Group #26021604 — subgroups

Subgroups — #26021604

Order 11392.

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

normal subgroups only

This group has 314 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	#19151470	C₂	1	3	yes	maximal
2	#19151365	C₂	1	3	yes	maximal
3	#543×C₈₉	C₂	1	1	yes	maximal, characteristic
4	C₂³×Dic₈₉	C₂²	1	1	yes	characteristic
5	C₂×#13083553	—	2	3	no	—
6	C₂×C₂²⋊Dic₈₉	C₂²	1	3	yes	—
7	#15667919	—	2	3	no	—
8	C₂²⋊Q₈×C₈₉	—	2	3	no	—
9	C₂²⋊Q₈×C₈₉	—	2	3	no	—
10	C₂×C₂²⋊C₄×C₈₉	C₂²	1	3	yes	—
11	C₂²×Dic₈₉	—	2	3	no	—
12	C₂²×Dic₈₉	—	2	3	no	—
13	C₂²×Dic₈₉	—	4	2	no	—
14	#13083553	—	4	6	no	—
15	C₂²⋊Dic₈₉	—	2	3	no	—
16	C₂²⋊Dic₈₉	Q₈	1	6	yes	complemented
17	C₂⁴×C₈₉	C₂³	1	1	yes	characteristic
18	C₂×Q₈×C₈₉	—	2	1	no	—
19	C₂×Q₈×C₈₉	—	2	1	no	—
20	C₂²×C₄×C₈₉	D₄	1	3	yes	—
21	C₂²×C₄×C₈₉	D₄	1	3	yes	—
22	C₄⋊C₄×C₈₉	—	2	3	no	—
23	C₄⋊C₄×C₈₉	—	2	3	no	—
24	C₂²⋊C₄×C₈₉	—	2	6	no	—
25	C₂³×C₈₉	—	4	1	no	—
26	C₂³×C₈₉	—	4	1	no	—
27	C₂³×C₈₉	C₂×D₄	1	3	yes	—
28	C₂³×C₈₉	C₂×Q₈	1	3	yes	—
29	C₂³×C₈₉	#41	1	1	yes	characteristic
30	Q₈×C₈₉	—	8	1	no	—
31	Q₈×C₈₉	—	8	1	no	—
32	C₂×C₄×C₈₉	—	2	3	no	—
33	C₂×C₄×C₈₉	—	2	3	no	—
34	C₂×C₄×C₈₉	—	4	3	no	—
35	C₂×C₄×C₈₉	—	4	3	no	—
36	C₂×Dic₈₉	—	2	6	no	—
37	C₂×Dic₈₉	—	2	6	no	—
38	C₂×Dic₈₉	—	4	3	no	—
39	C₂×Dic₈₉	—	4	3	no	—
40	C₂×Dic₈₉	—	4	3	no	—
41	C₂×Dic₈₉	—	4	1	no	—
42	C₂²×C₈₉	—	2	3	no	—
43	C₂²×C₈₉	—	2	3	no	—
44	C₂²×C₈₉	—	4	3	no	—
45	C₂²×C₈₉	—	4	1	no	—
46	C₂²×C₈₉	#120	1	1	yes	characteristic
47	C₂²×C₈₉	C₂²⋊Q₈	1	3	yes	—
48	C₂²×C₈₉	C₄⋊Q₈	1	3	yes	—
49	C₄×C₈₉	—	4	3	no	—
50	C₄×C₈₉	—	4	3	no	—
51	Dic₈₉	—	4	6	no	—
52	Dic₈₉	—	4	2	no	—
53	C₂×C₈₉	—	2	3	no	—
54	C₂×C₈₉	—	2	3	no	—
55	C₂×C₈₉	#395	1	1	yes	characteristic
56	C₂×C₈₉	B(4,𝔽₂)	1	1	yes	characteristic
57	C₂×C₈₉	#458	1	1	yes	characteristic
58	#2062	—	89	1	no	maximal, Sylow
59	C₈₉	#2062	1	1	yes	characteristic, complemented, Sylow
60	#543	D₈₉	1	1	yes	characteristic
61	#385	—	89	3	no	—
62	#342	—	89	3	no	—
63	C₂³×C₄	—	89	1	no	—
64	C₂²⋊Q₈	—	2	3	no	—
65	C₂²⋊Q₈	—	2	3	no	—
66	C₂×C₄⋊C₄	—	178	3	no	—
67	C₂×C₂²⋊C₄	—	89	3	no	—
68	C₂×C₂²⋊C₄	C₂×D₈₉	1	3	yes	—
69	#95	—	178	3	no	—
70	C₂⁴	C₂²×D₈₉	1	1	yes	characteristic
71	C₂×Q₈	—	2	1	no	—
72	C₂×Q₈	—	2	1	no	—
73	C₂²×C₄	—	356	2	no	—
74	C₂²×C₄	—	178	3	no	—
75	C₂²×C₄	—	178	3	no	—
76	C₂²×C₄	C₂²⋊D₈₉	1	3	yes	—
77	C₂²×C₄	C₂²⋊D₈₉	1	3	yes	—
78	C₄⋊C₄	—	2	3	no	—
79	C₄⋊C₄	—	2	3	no	—
80	C₄⋊C₄	—	356	6	no	—
81	C₂²⋊C₄	—	2	6	no	—
82	C₂²⋊C₄	—	89	6	no	—
83	C₂²⋊C₄	—	178	3	no	—
84	C₂³	—	4	1	no	—
85	C₂³	—	4	1	no	—
86	C₂³	C₂×C₂²⋊D₈₉	1	3	yes	—
87	C₂³	Q₈×D₈₉	1	3	yes	—
88	C₂³	#13083575	1	1	yes	characteristic
89	Q₈	—	8	1	no	—
90	Q₈	—	8	1	no	—
91	C₂×C₄	—	2	3	no	—
92	C₂×C₄	—	2	3	no	—
93	C₂×C₄	—	4	3	no	—
94	C₂×C₄	—	4	3	no	—
95	C₂×C₄	—	356	3	no	—
96	C₂×C₄	—	356	3	no	—
97	C₂×C₄	—	356	3	no	—
98	C₂×C₄	—	356	1	no	—
99	C₂×C₄	—	178	6	no	—
100	C₂×C₄	—	178	6	no	—
101	C₂²	—	2	3	no	—
102	C₂²	—	2	3	no	—
103	C₂²	—	4	3	no	—
104	C₂²	—	4	1	no	—
105	C₂²	#15667981	1	1	yes	characteristic
106	C₂²	#15667988	1	3	yes	—
107	C₂²	#15668010	1	3	yes	—
108	C₄	—	4	3	no	—
109	C₄	—	4	3	no	—
110	C₄	—	356	6	no	—
111	C₄	—	356	2	no	—
112	C₂	—	2	3	no	—
113	C₂	—	2	3	no	—
114	C₂	#19151517	1	1	yes	characteristic
115	C₂	#19151785	1	1	yes	characteristic
116	C₂	#19151788	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19151470

C₂

yes

maximal

#19151365

C₂

yes

maximal

#543×C₈₉

C₂

yes

maximal, characteristic

C₂³×Dic₈₉

C₂²

yes

characteristic

C₂×#13083553

—

C₂×C₂²⋊Dic₈₉

C₂²

yes

—

#15667919

—

C₂²⋊Q₈×C₈₉

—

C₂²⋊Q₈×C₈₉

—

C₂×C₂²⋊C₄×C₈₉

C₂²

yes

—

C₂²×Dic₈₉

—

C₂²×Dic₈₉

—

C₂²×Dic₈₉

—

#13083553

—

C₂²⋊Dic₈₉

—

C₂²⋊Dic₈₉

Q₈

yes

complemented

C₂⁴×C₈₉

C₂³

yes

characteristic

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂²×C₄×C₈₉

D₄

yes

—

C₂²×C₄×C₈₉

D₄

yes

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₂²⋊C₄×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

C₂×D₄

yes

—

C₂³×C₈₉

C₂×Q₈

yes

—

C₂³×C₈₉

#41

yes

characteristic

Q₈×C₈₉

—

Q₈×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

#120

yes

characteristic

C₂²×C₈₉

C₂²⋊Q₈

yes

—

C₂²×C₈₉

C₄⋊Q₈

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₈₉

yes

characteristic, complemented, Sylow

#543

D₈₉

yes

characteristic

#385

—

#342

—

C₂³×C₄

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₂×C₄⋊C₄

—

178

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

C₂×D₈₉

yes

—

#95

—

178

—

C₂⁴

C₂²×D₈₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

356

—

C₂²×C₄

—

178

—

C₂²×C₄

—

178

—

C₂²×C₄

C₂²⋊D₈₉

yes

—

C₂²×C₄

C₂²⋊D₈₉

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

356

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

178

—

C₂³

—

C₂³

—

C₂³

C₂×C₂²⋊D₈₉

yes

—

C₂³

Q₈×D₈₉

yes

—

C₂³

#13083575

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

356

—

C₂×C₄

—

356

—

C₂×C₄

—

356

—

C₂×C₄

—

356

—

C₂×C₄

—

178

—

100

C₂×C₄

—

178

—

101

C₂²

—

102

C₂²

—

103

C₂²

—

104

C₂²

—

105

C₂²

#15667981

yes

characteristic

106

C₂²

#15667988

yes

—

107

C₂²

#15668010

yes

—

108

C₄

—

109

C₄

—

110

C₄

—

356

—

111

C₄

—

356

—

112

C₂

—

113

C₂

—

114

C₂

#19151517

yes

characteristic

115

C₂

#19151785

yes

characteristic

116

C₂

#19151788

yes

characteristic

	subgroup	G/H
center	C₂²	#15667981
commutator, prime radical, cosocle	C₂⁴×C₈₉	C₂³
Frattini	C₂⁴	C₂²×D₈₉
Fitting	#543×C₈₉	C₂
socle	C₂²×C₈₉	#120
hypercenter	#543	D₈₉
hypocenter	C₈₉	#2062
LCS	G ▷ C₂⁴×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ #543

subgroup

G/H

center

C₂²

#15667981

commutator, prime radical, cosocle

C₂⁴×C₈₉

C₂³

Frattini

C₂⁴

C₂²×D₈₉

Fitting

#543×C₈₉

C₂

C₂²×C₈₉

D₈₉

C₈₉

G ▷ C₂⁴×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂⁴ ◁ #543

N	H ≅ G/N	type
16. C₂²⋊Dic₈₉	Q₈	semidirect
59. C₈₉	#2062	semidirect

H ≅ G/N

type

16. C₂²⋊Dic₈₉

Q₈

semidirect

59. C₈₉

#2062

semidirect

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

subgroup	index
#19151470	2
#19151365	2
#543×C₈₉	2
#2062	89

subgroup

index

#19151470

#19151365

#543×C₈₉

#2062

Subgroups — #26021604

Conjugacy classes of subgroups

Complements

Maximal subgroups