Group #36469157 — subgroups

Subgroups — #36469157

Order 22784.

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

normal subgroups only

This group has 360 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	#26022501	C₂	1	2	yes	maximal, complemented
3	#26019867	C₂	1	1	yes	maximal, characteristic, complemented
4	#26018169	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈₉×#1709	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15667955	—	2	4	no	—
7	C₂×#15667953	—	2	4	no	—
8	#19151975	—	2	2	no	—
9	#19151975	C₂²	1	2	yes	—
10	C₄×#13083549	C₂²	1	2	yes	—
11	#494×C₈₉	C₂²	1	1	yes	characteristic
12	#493×C₈₉	C₂²	1	1	yes	characteristic
13	#486×C₈₉	—	2	2	no	—
14	C₈²×C₈₉	C₂²	1	1	yes	characteristic
15	C₂×D₇₁₂	—	2	2	no	—
16	C₂×D₇₁₂	—	2	2	no	—
17	C₂×D₇₁₂	—	4	4	no	—
18	#15668009	—	2	2	no	—
19	#15667955	—	4	8	no	—
20	#15667953	—	4	8	no	—
21	C₂×#13083549	D₄	1	4	yes	—
22	C₂×Q₁₆×C₈₉	—	2	2	no	—
23	C₂×D₈×C₈₉	—	2	2	no	—
24	C₄⋊Q₈×C₈₉	—	2	1	no	—
25	#127×C₈₉	—	2	1	no	—
26	#102×C₈₉	—	4	4	no	—
27	C₄×C₈×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	D₇₁₂	—	8	8	no	—
34	#13083549	D₈	1	8	yes	complemented
35	C₂×Q₈×C₈₉	—	4	1	no	—
36	C₂×D₄×C₈₉	—	4	1	no	—
37	C₂×D₄×C₈₉	—	4	2	no	—
38	Q₁₆×C₈₉	—	4	4	no	—
39	D₈×C₈₉	—	4	4	no	—
40	C₂×C₈×C₈₉	D₈	1	4	yes	—
41	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
42	C₄⋊C₄×C₈₉	—	4	2	no	—
43	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
44	C₂³×C₈₉	—	8	1	no	—
45	C₂²×D₈₉	—	8	2	no	—
46	Q₈×C₈₉	—	8	2	no	—
47	D₄×C₈₉	—	8	2	no	—
48	D₄×C₈₉	—	8	4	no	—
49	C₂×C₄×C₈₉	—	8	1	no	—
50	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
51	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
52	D₃₅₆	—	8	2	no	—
53	D₃₅₆	—	8	2	no	—
54	D₃₅₆	—	8	8	no	—
55	C₈×C₈₉	—	2	4	no	—
56	C₈×C₈₉	C₂×D₈	1	4	yes	—
57	C₂²×C₈₉	—	16	1	no	—
58	C₂²×C₈₉	—	16	2	no	—
59	C₂²×C₈₉	#493	1	1	yes	characteristic
60	C₂×D₈₉	—	16	2	no	—
61	C₂×D₈₉	—	16	4	no	—
62	C₄×C₈₉	—	16	1	no	—
63	C₄×C₈₉	#493	1	2	yes	—
64	C₄×C₈₉	#509	1	4	yes	—
65	#12519	—	89	1	no	maximal, Sylow
66	C₂×C₈₉	—	32	1	no	—
67	C₂×C₈₉	#1712	1	1	yes	characteristic
68	C₂×C₈₉	#2250	1	2	yes	—
69	D₈₉	—	32	2	no	—
70	#2248	—	89	2	no	—
71	#2246	—	89	2	no	—
72	#1712	—	89	1	no	—
73	#1709	D₈₉	1	1	yes	characteristic, complemented
74	#1311	—	89	1	no	—
75	C₈₉	#12519	1	1	yes	characteristic, complemented, Sylow
76	C₂×SD₃₂	—	178	4	no	—
77	C₂×D₁₆	—	178	4	no	—
78	#494	C₂×D₈₉	1	1	yes	characteristic
79	#493	—	89	2	no	—
80	#493	—	178	2	no	—
81	#493	C₂×D₈₉	1	1	yes	characteristic
82	#486	—	2	2	no	—
83	C₄×C₁₆	—	89	2	no	—
84	C₈²	C₂×D₈₉	1	1	yes	characteristic
85	C₂×Q₁₆	—	2	2	no	—
86	C₂×D₈	—	2	2	no	—
87	C₂×D₈	—	356	4	no	—
88	C₂×D₈	—	178	2	no	—
89	C₂×D₈	—	178	2	no	—
90	C₄⋊Q₈	—	2	1	no	—
91	#127	—	2	1	no	—
92	#127	—	178	2	no	—
93	SD₃₂	—	356	8	no	—
94	D₁₆	—	356	8	no	—
95	C₂×C₁₆	—	89	4	no	—
96	#102	—	4	4	no	—
97	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
98	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
99	C₂×Q₈	—	4	1	no	—
100	C₂×D₄	—	4	1	no	—
101	C₂×D₄	—	4	2	no	—
102	C₂×D₄	—	356	2	no	—
103	C₂×D₄	—	356	4	no	—
104	Q₁₆	—	4	4	no	—
105	D₈	—	4	4	no	—
106	D₈	—	356	4	no	—
107	D₈	—	356	4	no	—
108	D₈	—	712	8	no	—
109	C₂×C₈	#13083559	1	4	yes	—
110	C₂×C₈	D₄×D₈₉	1	2	yes	—
111	C₄⋊C₄	—	4	2	no	—
112	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
113	C₁₆	—	89	8	no	—
114	C₂³	—	8	1	no	—
115	C₂³	—	712	2	no	—
116	Q₈	—	8	2	no	—
117	D₄	—	8	2	no	—
118	D₄	—	8	4	no	—
119	D₄	—	712	2	no	—
120	D₄	—	712	2	no	—
121	D₄	—	712	8	no	—
122	C₂×C₄	—	8	1	no	—
123	C₂×C₄	#15668008	1	1	yes	characteristic
124	C₂×C₄	C₂×#13083559	1	2	yes	—
125	C₈	—	2	4	no	—
126	C₈	D₈×D₈₉	1	4	yes	—
127	C₂²	—	16	1	no	—
128	C₂²	—	16	2	no	—
129	C₂²	—	1424	2	no	—
130	C₂²	—	1424	4	no	—
131	C₂²	#19151974	1	1	yes	characteristic
132	C₄	—	16	1	no	—
133	C₄	#19151973	1	2	yes	—
134	C₄	#19152033	1	4	yes	—
135	C₂	—	32	1	no	—
136	C₂	—	2848	2	no	—
137	C₂	#26019866	1	1	yes	characteristic
138	C₂	#26022519	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26022509

C₂

yes

maximal, complemented

#26022501

C₂

yes

maximal, complemented

#26019867

C₂

yes

maximal, characteristic, complemented

#26018169

C₂

yes

maximal, characteristic, complemented

C₈₉×#1709

C₂

yes

maximal, characteristic, complemented

—

—

—

C₂²

yes

—

C₄×#13083549

C₂²

yes

—

#494×C₈₉

C₂²

yes

characteristic

#493×C₈₉

C₂²

yes

characteristic

#486×C₈₉

—

C₈²×C₈₉

C₂²

yes

characteristic

C₂×D₇₁₂

—

C₂×D₇₁₂

—

C₂×D₇₁₂

—

—

—

—

D₄

yes

—

C₂×Q₁₆×C₈₉

—

C₂×D₈×C₈₉

—

C₄⋊Q₈×C₈₉

—

#127×C₈₉

—

#102×C₈₉

—

C₄×C₈×C₈₉

D₄

yes

—

C₄×C₈×C₈₉

C₂³

yes

characteristic

C₂×D₃₅₆

—

C₂×D₃₅₆

—

D₇₁₂

—

D₇₁₂

—

D₇₁₂

—

#13083549

D₈

yes

complemented

C₂×Q₈×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

Q₁₆×C₈₉

—

D₈×C₈₉

—

C₂×C₈×C₈₉

D₈

yes

—

C₂×C₈×C₈₉

C₂×D₄

yes

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

C₂×D₄

yes

characteristic

C₂³×C₈₉

—

C₂²×D₈₉

—

Q₈×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

#127

yes

characteristic

C₂×C₄×C₈₉

C₂×D₈

yes

—

D₃₅₆

—

D₃₅₆

—

D₃₅₆

—

C₈×C₈₉

—

C₈×C₈₉

C₂×D₈

yes

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

#493

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#493

yes

—

C₄×C₈₉

#509

yes

—

#12519

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

#1712

yes

characteristic

C₂×C₈₉

#2250

yes

—

D₈₉

—

—

—

—

D₈₉

yes

characteristic, complemented

#1311

—

C₈₉

#12519

yes

characteristic, complemented, Sylow

C₂×SD₃₂

—

178

—

C₂×D₁₆

—

178

—

#494

C₂×D₈₉

yes

characteristic

#493

—

#493

—

178

—

#493

C₂×D₈₉

yes

characteristic

#486

—

C₄×C₁₆

—

C₈²

C₂×D₈₉

yes

characteristic

C₂×Q₁₆

—

C₂×D₈

—

C₂×D₈

—

356

—

C₂×D₈

—

178

—

C₂×D₈

—

178

—

C₄⋊Q₈

—

#127

—

#127

—

178

—

SD₃₂

—

356

—

D₁₆

—

356

—

C₂×C₁₆

—

#102

—

C₄×C₈

C₂²⋊D₈₉

yes

—

C₄×C₈

C₂²×D₈₉

yes

characteristic

C₂×Q₈

—

100

C₂×D₄

—

101

C₂×D₄

—

102

C₂×D₄

—

356

—

103

C₂×D₄

—

356

—

104

Q₁₆

—

105

D₈

—

106

D₈

—

356

—

107

D₈

—

356

—

108

D₈

—

712

—

109

C₂×C₈

#13083559

yes

—

110

C₂×C₈

D₄×D₈₉

yes

—

111

C₄⋊C₄

—

112

C₄²

C₂×C₂²⋊D₈₉

yes

characteristic

113

C₁₆

—

114

C₂³

—

115

C₂³

—

712

—

116

Q₈

—

117

D₄

—

118

D₄

—

119

D₄

—

712

—

120

D₄

—

712

—

121

D₄

—

712

—

122

C₂×C₄

—

123

C₂×C₄

#15668008

yes

characteristic

124

C₂×C₄

C₂×#13083559

yes

—

125

C₈

—

126

C₈

D₈×D₈₉

yes

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

1424

—

130

C₂²

—

1424

—

131

C₂²

#19151974

yes

characteristic

132

C₄

—

133

C₄

#19151973

yes

—

134

C₄

#19152033

yes

—

135

C₂

—

136

C₂

—

2848

—

137

C₂

#26019866

yes

characteristic

138

C₂

#26022519

yes

—

	subgroup	G/H
center	C₂²	#19151974
commutator, cosocle	C₄×C₈×C₈₉	C₂³
Frattini	C₄×C₈	C₂²×D₈₉
Fitting	C₈₉×#1709	C₂
socle	C₂²×C₈₉	#493
hypercenter	#1709	D₈₉
hypocenter	C₈₉	#12519
LCS	G ▷ C₄×C₈×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1709

subgroup

G/H

center

C₂²

#19151974

commutator, cosocle

C₄×C₈×C₈₉

C₂³

Frattini

C₄×C₈

C₂²×D₈₉

Fitting

C₈₉×#1709

C₂

C₂²×C₈₉

D₈₉

C₈₉

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1709

N	H ≅ G/N	type
1. #26022509	C₂	semidirect
2. #26022501	C₂	semidirect
3. #26019867	C₂	semidirect
4. #26018169	C₂	semidirect
5. C₈₉×#1709	C₂	semidirect
34. #13083549	D₈	semidirect
73. #1709	D₈₉	semidirect
75. C₈₉	#12519	semidirect

H ≅ G/N

type

1. #26022509

C₂

semidirect

2. #26022501

C₂

semidirect

3. #26019867

C₂

semidirect

4. #26018169

C₂

semidirect

5. C₈₉×#1709

C₂

semidirect

34. #13083549

D₈

semidirect

73. #1709

D₈₉

semidirect

75. C₈₉

#12519

semidirect

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

subgroup	index
#26022509	2
#26022501	2
#26019867	2
#26018169	2
C₈₉×#1709	2
#12519	89

subgroup

index

C₈₉×#1709

Subgroups — #36469157

Conjugacy classes of subgroups

Complements

Maximal subgroups