Group #26030763 — subgroups

Subgroups — #26030763

Order 11392.

356 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 145 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	#19152153	C₂	1	2	yes	maximal
2	#19152003	C₂	1	1	yes	maximal, characteristic, complemented
3	#19151999	C₂	1	4	yes	maximal, complemented
4	#19151994	C₂	1	2	yes	maximal, complemented
5	#19151635	C₂	1	4	yes	maximal
6	C₄×C₈×D₈₉	C₂	1	1	yes	maximal, characteristic
7	#500×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₈×D₈₉	C₂²	1	2	yes	—
9	#15668012	C₂²	1	2	yes	—
10	#15668003	C₂²	1	4	yes	—
11	#15667992	—	2	2	no	—
12	#15667988	—	2	2	no	—
13	Q₈×Dic₈₉	—	2	2	no	—
14	#15667977	—	2	2	no	—
15	#15667970	C₂²	1	4	yes	—
16	C₄²×D₈₉	C₂²	1	1	yes	characteristic
17	#15667946	C₂²	1	4	yes	—
18	#15667943	C₂²	1	8	yes	—
19	C₄×#10623494	C₂²	1	1	yes	characteristic
20	C₈×Dic₈₉	C₂²	1	2	yes	—
21	C₄⋊Q₈×C₈₉	C₂²	1	2	yes	—
22	#107×C₈₉	C₂²	1	4	yes	—
23	C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
24	C₂×Dic₁₇₈	—	2	2	no	—
25	C₂×C₄×D₈₉	D₄	1	1	yes	characteristic
26	C₂×C₄×D₈₉	D₄	1	1	yes	characteristic
27	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
28	C₂×#10623494	C₂³	1	2	yes	—
29	C₈×D₈₉	Q₈	1	8	yes	complemented
30	C₄⋊Dic₈₉	—	2	2	no	—
31	C₄⋊Dic₈₉	—	2	4	no	—
32	C₄⋊Dic₈₉	C₂³	1	4	yes	—
33	#13083553	—	2	4	no	—
34	#13083551	—	2	4	no	—
35	C₄×Dic₈₉	—	2	4	no	—
36	C₄×Dic₈₉	D₄	1	1	yes	characteristic
37	C₄×Dic₈₉	D₄	1	1	yes	characteristic
38	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
39	C₂×Q₈×C₈₉	—	2	2	no	—
40	C₂×C₈×C₈₉	C₂³	1	2	yes	—
41	C₄⋊C₄×C₈₉	—	2	2	no	—
42	C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
43	C₄²×C₈₉	C₂³	1	1	yes	characteristic
44	C₂²×D₈₉	C₂×D₄	1	1	yes	characteristic
45	Q₈×C₈₉	—	4	4	no	—
46	C₂×C₄×C₈₉	—	2	4	no	—
47	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
50	C₂×Dic₈₉	—	2	4	no	—
51	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
52	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
53	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
54	C₄×D₈₉	—	2	2	no	—
55	C₄×D₈₉	—	2	2	no	—
56	C₄×D₈₉	C₂×Q₈	1	4	yes	—
57	Dic₁₇₈	—	4	4	no	—
58	C₈×C₈₉	C₂×Q₈	1	4	yes	—
59	#10623494	C₂×Q₈	1	4	yes	—
60	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
61	C₂×D₈₉	—	2	1	no	—
62	C₂×D₈₉	—	2	1	no	—
63	C₂×D₈₉	C₄⋊Q₈	1	2	yes	—
64	C₄×C₈₉	—	4	4	no	—
65	C₄×C₈₉	#135	1	2	yes	—
66	C₄×C₈₉	#135	1	2	yes	—
67	C₄×C₈₉	C₂²×Q₈	1	2	yes	—
68	Dic₈₉	—	2	1	no	—
69	Dic₈₉	—	2	1	no	—
70	Dic₈₉	—	4	4	no	—
71	Dic₈₉	C₄⋊Q₈	1	2	yes	—
72	C₂×C₈₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
73	C₂×C₈₉	C₂×#135	1	1	yes	characteristic
74	C₂×C₈₉	C₂×#135	1	1	yes	characteristic
75	D₈₉	—	2	2	no	—
76	#3160	—	89	1	no	maximal, Sylow
77	C₈₉	#3160	1	1	yes	characteristic, complemented, Sylow
78	#533	—	89	2	no	—
79	#500	—	89	1	no	—
80	#500	D₈₉	1	1	yes	characteristic, complemented
81	#499	—	89	4	no	—
82	#498	—	89	2	no	—
83	#427	—	89	4	no	—
84	C₂×C₄×C₈	—	89	1	no	—
85	C₂²×C₈	—	89	2	no	—
86	C₄⋊Q₈	—	89	2	no	—
87	C₄⋊Q₈	C₂×D₈₉	1	2	yes	—
88	#125	—	89	4	no	—
89	C₂²⋊Q₈	—	178	2	no	—
90	C₂²⋊Q₈	—	178	2	no	—
91	C₄×Q₈	—	178	2	no	—
92	C₄×Q₈	—	178	2	no	—
93	#117	—	89	4	no	—
94	C₂×C₄²	—	89	1	no	—
95	#107	—	89	4	no	—
96	#107	C₂×D₈₉	1	4	yes	—
97	#106	—	89	8	no	—
98	C₄×C₈	—	89	2	no	—
99	C₄×C₈	—	89	1	no	—
100	C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
101	C₂×Q₈	—	2	2	no	—
102	C₂×Q₈	—	178	2	no	—
103	C₂²×C₄	—	89	1	no	—
104	C₂²×C₄	—	89	1	no	—
105	C₂²×C₄	—	89	1	no	—
106	C₂×C₈	—	89	2	no	—
107	C₂×C₈	—	89	8	no	—
108	C₂×C₈	C₂²×D₈₉	1	2	yes	—
109	C₄⋊C₄	—	2	2	no	—
110	C₄⋊C₄	—	89	4	no	—
111	C₄⋊C₄	—	178	4	no	—
112	C₄⋊C₄	—	178	4	no	—
113	C₄⋊C₄	—	178	2	no	—
114	C₄⋊C₄	C₂²×D₈₉	1	4	yes	—
115	C₂²⋊C₄	—	178	4	no	—
116	C₄²	—	89	1	no	—
117	C₄²	—	89	1	no	—
118	C₄²	—	89	1	no	—
119	C₄²	—	178	4	no	—
120	C₄²	C₂²×D₈₉	1	1	yes	characteristic
121	C₂³	—	89	1	no	—
122	Q₈	—	4	4	no	—
123	Q₈	—	356	4	no	—
124	C₂×C₄	—	2	4	no	—
125	C₂×C₄	—	89	1	no	—
126	C₂×C₄	—	89	4	no	—
127	C₂×C₄	—	89	1	no	—
128	C₂×C₄	—	89	1	no	—
129	C₂×C₄	—	178	4	no	—
130	C₂×C₄	—	178	2	no	—
131	C₂×C₄	—	178	2	no	—
132	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
133	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
134	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
135	C₈	—	89	4	no	—
136	C₈	Q₈×D₈₉	1	4	yes	—
137	C₂²	—	89	2	no	—
138	C₂²	—	178	1	no	—
139	C₂²	—	178	1	no	—
140	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
141	C₄	—	4	4	no	—
142	C₄	—	89	2	no	—
143	C₄	—	356	4	no	—
144	C₄	—	178	1	no	—
145	C₄	—	178	1	no	—
146	C₄	#15668031	1	2	yes	—
147	C₄	#15668035	1	2	yes	—
148	C₄	C₂×Q₈×D₈₉	1	2	yes	—
149	C₂	—	178	2	no	—
150	C₂	C₄⋊Q₈×D₈₉	1	1	yes	characteristic
151	C₂	C₂×#15668031	1	1	yes	characteristic
152	C₂	C₂×#15668035	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19152153

C₂

yes

maximal

#19152003

C₂

yes

maximal, characteristic, complemented

#19151999

C₂

yes

maximal, complemented

#19151994

C₂

yes

maximal, complemented

#19151635

C₂

yes

maximal

C₄×C₈×D₈₉

C₂

yes

maximal, characteristic

#500×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂×C₈×D₈₉

C₂²

yes

—

#15668012

C₂²

yes

—

#15668003

C₂²

yes

—

#15667992

—

#15667988

—

Q₈×Dic₈₉

—

#15667977

—

#15667970

C₂²

yes

—

C₄²×D₈₉

C₂²

yes

characteristic

#15667946

C₂²

yes

—

#15667943

C₂²

yes

—

C₄×#10623494

C₂²

yes

characteristic

C₈×Dic₈₉

C₂²

yes

—

C₄⋊Q₈×C₈₉

C₂²

yes

—

#107×C₈₉

C₂²

yes

—

C₄×C₈×C₈₉

C₂²

yes

characteristic

C₂×Dic₁₇₈

—

C₂×C₄×D₈₉

D₄

yes

characteristic

C₂×C₄×D₈₉

D₄

yes

characteristic

C₂×C₄×D₈₉

C₂³

yes

characteristic

C₂×#10623494

C₂³

yes

—

C₈×D₈₉

Q₈

yes

complemented

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

C₂³

yes

—

#13083553

—

#13083551

—

C₄×Dic₈₉

—

C₄×Dic₈₉

D₄

yes

characteristic

C₄×Dic₈₉

D₄

yes

characteristic

C₄×Dic₈₉

C₂³

yes

characteristic

C₂×Q₈×C₈₉

—

C₂×C₈×C₈₉

C₂³

yes

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

C₂³

yes

—

C₄²×C₈₉

C₂³

yes

characteristic

C₂²×D₈₉

C₂×D₄

yes

characteristic

Q₈×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂⁴

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

C₂×D₄

yes

characteristic

C₂×Dic₈₉

C₂×D₄

yes

characteristic

C₂×Dic₈₉

C₂×D₄

yes

characteristic

C₄×D₈₉

—

C₄×D₈₉

—

C₄×D₈₉

C₂×Q₈

yes

—

Dic₁₇₈

—

C₈×C₈₉

C₂×Q₈

yes

—

#10623494

C₂×Q₈

yes

—

C₂²×C₈₉

C₂²×D₄

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

C₄⋊Q₈

yes

—

C₄×C₈₉

—

C₄×C₈₉

#135

yes

—

C₄×C₈₉

#135

yes

—

C₄×C₈₉

C₂²×Q₈

yes

—

Dic₈₉

—

Dic₈₉

—

Dic₈₉

—

Dic₈₉

C₄⋊Q₈

yes

—

C₂×C₈₉

C₂×C₄⋊Q₈

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

D₈₉

—

#3160

—

maximal, Sylow

C₈₉

#3160

yes

characteristic, complemented, Sylow

#533

—

#500

—

#500

D₈₉

yes

characteristic, complemented

#499

—

#498

—

#427

—

C₂×C₄×C₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₈₉

yes

—

#125

—

C₂²⋊Q₈

—

178

—

C₂²⋊Q₈

—

178

—

C₄×Q₈

—

178

—

C₄×Q₈

—

178

—

#117

—

C₂×C₄²

—

#107

—

#107

C₂×D₈₉

yes

—

#106

—

C₄×C₈

—

C₄×C₈

—

100

C₄×C₈

C₂×D₈₉

yes

characteristic

101

C₂×Q₈

—

102

C₂×Q₈

—

178

—

103

C₂²×C₄

—

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

C₂²×D₈₉

yes

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

178

—

112

C₄⋊C₄

—

178

—

113

C₄⋊C₄

—

178

—

114

C₄⋊C₄

C₂²×D₈₉

yes

—

115

C₂²⋊C₄

—

178

—

116

C₄²

—

117

C₄²

—

118

C₄²

—

119

C₄²

—

178

—

120

C₄²

C₂²×D₈₉

yes

characteristic

121

C₂³

—

122

Q₈

—

123

Q₈

—

356

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

178

—

130

C₂×C₄

—

178

—

131

C₂×C₄

—

178

—

132

C₂×C₄

D₄×D₈₉

yes

characteristic

133

C₂×C₄

D₄×D₈₉

yes

characteristic

134

C₂×C₄

C₂³×D₈₉

yes

characteristic

135

C₈

—

136

C₈

Q₈×D₈₉

yes

—

137

C₂²

—

138

C₂²

—

178

—

139

C₂²

—

178

—

140

C₂²

C₂×D₄×D₈₉

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

—

356

—

144

C₄

—

178

—

145

C₄

—

178

—

146

C₄

#15668031

yes

—

147

C₄

#15668035

yes

—

148

C₄

C₂×Q₈×D₈₉

yes

—

149

C₂

—

178

—

150

C₂

C₄⋊Q₈×D₈₉

yes

characteristic

151

C₂

C₂×#15668031

yes

characteristic

152

C₂

C₂×#15668035

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#500×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	C₂²×D₈₉	C₂×D₄
hypercenter	#500	D₈₉
hypocenter	C₈₉	#3160
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

subgroup

G/H

center

C₂²

C₂×D₄×D₈₉

commutator, cosocle

C₂×C₄×C₈₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₈₉

Fitting

#500×C₈₉

C₂

socle

C₂²×C₈₉

C₂²×D₄

prime radical

C₂²×D₈₉

C₂×D₄

D₈₉

C₈₉

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #500

N	H ≅ G/N	type
2. #19152003	C₂	semidirect
3. #19151999	C₂	semidirect
4. #19151994	C₂	semidirect
7. #500×C₈₉	C₂	semidirect
29. C₈×D₈₉	Q₈	semidirect
77. C₈₉	#3160	semidirect
80. #500	D₈₉	semidirect

H ≅ G/N

type

2. #19152003

C₂

semidirect

3. #19151999

C₂

semidirect

4. #19151994

C₂

semidirect

7. #500×C₈₉

C₂

semidirect

29. C₈×D₈₉

Q₈

semidirect

77. C₈₉

#3160

semidirect

80. #500

D₈₉

semidirect

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

subgroup	index
#19152153	2
#19152003	2
#19151999	2
#19151994	2
#19151635	2
C₄×C₈×D₈₉	2
#500×C₈₉	2
#3160	89

subgroup

index

C₄×C₈×D₈₉

#500×C₈₉

#3160

Subgroups — #26030763

Conjugacy classes of subgroups

Complements

Maximal subgroups