Group #46811636 — subgroups

Subgroups — #46811636

Order 34432.

200 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 89 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	#31770643	C₂	1	1	yes	maximal, characteristic
2	#31770633	C₂	1	2	yes	maximal
3	C₂×#22575363	C₂	1	1	yes	maximal, characteristic
4	#31770366	C₂	1	2	yes	maximal
5	C₂×#108×C₂₆₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×#14524130	C₂²	1	1	yes	characteristic
7	#22575412	C₂²	1	2	yes	—
8	C₂×C₄×Dic₂₆₉	C₂²	1	1	yes	characteristic
9	#22575387	C₄	1	2	yes	complemented
10	#22575385	C₄	1	2	yes	complemented
11	#22575382	—	2	4	no	—
12	#22575363	—	2	2	no	—
13	C₂×M₁₆×C₂₆₉	C₂²	1	2	yes	—
14	C₂²×C₈×C₂₆₉	C₂²	1	1	yes	characteristic
15	#108×C₂₆₉	—	2	2	no	—
16	C₂²×Dic₂₆₉	—	2	1	no	—
17	C₂×#14524130	—	2	1	no	—
18	C₂×#14524130	—	2	1	no	—
19	C₂×#14524130	D₄	1	2	yes	—
20	C₄⋊Dic₂₆₉	C₂×C₄	1	4	yes	—
21	C₂²⋊Dic₂₆₉	—	2	2	no	—
22	C₄×Dic₂₆₉	—	2	1	no	—
23	C₄×Dic₂₆₉	—	2	2	no	—
24	C₄×Dic₂₆₉	Q₈	1	2	yes	—
25	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
26	M₁₆×C₂₆₉	—	2	2	no	—
27	M₁₆×C₂₆₉	—	2	2	no	—
28	C₂×C₈×C₂₆₉	—	2	2	no	—
29	C₂×C₈×C₂₆₉	C₂×C₄	1	1	yes	characteristic
30	C₂×C₈×C₂₆₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₈×C₂₆₉	D₄	1	2	yes	—
32	C₂×C₈×C₂₆₉	Q₈	1	2	yes	—
33	C₂×Dic₂₆₉	—	2	4	no	—
34	C₂×Dic₂₆₉	—	2	2	no	—
35	C₂×Dic₂₆₉	—	4	2	no	—
36	#14524130	—	2	2	no	—
37	C₂³×C₂₆₉	#41	1	1	yes	characteristic
38	C₂×C₄×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₂₆₉	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₂₆₉	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₂₆₉	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₄×C₂₆₉	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₄×C₂₆₉	#41	1	1	yes	characteristic
44	C₈×C₂₆₉	—	4	2	no	—
45	C₈×C₂₆₉	C₄⋊C₄	1	2	yes	—
46	C₈×C₂₆₉	C₄⋊C₄	1	2	yes	—
47	C₂²×C₂₆₉	—	2	2	no	—
48	C₂²×C₂₆₉	C₄×D₄	1	1	yes	characteristic
49	C₂²×C₂₆₉	C₂²⋊Q₈	1	1	yes	characteristic
50	C₂²×C₂₆₉	#125	1	1	yes	characteristic
51	C₄×C₂₆₉	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₄×C₂₆₉	C₄×Q₈	1	1	yes	characteristic
53	C₄×C₂₆₉	#121	1	1	yes	characteristic
54	C₄×C₂₆₉	C₄⋊Q₈	1	1	yes	characteristic
55	Dic₂₆₉	—	4	2	no	—
56	Dic₂₆₉	—	4	4	no	—
57	C₂×C₂₆₉	—	2	1	no	—
58	C₂×C₂₆₉	—	2	1	no	—
59	C₂×C₂₆₉	#389	1	1	yes	characteristic
60	C₂×C₂₆₉	#444	1	2	yes	—
61	C₂₆₉	#1946	1	1	yes	characteristic, complemented, Sylow
62	#1946	—	269	1	no	maximal, Sylow
63	C₂×#108	D₂₆₉	1	1	yes	characteristic
64	#427	—	269	1	no	—
65	#424	—	269	2	no	—
66	C₂×#97	—	269	1	no	—
67	#339	—	269	2	no	—
68	C₂×M₁₆	C₂×D₂₆₉	1	2	yes	—
69	C₂²×C₈	—	269	1	no	—
70	C₂²×C₈	C₂×D₂₆₉	1	1	yes	characteristic
71	#117	—	269	2	no	—
72	C₂×C₄²	—	269	1	no	—
73	#108	—	2	2	no	—
74	#107	—	269	2	no	—
75	#106	—	269	2	no	—
76	C₄⋊C₈	—	538	4	no	—
77	#97	—	538	2	no	—
78	C₂²×C₄	—	538	1	no	—
79	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
80	M₁₆	—	2	2	no	—
81	M₁₆	—	2	2	no	—
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	269	2	no	—
84	C₂×C₈	—	538	1	no	—
85	C₂×C₈	—	538	1	no	—
86	C₂×C₈	C₄×D₂₆₉	1	1	yes	characteristic
87	C₂×C₈	C₄×D₂₆₉	1	1	yes	characteristic
88	C₂×C₈	C₂²⋊D₂₆₉	1	2	yes	—
89	C₂×C₈	Dic₅₃₈	1	2	yes	—
90	C₄⋊C₄	—	269	4	no	—
91	C₂²⋊C₄	—	538	2	no	—
92	C₄²	—	269	2	no	—
93	C₄²	—	538	2	no	—
94	C₄²	—	538	1	no	—
95	C₂³	#16864403	1	1	yes	characteristic
96	C₂×C₄	—	538	4	no	—
97	C₂×C₄	—	538	2	no	—
98	C₂×C₄	—	1076	2	no	—
99	C₂×C₄	C₂×C₄×D₂₆₉	1	1	yes	characteristic
100	C₂×C₄	D₄×D₂₆₉	1	1	yes	characteristic
101	C₂×C₄	C₂×C₂²⋊D₂₆₉	1	1	yes	characteristic
102	C₂×C₄	Q₈×D₂₆₉	1	1	yes	characteristic
103	C₂×C₄	C₂×Dic₅₃₈	1	1	yes	characteristic
104	C₂×C₄	#16864401	1	1	yes	characteristic
105	C₈	—	4	2	no	—
106	C₈	—	538	2	no	—
107	C₈	#16864381	1	2	yes	—
108	C₈	#16864381	1	2	yes	—
109	C₂²	—	2	2	no	—
110	C₂²	#22575415	1	1	yes	characteristic
111	C₂²	#22575429	1	1	yes	characteristic
112	C₂²	#22575444	1	1	yes	characteristic
113	C₄	—	1076	2	no	—
114	C₄	—	1076	4	no	—
115	C₄	C₂×#16864381	1	1	yes	characteristic
116	C₄	#22575418	1	1	yes	characteristic
117	C₄	#22575424	1	1	yes	characteristic
118	C₄	#22575452	1	1	yes	characteristic
119	C₂	—	2	1	no	—
120	C₂	—	2	1	no	—
121	C₂	#31770501	1	1	yes	characteristic
122	C₂	#31770726	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#31770643

C₂

yes

maximal, characteristic

#31770633

C₂

yes

maximal

C₂×#22575363

C₂

yes

maximal, characteristic

#31770366

C₂

yes

maximal

C₂×#108×C₂₆₉

C₂

yes

maximal, characteristic

C₂²×#14524130

C₂²

yes

characteristic

#22575412

C₂²

yes

—

C₂×C₄×Dic₂₆₉

C₂²

yes

characteristic

#22575387

C₄

yes

complemented

#22575385

C₄

yes

complemented

#22575382

—

#22575363

—

C₂×M₁₆×C₂₆₉

C₂²

yes

—

C₂²×C₈×C₂₆₉

C₂²

yes

characteristic

#108×C₂₆₉

—

C₂²×Dic₂₆₉

—

C₂×#14524130

—

C₂×#14524130

—

C₂×#14524130

D₄

yes

—

C₄⋊Dic₂₆₉

C₂×C₄

yes

—

C₂²⋊Dic₂₆₉

—

C₄×Dic₂₆₉

—

C₄×Dic₂₆₉

—

C₄×Dic₂₆₉

Q₈

yes

—

C₂²×C₄×C₂₆₉

C₂³

yes

characteristic

M₁₆×C₂₆₉

—

M₁₆×C₂₆₉

—

C₂×C₈×C₂₆₉

—

C₂×C₈×C₂₆₉

C₂×C₄

yes

characteristic

C₂×C₈×C₂₆₉

C₂×C₄

yes

characteristic

C₂×C₈×C₂₆₉

D₄

yes

—

C₂×C₈×C₂₆₉

Q₈

yes

—

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

—

#14524130

—

C₂³×C₂₆₉

#41

yes

characteristic

C₂×C₄×C₂₆₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₆₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₆₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₆₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₆₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₆₉

#41

yes

characteristic

C₈×C₂₆₉

—

C₈×C₂₆₉

C₄⋊C₄

yes

—

C₈×C₂₆₉

C₄⋊C₄

yes

—

C₂²×C₂₆₉

—

C₂²×C₂₆₉

C₄×D₄

yes

characteristic

C₂²×C₂₆₉

C₂²⋊Q₈

yes

characteristic

C₂²×C₂₆₉

#125

yes

characteristic

C₄×C₂₆₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₂₆₉

C₄×Q₈

yes

characteristic

C₄×C₂₆₉

#121

yes

characteristic

C₄×C₂₆₉

C₄⋊Q₈

yes

characteristic

Dic₂₆₉

—

Dic₂₆₉

—

C₂×C₂₆₉

—

C₂×C₂₆₉

—

C₂×C₂₆₉

#389

yes

characteristic

C₂×C₂₆₉

#444

yes

—

C₂₆₉

#1946

yes

characteristic, complemented, Sylow

#1946

—

269

maximal, Sylow

C₂×#108

D₂₆₉

yes

characteristic

#427

—

269

—

#424

—

269

—

C₂×#97

—

269

—

#339

—

269

—

C₂×M₁₆

C₂×D₂₆₉

yes

—

C₂²×C₈

—

269

—

C₂²×C₈

C₂×D₂₆₉

yes

characteristic

#117

—

269

—

C₂×C₄²

—

269

—

#108

—

#107

—

269

—

#106

—

269

—

C₄⋊C₈

—

538

—

#97

—

538

—

C₂²×C₄

—

538

—

C₂²×C₄

C₂²×D₂₆₉

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

269

—

C₂×C₈

—

538

—

C₂×C₈

—

538

—

C₂×C₈

C₄×D₂₆₉

yes

characteristic

C₂×C₈

C₄×D₂₆₉

yes

characteristic

C₂×C₈

C₂²⋊D₂₆₉

yes

—

C₂×C₈

Dic₅₃₈

yes

—

C₄⋊C₄

—

269

—

C₂²⋊C₄

—

538

—

C₄²

—

269

—

C₄²

—

538

—

C₄²

—

538

—

C₂³

#16864403

yes

characteristic

C₂×C₄

—

538

—

C₂×C₄

—

538

—

C₂×C₄

—

1076

—

C₂×C₄

C₂×C₄×D₂₆₉

yes

characteristic

100

C₂×C₄

D₄×D₂₆₉

yes

characteristic

101

C₂×C₄

C₂×C₂²⋊D₂₆₉

yes

characteristic

102

C₂×C₄

Q₈×D₂₆₉

yes

characteristic

103

C₂×C₄

C₂×Dic₅₃₈

yes

characteristic

104

C₂×C₄

#16864401

yes

characteristic

105

C₈

—

106

C₈

—

538

—

107

C₈

#16864381

yes

—

108

C₈

#16864381

yes

—

109

C₂²

—

110

C₂²

#22575415

yes

characteristic

111

C₂²

#22575429

yes

characteristic

112

C₂²

#22575444

yes

characteristic

113

C₄

—

1076

—

114

C₄

—

1076

—

115

C₄

C₂×#16864381

yes

characteristic

116

C₄

#22575418

yes

characteristic

117

C₄

#22575424

yes

characteristic

118

C₄

#22575452

yes

characteristic

119

C₂

—

120

C₂

—

121

C₂

#31770501

yes

characteristic

122

C₂

#31770726

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₂₆₉
commutator	C₂×C₄×C₂₆₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₆₉
Fitting	C₂×#108×C₂₆₉	C₂
socle	C₂²×C₂₆₉	C₄×D₄
prime radical	C₂³×C₂₆₉	#41
cosocle	C₂²×C₄×C₂₆₉	C₂³
hypercenter	C₂×#108	D₂₆₉
hypocenter	C₂₆₉	#1946
LCS	G ▷ C₂×C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₂×C₄

D₄×D₂₆₉

commutator

C₂×C₄×C₂₆₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₆₉

Fitting

C₂×#108×C₂₆₉

C₂

socle

C₂²×C₂₆₉

C₄×D₄

prime radical

C₂³×C₂₆₉

#41

cosocle

C₂²×C₄×C₂₆₉

C₂³

D₂₆₉

C₂₆₉

G ▷ C₂×C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
9. #22575387	C₄	semidirect
10. #22575385	C₄	semidirect
61. C₂₆₉	#1946	semidirect

H ≅ G/N

type

9. #22575387

C₄

semidirect

10. #22575385

C₄

semidirect

61. C₂₆₉

#1946

semidirect

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

subgroup	index
#31770643	2
#31770633	2
C₂×#22575363	2
#31770366	2
C₂×#108×C₂₆₉	2
#1946	269

subgroup

index

C₂×#108×C₂₆₉

269

Subgroups — #46811636

Conjugacy classes of subgroups

Complements

Maximal subgroups