Group #36445590 — subgroups

Subgroups — #36445590

Order 22784.

390 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 179 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	#26019000	C₂	1	2	yes	maximal
2	#26018544	C₂	1	2	yes	maximal
3	#26018085	C₂	1	2	yes	maximal
4	C₄²×C₈×C₈₉	C₂	1	1	yes	maximal, characteristic
5	#19151651	C₂²	1	4	yes	—
6	#19151351	C₄	1	4	yes	complemented
7	#19151349	C₄	1	4	yes	complemented
8	#19151320	—	2	4	no	—
9	C₂×C₄×C₈×C₈₉	C₄	1	2	yes	—
10	C₂×C₄×C₈×C₈₉	C₄	1	2	yes	—
11	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
13	C₄³×C₈₉	C₂²	1	1	yes	characteristic
14	C₄⋊#10623494	C₂×C₄	1	8	yes	—
15	C₂²⋊#10623494	—	2	4	no	—
16	C₄×#10623494	—	2	4	no	—
17	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
19	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄²×C₈₉	—	2	2	no	—
21	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
22	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
23	C₄×C₈×C₈₉	—	2	4	no	—
24	C₄×C₈×C₈₉	—	2	4	no	—
25	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
26	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
27	C₄×C₈×C₈₉	D₄	1	2	yes	—
28	C₄×C₈×C₈₉	Q₈	1	2	yes	—
29	C₂×#10623494	—	2	8	no	—
30	C₂²×C₄×C₈₉	—	2	2	no	—
31	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₈₉	#41	1	2	yes	—
33	C₂×C₈×C₈₉	—	2	2	no	—
34	C₂×C₈×C₈₉	—	2	2	no	—
35	C₂×C₈×C₈₉	—	2	2	no	—
36	C₂×C₈×C₈₉	C₄²	1	2	yes	—
37	C₂×C₈×C₈₉	C₄²	1	1	yes	characteristic
38	C₂×C₈×C₈₉	C₄²	1	1	yes	characteristic
39	C₂×C₈×C₈₉	C₄⋊C₄	1	4	yes	—
40	C₂×C₈×C₈₉	M₁₆	1	4	yes	—
41	C₄²×C₈₉	—	2	8	no	—
42	C₄²×C₈₉	—	2	2	no	—
43	C₄²×C₈₉	M₁₆	1	4	yes	—
44	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
45	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
46	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
47	C₂³×C₈₉	#117	1	1	yes	characteristic
48	C₂×C₄×C₈₉	—	2	8	no	—
49	C₂×C₄×C₈₉	—	2	4	no	—
50	C₂×C₄×C₈₉	—	2	4	no	—
51	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
52	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
54	C₂×C₄×C₈₉	C₄×Q₈	1	2	yes	—
55	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
56	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
57	C₈×C₈₉	—	2	2	no	—
58	C₈×C₈₉	—	2	4	no	—
59	C₈×C₈₉	#97	1	2	yes	—
60	C₈×C₈₉	#97	1	2	yes	—
61	#10623494	—	4	8	no	—
62	C₂²×C₈₉	—	2	2	no	—
63	C₂²×C₈₉	C₄×C₄⋊C₄	1	1	yes	characteristic
64	C₂²×C₈₉	#432	1	2	yes	—
65	C₄×C₈₉	—	2	8	no	—
66	C₄×C₈₉	—	2	2	no	—
67	C₄×C₈₉	C₂×#97	1	1	yes	characteristic
68	C₄×C₈₉	C₄×M₁₆	1	1	yes	characteristic
69	C₄×C₈₉	#423	1	2	yes	—
70	C₄×C₈₉	#443	1	4	yes	—
71	#8267	—	89	1	no	maximal, Sylow
72	C₂×C₈₉	—	2	2	no	—
73	C₂×C₈₉	#1767	1	1	yes	characteristic
74	C₂×C₈₉	#1772	1	2	yes	—
75	C₄²×C₈	D₈₉	1	1	yes	characteristic
76	#1564	—	89	2	no	—
77	#1452	—	89	2	no	—
78	#1276	—	89	2	no	—
79	C₈₉	#8267	1	1	yes	characteristic, complemented, Sylow
80	#431	—	89	4	no	—
81	C₂×C₄×C₈	Dic₈₉	1	2	yes	—
82	C₂×C₄×C₈	Dic₈₉	1	2	yes	—
83	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
84	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
85	C₄³	C₂×D₈₉	1	1	yes	characteristic
86	#335	—	89	4	no	—
87	#334	—	89	4	no	—
88	#322	—	178	4	no	—
89	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
90	C₂²×C₈	C₂×Dic₈₉	1	2	yes	—
91	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
92	C₂×C₄²	—	2	2	no	—
93	C₂×C₄²	C₂×Dic₈₉	1	2	yes	—
94	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
95	C₄⋊C₈	—	89	8	no	—
96	C₂²⋊C₈	—	178	4	no	—
97	C₄×C₈	—	2	4	no	—
98	C₄×C₈	—	2	4	no	—
99	C₄×C₈	—	178	4	no	—
100	C₄×C₈	Dic₁₇₈	1	2	yes	—
101	C₄×C₈	C₄×D₈₉	1	2	yes	—
102	C₄×C₈	C₄×D₈₉	1	2	yes	—
103	C₄×C₈	D₃₅₆	1	2	yes	—
104	C₂²×C₄	—	2	2	no	—
105	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
106	C₂²×C₄	#13083574	1	2	yes	—
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	2	2	no	—
109	C₂×C₈	—	2	2	no	—
110	C₂×C₈	—	178	8	no	—
111	C₂×C₈	C₄×Dic₈₉	1	2	yes	—
112	C₂×C₈	C₄×Dic₈₉	1	1	yes	characteristic
113	C₂×C₈	C₄×Dic₈₉	1	1	yes	characteristic
114	C₂×C₈	C₄⋊Dic₈₉	1	4	yes	—
115	C₂×C₈	#13083558	1	4	yes	—
116	C₄²	—	2	8	no	—
117	C₄²	—	2	2	no	—
118	C₄²	#13083558	1	4	yes	—
119	C₄²	C₂×C₄×D₈₉	1	2	yes	—
120	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
121	C₄²	C₂×Dic₁₇₈	1	1	yes	characteristic
122	C₂³	#15667971	1	1	yes	characteristic
123	C₂×C₄	—	2	8	no	—
124	C₂×C₄	—	2	4	no	—
125	C₂×C₄	—	2	4	no	—
126	C₂×C₄	C₂×C₄×Dic₈₉	1	1	yes	characteristic
127	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
128	C₂×C₄	C₄×D₃₅₆	1	2	yes	—
129	C₂×C₄	C₄×Dic₁₇₈	1	2	yes	—
130	C₂×C₄	C₂×#13083558	1	2	yes	—
131	C₂×C₄	C₂×#13083558	1	2	yes	—
132	C₈	—	2	4	no	—
133	C₈	—	2	2	no	—
134	C₈	—	356	8	no	—
135	C₈	#15667923	1	2	yes	—
136	C₈	#15667923	1	2	yes	—
137	C₂²	—	2	2	no	—
138	C₂²	C₄×C₄⋊Dic₈₉	1	1	yes	characteristic
139	C₂²	#19151655	1	2	yes	—
140	C₄	—	2	8	no	—
141	C₄	—	2	2	no	—
142	C₄	C₂×#15667923	1	1	yes	characteristic
143	C₄	C₄×#13083558	1	1	yes	characteristic
144	C₄	#19151622	1	2	yes	—
145	C₄	#19151716	1	4	yes	—
146	C₂	—	2	2	no	—
147	C₂	#26020105	1	1	yes	characteristic
148	C₂	#26020131	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26019000

C₂

yes

maximal

#26018544

C₂

yes

maximal

#26018085

C₂

yes

maximal

C₄²×C₈×C₈₉

C₂

yes

maximal, characteristic

#19151651

C₂²

yes

—

#19151351

C₄

yes

complemented

#19151349

C₄

yes

complemented

#19151320

—

C₂×C₄×C₈×C₈₉

C₄

yes

—

C₂×C₄×C₈×C₈₉

C₄

yes

—

C₂×C₄×C₈×C₈₉

C₂²

yes

characteristic

C₂×C₄×C₈×C₈₉

C₂²

yes

characteristic

C₄³×C₈₉

C₂²

yes

characteristic

C₄⋊#10623494

C₂×C₄

yes

—

C₂²⋊#10623494

—

C₄×#10623494

—

C₂²×C₈×C₈₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₈₉

C₂×C₄

yes

—

C₂²×C₈×C₈₉

C₂×C₄

yes

characteristic

C₂×C₄²×C₈₉

—

C₂×C₄²×C₈₉

C₂×C₄

yes

—

C₂×C₄²×C₈₉

C₂³

yes

characteristic

C₄×C₈×C₈₉

—

C₄×C₈×C₈₉

—

C₄×C₈×C₈₉

C₂×C₄

yes

—

C₄×C₈×C₈₉

C₂×C₄

yes

—

C₄×C₈×C₈₉

D₄

yes

—

C₄×C₈×C₈₉

Q₈

yes

—

C₂×#10623494

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₈₉

#41

yes

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₄²

yes

—

C₂×C₈×C₈₉

C₄²

yes

characteristic

C₂×C₈×C₈₉

C₄²

yes

characteristic

C₂×C₈×C₈₉

C₄⋊C₄

yes

—

C₂×C₈×C₈₉

M₁₆

yes

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

M₁₆

yes

—

C₄²×C₈₉

C₂²×C₄

yes

—

C₄²×C₈₉

C₂×D₄

yes

characteristic

C₄²×C₈₉

C₂×Q₈

yes

characteristic

C₂³×C₈₉

#117

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×C₄²

yes

characteristic

C₂×C₄×C₈₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₉

C₄×D₄

yes

—

C₂×C₄×C₈₉

C₄×Q₈

yes

—

C₂×C₄×C₈₉

C₂×M₁₆

yes

—

C₂×C₄×C₈₉

C₂×M₁₆

yes

—

C₈×C₈₉

—

C₈×C₈₉

—

C₈×C₈₉

#97

yes

—

C₈×C₈₉

#97

yes

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

C₄×C₄⋊C₄

yes

characteristic

C₂²×C₈₉

#432

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₂×#97

yes

characteristic

C₄×C₈₉

C₄×M₁₆

yes

characteristic

C₄×C₈₉

#423

yes

—

C₄×C₈₉

#443

yes

—

#8267

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

#1767

yes

characteristic

C₂×C₈₉

#1772

yes

—

C₄²×C₈

D₈₉

yes

—

—

—

C₈₉

#8267

yes

characteristic, complemented, Sylow

#431

—

C₂×C₄×C₈

Dic₈₉

yes

—

C₂×C₄×C₈

Dic₈₉

yes

—

C₂×C₄×C₈

C₂×D₈₉

yes

characteristic

C₂×C₄×C₈

C₂×D₈₉

yes

characteristic

C₄³

C₂×D₈₉

yes

—

—

—

178

—

C₂²×C₈

C₂×Dic₈₉

yes

characteristic

C₂²×C₈

C₂×Dic₈₉

yes

—

C₂²×C₈

C₂×Dic₈₉

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₂×Dic₈₉

yes

—

C₂×C₄²

C₂²×D₈₉

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

178

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

178

—

100

C₄×C₈

Dic₁₇₈

yes

—

101

C₄×C₈

C₄×D₈₉

yes

—

102

C₄×C₈

C₄×D₈₉

yes

—

103

C₄×C₈

D₃₅₆

yes

—

104

C₂²×C₄

—

105

C₂²×C₄

C₂²×Dic₈₉

yes

characteristic

106

C₂²×C₄

#13083574

yes

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

178

—

111

C₂×C₈

C₄×Dic₈₉

yes

—

112

C₂×C₈

C₄×Dic₈₉

yes

characteristic

113

C₂×C₈

C₄×Dic₈₉

yes

characteristic

114

C₂×C₈

C₄⋊Dic₈₉

yes

—

115

C₂×C₈

#13083558

yes

—

116

C₄²

—

117

C₄²

—

118

C₄²

#13083558

yes

—

119

C₄²

C₂×C₄×D₈₉

yes

—

120

C₄²

C₂×D₃₅₆

yes

characteristic

121

C₄²

C₂×Dic₁₇₈

yes

characteristic

122

C₂³

#15667971

yes

characteristic

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

C₂×C₄×Dic₈₉

yes

characteristic

127

C₂×C₄

C₂×C₄⋊Dic₈₉

yes

characteristic

128

C₂×C₄

C₄×D₃₅₆

yes

—

129

C₂×C₄

C₄×Dic₁₇₈

yes

—

130

C₂×C₄

C₂×#13083558

yes

—

131

C₂×C₄

C₂×#13083558

yes

—

132

C₈

—

133

C₈

—

134

C₈

—

356

—

135

C₈

#15667923

yes

—

136

C₈

#15667923

yes

—

137

C₂²

—

138

C₂²

C₄×C₄⋊Dic₈₉

yes

characteristic

139

C₂²

#19151655

yes

—

140

C₄

—

141

C₄

—

142

C₄

C₂×#15667923

yes

characteristic

143

C₄

C₄×#13083558

yes

characteristic

144

C₄

#19151622

yes

—

145

C₄

#19151716

yes

—

146

C₂

—

147

C₂

#26020105

yes

characteristic

148

C₂

#26020131

yes

—

	subgroup	G/H
center	C₄²	C₂×D₃₅₆
commutator	C₂×C₄×C₈₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₄²×C₈×C₈₉	C₂
socle	C₂²×C₈₉	C₄×C₄⋊C₄
prime radical	C₂³×C₈₉	#117
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₄²×C₈	D₈₉
hypocenter	C₈₉	#8267
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ C₄²×C₈

subgroup

G/H

center

C₄²

C₂×D₃₅₆

commutator

C₂×C₄×C₈₉

C₂×C₄²

Frattini

C₂×C₄²

C₂²×D₈₉

Fitting

C₄²×C₈×C₈₉

C₂

socle

C₂²×C₈₉

C₄×C₄⋊C₄

prime radical

C₂³×C₈₉

#117

cosocle

C₂×C₄²×C₈₉

C₂³

hypercenter

C₄²×C₈

D₈₉

hypocenter

C₈₉

#8267

LCS

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ C₄²×C₈

N	H ≅ G/N	type
6. #19151351	C₄	semidirect
7. #19151349	C₄	semidirect
79. C₈₉	#8267	semidirect

H ≅ G/N

type

6. #19151351

C₄

semidirect

7. #19151349

C₄

semidirect

79. C₈₉

#8267

semidirect

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

subgroup	index
#26019000	2
#26018544	2
#26018085	2
C₄²×C₈×C₈₉	2
#8267	89

subgroup

index

C₄²×C₈×C₈₉

Subgroups — #36445590

Conjugacy classes of subgroups

Complements

Maximal subgroups