Group #26030792 — subgroups

Subgroups — #26030792

Order 11392.

340 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 149 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	#19152226	C₂	1	2	yes	maximal
2	#19152005	C₂	1	4	yes	maximal, complemented
3	#19152002	C₂	1	2	yes	maximal, complemented
4	#19151999	C₂	1	2	yes	maximal, complemented
5	#19151641	C₂	1	2	yes	maximal
6	C₂×#15667946	C₂	1	1	yes	maximal, characteristic
7	#19151553	C₂	1	1	yes	maximal, characteristic
8	#427×C₈₉	C₂	1	1	yes	maximal, characteristic
9	C₂×#13083558	C₂²	1	1	yes	characteristic
10	#15668012	—	2	2	no	—
11	#15668011	C₂²	1	4	yes	—
12	#15668004	—	2	2	no	—
13	#15668003	C₂²	1	4	yes	—
14	#15667989	—	2	4	no	—
15	#15667971	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₈₉	C₂²	1	2	yes	—
17	#15667946	C₂²	1	2	yes	—
18	#15667946	C₂²	1	2	yes	—
19	#15667945	C₂²	1	4	yes	—
20	#15667943	C₂²	1	4	yes	—
21	#15667922	C₂²	1	2	yes	—
22	C₈×Dic₈₉	C₂²	1	2	yes	—
23	C₂²×C₈×C₈₉	C₂²	1	1	yes	characteristic
24	#117×C₈₉	C₂²	1	2	yes	—
25	#107×C₈₉	C₂²	1	1	yes	characteristic
26	#107×C₈₉	C₂²	1	1	yes	characteristic
27	#106×C₈₉	C₂²	1	2	yes	—
28	C₂×Dic₁₇₈	—	2	4	no	—
29	C₂²×Dic₈₉	—	2	2	no	—
30	#13083558	Q₈	1	4	yes	complemented
31	C₂×#10623494	C₂³	1	2	yes	—
32	C₄⋊Dic₈₉	—	2	2	no	—
33	C₄⋊Dic₈₉	—	2	4	no	—
34	C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
35	C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
36	C₄⋊Dic₈₉	C₂³	1	4	yes	—
37	#13083553	—	2	4	no	—
38	#13083553	—	2	4	no	—
39	C₂²⋊Dic₈₉	D₄	1	2	yes	—
40	C₄×Dic₈₉	C₂³	1	2	yes	—
41	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
42	C₂×C₈×C₈₉	Q₈	1	4	yes	complemented
43	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
44	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
45	C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
46	C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
47	C₂²⋊C₄×C₈₉	—	2	2	no	—
48	C₄²×C₈₉	—	2	2	no	—
49	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₈₉	—	2	2	no	—
51	C₂×C₄×C₈₉	—	2	2	no	—
52	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₈₉	C₂×Q₈	1	2	yes	—
54	C₂×C₄×C₈₉	C₂×Q₈	1	2	yes	—
55	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
56	C₂×Dic₈₉	—	2	4	no	—
57	C₂×Dic₈₉	—	8	2	no	—
58	C₂×Dic₈₉	C₂×D₄	1	2	yes	—
59	C₂×Dic₈₉	C₂×D₄	1	2	yes	—
60	Dic₁₇₈	—	8	4	no	—
61	C₈×C₈₉	C₂×Q₈	1	2	yes	—
62	C₈×C₈₉	C₂×Q₈	1	2	yes	—
63	#10623494	C₂×Q₈	1	4	yes	—
64	C₂²×C₈₉	—	2	1	no	—
65	C₂²×C₈₉	—	2	1	no	—
66	C₂²×C₈₉	C₄⋊Q₈	1	2	yes	—
67	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
68	C₄×C₈₉	—	4	2	no	—
69	C₄×C₈₉	—	4	2	no	—
70	C₄×C₈₉	C₄⋊Q₈	1	2	yes	—
71	C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
72	C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
73	Dic₈₉	—	2	2	no	—
74	Dic₈₉	—	2	2	no	—
75	Dic₈₉	—	4	4	no	—
76	C₂×C₈₉	—	2	2	no	—
77	C₂×C₈₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
78	C₂×C₈₉	#576	1	1	yes	characteristic
79	C₂×C₈₉	#578	1	1	yes	characteristic
80	#3164	—	89	1	no	maximal, Sylow
81	C₈₉	#3164	1	1	yes	characteristic, complemented, Sylow
82	#544	—	89	2	no	—
83	#501	—	89	4	no	—
84	#500	—	89	2	no	—
85	#499	—	89	2	no	—
86	#428	—	89	2	no	—
87	#427	D₈₉	1	1	yes	characteristic
88	C₂×#107	—	89	1	no	—
89	#405	—	89	1	no	—
90	C₂×M₁₆	—	89	1	no	—
91	C₂²×C₈	C₂×D₈₉	1	1	yes	characteristic
92	C₄⋊Q₈	—	89	4	no	—
93	C₄⋊Q₈	—	178	2	no	—
94	#125	—	89	4	no	—
95	#125	—	178	2	no	—
96	C₂²⋊Q₈	—	178	4	no	—
97	#117	—	89	1	no	—
98	#117	C₂×D₈₉	1	2	yes	—
99	C₂×C₄⋊C₄	—	89	2	no	—
100	#107	—	89	2	no	—
101	#107	—	89	2	no	—
102	#107	—	89	4	no	—
103	#107	C₂×D₈₉	1	1	yes	characteristic
104	#107	C₂×D₈₉	1	1	yes	characteristic
105	#106	—	89	4	no	—
106	#106	C₂×D₈₉	1	2	yes	—
107	#97	—	89	2	no	—
108	C₄×C₈	—	89	2	no	—
109	C₂×Q₈	—	178	4	no	—
110	C₂²×C₄	—	178	2	no	—
111	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
112	M₁₆	—	89	4	no	—
113	C₂×C₈	—	89	2	no	—
114	C₂×C₈	Dic₁₇₈	1	4	yes	complemented
115	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
116	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
117	C₄⋊C₄	—	89	1	no	—
118	C₄⋊C₄	—	89	1	no	—
119	C₄⋊C₄	—	89	4	no	—
120	C₄⋊C₄	—	178	4	no	—
121	C₄⋊C₄	—	178	4	no	—
122	C₄⋊C₄	—	178	4	no	—
123	C₄⋊C₄	—	178	2	no	—
124	C₄⋊C₄	C₂²×D₈₉	1	2	yes	—
125	C₄⋊C₄	C₂²×D₈₉	1	2	yes	—
126	C₂²⋊C₄	—	2	2	no	—
127	C₂²⋊C₄	—	89	2	no	—
128	C₄²	—	2	2	no	—
129	C₄²	—	89	2	no	—
130	C₂³	D₄×D₈₉	1	1	yes	characteristic
131	Q₈	—	712	4	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	89	2	no	—
135	C₂×C₄	—	89	2	no	—
136	C₂×C₄	—	178	4	no	—
137	C₂×C₄	—	712	2	no	—
138	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
139	C₂×C₄	Q₈×D₈₉	1	2	yes	—
140	C₂×C₄	C₂×Dic₁₇₈	1	2	yes	—
141	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
142	C₈	—	89	4	no	—
143	C₈	C₂×Dic₁₇₈	1	2	yes	—
144	C₈	C₂×Dic₁₇₈	1	2	yes	—
145	C₂²	—	2	1	no	—
146	C₂²	—	2	1	no	—
147	C₂²	#15668011	1	2	yes	—
148	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
149	C₄	—	4	2	no	—
150	C₄	—	4	2	no	—
151	C₄	—	356	4	no	—
152	C₄	—	178	2	no	—
153	C₄	—	178	2	no	—
154	C₄	#15668011	1	2	yes	—
155	C₄	C₂×Q₈×D₈₉	1	1	yes	characteristic
156	C₄	C₂²×Dic₁₇₈	1	1	yes	characteristic
157	C₂	—	2	2	no	—
158	C₂	C₂×#15668011	1	1	yes	characteristic
159	C₂	#19152400	1	1	yes	characteristic
160	C₂	#19152412	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19152226

C₂

yes

maximal

#19152005

C₂

yes

maximal, complemented

#19152002

C₂

yes

maximal, complemented

#19151999

C₂

yes

maximal, complemented

#19151641

C₂

yes

maximal

C₂×#15667946

C₂

yes

maximal, characteristic

#19151553

C₂

yes

maximal, characteristic

#427×C₈₉

C₂

yes

maximal, characteristic

C₂×#13083558

C₂²

yes

characteristic

#15668012

—

#15668011

C₂²

yes

—

#15668004

—

#15668003

C₂²

yes

—

#15667989

—

#15667971

C₂²

yes

characteristic

C₂×C₄⋊Dic₈₉

C₂²

yes

—

#15667946

C₂²

yes

—

#15667946

C₂²

yes

—

#15667945

C₂²

yes

—

#15667943

C₂²

yes

—

#15667922

C₂²

yes

—

C₈×Dic₈₉

C₂²

yes

—

C₂²×C₈×C₈₉

C₂²

yes

characteristic

#117×C₈₉

C₂²

yes

—

#107×C₈₉

C₂²

yes

characteristic

#107×C₈₉

C₂²

yes

characteristic

#106×C₈₉

C₂²

yes

—

C₂×Dic₁₇₈

—

C₂²×Dic₈₉

—

#13083558

Q₈

yes

complemented

C₂×#10623494

C₂³

yes

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

D₄

yes

characteristic

C₄⋊Dic₈₉

D₄

yes

characteristic

C₄⋊Dic₈₉

C₂³

yes

—

#13083553

—

#13083553

—

C₂²⋊Dic₈₉

D₄

yes

—

C₄×Dic₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₂×C₈×C₈₉

Q₈

yes

complemented

C₂×C₈×C₈₉

C₂³

yes

characteristic

C₂×C₈×C₈₉

C₂³

yes

characteristic

C₄⋊C₄×C₈₉

C₂³

yes

—

C₄⋊C₄×C₈₉

C₂³

yes

—

C₂²⋊C₄×C₈₉

—

C₄²×C₈₉

—

C₂³×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×Q₈

yes

—

C₂×C₄×C₈₉

C₂×Q₈

yes

—

C₂×C₄×C₈₉

C₂⁴

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

C₂×D₄

yes

—

C₂×Dic₈₉

C₂×D₄

yes

—

Dic₁₇₈

—

C₈×C₈₉

C₂×Q₈

yes

—

C₈×C₈₉

C₂×Q₈

yes

—

#10623494

C₂×Q₈

yes

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

C₄⋊Q₈

yes

—

C₂²×C₈₉

C₂²×D₄

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₄⋊Q₈

yes

—

C₄×C₈₉

C₂²×Q₈

yes

characteristic

C₄×C₈₉

C₂²×Q₈

yes

characteristic

Dic₈₉

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

C₂×C₄⋊Q₈

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₈₉

yes

characteristic, complemented, Sylow

—

—

—

—

—

D₈₉

yes

characteristic

C₂×#107

—

#405

—

C₂×M₁₆

—

C₂²×C₈

C₂×D₈₉

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

178

—

#125

—

#125

—

178

—

C₂²⋊Q₈

—

178

—

#117

—

#117

C₂×D₈₉

yes

—

C₂×C₄⋊C₄

—

100

#107

—

101

#107

—

102

#107

—

103

#107

C₂×D₈₉

yes

characteristic

104

#107

C₂×D₈₉

yes

characteristic

105

#106

—

106

#106

C₂×D₈₉

yes

—

107

#97

—

108

C₄×C₈

—

109

C₂×Q₈

—

178

—

110

C₂²×C₄

—

178

—

111

C₂²×C₄

C₂²×D₈₉

yes

characteristic

112

M₁₆

—

113

C₂×C₈

—

114

C₂×C₈

Dic₁₇₈

yes

complemented

115

C₂×C₈

C₂²×D₈₉

yes

characteristic

116

C₂×C₈

C₂²×D₈₉

yes

characteristic

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

178

—

121

C₄⋊C₄

—

178

—

122

C₄⋊C₄

—

178

—

123

C₄⋊C₄

—

178

—

124

C₄⋊C₄

C₂²×D₈₉

yes

—

125

C₄⋊C₄

C₂²×D₈₉

yes

—

126

C₂²⋊C₄

—

127

C₂²⋊C₄

—

128

C₄²

—

129

C₄²

—

130

C₂³

D₄×D₈₉

yes

characteristic

131

Q₈

—

712

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

178

—

137

C₂×C₄

—

712

—

138

C₂×C₄

D₄×D₈₉

yes

characteristic

139

C₂×C₄

Q₈×D₈₉

yes

—

140

C₂×C₄

C₂×Dic₁₇₈

yes

—

141

C₂×C₄

C₂³×D₈₉

yes

characteristic

142

C₈

—

143

C₈

C₂×Dic₁₇₈

yes

—

144

C₈

C₂×Dic₁₇₈

yes

—

145

C₂²

—

146

C₂²

—

147

C₂²

#15668011

yes

—

148

C₂²

C₂×D₄×D₈₉

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

356

—

152

C₄

—

178

—

153

C₄

—

178

—

154

C₄

#15668011

yes

—

155

C₄

C₂×Q₈×D₈₉

yes

characteristic

156

C₄

C₂²×Dic₁₇₈

yes

characteristic

157

C₂

—

158

C₂

C₂×#15668011

yes

characteristic

159

C₂

#19152400

yes

characteristic

160

C₂

#19152412

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#427×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	C₂³×C₈₉	C₂×D₄
hypercenter	#427	D₈₉
hypocenter	C₈₉	#3164
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #427

subgroup

G/H

center

C₂²

C₂×D₄×D₈₉

commutator, cosocle

C₂×C₄×C₈₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₈₉

Fitting

#427×C₈₉

C₂

socle

C₂²×C₈₉

C₂²×D₄

prime radical

C₂³×C₈₉

C₂×D₄

D₈₉

C₈₉

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #427

N	H ≅ G/N	type
2. #19152005	C₂	semidirect
3. #19152002	C₂	semidirect
4. #19151999	C₂	semidirect
30. #13083558	Q₈	semidirect
42. C₂×C₈×C₈₉	Q₈	semidirect
81. C₈₉	#3164	semidirect
114. C₂×C₈	Dic₁₇₈	semidirect

H ≅ G/N

type

2. #19152005

C₂

semidirect

3. #19152002

C₂

semidirect

4. #19151999

C₂

semidirect

30. #13083558

Q₈

semidirect

42. C₂×C₈×C₈₉

Q₈

semidirect

81. C₈₉

#3164

semidirect

114. C₂×C₈

Dic₁₇₈

semidirect

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

subgroup	index
#19152226	2
#19152005	2
#19152002	2
#19151999	2
#19151641	2
C₂×#15667946	2
#19151553	2
#427×C₈₉	2
#3164	89

subgroup

index

#427×C₈₉

Subgroups — #26030792

Conjugacy classes of subgroups

Complements

Maximal subgroups