Group #26030693 — subgroups

Subgroups — C₂×#19151993 #26030693

Order 11392.

564 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 165 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	C₂²×#13083564	C₂	1	2	yes	maximal, complemented
2	C₂²×#13083562	C₂	1	2	yes	maximal
3	C₂×#15668012	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151993	C₂	1	8	yes	maximal, direct product factor
5	C₂×#15667923	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#124×C₈₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₁₇₈	C₂²	1	2	yes	—
8	C₂×#13083564	—	2	8	no	—
9	C₂×#13083564	C₂²	1	8	yes	complemented
10	C₂×#13083562	—	2	8	no	—
11	C₂×#13083562	C₂²	1	8	yes	—
12	C₂²×#10623494	C₂²	1	2	yes	—
13	#15668012	—	2	2	no	—
14	#15668012	C₂²	1	4	yes	complemented
15	C₂×C₄⋊Dic₈₉	—	2	2	no	—
16	#15667923	C₂²	1	4	yes	complemented
17	C₂²×Q₈×C₈₉	C₂²	1	1	yes	characteristic
18	C₂²×D₄×C₈₉	C₂²	1	1	yes	characteristic
19	#124×C₈₉	—	2	2	no	—
20	#124×C₈₉	C₂²	1	4	yes	—
21	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
22	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
23	C₂×Dic₁₇₈	—	2	4	no	—
24	C₂×Dic₁₇₈	—	2	4	no	—
25	C₂×Dic₁₇₈	—	2	2	no	—
26	C₂×Dic₁₇₈	C₂³	1	4	yes	—
27	C₂²×Dic₈₉	—	2	2	no	—
28	#13083564	—	2	16	no	—
29	#13083562	—	2	16	no	—
30	C₂×#10623494	D₄	1	8	yes	—
31	C₂×#10623494	C₂³	1	4	yes	—
32	C₄⋊Dic₈₉	—	2	8	no	—
33	C₂⁴×C₈₉	—	2	1	no	—
34	C₂×Q₈×C₈₉	—	2	1	no	—
35	C₂×Q₈×C₈₉	—	2	4	no	—
36	C₂×Q₈×C₈₉	C₂³	1	2	yes	—
37	C₂×D₄×C₈₉	—	2	4	no	—
38	C₂×D₄×C₈₉	—	2	1	no	—
39	C₂×D₄×C₈₉	C₂³	1	2	yes	—
40	C₂²×C₄×C₈₉	—	2	1	no	—
41	C₂²×C₄×C₈₉	D₄	1	2	yes	—
42	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
43	C₂²⋊C₄×C₈₉	—	2	8	no	—
44	C₄²×C₈₉	D₄	1	2	yes	—
45	C₄²×C₈₉	C₂³	1	2	yes	—
46	C₂³×C₈₉	—	2	2	no	—
47	C₂³×C₈₉	—	4	2	no	—
48	C₂³×C₈₉	—	4	4	no	—
49	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
50	Q₈×C₈₉	—	2	4	no	—
51	Q₈×C₈₉	—	4	2	no	—
52	D₄×C₈₉	—	2	4	no	—
53	D₄×C₈₉	—	4	2	no	—
54	C₂×C₄×C₈₉	—	2	4	no	—
55	C₂×C₄×C₈₉	—	2	2	no	—
56	C₂×C₄×C₈₉	—	4	2	no	—
57	C₂×C₄×C₈₉	C₂×D₄	1	4	yes	—
58	C₂×C₄×C₈₉	C₂×D₄	1	4	yes	—
59	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
61	C₂×Dic₈₉	—	2	4	no	—
62	C₂×Dic₈₉	—	4	4	no	—
63	Dic₁₇₈	—	2	4	no	—
64	Dic₁₇₈	—	2	4	no	—
65	Dic₁₇₈	—	4	4	no	—
66	#10623494	C₂×D₄	1	8	yes	—
67	C₂²×C₈₉	—	4	2	no	—
68	C₂²×C₈₉	—	4	4	no	—
69	C₂²×C₈₉	—	8	4	no	—
70	C₂²×C₈₉	#127	1	2	yes	—
71	C₂²×C₈₉	#137	1	4	yes	—
72	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
73	C₄×C₈₉	—	2	4	no	—
74	C₄×C₈₉	—	4	2	no	—
75	C₄×C₈₉	#127	1	2	yes	—
76	C₄×C₈₉	C₂²×D₄	1	2	yes	—
77	Dic₈₉	—	4	4	no	—
78	C₂×C₈₉	—	8	2	no	—
79	C₂×C₈₉	#497	1	4	yes	complemented
80	C₂×C₈₉	C₂×#127	1	1	yes	characteristic
81	C₂×C₈₉	C₂×#137	1	2	yes	—
82	C₂×#497	—	89	1	no	maximal, Sylow
83	C₈₉	C₂×#497	1	1	yes	characteristic, complemented, Sylow
84	C₂²×Q₁₆	—	89	2	no	—
85	C₂²×SD₁₆	—	89	2	no	—
86	C₂×C₄⋊Q₈	—	89	1	no	—
87	C₂×#124	D₈₉	1	1	yes	characteristic
88	#497	—	89	8	no	—
89	C₂×#97	—	89	1	no	—
90	C₂²×Q₈	—	89	2	no	—
91	C₂²×Q₈	C₂×D₈₉	1	1	yes	characteristic
92	C₂²×D₄	C₂×D₈₉	1	1	yes	characteristic
93	C₂×Q₁₆	—	89	8	no	—
94	C₂×Q₁₆	—	178	8	no	—
95	C₂×SD₁₆	—	89	8	no	—
96	C₂×SD₁₆	—	178	8	no	—
97	C₂²×C₈	—	89	2	no	—
98	C₄⋊Q₈	—	89	4	no	—
99	C₄⋊Q₈	—	178	2	no	—
100	#124	—	2	2	no	—
101	#124	C₂×D₈₉	1	4	yes	—
102	C₂×C₄⋊C₄	—	178	2	no	—
103	C₂×C₂²⋊C₄	—	2	2	no	—
104	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
105	#97	—	89	4	no	—
106	C₂⁴	—	2	1	no	—
107	C₂×Q₈	—	2	1	no	—
108	C₂×Q₈	—	2	4	no	—
109	C₂×Q₈	—	89	4	no	—
110	C₂×Q₈	—	178	4	no	—
111	C₂×Q₈	—	178	4	no	—
112	C₂×Q₈	—	178	2	no	—
113	C₂×Q₈	C₂²×D₈₉	1	2	yes	—
114	C₂×D₄	—	2	4	no	—
115	C₂×D₄	—	2	1	no	—
116	C₂×D₄	C₂²×D₈₉	1	2	yes	—
117	C₂²×C₄	—	2	1	no	—
118	C₂²×C₄	—	178	2	no	—
119	C₂²×C₄	C₂²⋊D₈₉	1	2	yes	—
120	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
121	Q₁₆	—	178	16	no	—
122	SD₁₆	—	178	16	no	—
123	C₂×C₈	—	89	8	no	—
124	C₂×C₈	—	89	4	no	—
125	C₄⋊C₄	—	178	8	no	—
126	C₂²⋊C₄	—	2	8	no	—
127	C₄²	C₂²⋊D₈₉	1	2	yes	—
128	C₄²	C₂²×D₈₉	1	2	yes	—
129	C₂³	—	2	2	no	—
130	C₂³	—	4	2	no	—
131	C₂³	—	4	4	no	—
132	C₂³	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
133	Q₈	—	2	4	no	—
134	Q₈	—	4	2	no	—
135	Q₈	—	356	4	no	—
136	Q₈	—	178	4	no	—
137	Q₈	—	178	4	no	—
138	D₄	—	2	4	no	—
139	D₄	—	4	2	no	—
140	C₂×C₄	—	2	4	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	4	2	no	—
143	C₂×C₄	—	356	4	no	—
144	C₂×C₄	—	178	4	no	—
145	C₂×C₄	D₄×D₈₉	1	4	yes	—
146	C₂×C₄	C₂×C₂²⋊D₈₉	1	4	yes	—
147	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
148	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
149	C₈	—	89	8	no	—
150	C₂²	—	4	4	no	—
151	C₂²	—	4	2	no	—
152	C₂²	—	8	4	no	—
153	C₂²	#15668008	1	2	yes	—
154	C₂²	#15668042	1	4	yes	—
155	C₂²	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
156	C₄	—	2	4	no	—
157	C₄	—	4	2	no	—
158	C₄	—	356	4	no	—
159	C₄	#15668008	1	2	yes	—
160	C₄	C₂×D₄×D₈₉	1	2	yes	—
161	C₂	—	8	2	no	—
162	C₂	#19151993	1	4	yes	direct product factor
163	C₂	C₂×#15668008	1	1	yes	characteristic
164	C₂	C₂×#15668042	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#13083564

C₂

yes

maximal, complemented

C₂²×#13083562

C₂

yes

maximal

C₂×#15668012

C₂

yes

maximal, characteristic, complemented

#19151993

C₂

yes

maximal, direct product factor

C₂×#15667923

C₂

yes

maximal, characteristic, complemented

C₂×#124×C₈₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₇₈

C₂²

yes

—

C₂×#13083564

—

C₂×#13083564

C₂²

yes

complemented

C₂×#13083562

—

C₂×#13083562

C₂²

yes

—

C₂²×#10623494

C₂²

yes

—

#15668012

—

#15668012

C₂²

yes

complemented

C₂×C₄⋊Dic₈₉

—

#15667923

C₂²

yes

complemented

C₂²×Q₈×C₈₉

C₂²

yes

characteristic

C₂²×D₄×C₈₉

C₂²

yes

characteristic

#124×C₈₉

—

#124×C₈₉

C₂²

yes

—

C₂×C₂²⋊C₄×C₈₉

—

C₂×C₄²×C₈₉

C₂²

yes

characteristic

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

C₂³

yes

—

C₂²×Dic₈₉

—

#13083564

—

#13083562

—

C₂×#10623494

D₄

yes

—

C₂×#10623494

C₂³

yes

—

C₄⋊Dic₈₉

—

C₂⁴×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×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

C₂²⋊C₄×C₈₉

—

C₄²×C₈₉

D₄

yes

—

C₄²×C₈₉

C₂³

yes

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

C₂×D₄

yes

characteristic

Q₈×C₈₉

—

Q₈×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

C₂⁴

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

Dic₁₇₈

—

#10623494

C₂×D₄

yes

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

#127

yes

—

C₂²×C₈₉

#137

yes

—

C₂²×C₈₉

C₂²×D₄

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#127

yes

—

C₄×C₈₉

C₂²×D₄

yes

—

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₂×#497

—

maximal, Sylow

C₈₉

C₂×#497

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

—

C₂²×SD₁₆

—

C₂×C₄⋊Q₈

—

C₂×#124

D₈₉

yes

characteristic

#497

—

C₂×#97

—

C₂²×Q₈

—

C₂²×Q₈

C₂×D₈₉

yes

characteristic

C₂²×D₄

C₂×D₈₉

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

—

178

—

C₂×SD₁₆

—

C₂×SD₁₆

—

178

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

178

—

100

#124

—

101

#124

C₂×D₈₉

yes

—

102

C₂×C₄⋊C₄

—

178

—

103

C₂×C₂²⋊C₄

—

104

C₂×C₄²

C₂×D₈₉

yes

characteristic

105

#97

—

106

C₂⁴

—

107

C₂×Q₈

—

108

C₂×Q₈

—

109

C₂×Q₈

—

110

C₂×Q₈

—

178

—

111

C₂×Q₈

—

178

—

112

C₂×Q₈

—

178

—

113

C₂×Q₈

C₂²×D₈₉

yes

—

114

C₂×D₄

—

115

C₂×D₄

—

116

C₂×D₄

C₂²×D₈₉

yes

—

117

C₂²×C₄

—

118

C₂²×C₄

—

178

—

119

C₂²×C₄

C₂²⋊D₈₉

yes

—

120

C₂²×C₄

C₂²×D₈₉

yes

characteristic

121

Q₁₆

—

178

—

122

SD₁₆

—

178

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₄⋊C₄

—

178

—

126

C₂²⋊C₄

—

127

C₄²

C₂²⋊D₈₉

yes

—

128

C₄²

C₂²×D₈₉

yes

—

129

C₂³

—

130

C₂³

—

131

C₂³

—

132

C₂³

C₂×C₂²⋊D₈₉

yes

characteristic

133

Q₈

—

134

Q₈

—

135

Q₈

—

356

—

136

Q₈

—

178

—

137

Q₈

—

178

—

138

D₄

—

139

D₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

356

—

144

C₂×C₄

—

178

—

145

C₂×C₄

D₄×D₈₉

yes

—

146

C₂×C₄

C₂×C₂²⋊D₈₉

yes

—

147

C₂×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

148

C₂×C₄

C₂³×D₈₉

yes

characteristic

149

C₈

—

150

C₂²

—

151

C₂²

—

152

C₂²

—

153

C₂²

#15668008

yes

—

154

C₂²

#15668042

yes

—

155

C₂²

C₂²×C₂²⋊D₈₉

yes

characteristic

156

C₄

—

157

C₄

—

158

C₄

—

356

—

159

C₄

#15668008

yes

—

160

C₄

C₂×D₄×D₈₉

yes

—

161

C₂

—

162

C₂

#19151993

yes

direct product factor

163

C₂

C₂×#15668008

yes

characteristic

164

C₂

C₂×#15668042

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	C₂×#124×C₈₉	C₂
socle	C₂³×C₈₉	C₂×D₄
prime radical	C₂²×D₄×C₈₉	C₂²
hypercenter	C₂×#124	D₈₉
hypocenter	C₈₉	C₂×#497
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#124

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₈₉

commutator, cosocle

C₂×C₄×C₈₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₈₉

Fitting

C₂×#124×C₈₉

C₂

socle

C₂³×C₈₉

C₂×D₄

prime radical

C₂²×D₄×C₈₉

C₂²

D₈₉

C₈₉

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#124

N	H ≅ G/N	type
1. C₂²×#13083564	C₂	semidirect
3. C₂×#15668012	C₂	semidirect
4. #19151993	C₂	direct (N × H)
5. C₂×#15667923	C₂	semidirect
9. C₂×#13083564	C₂²	semidirect
14. #15668012	C₂²	semidirect
16. #15667923	C₂²	semidirect
79. C₂×C₈₉	#497	semidirect
83. C₈₉	C₂×#497	semidirect
162. C₂	#19151993	direct (N × H)

H ≅ G/N

type

1. C₂²×#13083564

C₂

semidirect

3. C₂×#15668012

C₂

semidirect

4. #19151993

C₂

direct (N × H)

5. C₂×#15667923

C₂

semidirect

9. C₂×#13083564

C₂²

semidirect

14. #15668012

C₂²

semidirect

16. #15667923

C₂²

semidirect

79. C₂×C₈₉

#497

semidirect

83. C₈₉

C₂×#497

semidirect

162. C₂

#19151993

direct (N × H)

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

subgroup	index
C₂²×#13083564	2
C₂²×#13083562	2
C₂×#15668012	2
#19151993	2
C₂×#15667923	2
C₂×#124×C₈₉	2
C₂×#497	89

subgroup

index

C₂×#124×C₈₉

Subgroups — C₂×#19151993 #26030693

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#19151993 #26030693

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#19151993 #26030693