Group #36466319 — subgroups

Subgroups — #36466319

Order 22784.

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

normal subgroups only

This group has 204 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	#26022139	C₂	1	1	yes	maximal, characteristic
2	#26022095	C₂	1	2	yes	maximal, complemented
3	#26018320	C₂	1	1	yes	maximal, characteristic, complemented
4	#26018229	C₂	1	2	yes	maximal, complemented
5	C₈₉×#1313	C₂	1	1	yes	maximal, characteristic, complemented
6	#19152030	C₄	1	1	yes	characteristic
7	#19152014	—	2	2	no	—
8	#19151716	C₂²	1	2	yes	—
9	#19151428	C₄	1	1	yes	characteristic
10	#19151419	C₂²	1	2	yes	—
11	#19151406	C₄	1	2	yes	—
12	#19151381	—	2	2	no	—
13	#346×C₈₉	C₂²	1	1	yes	characteristic
14	#346×C₈₉	C₂²	1	1	yes	characteristic
15	#322×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	#15667952	—	2	2	no	—
22	#15667951	—	4	2	no	—
23	C₁₆×D₈₉	—	4	2	no	—
24	#15667948	C₂×C₄	1	2	yes	—
25	#15667940	—	2	2	no	—
26	C₂×C₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
27	C₂×C₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₁₆×C₈₉	D₄	1	2	yes	—
29	C₄×C₈×C₈₉	—	2	1	no	—
30	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
31	#13083574	—	2	2	no	—
32	Dic₃₅₆	C₂×C₈	1	2	yes	—
33	#13083563	—	2	2	no	—
34	D₇₁₂	C₂×C₈	1	2	yes	—
35	#13083558	—	2	2	no	—
36	#13083557	—	4	2	no	—
37	C₈×D₈₉	—	4	2	no	—
38	#13083549	—	4	2	no	—
39	C₂×C₈×C₈₉	—	4	1	no	—
40	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
42	C₄²×C₈₉	#41	1	1	yes	characteristic
43	C₁₆×C₈₉	—	2	1	no	—
44	C₁₆×C₈₉	—	2	2	no	—
45	C₁₆×C₈₉	C₂×C₈	1	1	yes	characteristic
46	C₁₆×C₈₉	C₂×C₈	1	1	yes	characteristic
47	C₂×C₄×C₈₉	—	2	1	no	—
48	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
49	C₂²⋊D₈₉	—	4	2	no	—
50	D₃₅₆	—	2	2	no	—
51	C₄×D₈₉	—	4	2	no	—
52	Dic₁₇₈	—	2	2	no	—
53	C₈×C₈₉	—	8	1	no	—
54	C₈×C₈₉	C₂²×C₈	1	1	yes	characteristic
55	C₈×C₈₉	#131	1	1	yes	characteristic
56	C₈×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
57	#10623494	—	4	2	no	—
58	C₂²×C₈₉	#444	1	1	yes	characteristic
59	C₂×D₈₉	—	4	2	no	—
60	C₄×C₈₉	—	4	1	no	—
61	C₄×C₈₉	C₈×D₄	1	1	yes	characteristic
62	C₄×C₈₉	#442	1	1	yes	characteristic
63	Dic₈₉	—	4	2	no	—
64	#12068	—	89	1	no	maximal, Sylow
65	C₂×C₈₉	—	2	1	no	—
66	C₂×C₈₉	#1580	1	1	yes	characteristic
67	D₈₉	—	8	2	no	—
68	#2179	—	89	1	no	—
69	#2171	—	89	2	no	—
70	#1369	—	89	1	no	—
71	#1336	—	89	2	no	—
72	#1313	D₈₉	1	1	yes	characteristic, complemented
73	C₈₉	#12068	1	1	yes	characteristic, complemented, Sylow
74	#508	—	89	1	no	—
75	#504	—	178	2	no	—
76	#443	—	89	2	no	—
77	#368	—	89	1	no	—
78	#364	—	89	2	no	—
79	#359	—	89	2	no	—
80	#350	—	178	2	no	—
81	#346	C₂×D₈₉	1	1	yes	characteristic
82	#346	C₂×D₈₉	1	1	yes	characteristic
83	#322	C₂×D₈₉	1	1	yes	characteristic
84	#135	—	89	2	no	—
85	#131	—	178	2	no	—
86	Q₃₂	—	89	1	no	—
87	SD₃₂	—	89	2	no	—
88	D₁₆	—	89	1	no	—
89	M₃₂	—	356	2	no	—
90	M₃₂	—	178	2	no	—
91	C₂×C₁₆	—	356	2	no	—
92	C₂×C₁₆	C₄×D₈₉	1	1	yes	characteristic
93	C₂×C₁₆	C₄×D₈₉	1	1	yes	characteristic
94	C₂×C₁₆	D₃₅₆	1	2	yes	—
95	#108	—	89	2	no	—
96	#104	—	178	2	no	—
97	C₄×C₈	—	2	1	no	—
98	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
99	#41	—	178	2	no	—
100	Q₁₆	—	89	2	no	—
101	SD₁₆	—	178	2	no	—
102	D₈	—	89	2	no	—
103	M₁₆	—	356	2	no	—
104	M₁₆	—	178	2	no	—
105	C₂×C₈	—	4	1	no	—
106	C₂×C₈	—	356	2	no	—
107	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
108	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
109	C₄²	#13083574	1	1	yes	characteristic
110	C₁₆	—	2	1	no	—
111	C₁₆	—	2	2	no	—
112	C₁₆	—	356	2	no	—
113	C₁₆	C₈×D₈₉	1	1	yes	characteristic
114	C₁₆	C₈×D₈₉	1	1	yes	characteristic
115	Q₈	—	178	2	no	—
116	D₄	—	356	2	no	—
117	D₄	—	178	2	no	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	356	2	no	—
120	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
121	C₈	—	8	1	no	—
122	C₈	—	356	2	no	—
123	C₈	C₂×C₈×D₈₉	1	1	yes	characteristic
124	C₈	#15668018	1	1	yes	characteristic
125	C₈	#15668040	1	2	yes	—
126	C₂²	—	356	2	no	—
127	C₂²	#19151721	1	1	yes	characteristic
128	C₄	—	4	1	no	—
129	C₄	—	356	2	no	—
130	C₄	C₈×D₃₅₆	1	1	yes	characteristic
131	C₄	#19151711	1	1	yes	characteristic
132	C₂	—	2	1	no	—
133	C₂	—	712	2	no	—
134	C₂	#26019077	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26022139

C₂

yes

maximal, characteristic

#26022095

C₂

yes

maximal, complemented

#26018320

C₂

yes

maximal, characteristic, complemented

#26018229

C₂

yes

maximal, complemented

C₈₉×#1313

C₂

yes

maximal, characteristic, complemented

#19152030

C₄

yes

characteristic

#19152014

—

#19151716

C₂²

yes

—

#19151428

C₄

yes

characteristic

#19151419

C₂²

yes

—

#19151406

C₄

yes

—

#19151381

—

#346×C₈₉

C₂²

yes

characteristic

#346×C₈₉

C₂²

yes

characteristic

#322×C₈₉

C₂²

yes

characteristic

#15668034

C₂×C₄

yes

—

#15668018

—

Dic₇₁₂

C₈

yes

characteristic, complemented

#15667957

C₈

yes

complemented

D₁₄₂₄

C₈

yes

characteristic, complemented

#15667952

—

#15667951

—

C₁₆×D₈₉

—

#15667948

C₂×C₄

yes

—

#15667940

—

C₂×C₁₆×C₈₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₈₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₈₉

D₄

yes

—

C₄×C₈×C₈₉

—

C₄×C₈×C₈₉

C₂³

yes

characteristic

#13083574

—

Dic₃₅₆

C₂×C₈

yes

—

#13083563

—

D₇₁₂

C₂×C₈

yes

—

#13083558

—

#13083557

—

C₈×D₈₉

—

#13083549

—

C₂×C₈×C₈₉

—

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

—

C₈×C₈₉

C₂²×C₈

yes

C₈×C₈₉

yes

C₈×C₈₉

yes

—

#10623494

—

C₂²×C₈₉

#444

yes

characteristic

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₈×D₄

yes

characteristic

C₄×C₈₉

#442

yes

characteristic

Dic₈₉

—

#12068

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

#1580

yes

characteristic

D₈₉

—

—

—

—

—

D₈₉

yes

characteristic, complemented

C₈₉

#12068

yes

characteristic, complemented, Sylow

#508

—

#504

—

178

—

—

—

—

—

—

178

—

#346

C₂×D₈₉

yes

characteristic

#346

C₂×D₈₉

yes

characteristic

#322

C₂×D₈₉

yes

characteristic

#135

—

#131

—

178

—

Q₃₂

—

SD₃₂

—

D₁₆

—

M₃₂

—

356

—

M₃₂

—

178

—

C₂×C₁₆

—

356

—

C₂×C₁₆

C₄×D₈₉

yes

characteristic

C₂×C₁₆

C₄×D₈₉

yes

characteristic

C₂×C₁₆

D₃₅₆

yes

—

#108

—

#104

—

178

—

C₄×C₈

—

C₄×C₈

C₂²×D₈₉

yes

characteristic

#41

—

178

—

100

Q₁₆

—

101

SD₁₆

—

178

—

102

D₈

—

103

M₁₆

—

356

—

104

M₁₆

—

178

—

105

C₂×C₈

—

106

C₂×C₈

—

356

—

107

C₂×C₈

C₂×C₄×D₈₉

yes

characteristic

108

C₂×C₈

C₂×D₃₅₆

yes

characteristic

109

C₄²

#13083574

yes

characteristic

110

C₁₆

—

111

C₁₆

—

112

C₁₆

—

356

—

113

C₁₆

C₈×D₈₉

yes

characteristic

114

C₁₆

C₈×D₈₉

yes

characteristic

115

Q₈

—

178

—

116

D₄

—

356

—

117

D₄

—

178

—

118

C₂×C₄

—

119

C₂×C₄

—

356

—

120

C₂×C₄

C₄×D₃₅₆

yes

characteristic

121

C₈

—

122

C₈

—

356

—

123

C₈

C₂×C₈×D₈₉

yes

characteristic

124

C₈

#15668018

yes

characteristic

125

C₈

#15668040

yes

—

126

C₂²

—

356

—

127

C₂²

#19151721

yes

characteristic

128

C₄

—

129

C₄

—

356

—

130

C₄

C₈×D₃₅₆

yes

characteristic

131

C₄

#19151711

yes

characteristic

132

C₂

—

133

C₂

—

712

—

134

C₂

#26019077

yes

characteristic

	subgroup	G/H
center	C₄	#19151711
commutator	C₈×C₈₉	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₈₉
Fitting	C₈₉×#1313	C₂
socle	C₂×C₈₉	#1580
prime radical	#19152030	C₄
cosocle	C₄×C₈×C₈₉	C₂³
hypercenter	#1313	D₈₉
hypocenter	C₈₉	#12068
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #346 ◁ #1313

subgroup

G/H

center

C₄

#19151711

commutator

C₈×C₈₉

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₈₉

Fitting

C₈₉×#1313

C₂

C₂×C₈₉

C₄

C₄×C₈×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₄ ◁ C₂×C₈ ◁ #346 ◁ #1313

N	H ≅ G/N	type
2. #26022095	C₂	semidirect
3. #26018320	C₂	semidirect
4. #26018229	C₂	semidirect
5. C₈₉×#1313	C₂	semidirect
18. Dic₇₁₂	C₈	semidirect
19. #15667957	C₈	semidirect
20. D₁₄₂₄	C₈	semidirect
72. #1313	D₈₉	semidirect
73. C₈₉	#12068	semidirect

H ≅ G/N

type

2. #26022095

C₂

semidirect

3. #26018320

C₂

semidirect

4. #26018229

C₂

semidirect

5. C₈₉×#1313

C₂

semidirect

18. Dic₇₁₂

C₈

semidirect

19. #15667957

C₈

semidirect

20. D₁₄₂₄

C₈

semidirect

72. #1313

D₈₉

semidirect

73. C₈₉

#12068

semidirect

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

subgroup	index
#26022139	2
#26022095	2
#26018320	2
#26018229	2
C₈₉×#1313	2
#12068	89

subgroup

index

C₈₉×#1313

Subgroups — #36466319

Conjugacy classes of subgroups

Complements

Maximal subgroups