Group #26021603 — subgroups

Subgroups — #26021603

Order 11392.

230 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 75 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	#19152229	C₂	1	1	yes	maximal, characteristic
2	#19151640	C₂	1	2	yes	maximal
3	#19151463	C₂	1	1	yes	maximal, characteristic
4	#19151359	C₂	1	2	yes	maximal
5	#343×C₈₉	C₂	1	1	yes	maximal, characteristic
6	#15668011	—	2	2	no	—
7	#15668003	—	2	2	no	—
8	#15667991	—	2	2	no	—
9	#15667971	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₈₉	C₂²	1	1	yes	characteristic
12	#15667946	—	2	2	no	—
13	#15667944	—	2	2	no	—
14	#15667919	—	2	2	no	—
15	C₂×M₁₆×C₈₉	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₇₈	—	2	2	no	—
18	C₂²×Dic₈₉	—	2	1	no	—
19	C₂²×Dic₈₉	—	2	1	no	—
20	C₄⋊Dic₈₉	—	2	2	no	—
21	C₄⋊Dic₈₉	—	2	2	no	—
22	C₄⋊Dic₈₉	D₄	1	2	yes	—
23	C₄⋊Dic₈₉	D₄	1	2	yes	—
24	#13083553	—	2	4	no	—
25	C₂²⋊Dic₈₉	—	2	2	no	—
26	C₄×Dic₈₉	—	2	2	no	—
27	C₄×Dic₈₉	—	2	1	no	—
28	C₄×Dic₈₉	Q₈	1	2	yes	complemented
29	C₂²×C₄×C₈₉	D₄	1	2	yes	—
30	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
31	M₁₆×C₈₉	—	2	2	no	—
32	M₁₆×C₈₉	Q₈	1	4	yes	complemented
33	C₂×C₈×C₈₉	—	2	2	no	—
34	C₄⋊C₄×C₈₉	—	2	2	no	—
35	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₈₉	—	2	2	no	—
37	C₂×C₄×C₈₉	—	4	2	no	—
38	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
39	C₂×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₄×C₈₉	C₂×Q₈	1	2	yes	—
41	C₂×C₄×C₈₉	#41	1	1	yes	characteristic
42	C₂×Dic₈₉	—	2	2	no	—
43	C₂×Dic₈₉	—	2	2	no	—
44	C₂×Dic₈₉	—	2	2	no	—
45	C₂×Dic₈₉	—	4	1	no	—
46	C₂×Dic₈₉	—	4	1	no	—
47	C₂×Dic₈₉	—	4	2	no	—
48	C₂×Dic₈₉	—	4	1	no	—
49	Dic₁₇₈	—	8	2	no	—
50	C₈×C₈₉	—	2	4	no	—
51	C₂²×C₈₉	—	2	1	no	—
52	C₂²×C₈₉	—	2	1	no	—
53	C₂²×C₈₉	#120	1	1	yes	characteristic
54	C₂²×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
55	C₂²×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
56	C₄×C₈₉	—	4	2	no	—
57	C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
58	C₄×C₈₉	C₄⋊Q₈	1	2	yes	—
59	Dic₈₉	—	4	2	no	—
60	Dic₈₉	—	4	2	no	—
61	Dic₈₉	—	4	2	no	—
62	Dic₈₉	—	8	1	no	—
63	C₂×C₈₉	—	2	1	no	—
64	C₂×C₈₉	—	2	1	no	—
65	C₂×C₈₉	#395	1	1	yes	characteristic
66	C₂×C₈₉	#454	1	1	yes	characteristic
67	C₂×C₈₉	#455	1	1	yes	characteristic
68	#2061	—	89	1	no	maximal, Sylow
69	C₈₉	#2061	1	1	yes	characteristic, complemented, Sylow
70	#544	—	89	1	no	—
71	#428	—	89	2	no	—
72	#382	—	89	1	no	—
73	#343	D₈₉	1	1	yes	characteristic
74	#339	—	89	2	no	—
75	C₂×M₁₆	C₂×D₈₉	1	2	yes	—
76	C₄⋊Q₈	—	178	2	no	—
77	#125	—	178	2	no	—
78	C₂²⋊Q₈	—	178	2	no	—
79	#117	—	89	2	no	—
80	C₂×C₄⋊C₄	—	89	1	no	—
81	C₂×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
82	C₂×C₄²	—	89	1	no	—
83	#107	—	178	2	no	—
84	#106	—	178	2	no	—
85	#95	—	178	2	no	—
86	C₂×Q₈	—	178	2	no	—
87	C₂²×C₄	—	178	1	no	—
88	C₂²×C₄	—	178	1	no	—
89	C₂²×C₄	D₃₅₆	1	2	yes	—
90	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
91	M₁₆	—	2	2	no	—
92	M₁₆	Dic₁₇₈	1	4	yes	complemented
93	C₂×C₈	—	2	2	no	—
94	C₄⋊C₄	—	2	2	no	—
95	C₄⋊C₄	—	89	2	no	—
96	C₄⋊C₄	—	89	2	no	—
97	C₄⋊C₄	—	178	2	no	—
98	C₄⋊C₄	—	178	4	no	—
99	C₄⋊C₄	—	178	2	no	—
100	C₂²⋊C₄	—	178	2	no	—
101	C₄²	—	89	2	no	—
102	C₄²	—	178	2	no	—
103	C₄²	—	178	1	no	—
104	C₂³	C₂×D₃₅₆	1	1	yes	characteristic
105	Q₈	—	712	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	4	2	no	—
108	C₂×C₄	—	356	1	no	—
109	C₂×C₄	—	356	1	no	—
110	C₂×C₄	—	356	2	no	—
111	C₂×C₄	—	356	1	no	—
112	C₂×C₄	—	178	2	no	—
113	C₂×C₄	—	178	2	no	—
114	C₂×C₄	—	178	2	no	—
115	C₂×C₄	D₄×D₈₉	1	2	yes	—
116	C₂×C₄	Q₈×D₈₉	1	1	yes	characteristic
117	C₂×C₄	C₂×Dic₁₇₈	1	2	yes	—
118	C₂×C₄	#13083573	1	1	yes	characteristic
119	C₈	—	2	4	no	—
120	C₂²	—	2	1	no	—
121	C₂²	—	2	1	no	—
122	C₂²	#15667982	1	1	yes	characteristic
123	C₂²	#15667992	1	1	yes	characteristic
124	C₂²	#15668012	1	1	yes	characteristic
125	C₄	—	4	2	no	—
126	C₄	—	356	2	no	—
127	C₄	—	356	2	no	—
128	C₄	—	356	2	no	—
129	C₄	—	712	1	no	—
130	C₄	#15667989	1	2	yes	—
131	C₄	#15668011	1	2	yes	—
132	C₂	—	2	1	no	—
133	C₂	—	2	1	no	—
134	C₂	#19151519	1	1	yes	characteristic
135	C₂	#19151773	1	1	yes	characteristic
136	C₂	#19151778	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19152229

C₂

yes

maximal, characteristic

#19151640

C₂

yes

maximal

#19151463

C₂

yes

maximal, characteristic

#19151359

C₂

yes

maximal

#343×C₈₉

C₂

yes

maximal, characteristic

—

—

—

C₂²

yes

—

C₂×C₄⋊Dic₈₉

C₂²

yes

characteristic

C₂×C₄×Dic₈₉

C₂²

yes

—

—

—

C₂×M₁₆×C₈₉

C₂²

yes

—

C₂×C₄⋊C₄×C₈₉

C₂²

yes

characteristic

C₂×Dic₁₇₈

—

C₂²×Dic₈₉

—

C₂²×Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

D₄

yes

—

C₄⋊Dic₈₉

D₄

yes

—

#13083553

—

C₂²⋊Dic₈₉

—

C₄×Dic₈₉

—

C₄×Dic₈₉

—

C₄×Dic₈₉

Q₈

yes

complemented

C₂²×C₄×C₈₉

D₄

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

M₁₆×C₈₉

—

M₁₆×C₈₉

Q₈

yes

complemented

C₂×C₈×C₈₉

—

C₄⋊C₄×C₈₉

—

C₂³×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₈₉

C₂×Q₈

yes

—

C₂×C₄×C₈₉

#41

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

C₈×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

#120

yes

characteristic

C₂²×C₈₉

C₂²⋊Q₈

yes

characteristic

C₂²×C₈₉

C₄⋊Q₈

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

C₂²⋊Q₈

yes

—

C₄×C₈₉

C₄⋊Q₈

yes

—

Dic₈₉

—

Dic₈₉

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₈₉

yes

characteristic, complemented, Sylow

—

—

—

D₈₉

yes

characteristic

#339

—

C₂×M₁₆

C₂×D₈₉

yes

—

C₄⋊Q₈

—

178

—

#125

—

178

—

C₂²⋊Q₈

—

178

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₈₉

yes

characteristic

C₂×C₄²

—

#107

—

178

—

#106

—

178

—

#95

—

178

—

C₂×Q₈

—

178

—

C₂²×C₄

—

178

—

C₂²×C₄

—

178

—

C₂²×C₄

D₃₅₆

yes

—

C₂²×C₄

C₂²×D₈₉

yes

characteristic

M₁₆

—

M₁₆

Dic₁₇₈

yes

complemented

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

178

—

C₄⋊C₄

—

178

—

C₄⋊C₄

—

178

—

100

C₂²⋊C₄

—

178

—

101

C₄²

—

102

C₄²

—

178

—

103

C₄²

—

178

—

104

C₂³

C₂×D₃₅₆

yes

characteristic

105

Q₈

—

712

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

356

—

109

C₂×C₄

—

356

—

110

C₂×C₄

—

356

—

111

C₂×C₄

—

356

—

112

C₂×C₄

—

178

—

113

C₂×C₄

—

178

—

114

C₂×C₄

—

178

—

115

C₂×C₄

D₄×D₈₉

yes

—

116

C₂×C₄

Q₈×D₈₉

yes

characteristic

117

C₂×C₄

C₂×Dic₁₇₈

yes

—

118

C₂×C₄

#13083573

yes

characteristic

119

C₈

—

120

C₂²

—

121

C₂²

—

122

C₂²

#15667982

yes

characteristic

123

C₂²

#15667992

yes

characteristic

124

C₂²

#15668012

yes

characteristic

125

C₄

—

126

C₄

—

356

—

127

C₄

—

356

—

128

C₄

—

356

—

129

C₄

—

712

—

130

C₄

#15667989

yes

—

131

C₄

#15668011

yes

—

132

C₂

—

133

C₂

—

134

C₂

#19151519

yes

characteristic

135

C₂

#19151773

yes

characteristic

136

C₂

#19151778

yes

characteristic

	subgroup	G/H
center	C₂²	#15667982
commutator, cosocle	C₂²×C₄×C₈₉	C₂³
Frattini	C₂²×C₄	C₂²×D₈₉
Fitting	#343×C₈₉	C₂
socle	C₂²×C₈₉	#120
prime radical	C₂³×C₈₉	C₂×D₄
hypercenter	#343	D₈₉
hypocenter	C₈₉	#2061
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #343

subgroup

G/H

center

C₂²

#15667982

commutator, cosocle

C₂²×C₄×C₈₉

C₂³

Frattini

C₂²×C₄

C₂²×D₈₉

Fitting

#343×C₈₉

C₂

socle

C₂²×C₈₉

#120

prime radical

C₂³×C₈₉

C₂×D₄

D₈₉

C₈₉

G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #343

N	H ≅ G/N	type
28. C₄×Dic₈₉	Q₈	semidirect
32. M₁₆×C₈₉	Q₈	semidirect
69. C₈₉	#2061	semidirect
92. M₁₆	Dic₁₇₈	semidirect

H ≅ G/N

type

28. C₄×Dic₈₉

Q₈

semidirect

32. M₁₆×C₈₉

Q₈

semidirect

69. C₈₉

#2061

semidirect

92. M₁₆

Dic₁₇₈

semidirect

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

subgroup	index
#19152229	2
#19151640	2
#19151463	2
#19151359	2
#343×C₈₉	2
#2061	89

subgroup

index

#343×C₈₉

Subgroups — #26021603

Conjugacy classes of subgroups

Complements

Maximal subgroups