Group #26022478 — subgroups

Subgroups — #26022478

Order 11392.

140 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 140 conjugacy classes of subgroups — too many to draw. Only its 51 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	#19152002	C₂	1	1	yes	maximal, characteristic
2	#19151976	C₂	1	1	yes	maximal, characteristic
3	#19151403	C₂	1	1	yes	maximal, characteristic
4	#19151402	C₂	1	1	yes	maximal, characteristic
5	#19151401	C₂	1	1	yes	maximal, characteristic
6	C₁₆×Dic₈₉	C₂	1	1	yes	maximal, characteristic
7	#358×C₈₉	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₃₅₆	C₂²	1	1	yes	characteristic
9	C₂×#13083564	—	2	1	no	—
10	#15668011	—	2	1	no	—
11	#15668010	—	2	1	no	—
12	C₂×#13083549	C₂²	1	1	yes	characteristic
13	#15667946	C₂²	1	1	yes	characteristic
14	#15667945	—	2	1	no	—
15	C₈×Dic₈₉	C₂²	1	1	yes	characteristic
16	C₂×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
17	C₂×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
18	#107×C₈₉	C₂²	1	1	yes	characteristic
19	C₂×Dic₁₇₈	—	2	1	no	—
20	C₂×Dic₁₇₈	—	4	1	no	—
21	Dic₃₅₆	—	2	1	no	—
22	Dic₃₅₆	—	2	1	no	—
23	#13083564	—	4	2	no	—
24	C₂×#10623494	D₄	1	1	yes	characteristic
25	C₄⋊Dic₈₉	—	2	1	no	—
26	#13083553	—	4	1	no	—
27	#13083553	—	4	1	no	—
28	#13083553	—	4	1	no	—
29	C₄×Dic₈₉	D₄	1	1	yes	characteristic
30	#13083549	—	2	1	no	—
31	C₂×Q₈×C₈₉	—	2	1	no	—
32	Q₁₆×C₈₉	—	2	1	no	—
33	Q₁₆×C₈₉	—	2	1	no	—
34	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
35	C₄⋊C₄×C₈₉	—	2	1	no	—
36	C₁₆×C₈₉	—	2	1	no	—
37	Q₈×C₈₉	—	4	1	no	—
38	Q₈×C₈₉	—	4	1	no	—
39	C₂×C₄×C₈₉	—	4	1	no	—
40	C₂×C₄×C₈₉	—	4	1	no	—
41	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
42	C₂×Dic₈₉	—	4	1	no	—
43	C₂×Dic₈₉	—	4	1	no	—
44	C₂×Dic₈₉	D₈	1	1	yes	characteristic
45	C₂×Dic₈₉	D₈	1	1	yes	characteristic
46	Dic₁₇₈	—	4	1	no	—
47	Dic₁₇₈	—	4	1	no	—
48	Dic₁₇₈	—	8	2	no	—
49	C₈×C₈₉	#41	1	1	yes	characteristic
50	C₈×C₈₉	#41	1	1	yes	characteristic
51	#10623494	SD₁₆	1	2	yes	—
52	C₂²×C₈₉	C₂×D₈	1	1	yes	characteristic
53	C₄×C₈₉	—	8	1	no	—
54	C₄×C₈₉	—	8	1	no	—
55	C₄×C₈₉	#124	1	1	yes	characteristic
56	C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
57	Dic₈₉	—	8	1	no	—
58	Dic₈₉	—	8	1	no	—
59	Dic₈₉	SD₃₂	1	2	yes	—
60	Dic₈₉	Q₃₂	1	2	yes	—
61	C₂×C₈₉	#486	1	1	yes	characteristic
62	C₂×C₈₉	C₂×SD₃₂	1	1	yes	characteristic
63	C₂×C₈₉	C₂×Q₃₂	1	1	yes	characteristic
64	#2241	—	89	1	no	maximal, Sylow
65	C₈₉	#2241	1	1	yes	characteristic, complemented, Sylow
66	#500	—	89	1	no	—
67	#494	—	89	1	no	—
68	#358	—	89	1	no	—
69	#358	—	89	1	no	—
70	#358	—	89	1	no	—
71	#358	D₈₉	1	1	yes	characteristic
72	C₄×C₁₆	—	89	1	no	—
73	C₂×Q₁₆	—	89	1	no	—
74	C₂×Q₁₆	—	178	1	no	—
75	C₂×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
76	C₄⋊Q₈	—	178	1	no	—
77	C₄⋊Q₈	—	178	1	no	—
78	C₂×C₁₆	—	89	1	no	—
79	C₂×C₁₆	C₂×D₈₉	1	1	yes	characteristic
80	#107	—	89	1	no	—
81	#107	—	178	1	no	—
82	#107	C₂×D₈₉	1	1	yes	characteristic
83	C₄×C₈	—	89	1	no	—
84	C₂×Q₈	—	2	1	no	—
85	C₂×Q₈	—	356	1	no	—
86	C₂×Q₈	—	178	1	no	—
87	Q₁₆	—	2	1	no	—
88	Q₁₆	—	2	1	no	—
89	Q₁₆	—	356	2	no	—
90	Q₁₆	—	178	1	no	—
91	Q₁₆	—	178	1	no	—
92	C₂×C₈	—	89	1	no	—
93	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
94	C₄⋊C₄	—	2	1	no	—
95	C₄⋊C₄	—	356	1	no	—
96	C₄⋊C₄	—	356	1	no	—
97	C₄⋊C₄	—	356	1	no	—
98	C₄⋊C₄	—	178	1	no	—
99	C₄²	—	89	1	no	—
100	C₁₆	—	2	1	no	—
101	C₁₆	—	178	1	no	—
102	Q₈	—	4	1	no	—
103	Q₈	—	4	1	no	—
104	Q₈	—	356	1	no	—
105	Q₈	—	356	1	no	—
106	Q₈	—	712	2	no	—
107	C₂×C₄	—	4	1	no	—
108	C₂×C₄	—	4	1	no	—
109	C₂×C₄	—	89	1	no	—
110	C₂×C₄	—	89	1	no	—
111	C₂×C₄	—	356	1	no	—
112	C₂×C₄	—	356	1	no	—
113	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
114	C₈	—	89	2	no	—
115	C₈	#13083573	1	1	yes	characteristic
116	C₈	#13083574	1	1	yes	characteristic
117	C₂²	D₈×D₈₉	1	1	yes	characteristic
118	C₄	—	8	1	no	—
119	C₄	—	8	1	no	—
120	C₄	—	89	2	no	—
121	C₄	—	89	2	no	—
122	C₄	—	712	1	no	—
123	C₄	—	712	1	no	—
124	C₄	#15667998	1	1	yes	characteristic
125	C₄	SD₁₆×D₈₉	1	1	yes	characteristic
126	C₂	#19151943	1	1	yes	characteristic
127	C₂	SD₃₂×D₈₉	1	1	yes	characteristic
128	C₂	Q₃₂×D₈₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19152002

C₂

yes

maximal, characteristic

#19151976

C₂

yes

maximal, characteristic

#19151403

C₂

yes

maximal, characteristic

#19151402

C₂

yes

maximal, characteristic

#19151401

C₂

yes

maximal, characteristic

C₁₆×Dic₈₉

C₂

yes

maximal, characteristic

#358×C₈₉

C₂

yes

maximal, characteristic

C₂×Dic₃₅₆

C₂²

yes

—

—

—

C₂²

yes

characteristic

#15667946

C₂²

yes

characteristic

#15667945

—

C₈×Dic₈₉

C₂²

yes

characteristic

C₂×Q₁₆×C₈₉

C₂²

yes

characteristic

C₂×C₁₆×C₈₉

C₂²

yes

characteristic

#107×C₈₉

C₂²

yes

characteristic

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

Dic₃₅₆

—

Dic₃₅₆

—

#13083564

—

C₂×#10623494

D₄

yes

characteristic

C₄⋊Dic₈₉

—

#13083553

—

#13083553

—

#13083553

—

C₄×Dic₈₉

D₄

yes

characteristic

#13083549

—

C₂×Q₈×C₈₉

—

Q₁₆×C₈₉

—

Q₁₆×C₈₉

—

C₂×C₈×C₈₉

C₂³

yes

characteristic

C₄⋊C₄×C₈₉

—

C₁₆×C₈₉

—

Q₈×C₈₉

—

Q₈×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

D₈

yes

characteristic

C₂×Dic₈₉

D₈

yes

characteristic

Dic₁₇₈

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

yes

C₈×C₈₉

yes

SD₁₆

yes

—

C₂²×C₈₉

C₂×D₈

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#124

yes

characteristic

C₄×C₈₉

C₂×SD₁₆

yes

characteristic

Dic₈₉

—

Dic₈₉

—

Dic₈₉

SD₃₂

yes

—

Dic₈₉

Q₃₂

yes

—

C₂×C₈₉

#486

yes

characteristic

C₂×C₈₉

C₂×SD₃₂

yes

characteristic

C₂×C₈₉

C₂×Q₃₂

yes

—

C₈₉

yes

characteristic, complemented, Sylow

—

—

—

—

—

D₈₉

yes

characteristic

C₄×C₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

178

—

C₂×Q₁₆

C₂×D₈₉

yes

characteristic

C₄⋊Q₈

—

178

—

C₄⋊Q₈

—

178

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₈₉

yes

characteristic

#107

—

#107

—

178

—

#107

C₂×D₈₉

yes

characteristic

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

356

—

C₂×Q₈

—

178

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

356

—

Q₁₆

—

178

—

Q₁₆

—

178

—

C₂×C₈

—

C₂×C₈

C₂²×D₈₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

356

—

C₄⋊C₄

—

356

—

C₄⋊C₄

—

356

—

C₄⋊C₄

—

178

—

C₄²

—

100

C₁₆

—

101

C₁₆

—

178

—

102

Q₈

—

103

Q₈

—

104

Q₈

—

356

—

105

Q₈

—

356

—

106

Q₈

—

712

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

356

—

112

C₂×C₄

—

356

—

113

C₂×C₄

D₄×D₈₉

yes

characteristic

114

C₈

—

115

C₈

#13083573

yes

characteristic

116

C₈

#13083574

yes

characteristic

117

C₂²

D₈×D₈₉

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

—

712

—

123

C₄

—

712

—

124

C₄

#15667998

yes

characteristic

125

C₄

SD₁₆×D₈₉

yes

characteristic

126

C₂

#19151943

yes

characteristic

127

C₂

SD₃₂×D₈₉

yes

characteristic

128

C₂

Q₃₂×D₈₉

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₈₉
commutator, cosocle	C₂×C₈×C₈₉	C₂³
Frattini	C₂×C₈	C₂²×D₈₉
Fitting	#358×C₈₉	C₂
socle, prime radical	C₂²×C₈₉	C₂×D₈
hypercenter	#358	D₈₉
hypocenter	C₈₉	#2241
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

subgroup

G/H

center

C₂²

D₈×D₈₉

commutator, cosocle

C₂×C₈×C₈₉

C₂³

Frattini

C₂×C₈

C₂²×D₈₉

Fitting

#358×C₈₉

C₂

socle, prime radical

C₂²×C₈₉

C₂×D₈

D₈₉

C₈₉

G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

N	H ≅ G/N	type
65. C₈₉	#2241	semidirect

H ≅ G/N

type

65. C₈₉

#2241

semidirect

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

subgroup	index
#19152002	2
#19151976	2
#19151403	2
#19151402	2
#19151401	2
C₁₆×Dic₈₉	2
#358×C₈₉	2
#2241	89

subgroup

index

C₁₆×Dic₈₉

#358×C₈₉

#2241

Subgroups — #26022478

Conjugacy classes of subgroups

Complements

Maximal subgroups