Group #26030712 — subgroups

Subgroups — #26030712

Order 11392.

552 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 157 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	C₂×#15668042	C₂	1	4	yes	maximal, complemented
2	C₂×#15668012	C₂	1	1	yes	maximal, characteristic, complemented
3	#19151993	C₂	1	4	yes	maximal, complemented
4	#19151977	C₂	1	2	yes	maximal, complemented
5	#19151970	C₂	1	2	yes	maximal, complemented
6	C₄×#13083558	C₂	1	1	yes	maximal, characteristic, complemented
7	#532×C₈₉	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₇₈	C₂²	1	2	yes	—
9	#15668042	—	2	16	no	—
10	C₂×#13083564	C₂²	1	8	yes	—
11	C₂×#13083562	C₂²	1	8	yes	—
12	C₂×#13083558	C₂²	1	2	yes	—
13	#15668012	—	2	2	no	—
14	#15668012	C₂²	1	2	yes	—
15	#15668012	C₂²	1	2	yes	—
16	C₂×C₄⋊Dic₈₉	—	2	2	no	—
17	#15667923	C₂²	1	2	yes	—
18	C₄×#10623494	C₂²	1	2	yes	—
19	C₂×#41×C₈₉	C₂²	1	2	yes	—
20	C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
21	#127×C₈₉	C₂²	1	1	yes	characteristic
22	#124×C₈₉	C₂²	1	2	yes	—
23	#121×C₈₉	—	2	2	no	—
24	C₄×D₄×C₈₉	—	2	2	no	—
25	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
26	C₂×Dic₁₇₈	—	2	8	no	—
27	C₂×Dic₁₇₈	—	2	2	no	—
28	C₂×Dic₁₇₈	C₂³	1	4	yes	—
29	C₂²×Dic₈₉	—	2	2	no	—
30	#13083564	—	2	16	no	—
31	#13083562	—	2	16	no	—
32	#13083558	D₄	1	8	yes	complemented
33	C₂×#10623494	C₂³	1	4	yes	—
34	C₄⋊Dic₈₉	—	2	4	no	—
35	C₄⋊Dic₈₉	—	2	4	no	—
36	#41×C₈₉	—	2	8	no	—
37	C₂×Q₈×C₈₉	C₂³	1	2	yes	—
38	C₂×D₄×C₈₉	—	2	2	no	—
39	C₂×D₄×C₈₉	—	2	2	no	—
40	C₂×D₄×C₈₉	C₂³	1	2	yes	—
41	C₂²×C₄×C₈₉	—	2	2	no	—
42	C₂²×C₄×C₈₉	D₄	1	2	yes	—
43	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
44	C₄⋊C₄×C₈₉	—	2	2	no	—
45	C₂²⋊C₄×C₈₉	—	2	4	no	—
46	C₄²×C₈₉	D₄	1	2	yes	—
47	C₄²×C₈₉	C₂³	1	1	yes	characteristic
48	C₄²×C₈₉	C₂³	1	1	yes	characteristic
49	C₂³×C₈₉	—	2	2	no	—
50	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
51	Q₈×C₈₉	—	2	4	no	—
52	D₄×C₈₉	—	2	4	no	—
53	D₄×C₈₉	—	4	4	no	—
54	D₄×C₈₉	—	4	4	no	—
55	C₂×C₄×C₈₉	—	2	4	no	—
56	C₂×C₄×C₈₉	—	2	2	no	—
57	C₂×C₄×C₈₉	—	4	4	no	—
58	C₂×C₄×C₈₉	C₂×D₄	1	4	yes	—
59	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
62	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
63	C₂×Dic₈₉	—	2	4	no	—
64	C₂×Dic₈₉	—	4	4	no	—
65	Dic₁₇₈	—	2	8	no	—
66	Dic₁₇₈	—	4	4	no	—
67	#10623494	C₂×D₄	1	8	yes	—
68	C₂²×C₈₉	—	2	2	no	—
69	C₂²×C₈₉	—	4	2	no	—
70	C₂²×C₈₉	—	4	4	no	—
71	C₂²×C₈₉	#127	1	2	yes	—
72	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
73	C₄×C₈₉	—	2	2	no	—
74	C₄×C₈₉	—	4	2	no	—
75	C₄×C₈₉	#127	1	2	yes	—
76	C₄×C₈₉	#137	1	4	yes	complemented
77	C₄×C₈₉	C₂²×D₄	1	2	yes	—
78	Dic₈₉	—	4	4	no	—
79	C₂×C₈₉	—	2	2	no	—
80	C₂×C₈₉	—	8	2	no	—
81	C₂×C₈₉	C₂×#127	1	1	yes	characteristic
82	C₂×C₈₉	C₂×#137	1	2	yes	—
83	#3152	—	89	1	no	maximal, Sylow
84	C₈₉	#3152	1	1	yes	characteristic, complemented, Sylow
85	C₂×#137	—	89	4	no	—
86	#532	D₈₉	1	1	yes	characteristic
87	C₂×C₄⋊Q₈	—	89	1	no	—
88	#497	—	89	4	no	—
89	#494	—	89	2	no	—
90	#492	—	89	2	no	—
91	C₄×M₁₆	—	89	1	no	—
92	C₂×#41	C₂×D₈₉	1	2	yes	—
93	C₂²×Q₈	—	89	2	no	—
94	#137	—	178	16	no	—
95	C₂×Q₁₆	—	89	8	no	—
96	C₂×SD₁₆	—	89	8	no	—
97	C₂×M₁₆	—	89	2	no	—
98	C₄⋊Q₈	—	89	2	no	—
99	C₄⋊Q₈	—	89	2	no	—
100	C₄⋊Q₈	—	178	2	no	—
101	C₄⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
102	#127	C₂×D₈₉	1	1	yes	characteristic
103	#124	C₂×D₈₉	1	2	yes	—
104	#121	—	2	2	no	—
105	C₄×D₄	—	2	2	no	—
106	C₂×C₄⋊C₄	—	178	2	no	—
107	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
108	#97	—	89	2	no	—
109	C₄×C₈	—	89	2	no	—
110	#41	—	2	8	no	—
111	C₂×Q₈	—	89	4	no	—
112	C₂×Q₈	—	178	8	no	—
113	C₂×Q₈	—	178	2	no	—
114	C₂×Q₈	C₂²×D₈₉	1	2	yes	—
115	C₂×D₄	—	2	2	no	—
116	C₂×D₄	—	2	2	no	—
117	C₂×D₄	C₂²×D₈₉	1	2	yes	—
118	C₂²×C₄	—	2	2	no	—
119	C₂²×C₄	—	178	2	no	—
120	C₂²×C₄	C₂²⋊D₈₉	1	2	yes	—
121	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
122	Q₁₆	—	178	16	no	—
123	SD₁₆	—	178	16	no	—
124	M₁₆	—	89	8	no	—
125	C₂×C₈	—	89	4	no	—
126	C₄⋊C₄	—	2	2	no	—
127	C₄⋊C₄	—	178	4	no	—
128	C₄⋊C₄	—	178	4	no	—
129	C₂²⋊C₄	—	2	4	no	—
130	C₄²	C₂²⋊D₈₉	1	2	yes	—
131	C₄²	C₂²×D₈₉	1	1	yes	characteristic
132	C₄²	C₂²×D₈₉	1	1	yes	characteristic
133	C₂³	—	2	2	no	—
134	C₂³	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
135	Q₈	—	2	4	no	—
136	Q₈	—	356	4	no	—
137	Q₈	—	178	8	no	—
138	D₄	—	2	4	no	—
139	D₄	—	4	4	no	—
140	D₄	—	4	4	no	—
141	C₂×C₄	—	2	4	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	4	4	no	—
144	C₂×C₄	—	356	4	no	—
145	C₂×C₄	—	178	4	no	—
146	C₂×C₄	D₄×D₈₉	1	4	yes	—
147	C₂×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
148	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
149	C₂×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
150	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
151	C₈	—	89	8	no	—
152	C₂²	—	2	2	no	—
153	C₂²	—	4	4	no	—
154	C₂²	—	4	2	no	—
155	C₂²	#15668008	1	2	yes	—
156	C₂²	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
157	C₄	—	2	2	no	—
158	C₄	—	4	2	no	—
159	C₄	—	356	4	no	—
160	C₄	#15668008	1	2	yes	—
161	C₄	#15668042	1	4	yes	complemented
162	C₄	C₂×D₄×D₈₉	1	2	yes	—
163	C₂	—	2	2	no	—
164	C₂	—	8	2	no	—
165	C₂	C₂×#15668008	1	1	yes	characteristic
166	C₂	C₂×#15668042	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15668042

C₂

yes

maximal, complemented

C₂×#15668012

C₂

yes

maximal, characteristic, complemented

#19151993

C₂

yes

maximal, complemented

#19151977

C₂

yes

maximal, complemented

#19151970

C₂

yes

maximal, complemented

C₄×#13083558

C₂

yes

maximal, characteristic, complemented

#532×C₈₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₇₈

C₂²

yes

—

#15668042

—

C₂×#13083564

C₂²

yes

—

C₂×#13083562

C₂²

yes

—

C₂×#13083558

C₂²

yes

—

#15668012

—

#15668012

C₂²

yes

—

#15668012

C₂²

yes

—

C₂×C₄⋊Dic₈₉

—

#15667923

C₂²

yes

—

C₄×#10623494

C₂²

yes

—

C₂×#41×C₈₉

C₂²

yes

—

C₄⋊Q₈×C₈₉

C₂²

yes

characteristic

#127×C₈₉

C₂²

yes

characteristic

#124×C₈₉

C₂²

yes

—

#121×C₈₉

—

C₄×D₄×C₈₉

—

C₂×C₄²×C₈₉

C₂²

yes

characteristic

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

C₂³

yes

—

C₂²×Dic₈₉

—

#13083564

—

#13083562

—

#13083558

D₄

yes

complemented

C₂×#10623494

C₂³

yes

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

#41×C₈₉

—

C₂×Q₈×C₈₉

C₂³

yes

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

D₄

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₄⋊C₄×C₈₉

—

C₂²⋊C₄×C₈₉

—

C₄²×C₈₉

D₄

yes

—

C₄²×C₈₉

C₂³

yes

characteristic

C₄²×C₈₉

C₂³

yes

characteristic

C₂³×C₈₉

—

C₂³×C₈₉

C₂×D₄

yes

characteristic

Q₈×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂⁴

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

#10623494

C₂×D₄

yes

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

#127

yes

—

C₂²×C₈₉

C₂²×D₄

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#127

yes

—

C₄×C₈₉

#137

yes

complemented

C₄×C₈₉

C₂²×D₄

yes

—

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

C₂×#127

yes

characteristic

C₂×C₈₉

C₂×#137

yes

—

#3152

—

maximal, Sylow

C₈₉

#3152

yes

characteristic, complemented, Sylow

C₂×#137

—

#532

D₈₉

yes

characteristic

C₂×C₄⋊Q₈

—

#497

—

#494

—

#492

—

C₄×M₁₆

—

C₂×#41

C₂×D₈₉

yes

—

C₂²×Q₈

—

#137

—

178

—

C₂×Q₁₆

—

C₂×SD₁₆

—

C₂×M₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

100

C₄⋊Q₈

—

178

—

101

C₄⋊Q₈

C₂×D₈₉

yes

characteristic

102

#127

C₂×D₈₉

yes

characteristic

103

#124

C₂×D₈₉

yes

—

104

#121

—

105

C₄×D₄

—

106

C₂×C₄⋊C₄

—

178

—

107

C₂×C₄²

C₂×D₈₉

yes

characteristic

108

#97

—

109

C₄×C₈

—

110

#41

—

111

C₂×Q₈

—

112

C₂×Q₈

—

178

—

113

C₂×Q₈

—

178

—

114

C₂×Q₈

C₂²×D₈₉

yes

—

115

C₂×D₄

—

116

C₂×D₄

—

117

C₂×D₄

C₂²×D₈₉

yes

—

118

C₂²×C₄

—

119

C₂²×C₄

—

178

—

120

C₂²×C₄

C₂²⋊D₈₉

yes

—

121

C₂²×C₄

C₂²×D₈₉

yes

characteristic

122

Q₁₆

—

178

—

123

SD₁₆

—

178

—

124

M₁₆

—

125

C₂×C₈

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

178

—

128

C₄⋊C₄

—

178

—

129

C₂²⋊C₄

—

130

C₄²

C₂²⋊D₈₉

yes

—

131

C₄²

C₂²×D₈₉

yes

characteristic

132

C₄²

C₂²×D₈₉

yes

characteristic

133

C₂³

—

134

C₂³

C₂×C₂²⋊D₈₉

yes

characteristic

135

Q₈

—

136

Q₈

—

356

—

137

Q₈

—

178

—

138

D₄

—

139

D₄

—

140

D₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

356

—

145

C₂×C₄

—

178

—

146

C₂×C₄

D₄×D₈₉

yes

—

147

C₂×C₄

C₂×C₂²⋊D₈₉

yes

—

148

C₂×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

149

C₂×C₄

C₂×C₂²⋊D₈₉

yes

—

150

C₂×C₄

C₂³×D₈₉

yes

characteristic

151

C₈

—

152

C₂²

—

153

C₂²

—

154

C₂²

—

155

C₂²

#15668008

yes

—

156

C₂²

C₂²×C₂²⋊D₈₉

yes

characteristic

157

C₄

—

158

C₄

—

159

C₄

—

356

—

160

C₄

#15668008

yes

—

161

C₄

#15668042

yes

complemented

162

C₄

C₂×D₄×D₈₉

yes

—

163

C₂

—

164

C₂

—

165

C₂

C₂×#15668008

yes

characteristic

166

C₂

C₂×#15668042

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting, prime radical	#532×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
hypercenter	#532	D₈₉
hypocenter	C₈₉	#3152
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #532

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₈₉

commutator, cosocle

C₂×C₄×C₈₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₈₉

Fitting, prime radical

#532×C₈₉

C₂

socle

C₂²×C₈₉

C₂²×D₄

D₈₉

C₈₉

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #532

N	H ≅ G/N	type
1. C₂×#15668042	C₂	semidirect
2. C₂×#15668012	C₂	semidirect
3. #19151993	C₂	semidirect
4. #19151977	C₂	semidirect
5. #19151970	C₂	semidirect
6. C₄×#13083558	C₂	semidirect
32. #13083558	D₄	semidirect
76. C₄×C₈₉	#137	semidirect
84. C₈₉	#3152	semidirect
161. C₄	#15668042	semidirect

H ≅ G/N

type

1. C₂×#15668042

C₂

semidirect

2. C₂×#15668012

C₂

semidirect

3. #19151993

C₂

semidirect

4. #19151977

C₂

semidirect

5. #19151970

C₂

semidirect

6. C₄×#13083558

C₂

semidirect

32. #13083558

D₄

semidirect

76. C₄×C₈₉

#137

semidirect

84. C₈₉

#3152

semidirect

161. C₄

#15668042

semidirect

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

subgroup	index
C₂×#15668042	2
C₂×#15668012	2
#19151993	2
#19151977	2
#19151970	2
C₄×#13083558	2
#532×C₈₉	2
#3152	89

subgroup

index

#532×C₈₉

Subgroups — #26030712

Conjugacy classes of subgroups

Complements

Maximal subgroups