Group #26022511 — subgroups

Subgroups — #26022511

Order 11392.

184 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 57 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×#15667958	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#15667956	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#15667955	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#15667953	C₂	1	1	yes	maximal, characteristic
5	#19151982	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#13083549	C₂	1	1	yes	maximal, characteristic, complemented
7	#495×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₃₅₆	C₂²	1	1	yes	characteristic
9	C₂×#13083563	—	2	1	no	—
10	C₂×D₇₁₂	C₂²	1	1	yes	characteristic
11	#15668000	—	2	1	no	—
12	#15667958	—	2	2	no	—
13	#15667956	—	2	2	no	—
14	#15667955	—	2	2	no	—
15	#15667953	—	2	2	no	—
16	C₂×#13083549	C₂²	1	1	yes	characteristic
17	C₂×#13083549	C₂²	1	1	yes	characteristic
18	C₂×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
19	C₂×SD₁₆×C₈₉	—	2	1	no	—
20	C₂×D₈×C₈₉	C₂²	1	1	yes	characteristic
21	#124×C₈₉	—	2	1	no	—
22	C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
23	C₂×Dic₁₇₈	—	2	1	no	—
24	C₂×D₃₅₆	—	2	1	no	—
25	Dic₃₅₆	—	2	1	no	—
26	Dic₃₅₆	—	2	1	no	—
27	#13083563	—	4	2	no	—
28	D₇₁₂	—	2	1	no	—
29	D₇₁₂	—	2	1	no	—
30	#13083551	—	4	2	no	—
31	#13083549	D₄	1	2	yes	—
32	#13083549	D₄	1	2	yes	—
33	C₂×Q₈×C₈₉	—	2	1	no	—
34	C₂×D₄×C₈₉	—	2	1	no	—
35	Q₁₆×C₈₉	—	2	1	no	—
36	Q₁₆×C₈₉	—	2	1	no	—
37	SD₁₆×C₈₉	—	4	2	no	—
38	D₈×C₈₉	—	2	1	no	—
39	D₈×C₈₉	—	2	1	no	—
40	C₂×C₈×C₈₉	D₄	1	1	yes	characteristic
41	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
42	C₂²⋊C₄×C₈₉	—	4	2	no	—
43	C₄²×C₈₉	D₄	1	1	yes	characteristic
44	C₂³×C₈₉	—	4	1	no	—
45	C₂²×D₈₉	—	4	1	no	—
46	Q₈×C₈₉	—	4	1	no	—
47	Q₈×C₈₉	—	4	1	no	—
48	D₄×C₈₉	—	4	1	no	—
49	D₄×C₈₉	—	4	1	no	—
50	C₂×C₄×C₈₉	—	4	1	no	—
51	C₂×C₄×C₈₉	D₈	1	2	yes	—
52	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
53	C₂×Dic₈₉	—	4	1	no	—
54	D₃₅₆	—	4	1	no	—
55	D₃₅₆	—	4	1	no	—
56	Dic₁₇₈	—	4	1	no	—
57	Dic₁₇₈	—	4	1	no	—
58	C₈×C₈₉	D₈	1	2	yes	—
59	C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
60	C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₈₉	—	8	1	no	—
62	C₂²×C₈₉	—	8	2	no	—
63	C₂²×C₈₉	C₂×D₈	1	1	yes	characteristic
64	C₂×D₈₉	—	8	1	no	—
65	C₂×D₈₉	—	8	2	no	—
66	C₄×C₈₉	—	2	2	no	—
67	C₄×C₈₉	—	8	1	no	—
68	C₄×C₈₉	#127	1	1	yes	characteristic
69	C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
70	Dic₈₉	—	8	1	no	—
71	C₂×C₈₉	—	16	1	no	—
72	C₂×C₈₉	#493	1	1	yes	characteristic
73	C₂×C₈₉	#508	1	2	yes	—
74	D₈₉	—	16	1	no	—
75	#2249	—	89	1	no	maximal, Sylow
76	C₈₉	#2249	1	1	yes	characteristic, complemented, Sylow
77	C₂×Q₃₂	—	89	1	no	—
78	C₂×SD₃₂	—	89	1	no	—
79	C₂×SD₃₂	—	89	1	no	—
80	C₂×D₁₆	—	89	1	no	—
81	#495	—	89	1	no	—
82	#495	D₈₉	1	1	yes	characteristic, complemented
83	C₄×C₁₆	—	89	1	no	—
84	C₂×Q₁₆	—	89	1	no	—
85	C₂×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
86	C₂×SD₁₆	—	2	1	no	—
87	C₂×SD₁₆	—	178	1	no	—
88	C₂×D₈	—	89	1	no	—
89	C₂×D₈	C₂×D₈₉	1	1	yes	characteristic
90	#124	—	2	1	no	—
91	#124	—	178	1	no	—
92	Q₃₂	—	178	2	no	—
93	SD₃₂	—	178	2	no	—
94	SD₃₂	—	178	2	no	—
95	D₁₆	—	178	2	no	—
96	C₂×C₁₆	—	89	1	no	—
97	C₂×C₁₆	—	89	1	no	—
98	C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
99	C₂×Q₈	—	2	1	no	—
100	C₂×Q₈	—	178	1	no	—
101	C₂×D₄	—	2	1	no	—
102	C₂×D₄	—	178	1	no	—
103	Q₁₆	—	2	1	no	—
104	Q₁₆	—	2	1	no	—
105	Q₁₆	—	178	1	no	—
106	Q₁₆	—	178	1	no	—
107	SD₁₆	—	4	2	no	—
108	SD₁₆	—	356	2	no	—
109	D₈	—	2	1	no	—
110	D₈	—	2	1	no	—
111	D₈	—	178	1	no	—
112	D₈	—	178	1	no	—
113	C₂×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
114	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
115	C₂²⋊C₄	—	4	2	no	—
116	C₂²⋊C₄	—	356	2	no	—
117	C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
118	C₁₆	—	89	2	no	—
119	C₁₆	—	89	2	no	—
120	C₂³	—	4	1	no	—
121	C₂³	—	356	1	no	—
122	Q₈	—	4	1	no	—
123	Q₈	—	4	1	no	—
124	Q₈	—	356	1	no	—
125	Q₈	—	356	1	no	—
126	D₄	—	4	1	no	—
127	D₄	—	4	1	no	—
128	D₄	—	356	1	no	—
129	D₄	—	356	1	no	—
130	C₂×C₄	—	4	1	no	—
131	C₂×C₄	—	356	1	no	—
132	C₂×C₄	#13083559	1	2	yes	—
133	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
134	C₈	#13083559	1	2	yes	—
135	C₈	D₄×D₈₉	1	1	yes	characteristic
136	C₈	D₄×D₈₉	1	1	yes	characteristic
137	C₂²	—	8	1	no	—
138	C₂²	—	8	2	no	—
139	C₂²	—	712	1	no	—
140	C₂²	—	712	2	no	—
141	C₂²	C₂×#13083559	1	1	yes	characteristic
142	C₄	—	2	2	no	—
143	C₄	—	8	1	no	—
144	C₄	—	712	1	no	—
145	C₄	#15668008	1	1	yes	characteristic
146	C₄	C₂×#13083559	1	1	yes	characteristic
147	C₂	—	16	1	no	—
148	C₂	—	1424	1	no	—
149	C₂	#19151974	1	1	yes	characteristic
150	C₂	#19152028	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15667958

C₂

yes

maximal, characteristic, complemented

C₂×#15667956

C₂

yes

maximal, characteristic, complemented

C₂×#15667955

C₂

yes

maximal, characteristic, complemented

C₂×#15667953

C₂

yes

maximal, characteristic

#19151982

C₂

yes

maximal, characteristic, complemented

C₄×#13083549

C₂

yes

maximal, characteristic, complemented

#495×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂×Dic₃₅₆

C₂²

yes

characteristic

C₂×#13083563

—

C₂×D₇₁₂

C₂²

yes

—

—

—

—

—

C₂²

yes

characteristic

C₂×#13083549

C₂²

yes

characteristic

C₂×Q₁₆×C₈₉

C₂²

yes

characteristic

C₂×SD₁₆×C₈₉

—

C₂×D₈×C₈₉

C₂²

yes

characteristic

#124×C₈₉

—

C₄×C₈×C₈₉

C₂²

yes

characteristic

C₂×Dic₁₇₈

—

C₂×D₃₅₆

—

Dic₃₅₆

—

Dic₃₅₆

—

#13083563

—

D₇₁₂

—

D₇₁₂

—

#13083551

—

#13083549

D₄

yes

—

#13083549

D₄

yes

—

C₂×Q₈×C₈₉

—

C₂×D₄×C₈₉

—

Q₁₆×C₈₉

—

Q₁₆×C₈₉

—

SD₁₆×C₈₉

—

D₈×C₈₉

—

D₈×C₈₉

—

C₂×C₈×C₈₉

D₄

yes

characteristic

C₂×C₈×C₈₉

C₂³

yes

characteristic

C₂²⋊C₄×C₈₉

—

C₄²×C₈₉

D₄

yes

characteristic

C₂³×C₈₉

—

C₂²×D₈₉

—

Q₈×C₈₉

—

Q₈×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

D₈

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×Dic₈₉

—

D₃₅₆

—

D₃₅₆

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

D₈

yes

—

C₈×C₈₉

C₂×D₄

yes

characteristic

C₈×C₈₉

C₂×D₄

yes

characteristic

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂×D₈

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#127

yes

characteristic

C₄×C₈₉

C₂×D₈

yes

characteristic

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#493

yes

characteristic

C₂×C₈₉

#508

yes

—

D₈₉

—

#2249

—

maximal, Sylow

C₈₉

#2249

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

C₂×SD₃₂

—

C₂×SD₃₂

—

C₂×D₁₆

—

#495

—

#495

D₈₉

yes

characteristic, complemented

C₄×C₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

C₂×D₈₉

yes

characteristic

C₂×SD₁₆

—

C₂×SD₁₆

—

178

—

C₂×D₈

—

C₂×D₈

C₂×D₈₉

yes

characteristic

#124

—

#124

—

178

—

Q₃₂

—

178

—

SD₃₂

—

178

—

SD₃₂

—

178

—

D₁₆

—

178

—

C₂×C₁₆

—

C₂×C₁₆

—

C₄×C₈

C₂×D₈₉

yes

characteristic

C₂×Q₈

—

100

C₂×Q₈

—

178

—

101

C₂×D₄

—

102

C₂×D₄

—

178

—

103

Q₁₆

—

104

Q₁₆

—

105

Q₁₆

—

178

—

106

Q₁₆

—

178

—

107

SD₁₆

—

108

SD₁₆

—

356

—

109

D₈

—

110

D₈

—

111

D₈

—

178

—

112

D₈

—

178

—

113

C₂×C₈

C₂²⋊D₈₉

yes

characteristic

114

C₂×C₈

C₂²×D₈₉

yes

characteristic

115

C₂²⋊C₄

—

116

C₂²⋊C₄

—

356

—

117

C₄²

C₂²⋊D₈₉

yes

characteristic

118

C₁₆

—

119

C₁₆

—

120

C₂³

—

121

C₂³

—

356

—

122

Q₈

—

123

Q₈

—

124

Q₈

—

356

—

125

Q₈

—

356

—

126

D₄

—

127

D₄

—

128

D₄

—

356

—

129

D₄

—

356

—

130

C₂×C₄

—

131

C₂×C₄

—

356

—

132

C₂×C₄

#13083559

yes

—

133

C₂×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

134

C₈

#13083559

yes

—

135

C₈

D₄×D₈₉

yes

characteristic

136

C₈

D₄×D₈₉

yes

characteristic

137

C₂²

—

138

C₂²

—

139

C₂²

—

712

—

140

C₂²

—

712

—

141

C₂²

C₂×#13083559

yes

characteristic

142

C₄

—

143

C₄

—

144

C₄

—

712

—

145

C₄

#15668008

yes

characteristic

146

C₄

C₂×#13083559

yes

characteristic

147

C₂

—

148

C₂

—

1424

—

149

C₂

#19151974

yes

characteristic

150

C₂

#19152028

yes

—

	subgroup	G/H
center	C₂²	C₂×#13083559
commutator, cosocle	C₂×C₈×C₈₉	C₂³
Frattini	C₂×C₈	C₂²×D₈₉
Fitting	#495×C₈₉	C₂
socle	C₂²×C₈₉	C₂×D₈
prime radical	C₂×#15667953	C₂
hypercenter	#495	D₈₉
hypocenter	C₈₉	#2249
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #495

subgroup

G/H

center

C₂²

C₂×#13083559

commutator, cosocle

C₂×C₈×C₈₉

C₂³

Frattini

C₂×C₈

C₂²×D₈₉

Fitting

#495×C₈₉

C₂

socle

C₂²×C₈₉

C₂×D₈

C₂

D₈₉

C₈₉

G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #495

N	H ≅ G/N	type
1. C₂×#15667958	C₂	semidirect
2. C₂×#15667956	C₂	semidirect
3. C₂×#15667955	C₂	semidirect
5. #19151982	C₂	semidirect
6. C₄×#13083549	C₂	semidirect
7. #495×C₈₉	C₂	semidirect
76. C₈₉	#2249	semidirect
82. #495	D₈₉	semidirect

H ≅ G/N

type

1. C₂×#15667958

C₂

semidirect

2. C₂×#15667956

C₂

semidirect

3. C₂×#15667955

C₂

semidirect

5. #19151982

C₂

semidirect

6. C₄×#13083549

C₂

semidirect

7. #495×C₈₉

C₂

semidirect

76. C₈₉

#2249

semidirect

82. #495

D₈₉

semidirect

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

subgroup	index
C₂×#15667958	2
C₂×#15667956	2
C₂×#15667955	2
C₂×#15667953	2
#19151982	2
C₄×#13083549	2
#495×C₈₉	2
#2249	89

subgroup

index

#495×C₈₉

Subgroups — #26022511

Conjugacy classes of subgroups

Complements

Maximal subgroups