Group #26021205 — subgroups

Subgroups — #26021205

Order 11392.

292 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 129 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×C₄⋊#10623494	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊Dic₈₉	C₂	1	1	yes	maximal, characteristic
3	#19151360	C₂	1	4	yes	maximal
4	C₂×C₄⋊Q₈×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×#10623494	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₈₉	C₂²	1	2	yes	—
7	C₂×C₄×Dic₈₉	—	2	1	no	—
8	C₄⋊#10623494	—	2	2	no	—
9	#15667919	—	2	1	no	—
10	C₂²×Q₈×C₈₉	—	2	1	no	—
11	C₄⋊Q₈×C₈₉	—	2	2	no	—
12	C₄⋊Q₈×C₈₉	C₄	1	4	yes	complemented
13	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
14	C₂×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
15	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
16	C₂²×Dic₈₉	—	2	2	no	—
17	C₂×#10623494	—	2	2	no	—
18	C₂×#10623494	—	2	4	no	—
19	C₄⋊Dic₈₉	—	2	2	no	—
20	C₄⋊Dic₈₉	Q₈	1	4	yes	complemented
21	C₄×Dic₈₉	—	4	2	no	—
22	C₂×Q₈×C₈₉	—	2	4	no	—
23	C₂×Q₈×C₈₉	—	4	2	no	—
24	C₂×Q₈×C₈₉	—	4	2	no	—
25	C₂²×C₄×C₈₉	—	2	2	no	—
26	C₂²×C₄×C₈₉	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₈₉	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
29	C₄⋊C₄×C₈₉	—	2	4	no	—
30	C₄⋊C₄×C₈₉	—	2	2	no	—
31	C₄⋊C₄×C₈₉	C₂×C₄	1	4	yes	—
32	C₄²×C₈₉	C₂×C₄	1	2	yes	—
33	C₄²×C₈₉	D₄	1	2	yes	—
34	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
35	Q₈×C₈₉	—	4	8	no	—
36	C₂×C₄×C₈₉	—	2	2	no	—
37	C₂×C₄×C₈₉	—	2	4	no	—
38	C₂×C₄×C₈₉	—	4	2	no	—
39	C₂×C₄×C₈₉	—	4	2	no	—
40	C₂×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
41	C₂×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
42	C₂×C₄×C₈₉	D₈	1	2	yes	—
43	C₂×C₄×C₈₉	SD₁₆	1	2	yes	—
44	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₈₉	C₂×Q₈	1	2	yes	—
47	C₂×C₄×C₈₉	#41	1	2	yes	—
48	C₂×Dic₈₉	—	2	4	no	—
49	C₂×Dic₈₉	—	4	2	no	—
50	C₂×Dic₈₉	—	4	2	no	—
51	#10623494	—	4	2	no	—
52	C₂²×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₈₉	C₂²⋊Q₈	1	2	yes	—
54	C₂²×C₈₉	C₂×D₈	1	1	yes	characteristic
55	C₂²×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
56	C₂²×C₈₉	#137	1	2	yes	—
57	C₄×C₈₉	—	2	2	no	—
58	C₄×C₈₉	—	4	4	no	—
59	C₄×C₈₉	#102	1	4	yes	—
60	C₄×C₈₉	C₄×Q₈	1	2	yes	—
61	C₄×C₈₉	#124	1	1	yes	characteristic
62	C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
63	Dic₈₉	—	4	4	no	—
64	C₂×C₈₉	#391	1	1	yes	characteristic
65	C₂×C₈₉	C₂×#102	1	1	yes	characteristic
66	C₂×C₈₉	#419	1	1	yes	characteristic
67	C₂×C₈₉	#474	1	2	yes	—
68	C₂×C₈₉	#475	1	2	yes	—
69	#1988	—	89	1	no	maximal, Sylow
70	C₈₉	#1988	1	1	yes	characteristic, complemented, Sylow
71	C₂×C₄⋊Q₈	D₈₉	1	1	yes	characteristic
72	C₂×C₄⋊C₈	—	89	1	no	—
73	C₄×C₄⋊C₄	—	89	1	no	—
74	#340	—	89	4	no	—
75	C₂²×Q₈	—	2	1	no	—
76	C₂²×C₈	—	89	2	no	—
77	C₄⋊Q₈	—	2	2	no	—
78	C₄⋊Q₈	Dic₈₉	1	4	yes	complemented
79	C₂×C₄⋊C₄	—	2	1	no	—
80	C₂×C₄⋊C₄	—	89	2	no	—
81	C₂×C₄⋊C₄	C₂×D₈₉	1	2	yes	—
82	C₂×C₄²	—	178	1	no	—
83	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
84	C₄⋊C₈	—	178	2	no	—
85	#95	—	178	1	no	—
86	C₂×Q₈	—	2	4	no	—
87	C₂×Q₈	—	4	2	no	—
88	C₂×Q₈	—	4	2	no	—
89	C₂²×C₄	—	2	2	no	—
90	C₂²×C₄	—	178	2	no	—
91	C₂²×C₄	C₂²⋊D₈₉	1	1	yes	characteristic
92	C₂²×C₄	C₂²⋊D₈₉	1	1	yes	characteristic
93	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
94	C₂×C₈	—	178	2	no	—
95	C₂×C₈	—	178	4	no	—
96	C₄⋊C₄	—	2	4	no	—
97	C₄⋊C₄	—	2	2	no	—
98	C₄⋊C₄	—	89	4	no	—
99	C₄⋊C₄	—	178	2	no	—
100	C₄⋊C₄	C₂×Dic₈₉	1	4	yes	—
101	C₄²	—	356	2	no	—
102	C₄²	C₂×Dic₈₉	1	2	yes	—
103	C₄²	C₂²⋊D₈₉	1	2	yes	—
104	C₂³	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
105	Q₈	—	4	8	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	2	4	no	—
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	—	4	2	no	—
110	C₂×C₄	—	356	2	no	—
111	C₂×C₄	—	356	2	no	—
112	C₂×C₄	—	178	4	no	—
113	C₂×C₄	C₂²⋊Dic₈₉	1	2	yes	—
114	C₂×C₄	C₂²⋊Dic₈₉	1	2	yes	—
115	C₂×C₄	#13083559	1	2	yes	—
116	C₂×C₄	#13083562	1	2	yes	—
117	C₂×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
118	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
119	C₂×C₄	Q₈×D₈₉	1	2	yes	—
120	C₂×C₄	#13083575	1	2	yes	—
121	C₈	—	356	2	no	—
122	C₂²	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
123	C₂²	#15667988	1	2	yes	—
124	C₂²	C₂×#13083559	1	1	yes	characteristic
125	C₂²	C₂×#13083562	1	1	yes	characteristic
126	C₂²	#15668043	1	2	yes	—
127	C₄	—	2	2	no	—
128	C₄	—	4	4	no	—
129	C₄	—	356	4	no	—
130	C₄	#15667933	1	4	yes	—
131	C₄	Q₈×Dic₈₉	1	2	yes	—
132	C₄	#15668001	1	1	yes	characteristic
133	C₄	#15668010	1	1	yes	characteristic
134	C₂	#19151505	1	1	yes	characteristic
135	C₂	C₂×#15667933	1	1	yes	characteristic
136	C₂	#19151607	1	1	yes	characteristic
137	C₂	#19151875	1	2	yes	—
138	C₂	#19151880	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#10623494

C₂

yes

maximal, characteristic

C₄×C₄⋊Dic₈₉

C₂

yes

maximal, characteristic

#19151360

C₂

yes

maximal

C₂×C₄⋊Q₈×C₈₉

C₂

yes

maximal, characteristic

C₂²×#10623494

C₂²

yes

—

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×C₄×Dic₈₉

—

C₄⋊#10623494

—

#15667919

—

C₂²×Q₈×C₈₉

—

C₄⋊Q₈×C₈₉

—

C₄⋊Q₈×C₈₉

C₄

yes

complemented

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄⋊C₄×C₈₉

C₂²

yes

—

C₂×C₄²×C₈₉

C₂²

yes

characteristic

C₂²×Dic₈₉

—

C₂×#10623494

—

C₂×#10623494

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

Q₈

yes

complemented

C₄×Dic₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

D₄

yes

characteristic

C₂²×C₄×C₈₉

D₄

yes

characteristic

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

C₂×C₄

yes

—

C₄²×C₈₉

C₂×C₄

yes

—

C₄²×C₈₉

D₄

yes

—

C₂³×C₈₉

C₂×D₄

yes

characteristic

Q₈×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

—

C₂×C₄×C₈₉

D₈

yes

—

C₂×C₄×C₈₉

SD₁₆

yes

—

C₂×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×Q₈

yes

—

C₂×C₄×C₈₉

#41

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

#10623494

—

C₂²×C₈₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₈₉

C₂²⋊Q₈

yes

—

C₂²×C₈₉

C₂×D₈

yes

characteristic

C₂²×C₈₉

C₂×SD₁₆

yes

characteristic

C₂²×C₈₉

#137

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#102

yes

—

C₄×C₈₉

C₄×Q₈

yes

—

C₄×C₈₉

#124

yes

characteristic

C₄×C₈₉

C₄⋊Q₈

yes

characteristic

Dic₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₂×C₈₉

#475

yes

—

#1988

—

maximal, Sylow

C₈₉

#1988

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

D₈₉

yes

characteristic

C₂×C₄⋊C₈

—

C₄×C₄⋊C₄

—

#340

—

C₂²×Q₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

Dic₈₉

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₈₉

yes

—

C₂×C₄²

—

178

—

C₂×C₄²

C₂×D₈₉

yes

characteristic

C₄⋊C₈

—

178

—

#95

—

178

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

178

—

C₂²×C₄

C₂²⋊D₈₉

yes

characteristic

C₂²×C₄

C₂²⋊D₈₉

yes

characteristic

C₂²×C₄

C₂²×D₈₉

yes

characteristic

C₂×C₈

—

178

—

C₂×C₈

—

178

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

178

—

100

C₄⋊C₄

C₂×Dic₈₉

yes

—

101

C₄²

—

356

—

102

C₄²

C₂×Dic₈₉

yes

—

103

C₄²

C₂²⋊D₈₉

yes

—

104

C₂³

C₂×C₂²⋊D₈₉

yes

characteristic

105

Q₈

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

356

—

111

C₂×C₄

—

356

—

112

C₂×C₄

—

178

—

113

C₂×C₄

C₂²⋊Dic₈₉

yes

—

114

C₂×C₄

C₂²⋊Dic₈₉

yes

—

115

C₂×C₄

#13083559

yes

—

116

C₂×C₄

#13083562

yes

—

117

C₂×C₄

C₂²×Dic₈₉

yes

characteristic

118

C₂×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

119

C₂×C₄

Q₈×D₈₉

yes

—

120

C₂×C₄

#13083575

yes

—

121

C₈

—

356

—

122

C₂²

C₂×C₂²⋊Dic₈₉

yes

characteristic

123

C₂²

#15667988

yes

—

124

C₂²

C₂×#13083559

yes

characteristic

125

C₂²

C₂×#13083562

yes

characteristic

126

C₂²

#15668043

yes

—

127

C₄

—

128

C₄

—

129

C₄

—

356

—

130

C₄

#15667933

yes

—

131

C₄

Q₈×Dic₈₉

yes

—

132

C₄

#15668001

yes

characteristic

133

C₄

#15668010

yes

characteristic

134

C₂

#19151505

yes

characteristic

135

C₂

C₂×#15667933

yes

characteristic

136

C₂

#19151607

yes

characteristic

137

C₂

#19151875

yes

—

138

C₂

#19151880

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₈₉
commutator	C₂×C₄×C₈₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₈₉
Fitting	C₂×C₄⋊Q₈×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂×D₄
cosocle	C₂²×C₄×C₈₉	C₂³
hypercenter	C₂×C₄⋊Q₈	D₈₉
hypocenter	C₈₉	#1988
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₈₉

commutator

C₂×C₄×C₈₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₈₉

Fitting

C₂×C₄⋊Q₈×C₈₉

C₂

socle, prime radical

C₂³×C₈₉

C₂×D₄

cosocle

C₂²×C₄×C₈₉

C₂³

hypercenter

C₂×C₄⋊Q₈

D₈₉

hypocenter

C₈₉

#1988

LCS

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
12. C₄⋊Q₈×C₈₉	C₄	semidirect
20. C₄⋊Dic₈₉	Q₈	semidirect
70. C₈₉	#1988	semidirect
78. C₄⋊Q₈	Dic₈₉	semidirect

H ≅ G/N

type

12. C₄⋊Q₈×C₈₉

C₄

semidirect

20. C₄⋊Dic₈₉

Q₈

semidirect

70. C₈₉

#1988

semidirect

78. C₄⋊Q₈

Dic₈₉

semidirect

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

subgroup	index
C₂×C₄⋊#10623494	2
C₄×C₄⋊Dic₈₉	2
#19151360	2
C₂×C₄⋊Q₈×C₈₉	2
#1988	89

subgroup

index

C₂×C₄⋊#10623494

C₄×C₄⋊Dic₈₉

#19151360

C₂×C₄⋊Q₈×C₈₉

#1988

Subgroups — #26021205

Conjugacy classes of subgroups

Complements

Maximal subgroups