Group #26021667 — subgroups

Subgroups — #26021667

Order 11392.

258 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 79 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×#15667934	C₂	1	2	yes	maximal
2	#19151500	C₂	1	1	yes	maximal, characteristic
3	#19151464	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151361	C₂	1	2	yes	maximal, complemented
5	#336×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₃₅₆	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₈₉	C₂²	1	2	yes	—
9	C₂×#13083551	—	2	2	no	—
10	#15667934	—	2	4	no	—
11	#15667919	—	2	2	no	—
12	C₂²×C₈×C₈₉	C₂²	1	2	yes	—
13	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
14	C₂³×D₈₉	—	2	1	no	—
15	C₂×D₃₅₆	—	2	2	no	—
16	C₂×D₃₅₆	—	2	2	no	—
17	C₂×D₃₅₆	—	2	1	no	—
18	C₂×D₃₅₆	D₄	1	2	yes	—
19	C₂²×Dic₈₉	—	2	2	no	—
20	C₂²×Dic₈₉	—	2	1	no	—
21	C₄⋊Dic₈₉	—	2	2	no	—
22	C₄⋊Dic₈₉	—	2	2	no	—
23	C₄⋊Dic₈₉	—	2	2	no	—
24	#13083551	—	4	4	no	—
25	C₂²×C₄×C₈₉	D₄	1	2	yes	—
26	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
27	C₂×C₈×C₈₉	—	2	2	no	—
28	C₂×C₈×C₈₉	—	2	2	no	—
29	C₂×C₈×C₈₉	—	2	2	no	—
30	C₄²×C₈₉	—	2	2	no	—
31	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
32	C₂²×D₈₉	—	2	2	no	—
33	C₂²×D₈₉	—	4	1	no	—
34	C₂²×D₈₉	—	4	1	no	—
35	C₂²×D₈₉	—	4	2	no	—
36	C₂²×D₈₉	—	4	2	no	—
37	C₂×C₄×C₈₉	—	2	2	no	—
38	C₂×C₄×C₈₉	—	2	2	no	—
39	C₂×C₄×C₈₉	—	2	2	no	—
40	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₈₉	#41	1	2	yes	—
42	C₂×C₄×C₈₉	#41	1	2	yes	—
43	C₂×C₄×C₈₉	#41	1	1	yes	characteristic
44	C₂×Dic₈₉	—	4	2	no	—
45	C₂×Dic₈₉	—	4	1	no	—
46	C₂×Dic₈₉	—	4	2	no	—
47	C₂×Dic₈₉	—	4	2	no	—
48	C₂×Dic₈₉	—	4	2	no	—
49	D₃₅₆	—	2	4	no	—
50	D₃₅₆	—	4	2	no	—
51	C₈×C₈₉	—	4	2	no	—
52	C₂²×C₈₉	#121	1	1	yes	characteristic
53	C₂²×C₈₉	#123	1	1	yes	characteristic
54	C₂²×C₈₉	#124	1	1	yes	characteristic
55	C₂²×C₈₉	#135	1	2	yes	—
56	C₂²×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
57	C₂×D₈₉	—	4	2	no	—
58	C₂×D₈₉	—	4	4	no	—
59	C₂×D₈₉	—	8	2	no	—
60	C₂×D₈₉	—	8	1	no	—
61	C₂×D₈₉	—	8	1	no	—
62	C₄×C₈₉	—	4	2	no	—
63	C₄×C₈₉	#123	1	2	yes	—
64	C₄×C₈₉	#126	1	2	yes	—
65	Dic₈₉	—	8	2	no	—
66	Dic₈₉	—	8	1	no	—
67	C₂×C₈₉	#397	1	1	yes	characteristic
68	C₂×C₈₉	#463	1	2	yes	—
69	C₂×C₈₉	#482	1	2	yes	—
70	C₂×C₈₉	#488	1	1	yes	characteristic
71	C₂×C₈₉	#490	1	1	yes	characteristic
72	D₈₉	—	8	2	no	—
73	#2073	—	89	1	no	maximal, Sylow
74	C₈₉	#2073	1	1	yes	characteristic, complemented, Sylow
75	C₂×#102	—	89	2	no	—
76	#390	—	89	1	no	—
77	#382	—	89	1	no	—
78	#340	—	89	2	no	—
79	#336	D₈₉	1	1	yes	characteristic, complemented
80	C₂²×D₄	—	89	1	no	—
81	C₂²×C₈	C₂×D₈₉	1	2	yes	—
82	C₂×C₄⋊C₄	—	89	1	no	—
83	C₂×C₄⋊C₄	—	89	2	no	—
84	C₂×C₂²⋊C₄	—	178	2	no	—
85	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
86	#102	—	178	4	no	—
87	#95	—	178	2	no	—
88	C₂⁴	—	178	1	no	—
89	C₂×D₄	—	89	2	no	—
90	C₂×D₄	—	178	2	no	—
91	C₂×D₄	—	178	2	no	—
92	C₂×D₄	—	178	1	no	—
93	C₂²×C₄	—	178	2	no	—
94	C₂²×C₄	—	178	1	no	—
95	C₂²×C₄	D₃₅₆	1	2	yes	—
96	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
97	C₂×C₈	—	2	2	no	—
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	—	2	2	no	—
100	C₄⋊C₄	—	178	2	no	—
101	C₄⋊C₄	—	178	2	no	—
102	C₄⋊C₄	—	178	2	no	—
103	C₂²⋊C₄	—	356	4	no	—
104	C₄²	—	2	2	no	—
105	C₂³	—	356	1	no	—
106	C₂³	—	356	1	no	—
107	C₂³	—	356	2	no	—
108	C₂³	—	356	2	no	—
109	C₂³	—	178	2	no	—
110	C₂³	C₂×D₃₅₆	1	1	yes	characteristic
111	D₄	—	356	2	no	—
112	D₄	—	178	4	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	356	2	no	—
117	C₂×C₄	—	356	1	no	—
118	C₂×C₄	—	356	2	no	—
119	C₂×C₄	—	356	2	no	—
120	C₂×C₄	—	356	2	no	—
121	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
122	C₂×C₄	#13083573	1	2	yes	—
123	C₂×C₄	#13083574	1	2	yes	—
124	C₂×C₄	#13083575	1	1	yes	characteristic
125	C₈	—	4	2	no	—
126	C₂²	—	356	2	no	—
127	C₂²	—	356	4	no	—
128	C₂²	—	712	2	no	—
129	C₂²	—	712	1	no	—
130	C₂²	—	712	1	no	—
131	C₂²	#15667987	1	1	yes	characteristic
132	C₂²	#15667996	1	1	yes	characteristic
133	C₂²	#15668000	1	1	yes	characteristic
134	C₂²	#15668034	1	2	yes	—
135	C₂²	#15668040	1	2	yes	—
136	C₄	—	4	2	no	—
137	C₄	—	712	1	no	—
138	C₄	—	712	2	no	—
139	C₄	#15667994	1	2	yes	—
140	C₄	#15668007	1	2	yes	—
141	C₂	—	712	2	no	—
142	C₂	#19151525	1	1	yes	characteristic
143	C₂	#19151814	1	2	yes	—
144	C₂	#19151923	1	2	yes	—
145	C₂	#19151953	1	1	yes	characteristic
146	C₂	#19151963	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15667934

C₂

yes

maximal

#19151500

C₂

yes

maximal, characteristic

#19151464

C₂

yes

maximal, characteristic, complemented

#19151361

C₂

yes

maximal, complemented

#336×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂²×D₃₅₆

C₂²

yes

characteristic

C₂×C₄⋊Dic₈₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×#13083551

—

#15667934

—

#15667919

—

C₂²×C₈×C₈₉

C₂²

yes

—

C₂×C₄²×C₈₉

C₂²

yes

characteristic

C₂³×D₈₉

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

D₄

yes

—

C₂²×Dic₈₉

—

C₂²×Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

#13083551

—

C₂²×C₄×C₈₉

D₄

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₄²×C₈₉

—

C₂³×C₈₉

C₂×D₄

yes

characteristic

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

#41

yes

—

C₂×C₄×C₈₉

#41

yes

—

C₂×C₄×C₈₉

#41

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

D₃₅₆

—

D₃₅₆

—

C₈×C₈₉

—

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

—

C₂²×C₈₉

AGL(1,ℤ/8)

yes

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#123

yes

—

C₄×C₈₉

#126

yes

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

#397

yes

characteristic

C₂×C₈₉

#463

yes

—

C₂×C₈₉

#482

yes

—

C₂×C₈₉

yes

C₂×C₈₉

yes

D₈₉

—

#2073

—

maximal, Sylow

C₈₉

#2073

yes

characteristic, complemented, Sylow

—

—

—

—

D₈₉

yes

characteristic, complemented

C₂²×D₄

—

C₂²×C₈

C₂×D₈₉

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

178

—

C₂×C₄²

C₂×D₈₉

yes

characteristic

#102

—

178

—

#95

—

178

—

C₂⁴

—

178

—

C₂×D₄

—

C₂×D₄

—

178

—

C₂×D₄

—

178

—

C₂×D₄

—

178

—

C₂²×C₄

—

178

—

C₂²×C₄

—

178

—

C₂²×C₄

D₃₅₆

yes

—

C₂²×C₄

C₂²×D₈₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₄⋊C₄

—

178

—

101

C₄⋊C₄

—

178

—

102

C₄⋊C₄

—

178

—

103

C₂²⋊C₄

—

356

—

104

C₄²

—

105

C₂³

—

356

—

106

C₂³

—

356

—

107

C₂³

—

356

—

108

C₂³

—

356

—

109

C₂³

—

178

—

110

C₂³

C₂×D₃₅₆

yes

characteristic

111

D₄

—

356

—

112

D₄

—

178

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

356

—

117

C₂×C₄

—

356

—

118

C₂×C₄

—

356

—

119

C₂×C₄

—

356

—

120

C₂×C₄

—

356

—

121

C₂×C₄

D₄×D₈₉

yes

characteristic

122

C₂×C₄

#13083573

yes

—

123

C₂×C₄

#13083574

yes

—

124

C₂×C₄

#13083575

yes

characteristic

125

C₈

—

126

C₂²

—

356

—

127

C₂²

—

356

—

128

C₂²

—

712

—

129

C₂²

—

712

—

130

C₂²

—

712

—

131

C₂²

#15667987

yes

characteristic

132

C₂²

#15667996

yes

characteristic

133

C₂²

#15668000

yes

characteristic

134

C₂²

#15668034

yes

—

135

C₂²

#15668040

yes

—

136

C₄

—

137

C₄

—

712

—

138

C₄

—

712

—

139

C₄

#15667994

yes

—

140

C₄

#15668007

yes

—

141

C₂

—

712

—

142

C₂

#19151525

yes

characteristic

143

C₂

#19151814

yes

—

144

C₂

#19151923

yes

—

145

C₂

#19151953

yes

characteristic

146

C₂

#19151963

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₃₅₆
commutator, cosocle	C₂²×C₄×C₈₉	C₂³
Frattini	C₂²×C₄	C₂²×D₈₉
Fitting	#336×C₈₉	C₂
socle	C₂³×C₈₉	C₂×D₄
prime radical	C₂²×D₃₅₆	C₂²
hypercenter	#336	D₈₉
hypocenter	C₈₉	#2073
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

subgroup

G/H

center

C₂³

C₂×D₃₅₆

commutator, cosocle

C₂²×C₄×C₈₉

C₂³

Frattini

C₂²×C₄

C₂²×D₈₉

Fitting

#336×C₈₉

C₂

socle

C₂³×C₈₉

C₂×D₄

prime radical

C₂²×D₃₅₆

C₂²

D₈₉

C₈₉

G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

N	H ≅ G/N	type
3. #19151464	C₂	semidirect
4. #19151361	C₂	semidirect
5. #336×C₈₉	C₂	semidirect
74. C₈₉	#2073	semidirect
79. #336	D₈₉	semidirect

H ≅ G/N

type

3. #19151464

C₂

semidirect

4. #19151361

C₂

semidirect

5. #336×C₈₉

C₂

semidirect

74. C₈₉

#2073

semidirect

79. #336

D₈₉

semidirect

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

subgroup	index
C₂×#15667934	2
#19151500	2
#19151464	2
#19151361	2
#336×C₈₉	2
#2073	89

subgroup

index

#336×C₈₉

Subgroups — #26021667

Conjugacy classes of subgroups

Complements

Maximal subgroups