Group #26026193 — subgroups

Subgroups — #26026193

Order 11392.

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

normal subgroups only

This group has 500 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	C₂×#15668000	C₂	1	1	yes	maximal, characteristic
2	#19151524	C₂	1	4	yes	maximal, complemented
3	#19151509	C₂	1	4	yes	maximal, complemented
4	#19151503	C₂	1	2	yes	maximal, complemented
5	#19151499	C₂	1	2	yes	maximal
6	#19151444	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄⋊Q₈×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Dic₁₇₈	C₂²	1	1	yes	characteristic
9	C₂²×D₃₅₆	C₂²	1	1	yes	characteristic
10	#15668000	—	2	4	no	—
11	C₂×#13083551	C₂²	1	2	yes	—
12	C₂×#13083551	C₂²	1	2	yes	—
13	C₂×#13083551	C₂²	1	8	yes	—
14	C₂×C₄×Dic₈₉	C₂²	1	2	yes	—
15	#15667919	C₂²	1	2	yes	—
16	#15667919	C₂²	1	2	yes	—
17	#15667919	C₂²	1	8	yes	—
18	C₂²×Q₈×C₈₉	C₂²	1	2	yes	—
19	C₄⋊Q₈×C₈₉	—	2	4	no	—
20	C₂×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
21	C₂×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
22	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
23	C₂³×D₈₉	C₂³	1	2	yes	—
24	C₂×Dic₁₇₈	—	2	2	no	—
25	C₂×Dic₁₇₈	—	2	4	no	—
26	C₂×D₃₅₆	—	2	2	no	—
27	C₂×D₃₅₆	—	2	4	no	—
28	C₂²×Dic₈₉	C₂³	1	2	yes	—
29	C₂²×Dic₈₉	C₂³	1	4	yes	—
30	#13083551	—	2	4	no	—
31	#13083551	—	2	4	no	—
32	#13083551	—	4	8	no	—
33	C₄×Dic₈₉	—	2	4	no	—
34	C₂×Q₈×C₈₉	—	2	4	no	—
35	C₂×Q₈×C₈₉	—	4	4	no	—
36	C₂²×C₄×C₈₉	C₂³	1	4	yes	—
37	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
38	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
40	C₄⋊C₄×C₈₉	—	2	4	no	—
41	C₄⋊C₄×C₈₉	—	2	4	no	—
42	C₄²×C₈₉	D₄	1	4	yes	—
43	C₂³×C₈₉	C₂⁴	1	1	yes	characteristic
44	C₂²×D₈₉	—	2	2	no	—
45	C₂²×D₈₉	—	2	4	no	—
46	C₂²×D₈₉	—	2	4	no	—
47	C₂²×D₈₉	—	2	4	no	—
48	Q₈×C₈₉	—	4	8	no	—
49	C₂×C₄×C₈₉	—	2	2	no	—
50	C₂×C₄×C₈₉	—	2	2	no	—
51	C₂×C₄×C₈₉	—	2	4	no	—
52	C₂×C₄×C₈₉	—	2	8	no	—
53	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
56	C₂×C₄×C₈₉	#41	1	4	yes	—
57	C₂×Dic₈₉	—	2	2	no	—
58	C₂×Dic₈₉	—	2	4	no	—
59	C₂×Dic₈₉	—	2	4	no	—
60	C₂×Dic₈₉	—	2	4	no	—
61	C₂×Dic₈₉	—	2	4	no	—
62	D₃₅₆	—	4	4	no	—
63	Dic₁₇₈	—	4	4	no	—
64	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
65	C₂²×C₈₉	C₂×#41	1	2	yes	—
66	C₂²×C₈₉	H(5,𝔽₂)	1	2	yes	—
67	C₂²×C₈₉	2₋¹⁺⁴	1	2	yes	—
68	C₂×D₈₉	—	4	2	no	—
69	C₂×D₈₉	—	4	2	no	—
70	C₂×D₈₉	—	4	2	no	—
71	C₂×D₈₉	—	4	8	no	—
72	C₄×C₈₉	—	2	2	no	—
73	C₄×C₈₉	—	2	2	no	—
74	C₄×C₈₉	—	4	4	no	—
75	C₄×C₈₉	#124	1	4	yes	—
76	Dic₈₉	—	4	2	no	—
77	Dic₈₉	—	4	4	no	—
78	C₂×C₈₉	C₂×#124	1	1	yes	characteristic
79	C₂×C₈₉	#535	1	1	yes	characteristic
80	C₂×C₈₉	#536	1	1	yes	characteristic
81	C₂×C₈₉	#542	1	4	yes	—
82	D₈₉	—	8	2	no	—
83	#2640	—	89	1	no	maximal, Sylow
84	C₈₉	#2640	1	1	yes	characteristic, complemented, Sylow
85	C₂×C₄⋊Q₈	D₈₉	1	1	yes	characteristic, complemented
86	C₂×#124	—	89	1	no	—
87	#397	—	89	4	no	—
88	#393	—	89	4	no	—
89	#391	—	89	2	no	—
90	#390	—	89	2	no	—
91	#376	—	89	1	no	—
92	C₂²×Q₈	—	89	1	no	—
93	C₂²×Q₈	C₂×D₈₉	1	2	yes	—
94	C₂²×D₄	—	89	1	no	—
95	C₄⋊Q₈	—	2	4	no	—
96	#124	—	178	4	no	—
97	C₂×C₄⋊C₄	C₂×D₈₉	1	2	yes	—
98	C₂×C₄⋊C₄	C₂×D₈₉	1	2	yes	—
99	C₂×C₂²⋊C₄	—	89	8	no	—
100	C₂×C₂²⋊C₄	—	89	2	no	—
101	C₂×C₂²⋊C₄	—	89	2	no	—
102	C₂×C₄²	—	89	2	no	—
103	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
104	#95	—	89	2	no	—
105	#95	—	89	2	no	—
106	#95	—	89	8	no	—
107	C₂⁴	—	89	2	no	—
108	C₂×Q₈	—	2	4	no	—
109	C₂×Q₈	—	4	4	no	—
110	C₂×Q₈	—	178	2	no	—
111	C₂×Q₈	—	178	4	no	—
112	C₂×D₄	—	178	2	no	—
113	C₂×D₄	—	178	4	no	—
114	C₂²×C₄	—	89	4	no	—
115	C₂²×C₄	—	89	2	no	—
116	C₂²×C₄	C₂²×D₈₉	1	4	yes	—
117	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
118	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
119	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
120	C₄⋊C₄	—	2	4	no	—
121	C₄⋊C₄	—	2	4	no	—
122	C₂²⋊C₄	—	356	8	no	—
123	C₂²⋊C₄	—	178	4	no	—
124	C₂²⋊C₄	—	178	4	no	—
125	C₄²	—	178	4	no	—
126	C₄²	C₂²⋊D₈₉	1	4	yes	—
127	C₂³	—	178	2	no	—
128	C₂³	—	178	4	no	—
129	C₂³	—	178	4	no	—
130	C₂³	—	178	4	no	—
131	C₂³	C₂³×D₈₉	1	1	yes	characteristic
132	Q₈	—	4	8	no	—
133	Q₈	—	356	4	no	—
134	D₄	—	356	4	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	4	no	—
138	C₂×C₄	—	2	8	no	—
139	C₂×C₄	—	178	2	no	—
140	C₂×C₄	—	178	4	no	—
141	C₂×C₄	—	178	4	no	—
142	C₂×C₄	—	178	4	no	—
143	C₂×C₄	—	178	4	no	—
144	C₂×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
145	C₂×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
146	C₂×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
147	C₂×C₄	#13083575	1	4	yes	—
148	C₂²	—	356	2	no	—
149	C₂²	—	356	2	no	—
150	C₂²	—	356	8	no	—
151	C₂²	—	356	2	no	—
152	C₂²	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
153	C₂²	C₂×#13083575	1	2	yes	—
154	C₂²	#15668057	1	2	yes	—
155	C₂²	#15668060	1	2	yes	—
156	C₄	—	2	2	no	—
157	C₄	—	2	2	no	—
158	C₄	—	4	4	no	—
159	C₄	—	356	4	no	—
160	C₄	—	356	2	no	—
161	C₄	#15668001	1	4	yes	—
162	C₂	—	712	2	no	—
163	C₂	C₂×#15668001	1	1	yes	characteristic
164	C₂	#19152165	1	1	yes	characteristic
165	C₂	#19152172	1	1	yes	characteristic
166	C₂	#19152222	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15668000

C₂

yes

maximal, characteristic

#19151524

C₂

yes

maximal, complemented

#19151509

C₂

yes

maximal, complemented

#19151503

C₂

yes

maximal, complemented

#19151499

C₂

yes

maximal

#19151444

C₂

yes

maximal, characteristic, complemented

C₂×C₄⋊Q₈×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂²×Dic₁₇₈

C₂²

yes

characteristic

C₂²×D₃₅₆

C₂²

yes

characteristic

#15668000

—

C₂×#13083551

C₂²

yes

—

C₂×#13083551

C₂²

yes

—

C₂×#13083551

C₂²

yes

—

C₂×C₄×Dic₈₉

C₂²

yes

—

#15667919

C₂²

yes

—

#15667919

C₂²

yes

—

#15667919

C₂²

yes

—

C₂²×Q₈×C₈₉

C₂²

yes

—

C₄⋊Q₈×C₈₉

—

C₂×C₄⋊C₄×C₈₉

C₂²

yes

—

C₂×C₄⋊C₄×C₈₉

C₂²

yes

—

C₂×C₄²×C₈₉

C₂²

yes

characteristic

C₂³×D₈₉

C₂³

yes

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂²×Dic₈₉

C₂³

yes

—

C₂²×Dic₈₉

C₂³

yes

—

#13083551

—

#13083551

—

#13083551

—

C₄×Dic₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂²×C₄×C₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

D₄

yes

—

C₂³×C₈₉

C₂⁴

yes

characteristic

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂²×D₈₉

—

Q₈×C₈₉

—

C₂×C₄×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

—

C₂×C₄×C₈₉

#41

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

D₃₅₆

—

Dic₁₇₈

—

C₂²×C₈₉

C₂²×D₄

yes

characteristic

C₂²×C₈₉

C₂×#41

yes

—

C₂²×C₈₉

H(5,𝔽₂)

yes

—

C₂²×C₈₉

2₋¹⁺⁴

yes

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#124

yes

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

D₈₉

—

#2640

—

maximal, Sylow

C₈₉

#2640

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

D₈₉

yes

characteristic, complemented

—

—

—

—

—

—

C₂²×Q₈

—

C₂²×Q₈

C₂×D₈₉

yes

—

C₂²×D₄

—

C₄⋊Q₈

—

#124

—

178

—

C₂×C₄⋊C₄

C₂×D₈₉

yes

—

C₂×C₄⋊C₄

C₂×D₈₉

yes

—

C₂×C₂²⋊C₄

—

100

C₂×C₂²⋊C₄

—

101

C₂×C₂²⋊C₄

—

102

C₂×C₄²

—

103

C₂×C₄²

C₂×D₈₉

yes

characteristic

104

#95

—

105

#95

—

106

#95

—

107

C₂⁴

—

108

C₂×Q₈

—

109

C₂×Q₈

—

110

C₂×Q₈

—

178

—

111

C₂×Q₈

—

178

—

112

C₂×D₄

—

178

—

113

C₂×D₄

—

178

—

114

C₂²×C₄

—

115

C₂²×C₄

—

116

C₂²×C₄

C₂²×D₈₉

yes

—

117

C₂²×C₄

C₂²×D₈₉

yes

characteristic

118

C₂²×C₄

C₂²×D₈₉

yes

characteristic

119

C₂²×C₄

C₂²×D₈₉

yes

characteristic

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₂²⋊C₄

—

356

—

123

C₂²⋊C₄

—

178

—

124

C₂²⋊C₄

—

178

—

125

C₄²

—

178

—

126

C₄²

C₂²⋊D₈₉

yes

—

127

C₂³

—

178

—

128

C₂³

—

178

—

129

C₂³

—

178

—

130

C₂³

—

178

—

131

C₂³

C₂³×D₈₉

yes

characteristic

132

Q₈

—

133

Q₈

—

356

—

134

D₄

—

356

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

178

—

140

C₂×C₄

—

178

—

141

C₂×C₄

—

178

—

142

C₂×C₄

—

178

—

143

C₂×C₄

—

178

—

144

C₂×C₄

C₂×C₂²⋊D₈₉

yes

—

145

C₂×C₄

C₂×C₂²⋊D₈₉

yes

—

146

C₂×C₄

C₂×C₂²⋊D₈₉

yes

—

147

C₂×C₄

#13083575

yes

—

148

C₂²

—

356

—

149

C₂²

—

356

—

150

C₂²

—

356

—

151

C₂²

—

356

—

152

C₂²

C₂²×C₂²⋊D₈₉

yes

characteristic

153

C₂²

C₂×#13083575

yes

—

154

C₂²

#15668057

yes

—

155

C₂²

#15668060

yes

—

156

C₄

—

157

C₄

—

158

C₄

—

159

C₄

—

356

—

160

C₄

—

356

—

161

C₄

#15668001

yes

—

162

C₂

—

712

—

163

C₂

C₂×#15668001

yes

characteristic

164

C₂

#19152165

yes

characteristic

165

C₂

#19152172

yes

characteristic

166

C₂

#19152222

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₈₉
commutator, socle, cosocle	C₂³×C₈₉	C₂⁴
Fitting	C₂×C₄⋊Q₈×C₈₉	C₂
prime radical	C₂²×D₃₅₆	C₂²
hypercenter	C₂×C₄⋊Q₈	D₈₉
hypocenter	C₈₉	#2640
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₈₉

commutator, socle, cosocle

C₂³×C₈₉

C₂⁴

Fitting

C₂×C₄⋊Q₈×C₈₉

C₂

prime radical

C₂²×D₃₅₆

C₂²

hypercenter

C₂×C₄⋊Q₈

D₈₉

C₈₉

G ▷ C₂³×C₈₉ ▷ C₈₉

C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
2. #19151524	C₂	semidirect
3. #19151509	C₂	semidirect
4. #19151503	C₂	semidirect
6. #19151444	C₂	semidirect
7. C₂×C₄⋊Q₈×C₈₉	C₂	semidirect
84. C₈₉	#2640	semidirect
85. C₂×C₄⋊Q₈	D₈₉	semidirect

H ≅ G/N

type

2. #19151524

C₂

semidirect

3. #19151509

C₂

semidirect

4. #19151503

C₂

semidirect

6. #19151444

C₂

semidirect

7. C₂×C₄⋊Q₈×C₈₉

C₂

semidirect

84. C₈₉

#2640

semidirect

85. C₂×C₄⋊Q₈

D₈₉

semidirect

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

subgroup	index
C₂×#15668000	2
#19151524	2
#19151509	2
#19151503	2
#19151499	2
#19151444	2
C₂×C₄⋊Q₈×C₈₉	2
#2640	89

subgroup

index

C₂×C₄⋊Q₈×C₈₉

Subgroups — #26026193

Conjugacy classes of subgroups

Complements

Maximal subgroups