Group #26021924 — subgroups

Subgroups — #26021924

Order 11392.

270 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 73 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	#19151572	C₂	1	2	yes	maximal, complemented
2	#19151391	C₂	1	4	yes	maximal, complemented
3	#535×C₈₉	C₂	1	1	yes	maximal, characteristic
4	#15667971	—	2	2	no	—
5	#15667927	—	2	2	no	—
6	#15667927	C₂²	1	4	yes	complemented
7	C₂×#41×C₈₉	C₂²	1	1	yes	characteristic
8	C₂²×D₄×C₈₉	C₄	1	1	yes	characteristic, complemented
9	#127×C₈₉	C₂²	1	2	yes	—
10	#124×C₈₉	C₄	1	2	yes	complemented
11	#121×C₈₉	—	2	2	no	—
12	#120×C₈₉	—	2	2	no	—
13	#117×C₈₉	C₄	1	1	yes	characteristic, complemented
14	C₄⋊Dic₈₉	—	4	2	no	—
15	C₂²⋊Dic₈₉	—	2	4	no	—
16	C₄×Dic₈₉	—	4	2	no	—
17	C₂⁴×C₈₉	—	2	1	no	—
18	#41×C₈₉	—	4	2	no	—
19	C₂×Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
20	C₂×D₄×C₈₉	—	2	4	no	—
21	C₂×D₄×C₈₉	—	2	1	no	—
22	C₂×D₄×C₈₉	—	4	2	no	—
23	C₂×D₄×C₈₉	C₂×C₄	1	2	yes	—
24	C₂×D₄×C₈₉	D₄	1	2	yes	—
25	C₂×D₄×C₈₉	C₂³	1	1	yes	characteristic
26	C₂²×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₈₉	D₄	1	2	yes	—
28	C₄⋊C₄×C₈₉	—	2	1	no	—
29	C₂²⋊C₄×C₈₉	—	2	1	no	—
30	C₂²⋊C₄×C₈₉	—	2	4	no	—
31	C₄²×C₈₉	C₂×C₄	1	2	yes	—
32	C₂³×C₈₉	—	2	2	no	—
33	C₂³×C₈₉	—	4	2	no	—
34	C₂³×C₈₉	—	8	2	no	—
35	C₂³×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
36	C₂³×C₈₉	C₂×D₄	1	2	yes	—
37	Q₈×C₈₉	—	4	1	no	—
38	D₄×C₈₉	—	4	2	no	—
39	D₄×C₈₉	—	4	1	no	—
40	D₄×C₈₉	—	4	2	no	—
41	D₄×C₈₉	—	4	2	no	—
42	D₄×C₈₉	—	8	4	no	—
43	C₂×C₄×C₈₉	—	2	2	no	—
44	C₂×C₄×C₈₉	—	2	2	no	—
45	C₂×C₄×C₈₉	—	2	2	no	—
46	C₂×C₄×C₈₉	—	2	2	no	—
47	C₂×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
48	C₂×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
50	C₂×Dic₈₉	—	4	4	no	—
51	C₂²×C₈₉	—	2	2	no	—
52	C₂²×C₈₉	—	2	1	no	—
53	C₂²×C₈₉	—	4	2	no	—
54	C₂²×C₈₉	—	4	1	no	—
55	C₂²×C₈₉	—	4	4	no	—
56	C₂²×C₈₉	—	4	2	no	—
57	C₂²×C₈₉	—	8	4	no	—
58	C₂²×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₄×C₈₉	—	2	2	no	—
60	C₄×C₈₉	—	2	1	no	—
61	C₄×C₈₉	—	4	2	no	—
62	C₄×C₈₉	#99	1	2	yes	—
63	Dic₈₉	—	8	4	no	—
64	C₂×C₈₉	—	2	1	no	—
65	C₂×C₈₉	—	4	2	no	—
66	C₂×C₈₉	—	4	1	no	—
67	C₂×C₈₉	—	8	2	no	—
68	C₂×C₈₉	C₂×#99	1	1	yes	characteristic
69	#2128	—	89	1	no	maximal, Sylow
70	C₈₉	#2128	1	1	yes	characteristic, complemented, Sylow
71	#535	D₈₉	1	1	yes	characteristic
72	#410	—	89	2	no	—
73	#353	—	89	4	no	—
74	C₂×#41	C₂×D₈₉	1	1	yes	characteristic
75	C₂²×D₄	Dic₈₉	1	1	yes	characteristic, complemented
76	#127	C₂×D₈₉	1	2	yes	—
77	#124	Dic₈₉	1	2	yes	complemented
78	#121	—	2	2	no	—
79	#120	—	2	2	no	—
80	#117	—	178	2	no	—
81	#117	Dic₈₉	1	1	yes	characteristic, complemented
82	#99	—	89	4	no	—
83	#99	—	178	2	no	—
84	C₂⁴	—	2	1	no	—
85	#41	—	4	2	no	—
86	C₂×Q₈	C₂×Dic₈₉	1	1	yes	characteristic
87	C₂×D₄	—	2	4	no	—
88	C₂×D₄	—	2	1	no	—
89	C₂×D₄	—	4	2	no	—
90	C₂×D₄	C₂×Dic₈₉	1	2	yes	—
91	C₂×D₄	C₂²⋊D₈₉	1	2	yes	—
92	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
93	C₂²×C₄	C₂×Dic₈₉	1	1	yes	characteristic
94	C₂²×C₄	C₂²⋊D₈₉	1	2	yes	—
95	C₄⋊C₄	—	2	1	no	—
96	C₄⋊C₄	—	356	2	no	—
97	C₂²⋊C₄	—	2	1	no	—
98	C₂²⋊C₄	—	2	4	no	—
99	C₂²⋊C₄	—	178	4	no	—
100	C₄²	—	356	2	no	—
101	C₄²	C₂×Dic₈₉	1	2	yes	—
102	C₂³	—	2	2	no	—
103	C₂³	—	4	2	no	—
104	C₂³	—	8	2	no	—
105	C₂³	C₂²⋊Dic₈₉	1	1	yes	characteristic
106	C₂³	C₂×C₂²⋊D₈₉	1	2	yes	—
107	Q₈	—	4	1	no	—
108	D₄	—	4	2	no	—
109	D₄	—	4	1	no	—
110	D₄	—	4	2	no	—
111	D₄	—	4	2	no	—
112	D₄	—	8	4	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	356	4	no	—
118	C₂×C₄	C₂²⋊Dic₈₉	1	2	yes	—
119	C₂×C₄	C₂²⋊Dic₈₉	1	1	yes	characteristic
120	C₂×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
121	C₂²	—	2	2	no	—
122	C₂²	—	2	1	no	—
123	C₂²	—	4	2	no	—
124	C₂²	—	4	1	no	—
125	C₂²	—	4	4	no	—
126	C₂²	—	4	2	no	—
127	C₂²	—	8	4	no	—
128	C₂²	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
129	C₄	—	2	2	no	—
130	C₄	—	2	1	no	—
131	C₄	—	4	2	no	—
132	C₄	—	712	4	no	—
133	C₄	#15667927	1	2	yes	—
134	C₂	—	2	1	no	—
135	C₂	—	4	2	no	—
136	C₂	—	4	1	no	—
137	C₂	—	8	2	no	—
138	C₂	C₂×#15667927	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19151572

C₂

yes

maximal, complemented

#19151391

C₂

yes

maximal, complemented

#535×C₈₉

C₂

yes

maximal, characteristic

#15667971

—

#15667927

—

#15667927

C₂²

yes

complemented

C₂×#41×C₈₉

C₂²

yes

characteristic

C₂²×D₄×C₈₉

C₄

yes

characteristic, complemented

#127×C₈₉

C₂²

yes

—

#124×C₈₉

C₄

yes

complemented

#121×C₈₉

—

#120×C₈₉

—

#117×C₈₉

C₄

yes

characteristic, complemented

C₄⋊Dic₈₉

—

C₂²⋊Dic₈₉

—

C₄×Dic₈₉

—

C₂⁴×C₈₉

—

#41×C₈₉

—

C₂×Q₈×C₈₉

C₂×C₄

yes

characteristic

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

C₂×C₄

yes

—

C₂×D₄×C₈₉

D₄

yes

—

C₂×D₄×C₈₉

C₂³

yes

characteristic

C₂²×C₄×C₈₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₉

D₄

yes

—

C₄⋊C₄×C₈₉

—

C₂²⋊C₄×C₈₉

—

C₂²⋊C₄×C₈₉

—

C₄²×C₈₉

C₂×C₄

yes

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

C₂²⋊C₄

yes

characteristic

C₂³×C₈₉

C₂×D₄

yes

—

Q₈×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂²⋊C₄

yes

—

C₂×C₄×C₈₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂×Dic₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#99

yes

—

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

C₂×#99

yes

—

C₈₉

yes

characteristic, complemented, Sylow

#535

D₈₉

yes

characteristic

#410

—

#353

—

C₂×#41

C₂×D₈₉

yes

characteristic

C₂²×D₄

Dic₈₉

yes

characteristic, complemented

#127

C₂×D₈₉

yes

—

#124

Dic₈₉

yes

—

—

—

178

—

#117

Dic₈₉

yes

characteristic, complemented

#99

—

#99

—

178

—

C₂⁴

—

#41

—

C₂×Q₈

C₂×Dic₈₉

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×Dic₈₉

yes

—

C₂×D₄

C₂²⋊D₈₉

yes

—

C₂×D₄

C₂²×D₈₉

yes

characteristic

C₂²×C₄

C₂×Dic₈₉

yes

characteristic

C₂²×C₄

C₂²⋊D₈₉

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

356

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

178

—

100

C₄²

—

356

—

101

C₄²

C₂×Dic₈₉

yes

—

102

C₂³

—

103

C₂³

—

104

C₂³

—

105

C₂³

C₂²⋊Dic₈₉

yes

characteristic

106

C₂³

C₂×C₂²⋊D₈₉

yes

—

107

Q₈

—

108

D₄

—

109

D₄

—

110

D₄

—

111

D₄

—

112

D₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

356

—

118

C₂×C₄

C₂²⋊Dic₈₉

yes

—

119

C₂×C₄

C₂²⋊Dic₈₉

yes

characteristic

120

C₂×C₄

C₂²×Dic₈₉

yes

characteristic

121

C₂²

—

122

C₂²

—

123

C₂²

—

124

C₂²

—

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

C₂×C₂²⋊Dic₈₉

yes

characteristic

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

—

712

—

133

C₄

#15667927

yes

—

134

C₂

—

135

C₂

—

136

C₂

—

137

C₂

—

138

C₂

C₂×#15667927

yes

characteristic

	subgroup	G/H
center	C₂	C₂×#15667927
commutator	C₂×C₄×C₈₉	C₂²×C₄
Frattini	C₂×D₄	C₂²×D₈₉
Fitting, prime radical	#535×C₈₉	C₂
socle	C₂×C₈₉	C₂×#99
cosocle	C₂×D₄×C₈₉	C₂³
hypercenter	#535	D₈₉
hypocenter	C₈₉	#2128
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#41 ◁ #535

subgroup

G/H

center

C₂

C₂×#15667927

commutator

C₂×C₄×C₈₉

C₂²×C₄

Frattini

C₂×D₄

C₂²×D₈₉

Fitting, prime radical

#535×C₈₉

C₂

socle

C₂×C₈₉

C₂×#99

cosocle

C₂×D₄×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#41 ◁ #535

N	H ≅ G/N	type
1. #19151572	C₂	semidirect
2. #19151391	C₂	semidirect
6. #15667927	C₂²	semidirect
8. C₂²×D₄×C₈₉	C₄	semidirect
10. #124×C₈₉	C₄	semidirect
13. #117×C₈₉	C₄	semidirect
70. C₈₉	#2128	semidirect
75. C₂²×D₄	Dic₈₉	semidirect
77. #124	Dic₈₉	semidirect
81. #117	Dic₈₉	semidirect

H ≅ G/N

type

1. #19151572

C₂

semidirect

2. #19151391

C₂

semidirect

6. #15667927

C₂²

semidirect

8. C₂²×D₄×C₈₉

C₄

semidirect

10. #124×C₈₉

C₄

semidirect

13. #117×C₈₉

C₄

semidirect

70. C₈₉

#2128

semidirect

75. C₂²×D₄

Dic₈₉

semidirect

77. #124

Dic₈₉

semidirect

81. #117

Dic₈₉

semidirect

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

subgroup	index
#19151572	2
#19151391	2
#535×C₈₉	2
#2128	89

subgroup

index

#19151572

#19151391

#535×C₈₉

#2128

Subgroups — #26021924

Conjugacy classes of subgroups

Complements

Maximal subgroups