Group #26033145 — subgroups

Subgroups — #26033145

Order 11392.

324 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 125 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	#19152323	C₂	1	2	yes	maximal
2	#19151999	C₂	1	2	yes	maximal
3	#19151904	C₂	1	4	yes	maximal
4	#19151880	C₂	1	2	yes	maximal
5	C₈×Dic₁₇₈	C₂	1	1	yes	maximal, characteristic
6	C₄×Dic₃₅₆	C₂	1	1	yes	maximal, characteristic
7	#19151692	C₂	1	2	yes	maximal
8	#500×C₈₉	C₂	1	1	yes	maximal, characteristic
9	C₂×Dic₃₅₆	C₂²	1	1	yes	characteristic
10	#15668010	—	2	2	no	—
11	#15668003	C₂²	1	4	yes	—
12	#15668002	—	2	4	no	—
13	Q₈×Dic₈₉	—	2	2	no	—
14	C₄×Dic₁₇₈	C₂²	1	2	yes	—
15	C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
16	#15667977	C₂²	1	4	yes	—
17	#15667946	C₂²	1	1	yes	characteristic
18	#15667943	C₂²	1	4	yes	—
19	C₄⋊#10623494	C₂²	1	1	yes	characteristic
20	#15667941	C₂²	1	2	yes	—
21	#15667937	C₂²	1	2	yes	—
22	#15667935	C₂²	1	4	yes	—
23	C₈×Dic₈₉	C₂²	1	2	yes	—
24	C₄⋊Q₈×C₈₉	C₂²	1	2	yes	—
25	#107×C₈₉	C₂²	1	2	yes	—
26	#107×C₈₉	C₂²	1	2	yes	—
27	C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
28	C₂×Dic₁₇₈	D₄	1	1	yes	characteristic
29	C₂×Dic₁₇₈	C₂³	1	2	yes	—
30	Dic₃₅₆	Q₈	1	4	yes	complemented
31	C₂×#10623494	C₂³	1	2	yes	—
32	C₄⋊Dic₈₉	—	2	4	no	—
33	C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
34	C₄⋊Dic₈₉	C₂³	1	2	yes	—
35	#13083553	—	2	2	no	—
36	#13083553	—	2	4	no	—
37	#13083553	—	2	4	no	—
38	#13083553	D₄	1	2	yes	—
39	C₄×Dic₈₉	—	2	2	no	—
40	C₄×Dic₈₉	—	2	4	no	—
41	C₄×Dic₈₉	C₂³	1	2	yes	—
42	C₂×Q₈×C₈₉	—	2	2	no	—
43	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
44	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
45	C₄⋊C₄×C₈₉	—	2	2	no	—
46	C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
47	C₄²×C₈₉	C₂³	1	1	yes	characteristic
48	Q₈×C₈₉	—	4	4	no	—
49	C₂×C₄×C₈₉	—	2	4	no	—
50	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
53	C₂×Dic₈₉	—	2	2	no	—
54	C₂×Dic₈₉	—	2	2	no	—
55	C₂×Dic₈₉	C₂×D₄	1	2	yes	—
56	C₂×Dic₈₉	C₂×D₄	1	2	yes	—
57	Dic₁₇₈	—	2	2	no	—
58	Dic₁₇₈	—	2	2	no	—
59	Dic₁₇₈	C₂×Q₈	1	4	yes	—
60	C₈×C₈₉	—	2	1	no	—
61	C₈×C₈₉	C₂×Q₈	1	2	yes	—
62	#10623494	—	2	2	no	—
63	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
64	C₄×C₈₉	—	2	1	no	—
65	C₄×C₈₉	—	4	4	no	—
66	C₄×C₈₉	#135	1	2	yes	—
67	C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
68	C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
69	Dic₈₉	—	2	4	no	—
70	Dic₈₉	—	2	2	no	—
71	Dic₈₉	—	2	2	no	—
72	Dic₈₉	—	4	2	no	—
73	C₂×C₈₉	D₄×Q₈	1	1	yes	characteristic
74	C₂×C₈₉	C₂×#135	1	1	yes	characteristic
75	C₂×C₈₉	#578	1	1	yes	characteristic
76	#3383	—	89	1	no	maximal, Sylow
77	C₈₉	#3383	1	1	yes	characteristic, complemented, Sylow
78	#557	—	89	2	no	—
79	#500	D₈₉	1	1	yes	characteristic
80	#499	—	89	2	no	—
81	#479	—	89	4	no	—
82	#475	—	89	2	no	—
83	C₈×Q₈	—	89	1	no	—
84	C₄×Q₁₆	—	89	2	no	—
85	C₄×Q₁₆	—	89	1	no	—
86	C₂×Q₁₆	—	89	1	no	—
87	C₄⋊Q₈	—	178	2	no	—
88	C₄⋊Q₈	C₂×D₈₉	1	2	yes	—
89	#125	—	89	4	no	—
90	#125	—	178	4	no	—
91	C₄×Q₈	—	89	1	no	—
92	C₄×Q₈	—	89	4	no	—
93	C₄×Q₈	—	89	2	no	—
94	C₄×Q₈	—	178	2	no	—
95	#107	—	89	1	no	—
96	#107	C₂×D₈₉	1	2	yes	—
97	#107	C₂×D₈₉	1	2	yes	—
98	#106	—	89	4	no	—
99	C₄⋊C₈	—	89	2	no	—
100	C₄⋊C₈	—	89	1	no	—
101	#103	—	89	2	no	—
102	#103	—	89	4	no	—
103	C₄×C₈	—	89	2	no	—
104	C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
105	C₂×Q₈	—	2	2	no	—
106	C₂×Q₈	—	89	1	no	—
107	C₂×Q₈	—	89	2	no	—
108	Q₁₆	—	89	4	no	—
109	C₂×C₈	—	89	2	no	—
110	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
111	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
112	C₄⋊C₄	—	2	2	no	—
113	C₄⋊C₄	—	89	1	no	—
114	C₄⋊C₄	—	89	2	no	—
115	C₄⋊C₄	—	89	2	no	—
116	C₄⋊C₄	—	178	4	no	—
117	C₄⋊C₄	—	178	4	no	—
118	C₄⋊C₄	—	178	4	no	—
119	C₄⋊C₄	—	178	2	no	—
120	C₄⋊C₄	C₂²×D₈₉	1	4	yes	—
121	C₄²	—	89	2	no	—
122	C₄²	—	178	4	no	—
123	C₄²	—	178	2	no	—
124	C₄²	C₂²×D₈₉	1	1	yes	characteristic
125	Q₈	—	4	4	no	—
126	Q₈	—	89	4	no	—
127	Q₈	—	178	2	no	—
128	Q₈	—	178	2	no	—
129	C₂×C₄	—	2	4	no	—
130	C₂×C₄	—	89	2	no	—
131	C₂×C₄	—	89	2	no	—
132	C₂×C₄	—	178	2	no	—
133	C₂×C₄	—	178	2	no	—
134	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
135	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
136	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
137	C₈	—	2	1	no	—
138	C₈	—	178	2	no	—
139	C₈	Q₈×D₈₉	1	2	yes	—
140	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
141	C₄	—	2	1	no	—
142	C₄	—	4	4	no	—
143	C₄	—	356	2	no	—
144	C₄	—	178	2	no	—
145	C₄	—	178	4	no	—
146	C₄	—	178	2	no	—
147	C₄	#15668031	1	2	yes	—
148	C₄	C₂×Q₈×D₈₉	1	1	yes	characteristic
149	C₄	#15668060	1	1	yes	characteristic
150	C₂	#19152263	1	1	yes	characteristic
151	C₂	C₂×#15668031	1	1	yes	characteristic
152	C₂	#19152411	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19152323

C₂

yes

maximal

#19151999

C₂

yes

maximal

#19151904

C₂

yes

maximal

#19151880

C₂

yes

maximal

C₈×Dic₁₇₈

C₂

yes

maximal, characteristic

C₄×Dic₃₅₆

C₂

yes

maximal, characteristic

#19151692

C₂

yes

maximal

#500×C₈₉

C₂

yes

maximal, characteristic

C₂×Dic₃₅₆

C₂²

yes

characteristic

#15668010

—

#15668003

C₂²

yes

—

#15668002

—

Q₈×Dic₈₉

—

C₄×Dic₁₇₈

C₂²

yes

—

C₄×Dic₁₇₈

C₂²

yes

characteristic

#15667977

C₂²

yes

—

#15667946

C₂²

yes

characteristic

#15667943

C₂²

yes

—

C₄⋊#10623494

C₂²

yes

characteristic

#15667941

C₂²

yes

—

#15667937

C₂²

yes

—

#15667935

C₂²

yes

—

C₈×Dic₈₉

C₂²

yes

—

C₄⋊Q₈×C₈₉

C₂²

yes

—

#107×C₈₉

C₂²

yes

—

#107×C₈₉

C₂²

yes

—

C₄×C₈×C₈₉

C₂²

yes

characteristic

C₂×Dic₁₇₈

D₄

yes

characteristic

C₂×Dic₁₇₈

C₂³

yes

—

Dic₃₅₆

Q₈

yes

complemented

C₂×#10623494

C₂³

yes

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

D₄

yes

characteristic

C₄⋊Dic₈₉

C₂³

yes

—

—

—

—

D₄

yes

—

C₄×Dic₈₉

—

C₄×Dic₈₉

—

C₄×Dic₈₉

C₂³

yes

—

C₂×Q₈×C₈₉

—

C₂×C₈×C₈₉

C₂³

yes

characteristic

C₂×C₈×C₈₉

C₂³

yes

characteristic

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

C₂³

yes

—

C₄²×C₈₉

C₂³

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×Dic₈₉

C₂×D₄

yes

—

C₂×Dic₈₉

C₂×D₄

yes

—

Dic₁₇₈

—

Dic₁₇₈

—

Dic₁₇₈

C₂×Q₈

yes

—

C₈×C₈₉

—

C₈×C₈₉

C₂×Q₈

yes

—

#10623494

—

C₂²×C₈₉

C₂²×D₄

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#135

yes

—

C₄×C₈₉

C₂²×Q₈

yes

characteristic

C₄×C₈₉

2₋¹⁺⁴

yes

characteristic

Dic₈₉

—

Dic₈₉

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

D₄×Q₈

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₈₉

yes

characteristic, complemented, Sylow

#557

—

#500

D₈₉

yes

—

—

—

C₈×Q₈

—

C₄×Q₁₆

—

C₄×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

178

—

C₄⋊Q₈

C₂×D₈₉

yes

—

#125

—

#125

—

178

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

178

—

#107

—

#107

C₂×D₈₉

yes

—

#107

C₂×D₈₉

yes

—

#106

—

C₄⋊C₈

—

100

C₄⋊C₈

—

101

#103

—

102

#103

—

103

C₄×C₈

—

104

C₄×C₈

C₂×D₈₉

yes

characteristic

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×Q₈

—

108

Q₁₆

—

109

C₂×C₈

—

110

C₂×C₈

C₂²×D₈₉

yes

characteristic

111

C₂×C₈

C₂²×D₈₉

yes

characteristic

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

178

—

117

C₄⋊C₄

—

178

—

118

C₄⋊C₄

—

178

—

119

C₄⋊C₄

—

178

—

120

C₄⋊C₄

C₂²×D₈₉

yes

—

121

C₄²

—

122

C₄²

—

178

—

123

C₄²

—

178

—

124

C₄²

C₂²×D₈₉

yes

characteristic

125

Q₈

—

126

Q₈

—

127

Q₈

—

178

—

128

Q₈

—

178

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

178

—

133

C₂×C₄

—

178

—

134

C₂×C₄

D₄×D₈₉

yes

characteristic

135

C₂×C₄

D₄×D₈₉

yes

characteristic

136

C₂×C₄

C₂³×D₈₉

yes

characteristic

137

C₈

—

138

C₈

—

178

—

139

C₈

Q₈×D₈₉

yes

—

140

C₂²

C₂×D₄×D₈₉

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

—

356

—

144

C₄

—

178

—

145

C₄

—

178

—

146

C₄

—

178

—

147

C₄

#15668031

yes

—

148

C₄

C₂×Q₈×D₈₉

yes

characteristic

149

C₄

#15668060

yes

characteristic

150

C₂

#19152263

yes

characteristic

151

C₂

C₂×#15668031

yes

characteristic

152

C₂

#19152411

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, prime radical	C₂²×C₈₉	C₂²×D₄
hypercenter	#500	D₈₉
hypocenter	C₈₉	#3383
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, prime radical

C₂²×C₈₉

C₂²×D₄

D₈₉

C₈₉

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #500

N	H ≅ G/N	type
30. Dic₃₅₆	Q₈	semidirect
77. C₈₉	#3383	semidirect

H ≅ G/N

type

30. Dic₃₅₆

Q₈

semidirect

77. C₈₉

#3383

semidirect

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

subgroup	index
#19152323	2
#19151999	2
#19151904	2
#19151880	2
C₈×Dic₁₇₈	2
C₄×Dic₃₅₆	2
#19151692	2
#500×C₈₉	2
#3383	89

subgroup

index

C₈×Dic₁₇₈

C₄×Dic₃₅₆

#19151692

#500×C₈₉

#3383

Subgroups — #26033145

Conjugacy classes of subgroups

Complements

Maximal subgroups