Group #26031597 — subgroups

Subgroups — #26031597

Order 11392.

348 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 137 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	#19152155	C₂	1	2	yes	maximal
2	#19151894	C₂	1	4	yes	maximal, complemented
3	#19151839	C₂	1	4	yes	maximal
4	#19151809	C₂	1	4	yes	maximal, complemented
5	C₂×C₄⋊C₈×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₃₅₆	C₂²	1	4	yes	—
7	#15668012	C₂²	1	2	yes	—
8	#15668012	C₂²	1	2	yes	—
9	#15668004	—	2	2	no	—
10	#15667991	—	2	2	no	—
11	#15667991	C₂²	1	4	yes	—
12	C₄×Dic₁₇₈	—	2	2	no	—
13	C₄×Dic₁₇₈	C₂²	1	4	yes	—
14	#15667971	—	2	2	no	—
15	#15667946	C₂²	1	4	yes	—
16	#15667937	C₂²	1	8	yes	—
17	C₂²×C₈×C₈₉	C₂²	1	2	yes	—
18	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
19	C₄⋊C₈×C₈₉	C₂²	1	4	yes	—
20	C₂×Dic₁₇₈	—	2	2	no	—
21	C₂×Dic₁₇₈	C₂³	1	4	yes	—
22	Dic₃₅₆	—	4	4	no	—
23	C₄⋊Dic₈₉	—	2	2	no	—
24	C₄⋊Dic₈₉	—	2	2	no	—
25	C₄⋊Dic₈₉	—	2	2	no	—
26	C₄⋊Dic₈₉	C₂³	1	4	yes	—
27	#13083553	—	2	4	no	—
28	#13083553	—	2	4	no	—
29	C₂²⋊Dic₈₉	—	2	4	no	—
30	C₄×Dic₈₉	—	2	4	no	—
31	C₂²×C₄×C₈₉	D₄	1	1	yes	characteristic
32	C₂²×C₄×C₈₉	D₄	1	1	yes	characteristic
33	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
34	C₂×C₈×C₈₉	—	4	2	no	—
35	C₂×C₈×C₈₉	C₂³	1	4	yes	—
36	C₄²×C₈₉	D₄	1	2	yes	—
37	C₄²×C₈₉	C₂³	1	2	yes	—
38	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₈₉	—	2	1	no	—
40	C₂×C₄×C₈₉	—	2	1	no	—
41	C₂×C₄×C₈₉	—	4	1	no	—
42	C₂×C₄×C₈₉	Q₁₆	1	4	yes	complemented
43	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
47	C₂×Dic₈₉	—	2	4	no	—
48	C₂×Dic₈₉	—	2	4	no	—
49	Dic₁₇₈	—	2	4	no	—
50	Dic₁₇₈	—	2	4	no	—
51	Dic₁₇₈	—	4	4	no	—
52	C₈×C₈₉	—	2	4	no	—
53	C₂²×C₈₉	—	2	1	no	—
54	C₂²×C₈₉	—	2	1	no	—
55	C₂²×C₈₉	C₂×Q₁₆	1	2	yes	—
56	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
57	C₄×C₈₉	—	2	1	no	—
58	C₄×C₈₉	—	2	2	no	—
59	C₄×C₈₉	C₂×Q₁₆	1	4	yes	—
60	C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
61	C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
62	Dic₈₉	—	4	4	no	—
63	Dic₈₉	—	4	4	no	—
64	C₂×C₈₉	—	2	2	no	—
65	C₂×C₈₉	#537	1	1	yes	characteristic
66	C₂×C₈₉	C₂²×Q₁₆	1	1	yes	characteristic
67	C₂×C₈₉	#575	1	1	yes	characteristic
68	#3249	—	89	1	no	maximal, Sylow
69	C₈₉	#3249	1	1	yes	characteristic, complemented, Sylow
70	#533	—	89	2	no	—
71	#477	—	89	4	no	—
72	#467	—	89	4	no	—
73	#462	—	89	4	no	—
74	C₂×C₄⋊C₈	D₈₉	1	1	yes	characteristic
75	C₂×Q₁₆	—	89	4	no	—
76	C₂²×C₈	C₂×D₈₉	1	2	yes	—
77	C₄⋊Q₈	—	89	2	no	—
78	C₄⋊Q₈	—	89	2	no	—
79	#125	—	178	2	no	—
80	C₂²⋊Q₈	—	89	4	no	—
81	C₂²⋊Q₈	—	178	2	no	—
82	C₄×Q₈	—	89	4	no	—
83	C₄×Q₈	—	178	2	no	—
84	#117	—	178	2	no	—
85	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
86	#107	—	89	4	no	—
87	C₄⋊C₈	C₂×D₈₉	1	4	yes	—
88	#103	—	89	8	no	—
89	C₂×Q₈	—	89	4	no	—
90	C₂×Q₈	—	178	2	no	—
91	C₂²×C₄	D₃₅₆	1	1	yes	characteristic
92	C₂²×C₄	D₃₅₆	1	1	yes	characteristic
93	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
94	Q₁₆	—	356	4	no	—
95	C₂×C₈	—	4	2	no	—
96	C₂×C₈	C₂²×D₈₉	1	4	yes	—
97	C₄⋊C₄	—	89	4	no	—
98	C₄⋊C₄	—	178	2	no	—
99	C₄⋊C₄	—	178	2	no	—
100	C₄⋊C₄	—	178	4	no	—
101	C₄⋊C₄	—	178	4	no	—
102	C₄⋊C₄	—	178	2	no	—
103	C₂²⋊C₄	—	178	4	no	—
104	C₄²	—	178	4	no	—
105	C₄²	D₃₅₆	1	2	yes	—
106	C₄²	C₂²×D₈₉	1	2	yes	—
107	C₂³	C₂×D₃₅₆	1	1	yes	characteristic
108	Q₈	—	356	4	no	—
109	Q₈	—	178	4	no	—
110	Q₈	—	178	4	no	—
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	4	1	no	—
114	C₂×C₄	—	178	4	no	—
115	C₂×C₄	—	178	4	no	—
116	C₂×C₄	Dic₃₅₆	1	4	yes	complemented
117	C₂×C₄	C₂×D₃₅₆	1	1	yes	characteristic
118	C₂×C₄	C₂×D₃₅₆	1	2	yes	—
119	C₂×C₄	C₂×D₃₅₆	1	2	yes	—
120	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
121	C₈	—	2	4	no	—
122	C₂²	—	2	1	no	—
123	C₂²	—	2	1	no	—
124	C₂²	C₂×Dic₃₅₆	1	2	yes	—
125	C₂²	C₂²×D₃₅₆	1	1	yes	characteristic
126	C₄	—	2	2	no	—
127	C₄	—	2	1	no	—
128	C₄	—	356	4	no	—
129	C₄	—	356	4	no	—
130	C₄	C₂×Dic₃₅₆	1	4	yes	—
131	C₄	#15668057	1	1	yes	characteristic
132	C₄	#15668060	1	1	yes	characteristic
133	C₂	—	2	2	no	—
134	C₂	#19152179	1	1	yes	characteristic
135	C₂	C₂²×Dic₃₅₆	1	1	yes	characteristic
136	C₂	#19152398	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19152155

C₂

yes

maximal

#19151894

C₂

yes

maximal, complemented

#19151839

C₂

yes

maximal

#19151809

C₂

yes

maximal, complemented

C₂×C₄⋊C₈×C₈₉

C₂

yes

maximal, characteristic

C₂×Dic₃₅₆

C₂²

yes

—

#15668012

C₂²

yes

—

#15668012

C₂²

yes

—

#15668004

—

#15667991

—

#15667991

C₂²

yes

—

C₄×Dic₁₇₈

—

C₄×Dic₁₇₈

C₂²

yes

—

#15667971

—

#15667946

C₂²

yes

—

#15667937

C₂²

yes

—

C₂²×C₈×C₈₉

C₂²

yes

—

C₂×C₄²×C₈₉

C₂²

yes

characteristic

C₄⋊C₈×C₈₉

C₂²

yes

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

C₂³

yes

—

Dic₃₅₆

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

C₂³

yes

—

#13083553

—

#13083553

—

C₂²⋊Dic₈₉

—

C₄×Dic₈₉

—

C₂²×C₄×C₈₉

D₄

yes

characteristic

C₂²×C₄×C₈₉

D₄

yes

characteristic

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂³

yes

—

C₄²×C₈₉

D₄

yes

—

C₄²×C₈₉

C₂³

yes

—

C₂³×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

Q₁₆

yes

complemented

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂⁴

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂×Q₁₆

yes

—

C₂²×C₈₉

C₂²×D₄

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₂×Q₁₆

yes

—

C₄×C₈₉

H(5,𝔽₂)

yes

characteristic

C₄×C₈₉

2₋¹⁺⁴

yes

characteristic

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#537

yes

characteristic

C₂×C₈₉

C₂²×Q₁₆

yes

C₂×C₈₉

yes

—

C₈₉

yes

characteristic, complemented, Sylow

—

—

—

—

C₂×C₄⋊C₈

D₈₉

yes

characteristic

C₂×Q₁₆

—

C₂²×C₈

C₂×D₈₉

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

178

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

178

—

C₄×Q₈

—

C₄×Q₈

—

178

—

#117

—

178

—

C₂×C₄²

C₂×D₈₉

yes

characteristic

#107

—

C₄⋊C₈

C₂×D₈₉

yes

—

#103

—

C₂×Q₈

—

C₂×Q₈

—

178

—

C₂²×C₄

D₃₅₆

yes

characteristic

C₂²×C₄

D₃₅₆

yes

characteristic

C₂²×C₄

C₂²×D₈₉

yes

characteristic

Q₁₆

—

356

—

C₂×C₈

—

C₂×C₈

C₂²×D₈₉

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

178

—

C₄⋊C₄

—

178

—

100

C₄⋊C₄

—

178

—

101

C₄⋊C₄

—

178

—

102

C₄⋊C₄

—

178

—

103

C₂²⋊C₄

—

178

—

104

C₄²

—

178

—

105

C₄²

D₃₅₆

yes

—

106

C₄²

C₂²×D₈₉

yes

—

107

C₂³

C₂×D₃₅₆

yes

characteristic

108

Q₈

—

356

—

109

Q₈

—

178

—

110

Q₈

—

178

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

178

—

115

C₂×C₄

—

178

—

116

C₂×C₄

Dic₃₅₆

yes

complemented

117

C₂×C₄

C₂×D₃₅₆

yes

characteristic

118

C₂×C₄

C₂×D₃₅₆

yes

—

119

C₂×C₄

C₂×D₃₅₆

yes

—

120

C₂×C₄

C₂³×D₈₉

yes

characteristic

121

C₈

—

122

C₂²

—

123

C₂²

—

124

C₂²

C₂×Dic₃₅₆

yes

—

125

C₂²

C₂²×D₃₅₆

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

—

356

—

129

C₄

—

356

—

130

C₄

C₂×Dic₃₅₆

yes

—

131

C₄

#15668057

yes

characteristic

132

C₄

#15668060

yes

characteristic

133

C₂

—

134

C₂

#19152179

yes

characteristic

135

C₂

C₂²×Dic₃₅₆

yes

characteristic

136

C₂

#19152398

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×D₃₅₆
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	C₂×C₄⋊C₈×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	C₂³×C₈₉	C₂×D₄
hypercenter	C₂×C₄⋊C₈	D₈₉
hypocenter	C₈₉	#3249
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂²

C₂²×D₃₅₆

commutator, cosocle

C₂×C₄×C₈₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₈₉

Fitting

C₂×C₄⋊C₈×C₈₉

C₂

socle

C₂²×C₈₉

C₂²×D₄

prime radical

C₂³×C₈₉

C₂×D₄

hypercenter

C₂×C₄⋊C₈

D₈₉

hypocenter

C₈₉

#3249

LCS

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
2. #19151894	C₂	semidirect
4. #19151809	C₂	semidirect
42. C₂×C₄×C₈₉	Q₁₆	semidirect
69. C₈₉	#3249	semidirect
116. C₂×C₄	Dic₃₅₆	semidirect

H ≅ G/N

type

2. #19151894

C₂

semidirect

4. #19151809

C₂

semidirect

42. C₂×C₄×C₈₉

Q₁₆

semidirect

69. C₈₉

#3249

semidirect

116. C₂×C₄

Dic₃₅₆

semidirect

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

subgroup	index
#19152155	2
#19151894	2
#19151839	2
#19151809	2
C₂×C₄⋊C₈×C₈₉	2
#3249	89

subgroup

index

C₂×C₄⋊C₈×C₈₉

Subgroups — #26031597

Conjugacy classes of subgroups

Complements

Maximal subgroups