Group #36448227 — subgroups

Subgroups — #36448227

Order 22784.

354 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 171 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	#26020424	C₂	1	1	yes	maximal, characteristic
2	#26019007	C₂	1	4	yes	maximal
3	C₂×C₈×#10623494	C₂	1	1	yes	maximal, characteristic
4	C₈₉×#1848	C₂	1	1	yes	maximal, characteristic
5	C₂×#15667948	—	2	2	no	—
6	#19151622	C₂²	1	2	yes	—
7	C₂×C₄×#10623494	C₂²	1	2	yes	—
8	#19151348	—	2	8	no	—
9	C₈×#10623494	C₄	1	1	yes	characteristic
10	C₈×#10623494	C₄	1	2	yes	—
11	C₈×#10623494	C₄	1	1	yes	characteristic
12	C₂×#108×C₈₉	—	2	2	no	—
13	#423×C₈₉	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
15	C₂×#13083558	—	2	2	no	—
16	C₂²×#10623494	Q₈	1	4	yes	—
17	#15667948	—	4	4	no	—
18	C₄⋊#10623494	—	2	4	no	—
19	C₄×#10623494	C₂×C₄	1	2	yes	—
20	C₄×#10623494	C₂×C₄	1	2	yes	—
21	C₄×#10623494	D₄	1	4	yes	—
22	C₂×M₁₆×C₈₉	—	2	2	no	—
23	C₂²×C₈×C₈₉	D₄	1	2	yes	—
24	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
25	#108×C₈₉	—	4	4	no	—
26	C₄⋊C₈×C₈₉	—	2	4	no	—
27	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₈₉	Q₈	1	2	yes	—
30	#13083558	—	4	4	no	—
31	C₂×#10623494	—	2	8	no	—
32	C₂×#10623494	—	4	2	no	—
33	C₂×#10623494	C₄⋊C₄	1	4	yes	—
34	C₂×#10623494	C₄⋊C₄	1	4	yes	—
35	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
36	C₂²×C₄×C₈₉	C₂×Q₈	1	2	yes	—
37	M₁₆×C₈₉	—	4	4	no	—
38	C₂×C₈×C₈₉	—	4	2	no	—
39	C₂×C₈×C₈₉	C₄⋊C₄	1	2	yes	—
40	C₂×C₈×C₈₉	C₄⋊C₄	1	2	yes	—
41	C₂×C₈×C₈₉	SD₁₆	1	4	yes	—
42	C₂×C₈×C₈₉	SD₁₆	1	4	yes	—
43	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
44	C₄²×C₈₉	C₂×D₄	1	2	yes	—
45	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
46	C₂³×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
47	C₂×C₄×C₈₉	—	2	4	no	—
48	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
49	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
51	C₂×C₄×C₈₉	C₄⋊Q₈	1	2	yes	—
52	C₂×C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
53	C₂×C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
54	C₈×C₈₉	—	8	2	no	—
55	C₈×C₈₉	#106	1	4	yes	—
56	C₈×C₈₉	#106	1	4	yes	—
57	#10623494	—	2	4	no	—
58	#10623494	—	8	2	no	—
59	#10623494	#108	1	8	yes	complemented
60	C₂²×C₈₉	—	2	2	no	—
61	C₂²×C₈₉	#384	1	1	yes	characteristic
62	C₂²×C₈₉	#498	1	2	yes	—
63	C₄×C₈₉	—	2	2	no	—
64	C₄×C₈₉	C₂×#106	1	2	yes	—
65	C₄×C₈₉	C₂×#108	1	4	yes	—
66	C₄×C₈₉	#492	1	1	yes	characteristic
67	C₄×C₈₉	#492	1	1	yes	characteristic
68	#8822	—	89	1	no	maximal, Sylow
69	C₂×C₈₉	—	2	2	no	—
70	C₂×C₈₉	#1844	1	1	yes	characteristic
71	C₂×C₈₉	#1848	1	2	yes	—
72	#1848	—	89	1	no	—
73	#1848	D₈₉	1	1	yes	characteristic
74	#1566	—	89	4	no	—
75	C₂×C₈²	—	89	1	no	—
76	C₈₉	#8822	1	1	yes	characteristic, complemented, Sylow
77	C₂×#108	—	2	2	no	—
78	C₂×#108	—	178	2	no	—
79	#423	—	89	2	no	—
80	#423	C₂×D₈₉	1	2	yes	—
81	C₂×C₄×C₈	—	89	2	no	—
82	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
83	#334	—	178	8	no	—
84	C₈²	—	89	1	no	—
85	C₈²	—	89	2	no	—
86	C₈²	—	89	1	no	—
87	C₂×M₁₆	—	2	2	no	—
88	C₂×M₁₆	—	178	2	no	—
89	C₂²×C₈	—	89	4	no	—
90	C₂²×C₈	C₂²⋊D₈₉	1	2	yes	—
91	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
92	#108	—	4	4	no	—
93	#108	—	356	4	no	—
94	C₄⋊C₈	—	2	4	no	—
95	C₄⋊C₈	—	178	4	no	—
96	C₄×C₈	—	89	2	no	—
97	C₄×C₈	—	89	2	no	—
98	C₄×C₈	—	89	4	no	—
99	C₄×C₈	Dic₁₇₈	1	2	yes	—
100	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
101	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
102	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
103	C₂²×C₄	Q₈×D₈₉	1	2	yes	—
104	M₁₆	—	4	4	no	—
105	M₁₆	—	356	4	no	—
106	C₂×C₈	—	4	2	no	—
107	C₂×C₈	—	89	4	no	—
108	C₂×C₈	—	89	4	no	—
109	C₂×C₈	—	356	2	no	—
110	C₂×C₈	—	178	8	no	—
111	C₂×C₈	#13083553	1	2	yes	—
112	C₂×C₈	#13083553	1	2	yes	—
113	C₂×C₈	#13083561	1	4	yes	—
114	C₂×C₈	#13083562	1	4	yes	—
115	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
116	C₄²	D₄×D₈₉	1	2	yes	—
117	C₄²	C₂×Dic₁₇₈	1	1	yes	characteristic
118	C₂³	#15668010	1	1	yes	characteristic
119	C₂×C₄	—	2	4	no	—
120	C₂×C₄	C₄⋊C₄×D₈₉	1	2	yes	—
121	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
122	C₂×C₄	#15668008	1	1	yes	characteristic
123	C₂×C₄	#15668011	1	2	yes	—
124	C₂×C₄	C₂×#13083561	1	2	yes	—
125	C₂×C₄	C₂×#13083562	1	2	yes	—
126	C₈	—	8	2	no	—
127	C₈	—	89	8	no	—
128	C₈	—	178	4	no	—
129	C₈	—	712	2	no	—
130	C₈	#15667943	1	4	yes	—
131	C₈	#15667943	1	4	yes	—
132	C₂²	—	2	2	no	—
133	C₂²	#19151467	1	1	yes	characteristic
134	C₂²	#19151994	1	2	yes	—
135	C₄	—	2	2	no	—
136	C₄	C₂×#15667943	1	2	yes	—
137	C₄	#108×D₈₉	1	4	yes	—
138	C₄	#19151970	1	1	yes	characteristic
139	C₄	#19151972	1	1	yes	characteristic
140	C₂	—	2	2	no	—
141	C₂	#26020408	1	1	yes	characteristic
142	C₂	#26020422	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26020424

C₂

yes

maximal, characteristic

#26019007

C₂

yes

maximal

C₂×C₈×#10623494

C₂

yes

maximal, characteristic

C₈₉×#1848

C₂

yes

maximal, characteristic

C₂×#15667948

—

#19151622

C₂²

yes

—

C₂×C₄×#10623494

C₂²

yes

—

#19151348

—

C₈×#10623494

C₄

yes

characteristic

C₈×#10623494

C₄

yes

—

C₈×#10623494

C₄

yes

characteristic

C₂×#108×C₈₉

—

#423×C₈₉

C₂²

yes

—

C₂×C₄×C₈×C₈₉

C₂²

yes

characteristic

C₂×#13083558

—

C₂²×#10623494

Q₈

yes

—

#15667948

—

C₄⋊#10623494

—

C₄×#10623494

C₂×C₄

yes

—

C₄×#10623494

C₂×C₄

yes

—

C₄×#10623494

D₄

yes

—

C₂×M₁₆×C₈₉

—

C₂²×C₈×C₈₉

D₄

yes

—

C₂×C₄²×C₈₉

C₂³

yes

characteristic

#108×C₈₉

—

C₄⋊C₈×C₈₉

—

C₄×C₈×C₈₉

C₂×C₄

yes

characteristic

C₄×C₈×C₈₉

C₂×C₄

yes

characteristic

C₄×C₈×C₈₉

Q₈

yes

—

—

—

—

C₄⋊C₄

yes

—

C₂×#10623494

C₄⋊C₄

yes

—

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₉

C₂×Q₈

yes

—

M₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₄⋊C₄

yes

—

C₂×C₈×C₈₉

C₄⋊C₄

yes

—

C₂×C₈×C₈₉

SD₁₆

yes

—

C₂×C₈×C₈₉

SD₁₆

yes

—

C₄²×C₈₉

C₂²×C₄

yes

characteristic

C₄²×C₈₉

C₂×D₄

yes

—

C₄²×C₈₉

C₂×Q₈

yes

characteristic

C₂³×C₈₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₈₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₉

#127

yes

characteristic

C₂×C₄×C₈₉

C₄⋊Q₈

yes

—

C₂×C₄×C₈₉

C₂×SD₁₆

yes

—

C₂×C₄×C₈₉

C₂×SD₁₆

yes

—

C₈×C₈₉

—

C₈×C₈₉

#106

yes

—

C₈×C₈₉

#106

yes

—

—

—

yes

C₂²×C₈₉

—

C₂²×C₈₉

#384

yes

characteristic

C₂²×C₈₉

#498

yes

—

C₄×C₈₉

—

C₄×C₈₉

C₂×#106

yes

—

C₄×C₈₉

C₂×#108

yes

—

C₄×C₈₉

yes

C₄×C₈₉

yes

—

C₂×C₈₉

—

C₂×C₈₉

#1844

yes

characteristic

C₂×C₈₉

#1848

yes

—

#1848

—

#1848

D₈₉

yes

characteristic

#1566

—

C₂×C₈²

—

C₈₉

#8822

yes

characteristic, complemented, Sylow

C₂×#108

—

C₂×#108

—

178

—

#423

—

#423

C₂×D₈₉

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₈₉

yes

characteristic

#334

—

178

—

C₈²

—

C₈²

—

C₈²

—

C₂×M₁₆

—

C₂×M₁₆

—

178

—

C₂²×C₈

—

C₂²×C₈

C₂²⋊D₈₉

yes

—

C₂×C₄²

C₂²×D₈₉

yes

characteristic

#108

—

#108

—

356

—

C₄⋊C₈

—

C₄⋊C₈

—

178

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

Dic₁₇₈

yes

—

100

C₄×C₈

C₄×D₈₉

yes

characteristic

101

C₄×C₈

C₄×D₈₉

yes

characteristic

102

C₂²×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

103

C₂²×C₄

Q₈×D₈₉

yes

—

104

M₁₆

—

105

M₁₆

—

356

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

356

—

110

C₂×C₈

—

178

—

111

C₂×C₈

#13083553

yes

—

112

C₂×C₈

#13083553

yes

—

113

C₂×C₈

#13083561

yes

—

114

C₂×C₈

#13083562

yes

—

115

C₄²

C₂×C₄×D₈₉

yes

characteristic

116

C₄²

D₄×D₈₉

yes

—

117

C₄²

C₂×Dic₁₇₈

yes

characteristic

118

C₂³

#15668010

yes

characteristic

119

C₂×C₄

—

120

C₂×C₄

C₄⋊C₄×D₈₉

yes

—

121

C₂×C₄

C₂×#13083553

yes

characteristic

122

C₂×C₄

#15668008

yes

characteristic

123

C₂×C₄

#15668011

yes

—

124

C₂×C₄

C₂×#13083561

yes

—

125

C₂×C₄

C₂×#13083562

yes

—

126

C₈

—

127

C₈

—

128

C₈

—

178

—

129

C₈

—

712

—

130

C₈

#15667943

yes

—

131

C₈

#15667943

yes

—

132

C₂²

—

133

C₂²

#19151467

yes

characteristic

134

C₂²

#19151994

yes

—

135

C₄

—

136

C₄

C₂×#15667943

yes

—

137

C₄

#108×D₈₉

yes

—

138

C₄

#19151970

yes

characteristic

139

C₄

#19151972

yes

characteristic

140

C₂

—

141

C₂

#26020408

yes

characteristic

142

C₂

#26020422

yes

—

	subgroup	G/H
center	C₂×C₄	#15668008
commutator	C₄²×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1848	C₂
socle	C₂²×C₈₉	#384
prime radical	C₂³×C₈₉	C₄⋊Q₈
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1848	D₈₉
hypocenter	C₈₉	#8822
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1848

subgroup

G/H

center

C₂×C₄

#15668008

commutator

C₄²×C₈₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₉

Fitting

C₈₉×#1848

C₂

socle

C₂²×C₈₉

#384

prime radical

C₂³×C₈₉

C₄⋊Q₈

cosocle

C₂×C₄²×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1848

N	H ≅ G/N	type
59. #10623494	#108	semidirect
76. C₈₉	#8822	semidirect

H ≅ G/N

type

59. #10623494

#108

semidirect

76. C₈₉

#8822

semidirect

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

subgroup	index
#26020424	2
#26019007	2
C₂×C₈×#10623494	2
C₈₉×#1848	2
#8822	89

subgroup

index

C₈₉×#1848

Subgroups — #36448227

Conjugacy classes of subgroups

Complements

Maximal subgroups