Group #26019835 — subgroups

Subgroups — #26019835

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	#19151949	C₂	1	1	yes	maximal, characteristic
2	#19151894	C₂	1	1	yes	maximal, characteristic
3	#19151884	C₂	1	1	yes	maximal, characteristic
4	#19151728	C₂	1	1	yes	maximal, characteristic
5	#19151346	C₂	1	1	yes	maximal, characteristic
6	#19151343	C₂	1	1	yes	maximal, characteristic
7	#334×C₈₉	C₂	1	1	yes	maximal, characteristic
8	#15668012	C₂²	1	1	yes	characteristic
9	#15668012	C₂²	1	1	yes	characteristic
10	C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
11	#15667946	—	2	1	no	—
12	#15667944	—	2	1	no	—
13	C₄⋊#10623494	C₂²	1	1	yes	characteristic
14	#15667941	—	2	1	no	—
15	#15667937	—	2	1	no	—
16	#15667937	—	2	1	no	—
17	#15667937	—	2	1	no	—
18	#15667937	—	2	1	no	—
19	#15667922	—	2	1	no	—
20	C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
22	C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
23	C₂×Dic₁₇₈	—	2	1	no	—
24	C₂×Dic₁₇₈	—	2	1	no	—
25	C₂×Dic₁₇₈	D₄	1	1	yes	characteristic
26	C₂×#10623494	—	2	1	no	—
27	C₄⋊Dic₈₉	—	2	1	no	—
28	C₄⋊Dic₈₉	—	2	1	no	—
29	C₄⋊Dic₈₉	—	2	1	no	—
30	C₄⋊Dic₈₉	—	2	1	no	—
31	C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
32	#13083553	—	2	1	no	—
33	C₄×Dic₈₉	—	2	1	no	—
34	C₂×C₈×C₈₉	—	2	1	no	—
35	C₂×C₈×C₈₉	—	2	1	no	—
36	C₂×C₈×C₈₉	D₄	1	1	yes	characteristic
37	C₂×C₈×C₈₉	D₄	1	1	yes	characteristic
38	C₄²×C₈₉	C₂³	1	1	yes	characteristic
39	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₈₉	#41	1	1	yes	characteristic
42	C₂×Dic₈₉	—	2	1	no	—
43	C₂×Dic₈₉	—	2	1	no	—
44	C₂×Dic₈₉	—	4	1	no	—
45	C₂×Dic₈₉	—	4	1	no	—
46	Dic₁₇₈	—	2	1	no	—
47	Dic₁₇₈	—	2	1	no	—
48	Dic₁₇₈	—	4	1	no	—
49	Dic₁₇₈	—	4	1	no	—
50	Dic₁₇₈	—	8	1	no	—
51	C₈×C₈₉	—	2	1	no	—
52	C₈×C₈₉	—	2	1	no	—
53	C₈×C₈₉	—	4	1	no	—
54	C₈×C₈₉	—	4	1	no	—
55	#10623494	—	4	1	no	—
56	C₂²×C₈₉	#121	1	1	yes	characteristic
57	C₄×C₈₉	—	2	1	no	—
58	C₄×C₈₉	#135	1	1	yes	characteristic
59	C₄×C₈₉	#137	1	1	yes	characteristic
60	C₄×C₈₉	#137	1	1	yes	characteristic
61	C₄×C₈₉	#137	1	1	yes	characteristic
62	Dic₈₉	—	4	1	no	—
63	Dic₈₉	—	4	1	no	—
64	Dic₈₉	—	8	1	no	—
65	Dic₈₉	—	8	1	no	—
66	C₂×C₈₉	#464	1	1	yes	characteristic
67	C₂×C₈₉	#470	1	1	yes	characteristic
68	C₂×C₈₉	#471	1	1	yes	characteristic
69	#1704	—	89	1	no	maximal, Sylow
70	C₈₉	#1704	1	1	yes	characteristic, complemented, Sylow
71	#487	—	89	1	no	—
72	#477	—	89	1	no	—
73	#475	—	89	1	no	—
74	#446	—	89	1	no	—
75	#334	D₈₉	1	1	yes	characteristic
76	#333	—	89	1	no	—
77	#332	—	89	1	no	—
78	C₄⋊Q₈	—	89	1	no	—
79	C₄⋊Q₈	—	89	1	no	—
80	C₄×Q₈	—	89	1	no	—
81	#107	—	178	1	no	—
82	#106	—	178	1	no	—
83	C₄⋊C₈	—	89	1	no	—
84	C₄⋊C₈	—	178	1	no	—
85	C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
86	C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
87	#103	—	178	1	no	—
88	#103	—	178	1	no	—
89	#103	—	178	1	no	—
90	#103	—	178	1	no	—
91	#97	—	178	1	no	—
92	C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
93	C₂×Q₈	—	89	1	no	—
94	C₂×Q₈	—	178	1	no	—
95	C₂×Q₈	—	178	1	no	—
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	—	2	1	no	—
98	C₂×C₈	—	178	1	no	—
99	C₂×C₈	D₃₅₆	1	1	yes	characteristic
100	C₂×C₈	D₃₅₆	1	1	yes	characteristic
101	C₄⋊C₄	—	89	1	no	—
102	C₄⋊C₄	—	178	1	no	—
103	C₄⋊C₄	—	178	1	no	—
104	C₄⋊C₄	—	178	1	no	—
105	C₄⋊C₄	—	178	1	no	—
106	C₄⋊C₄	—	178	1	no	—
107	C₄²	—	178	1	no	—
108	C₄²	C₂²×D₈₉	1	1	yes	characteristic
109	Q₈	—	356	1	no	—
110	Q₈	—	356	1	no	—
111	Q₈	—	178	1	no	—
112	Q₈	—	178	1	no	—
113	Q₈	—	712	1	no	—
114	C₂×C₄	—	356	1	no	—
115	C₂×C₄	—	356	1	no	—
116	C₂×C₄	—	178	1	no	—
117	C₂×C₄	—	178	1	no	—
118	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
119	C₂×C₄	C₂×D₃₅₆	1	1	yes	characteristic
120	C₂×C₄	#13083575	1	1	yes	characteristic
121	C₈	—	2	1	no	—
122	C₈	—	2	1	no	—
123	C₈	—	4	1	no	—
124	C₈	—	4	1	no	—
125	C₈	—	356	1	no	—
126	C₂²	#15667987	1	1	yes	characteristic
127	C₄	—	2	1	no	—
128	C₄	—	356	1	no	—
129	C₄	—	356	1	no	—
130	C₄	—	712	1	no	—
131	C₄	—	712	1	no	—
132	C₄	#15668034	1	1	yes	characteristic
133	C₄	#15668041	1	1	yes	characteristic
134	C₄	#15668041	1	1	yes	characteristic
135	C₄	#15668045	1	1	yes	characteristic
136	C₂	#19151821	1	1	yes	characteristic
137	C₂	#19151857	1	1	yes	characteristic
138	C₂	#19151862	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19151949

C₂

yes

maximal, characteristic

#19151894

C₂

yes

maximal, characteristic

#19151884

C₂

yes

maximal, characteristic

#19151728

C₂

yes

maximal, characteristic

#19151346

C₂

yes

maximal, characteristic

#19151343

C₂

yes

maximal, characteristic

#334×C₈₉

C₂

yes

maximal, characteristic

#15668012

C₂²

yes

characteristic

#15668012

C₂²

yes

characteristic

C₄×Dic₁₇₈

C₂²

yes

—

—

C₂²

yes

—

—

—

—

—

—

C₄⋊C₈×C₈₉

C₂²

yes

characteristic

C₄⋊C₈×C₈₉

C₂²

yes

characteristic

C₄×C₈×C₈₉

C₂²

yes

characteristic

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

D₄

yes

characteristic

C₂×#10623494

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

D₄

yes

characteristic

#13083553

—

C₄×Dic₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

D₄

yes

characteristic

C₂×C₈×C₈₉

D₄

yes

characteristic

C₄²×C₈₉

C₂³

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

#41

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

Dic₁₇₈

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

—

C₈×C₈₉

—

C₈×C₈₉

—

#10623494

—

C₂²×C₈₉

#121

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

Dic₈₉

—

Dic₈₉

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₈₉

yes

characteristic, complemented, Sylow

—

—

—

—

D₈₉

yes

characteristic

#333

—

#332

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄×Q₈

—

#107

—

178

—

#106

—

178

—

C₄⋊C₈

—

C₄⋊C₈

—

178

—

C₄⋊C₈

C₂×D₈₉

yes

characteristic

C₄⋊C₈

C₂×D₈₉

yes

characteristic

#103

—

178

—

#103

—

178

—

#103

—

178

—

#103

—

178

—

#97

—

178

—

C₄×C₈

C₂×D₈₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

178

—

C₂×Q₈

—

178

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

178

—

C₂×C₈

D₃₅₆

yes

characteristic

100

C₂×C₈

D₃₅₆

yes

characteristic

101

C₄⋊C₄

—

102

C₄⋊C₄

—

178

—

103

C₄⋊C₄

—

178

—

104

C₄⋊C₄

—

178

—

105

C₄⋊C₄

—

178

—

106

C₄⋊C₄

—

178

—

107

C₄²

—

178

—

108

C₄²

C₂²×D₈₉

yes

characteristic

109

Q₈

—

356

—

110

Q₈

—

356

—

111

Q₈

—

178

—

112

Q₈

—

178

—

113

Q₈

—

712

—

114

C₂×C₄

—

356

—

115

C₂×C₄

—

356

—

116

C₂×C₄

—

178

—

117

C₂×C₄

—

178

—

118

C₂×C₄

D₄×D₈₉

yes

characteristic

119

C₂×C₄

C₂×D₃₅₆

yes

characteristic

120

C₂×C₄

#13083575

yes

characteristic

121

C₈

—

122

C₈

—

123

C₈

—

124

C₈

—

125

C₈

—

356

—

126

C₂²

#15667987

yes

characteristic

127

C₄

—

128

C₄

—

356

—

129

C₄

—

356

—

130

C₄

—

712

—

131

C₄

—

712

—

132

C₄

#15668034

yes

characteristic

133

C₄

#15668041

yes

characteristic

134

C₄

#15668041

yes

characteristic

135

C₄

#15668045

yes

characteristic

136

C₂

#19151821

yes

characteristic

137

C₂

#19151857

yes

characteristic

138

C₂

#19151862

yes

characteristic

	subgroup	G/H
center	C₂²	#15667987
commutator, cosocle	C₄²×C₈₉	C₂³
Frattini	C₄²	C₂²×D₈₉
Fitting	#334×C₈₉	C₂
socle, prime radical	C₂²×C₈₉	#121
hypercenter	#334	D₈₉
hypocenter	C₈₉	#1704
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #334

subgroup

G/H

center

C₂²

#15667987

commutator, cosocle

C₄²×C₈₉

C₂³

Frattini

C₄²

C₂²×D₈₉

Fitting

#334×C₈₉

C₂

C₂²×C₈₉

D₈₉

C₈₉

G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #334

N	H ≅ G/N	type
70. C₈₉	#1704	semidirect

H ≅ G/N

type

70. C₈₉

#1704

semidirect

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

subgroup	index
#19151949	2
#19151894	2
#19151884	2
#19151728	2
#19151346	2
#19151343	2
#334×C₈₉	2
#1704	89

subgroup

index

#334×C₈₉

Subgroups — #26019835

Conjugacy classes of subgroups

Complements

Maximal subgroups