Group #26031706 — subgroups

Subgroups — #26031706

Order 11392.

322 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 117 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	#19152210	C₂	1	2	yes	maximal
2	#19151906	C₂	1	4	yes	maximal, complemented
3	#19151856	C₂	1	2	yes	maximal
4	#19151839	C₂	1	2	yes	maximal
5	#19151809	C₂	1	4	yes	maximal, complemented
6	#424×C₈₉	C₂	1	1	yes	maximal, characteristic
7	C₂×Dic₃₅₆	C₂²	1	2	yes	—
8	C₂×Dic₃₅₆	C₂²	1	2	yes	—
9	#15668012	C₂²	1	2	yes	—
10	#15668005	—	2	4	no	—
11	#15668004	C₂²	1	2	yes	—
12	#15668002	—	2	4	no	—
13	#15667991	C₂²	1	4	yes	—
14	C₄×Dic₁₇₈	C₂²	1	4	yes	—
15	#15667946	C₂²	1	2	yes	—
16	#15667944	C₂²	1	2	yes	—
17	#15667937	C₂²	1	4	yes	—
18	#15667937	C₂²	1	4	yes	—
19	C₂×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
20	C₂²×C₈×C₈₉	C₂²	1	1	yes	characteristic
21	#117×C₈₉	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₈₉	C₂²	1	4	yes	—
23	C₂×Dic₁₇₈	C₂³	1	4	yes	—
24	Dic₃₅₆	—	4	2	no	—
25	Dic₃₅₆	—	4	2	no	—
26	C₄⋊Dic₈₉	—	2	4	no	—
27	C₄⋊Dic₈₉	C₂³	1	4	yes	—
28	#13083553	—	2	4	no	—
29	#13083553	—	2	4	no	—
30	#13083553	—	2	4	no	—
31	C₂²⋊Dic₈₉	—	2	4	no	—
32	C₄×Dic₈₉	—	2	4	no	—
33	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
34	M₁₆×C₈₉	—	4	1	no	—
35	C₂×C₈×C₈₉	—	4	1	no	—
36	C₂×C₈×C₈₉	C₂³	1	2	yes	—
37	C₂×C₈×C₈₉	C₂³	1	2	yes	—
38	C₄⋊C₄×C₈₉	D₄	1	2	yes	—
39	C₂²⋊C₄×C₈₉	D₄	1	2	yes	—
40	C₄²×C₈₉	C₂³	1	2	yes	—
41	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₈₉	—	2	1	no	—
43	C₂×C₄×C₈₉	—	2	1	no	—
44	C₂×C₄×C₈₉	C₂×D₄	1	4	yes	—
45	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
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	C₈×C₈₉	—	2	2	no	—
52	C₈×C₈₉	—	2	2	no	—
53	C₂²×C₈₉	—	2	1	no	—
54	C₂²×C₈₉	—	2	2	no	—
55	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
56	C₄×C₈₉	—	2	1	no	—
57	C₄×C₈₉	—	2	4	no	—
58	C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
59	C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
60	Dic₈₉	—	4	4	no	—
61	Dic₈₉	—	4	4	no	—
62	C₂×C₈₉	—	4	1	no	—
63	C₂×C₈₉	#537	1	1	yes	characteristic
64	C₂×C₈₉	#578	1	2	yes	—
65	#3261	—	89	1	no	maximal, Sylow
66	C₈₉	#3261	1	1	yes	characteristic, complemented, Sylow
67	#541	—	89	2	no	—
68	#479	—	89	4	no	—
69	#470	—	89	2	no	—
70	#467	—	89	2	no	—
71	#462	—	89	4	no	—
72	#424	D₈₉	1	1	yes	characteristic
73	C₂×Q₁₆	—	89	2	no	—
74	C₂×Q₁₆	—	89	2	no	—
75	C₂×M₁₆	C₂×D₈₉	1	1	yes	characteristic
76	C₂²×C₈	C₂×D₈₉	1	1	yes	characteristic
77	C₄⋊Q₈	—	89	2	no	—
78	#126	—	178	4	no	—
79	#125	—	89	2	no	—
80	#125	—	178	4	no	—
81	C₂²⋊Q₈	—	89	4	no	—
82	C₄×Q₈	—	89	4	no	—
83	#117	C₂×D₈₉	1	1	yes	characteristic
84	#107	—	89	2	no	—
85	#106	—	89	2	no	—
86	C₄⋊C₈	C₂×D₈₉	1	4	yes	—
87	#103	—	89	4	no	—
88	#103	—	89	4	no	—
89	C₂×Q₈	—	89	4	no	—
90	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
91	Q₁₆	—	356	2	no	—
92	Q₁₆	—	356	2	no	—
93	M₁₆	—	4	1	no	—
94	C₂×C₈	—	4	1	no	—
95	C₂×C₈	C₂²×D₈₉	1	2	yes	—
96	C₂×C₈	C₂²×D₈₉	1	2	yes	—
97	C₄⋊C₄	—	89	4	no	—
98	C₄⋊C₄	—	178	4	no	—
99	C₄⋊C₄	—	178	4	no	—
100	C₄⋊C₄	—	178	4	no	—
101	C₄⋊C₄	—	178	4	no	—
102	C₄⋊C₄	D₃₅₆	1	2	yes	—
103	C₂²⋊C₄	—	178	4	no	—
104	C₂²⋊C₄	D₃₅₆	1	2	yes	—
105	C₄²	—	178	4	no	—
106	C₄²	C₂²×D₈₉	1	2	yes	—
107	C₂³	C₂×D₃₅₆	1	1	yes	characteristic
108	Q₈	—	178	4	no	—
109	Q₈	—	178	4	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	178	4	no	—
113	C₂×C₄	—	178	4	no	—
114	C₂×C₄	C₂×D₃₅₆	1	4	yes	—
115	C₂×C₄	C₂×D₃₅₆	1	1	yes	characteristic
116	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
117	C₈	—	2	2	no	—
118	C₈	—	2	2	no	—
119	C₂²	—	2	2	no	—
120	C₂²	—	2	1	no	—
121	C₂²	C₂²×D₃₅₆	1	1	yes	characteristic
122	C₄	—	2	4	no	—
123	C₄	—	2	1	no	—
124	C₄	—	356	4	no	—
125	C₄	—	356	4	no	—
126	C₄	#15668057	1	1	yes	characteristic
127	C₄	#15668060	1	1	yes	characteristic
128	C₂	—	4	1	no	—
129	C₂	#19152179	1	1	yes	characteristic
130	C₂	#19152415	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19152210

C₂

yes

maximal

#19151906

C₂

yes

maximal, complemented

#19151856

C₂

yes

maximal

#19151839

C₂

yes

maximal

#19151809

C₂

yes

maximal, complemented

#424×C₈₉

C₂

yes

maximal, characteristic

C₂×Dic₃₅₆

C₂²

yes

—

C₂×Dic₃₅₆

C₂²

yes

—

#15668012

C₂²

yes

—

#15668005

—

#15668004

C₂²

yes

—

#15668002

—

#15667991

C₂²

yes

—

C₄×Dic₁₇₈

C₂²

yes

—

#15667946

C₂²

yes

—

#15667944

C₂²

yes

—

#15667937

C₂²

yes

—

#15667937

C₂²

yes

—

C₂×M₁₆×C₈₉

C₂²

yes

characteristic

C₂²×C₈×C₈₉

C₂²

yes

characteristic

#117×C₈₉

C₂²

yes

characteristic

C₄⋊C₈×C₈₉

C₂²

yes

—

C₂×Dic₁₇₈

C₂³

yes

—

Dic₃₅₆

—

Dic₃₅₆

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

C₂³

yes

—

#13083553

—

#13083553

—

#13083553

—

C₂²⋊Dic₈₉

—

C₄×Dic₈₉

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

M₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂³

yes

—

C₂×C₈×C₈₉

C₂³

yes

—

C₄⋊C₄×C₈₉

D₄

yes

—

C₂²⋊C₄×C₈₉

D₄

yes

—

C₄²×C₈₉

C₂³

yes

—

C₂³×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂⁴

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂²×D₄

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

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

#578

yes

—

#3261

—

maximal, Sylow

C₈₉

#3261

yes

characteristic, complemented, Sylow

—

—

—

—

—

D₈₉

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×M₁₆

C₂×D₈₉

yes

characteristic

C₂²×C₈

C₂×D₈₉

yes

characteristic

C₄⋊Q₈

—

#126

—

178

—

#125

—

#125

—

178

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

C₂×D₈₉

yes

characteristic

#107

—

#106

—

C₄⋊C₈

C₂×D₈₉

yes

—

#103

—

#103

—

C₂×Q₈

—

C₂²×C₄

C₂²×D₈₉

yes

characteristic

Q₁₆

—

356

—

Q₁₆

—

356

—

M₁₆

—

C₂×C₈

—

C₂×C₈

C₂²×D₈₉

yes

—

C₂×C₈

C₂²×D₈₉

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

178

—

C₄⋊C₄

—

178

—

100

C₄⋊C₄

—

178

—

101

C₄⋊C₄

—

178

—

102

C₄⋊C₄

D₃₅₆

yes

—

103

C₂²⋊C₄

—

178

—

104

C₂²⋊C₄

D₃₅₆

yes

—

105

C₄²

—

178

—

106

C₄²

C₂²×D₈₉

yes

—

107

C₂³

C₂×D₃₅₆

yes

characteristic

108

Q₈

—

178

—

109

Q₈

—

178

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

178

—

113

C₂×C₄

—

178

—

114

C₂×C₄

C₂×D₃₅₆

yes

—

115

C₂×C₄

C₂×D₃₅₆

yes

characteristic

116

C₂×C₄

C₂³×D₈₉

yes

characteristic

117

C₈

—

118

C₈

—

119

C₂²

—

120

C₂²

—

121

C₂²

C₂²×D₃₅₆

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

356

—

125

C₄

—

356

—

126

C₄

#15668057

yes

characteristic

127

C₄

#15668060

yes

characteristic

128

C₂

—

129

C₂

#19152179

yes

characteristic

130

C₂

#19152415

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₃₅₆
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#424×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	C₂³×C₈₉	C₂×D₄
hypercenter	#424	D₈₉
hypocenter	C₈₉	#3261
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ #117 ◁ #424

subgroup

G/H

center

C₂²

C₂²×D₃₅₆

commutator, cosocle

C₂×C₄×C₈₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₈₉

Fitting

#424×C₈₉

C₂

socle

C₂²×C₈₉

C₂²×D₄

prime radical

C₂³×C₈₉

C₂×D₄

D₈₉

C₈₉

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ #117 ◁ #424

N	H ≅ G/N	type
2. #19151906	C₂	semidirect
5. #19151809	C₂	semidirect
66. C₈₉	#3261	semidirect

H ≅ G/N

type

2. #19151906

C₂

semidirect

5. #19151809

C₂

semidirect

66. C₈₉

#3261

semidirect

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

subgroup	index
#19152210	2
#19151906	2
#19151856	2
#19151839	2
#19151809	2
#424×C₈₉	2
#3261	89

subgroup

index

#424×C₈₉

Subgroups — #26031706

Conjugacy classes of subgroups

Complements

Maximal subgroups