Group #26029658 — subgroups

Subgroups — #26029658

Order 11392.

716 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 161 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	#19152169	C₂	1	3	yes	maximal, complemented
2	#19151788	C₂	1	8	yes	maximal, complemented
3	#19151572	C₂	1	3	yes	maximal, complemented
4	#585×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₁₇₈	C₂²	1	1	yes	characteristic, complemented
6	#15668010	—	2	3	no	—
7	#15667999	—	2	3	no	—
8	#15667993	C₂²	1	12	yes	complemented
9	#15667991	—	2	6	no	—
10	#15667971	C₂²	1	3	yes	complemented
11	#15667927	C₂²	1	12	yes	complemented
12	2₋¹⁺⁴×C₈₉	—	2	3	no	—
13	H(5,𝔽₂)×C₈₉	—	2	3	no	—
14	H(5,𝔽₂)×C₈₉	C₂²	1	4	yes	—
15	C₂×#41×C₈₉	—	2	6	no	—
16	C₂×#41×C₈₉	C₂²	1	3	yes	—
17	C₂×Dic₁₇₈	—	2	3	no	—
18	C₂×Dic₁₇₈	—	2	3	no	—
19	C₂²×Dic₈₉	C₂³	1	2	yes	—
20	C₄⋊Dic₈₉	—	2	3	no	—
21	#13083553	—	2	12	no	—
22	C₂²⋊Dic₈₉	—	2	12	no	—
23	C₂²⋊Dic₈₉	C₂³	1	6	yes	—
24	C₄×Dic₈₉	—	2	3	no	—
25	#41×C₈₉	—	2	4	no	—
26	#41×C₈₉	—	2	12	no	—
27	#41×C₈₉	—	2	6	no	—
28	#41×C₈₉	—	2	12	no	—
29	#41×C₈₉	—	2	6	no	—
30	C₂×Q₈×C₈₉	—	2	6	no	—
31	C₂×Q₈×C₈₉	D₄	1	3	yes	—
32	C₂×D₄×C₈₉	—	2	6	no	—
33	C₂×D₄×C₈₉	—	2	12	no	—
34	C₂×D₄×C₈₉	D₄	1	3	yes	—
35	C₂×D₄×C₈₉	C₂³	1	6	yes	—
36	C₂²×C₄×C₈₉	—	2	4	no	—
37	C₂²×C₄×C₈₉	D₄	1	6	yes	—
38	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
39	C₂³×C₈₉	—	2	4	no	—
40	C₂³×C₈₉	C₂×D₄	1	6	yes	—
41	C₂³×C₈₉	C₂⁴	1	1	yes	characteristic
42	Q₈×C₈₉	—	2	4	no	—
43	Q₈×C₈₉	—	2	6	no	—
44	D₄×C₈₉	—	2	12	no	—
45	D₄×C₈₉	—	2	12	no	—
46	D₄×C₈₉	—	2	6	no	—
47	C₂×C₄×C₈₉	—	2	12	no	—
48	C₂×C₄×C₈₉	—	2	6	no	—
49	C₂×C₄×C₈₉	—	2	6	no	—
50	C₂×C₄×C₈₉	C₂×D₄	1	6	yes	—
51	C₂×C₄×C₈₉	C₂×D₄	1	3	yes	—
52	C₂×C₄×C₈₉	C₂×D₄	1	3	yes	—
53	C₂×Dic₈₉	—	2	6	no	—
54	C₂×Dic₈₉	—	2	6	no	—
55	Dic₁₇₈	—	4	1	no	—
56	Dic₁₇₈	—	4	3	no	—
57	C₂²×C₈₉	—	2	6	no	—
58	C₂²×C₈₉	—	4	1	no	—
59	C₂²×C₈₉	—	4	6	no	—
60	C₂²×C₈₉	—	8	4	no	—
61	C₂²×C₈₉	C₂²×D₄	1	3	yes	—
62	C₄×C₈₉	—	2	6	no	—
63	C₄×C₈₉	#120	1	1	yes	characteristic
64	C₄×C₈₉	#120	1	3	yes	—
65	Dic₈₉	—	4	6	no	—
66	Dic₈₉	—	4	2	no	—
67	C₂×C₈₉	—	2	3	no	—
68	C₂×C₈₉	—	4	6	no	—
69	C₂×C₈₉	C₂×#120	1	1	yes	characteristic
70	#3028	—	89	1	no	maximal, Sylow
71	C₈₉	#3028	1	1	yes	characteristic, complemented, Sylow
72	#585	D₈₉	1	1	yes	characteristic
73	#536	—	89	3	no	—
74	#458	—	89	8	no	—
75	#410	—	89	3	no	—
76	2₋¹⁺⁴	—	2	3	no	—
77	H(5,𝔽₂)	—	2	3	no	—
78	H(5,𝔽₂)	C₂×D₈₉	1	4	yes	—
79	C₂×#41	—	2	6	no	—
80	C₂×#41	C₂×D₈₉	1	3	yes	—
81	C₂²×Q₈	—	89	1	no	—
82	C₄⋊Q₈	—	178	3	no	—
83	#124	—	178	3	no	—
84	#123	—	89	12	no	—
85	C₂²⋊Q₈	—	178	6	no	—
86	#117	—	89	3	no	—
87	#99	—	89	12	no	—
88	#41	—	2	4	no	—
89	#41	—	2	12	no	—
90	#41	—	2	6	no	—
91	#41	—	2	12	no	—
92	#41	—	2	6	no	—
93	C₂×Q₈	—	2	6	no	—
94	C₂×Q₈	—	178	3	no	—
95	C₂×Q₈	—	178	3	no	—
96	C₂×Q₈	C₂²⋊D₈₉	1	3	yes	—
97	C₂×D₄	—	2	6	no	—
98	C₂×D₄	—	2	12	no	—
99	C₂×D₄	C₂²⋊D₈₉	1	3	yes	—
100	C₂×D₄	C₂²×D₈₉	1	6	yes	—
101	C₂²×C₄	—	2	4	no	—
102	C₂²×C₄	—	89	2	no	—
103	C₂²×C₄	C₂²⋊D₈₉	1	6	yes	—
104	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
105	C₄⋊C₄	—	178	3	no	—
106	C₄⋊C₄	—	178	12	no	—
107	C₂²⋊C₄	—	89	6	no	—
108	C₂²⋊C₄	—	178	12	no	—
109	C₄²	—	178	3	no	—
110	C₂³	—	2	4	no	—
111	C₂³	C₂×C₂²⋊D₈₉	1	6	yes	—
112	C₂³	C₂³×D₈₉	1	1	yes	characteristic
113	Q₈	—	2	4	no	—
114	Q₈	—	2	6	no	—
115	Q₈	—	356	1	no	—
116	Q₈	—	356	3	no	—
117	D₄	—	2	12	no	—
118	D₄	—	2	12	no	—
119	D₄	—	2	6	no	—
120	C₂×C₄	—	2	12	no	—
121	C₂×C₄	—	2	6	no	—
122	C₂×C₄	—	2	6	no	—
123	C₂×C₄	—	178	6	no	—
124	C₂×C₄	—	178	6	no	—
125	C₂×C₄	C₂×C₂²⋊D₈₉	1	6	yes	—
126	C₂×C₄	C₂×C₂²⋊D₈₉	1	3	yes	—
127	C₂×C₄	C₂×C₂²⋊D₈₉	1	3	yes	—
128	C₂²	—	2	6	no	—
129	C₂²	—	4	1	no	—
130	C₂²	—	4	6	no	—
131	C₂²	—	8	4	no	—
132	C₂²	C₂²×C₂²⋊D₈₉	1	3	yes	—
133	C₄	—	2	6	no	—
134	C₄	—	356	6	no	—
135	C₄	—	356	2	no	—
136	C₄	#15667981	1	1	yes	characteristic
137	C₄	#15667981	1	3	yes	—
138	C₂	—	2	3	no	—
139	C₂	—	4	6	no	—
140	C₂	C₂×#15667981	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19152169

C₂

yes

maximal, complemented

#19151788

C₂

yes

maximal, complemented

#19151572

C₂

yes

maximal, complemented

#585×C₈₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₇₈

C₂²

yes

characteristic, complemented

#15668010

—

#15667999

—

#15667993

C₂²

yes

complemented

#15667991

—

#15667971

C₂²

yes

complemented

#15667927

C₂²

yes

complemented

2₋¹⁺⁴×C₈₉

—

H(5,𝔽₂)×C₈₉

—

H(5,𝔽₂)×C₈₉

C₂²

yes

—

C₂×#41×C₈₉

—

C₂×#41×C₈₉

C₂²

yes

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂²×Dic₈₉

C₂³

yes

—

C₄⋊Dic₈₉

—

#13083553

—

C₂²⋊Dic₈₉

—

C₂²⋊Dic₈₉

C₂³

yes

—

C₄×Dic₈₉

—

#41×C₈₉

—

#41×C₈₉

—

#41×C₈₉

—

#41×C₈₉

—

#41×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

D₄

yes

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

D₄

yes

—

C₂×D₄×C₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

D₄

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₂³×C₈₉

—

C₂³×C₈₉

C₂×D₄

yes

—

C₂³×C₈₉

C₂⁴

yes

characteristic

Q₈×C₈₉

—

Q₈×C₈₉

—

D₄×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

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂²×D₄

yes

—

C₄×C₈₉

—

C₄×C₈₉

#120

yes

characteristic

C₄×C₈₉

#120

yes

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

yes

—

C₈₉

yes

characteristic, complemented, Sylow

#585

D₈₉

yes

—

—

—

2₋¹⁺⁴

—

H(5,𝔽₂)

—

H(5,𝔽₂)

C₂×D₈₉

yes

—

C₂×#41

—

C₂×#41

C₂×D₈₉

yes

—

C₂²×Q₈

—

C₄⋊Q₈

—

178

—

#124

—

178

—

#123

—

C₂²⋊Q₈

—

178

—

—

—

—

—

—

—

—

C₂×Q₈

—

C₂×Q₈

—

178

—

C₂×Q₈

—

178

—

C₂×Q₈

C₂²⋊D₈₉

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂²⋊D₈₉

yes

—

100

C₂×D₄

C₂²×D₈₉

yes

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

C₂²⋊D₈₉

yes

—

104

C₂²×C₄

C₂²×D₈₉

yes

characteristic

105

C₄⋊C₄

—

178

—

106

C₄⋊C₄

—

178

—

107

C₂²⋊C₄

—

108

C₂²⋊C₄

—

178

—

109

C₄²

—

178

—

110

C₂³

—

111

C₂³

C₂×C₂²⋊D₈₉

yes

—

112

C₂³

C₂³×D₈₉

yes

characteristic

113

Q₈

—

114

Q₈

—

115

Q₈

—

356

—

116

Q₈

—

356

—

117

D₄

—

118

D₄

—

119

D₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

178

—

124

C₂×C₄

—

178

—

125

C₂×C₄

C₂×C₂²⋊D₈₉

yes

—

126

C₂×C₄

C₂×C₂²⋊D₈₉

yes

—

127

C₂×C₄

C₂×C₂²⋊D₈₉

yes

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

C₂²×C₂²⋊D₈₉

yes

—

133

C₄

—

134

C₄

—

356

—

135

C₄

—

356

—

136

C₄

#15667981

yes

characteristic

137

C₄

#15667981

yes

—

138

C₂

—

139

C₂

—

140

C₂

C₂×#15667981

yes

characteristic

	subgroup	G/H
center	C₂	C₂×#15667981
commutator, cosocle	C₂³×C₈₉	C₂⁴
Frattini	C₂³	C₂³×D₈₉
Fitting, prime radical	#585×C₈₉	C₂
socle	C₂×C₈₉	C₂×#120
hypercenter	#585	D₈₉
hypocenter	C₈₉	#3028
LCS	G ▷ C₂³×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #585

subgroup

G/H

center

C₂

C₂×#15667981

commutator, cosocle

C₂³×C₈₉

C₂⁴

Frattini

C₂³

C₂³×D₈₉

Fitting, prime radical

#585×C₈₉

C₂

C₂×C₈₉

D₈₉

C₈₉

G ▷ C₂³×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #585

N	H ≅ G/N	type
1. #19152169	C₂	semidirect
2. #19151788	C₂	semidirect
3. #19151572	C₂	semidirect
5. C₂²×Dic₁₇₈	C₂²	semidirect
8. #15667993	C₂²	semidirect
10. #15667971	C₂²	semidirect
11. #15667927	C₂²	semidirect
71. C₈₉	#3028	semidirect

H ≅ G/N

type

1. #19152169

C₂

semidirect

2. #19151788

C₂

semidirect

3. #19151572

C₂

semidirect

5. C₂²×Dic₁₇₈

C₂²

semidirect

8. #15667993

C₂²

semidirect

10. #15667971

C₂²

semidirect

11. #15667927

C₂²

semidirect

71. C₈₉

#3028

semidirect

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

subgroup	index
#19152169	2
#19151788	2
#19151572	2
#585×C₈₉	2
#3028	89

subgroup

index

#585×C₈₉

Subgroups — #26029658

Conjugacy classes of subgroups

Complements

Maximal subgroups