Group #26028734 — subgroups

Subgroups — #26028734

Order 11392.

704 conjugacy classes of subgroups; only the 249 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 249 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×#15668059	C₂	1	1	yes	maximal, characteristic
2	#19152365	C₂	1	1	yes	maximal, characteristic, complemented
3	#19151580	C₂	1	6	yes	maximal, complemented
4	C₂×#15667930	C₂	1	6	yes	maximal, complemented
5	#568×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#15668059	—	2	4	no	—
7	#15668059	—	2	4	no	—
8	C₂×#13083574	C₂²	1	3	yes	—
9	C₂×#13083573	C₄	1	6	yes	—
10	C₂²×Dic₁₇₈	C₂²	1	3	yes	—
11	C₂×Q₈×D₈₉	C₄	1	2	yes	—
12	#15668019	—	2	4	no	—
13	C₂×#13083558	C₂²	1	6	yes	—
14	#15667930	C₂²	1	12	yes	—
15	#15667928	C₂²	1	4	yes	—
16	C₂×#41×C₈₉	C₂²	1	1	yes	characteristic
17	#131×C₈₉	—	2	4	no	—
18	C₂×M₁₆×C₈₉	C₂²	1	6	yes	—
19	#13083574	—	2	6	no	—
20	#13083574	—	2	6	no	—
21	#13083573	—	2	24	no	—
22	C₂×Dic₁₇₈	—	2	6	no	—
23	C₂×Dic₁₇₈	—	2	6	no	—
24	C₂×Dic₁₇₈	C₂×C₄	1	6	yes	—
25	C₂×Dic₁₇₈	C₂³	1	3	yes	—
26	Q₈×D₈₉	—	2	8	no	—
27	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
28	C₂×C₂²⋊D₈₉	C₂×C₄	1	6	yes	—
29	C₂²×Dic₈₉	C₂×C₄	1	6	yes	—
30	C₂×C₄×D₈₉	C₂×C₄	1	6	yes	—
31	#13083558	—	2	6	no	—
32	#13083558	C₂³	1	4	yes	—
33	C₂×#10623494	—	2	6	no	—
34	#41×C₈₉	D₄	1	4	yes	—
35	#41×C₈₉	D₄	1	4	yes	—
36	C₂×Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
37	C₂×D₄×C₈₉	C₂×C₄	1	3	yes	—
38	C₂²×C₄×C₈₉	C₂³	1	3	yes	—
39	M₁₆×C₈₉	—	2	6	no	—
40	M₁₆×C₈₉	C₂³	1	4	yes	—
41	C₂×C₈×C₈₉	—	2	6	no	—
42	C₂³×C₈₉	C₂²×C₄	1	3	yes	—
43	C₂²×D₈₉	C₂²×C₄	1	2	yes	—
44	Q₈×C₈₉	C₂²⋊C₄	1	4	yes	—
45	D₄×C₈₉	C₂²⋊C₄	1	12	yes	—
46	C₂×C₄×C₈₉	C₂²×C₄	1	3	yes	—
47	C₂×C₄×C₈₉	C₂×D₄	1	6	yes	—
48	C₂×C₄×C₈₉	C₂×D₄	1	6	yes	—
49	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
50	C₂²⋊D₈₉	—	2	12	no	—
51	C₂×Dic₈₉	—	2	12	no	—
52	C₂×Dic₈₉	C₂²×C₄	1	6	yes	—
53	D₃₅₆	—	2	1	no	—
54	D₃₅₆	—	2	1	no	—
55	C₄×D₈₉	—	2	12	no	—
56	Dic₁₇₈	—	2	12	no	—
57	Dic₁₇₈	—	2	3	no	—
58	Dic₁₇₈	—	2	3	no	—
59	C₈×C₈₉	—	2	4	no	—
60	#10623494	—	2	4	no	—
61	C₂²×C₈₉	—	4	3	no	—
62	C₂²×C₈₉	C₂×C₂²⋊C₄	1	6	yes	—
63	C₂²×C₈₉	C₂³×C₄	1	1	yes	characteristic
64	C₂×D₈₉	—	2	2	no	—
65	C₂×D₈₉	—	4	2	no	—
66	C₄×C₈₉	C₂×C₂²⋊C₄	1	6	yes	—
67	C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
68	C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
69	Dic₈₉	—	2	6	no	—
70	C₂×C₈₉	—	2	1	no	—
71	C₂×C₈₉	—	2	6	no	—
72	C₂×C₈₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
73	D₈₉	—	4	2	no	—
74	#2889	—	89	1	no	maximal, Sylow
75	C₈₉	#2889	1	1	yes	characteristic, complemented, Sylow
76	C₂×2₋¹⁺⁴	—	89	1	no	—
77	#568	—	89	1	no	—
78	#568	D₈₉	1	1	yes	characteristic, complemented
79	#413	—	89	6	no	—
80	C₂×#101	—	89	6	no	—
81	2₋¹⁺⁴	—	178	4	no	—
82	2₋¹⁺⁴	—	178	4	no	—
83	C₂×#41	—	89	6	no	—
84	C₂×#41	—	89	3	no	—
85	C₂×#41	C₂×D₈₉	1	1	yes	characteristic
86	C₂²×Q₈	—	89	2	no	—
87	C₂²×Q₈	—	89	3	no	—
88	#131	—	2	4	no	—
89	#131	—	178	4	no	—
90	C₂×M₁₆	—	89	6	no	—
91	C₂×M₁₆	C₂×D₈₉	1	6	yes	—
92	#101	—	89	12	no	—
93	#100	—	89	4	no	—
94	#41	—	178	24	no	—
95	#41	—	178	6	no	—
96	#41	—	178	6	no	—
97	#41	D₃₅₆	1	4	yes	—
98	#41	C₂²⋊D₈₉	1	4	yes	—
99	C₂×Q₈	—	89	6	no	—
100	C₂×Q₈	—	89	3	no	—
101	C₂×Q₈	—	178	8	no	—
102	C₂×Q₈	—	178	6	no	—
103	C₂×Q₈	—	178	6	no	—
104	C₂×Q₈	C₄×D₈₉	1	1	yes	characteristic
105	C₂×D₄	—	89	6	no	—
106	C₂×D₄	—	89	1	no	—
107	C₂×D₄	C₄×D₈₉	1	3	yes	—
108	C₂²×C₄	—	89	6	no	—
109	C₂²×C₄	—	89	6	no	—
110	C₂²×C₄	C₂²×D₈₉	1	3	yes	—
111	M₁₆	—	2	6	no	—
112	M₁₆	—	89	4	no	—
113	M₁₆	—	178	6	no	—
114	M₁₆	C₂²×D₈₉	1	4	yes	—
115	C₂×C₈	—	2	6	no	—
116	C₂×C₈	—	178	6	no	—
117	C₂³	—	89	2	no	—
118	C₂³	C₂×C₄×D₈₉	1	3	yes	—
119	Q₈	—	178	3	no	—
120	Q₈	—	178	3	no	—
121	Q₈	—	178	12	no	—
122	Q₈	#13083551	1	4	yes	—
123	D₄	—	178	12	no	—
124	D₄	—	178	1	no	—
125	D₄	—	178	1	no	—
126	D₄	#13083551	1	12	yes	—
127	C₂×C₄	—	89	6	no	—
128	C₂×C₄	—	178	12	no	—
129	C₂×C₄	—	178	12	no	—
130	C₂×C₄	C₂×C₄×D₈₉	1	3	yes	—
131	C₂×C₄	C₂×C₂²⋊D₈₉	1	6	yes	—
132	C₂×C₄	C₂×D₃₅₆	1	6	yes	—
133	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
134	C₈	—	2	4	no	—
135	C₈	—	178	4	no	—
136	C₂²	—	4	3	no	—
137	C₂²	—	356	2	no	—
138	C₂²	—	178	2	no	—
139	C₂²	C₂×#13083551	1	6	yes	—
140	C₂²	C₂²×C₄×D₈₉	1	1	yes	characteristic
141	C₄	—	178	6	no	—
142	C₄	C₂×#13083551	1	6	yes	—
143	C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
144	C₄	C₂²×D₃₅₆	1	1	yes	characteristic
145	C₂	—	2	1	no	—
146	C₂	—	2	6	no	—
147	C₂	—	356	2	no	—
148	C₂	C₂²×#13083551	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15668059

C₂

yes

maximal, characteristic

#19152365

C₂

yes

maximal, characteristic, complemented

#19151580

C₂

yes

maximal, complemented

C₂×#15667930

C₂

yes

maximal, complemented

#568×C₈₉

C₂

yes

maximal, characteristic, complemented

#15668059

—

#15668059

—

C₂×#13083574

C₂²

yes

—

C₂×#13083573

C₄

yes

—

C₂²×Dic₁₇₈

C₂²

yes

—

C₂×Q₈×D₈₉

C₄

yes

—

#15668019

—

C₂×#13083558

C₂²

yes

—

#15667930

C₂²

yes

—

#15667928

C₂²

yes

—

C₂×#41×C₈₉

C₂²

yes

characteristic

#131×C₈₉

—

C₂×M₁₆×C₈₉

C₂²

yes

—

#13083574

—

#13083574

—

#13083573

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

C₂×C₄

yes

—

C₂×Dic₁₇₈

C₂³

yes

—

Q₈×D₈₉

—

C₂×D₃₅₆

C₂³

yes

characteristic

C₂×C₂²⋊D₈₉

C₂×C₄

yes

—

C₂²×Dic₈₉

C₂×C₄

yes

—

C₂×C₄×D₈₉

C₂×C₄

yes

—

#13083558

—

#13083558

C₂³

yes

—

C₂×#10623494

—

#41×C₈₉

D₄

yes

—

#41×C₈₉

D₄

yes

—

C₂×Q₈×C₈₉

C₂×C₄

yes

characteristic

C₂×D₄×C₈₉

C₂×C₄

yes

—

C₂²×C₄×C₈₉

C₂³

yes

—

M₁₆×C₈₉

—

M₁₆×C₈₉

C₂³

yes

—

C₂×C₈×C₈₉

—

C₂³×C₈₉

C₂²×C₄

yes

—

C₂²×D₈₉

C₂²×C₄

yes

—

Q₈×C₈₉

C₂²⋊C₄

yes

—

D₄×C₈₉

C₂²⋊C₄

yes

—

C₂×C₄×C₈₉

C₂²×C₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂⁴

yes

characteristic

C₂²⋊D₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

C₂²×C₄

yes

—

D₃₅₆

—

D₃₅₆

—

C₄×D₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂×C₂²⋊C₄

yes

—

C₂²×C₈₉

C₂³×C₄

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

C₂×C₂²⋊C₄

yes

—

C₄×C₈₉

C₂²×D₄

yes

characteristic

C₄×C₈₉

C₂²×D₄

yes

characteristic

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

C₂²×C₂²⋊C₄

yes

characteristic

D₈₉

—

#2889

—

maximal, Sylow

C₈₉

#2889

yes

characteristic, complemented, Sylow

C₂×2₋¹⁺⁴

—

#568

—

#568

D₈₉

yes

characteristic, complemented

#413

—

C₂×#101

—

2₋¹⁺⁴

—

178

—

2₋¹⁺⁴

—

178

—

C₂×#41

—

C₂×#41

—

C₂×#41

C₂×D₈₉

yes

characteristic

C₂²×Q₈

—

C₂²×Q₈

—

#131

—

#131

—

178

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₈₉

yes

—

#101

—

#100

—

#41

—

178

—

#41

—

178

—

#41

—

178

—

#41

D₃₅₆

yes

—

#41

C₂²⋊D₈₉

yes

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂×Q₈

—

178

—

102

C₂×Q₈

—

178

—

103

C₂×Q₈

—

178

—

104

C₂×Q₈

C₄×D₈₉

yes

characteristic

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂×D₄

C₄×D₈₉

yes

—

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

C₂²×D₈₉

yes

—

111

M₁₆

—

112

M₁₆

—

113

M₁₆

—

178

—

114

M₁₆

C₂²×D₈₉

yes

—

115

C₂×C₈

—

116

C₂×C₈

—

178

—

117

C₂³

—

118

C₂³

C₂×C₄×D₈₉

yes

—

119

Q₈

—

178

—

120

Q₈

—

178

—

121

Q₈

—

178

—

122

Q₈

#13083551

yes

—

123

D₄

—

178

—

124

D₄

—

178

—

125

D₄

—

178

—

126

D₄

#13083551

yes

—

127

C₂×C₄

—

128

C₂×C₄

—

178

—

129

C₂×C₄

—

178

—

130

C₂×C₄

C₂×C₄×D₈₉

yes

—

131

C₂×C₄

C₂×C₂²⋊D₈₉

yes

—

132

C₂×C₄

C₂×D₃₅₆

yes

—

133

C₂×C₄

C₂³×D₈₉

yes

characteristic

134

C₈

—

135

C₈

—

178

—

136

C₂²

—

137

C₂²

—

356

—

138

C₂²

—

178

—

139

C₂²

C₂×#13083551

yes

—

140

C₂²

C₂²×C₄×D₈₉

yes

characteristic

141

C₄

—

178

—

142

C₄

C₂×#13083551

yes

—

143

C₄

C₂²×C₂²⋊D₈₉

yes

characteristic

144

C₄

C₂²×D₃₅₆

yes

characteristic

145

C₂

—

146

C₂

—

147

C₂

—

356

—

148

C₂

C₂²×#13083551

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×#13083551
commutator	C₂²×C₈₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#568×C₈₉	C₂
socle	C₂×C₈₉	C₂²×C₂²⋊C₄
prime radical	C₂×#15668059	C₂
cosocle	C₂×C₄×C₈₉	C₂⁴
hypercenter	#568	D₈₉
hypocenter	C₈₉	#2889
LCS	G ▷ C₂²×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂ ◁ C₂×#41 ◁ #568

subgroup

G/H

center

C₂

C₂²×#13083551

commutator

C₂²×C₈₉

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₈₉

Fitting

#568×C₈₉

C₂

socle

C₂×C₈₉

C₂²×C₂²⋊C₄

prime radical

C₂×#15668059

C₂

cosocle

C₂×C₄×C₈₉

C₂⁴

D₈₉

C₈₉

G ▷ C₂²×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂ ◁ C₂×#41 ◁ #568

N	H ≅ G/N	type
2. #19152365	C₂	semidirect
3. #19151580	C₂	semidirect
4. C₂×#15667930	C₂	semidirect
5. #568×C₈₉	C₂	semidirect
75. C₈₉	#2889	semidirect
78. #568	D₈₉	semidirect

H ≅ G/N

type

2. #19152365

C₂

semidirect

3. #19151580

C₂

semidirect

4. C₂×#15667930

C₂

semidirect

5. #568×C₈₉

C₂

semidirect

75. C₈₉

#2889

semidirect

78. #568

D₈₉

semidirect

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

subgroup	index
C₂×#15668059	2
#19152365	2
#19151580	2
C₂×#15667930	2
#568×C₈₉	2
#2889	89

subgroup

index

#568×C₈₉

Subgroups — #26028734

Conjugacy classes of subgroups

Complements

Maximal subgroups