Group #36469086 — subgroups

Subgroups — #36469086

Order 22784.

262 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 99 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	#26022527	C₂	1	1	yes	maximal, characteristic
2	#26022505	C₂	1	1	yes	maximal, characteristic
3	#26022480	C₂	1	2	yes	maximal
4	#26019849	C₂	1	2	yes	maximal
5	C₈×C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₇₁₂	—	2	2	no	—
7	#19152003	C₂²	1	2	yes	—
8	#19151996	—	2	2	no	—
9	#19151978	C₂²	1	2	yes	—
10	#19151949	—	2	2	no	—
11	#19151424	—	2	2	no	—
12	#19151403	—	2	4	no	—
13	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
14	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
15	C₈²×C₈₉	C₂²	1	1	yes	characteristic
16	C₂×Dic₃₅₆	—	2	2	no	—
17	C₂×Dic₃₅₆	—	2	2	no	—
18	#15668012	—	2	2	no	—
19	#15668012	—	2	2	no	—
20	Dic₇₁₂	—	4	4	no	—
21	#15667946	—	2	2	no	—
22	#15667946	—	2	2	no	—
23	#15667944	—	4	4	no	—
24	#15667937	—	4	4	no	—
25	C₂×C₁₆×C₈₉	D₄	1	2	yes	—
26	C₂×C₁₆×C₈₉	D₄	1	2	yes	—
27	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
28	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
29	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
30	C₂×Dic₁₇₈	—	4	2	no	—
31	C₂×Dic₁₇₈	—	4	2	no	—
32	Dic₃₅₆	—	4	4	no	—
33	Dic₃₅₆	—	4	4	no	—
34	C₄⋊Dic₈₉	—	4	2	no	—
35	C₄⋊Dic₈₉	—	4	2	no	—
36	C₄⋊Dic₈₉	—	4	2	no	—
37	C₄⋊Dic₈₉	—	4	2	no	—
38	C₂×C₈×C₈₉	D₈	1	2	yes	—
39	C₂×C₈×C₈₉	D₈	1	2	yes	—
40	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
41	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
42	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
43	C₁₆×C₈₉	—	2	2	no	—
44	C₁₆×C₈₉	SD₁₆	1	4	yes	—
45	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
46	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
47	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
48	C₂×Dic₈₉	—	8	2	no	—
49	C₂×Dic₈₉	—	8	2	no	—
50	Dic₁₇₈	—	8	2	no	—
51	Dic₁₇₈	—	8	2	no	—
52	Dic₁₇₈	—	8	4	no	—
53	C₈×C₈₉	—	2	2	no	—
54	C₈×C₈₉	Q₃₂	1	4	yes	complemented
55	C₈×C₈₉	C₂×SD₁₆	1	2	yes	—
56	C₈×C₈₉	#135	1	2	yes	—
57	C₂²×C₈₉	#493	1	1	yes	characteristic
58	C₄×C₈₉	#492	1	1	yes	characteristic
59	C₄×C₈₉	#495	1	1	yes	characteristic
60	C₄×C₈₉	C₂×Q₃₂	1	2	yes	—
61	C₄×C₈₉	#508	1	2	yes	—
62	Dic₈₉	—	16	2	no	—
63	Dic₈₉	—	16	2	no	—
64	#12504	—	89	1	no	maximal, Sylow
65	C₂×C₈₉	#1706	1	1	yes	characteristic
66	C₂×C₈₉	#2247	1	1	yes	characteristic
67	C₂×C₈₉	#2249	1	1	yes	characteristic
68	#2252	—	89	1	no	—
69	#2247	—	89	1	no	—
70	#2241	—	89	2	no	—
71	#1707	—	89	2	no	—
72	C₈×C₁₆	D₈₉	1	1	yes	characteristic
73	C₈₉	#12504	1	1	yes	characteristic, complemented, Sylow
74	C₂×Q₃₂	—	178	2	no	—
75	#500	—	89	2	no	—
76	#498	—	178	2	no	—
77	#494	—	89	2	no	—
78	#487	—	178	2	no	—
79	#366	—	178	2	no	—
80	#358	—	178	4	no	—
81	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
82	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
83	C₈²	C₂×D₈₉	1	1	yes	characteristic
84	C₂×Q₁₆	—	178	2	no	—
85	C₂×Q₁₆	—	178	2	no	—
86	C₄⋊Q₈	—	178	2	no	—
87	C₄⋊Q₈	—	178	2	no	—
88	Q₃₂	—	356	4	no	—
89	C₂×C₁₆	D₃₅₆	1	2	yes	—
90	C₂×C₁₆	D₃₅₆	1	2	yes	—
91	#107	—	178	2	no	—
92	#107	—	178	2	no	—
93	#106	—	356	4	no	—
94	#103	—	356	4	no	—
95	C₄×C₈	D₃₅₆	1	1	yes	characteristic
96	C₄×C₈	D₃₅₆	1	1	yes	characteristic
97	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
98	C₂×Q₈	—	356	2	no	—
99	C₂×Q₈	—	356	2	no	—
100	Q₁₆	—	356	4	no	—
101	Q₁₆	—	356	4	no	—
102	C₂×C₈	D₇₁₂	1	2	yes	—
103	C₂×C₈	D₇₁₂	1	2	yes	—
104	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
105	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
106	C₄⋊C₄	—	356	2	no	—
107	C₄⋊C₄	—	356	2	no	—
108	C₄⋊C₄	—	356	2	no	—
109	C₄⋊C₄	—	356	2	no	—
110	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
111	C₁₆	—	2	2	no	—
112	C₁₆	#13083563	1	4	yes	—
113	Q₈	—	712	2	no	—
114	Q₈	—	712	2	no	—
115	Q₈	—	712	4	no	—
116	C₂×C₄	—	712	2	no	—
117	C₂×C₄	—	712	2	no	—
118	C₂×C₄	#15668009	1	1	yes	characteristic
119	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
120	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
121	C₈	—	2	2	no	—
122	C₈	Dic₇₁₂	1	4	yes	complemented
123	C₈	C₂×#13083563	1	2	yes	—
124	C₈	#15668034	1	2	yes	—
125	C₂²	#19151975	1	1	yes	characteristic
126	C₄	—	1424	2	no	—
127	C₄	—	1424	2	no	—
128	C₄	#19151971	1	1	yes	characteristic
129	C₄	#19151982	1	1	yes	characteristic
130	C₄	C₂×Dic₇₁₂	1	2	yes	—
131	C₄	#19152030	1	2	yes	—
132	C₂	#26019844	1	1	yes	characteristic
133	C₂	#26022505	1	1	yes	characteristic
134	C₂	#26022513	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26022527

C₂

yes

maximal, characteristic

#26022505

C₂

yes

maximal, characteristic

#26022480

C₂

yes

maximal

#26019849

C₂

yes

maximal

C₈×C₁₆×C₈₉

C₂

yes

maximal, characteristic

C₂×Dic₇₁₂

—

#19152003

C₂²

yes

—

#19151996

—

#19151978

C₂²

yes

—

#19151949

—

#19151424

—

#19151403

—

C₄×C₁₆×C₈₉

C₂²

yes

characteristic

C₄×C₁₆×C₈₉

C₂²

yes

characteristic

C₈²×C₈₉

C₂²

yes

characteristic

C₂×Dic₃₅₆

—

C₂×Dic₃₅₆

—

#15668012

—

#15668012

—

Dic₇₁₂

—

—

—

—

—

C₂×C₁₆×C₈₉

D₄

yes

—

C₂×C₁₆×C₈₉

D₄

yes

—

C₄×C₈×C₈₉

D₄

yes

characteristic

C₄×C₈×C₈₉

D₄

yes

characteristic

C₄×C₈×C₈₉

C₂³

yes

characteristic

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

Dic₃₅₆

—

Dic₃₅₆

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₂×C₈×C₈₉

D₈

yes

—

C₂×C₈×C₈₉

D₈

yes

—

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₄²×C₈₉

C₂×D₄

yes

characteristic

C₁₆×C₈₉

—

C₁₆×C₈₉

SD₁₆

yes

—

C₂×C₄×C₈₉

#127

yes

characteristic

C₂×C₄×C₈₉

C₂×D₈

yes

characteristic

C₂×C₄×C₈₉

C₂×D₈

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

Q₃₂

yes

complemented

C₈×C₈₉

C₂×SD₁₆

yes

—

C₈×C₈₉

#135

yes

—

C₂²×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

C₂×Q₃₂

yes

—

C₄×C₈₉

#508

yes

—

Dic₈₉

—

Dic₈₉

—

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

—

—

—

C₈×C₁₆

D₈₉

yes

characteristic

C₈₉

#12504

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

178

—

#500

—

#498

—

178

—

#494

—

#487

—

178

—

#366

—

178

—

#358

—

178

—

C₄×C₁₆

C₂×D₈₉

yes

characteristic

C₄×C₁₆

C₂×D₈₉

yes

characteristic

C₈²

C₂×D₈₉

yes

characteristic

C₂×Q₁₆

—

178

—

C₂×Q₁₆

—

178

—

C₄⋊Q₈

—

178

—

C₄⋊Q₈

—

178

—

Q₃₂

—

356

—

C₂×C₁₆

D₃₅₆

yes

—

C₂×C₁₆

D₃₅₆

yes

—

#107

—

178

—

#107

—

178

—

#106

—

356

—

#103

—

356

—

C₄×C₈

D₃₅₆

yes

characteristic

C₄×C₈

D₃₅₆

yes

characteristic

C₄×C₈

C₂²×D₈₉

yes

characteristic

C₂×Q₈

—

356

—

C₂×Q₈

—

356

—

100

Q₁₆

—

356

—

101

Q₁₆

—

356

—

102

C₂×C₈

D₇₁₂

yes

—

103

C₂×C₈

D₇₁₂

yes

—

104

C₂×C₈

C₂×D₃₅₆

yes

characteristic

105

C₂×C₈

C₂×D₃₅₆

yes

characteristic

106

C₄⋊C₄

—

356

—

107

C₄⋊C₄

—

356

—

108

C₄⋊C₄

—

356

—

109

C₄⋊C₄

—

356

—

110

C₄²

C₂×D₃₅₆

yes

characteristic

111

C₁₆

—

112

C₁₆

#13083563

yes

—

113

Q₈

—

712

—

114

Q₈

—

712

—

115

Q₈

—

712

—

116

C₂×C₄

—

712

—

117

C₂×C₄

—

712

—

118

C₂×C₄

#15668009

yes

characteristic

119

C₂×C₄

C₂×D₇₁₂

yes

characteristic

120

C₂×C₄

C₂×D₇₁₂

yes

characteristic

121

C₈

—

122

C₈

Dic₇₁₂

yes

complemented

123

C₈

C₂×#13083563

yes

—

124

C₈

#15668034

yes

—

125

C₂²

#19151975

yes

characteristic

126

C₄

—

1424

—

127

C₄

—

1424

—

128

C₄

#19151971

yes

characteristic

129

C₄

#19151982

yes

characteristic

130

C₄

C₂×Dic₇₁₂

yes

—

131

C₄

#19152030

yes

—

132

C₂

#26019844

yes

characteristic

133

C₂

#26022505

yes

characteristic

134

C₂

#26022513

yes

characteristic

	subgroup	G/H
center	C₂²	#19151975
commutator, cosocle	C₄×C₈×C₈₉	C₂³
Frattini	C₄×C₈	C₂²×D₈₉
Fitting	C₈×C₁₆×C₈₉	C₂
socle, prime radical	C₂²×C₈₉	#493
hypercenter	C₈×C₁₆	D₈₉
hypocenter	C₈₉	#12504
LCS	G ▷ C₄×C₈×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

subgroup

G/H

center

C₂²

#19151975

commutator, cosocle

C₄×C₈×C₈₉

C₂³

Frattini

C₄×C₈

C₂²×D₈₉

Fitting

C₈×C₁₆×C₈₉

C₂

socle, prime radical

C₂²×C₈₉

#493

hypercenter

C₈×C₁₆

D₈₉

hypocenter

C₈₉

#12504

LCS

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

N	H ≅ G/N	type
54. C₈×C₈₉	Q₃₂	semidirect
73. C₈₉	#12504	semidirect
122. C₈	Dic₇₁₂	semidirect

H ≅ G/N

type

54. C₈×C₈₉

Q₃₂

semidirect

73. C₈₉

#12504

semidirect

122. C₈

Dic₇₁₂

semidirect

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

subgroup	index
#26022527	2
#26022505	2
#26022480	2
#26019849	2
C₈×C₁₆×C₈₉	2
#12504	89

subgroup

index

C₈×C₁₆×C₈₉

Subgroups — #36469086

Conjugacy classes of subgroups

Complements

Maximal subgroups