Group #26028622 — subgroups

Subgroups — #26028622

Order 11392.

524 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 157 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×#15668004	C₂	1	1	yes	maximal, characteristic, complemented
2	#19151505	C₂	1	2	yes	maximal, complemented
3	#19151501	C₂	1	2	yes	maximal
4	#19151492	C₂	1	8	yes	maximal, complemented
5	C₄²×Dic₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#124×C₈₉	C₂	1	1	yes	maximal, characteristic
7	#15668004	—	2	4	no	—
8	C₂×C₄⋊Dic₈₉	C₂²	1	2	yes	—
9	C₂×#13083553	C₂²	1	4	yes	—
10	C₂×C₂²⋊Dic₈₉	C₂²	1	8	yes	—
11	C₂×C₄×Dic₈₉	C₂²	1	4	yes	—
12	C₂×C₄×Dic₈₉	C₂²	1	2	yes	—
13	#15667919	C₂²	1	8	yes	—
14	C₂²×Q₈×C₈₉	C₂²	1	1	yes	characteristic
15	C₂²×D₄×C₈₉	C₂²	1	1	yes	characteristic
16	#124×C₈₉	—	2	4	no	—
17	C₂×C₂²⋊C₄×C₈₉	C₂²	1	4	yes	—
18	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
19	C₂²×Dic₈₉	C₂³	1	4	yes	—
20	C₂²×Dic₈₉	C₂³	1	4	yes	—
21	C₄⋊Dic₈₉	—	2	4	no	—
22	#13083553	—	2	8	no	—
23	C₂²⋊Dic₈₉	—	2	16	no	—
24	C₄×Dic₈₉	—	2	4	no	—
25	C₄×Dic₈₉	—	2	8	no	—
26	C₂⁴×C₈₉	C₂³	1	2	yes	—
27	C₂×Q₈×C₈₉	—	2	2	no	—
28	C₂×Q₈×C₈₉	—	2	4	no	—
29	C₂×D₄×C₈₉	—	2	2	no	—
30	C₂×D₄×C₈₉	—	2	4	no	—
31	C₂²×C₄×C₈₉	C₂³	1	2	yes	—
32	C₂²×C₄×C₈₉	C₂³	1	2	yes	—
33	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
34	C₂²⋊C₄×C₈₉	—	2	8	no	—
35	C₄²×C₈₉	D₄	1	4	yes	—
36	C₂³×C₈₉	—	2	2	no	—
37	C₂³×C₈₉	—	2	4	no	—
38	C₂³×C₈₉	—	2	8	no	—
39	C₂³×C₈₉	C₂⁴	1	1	yes	characteristic
40	Q₈×C₈₉	—	4	4	no	—
41	D₄×C₈₉	—	4	4	no	—
42	C₂×C₄×C₈₉	—	2	4	no	—
43	C₂×C₄×C₈₉	—	2	2	no	—
44	C₂×C₄×C₈₉	—	2	4	no	—
45	C₂×C₄×C₈₉	C₂×D₄	1	4	yes	—
46	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
47	C₂×C₄×C₈₉	#41	1	4	yes	—
48	C₂×Dic₈₉	—	2	4	no	—
49	C₂×Dic₈₉	—	2	4	no	—
50	C₂×Dic₈₉	—	2	4	no	—
51	C₂×Dic₈₉	—	2	8	no	—
52	C₂×Dic₈₉	#41	1	8	yes	—
53	C₂²×C₈₉	—	4	2	no	—
54	C₂²×C₈₉	—	4	8	no	—
55	C₂²×C₈₉	—	4	4	no	—
56	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
57	C₂²×C₈₉	C₂×#41	1	2	yes	—
58	C₂²×C₈₉	C₂×#41	1	4	yes	—
59	C₄×C₈₉	—	2	4	no	—
60	C₄×C₈₉	—	4	2	no	—
61	C₄×C₈₉	#124	1	4	yes	—
62	Dic₈₉	—	2	8	no	—
63	Dic₈₉	—	4	4	no	—
64	C₂×C₈₉	—	8	2	no	—
65	C₂×C₈₉	C₂×#124	1	1	yes	characteristic
66	C₂×C₈₉	#529	1	4	yes	—
67	C₂×C₈₉	#532	1	2	yes	—
68	#2861	—	89	1	no	maximal, Sylow
69	C₈₉	#2861	1	1	yes	characteristic, complemented, Sylow
70	C₂×#125	—	89	1	no	—
71	C₂×#124	D₈₉	1	1	yes	characteristic
72	#391	—	89	2	no	—
73	#390	—	89	2	no	—
74	#388	—	89	8	no	—
75	C₄³	—	89	1	no	—
76	C₂²×Q₈	C₂×D₈₉	1	1	yes	characteristic
77	C₂²×D₄	C₂×D₈₉	1	1	yes	characteristic
78	#125	—	178	4	no	—
79	#124	—	2	4	no	—
80	C₂×C₄⋊C₄	—	89	4	no	—
81	C₂×C₄⋊C₄	—	89	2	no	—
82	C₂×C₂²⋊C₄	—	89	8	no	—
83	C₂×C₂²⋊C₄	C₂×D₈₉	1	4	yes	—
84	C₂×C₄²	—	89	4	no	—
85	C₂×C₄²	—	89	2	no	—
86	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
87	#95	—	89	8	no	—
88	C₂⁴	C₂²×D₈₉	1	2	yes	—
89	C₂×Q₈	—	2	2	no	—
90	C₂×Q₈	—	2	4	no	—
91	C₂×D₄	—	2	2	no	—
92	C₂×D₄	—	2	4	no	—
93	C₂²×C₄	—	89	4	no	—
94	C₂²×C₄	—	89	4	no	—
95	C₂²×C₄	C₂²×D₈₉	1	2	yes	—
96	C₂²×C₄	C₂²×D₈₉	1	2	yes	—
97	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
98	C₄⋊C₄	—	178	8	no	—
99	C₄⋊C₄	—	178	4	no	—
100	C₂²⋊C₄	—	2	8	no	—
101	C₂²⋊C₄	—	178	16	no	—
102	C₄²	—	178	8	no	—
103	C₄²	—	178	4	no	—
104	C₄²	C₂²⋊D₈₉	1	4	yes	—
105	C₂³	—	2	2	no	—
106	C₂³	—	2	8	no	—
107	C₂³	—	2	4	no	—
108	C₂³	C₂³×D₈₉	1	1	yes	characteristic
109	Q₈	—	4	4	no	—
110	D₄	—	4	4	no	—
111	C₂×C₄	—	2	4	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	89	8	no	—
115	C₂×C₄	—	178	4	no	—
116	C₂×C₄	—	178	8	no	—
117	C₂×C₄	—	178	4	no	—
118	C₂×C₄	—	178	4	no	—
119	C₂×C₄	C₂×C₂²⋊D₈₉	1	4	yes	—
120	C₂×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
121	C₂×C₄	#13083573	1	4	yes	—
122	C₂²	—	4	4	no	—
123	C₂²	—	4	2	no	—
124	C₂²	—	4	8	no	—
125	C₂²	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
126	C₂²	#41×D₈₉	1	4	yes	—
127	C₂²	C₂×#13083573	1	2	yes	—
128	C₄	—	2	4	no	—
129	C₄	—	4	2	no	—
130	C₄	—	356	4	no	—
131	C₄	—	178	8	no	—
132	C₄	#15667999	1	4	yes	—
133	C₂	—	8	2	no	—
134	C₂	C₂×#15667999	1	1	yes	characteristic
135	C₂	#19152127	1	4	yes	—
136	C₂	#19152144	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15668004

C₂

yes

maximal, characteristic, complemented

#19151505

C₂

yes

maximal, complemented

#19151501

C₂

yes

maximal

#19151492

C₂

yes

maximal, complemented

C₄²×Dic₈₉

C₂

yes

maximal, characteristic, complemented

C₂×#124×C₈₉

C₂

yes

maximal, characteristic

#15668004

—

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×#13083553

C₂²

yes

—

C₂×C₂²⋊Dic₈₉

C₂²

yes

—

C₂×C₄×Dic₈₉

C₂²

yes

—

C₂×C₄×Dic₈₉

C₂²

yes

—

#15667919

C₂²

yes

—

C₂²×Q₈×C₈₉

C₂²

yes

characteristic

C₂²×D₄×C₈₉

C₂²

yes

characteristic

#124×C₈₉

—

C₂×C₂²⋊C₄×C₈₉

C₂²

yes

—

C₂×C₄²×C₈₉

C₂²

yes

characteristic

C₂²×Dic₈₉

C₂³

yes

—

C₂²×Dic₈₉

C₂³

yes

—

C₄⋊Dic₈₉

—

#13083553

—

C₂²⋊Dic₈₉

—

C₄×Dic₈₉

—

C₄×Dic₈₉

—

C₂⁴×C₈₉

C₂³

yes

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂²×C₄×C₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₂²⋊C₄×C₈₉

—

C₄²×C₈₉

D₄

yes

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

C₂⁴

yes

characteristic

Q₈×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

#41

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

#41

yes

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂²×D₄

yes

characteristic

C₂²×C₈₉

C₂×#41

yes

—

C₂²×C₈₉

C₂×#41

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#124

yes

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

C₂×#124

yes

characteristic

C₂×C₈₉

#529

yes

—

C₂×C₈₉

#532

yes

—

#2861

—

maximal, Sylow

C₈₉

#2861

yes

characteristic, complemented, Sylow

C₂×#125

—

C₂×#124

D₈₉

yes

—

—

—

C₄³

—

C₂²×Q₈

C₂×D₈₉

yes

characteristic

C₂²×D₄

C₂×D₈₉

yes

characteristic

#125

—

178

—

#124

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

C₂×D₈₉

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₈₉

yes

characteristic

#95

—

C₂⁴

C₂²×D₈₉

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₈₉

yes

—

C₂²×C₄

C₂²×D₈₉

yes

—

C₂²×C₄

C₂²×D₈₉

yes

characteristic

C₄⋊C₄

—

178

—

C₄⋊C₄

—

178

—

100

C₂²⋊C₄

—

101

C₂²⋊C₄

—

178

—

102

C₄²

—

178

—

103

C₄²

—

178

—

104

C₄²

C₂²⋊D₈₉

yes

—

105

C₂³

—

106

C₂³

—

107

C₂³

—

108

C₂³

C₂³×D₈₉

yes

characteristic

109

Q₈

—

110

D₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

178

—

116

C₂×C₄

—

178

—

117

C₂×C₄

—

178

—

118

C₂×C₄

—

178

—

119

C₂×C₄

C₂×C₂²⋊D₈₉

yes

—

120

C₂×C₄

C₂×C₂²⋊D₈₉

yes

—

121

C₂×C₄

#13083573

yes

—

122

C₂²

—

123

C₂²

—

124

C₂²

—

125

C₂²

C₂²×C₂²⋊D₈₉

yes

characteristic

126

C₂²

#41×D₈₉

yes

—

127

C₂²

C₂×#13083573

yes

—

128

C₄

—

129

C₄

—

130

C₄

—

356

—

131

C₄

—

178

—

132

C₄

#15667999

yes

—

133

C₂

—

134

C₂

C₂×#15667999

yes

characteristic

135

C₂

#19152127

yes

—

136

C₂

#19152144

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₈₉
commutator, socle, cosocle	C₂³×C₈₉	C₂⁴
Fitting	C₂×#124×C₈₉	C₂
prime radical	C₂²×D₄×C₈₉	C₂²
hypercenter	C₂×#124	D₈₉
hypocenter	C₈₉	#2861
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×#124

subgroup

G/H

center, Frattini

C₂³

C₂³×D₈₉

commutator, socle, cosocle

C₂³×C₈₉

C₂⁴

Fitting

C₂×#124×C₈₉

C₂

prime radical

C₂²×D₄×C₈₉

C₂²

D₈₉

C₈₉

G ▷ C₂³×C₈₉ ▷ C₈₉

C₁ ◁ C₂³ ◁ C₂×#124

N	H ≅ G/N	type
1. C₂×#15668004	C₂	semidirect
2. #19151505	C₂	semidirect
4. #19151492	C₂	semidirect
5. C₄²×Dic₈₉	C₂	semidirect
69. C₈₉	#2861	semidirect

H ≅ G/N

type

1. C₂×#15668004

C₂

semidirect

2. #19151505

C₂

semidirect

4. #19151492

C₂

semidirect

5. C₄²×Dic₈₉

C₂

semidirect

69. C₈₉

#2861

semidirect

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

subgroup	index
C₂×#15668004	2
#19151505	2
#19151501	2
#19151492	2
C₄²×Dic₈₉	2
C₂×#124×C₈₉	2
#2861	89

subgroup

index

C₄²×Dic₈₉

C₂×#124×C₈₉

#2861

Subgroups — #26028622

Conjugacy classes of subgroups

Complements

Maximal subgroups