Group #36469152 — subgroups

Subgroups — #36469152

Order 22784.

330 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 87 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	#26022509	C₂	1	2	yes	maximal, complemented
2	#26022502	C₂	1	1	yes	maximal, characteristic
3	#26019859	C₂	1	2	yes	maximal, complemented
4	C₁₆×#10623494	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈₉×#2247	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15667955	—	2	4	no	—
7	C₂×D₁₄₂₄	—	2	2	no	—
8	#19151975	C₂²	1	2	yes	—
9	#19151946	—	2	4	no	—
10	C₄×#13083549	C₂²	1	1	yes	characteristic
11	C₈×#10623494	C₂²	1	1	yes	characteristic
12	C₂×Q₃₂×C₈₉	—	2	2	no	—
13	#494×C₈₉	C₂²	1	2	yes	—
14	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
15	C₂×D₇₁₂	—	2	2	no	—
16	C₂×D₇₁₂	—	2	2	no	—
17	#15668009	—	2	2	no	—
18	#15667955	—	4	8	no	—
19	D₁₄₂₄	—	4	4	no	—
20	C₂×#13083549	D₄	1	2	yes	—
21	#15667932	—	4	8	no	—
22	C₄×#10623494	D₄	1	2	yes	—
23	C₂×Q₁₆×C₈₉	—	2	2	no	—
24	C₂×Q₁₆×C₈₉	—	2	2	no	—
25	C₄⋊Q₈×C₈₉	—	2	2	no	—
26	Q₃₂×C₈₉	—	4	4	no	—
27	C₂×C₁₆×C₈₉	D₄	1	2	yes	—
28	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
29	C₂×D₃₅₆	—	4	2	no	—
30	C₂×D₃₅₆	—	4	4	no	—
31	D₇₁₂	—	4	4	no	—
32	D₇₁₂	—	4	4	no	—
33	C₂×#10623494	D₈	1	4	yes	—
34	#13083549	D₈	1	4	yes	—
35	C₂×Q₈×C₈₉	—	4	2	no	—
36	Q₁₆×C₈₉	—	4	4	no	—
37	Q₁₆×C₈₉	—	4	4	no	—
38	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
40	C₄⋊C₄×C₈₉	—	4	4	no	—
41	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
42	C₁₆×C₈₉	D₈	1	4	yes	—
43	C₂²×D₈₉	—	8	2	no	—
44	Q₈×C₈₉	—	8	2	no	—
45	Q₈×C₈₉	—	8	2	no	—
46	C₂×C₄×C₈₉	—	8	2	no	—
47	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
48	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
49	D₃₅₆	—	8	2	no	—
50	D₃₅₆	—	8	2	no	—
51	D₃₅₆	—	8	8	no	—
52	C₈×C₈₉	C₂×D₈	1	2	yes	—
53	C₈×C₈₉	C₂×D₈	1	2	yes	—
54	#10623494	—	2	4	no	—
55	C₂²×C₈₉	#493	1	1	yes	characteristic
56	C₂×D₈₉	—	16	2	no	—
57	C₂×D₈₉	—	16	4	no	—
58	C₄×C₈₉	—	16	2	no	—
59	C₄×C₈₉	#493	1	1	yes	characteristic
60	C₄×C₈₉	#493	1	1	yes	characteristic
61	C₄×C₈₉	#508	1	4	yes	—
62	#12518	—	89	1	no	maximal, Sylow
63	C₂×C₈₉	#1712	1	1	yes	characteristic
64	C₂×C₈₉	#2249	1	2	yes	—
65	D₈₉	—	32	2	no	—
66	#2248	—	89	2	no	—
67	#2247	D₈₉	1	1	yes	characteristic, complemented
68	#2246	—	89	1	no	—
69	#1709	—	89	2	no	—
70	C₈×C₁₆	—	89	1	no	—
71	C₈₉	#12518	1	1	yes	characteristic, complemented, Sylow
72	C₂×Q₃₂	—	2	2	no	—
73	C₂×SD₃₂	—	178	4	no	—
74	C₂×D₁₆	—	178	2	no	—
75	#494	C₂×D₈₉	1	2	yes	—
76	#493	—	89	2	no	—
77	#486	—	178	4	no	—
78	C₄×C₁₆	—	89	1	no	—
79	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
80	C₈²	—	89	1	no	—
81	C₂×Q₁₆	—	2	2	no	—
82	C₂×Q₁₆	—	2	2	no	—
83	C₂×D₈	—	178	2	no	—
84	C₂×D₈	—	178	2	no	—
85	C₄⋊Q₈	—	2	2	no	—
86	#127	—	178	2	no	—
87	Q₃₂	—	4	4	no	—
88	SD₃₂	—	356	8	no	—
89	D₁₆	—	356	4	no	—
90	C₂×C₁₆	—	89	2	no	—
91	C₂×C₁₆	C₂²⋊D₈₉	1	2	yes	—
92	#102	—	356	8	no	—
93	C₄×C₈	—	89	2	no	—
94	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
95	C₂×Q₈	—	4	2	no	—
96	C₂×D₄	—	356	2	no	—
97	C₂×D₄	—	356	4	no	—
98	Q₁₆	—	4	4	no	—
99	Q₁₆	—	4	4	no	—
100	D₈	—	356	4	no	—
101	D₈	—	356	4	no	—
102	C₂×C₈	—	89	4	no	—
103	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
104	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
105	C₄⋊C₄	—	4	4	no	—
106	C₄²	D₄×D₈₉	1	1	yes	characteristic
107	C₁₆	—	89	4	no	—
108	C₁₆	#13083559	1	4	yes	—
109	C₂³	—	712	2	no	—
110	Q₈	—	8	2	no	—
111	Q₈	—	8	2	no	—
112	D₄	—	712	2	no	—
113	D₄	—	712	2	no	—
114	D₄	—	712	8	no	—
115	C₂×C₄	—	8	2	no	—
116	C₂×C₄	#15668008	1	1	yes	characteristic
117	C₂×C₄	D₈×D₈₉	1	2	yes	—
118	C₈	—	178	4	no	—
119	C₈	D₈×D₈₉	1	2	yes	—
120	C₈	C₂×#13083559	1	2	yes	—
121	C₂²	—	1424	2	no	—
122	C₂²	—	1424	4	no	—
123	C₂²	#19151973	1	1	yes	characteristic
124	C₄	—	16	2	no	—
125	C₄	#19151973	1	1	yes	characteristic
126	C₄	#19151974	1	1	yes	characteristic
127	C₄	#19152031	1	4	yes	—
128	C₂	—	2848	2	no	—
129	C₂	#26019866	1	1	yes	characteristic
130	C₂	#26022514	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26022509

C₂

yes

maximal, complemented

#26022502

C₂

yes

maximal, characteristic

#26019859

C₂

yes

maximal, complemented

C₁₆×#10623494

C₂

yes

maximal, characteristic, complemented

C₈₉×#2247

C₂

yes

maximal, characteristic, complemented

C₂×#15667955

—

C₂×D₁₄₂₄

—

#19151975

C₂²

yes

—

#19151946

—

C₄×#13083549

C₂²

yes

characteristic

C₈×#10623494

C₂²

yes

characteristic

C₂×Q₃₂×C₈₉

—

#494×C₈₉

C₂²

yes

—

C₄×C₁₆×C₈₉

C₂²

yes

characteristic

C₂×D₇₁₂

—

C₂×D₇₁₂

—

#15668009

—

#15667955

—

D₁₄₂₄

—

C₂×#13083549

D₄

yes

—

#15667932

—

C₄×#10623494

D₄

yes

—

C₂×Q₁₆×C₈₉

—

C₂×Q₁₆×C₈₉

—

C₄⋊Q₈×C₈₉

—

Q₃₂×C₈₉

—

C₂×C₁₆×C₈₉

D₄

yes

—

C₄×C₈×C₈₉

C₂³

yes

characteristic

C₂×D₃₅₆

—

C₂×D₃₅₆

—

D₇₁₂

—

D₇₁₂

—

C₂×#10623494

D₈

yes

—

#13083549

D₈

yes

—

C₂×Q₈×C₈₉

—

Q₁₆×C₈₉

—

Q₁₆×C₈₉

—

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

C₂×D₄

yes

characteristic

C₁₆×C₈₉

D₈

yes

—

C₂²×D₈₉

—

Q₈×C₈₉

—

Q₈×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

#127

yes

characteristic

C₂×C₄×C₈₉

C₂×D₈

yes

—

D₃₅₆

—

D₃₅₆

—

D₃₅₆

—

C₈×C₈₉

C₂×D₈

yes

—

C₈×C₈₉

C₂×D₈

yes

—

#10623494

—

C₂²×C₈₉

#493

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

—

—

C₂×C₈₉

yes

C₂×C₈₉

yes

—

D₈₉

—

#2248

—

#2247

D₈₉

yes

characteristic, complemented

#2246

—

#1709

—

C₈×C₁₆

—

C₈₉

#12518

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

C₂×SD₃₂

—

178

—

C₂×D₁₆

—

178

—

#494

C₂×D₈₉

yes

—

#493

—

#486

—

178

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₈₉

yes

characteristic

C₈²

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×D₈

—

178

—

C₂×D₈

—

178

—

C₄⋊Q₈

—

#127

—

178

—

Q₃₂

—

SD₃₂

—

356

—

D₁₆

—

356

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₈₉

yes

—

#102

—

356

—

C₄×C₈

—

C₄×C₈

C₂²×D₈₉

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

356

—

C₂×D₄

—

356

—

Q₁₆

—

Q₁₆

—

100

D₈

—

356

—

101

D₈

—

356

—

102

C₂×C₈

—

103

C₂×C₈

D₄×D₈₉

yes

characteristic

104

C₂×C₈

C₂×C₂²⋊D₈₉

yes

characteristic

105

C₄⋊C₄

—

106

C₄²

D₄×D₈₉

yes

characteristic

107

C₁₆

—

108

C₁₆

#13083559

yes

—

109

C₂³

—

712

—

110

Q₈

—

111

Q₈

—

112

D₄

—

712

—

113

D₄

—

712

—

114

D₄

—

712

—

115

C₂×C₄

—

116

C₂×C₄

#15668008

yes

characteristic

117

C₂×C₄

D₈×D₈₉

yes

—

118

C₈

—

178

—

119

C₈

D₈×D₈₉

yes

—

120

C₈

C₂×#13083559

yes

—

121

C₂²

—

1424

—

122

C₂²

—

1424

—

123

C₂²

#19151973

yes

characteristic

124

C₄

—

125

C₄

#19151973

yes

characteristic

126

C₄

#19151974

yes

characteristic

127

C₄

#19152031

yes

—

128

C₂

—

2848

—

129

C₂

#26019866

yes

characteristic

130

C₂

#26022514

yes

—

	subgroup	G/H
center	C₂²	#19151973
commutator, cosocle	C₄×C₈×C₈₉	C₂³
Frattini	C₄×C₈	C₂²×D₈₉
Fitting	C₈₉×#2247	C₂
socle	C₂²×C₈₉	#493
prime radical	#26022502	C₂
hypercenter	#2247	D₈₉
hypocenter	C₈₉	#12518
LCS	G ▷ C₄×C₈×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2247

subgroup

G/H

center

C₂²

#19151973

commutator, cosocle

C₄×C₈×C₈₉

C₂³

Frattini

C₄×C₈

C₂²×D₈₉

Fitting

C₈₉×#2247

C₂

C₂²×C₈₉

C₂

D₈₉

C₈₉

G ▷ C₄×C₈×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2247

N	H ≅ G/N	type
1. #26022509	C₂	semidirect
3. #26019859	C₂	semidirect
4. C₁₆×#10623494	C₂	semidirect
5. C₈₉×#2247	C₂	semidirect
67. #2247	D₈₉	semidirect
71. C₈₉	#12518	semidirect

H ≅ G/N

type

1. #26022509

C₂

semidirect

3. #26019859

C₂

semidirect

4. C₁₆×#10623494

C₂

semidirect

5. C₈₉×#2247

C₂

semidirect

67. #2247

D₈₉

semidirect

71. C₈₉

#12518

semidirect

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

subgroup	index
#26022509	2
#26022502	2
#26019859	2
C₁₆×#10623494	2
C₈₉×#2247	2
#12518	89

subgroup

index

#26022509

#26022502

#26019859

C₁₆×#10623494

C₈₉×#2247

#12518

Subgroups — #36469152

Conjugacy classes of subgroups

Complements

Maximal subgroups