Group #26019895 — subgroups

Subgroups — #26019895

Order 11392.

164 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 164 conjugacy classes of subgroups — too many to draw. Only its 57 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	#19152001	C₂	1	1	yes	maximal, characteristic
2	#19151994	C₂	1	1	yes	maximal, characteristic
3	#19151950	C₂	1	1	yes	maximal, characteristic
4	#19151949	C₂	1	1	yes	maximal, characteristic
5	#19151948	C₂	1	1	yes	maximal, characteristic
6	#19151319	C₂	1	1	yes	maximal, characteristic
7	#487×C₈₉	C₂	1	1	yes	maximal, characteristic
8	#15668012	C₂²	1	1	yes	characteristic
9	#15668012	C₂²	1	1	yes	characteristic
10	#15667945	—	2	2	no	—
11	#15667943	—	2	2	no	—
12	#15667937	—	2	2	no	—
13	#15667936	—	2	2	no	—
14	#15667935	—	2	2	no	—
15	C₄×#10623494	C₂²	1	1	yes	characteristic
16	C₄×#10623494	C₂²	1	1	yes	characteristic
17	C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
18	C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
19	#103×C₈₉	—	2	2	no	—
20	C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
21	C₂×Dic₁₇₈	—	2	1	no	—
22	C₂×Dic₁₇₈	—	2	1	no	—
23	C₂×#10623494	D₄	1	2	yes	—
24	C₂×#10623494	D₄	1	2	yes	—
25	C₄⋊Dic₈₉	—	2	1	no	—
26	C₄⋊Dic₈₉	—	2	1	no	—
27	C₄⋊Dic₈₉	—	2	1	no	—
28	C₄⋊Dic₈₉	—	2	1	no	—
29	C₂×Q₈×C₈₉	—	2	1	no	—
30	C₂×Q₈×C₈₉	—	2	1	no	—
31	C₂×C₈×C₈₉	D₄	1	2	yes	—
32	C₄⋊C₄×C₈₉	—	2	1	no	—
33	C₄⋊C₄×C₈₉	—	2	1	no	—
34	C₄⋊C₄×C₈₉	—	2	1	no	—
35	C₄⋊C₄×C₈₉	—	2	1	no	—
36	C₄²×C₈₉	C₂³	1	1	yes	characteristic
37	Q₈×C₈₉	—	4	2	no	—
38	Q₈×C₈₉	—	4	2	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×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
44	C₂×Dic₈₉	—	4	1	no	—
45	C₂×Dic₈₉	—	4	1	no	—
46	Dic₁₇₈	—	4	2	no	—
47	Dic₁₇₈	—	4	2	no	—
48	C₈×C₈₉	—	2	2	no	—
49	#10623494	—	2	2	no	—
50	#10623494	—	2	2	no	—
51	C₂²×C₈₉	#127	1	1	yes	characteristic
52	C₄×C₈₉	—	8	1	no	—
53	C₄×C₈₉	—	8	1	no	—
54	C₄×C₈₉	#135	1	2	yes	—
55	C₄×C₈₉	#137	1	2	yes	—
56	C₄×C₈₉	#137	1	2	yes	—
57	Dic₈₉	—	8	1	no	—
58	Dic₈₉	—	8	1	no	—
59	C₂×C₈₉	#495	1	1	yes	characteristic
60	C₂×C₈₉	#497	1	1	yes	characteristic
61	C₂×C₈₉	#497	1	1	yes	characteristic
62	#1718	—	89	1	no	maximal, Sylow
63	C₈₉	#1718	1	1	yes	characteristic, complemented, Sylow
64	#500	—	89	1	no	—
65	#498	—	89	1	no	—
66	#487	—	89	1	no	—
67	#487	—	89	1	no	—
68	#487	—	89	1	no	—
69	#487	D₈₉	1	1	yes	characteristic
70	#322	—	89	1	no	—
71	C₄⋊Q₈	—	89	1	no	—
72	C₄⋊Q₈	—	89	1	no	—
73	C₄⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
74	C₄⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
75	#107	—	178	2	no	—
76	#106	—	178	2	no	—
77	#103	—	2	2	no	—
78	#103	—	178	2	no	—
79	#103	—	178	2	no	—
80	#103	—	178	2	no	—
81	C₄×C₈	—	89	1	no	—
82	C₄×C₈	—	89	1	no	—
83	C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
84	C₂×Q₈	—	2	1	no	—
85	C₂×Q₈	—	2	1	no	—
86	C₂×Q₈	—	178	1	no	—
87	C₂×Q₈	—	178	1	no	—
88	C₂×C₈	—	89	2	no	—
89	C₂×C₈	—	89	2	no	—
90	C₂×C₈	C₂²⋊D₈₉	1	2	yes	—
91	C₄⋊C₄	—	2	1	no	—
92	C₄⋊C₄	—	2	1	no	—
93	C₄⋊C₄	—	2	1	no	—
94	C₄⋊C₄	—	2	1	no	—
95	C₄⋊C₄	—	178	1	no	—
96	C₄⋊C₄	—	178	1	no	—
97	C₄⋊C₄	—	178	1	no	—
98	C₄⋊C₄	—	178	1	no	—
99	C₄²	C₂²×D₈₉	1	1	yes	characteristic
100	Q₈	—	4	2	no	—
101	Q₈	—	4	2	no	—
102	Q₈	—	356	2	no	—
103	Q₈	—	356	2	no	—
104	C₂×C₄	—	4	1	no	—
105	C₂×C₄	—	4	1	no	—
106	C₂×C₄	—	356	1	no	—
107	C₂×C₄	—	356	1	no	—
108	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
109	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
110	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
111	C₈	—	2	2	no	—
112	C₈	—	178	2	no	—
113	C₈	—	178	2	no	—
114	C₂²	#15668008	1	1	yes	characteristic
115	C₄	—	8	1	no	—
116	C₄	—	8	1	no	—
117	C₄	—	712	1	no	—
118	C₄	—	712	1	no	—
119	C₄	#15668032	1	2	yes	—
120	C₄	#15668041	1	2	yes	—
121	C₄	#15668044	1	2	yes	—
122	C₂	#19151980	1	1	yes	characteristic
123	C₂	#19151989	1	1	yes	characteristic
124	C₂	#19151991	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19152001

C₂

yes

maximal, characteristic

#19151994

C₂

yes

maximal, characteristic

#19151950

C₂

yes

maximal, characteristic

#19151949

C₂

yes

maximal, characteristic

#19151948

C₂

yes

maximal, characteristic

#19151319

C₂

yes

maximal, characteristic

#487×C₈₉

C₂

yes

maximal, characteristic

#15668012

C₂²

yes

characteristic

#15668012

C₂²

yes

—

—

—

—

—

C₂²

yes

characteristic

C₄×#10623494

C₂²

yes

characteristic

C₄⋊Q₈×C₈₉

C₂²

yes

characteristic

C₄⋊Q₈×C₈₉

C₂²

yes

characteristic

#103×C₈₉

—

C₄×C₈×C₈₉

C₂²

yes

characteristic

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×#10623494

D₄

yes

—

C₂×#10623494

D₄

yes

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×C₈×C₈₉

D₄

yes

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

C₂³

yes

characteristic

Q₈×C₈₉

—

Q₈×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

—

#10623494

—

#10623494

—

C₂²×C₈₉

#127

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#135

yes

—

C₄×C₈₉

#137

yes

—

C₄×C₈₉

#137

yes

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₈₉

yes

characteristic, complemented, Sylow

—

—

—

—

—

D₈₉

yes

characteristic

#322

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₈₉

yes

characteristic

C₄⋊Q₈

C₂×D₈₉

yes

characteristic

#107

—

178

—

#106

—

178

—

#103

—

#103

—

178

—

#103

—

178

—

#103

—

178

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₈₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

178

—

C₂×Q₈

—

178

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₈₉

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

178

—

C₄⋊C₄

—

178

—

C₄⋊C₄

—

178

—

C₄⋊C₄

—

178

—

C₄²

C₂²×D₈₉

yes

characteristic

100

Q₈

—

101

Q₈

—

102

Q₈

—

356

—

103

Q₈

—

356

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

356

—

107

C₂×C₄

—

356

—

108

C₂×C₄

D₄×D₈₉

yes

characteristic

109

C₂×C₄

D₄×D₈₉

yes

characteristic

110

C₂×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

111

C₈

—

112

C₈

—

178

—

113

C₈

—

178

—

114

C₂²

#15668008

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

—

712

—

118

C₄

—

712

—

119

C₄

#15668032

yes

—

120

C₄

#15668041

yes

—

121

C₄

#15668044

yes

—

122

C₂

#19151980

yes

characteristic

123

C₂

#19151989

yes

characteristic

124

C₂

#19151991

yes

characteristic

	subgroup	G/H
center	C₂²	#15668008
commutator, cosocle	C₄²×C₈₉	C₂³
Frattini	C₄²	C₂²×D₈₉
Fitting	#487×C₈₉	C₂
socle, prime radical	C₂²×C₈₉	#127
hypercenter	#487	D₈₉
hypocenter	C₈₉	#1718
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #487

subgroup

G/H

center

C₂²

#15668008

commutator, cosocle

C₄²×C₈₉

C₂³

Frattini

C₄²

C₂²×D₈₉

Fitting

#487×C₈₉

C₂

C₂²×C₈₉

D₈₉

C₈₉

G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #487

N	H ≅ G/N	type
63. C₈₉	#1718	semidirect

H ≅ G/N

type

63. C₈₉

#1718

semidirect

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

subgroup	index
#19152001	2
#19151994	2
#19151950	2
#19151949	2
#19151948	2
#19151319	2
#487×C₈₉	2
#1718	89

subgroup

index

#487×C₈₉

Subgroups — #26019895

Conjugacy classes of subgroups

Complements

Maximal subgroups