Group #36466436 — subgroups

Subgroups — #36466436

Order 22784.

234 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 83 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	#26019137	C₂	1	2	yes	maximal, complemented
2	#26018326	C₂	1	1	yes	maximal, characteristic, complemented
3	#26018213	C₂	1	2	yes	maximal, complemented
4	#26018168	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×D₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
6	#19151676	—	2	2	no	—
7	C₄×#13083549	C₂²	1	1	yes	characteristic
8	#19151351	C₂²	1	2	yes	—
9	D₄⋊#10623494	—	2	2	no	—
10	C₈×#10623494	C₂²	1	1	yes	characteristic
11	C₂×D₁₆×C₈₉	C₄	1	1	yes	characteristic
12	C₄×D₈×C₈₉	C₂²	1	2	yes	—
13	#366×C₈₉	C₄	1	1	yes	characteristic
14	#357×C₈₉	C₄	1	2	yes	—
15	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
16	C₂×#13083558	—	4	2	no	—
17	C₂×#13083549	—	2	1	no	—
18	C₄⋊#10623494	—	2	2	no	—
19	C₂²⋊#10623494	—	4	2	no	—
20	C₄×#10623494	D₄	1	1	yes	characteristic
21	C₄×#10623494	D₄	1	1	yes	characteristic
22	C₂×D₈×C₈₉	C₂×C₄	1	2	yes	—
23	C₄×D₄×C₈₉	—	2	2	no	—
24	D₁₆×C₈₉	—	2	2	no	—
25	C₂×C₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
27	#107×C₈₉	C₂×C₄	1	2	yes	—
28	#102×C₈₉	—	2	2	no	—
29	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
30	#13083558	—	8	4	no	—
31	C₂×#10623494	—	2	1	no	—
32	C₂×#10623494	—	4	2	no	—
33	C₂×#10623494	D₈	1	2	yes	—
34	#13083549	—	2	2	no	—
35	C₂×D₄×C₈₉	—	2	2	no	—
36	C₂²×C₄×C₈₉	—	4	2	no	—
37	D₈×C₈₉	—	2	2	no	—
38	D₈×C₈₉	—	2	2	no	—
39	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
41	C₄⋊C₄×C₈₉	—	2	2	no	—
42	C₂²⋊C₄×C₈₉	—	4	2	no	—
43	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
44	C₁₆×C₈₉	—	2	1	no	—
45	C₁₆×C₈₉	M₁₆	1	2	yes	complemented
46	C₂³×C₈₉	—	4	2	no	—
47	D₄×C₈₉	—	4	2	no	—
48	D₄×C₈₉	—	4	2	no	—
49	C₂×C₄×C₈₉	—	4	2	no	—
50	C₂×C₄×C₈₉	—	8	2	no	—
51	C₂×C₄×C₈₉	—	8	2	no	—
52	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
53	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
54	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
55	C₈×C₈₉	—	2	1	no	—
56	C₈×C₈₉	C₂×M₁₆	1	1	yes	characteristic
57	C₈×C₈₉	#131	1	1	yes	characteristic
58	#10623494	—	2	2	no	—
59	#10623494	—	4	1	no	—
60	#10623494	—	8	2	no	—
61	C₂²×C₈₉	—	8	2	no	—
62	C₂²×C₈₉	—	8	2	no	—
63	C₂²×C₈₉	—	8	2	no	—
64	C₂²×C₈₉	C₄×D₈	1	1	yes	characteristic
65	C₄×C₈₉	—	2	1	no	—
66	C₄×C₈₉	—	8	2	no	—
67	C₄×C₈₉	#436	1	1	yes	characteristic
68	C₄×C₈₉	#443	1	1	yes	characteristic
69	C₄×C₈₉	#509	1	2	yes	—
70	#12086	—	89	1	no	maximal, Sylow
71	C₂×C₈₉	—	16	2	no	—
72	C₂×C₈₉	#1589	1	1	yes	characteristic
73	C₂×C₈₉	#2177	1	1	yes	characteristic
74	C₂×C₈₉	#2179	1	1	yes	characteristic
75	C₄×D₁₆	D₈₉	1	1	yes	characteristic
76	#1589	—	89	2	no	—
77	#1372	—	89	1	no	—
78	#1331	—	89	2	no	—
79	#1311	—	89	1	no	—
80	C₈₉	#12086	1	1	yes	characteristic, complemented, Sylow
81	C₂×D₁₆	Dic₈₉	1	1	yes	characteristic
82	C₄×D₈	C₂×D₈₉	1	2	yes	—
83	#436	—	178	2	no	—
84	#366	Dic₈₉	1	1	yes	characteristic
85	#357	Dic₈₉	1	2	yes	—
86	C₄×C₁₆	—	89	1	no	—
87	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
88	#335	—	89	2	no	—
89	D₄⋊C₈	—	178	2	no	—
90	C₈²	—	89	1	no	—
91	C₂×D₈	C₂×Dic₈₉	1	2	yes	—
92	C₂×M₁₆	—	356	2	no	—
93	C₄×D₄	—	2	2	no	—
94	D₁₆	—	2	2	no	—
95	C₂×C₁₆	—	178	1	no	—
96	C₂×C₁₆	C₂×Dic₈₉	1	1	yes	characteristic
97	C₂×C₁₆	C₂×Dic₈₉	1	1	yes	characteristic
98	#107	C₂×Dic₈₉	1	2	yes	—
99	C₄⋊C₈	—	178	2	no	—
100	#102	—	2	2	no	—
101	C₂²⋊C₈	—	356	2	no	—
102	C₄×C₈	—	89	1	no	—
103	C₄×C₈	—	89	1	no	—
104	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
105	C₂×D₄	—	2	2	no	—
106	C₂²×C₄	—	4	2	no	—
107	D₈	—	2	2	no	—
108	D₈	—	2	2	no	—
109	M₁₆	—	712	4	no	—
110	C₂×C₈	—	89	2	no	—
111	C₂×C₈	—	356	2	no	—
112	C₂×C₈	—	178	1	no	—
113	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
114	C₂×C₈	#13083573	1	1	yes	characteristic
115	C₄⋊C₄	—	2	2	no	—
116	C₂²⋊C₄	—	4	2	no	—
117	C₄²	D₄×D₈₉	1	1	yes	characteristic
118	C₁₆	—	2	1	no	—
119	C₁₆	—	178	2	no	—
120	C₁₆	#13083558	1	2	yes	complemented
121	C₂³	—	4	2	no	—
122	D₄	—	4	2	no	—
123	D₄	—	4	2	no	—
124	C₂×C₄	—	4	2	no	—
125	C₂×C₄	—	8	2	no	—
126	C₂×C₄	—	8	2	no	—
127	C₂×C₄	D₄×Dic₈₉	1	1	yes	characteristic
128	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
129	C₂×C₄	#15668031	1	1	yes	characteristic
130	C₈	—	2	1	no	—
131	C₈	—	356	1	no	—
132	C₈	—	178	2	no	—
133	C₈	—	712	2	no	—
134	C₈	C₂×#13083558	1	1	yes	characteristic
135	C₈	#15668019	1	1	yes	characteristic
136	C₂²	—	8	2	no	—
137	C₂²	—	8	2	no	—
138	C₂²	—	8	2	no	—
139	C₂²	D₈×Dic₈₉	1	1	yes	characteristic
140	C₄	—	2	1	no	—
141	C₄	—	8	2	no	—
142	C₄	#19151676	1	1	yes	characteristic
143	C₄	#19151715	1	1	yes	characteristic
144	C₄	#19152032	1	2	yes	—
145	C₂	—	16	2	no	—
146	C₂	#26019137	1	1	yes	characteristic
147	C₂	#26022128	1	1	yes	characteristic
148	C₂	#26022138	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26019137

C₂

yes

maximal, complemented

#26018326

C₂

yes

maximal, characteristic, complemented

#26018213

C₂

yes

maximal, complemented

#26018168

C₂

yes

maximal, characteristic, complemented

C₄×D₁₆×C₈₉

C₂

yes

maximal, characteristic

#19151676

—

C₄×#13083549

C₂²

yes

characteristic

#19151351

C₂²

yes

—

D₄⋊#10623494

—

C₈×#10623494

C₂²

yes

characteristic

C₂×D₁₆×C₈₉

C₄

yes

characteristic

C₄×D₈×C₈₉

C₂²

yes

—

#366×C₈₉

C₄

yes

characteristic

#357×C₈₉

C₄

yes

—

C₄×C₁₆×C₈₉

C₂²

yes

—

—

—

—

D₄

yes

characteristic

C₄×#10623494

D₄

yes

characteristic

C₂×D₈×C₈₉

C₂×C₄

yes

—

C₄×D₄×C₈₉

—

D₁₆×C₈₉

—

C₂×C₁₆×C₈₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₈₉

C₂×C₄

yes

characteristic

#107×C₈₉

C₂×C₄

yes

—

#102×C₈₉

—

C₄×C₈×C₈₉

C₂³

yes

—

—

—

D₈

yes

—

#13083549

—

C₂×D₄×C₈₉

—

C₂²×C₄×C₈₉

—

D₈×C₈₉

—

D₈×C₈₉

—

C₂×C₈×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₈₉

#41

yes

characteristic

C₄⋊C₄×C₈₉

—

C₂²⋊C₄×C₈₉

—

C₄²×C₈₉

C₂×D₄

yes

characteristic

C₁₆×C₈₉

—

C₁₆×C₈₉

M₁₆

yes

complemented

C₂³×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₄×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₈

yes

characteristic

C₂×C₄×C₈₉

#135

yes

characteristic

C₈×C₈₉

—

C₈×C₈₉

C₂×M₁₆

yes

C₈×C₈₉

yes

—

—

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

C₄×D₈

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

—

#12086

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

C₄×D₁₆

D₈₉

yes

—

—

—

—

C₈₉

#12086

yes

characteristic, complemented, Sylow

C₂×D₁₆

Dic₈₉

yes

characteristic

C₄×D₈

C₂×D₈₉

yes

—

#436

—

178

—

#366

Dic₈₉

yes

characteristic

#357

Dic₈₉

yes

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₈₉

yes

characteristic

#335

—

D₄⋊C₈

—

178

—

C₈²

—

C₂×D₈

C₂×Dic₈₉

yes

—

C₂×M₁₆

—

356

—

C₄×D₄

—

D₁₆

—

C₂×C₁₆

—

178

—

C₂×C₁₆

C₂×Dic₈₉

yes

characteristic

C₂×C₁₆

C₂×Dic₈₉

yes

characteristic

#107

C₂×Dic₈₉

yes

—

C₄⋊C₈

—

178

—

100

#102

—

101

C₂²⋊C₈

—

356

—

102

C₄×C₈

—

103

C₄×C₈

—

104

C₄×C₈

C₂²×D₈₉

yes

characteristic

105

C₂×D₄

—

106

C₂²×C₄

—

107

D₈

—

108

D₈

—

109

M₁₆

—

712

—

110

C₂×C₈

—

111

C₂×C₈

—

356

—

112

C₂×C₈

—

178

—

113

C₂×C₈

C₂²×Dic₈₉

yes

characteristic

114

C₂×C₈

#13083573

yes

characteristic

115

C₄⋊C₄

—

116

C₂²⋊C₄

—

117

C₄²

D₄×D₈₉

yes

characteristic

118

C₁₆

—

119

C₁₆

—

178

—

120

C₁₆

#13083558

yes

complemented

121

C₂³

—

122

D₄

—

123

D₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

D₄×Dic₈₉

yes

characteristic

128

C₂×C₄

D₈×D₈₉

yes

characteristic

129

C₂×C₄

#15668031

yes

characteristic

130

C₈

—

131

C₈

—

356

—

132

C₈

—

178

—

133

C₈

—

712

—

134

C₈

C₂×#13083558

yes

characteristic

135

C₈

#15668019

yes

characteristic

136

C₂²

—

137

C₂²

—

138

C₂²

—

139

C₂²

D₈×Dic₈₉

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

#19151676

yes

characteristic

143

C₄

#19151715

yes

characteristic

144

C₄

#19152032

yes

—

145

C₂

—

146

C₂

#26019137

yes

characteristic

147

C₂

#26022128

yes

characteristic

148

C₂

#26022138

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₈×D₈₉
commutator	C₂×C₈×C₈₉	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₈₉
Fitting	C₄×D₁₆×C₈₉	C₂
socle	C₂²×C₈₉	C₄×D₈
prime radical	C₂×D₁₆×C₈₉	C₄
cosocle	C₄×C₈×C₈₉	C₂³
hypercenter	C₄×D₁₆	D₈₉
hypocenter	C₈₉	#12086
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×D₁₆

subgroup

G/H

center

C₂×C₄

D₈×D₈₉

commutator

C₂×C₈×C₈₉

C₂²×C₄

Frattini

C₄×C₈

C₂²×D₈₉

Fitting

C₄×D₁₆×C₈₉

C₂

socle

C₂²×C₈₉

C₄×D₈

prime radical

C₂×D₁₆×C₈₉

C₄

cosocle

C₄×C₈×C₈₉

C₂³

hypercenter

C₄×D₁₆

D₈₉

hypocenter

C₈₉

#12086

LCS

G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×D₁₆

N	H ≅ G/N	type
1. #26019137	C₂	semidirect
2. #26018326	C₂	semidirect
3. #26018213	C₂	semidirect
4. #26018168	C₂	semidirect
45. C₁₆×C₈₉	M₁₆	semidirect
80. C₈₉	#12086	semidirect
120. C₁₆	#13083558	semidirect

H ≅ G/N

type

1. #26019137

C₂

semidirect

2. #26018326

C₂

semidirect

3. #26018213

C₂

semidirect

4. #26018168

C₂

semidirect

45. C₁₆×C₈₉

M₁₆

semidirect

80. C₈₉

#12086

semidirect

120. C₁₆

#13083558

semidirect

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

subgroup	index
#26019137	2
#26018326	2
#26018213	2
#26018168	2
C₄×D₁₆×C₈₉	2
#12086	89

subgroup

index

C₄×D₁₆×C₈₉

Subgroups — #36466436

Conjugacy classes of subgroups

Complements

Maximal subgroups