Group #36448204 — subgroups

Subgroups — #36448204

Order 22784.

422 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 203 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	#26020412	C₂	1	1	yes	maximal, characteristic
2	#26020379	C₂	1	4	yes	maximal
3	C₂×C₈×#10623494	C₂	1	1	yes	maximal, characteristic
4	C₈₉×#1845	C₂	1	1	yes	maximal, characteristic
5	C₂×#15667946	—	2	2	no	—
6	C₂×C₄×#10623494	C₂²	1	2	yes	—
7	#19151468	C₂²	1	2	yes	—
8	#19151360	—	2	8	no	—
9	C₈×#10623494	C₄	1	4	yes	complemented
10	C₂×#107×C₈₉	—	2	2	no	—
11	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
12	#384×C₈₉	C₂²	1	2	yes	—
13	C₂²×#10623494	D₄	1	4	yes	—
14	C₂×C₄⋊Dic₈₉	—	2	4	no	—
15	C₂×C₄⋊Dic₈₉	—	2	2	no	—
16	#15667946	—	4	4	no	—
17	C₄×#10623494	C₂×C₄	1	4	yes	—
18	C₄×#10623494	Q₈	1	4	yes	—
19	C₂²×C₈×C₈₉	D₄	1	2	yes	—
20	C₂×C₄⋊C₄×C₈₉	—	2	4	no	—
21	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
22	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
23	#107×C₈₉	—	4	4	no	—
24	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
25	C₄×C₈×C₈₉	Q₈	1	2	yes	—
26	C₂²×Dic₈₉	—	4	2	no	—
27	C₂×#10623494	—	2	4	no	—
28	C₂×#10623494	—	2	4	no	—
29	C₂×#10623494	C₄⋊C₄	1	8	yes	—
30	C₄⋊Dic₈₉	—	4	4	no	—
31	C₄⋊Dic₈₉	—	4	8	no	—
32	C₂²×C₄×C₈₉	—	4	2	no	—
33	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
35	C₂×C₈×C₈₉	C₄⋊C₄	1	4	yes	—
36	C₂×C₈×C₈₉	D₈	1	4	yes	—
37	C₂×C₈×C₈₉	Q₁₆	1	4	yes	—
38	C₄⋊C₄×C₈₉	—	4	4	no	—
39	C₄⋊C₄×C₈₉	—	4	8	no	—
40	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
41	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
42	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
43	C₂³×C₈₉	#127	1	1	yes	characteristic
44	C₂×C₄×C₈₉	—	8	2	no	—
45	C₂×C₄×C₈₉	—	8	4	no	—
46	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
47	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
49	C₂×C₄×C₈₉	C₄⋊Q₈	1	2	yes	—
50	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
51	C₂×C₄×C₈₉	C₂×Q₁₆	1	2	yes	—
52	C₂×C₄×C₈₉	#135	1	4	yes	—
53	C₂×C₄×C₈₉	#135	1	4	yes	—
54	C₂×Dic₈₉	—	8	2	no	—
55	C₂×Dic₈₉	—	8	4	no	—
56	C₈×C₈₉	#107	1	8	yes	—
57	#10623494	—	2	8	no	—
58	C₂²×C₈₉	#384	1	1	yes	characteristic
59	C₂²×C₈₉	#493	1	1	yes	characteristic
60	C₂²×C₈₉	#494	1	1	yes	characteristic
61	C₂²×C₈₉	#495	1	2	yes	—
62	C₂²×C₈₉	#495	1	2	yes	—
63	C₄×C₈₉	—	16	2	no	—
64	C₄×C₈₉	C₂×#107	1	2	yes	—
65	C₄×C₈₉	#427	1	4	yes	—
66	C₄×C₈₉	#499	1	4	yes	—
67	C₄×C₈₉	#500	1	2	yes	—
68	Dic₈₉	—	16	2	no	—
69	#8816	—	89	1	no	maximal, Sylow
70	C₂×C₈₉	#1709	1	1	yes	characteristic
71	C₂×C₈₉	#1709	1	1	yes	characteristic
72	C₂×C₈₉	#1710	1	2	yes	—
73	C₂×C₈₉	#1845	1	1	yes	characteristic
74	C₂×C₈₉	#1846	1	2	yes	—
75	#1845	—	89	1	no	—
76	#1845	D₈₉	1	1	yes	characteristic
77	#1834	—	89	4	no	—
78	C₂×C₈²	—	89	1	no	—
79	C₈₉	#8816	1	1	yes	characteristic, complemented, Sylow
80	C₂×#107	—	2	2	no	—
81	C₂×#107	—	178	2	no	—
82	C₂×C₄×C₈	—	89	2	no	—
83	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
84	#384	—	89	2	no	—
85	#384	C₂×D₈₉	1	2	yes	—
86	#340	—	178	8	no	—
87	C₈²	—	89	4	no	—
88	C₂²×C₈	—	89	4	no	—
89	C₂²×C₈	C₂²⋊D₈₉	1	2	yes	—
90	C₂×C₄⋊C₄	—	2	4	no	—
91	C₂×C₄⋊C₄	—	2	2	no	—
92	C₂×C₄⋊C₄	—	178	4	no	—
93	C₂×C₄⋊C₄	—	178	2	no	—
94	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
95	#107	—	4	4	no	—
96	#107	—	356	4	no	—
97	C₄×C₈	—	89	4	no	—
98	C₄×C₈	—	89	4	no	—
99	C₄×C₈	Dic₁₇₈	1	2	yes	—
100	C₄×C₈	C₄×D₈₉	1	2	yes	—
101	C₂²×C₄	—	4	2	no	—
102	C₂²×C₄	—	356	2	no	—
103	C₂²×C₄	D₄×D₈₉	1	2	yes	—
104	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
105	C₂×C₈	—	89	8	no	—
106	C₂×C₈	—	178	4	no	—
107	C₂×C₈	—	178	4	no	—
108	C₂×C₈	#13083553	1	4	yes	—
109	C₂×C₈	#13083559	1	4	yes	—
110	C₂×C₈	#13083564	1	4	yes	—
111	C₄⋊C₄	—	4	4	no	—
112	C₄⋊C₄	—	4	8	no	—
113	C₄⋊C₄	—	356	4	no	—
114	C₄⋊C₄	—	356	8	no	—
115	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
116	C₄²	Q₈×D₈₉	1	2	yes	—
117	C₄²	C₂×Dic₁₇₈	1	1	yes	characteristic
118	C₂³	#15668008	1	1	yes	characteristic
119	C₂×C₄	—	8	2	no	—
120	C₂×C₄	—	8	4	no	—
121	C₂×C₄	—	712	2	no	—
122	C₂×C₄	—	712	4	no	—
123	C₂×C₄	C₄⋊C₄×D₈₉	1	2	yes	—
124	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
125	C₂×C₄	#15668010	1	1	yes	characteristic
126	C₂×C₄	#15668011	1	2	yes	—
127	C₂×C₄	C₂×#13083559	1	2	yes	—
128	C₂×C₄	C₂×#13083564	1	2	yes	—
129	C₂×C₄	#15668031	1	4	yes	—
130	C₂×C₄	#15668035	1	4	yes	—
131	C₈	—	178	8	no	—
132	C₈	#15667945	1	8	yes	—
133	C₂²	#19151467	1	1	yes	characteristic
134	C₂²	#19151974	1	1	yes	characteristic
135	C₂²	#19151977	1	1	yes	characteristic
136	C₂²	#19151981	1	2	yes	—
137	C₂²	#19151983	1	2	yes	—
138	C₄	—	16	2	no	—
139	C₄	—	1424	2	no	—
140	C₄	C₂×#15667945	1	2	yes	—
141	C₄	#19151635	1	4	yes	—
142	C₄	#19151999	1	4	yes	—
143	C₄	#19152001	1	2	yes	—
144	C₂	#26019857	1	1	yes	characteristic
145	C₂	#26019859	1	1	yes	characteristic
146	C₂	#26019861	1	2	yes	—
147	C₂	#26020411	1	1	yes	characteristic
148	C₂	#26020415	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26020412

C₂

yes

maximal, characteristic

#26020379

C₂

yes

maximal

C₂×C₈×#10623494

C₂

yes

maximal, characteristic

C₈₉×#1845

C₂

yes

maximal, characteristic

C₂×#15667946

—

C₂×C₄×#10623494

C₂²

yes

—

#19151468

C₂²

yes

—

#19151360

—

C₈×#10623494

C₄

yes

complemented

C₂×#107×C₈₉

—

C₂×C₄×C₈×C₈₉

C₂²

yes

characteristic

#384×C₈₉

C₂²

yes

—

C₂²×#10623494

D₄

yes

—

C₂×C₄⋊Dic₈₉

—

C₂×C₄⋊Dic₈₉

—

#15667946

—

C₄×#10623494

C₂×C₄

yes

—

C₄×#10623494

Q₈

yes

—

C₂²×C₈×C₈₉

D₄

yes

—

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄²×C₈₉

C₂³

yes

characteristic

#107×C₈₉

—

C₄×C₈×C₈₉

C₂×C₄

yes

—

C₄×C₈×C₈₉

Q₈

yes

—

C₂²×Dic₈₉

—

C₂×#10623494

—

C₂×#10623494

—

C₂×#10623494

C₄⋊C₄

yes

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₈×C₈₉

C₄⋊C₄

yes

—

C₂×C₈×C₈₉

D₈

yes

—

C₂×C₈×C₈₉

Q₁₆

yes

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

C₂²×C₄

yes

characteristic

C₄²×C₈₉

C₂×Q₈

yes

—

C₄²×C₈₉

C₂×Q₈

yes

characteristic

C₂³×C₈₉

#127

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₈₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₈₉

C₄⋊Q₈

yes

—

C₂×C₄×C₈₉

C₂×D₈

yes

—

C₂×C₄×C₈₉

C₂×Q₁₆

yes

—

C₂×C₄×C₈₉

#135

yes

—

C₂×C₄×C₈₉

#135

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₈×C₈₉

#107

yes

—

#10623494

—

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

—

C₂²×C₈₉

#495

yes

—

C₄×C₈₉

—

C₄×C₈₉

C₂×#107

yes

—

C₄×C₈₉

#427

yes

—

C₄×C₈₉

#499

yes

—

C₄×C₈₉

#500

yes

—

Dic₈₉

—

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₂×C₈₉

#1845

yes

characteristic

C₂×C₈₉

#1846

yes

—

#1845

—

#1845

D₈₉

yes

characteristic

#1834

—

C₂×C₈²

—

C₈₉

#8816

yes

characteristic, complemented, Sylow

C₂×#107

—

C₂×#107

—

178

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₈₉

yes

characteristic

#384

—

#384

C₂×D₈₉

yes

—

#340

—

178

—

C₈²

—

C₂²×C₈

—

C₂²×C₈

C₂²⋊D₈₉

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

178

—

C₂×C₄⋊C₄

—

178

—

C₂×C₄²

C₂²×D₈₉

yes

characteristic

#107

—

#107

—

356

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

Dic₁₇₈

yes

—

100

C₄×C₈

C₄×D₈₉

yes

—

101

C₂²×C₄

—

102

C₂²×C₄

—

356

—

103

C₂²×C₄

D₄×D₈₉

yes

—

104

C₂²×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

105

C₂×C₈

—

106

C₂×C₈

—

178

—

107

C₂×C₈

—

178

—

108

C₂×C₈

#13083553

yes

—

109

C₂×C₈

#13083559

yes

—

110

C₂×C₈

#13083564

yes

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

356

—

114

C₄⋊C₄

—

356

—

115

C₄²

C₂×C₄×D₈₉

yes

characteristic

116

C₄²

Q₈×D₈₉

yes

—

117

C₄²

C₂×Dic₁₇₈

yes

characteristic

118

C₂³

#15668008

yes

characteristic

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

712

—

122

C₂×C₄

—

712

—

123

C₂×C₄

C₄⋊C₄×D₈₉

yes

—

124

C₂×C₄

C₂×#13083553

yes

characteristic

125

C₂×C₄

#15668010

yes

characteristic

126

C₂×C₄

#15668011

yes

—

127

C₂×C₄

C₂×#13083559

yes

—

128

C₂×C₄

C₂×#13083564

yes

—

129

C₂×C₄

#15668031

yes

—

130

C₂×C₄

#15668035

yes

—

131

C₈

—

178

—

132

C₈

#15667945

yes

—

133

C₂²

#19151467

yes

characteristic

134

C₂²

#19151974

yes

characteristic

135

C₂²

#19151977

yes

characteristic

136

C₂²

#19151981

yes

—

137

C₂²

#19151983

yes

—

138

C₄

—

139

C₄

—

1424

—

140

C₄

C₂×#15667945

yes

—

141

C₄

#19151635

yes

—

142

C₄

#19151999

yes

—

143

C₄

#19152001

yes

—

144

C₂

#26019857

yes

characteristic

145

C₂

#26019859

yes

characteristic

146

C₂

#26019861

yes

—

147

C₂

#26020411

yes

characteristic

148

C₂

#26020415

yes

—

	subgroup	G/H
center	C₂³	#15668008
commutator	C₄²×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1845	C₂
socle, prime radical	C₂³×C₈₉	#127
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1845	D₈₉
hypocenter	C₈₉	#8816
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

subgroup

G/H

center

C₂³

#15668008

commutator

C₄²×C₈₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₉

Fitting

C₈₉×#1845

C₂

socle, prime radical

C₂³×C₈₉

#127

cosocle

C₂×C₄²×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

N	H ≅ G/N	type
9. C₈×#10623494	C₄	semidirect
79. C₈₉	#8816	semidirect

H ≅ G/N

type

9. C₈×#10623494

C₄

semidirect

79. C₈₉

#8816

semidirect

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

subgroup	index
#26020412	2
#26020379	2
C₂×C₈×#10623494	2
C₈₉×#1845	2
#8816	89

subgroup

index

C₈₉×#1845

Subgroups — #36448204

Conjugacy classes of subgroups

Complements

Maximal subgroups