×C₈₉

—

C₁₆×C₈₉

C₂×C₈

yes

—

C₂×C₄×C₈₉

C₄×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₈

yes

characteristic

C₂×C₄×C₈₉

#135

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

—

C₈×C₈₉

C₂²×C₈

yes

C₈×C₈₉

yes

—

—

Q₃₂

yes

complemented

C₂²×C₈₉

C₄×D₈

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

C₈×D₄

yes

characteristic

C₄×C₈₉

#443

yes

characteristic

C₄×C₈₉

C₂×Q₃₂

yes

characteristic

C₄×C₈₉

#508

yes

characteristic

Dic₈₉

—

Dic₈₉

—

C₈×Q₃₂

—

maximal, Sylow

C₂×C₈₉

C₈×D₈

yes

characteristic

C₂×C₈₉

C₄×Q₃₂

yes

characteristic

C₂×C₈₉

#2178

yes

characteristic

C₄×Q₃₂

—

C₈×Q₁₆

—

#1371

D₈₉

yes

characteristic

#1332

—

C₈×C₁₆

—

C₈₉

C₈×Q₃₂

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

C₈×Q₈

—

178

—

C₄×Q₁₆

—

#366

—

#358

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₈₉

yes

characteristic

#335

C₂×D₈₉

yes

—

Q₈⋊C₈

—

178

—

C₈²

—

C₂×Q₁₆

—

C₄×Q₈

—

178

—

Q₃₂

—

C₂×C₁₆

—

178

—

C₂×C₁₆

C₄×D₈₉

yes

characteristic

C₂×C₁₆

C₄×D₈₉

yes

characteristic

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

356

—

#103

—

178

—

C₄×C₈

—

C₄×C₈

—

100

C₄×C₈

—

356

—

101

C₄×C₈

C₂²×D₈₉

yes

characteristic

102

C₂×Q₈

—

178

—

103

Q₁₆

—

104

Q₁₆

—

178

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

178

—

109

C₂×C₈

C₂×C₄×D₈₉

yes

characteristic

110

C₂×C₈

#13083575

yes

characteristic

111

C₄⋊C₄

—

356

—

112

C₄⋊C₄

—

178

—

113

C₄²

—

356

—

114

C₄²

D₄×D₈₉

yes

characteristic

115

C₁₆

—

116

C₁₆

—

178

—

117

C₁₆

C₈×D₈₉

yes

—

118

Q₈

—

356

—

119

Q₈

—

178

—

120

C₂×C₄

—

356

—

121

C₂×C₄

—

356

—

122

C₂×C₄

#15667974

yes

characteristic

123

C₂×C₄

D₈×D₈₉

yes

characteristic

124

C₂×C₄

#15668035

yes

characteristic

125

C₈

—

126

C₈

—

127

C₈

—

128

C₈

—

178

—

129

C₈

—

178

—

130

C₈

C₂×C₈×D₈₉

yes

characteristic

131

C₈

#15668020

yes

characteristic

132

C₂²

#19151680

yes

characteristic

133

C₄

—

134

C₄

—

356

—

135

C₄

—

712

—

136

C₄

#19151663

yes

characteristic

137

C₄

#19151714

yes

characteristic

138

C₄

Q₃₂×D₈₉

yes

characteristic

139

C₄

#19152027

yes

characteristic

140

C₂

#26019046

yes

characteristic

141

C₂

#26022110

yes

characteristic

142

C₂

#26022132

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₈×D₈₉
commutator	C₈×C₈₉	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₈₉
Fitting	C₈₉×#1371	C₂
socle, prime radical	C₂²×C₈₉	C₄×D₈
cosocle	C₄×C₈×C₈₉	C₂³
hypercenter	#1371	D₈₉
hypocenter	C₈₉	C₈×Q₃₂
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ #1371

subgroup

G/H

center

C₂×C₄

D₈×D₈₉

commutator

C₈×C₈₉

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₈₉

Fitting

C₈₉×#1371

C₂

socle, prime radical

C₂²×C₈₉

C₄×D₈

cosocle

C₄×C₈×C₈₉

C₂³

hypercenter

#1371

D₈₉

hypocenter

C₈₉

C₈×Q₃₂

LCS

G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ #1371

N	H ≅ G/N	type
18. Dic₇₁₂	C₈	semidirect
59. #10623494	Q₃₂	semidirect
77. C₈₉	C₈×Q₃₂	semidirect

H ≅ G/N

type

18. Dic₇₁₂

C₈

semidirect

59. #10623494

Q₃₂

semidirect

77. C₈₉

C₈×Q₃₂

semidirect

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

subgroup	index
C₄×Dic₇₁₂	2
#26019058	2
#26018215	2
C₁₆×#10623494	2
C₈₉×#1371	2
C₈×Q₃₂	89

subgroup

index

C₄×Dic₇₁₂

#26019058

#26018215

C₁₆×#10623494

C₈₉×#1371

C₈×Q₃₂

Subgroups — #36466161

Conjugacy classes of subgroups

Complements

Maximal subgroups