Group #26020403 — subgroups

Subgroups — #26020403

Order 11392.

300 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 153 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	C₂×#15667941	C₂	1	4	yes	maximal
3	C₄²×Dic₈₉	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊C₈×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×#10623494	C₂²	1	2	yes	—
6	C₂×C₄×Dic₈₉	C₄	1	2	yes	—
7	C₂×C₄×Dic₈₉	C₄	1	1	yes	characteristic
8	C₂×C₄×Dic₈₉	C₄	1	1	yes	characteristic
9	C₂×C₄×Dic₈₉	C₂²	1	2	yes	—
10	C₄⋊#10623494	—	2	2	no	—
11	#15667941	—	2	8	no	—
12	C₂²×C₈×C₈₉	C₂²	1	2	yes	—
13	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₈₉	—	2	2	no	—
15	C₂²×Dic₈₉	C₂×C₄	1	2	yes	—
16	C₂²×Dic₈₉	C₂×C₄	1	2	yes	—
17	C₂×#10623494	—	2	4	no	—
18	C₂×#10623494	—	2	2	no	—
19	C₄×Dic₈₉	—	2	2	no	—
20	C₄×Dic₈₉	—	2	4	no	—
21	C₄×Dic₈₉	C₈	1	4	yes	complemented
22	C₄×Dic₈₉	D₄	1	4	yes	—
23	C₄×Dic₈₉	Q₈	1	4	yes	—
24	C₂²×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
27	C₂×C₈×C₈₉	—	2	2	no	—
28	C₂×C₈×C₈₉	—	2	4	no	—
29	C₄²×C₈₉	D₄	1	2	yes	—
30	C₄²×C₈₉	Q₈	1	2	yes	—
31	C₂³×C₈₉	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₄×C₈₉	—	2	2	no	—
33	C₂×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
34	C₂×C₄×C₈₉	C₄⋊C₄	1	2	yes	—
35	C₂×C₄×C₈₉	C₂×C₈	1	2	yes	—
36	C₂×C₄×C₈₉	M₁₆	1	2	yes	—
37	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
38	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₈₉	C₂×Q₈	1	2	yes	—
40	C₂×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
41	C₂×Dic₈₉	—	2	2	no	—
42	C₂×Dic₈₉	—	2	2	no	—
43	C₂×Dic₈₉	C₄⋊C₄	1	4	yes	—
44	C₂×Dic₈₉	C₄⋊C₄	1	4	yes	—
45	C₂×Dic₈₉	C₂×C₈	1	4	yes	—
46	C₂×Dic₈₉	M₁₆	1	4	yes	—
47	C₈×C₈₉	—	4	2	no	—
48	#10623494	—	4	2	no	—
49	C₂²×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
50	C₂²×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
51	C₂²×C₈₉	C₂²×C₈	1	1	yes	characteristic
52	C₂²×C₈₉	C₂×M₁₆	1	2	yes	—
53	C₂²×C₈₉	C₂×M₁₆	1	1	yes	characteristic
54	C₄×C₈₉	—	2	2	no	—
55	C₄×C₈₉	C₄⋊C₈	1	4	yes	complemented
56	C₄×C₈₉	#127	1	1	yes	characteristic
57	C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
58	C₄×C₈₉	C₄⋊Q₈	1	2	yes	—
59	Dic₈₉	—	2	4	no	—
60	Dic₈₉	C₄⋊C₈	1	8	yes	complemented
61	C₂×C₈₉	#384	1	1	yes	characteristic
62	C₂×C₈₉	C₂×C₄⋊C₈	1	2	yes	—
63	C₂×C₈₉	C₂×C₄⋊C₈	1	1	yes	characteristic
64	C₂×C₈₉	#423	1	2	yes	—
65	C₂×C₈₉	#423	1	1	yes	characteristic
66	#1842	—	89	1	no	maximal, Sylow
67	C₈₉	#1842	1	1	yes	characteristic, complemented, Sylow
68	C₂×C₄⋊C₈	—	89	4	no	—
69	C₂×C₄⋊C₈	—	89	1	no	—
70	C₂×C₄⋊C₈	D₈₉	1	1	yes	characteristic
71	C₄³	—	89	1	no	—
72	C₂²×C₈	—	89	2	no	—
73	C₂²×C₈	C₂×D₈₉	1	2	yes	—
74	C₂×C₄²	—	89	1	no	—
75	C₂×C₄²	—	89	2	no	—
76	C₂×C₄²	—	89	2	no	—
77	C₂×C₄²	—	89	1	no	—
78	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
79	C₄⋊C₈	—	2	2	no	—
80	C₄⋊C₈	—	178	8	no	—
81	C₄⋊C₈	—	178	2	no	—
82	C₂²×C₄	—	89	2	no	—
83	C₂²×C₄	—	89	2	no	—
84	C₂²×C₄	C₄×D₈₉	1	1	yes	characteristic
85	C₂²×C₄	C₄×D₈₉	1	1	yes	characteristic
86	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
87	C₂×C₈	—	2	4	no	—
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	178	4	no	—
90	C₂×C₈	—	178	2	no	—
91	C₄²	—	89	4	no	—
92	C₄²	—	89	4	no	—
93	C₄²	—	89	4	no	—
94	C₄²	—	178	4	no	—
95	C₄²	—	178	2	no	—
96	C₄²	Dic₁₇₈	1	2	yes	—
97	C₄²	C₂²⋊D₈₉	1	2	yes	—
98	C₂³	C₂×C₄×D₈₉	1	1	yes	characteristic
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	89	4	no	—
101	C₂×C₄	—	89	4	no	—
102	C₂×C₄	—	89	4	no	—
103	C₂×C₄	—	89	4	no	—
104	C₂×C₄	—	178	2	no	—
105	C₂×C₄	—	178	2	no	—
106	C₂×C₄	#13083553	1	2	yes	—
107	C₂×C₄	#13083553	1	2	yes	—
108	C₂×C₄	C₈×D₈₉	1	2	yes	—
109	C₂×C₄	#13083557	1	2	yes	—
110	C₂×C₄	D₄×D₈₉	1	2	yes	—
111	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
112	C₂×C₄	Q₈×D₈₉	1	2	yes	—
113	C₂×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
114	C₈	—	4	2	no	—
115	C₈	—	356	2	no	—
116	C₂²	C₄⋊C₄×D₈₉	1	2	yes	—
117	C₂²	C₂×#13083553	1	1	yes	characteristic
118	C₂²	C₂×C₈×D₈₉	1	1	yes	characteristic
119	C₂²	M₁₆×D₈₉	1	2	yes	—
120	C₂²	C₂×#13083557	1	1	yes	characteristic
121	C₄	—	2	2	no	—
122	C₄	—	89	8	no	—
123	C₄	—	178	4	no	—
124	C₄	#15667941	1	4	yes	complemented
125	C₄	#15668008	1	1	yes	characteristic
126	C₄	#15668010	1	1	yes	characteristic
127	C₄	#15668011	1	2	yes	—
128	C₂	#19151467	1	1	yes	characteristic
129	C₂	C₄⋊C₈×D₈₉	1	2	yes	—
130	C₂	C₂×#15667941	1	1	yes	characteristic
131	C₂	#19151620	1	2	yes	—
132	C₂	#19151621	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#10623494

C₂

yes

maximal, characteristic

C₂×#15667941

C₂

yes

maximal

C₄²×Dic₈₉

C₂

yes

maximal, characteristic

C₂×C₄⋊C₈×C₈₉

C₂

yes

maximal, characteristic

C₂²×#10623494

C₂²

yes

—

C₂×C₄×Dic₈₉

C₄

yes

—

C₂×C₄×Dic₈₉

C₄

yes

characteristic

C₂×C₄×Dic₈₉

C₄

yes

characteristic

C₂×C₄×Dic₈₉

C₂²

yes

—

C₄⋊#10623494

—

#15667941

—

C₂²×C₈×C₈₉

C₂²

yes

—

C₂×C₄²×C₈₉

C₂²

yes

characteristic

C₄⋊C₈×C₈₉

—

C₂²×Dic₈₉

C₂×C₄

yes

—

C₂²×Dic₈₉

C₂×C₄

yes

—

C₂×#10623494

—

C₂×#10623494

—

C₄×Dic₈₉

—

C₄×Dic₈₉

—

C₄×Dic₈₉

C₈

yes

complemented

C₄×Dic₈₉

D₄

yes

—

C₄×Dic₈₉

Q₈

yes

—

C₂²×C₄×C₈₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₄²×C₈₉

D₄

yes

—

C₄²×C₈₉

Q₈

yes

—

C₂³×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₄⋊C₄

yes

—

C₂×C₄×C₈₉

C₄⋊C₄

yes

—

C₂×C₄×C₈₉

C₂×C₈

yes

—

C₂×C₄×C₈₉

M₁₆

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×Q₈

yes

—

C₂×C₄×C₈₉

C₂×Q₈

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

C₄⋊C₄

yes

—

C₂×Dic₈₉

C₄⋊C₄

yes

—

C₂×Dic₈₉

C₂×C₈

yes

—

C₂×Dic₈₉

M₁₆

yes

—

C₈×C₈₉

—

#10623494

—

C₂²×C₈₉

C₂×C₄⋊C₄

yes

—

C₂²×C₈₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₈₉

C₂²×C₈

yes

characteristic

C₂²×C₈₉

C₂×M₁₆

yes

—

C₂²×C₈₉

C₂×M₁₆

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

C₄⋊C₈

yes

complemented

C₄×C₈₉

#127

yes

characteristic

C₄×C₈₉

C₄⋊Q₈

yes

characteristic

C₄×C₈₉

C₄⋊Q₈

yes

—

Dic₈₉

—

Dic₈₉

C₄⋊C₈

yes

complemented

C₂×C₈₉

#384

yes

characteristic

C₂×C₈₉

C₂×C₄⋊C₈

yes

—

C₂×C₈₉

C₂×C₄⋊C₈

yes

characteristic

C₂×C₈₉

#423

yes

—

C₂×C₈₉

yes

—

C₈₉

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

D₈₉

yes

characteristic

C₄³

—

C₂²×C₈

—

C₂²×C₈

C₂×D₈₉

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₈₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

178

—

C₄⋊C₈

—

178

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₈₉

yes

characteristic

C₂²×C₄

C₄×D₈₉

yes

characteristic

C₂²×C₄

C₂²×D₈₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

178

—

C₂×C₈

—

178

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

178

—

C₄²

—

178

—

C₄²

Dic₁₇₈

yes

—

C₄²

C₂²⋊D₈₉

yes

—

C₂³

C₂×C₄×D₈₉

yes

characteristic

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

178

—

105

C₂×C₄

—

178

—

106

C₂×C₄

#13083553

yes

—

107

C₂×C₄

#13083553

yes

—

108

C₂×C₄

C₈×D₈₉

yes

—

109

C₂×C₄

#13083557

yes

—

110

C₂×C₄

D₄×D₈₉

yes

—

111

C₂×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

112

C₂×C₄

Q₈×D₈₉

yes

—

113

C₂×C₄

C₂×Dic₁₇₈

yes

characteristic

114

C₈

—

115

C₈

—

356

—

116

C₂²

C₄⋊C₄×D₈₉

yes

—

117

C₂²

C₂×#13083553

yes

characteristic

118

C₂²

C₂×C₈×D₈₉

yes

characteristic

119

C₂²

M₁₆×D₈₉

yes

—

120

C₂²

C₂×#13083557

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

178

—

124

C₄

#15667941

yes

complemented

125

C₄

#15668008

yes

characteristic

126

C₄

#15668010

yes

characteristic

127

C₄

#15668011

yes

—

128

C₂

#19151467

yes

characteristic

129

C₂

C₄⋊C₈×D₈₉

yes

—

130

C₂

C₂×#15667941

yes

characteristic

131

C₂

#19151620

yes

—

132

C₂

#19151621

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₈₉
commutator	C₂²×C₈₉	C₂²×C₈
Fitting	C₂×C₄⋊C₈×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂²×C₄
cosocle	C₂²×C₄×C₈₉	C₂³
hypercenter	C₂×C₄⋊C₈	D₈₉
hypocenter	C₈₉	#1842
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₈₉

commutator

C₂²×C₈₉

C₂²×C₈

Fitting

C₂×C₄⋊C₈×C₈₉

C₂

socle, prime radical

C₂³×C₈₉

C₂²×C₄

cosocle

C₂²×C₄×C₈₉

C₂³

hypercenter

C₂×C₄⋊C₈

D₈₉

C₈₉

G ▷ C₂²×C₈₉ ▷ C₈₉

C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
21. C₄×Dic₈₉	C₈	semidirect
55. C₄×C₈₉	C₄⋊C₈	semidirect
60. Dic₈₉	C₄⋊C₈	semidirect
67. C₈₉	#1842	semidirect
124. C₄	#15667941	semidirect

H ≅ G/N

type

21. C₄×Dic₈₉

C₈

semidirect

55. C₄×C₈₉

C₄⋊C₈

semidirect

60. Dic₈₉

C₄⋊C₈

semidirect

67. C₈₉

#1842

semidirect

124. C₄

#15667941

semidirect

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

subgroup	index
C₂×C₄⋊#10623494	2
C₂×#15667941	2
C₄²×Dic₈₉	2
C₂×C₄⋊C₈×C₈₉	2
#1842	89

subgroup

index

C₂×C₄⋊#10623494

C₂×#15667941

C₄²×Dic₈₉

C₂×C₄⋊C₈×C₈₉

#1842

Subgroups — #26020403

Conjugacy classes of subgroups

Complements

Maximal subgroups