Group #26020301 — subgroups

Subgroups — #26020301

Order 11392.

252 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 81 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×#15668020	C₂	1	1	yes	maximal, characteristic
2	C₂×#15667947	C₂	1	2	yes	maximal, complemented
3	#19151580	C₂	1	2	yes	maximal
4	#19151356	C₂	1	1	yes	maximal, characteristic, complemented
5	#338×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13083574	C₂²	1	1	yes	characteristic
7	#15668020	—	2	4	no	—
8	C₂×#13083558	C₂²	1	2	yes	—
9	C₂×#13083557	C₄	1	2	yes	—
10	C₂²×#10623494	C₂²	1	1	yes	characteristic
11	C₂×C₈×D₈₉	C₄	1	2	yes	—
12	#15667947	—	2	4	no	—
13	#15667930	—	2	2	no	—
14	#15667928	—	2	2	no	—
15	C₂×M₁₆×C₈₉	C₂²	1	2	yes	—
16	C₂×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
17	#13083574	—	2	2	no	—
18	#13083574	—	2	2	no	—
19	C₂×Dic₁₇₈	D₄	1	1	yes	characteristic
20	C₂×D₃₅₆	D₄	1	1	yes	characteristic
21	C₂×C₂²⋊D₈₉	Q₈	1	2	yes	—
22	C₂×C₄×D₈₉	C₂×C₄	1	2	yes	—
23	#13083558	—	2	2	no	—
24	#13083558	—	2	2	no	—
25	#13083557	—	2	4	no	—
26	C₂×#10623494	—	2	2	no	—
27	C₂×#10623494	C₂×C₄	1	1	yes	characteristic
28	C₂×#10623494	C₂×C₄	1	1	yes	characteristic
29	C₂×#10623494	D₄	1	4	yes	—
30	C₈×D₈₉	—	2	4	no	—
31	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
32	M₁₆×C₈₉	—	2	2	no	—
33	M₁₆×C₈₉	—	2	2	no	—
34	M₁₆×C₈₉	—	2	2	no	—
35	C₂×C₈×C₈₉	—	2	2	no	—
36	C₂×C₈×C₈₉	C₂×C₄	1	2	yes	—
37	C₂³×C₈₉	C₂×Q₈	1	1	yes	characteristic
38	C₂²×D₈₉	C₄⋊C₄	1	2	yes	—
39	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
41	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₈₉	#41	1	2	yes	—
43	C₂²⋊D₈₉	—	2	4	no	—
44	C₂×Dic₈₉	C₄⋊C₄	1	2	yes	—
45	D₃₅₆	—	2	2	no	—
46	C₄×D₈₉	—	2	2	no	—
47	C₄×D₈₉	—	2	2	no	—
48	Dic₁₇₈	—	2	2	no	—
49	C₈×C₈₉	—	2	2	no	—
50	C₈×C₈₉	—	4	2	no	—
51	#10623494	—	4	2	no	—
52	#10623494	C₂²⋊C₄	1	2	yes	—
53	#10623494	C₂²⋊C₄	1	2	yes	—
54	C₂²×C₈₉	—	4	1	no	—
55	C₂²×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₈₉	C₂²⋊Q₈	1	2	yes	—
57	C₂×D₈₉	—	2	2	no	—
58	C₂×D₈₉	—	4	2	no	—
59	C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₄×C₈₉	#117	1	1	yes	characteristic
61	C₄×C₈₉	#121	1	2	yes	—
62	Dic₈₉	—	2	2	no	—
63	C₂×C₈₉	—	2	1	no	—
64	C₂×C₈₉	—	2	2	no	—
65	C₂×C₈₉	#380	1	1	yes	characteristic
66	D₈₉	—	4	2	no	—
67	#1813	—	89	1	no	maximal, Sylow
68	C₈₉	#1813	1	1	yes	characteristic, complemented, Sylow
69	C₂×#131	—	89	1	no	—
70	C₂×#108	—	89	2	no	—
71	#413	—	89	2	no	—
72	#338	—	89	1	no	—
73	#338	D₈₉	1	1	yes	characteristic, complemented
74	C₂×#41	—	89	1	no	—
75	#131	—	178	4	no	—
76	C₂×M₁₆	—	89	2	no	—
77	C₂×M₁₆	—	89	2	no	—
78	C₂×M₁₆	C₂×D₈₉	1	2	yes	—
79	C₂×M₁₆	C₂×D₈₉	1	1	yes	characteristic
80	C₂²×C₈	—	89	1	no	—
81	C₂²×C₈	—	89	2	no	—
82	#108	—	178	4	no	—
83	#101	—	178	2	no	—
84	#100	—	178	2	no	—
85	#41	—	178	2	no	—
86	#41	—	178	2	no	—
87	C₂×Q₈	—	89	1	no	—
88	C₂×D₄	—	89	2	no	—
89	C₂×D₄	—	89	1	no	—
90	C₂²×C₄	—	89	2	no	—
91	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
92	M₁₆	—	2	2	no	—
93	M₁₆	—	2	2	no	—
94	M₁₆	—	2	2	no	—
95	M₁₆	—	178	4	no	—
96	M₁₆	—	178	2	no	—
97	M₁₆	—	178	2	no	—
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	—	89	1	no	—
100	C₂×C₈	—	89	1	no	—
101	C₂×C₈	—	89	4	no	—
102	C₂×C₈	—	178	4	no	—
103	C₂×C₈	—	178	2	no	—
104	C₂×C₈	C₄×D₈₉	1	2	yes	—
105	C₂³	—	89	2	no	—
106	C₂³	Q₈×D₈₉	1	1	yes	characteristic
107	Q₈	—	178	2	no	—
108	D₄	—	178	4	no	—
109	D₄	—	178	2	no	—
110	C₂×C₄	—	89	2	no	—
111	C₂×C₄	—	178	2	no	—
112	C₂×C₄	—	178	2	no	—
113	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
114	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
115	C₂×C₄	D₄×D₈₉	1	2	yes	—
116	C₂×C₄	#13083574	1	2	yes	—
117	C₈	—	2	2	no	—
118	C₈	—	4	2	no	—
119	C₈	—	89	2	no	—
120	C₈	—	89	2	no	—
121	C₈	—	356	2	no	—
122	C₂²	—	4	1	no	—
123	C₂²	—	356	2	no	—
124	C₂²	—	178	2	no	—
125	C₂²	C₄⋊C₄×D₈₉	1	1	yes	characteristic
126	C₂²	#15667990	1	2	yes	—
127	C₄	—	178	2	no	—
128	C₄	C₂²⋊C₄×D₈₉	1	1	yes	characteristic
129	C₄	#15667969	1	1	yes	characteristic
130	C₄	#15667985	1	2	yes	—
131	C₂	—	2	1	no	—
132	C₂	—	2	2	no	—
133	C₂	—	356	2	no	—
134	C₂	#19151457	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15668020

C₂

yes

maximal, characteristic

C₂×#15667947

C₂

yes

maximal, complemented

#19151580

C₂

yes

maximal

#19151356

C₂

yes

maximal, characteristic, complemented

#338×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂×#13083574

C₂²

yes

characteristic

#15668020

—

C₂×#13083558

C₂²

yes

—

C₂×#13083557

C₄

yes

—

C₂²×#10623494

C₂²

yes

characteristic

C₂×C₈×D₈₉

C₄

yes

—

#15667947

—

#15667930

—

#15667928

—

C₂×M₁₆×C₈₉

C₂²

yes

—

C₂×M₁₆×C₈₉

C₂²

yes

characteristic

#13083574

—

#13083574

—

C₂×Dic₁₇₈

D₄

yes

characteristic

C₂×D₃₅₆

D₄

yes

characteristic

C₂×C₂²⋊D₈₉

Q₈

yes

—

C₂×C₄×D₈₉

C₂×C₄

yes

—

—

—

—

—

C₂×C₄

yes

characteristic

C₂×#10623494

C₂×C₄

yes

characteristic

C₂×#10623494

D₄

yes

—

C₈×D₈₉

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

M₁₆×C₈₉

—

M₁₆×C₈₉

—

M₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂×C₄

yes

—

C₂³×C₈₉

C₂×Q₈

yes

characteristic

C₂²×D₈₉

C₄⋊C₄

yes

—

C₂×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

#41

yes

—

C₂²⋊D₈₉

—

C₂×Dic₈₉

C₄⋊C₄

yes

—

D₃₅₆

—

C₄×D₈₉

—

C₄×D₈₉

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

—

#10623494

—

#10623494

C₂²⋊C₄

yes

—

#10623494

C₂²⋊C₄

yes

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₈₉

C₂²⋊Q₈

yes

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

C₂×C₂²⋊C₄

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

—

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#380

yes

characteristic

D₈₉

—

#1813

—

maximal, Sylow

C₈₉

#1813

yes

characteristic, complemented, Sylow

—

—

—

—

D₈₉

yes

characteristic, complemented

C₂×#41

—

#131

—

178

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₈₉

yes

—

C₂×M₁₆

C₂×D₈₉

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

#108

—

178

—

#101

—

178

—

#100

—

178

—

#41

—

178

—

#41

—

178

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₈₉

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

178

—

M₁₆

—

178

—

M₁₆

—

178

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

178

—

103

C₂×C₈

—

178

—

104

C₂×C₈

C₄×D₈₉

yes

—

105

C₂³

—

106

C₂³

Q₈×D₈₉

yes

characteristic

107

Q₈

—

178

—

108

D₄

—

178

—

109

D₄

—

178

—

110

C₂×C₄

—

111

C₂×C₄

—

178

—

112

C₂×C₄

—

178

—

113

C₂×C₄

C₂×C₄×D₈₉

yes

characteristic

114

C₂×C₄

D₄×D₈₉

yes

characteristic

115

C₂×C₄

D₄×D₈₉

yes

—

116

C₂×C₄

#13083574

yes

—

117

C₈

—

118

C₈

—

119

C₈

—

120

C₈

—

121

C₈

—

356

—

122

C₂²

—

123

C₂²

—

356

—

124

C₂²

—

178

—

125

C₂²

C₄⋊C₄×D₈₉

yes

characteristic

126

C₂²

#15667990

yes

—

127

C₄

—

178

—

128

C₄

C₂²⋊C₄×D₈₉

yes

characteristic

129

C₄

#15667969

yes

characteristic

130

C₄

#15667985

yes

—

131

C₂

—

132

C₂

—

133

C₂

—

356

—

134

C₂

#19151457

yes

characteristic

	subgroup	G/H
center	C₄	#15667969
commutator	C₂×C₄×C₈₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₈₉
Fitting	#338×C₈₉	C₂
socle	C₂×C₈₉	#380
prime radical	C₂×#13083574	C₂²
cosocle	C₂²×C₄×C₈₉	C₂³
hypercenter	#338	D₈₉
hypocenter	C₈₉	#1813
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #338

subgroup

G/H

center

C₄

#15667969

commutator

C₂×C₄×C₈₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₈₉

Fitting

#338×C₈₉

C₂

C₂×C₈₉

C₂²

C₂²×C₄×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #338

N	H ≅ G/N	type
2. C₂×#15667947	C₂	semidirect
4. #19151356	C₂	semidirect
5. #338×C₈₉	C₂	semidirect
68. C₈₉	#1813	semidirect
73. #338	D₈₉	semidirect

H ≅ G/N

type

2. C₂×#15667947

C₂

semidirect

4. #19151356

C₂

semidirect

5. #338×C₈₉

C₂

semidirect

68. C₈₉

#1813

semidirect

73. #338

D₈₉

semidirect

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

subgroup	index
C₂×#15668020	2
C₂×#15667947	2
#19151580	2
#19151356	2
#338×C₈₉	2
#1813	89

subgroup

index

#338×C₈₉

Subgroups — #26020301

Conjugacy classes of subgroups

Complements

Maximal subgroups