Group #36466188 — subgroups

Subgroups — #36466188

Order 22784.

212 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 75 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	#26022134	C₂	1	1	yes	maximal, characteristic
2	#26022088	C₂	1	2	yes	maximal, complemented
3	#26018224	C₂	1	2	yes	maximal, complemented
4	#26018168	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈₉×#1373	C₂	1	1	yes	maximal, characteristic, complemented
6	#19152030	C₄	1	1	yes	characteristic
7	#19152016	—	2	2	no	—
8	#19151716	C₂²	1	2	yes	—
9	#19151428	C₄	1	1	yes	characteristic
10	#19151406	C₄	1	2	yes	—
11	#19151383	—	2	2	no	—
12	C₄×#13083549	C₂²	1	1	yes	characteristic
13	C₈×#10623494	C₂²	1	1	yes	characteristic
14	#364×C₈₉	C₂²	1	2	yes	—
15	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
16	#15668034	C₂×C₄	1	2	yes	—
17	#15668018	—	2	2	no	—
18	Dic₇₁₂	C₈	1	1	yes	characteristic, complemented
19	#15667957	C₈	1	2	yes	complemented
20	D₁₄₂₄	C₈	1	1	yes	characteristic, complemented
21	#15667951	—	4	2	no	—
22	C₂×#13083549	D₄	1	1	yes	characteristic
23	C₂×#13083549	D₄	1	1	yes	characteristic
24	C₁₆×D₈₉	—	4	2	no	—
25	#15667948	C₂×C₄	1	2	yes	—
26	#15667940	—	2	2	no	—
27	C₄×#10623494	—	2	1	no	—
28	M₃₂×C₈₉	—	2	2	no	—
29	C₂×C₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
30	C₂×C₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
32	#13083574	—	2	2	no	—
33	Dic₃₅₆	C₂×C₈	1	2	yes	—
34	#13083563	—	2	2	no	—
35	D₇₁₂	C₂×C₈	1	2	yes	—
36	#13083558	—	2	2	no	—
37	#13083557	—	4	2	no	—
38	C₂×#10623494	—	2	2	no	—
39	C₈×D₈₉	—	4	2	no	—
40	#13083549	—	2	1	no	—
41	#13083549	D₈	1	2	yes	—
42	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
44	C₄²×C₈₉	#41	1	1	yes	characteristic
45	C₁₆×C₈₉	—	2	1	no	—
46	C₁₆×C₈₉	—	4	2	no	—
47	C₁₆×C₈₉	C₂×C₈	1	1	yes	characteristic
48	C₁₆×C₈₉	C₂×C₈	1	1	yes	characteristic
49	C₂×C₄×C₈₉	—	2	1	no	—
50	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
51	C₂²⋊D₈₉	—	4	2	no	—
52	D₃₅₆	—	2	2	no	—
53	C₄×D₈₉	—	4	2	no	—
54	Dic₁₇₈	—	2	2	no	—
55	C₈×C₈₉	C₂²×C₈	1	1	yes	characteristic
56	C₈×C₈₉	#131	1	1	yes	characteristic
57	C₈×C₈₉	C₂×D₈	1	1	yes	characteristic
58	C₈×C₈₉	#135	1	1	yes	characteristic
59	#10623494	—	4	2	no	—
60	#10623494	—	4	2	no	—
61	C₂²×C₈₉	#443	1	1	yes	characteristic
62	C₂×D₈₉	—	4	2	no	—
63	C₄×C₈₉	—	2	2	no	—
64	C₄×C₈₉	C₈×D₄	1	1	yes	characteristic
65	C₄×C₈₉	C₄×D₈	1	1	yes	characteristic
66	Dic₈₉	—	4	2	no	—
67	#12047	—	89	1	no	maximal, Sylow
68	C₂×C₈₉	—	2	1	no	—
69	C₂×C₈₉	C₈×D₈	1	1	yes	characteristic
70	D₈₉	—	8	2	no	—
71	#2178	—	89	1	no	—
72	#2170	—	89	2	no	—
73	#1373	D₈₉	1	1	yes	characteristic, complemented
74	#1335	—	89	2	no	—
75	#1311	—	89	1	no	—
76	C₈₉	#12047	1	1	yes	characteristic, complemented, Sylow
77	#508	—	89	1	no	—
78	#504	—	178	2	no	—
79	#443	—	89	2	no	—
80	#368	—	89	1	no	—
81	#364	C₂×D₈₉	1	2	yes	—
82	#359	—	89	2	no	—
83	#350	—	178	2	no	—
84	C₄×C₁₆	—	89	1	no	—
85	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
86	C₈²	—	89	1	no	—
87	#135	—	89	2	no	—
88	#131	—	178	2	no	—
89	Q₃₂	—	89	1	no	—
90	SD₃₂	—	89	2	no	—
91	D₁₆	—	89	1	no	—
92	M₃₂	—	2	2	no	—
93	M₃₂	—	356	2	no	—
94	C₂×C₁₆	—	89	1	no	—
95	C₂×C₁₆	—	89	1	no	—
96	C₂×C₁₆	—	356	2	no	—
97	C₂×C₁₆	C₄×D₈₉	1	1	yes	characteristic
98	C₂×C₁₆	C₄×D₈₉	1	1	yes	characteristic
99	#108	—	89	2	no	—
100	#104	—	178	2	no	—
101	C₄×C₈	—	178	1	no	—
102	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
103	#41	—	178	2	no	—
104	Q₁₆	—	89	2	no	—
105	SD₁₆	—	178	2	no	—
106	D₈	—	89	2	no	—
107	M₁₆	—	356	2	no	—
108	M₁₆	—	178	2	no	—
109	C₂×C₈	—	356	2	no	—
110	C₂×C₈	—	178	2	no	—
111	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
112	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
113	C₄²	#13083575	1	1	yes	characteristic
114	C₁₆	—	2	1	no	—
115	C₁₆	—	4	2	no	—
116	C₁₆	—	89	2	no	—
117	C₁₆	—	178	1	no	—
118	C₁₆	C₈×D₈₉	1	1	yes	characteristic
119	C₁₆	C₈×D₈₉	1	1	yes	characteristic
120	Q₈	—	178	2	no	—
121	D₄	—	356	2	no	—
122	D₄	—	178	2	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	356	2	no	—
125	C₂×C₄	#15667974	1	1	yes	characteristic
126	C₈	—	356	2	no	—
127	C₈	—	356	2	no	—
128	C₈	C₂×C₈×D₈₉	1	1	yes	characteristic
129	C₈	#15668020	1	1	yes	characteristic
130	C₈	D₈×D₈₉	1	1	yes	characteristic
131	C₈	#15668035	1	1	yes	characteristic
132	C₂²	—	356	2	no	—
133	C₂²	#19151714	1	1	yes	characteristic
134	C₄	—	2	2	no	—
135	C₄	—	356	2	no	—
136	C₄	#19151663	1	1	yes	characteristic
137	C₄	#19151680	1	1	yes	characteristic
138	C₂	—	2	1	no	—
139	C₂	—	712	2	no	—
140	C₂	#26019046	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26022134

C₂

yes

maximal, characteristic

#26022088

C₂

yes

maximal, complemented

#26018224

C₂

yes

maximal, complemented

#26018168

C₂

yes

maximal, characteristic, complemented

C₈₉×#1373

C₂

yes

maximal, characteristic, complemented

#19152030

C₄

yes

characteristic

#19152016

—

#19151716

C₂²

yes

—

#19151428

C₄

yes

characteristic

#19151406

C₄

yes

—

#19151383

—

C₄×#13083549

C₂²

yes

characteristic

C₈×#10623494

C₂²

yes

characteristic

#364×C₈₉

C₂²

yes

—

C₄×C₁₆×C₈₉

C₂²

yes

characteristic

#15668034

C₂×C₄

yes

—

#15668018

—

Dic₇₁₂

C₈

yes

characteristic, complemented

#15667957

C₈

yes

complemented

D₁₄₂₄

C₈

yes

characteristic, complemented

#15667951

—

C₂×#13083549

D₄

yes

characteristic

C₂×#13083549

D₄

yes

characteristic

C₁₆×D₈₉

—

#15667948

C₂×C₄

yes

—

#15667940

—

C₄×#10623494

—

M₃₂×C₈₉

—

C₂×C₁₆×C₈₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₈₉

C₂×C₄

yes

characteristic

C₄×C₈×C₈₉

C₂³

yes

characteristic

#13083574

—

Dic₃₅₆

C₂×C₈

yes

—

#13083563

—

D₇₁₂

C₂×C₈

yes

—

#13083558

—

#13083557

—

C₂×#10623494

—

C₈×D₈₉

—

#13083549

—

#13083549

D₈

yes

—

C₂×C₈×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₄²×C₈₉

#41

yes

characteristic

C₁₆×C₈₉

—

C₁₆×C₈₉

—

C₁₆×C₈₉

C₂×C₈

yes

characteristic

C₁₆×C₈₉

C₂×C₈

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₄×D₄

yes

characteristic

C₂²⋊D₈₉

—

D₃₅₆

—

C₄×D₈₉

—

Dic₁₇₈

—

C₈×C₈₉

C₂²×C₈

yes

characteristic

C₈×C₈₉

#131

yes

characteristic

C₈×C₈₉

C₂×D₈

yes

C₈×C₈₉

yes

—

—

C₂²×C₈₉

#443

yes

characteristic

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₈×D₄

yes

characteristic

C₄×C₈₉

C₄×D₈

yes

characteristic

Dic₈₉

—

#12047

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

C₈×D₈

yes

characteristic

D₈₉

—

#2178

—

#2170

—

#1373

D₈₉

yes

characteristic, complemented

#1335

—

#1311

—

C₈₉

#12047

yes

characteristic, complemented, Sylow

#508

—

#504

—

178

—

#443

—

#368

—

#364

C₂×D₈₉

yes

—

#359

—

#350

—

178

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₈₉

yes

characteristic

C₈²

—

#135

—

#131

—

178

—

Q₃₂

—

SD₃₂

—

D₁₆

—

M₃₂

—

M₃₂

—

356

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

356

—

C₂×C₁₆

C₄×D₈₉

yes

characteristic

C₂×C₁₆

C₄×D₈₉

yes

characteristic

#108

—

100

#104

—

178

—

101

C₄×C₈

—

178

—

102

C₄×C₈

C₂²×D₈₉

yes

characteristic

103

#41

—

178

—

104

Q₁₆

—

105

SD₁₆

—

178

—

106

D₈

—

107

M₁₆

—

356

—

108

M₁₆

—

178

—

109

C₂×C₈

—

356

—

110

C₂×C₈

—

178

—

111

C₂×C₈

C₂×C₄×D₈₉

yes

characteristic

112

C₂×C₈

D₄×D₈₉

yes

characteristic

113

C₄²

#13083575

yes

characteristic

114

C₁₆

—

115

C₁₆

—

116

C₁₆

—

117

C₁₆

—

178

—

118

C₁₆

C₈×D₈₉

yes

characteristic

119

C₁₆

C₈×D₈₉

yes

characteristic

120

Q₈

—

178

—

121

D₄

—

356

—

122

D₄

—

178

—

123

C₂×C₄

—

124

C₂×C₄

—

356

—

125

C₂×C₄

#15667974

yes

characteristic

126

C₈

—

356

—

127

C₈

—

356

—

128

C₈

C₂×C₈×D₈₉

yes

characteristic

129

C₈

#15668020

yes

characteristic

130

C₈

D₈×D₈₉

yes

characteristic

131

C₈

#15668035

yes

characteristic

132

C₂²

—

356

—

133

C₂²

#19151714

yes

characteristic

134

C₄

—

135

C₄

—

356

—

136

C₄

#19151663

yes

characteristic

137

C₄

#19151680

yes

characteristic

138

C₂

—

139

C₂

—

712

—

140

C₂

#26019046

yes

characteristic

	subgroup	G/H
center	C₈	D₈×D₈₉
commutator	C₈×C₈₉	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₈₉
Fitting	C₈₉×#1373	C₂
socle	C₂×C₈₉	C₈×D₈
prime radical	#19152030	C₄
cosocle	C₄×C₈×C₈₉	C₂³
hypercenter	#1373	D₈₉
hypocenter	C₈₉	#12047
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1373

subgroup

G/H

center

C₈

D₈×D₈₉

commutator

C₈×C₈₉

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₈₉

Fitting

C₈₉×#1373

C₂

socle

C₂×C₈₉

C₈×D₈

prime radical

#19152030

C₄

cosocle

C₄×C₈×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1373

N	H ≅ G/N	type
2. #26022088	C₂	semidirect
3. #26018224	C₂	semidirect
4. #26018168	C₂	semidirect
5. C₈₉×#1373	C₂	semidirect
18. Dic₇₁₂	C₈	semidirect
19. #15667957	C₈	semidirect
20. D₁₄₂₄	C₈	semidirect
73. #1373	D₈₉	semidirect
76. C₈₉	#12047	semidirect

H ≅ G/N

type

2. #26022088

C₂

semidirect

3. #26018224

C₂

semidirect

4. #26018168

C₂

semidirect

5. C₈₉×#1373

C₂

semidirect

18. Dic₇₁₂

C₈

semidirect

19. #15667957

C₈

semidirect

20. D₁₄₂₄

C₈

semidirect

73. #1373

D₈₉

semidirect

76. C₈₉

#12047

semidirect

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

subgroup	index
#26022134	2
#26022088	2
#26018224	2
#26018168	2
C₈₉×#1373	2
#12047	89

subgroup

index

C₈₉×#1373

Subgroups — #36466188

Conjugacy classes of subgroups

Complements

Maximal subgroups