Group #26021521 — subgroups

Subgroups — #26021521

Order 11392.

256 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 67 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	C₂²×Dic₃₅₆	C₂	1	1	yes	maximal, characteristic
2	#19151624	C₂	1	1	yes	maximal, characteristic
3	#19151608	C₂	1	2	yes	maximal
4	C₂×#15667930	C₂	1	2	yes	maximal
5	#341×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₇₈	C₂²	1	2	yes	—
7	C₂×Dic₃₅₆	—	2	4	no	—
8	C₂×Dic₃₅₆	—	2	2	no	—
9	C₂×#13083558	C₂²	1	1	yes	characteristic
10	#15667971	C₂²	1	1	yes	characteristic
11	#15667941	—	2	2	no	—
12	#15667937	—	2	4	no	—
13	#15667930	—	2	4	no	—
14	C₂×M₁₆×C₈₉	C₂²	1	2	yes	—
15	C₂²×C₈×C₈₉	C₂²	1	1	yes	characteristic
16	C₂×Dic₁₇₈	—	2	2	no	—
17	C₂×Dic₁₇₈	—	2	2	no	—
18	C₂×Dic₁₇₈	—	2	2	no	—
19	C₂×Dic₁₇₈	—	2	2	no	—
20	C₂×Dic₁₇₈	—	2	2	no	—
21	C₂×Dic₁₇₈	D₄	1	4	yes	—
22	C₂²×Dic₈₉	—	2	2	no	—
23	Dic₃₅₆	—	2	8	no	—
24	#13083558	—	2	2	no	—
25	C₂×#10623494	—	2	1	no	—
26	C₄⋊Dic₈₉	D₄	1	2	yes	—
27	C₂²⋊Dic₈₉	—	2	1	no	—
28	C₄×Dic₈₉	—	2	1	no	—
29	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
30	M₁₆×C₈₉	—	2	2	no	—
31	M₁₆×C₈₉	—	2	2	no	—
32	C₂×C₈×C₈₉	—	2	2	no	—
33	C₂×C₈×C₈₉	—	2	2	no	—
34	C₂×C₈×C₈₉	D₄	1	2	yes	—
35	C₂³×C₈₉	#41	1	1	yes	characteristic
36	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
37	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₈₉	#41	1	2	yes	—
40	C₂×Dic₈₉	—	2	2	no	—
41	C₂×Dic₈₉	—	2	4	no	—
42	C₂×Dic₈₉	—	4	2	no	—
43	C₂×Dic₈₉	—	4	2	no	—
44	Dic₁₇₈	—	2	4	no	—
45	Dic₁₇₈	—	2	4	no	—
46	Dic₁₇₈	—	4	2	no	—
47	Dic₁₇₈	—	4	2	no	—
48	C₈×C₈₉	—	2	2	no	—
49	C₈×C₈₉	—	4	2	no	—
50	#10623494	—	4	1	no	—
51	C₂²×C₈₉	—	2	2	no	—
52	C₂²×C₈₉	#121	1	1	yes	characteristic
53	C₂²×C₈₉	#123	1	2	yes	—
54	C₄×C₈₉	#120	1	1	yes	characteristic
55	C₄×C₈₉	#121	1	2	yes	—
56	C₄×C₈₉	#124	1	1	yes	characteristic
57	Dic₈₉	—	4	4	no	—
58	Dic₈₉	—	4	2	no	—
59	C₂×C₈₉	—	2	2	no	—
60	C₂×C₈₉	#394	1	1	yes	characteristic
61	C₂×C₈₉	#473	1	2	yes	—
62	#2047	—	89	1	no	maximal, Sylow
63	C₈₉	#2047	1	1	yes	characteristic, complemented, Sylow
64	C₂²×Q₁₆	—	89	1	no	—
65	#424	—	89	1	no	—
66	#419	—	89	2	no	—
67	C₂×#101	—	89	2	no	—
68	#341	D₈₉	1	1	yes	characteristic
69	C₂²×Q₈	—	89	2	no	—
70	C₂×Q₁₆	—	178	4	no	—
71	C₂×Q₁₆	—	178	2	no	—
72	C₂×M₁₆	—	89	1	no	—
73	C₂×M₁₆	C₂×D₈₉	1	2	yes	—
74	C₂²×C₈	C₂×D₈₉	1	1	yes	characteristic
75	#117	—	89	1	no	—
76	C₄⋊C₈	—	178	2	no	—
77	#103	—	178	4	no	—
78	#101	—	178	4	no	—
79	C₂×Q₈	—	89	4	no	—
80	C₂×Q₈	—	178	2	no	—
81	C₂×Q₈	—	178	2	no	—
82	C₂×Q₈	—	178	2	no	—
83	C₂×Q₈	—	178	2	no	—
84	C₂×Q₈	—	178	2	no	—
85	C₂²×C₄	—	178	2	no	—
86	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
87	Q₁₆	—	178	8	no	—
88	M₁₆	—	2	2	no	—
89	M₁₆	—	2	2	no	—
90	M₁₆	—	178	2	no	—
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	178	1	no	—
94	C₂×C₈	D₃₅₆	1	2	yes	—
95	C₄⋊C₄	—	89	2	no	—
96	C₂²⋊C₄	—	178	1	no	—
97	C₄²	—	178	1	no	—
98	C₂³	#13083575	1	1	yes	characteristic
99	Q₈	—	356	2	no	—
100	Q₈	—	356	2	no	—
101	Q₈	—	178	4	no	—
102	Q₈	—	178	4	no	—
103	C₂×C₄	—	356	2	no	—
104	C₂×C₄	—	356	2	no	—
105	C₂×C₄	—	178	2	no	—
106	C₂×C₄	—	178	4	no	—
107	C₂×C₄	D₄×D₈₉	1	2	yes	—
108	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
109	C₂×C₄	C₂×D₃₅₆	1	1	yes	characteristic
110	C₂×C₄	#13083574	1	2	yes	—
111	C₈	—	2	2	no	—
112	C₈	—	4	2	no	—
113	C₈	—	356	1	no	—
114	C₂²	—	2	2	no	—
115	C₂²	#15667987	1	1	yes	characteristic
116	C₂²	#15667997	1	2	yes	—
117	C₄	—	356	2	no	—
118	C₄	—	356	4	no	—
119	C₄	#15667982	1	1	yes	characteristic
120	C₄	#15667985	1	2	yes	—
121	C₄	#15668000	1	1	yes	characteristic
122	C₂	—	2	2	no	—
123	C₂	#19151516	1	1	yes	characteristic
124	C₂	#19151874	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₃₅₆

C₂

yes

maximal, characteristic

#19151624

C₂

yes

maximal, characteristic

#19151608

C₂

yes

maximal

C₂×#15667930

C₂

yes

maximal

#341×C₈₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₇₈

C₂²

yes

—

C₂×Dic₃₅₆

—

C₂×Dic₃₅₆

—

C₂×#13083558

C₂²

yes

characteristic

#15667971

C₂²

yes

—

—

—

C₂×M₁₆×C₈₉

C₂²

yes

—

C₂²×C₈×C₈₉

C₂²

yes

characteristic

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

D₄

yes

—

C₂²×Dic₈₉

—

Dic₃₅₆

—

#13083558

—

C₂×#10623494

—

C₄⋊Dic₈₉

D₄

yes

—

C₂²⋊Dic₈₉

—

C₄×Dic₈₉

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

M₁₆×C₈₉

—

M₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

D₄

yes

—

C₂³×C₈₉

#41

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

#41

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

#121

yes

characteristic

C₂²×C₈₉

#123

yes

—

C₄×C₈₉

#120

yes

characteristic

C₄×C₈₉

#121

yes

—

C₄×C₈₉

#124

yes

characteristic

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#394

yes

characteristic

C₂×C₈₉

#473

yes

—

#2047

—

maximal, Sylow

C₈₉

#2047

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

—

—

—

—

D₈₉

yes

characteristic

C₂²×Q₈

—

C₂×Q₁₆

—

178

—

C₂×Q₁₆

—

178

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₈₉

yes

—

C₂²×C₈

C₂×D₈₉

yes

characteristic

#117

—

C₄⋊C₈

—

178

—

#103

—

178

—

#101

—

178

—

C₂×Q₈

—

C₂×Q₈

—

178

—

C₂×Q₈

—

178

—

C₂×Q₈

—

178

—

C₂×Q₈

—

178

—

C₂×Q₈

—

178

—

C₂²×C₄

—

178

—

C₂²×C₄

C₂²×D₈₉

yes

characteristic

Q₁₆

—

178

—

M₁₆

—

M₁₆

—

M₁₆

—

178

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

178

—

C₂×C₈

D₃₅₆

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

178

—

C₄²

—

178

—

C₂³

#13083575

yes

characteristic

Q₈

—

356

—

100

Q₈

—

356

—

101

Q₈

—

178

—

102

Q₈

—

178

—

103

C₂×C₄

—

356

—

104

C₂×C₄

—

356

—

105

C₂×C₄

—

178

—

106

C₂×C₄

—

178

—

107

C₂×C₄

D₄×D₈₉

yes

—

108

C₂×C₄

D₄×D₈₉

yes

characteristic

109

C₂×C₄

C₂×D₃₅₆

yes

characteristic

110

C₂×C₄

#13083574

yes

—

111

C₈

—

112

C₈

—

113

C₈

—

356

—

114

C₂²

—

115

C₂²

#15667987

yes

characteristic

116

C₂²

#15667997

yes

—

117

C₄

—

356

—

118

C₄

—

356

—

119

C₄

#15667982

yes

characteristic

120

C₄

#15667985

yes

—

121

C₄

#15668000

yes

characteristic

122

C₂

—

123

C₂

#19151516

yes

characteristic

124

C₂

#19151874

yes

—

	subgroup	G/H
center	C₂²	#15667987
commutator, cosocle	C₂²×C₄×C₈₉	C₂³
Frattini	C₂²×C₄	C₂²×D₈₉
Fitting	#341×C₈₉	C₂
socle	C₂²×C₈₉	#121
prime radical	C₂³×C₈₉	#41
hypercenter	#341	D₈₉
hypocenter	C₈₉	#2047
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂²

#15667987

commutator, cosocle

C₂²×C₄×C₈₉

C₂³

Frattini

C₂²×C₄

C₂²×D₈₉

Fitting

#341×C₈₉

C₂

C₂²×C₈₉

C₂³×C₈₉

D₈₉

C₈₉

G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
63. C₈₉	#2047	semidirect

H ≅ G/N

type

63. C₈₉

#2047

semidirect

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

subgroup	index
C₂²×Dic₃₅₆	2
#19151624	2
#19151608	2
C₂×#15667930	2
#341×C₈₉	2
#2047	89

subgroup

index

C₂²×Dic₃₅₆

#341×C₈₉

Subgroups — #26021521

Conjugacy classes of subgroups

Complements

Maximal subgroups