Group #36466452 — subgroups

Subgroups — #36466452

Order 22784.

226 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 69 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	#26022129	C₂	1	1	yes	maximal, characteristic
2	#26019165	C₂	1	2	yes	maximal, complemented
3	#26018218	C₂	1	2	yes	maximal, complemented
4	#26018172	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈₉×#1368	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₄₂₄	C₄	1	1	yes	characteristic
7	C₄×D₇₁₂	C₂²	1	2	yes	—
8	#19151675	—	2	2	no	—
9	#19151426	C₄	1	1	yes	characteristic
10	#19151400	C₄	1	2	yes	—
11	#19151374	C₂²	1	1	yes	characteristic
12	#19151325	—	2	2	no	—
13	#19151319	C₂²	1	1	yes	characteristic
14	#346×C₈₉	C₂²	1	1	yes	characteristic
15	#334×C₈₉	C₂²	1	2	yes	—
16	C₂×D₇₁₂	C₂×C₄	1	2	yes	—
17	C₂×#13083557	—	4	2	no	—
18	C₄×D₃₅₆	—	2	2	no	—
19	D₁₄₂₄	—	2	2	no	—
20	C₂×#13083549	—	2	1	no	—
21	#15667946	C₂×C₄	1	2	yes	—
22	#15667934	—	2	2	no	—
23	#15667924	—	4	2	no	—
24	C₄×#10623494	D₄	1	2	yes	—
25	C₂×C₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
27	C₄⋊C₈×C₈₉	—	2	2	no	—
28	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
29	C₂×D₃₅₆	—	2	2	no	—
30	C₂×C₄×D₈₉	—	4	2	no	—
31	D₇₁₂	—	2	2	no	—
32	D₇₁₂	—	2	2	no	—
33	#13083557	—	8	4	no	—
34	C₂×#10623494	—	2	2	no	—
35	C₄⋊Dic₈₉	—	2	2	no	—
36	#13083551	—	4	2	no	—
37	#13083549	—	4	1	no	—
38	C₂×C₈×C₈₉	—	4	2	no	—
39	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
41	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
42	C₁₆×C₈₉	—	2	1	no	—
43	C₁₆×C₈₉	M₁₆	1	2	yes	complemented
44	C₂²×D₈₉	—	4	2	no	—
45	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
46	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
47	C₂×Dic₈₉	—	4	2	no	—
48	D₃₅₆	—	4	2	no	—
49	D₃₅₆	—	4	2	no	—
50	C₄×D₈₉	—	8	2	no	—
51	C₄×D₈₉	—	8	2	no	—
52	C₈×C₈₉	—	2	1	no	—
53	C₈×C₈₉	—	8	2	no	—
54	C₈×C₈₉	C₂×M₁₆	1	1	yes	characteristic
55	C₈×C₈₉	#131	1	1	yes	characteristic
56	#10623494	—	4	2	no	—
57	C₂²×C₈₉	#442	1	1	yes	characteristic
58	C₂×D₈₉	—	8	2	no	—
59	C₂×D₈₉	—	8	2	no	—
60	C₂×D₈₉	—	8	2	no	—
61	C₄×C₈₉	—	2	2	no	—
62	C₄×C₈₉	#436	1	1	yes	characteristic
63	C₄×C₈₉	#444	1	1	yes	characteristic
64	Dic₈₉	—	8	2	no	—
65	#12089	—	89	1	no	maximal, Sylow
66	C₂×C₈₉	#1594	1	1	yes	characteristic
67	C₂×C₈₉	AGL(1,ℤ/16)	1	2	yes	—
68	D₈₉	—	16	2	no	—
69	#2177	—	89	1	no	—
70	#1594	—	89	2	no	—
71	#1368	D₈₉	1	1	yes	characteristic, complemented
72	#1333	—	89	2	no	—
73	#1313	—	89	1	no	—
74	C₈₉	#12089	1	1	yes	characteristic, complemented, Sylow
75	C₂×D₁₆	—	89	1	no	—
76	C₄×D₈	—	89	2	no	—
77	#436	—	178	2	no	—
78	#367	—	89	1	no	—
79	#357	—	89	2	no	—
80	#346	—	89	1	no	—
81	#346	C₂×D₈₉	1	1	yes	characteristic
82	#334	C₂×D₈₉	1	2	yes	—
83	D₄⋊C₈	—	178	2	no	—
84	#322	—	89	1	no	—
85	C₂×D₈	—	89	2	no	—
86	C₂×M₁₆	—	356	2	no	—
87	C₄×D₄	—	178	2	no	—
88	D₁₆	—	178	2	no	—
89	C₂×C₁₆	—	178	1	no	—
90	C₂×C₁₆	C₄×D₈₉	1	1	yes	characteristic
91	C₂×C₁₆	C₄×D₈₉	1	1	yes	characteristic
92	#107	—	89	2	no	—
93	C₄⋊C₈	—	2	2	no	—
94	#102	—	178	2	no	—
95	C₂²⋊C₈	—	356	2	no	—
96	C₄×C₈	—	89	2	no	—
97	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
98	C₂×D₄	—	178	2	no	—
99	C₂²×C₄	—	356	2	no	—
100	D₈	—	178	2	no	—
101	D₈	—	178	2	no	—
102	M₁₆	—	712	4	no	—
103	C₂×C₈	—	4	2	no	—
104	C₂×C₈	—	178	2	no	—
105	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
106	C₂×C₈	#13083575	1	1	yes	characteristic
107	C₄⋊C₄	—	178	2	no	—
108	C₂²⋊C₄	—	356	2	no	—
109	C₄²	D₄×D₈₉	1	1	yes	characteristic
110	C₁₆	—	2	1	no	—
111	C₁₆	—	356	1	no	—
112	C₁₆	#13083557	1	2	yes	complemented
113	C₂³	—	356	2	no	—
114	D₄	—	356	2	no	—
115	D₄	—	356	2	no	—
116	C₂×C₄	—	356	2	no	—
117	C₂×C₄	—	712	2	no	—
118	C₂×C₄	—	712	2	no	—
119	C₂×C₄	#15667974	1	1	yes	characteristic
120	C₂×C₄	#15668039	1	2	yes	—
121	C₈	—	2	1	no	—
122	C₈	—	8	2	no	—
123	C₈	—	356	2	no	—
124	C₈	C₂×#13083557	1	1	yes	characteristic
125	C₈	#15668020	1	1	yes	characteristic
126	C₂²	—	712	2	no	—
127	C₂²	—	712	2	no	—
128	C₂²	—	712	2	no	—
129	C₂²	#19151709	1	1	yes	characteristic
130	C₄	—	2	2	no	—
131	C₄	—	712	2	no	—
132	C₄	#19151675	1	1	yes	characteristic
133	C₄	#19151719	1	1	yes	characteristic
134	C₂	—	1424	2	no	—
135	C₂	#26019165	1	1	yes	characteristic
136	C₂	#26022149	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26022129

C₂

yes

maximal, characteristic

#26019165

C₂

yes

maximal, complemented

#26018218

C₂

yes

maximal, complemented

#26018172

C₂

yes

maximal, characteristic, complemented

C₈₉×#1368

C₂

yes

maximal, characteristic, complemented

C₂×D₁₄₂₄

C₄

yes

characteristic

C₄×D₇₁₂

C₂²

yes

—

#19151675

—

#19151426

C₄

yes

characteristic

#19151400

C₄

yes

—

#19151374

C₂²

yes

characteristic

#19151325

—

#19151319

C₂²

yes

characteristic

#346×C₈₉

C₂²

yes

characteristic

#334×C₈₉

C₂²

yes

—

C₂×D₇₁₂

C₂×C₄

yes

—

C₂×#13083557

—

C₄×D₃₅₆

—

D₁₄₂₄

—

C₂×#13083549

—

#15667946

C₂×C₄

yes

—

#15667934

—

#15667924

—

C₄×#10623494

D₄

yes

—

C₂×C₁₆×C₈₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₈₉

C₂×C₄

yes

characteristic

C₄⋊C₈×C₈₉

—

C₄×C₈×C₈₉

C₂³

yes

characteristic

C₂×D₃₅₆

—

C₂×C₄×D₈₉

—

D₇₁₂

—

D₇₁₂

—

#13083557

—

C₂×#10623494

—

C₄⋊Dic₈₉

—

#13083551

—

#13083549

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₈₉

#41

yes

characteristic

C₄²×C₈₉

C₂×D₄

yes

characteristic

C₁₆×C₈₉

—

C₁₆×C₈₉

M₁₆

yes

complemented

C₂²×D₈₉

—

C₂×C₄×C₈₉

C₄×D₄

yes

characteristic

C₂×C₄×C₈₉

AGL(1,ℤ/8)

yes

—

C₂×Dic₈₉

—

D₃₅₆

—

D₃₅₆

—

C₄×D₈₉

—

C₄×D₈₉

—

C₈×C₈₉

—

C₈×C₈₉

—

C₈×C₈₉

C₂×M₁₆

yes

C₈×C₈₉

yes

—

C₂²×C₈₉

#442

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

Dic₈₉

—

—

C₂×C₈₉

yes

C₂×C₈₉

yes

—

D₈₉

—

#2177

—

#1594

—

#1368

D₈₉

yes

characteristic, complemented

#1333

—

#1313

—

C₈₉

#12089

yes

characteristic, complemented, Sylow

C₂×D₁₆

—

C₄×D₈

—

#436

—

178

—

—

—

—

C₂×D₈₉

yes

characteristic

#334

C₂×D₈₉

yes

—

D₄⋊C₈

—

178

—

#322

—

C₂×D₈

—

C₂×M₁₆

—

356

—

C₄×D₄

—

178

—

D₁₆

—

178

—

C₂×C₁₆

—

178

—

C₂×C₁₆

C₄×D₈₉

yes

characteristic

C₂×C₁₆

C₄×D₈₉

yes

characteristic

#107

—

C₄⋊C₈

—

#102

—

178

—

C₂²⋊C₈

—

356

—

C₄×C₈

—

C₄×C₈

C₂²×D₈₉

yes

characteristic

C₂×D₄

—

178

—

C₂²×C₄

—

356

—

100

D₈

—

178

—

101

D₈

—

178

—

102

M₁₆

—

712

—

103

C₂×C₈

—

104

C₂×C₈

—

178

—

105

C₂×C₈

C₂×C₄×D₈₉

yes

characteristic

106

C₂×C₈

#13083575

yes

characteristic

107

C₄⋊C₄

—

178

—

108

C₂²⋊C₄

—

356

—

109

C₄²

D₄×D₈₉

yes

characteristic

110

C₁₆

—

111

C₁₆

—

356

—

112

C₁₆

#13083557

yes

complemented

113

C₂³

—

356

—

114

D₄

—

356

—

115

D₄

—

356

—

116

C₂×C₄

—

356

—

117

C₂×C₄

—

712

—

118

C₂×C₄

—

712

—

119

C₂×C₄

#15667974

yes

characteristic

120

C₂×C₄

#15668039

yes

—

121

C₈

—

122

C₈

—

123

C₈

—

356

—

124

C₈

C₂×#13083557

yes

characteristic

125

C₈

#15668020

yes

characteristic

126

C₂²

—

712

—

127

C₂²

—

712

—

128

C₂²

—

712

—

129

C₂²

#19151709

yes

characteristic

130

C₄

—

131

C₄

—

712

—

132

C₄

#19151675

yes

characteristic

133

C₄

#19151719

yes

characteristic

134

C₂

—

1424

—

135

C₂

#26019165

yes

characteristic

136

C₂

#26022149

yes

—

	subgroup	G/H
center	C₂²	#19151709
commutator	C₂×C₈×C₈₉	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₈₉
Fitting	C₈₉×#1368	C₂
socle	C₂²×C₈₉	#442
prime radical	C₂×D₁₄₂₄	C₄
cosocle	C₄×C₈×C₈₉	C₂³
hypercenter	#1368	D₈₉
hypocenter	C₈₉	#12089
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1368

subgroup

G/H

center

C₂²

#19151709

commutator

C₂×C₈×C₈₉

C₂²×C₄

Frattini

C₄×C₈

C₂²×D₈₉

Fitting

C₈₉×#1368

C₂

socle

C₂²×C₈₉

#442

prime radical

C₂×D₁₄₂₄

C₄

cosocle

C₄×C₈×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1368

N	H ≅ G/N	type
2. #26019165	C₂	semidirect
3. #26018218	C₂	semidirect
4. #26018172	C₂	semidirect
5. C₈₉×#1368	C₂	semidirect
43. C₁₆×C₈₉	M₁₆	semidirect
71. #1368	D₈₉	semidirect
74. C₈₉	#12089	semidirect
112. C₁₆	#13083557	semidirect

H ≅ G/N

type

2. #26019165

C₂

semidirect

3. #26018218

C₂

semidirect

4. #26018172

C₂

semidirect

5. C₈₉×#1368

C₂

semidirect

43. C₁₆×C₈₉

M₁₆

semidirect

71. #1368

D₈₉

semidirect

74. C₈₉

#12089

semidirect

112. C₁₆

#13083557

semidirect

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

subgroup	index
#26022129	2
#26019165	2
#26018218	2
#26018172	2
C₈₉×#1368	2
#12089	89

subgroup

index

C₈₉×#1368

Subgroups — #36466452

Conjugacy classes of subgroups

Complements

Maximal subgroups