Group #46812439 — subgroups

Subgroups — #46812439

Order 34432.

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

normal subgroups only

This group has 210 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	#31770654	C₂	1	1	yes	maximal, characteristic, complemented
2	#31770632	C₂	1	2	yes	maximal, complemented
3	#31770623	C₂	1	2	yes	maximal
4	#31770573	C₂	1	1	yes	maximal, characteristic
5	H(3,ℤ/4)×C₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16864402	C₂²	1	1	yes	characteristic
7	C₂×#16864386	C₂²	1	2	yes	—
8	C₂²×#14524130	C₂²	1	1	yes	characteristic
9	#22575389	—	2	2	no	—
10	C₄⋊#14524130	—	2	4	no	—
11	#22575381	—	2	4	no	—
12	#22575365	—	2	2	no	—
13	C₂×M₁₆×C₂₆₉	C₂²	1	1	yes	characteristic
14	#117×C₂₆₉	C₂²	1	2	yes	—
15	#16864402	—	2	2	no	—
16	#16864402	—	2	2	no	—
17	C₂×Dic₅₃₈	Q₈	1	1	yes	characteristic
18	C₂×D₁₀₇₆	Q₈	1	1	yes	characteristic
19	C₂×C₂²⋊D₂₆₉	—	2	1	no	—
20	C₂×C₄×D₂₆₉	—	2	1	no	—
21	#16864386	—	2	2	no	—
22	#16864386	—	2	2	no	—
23	C₂×#14524130	—	2	1	no	—
24	C₂×#14524130	—	2	2	no	—
25	C₂×#14524130	—	2	2	no	—
26	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
27	M₁₆×C₂₆₉	—	2	2	no	—
28	C₂×C₈×C₂₆₉	D₄	1	2	yes	—
29	C₄⋊C₄×C₂₆₉	—	2	2	no	—
30	C₂²⋊C₄×C₂₆₉	—	2	2	no	—
31	C₄²×C₂₆₉	D₄	1	4	yes	—
32	C₂²×D₂₆₉	—	2	1	no	—
33	Dic₅₃₈	—	2	2	no	—
34	D₁₀₇₆	—	2	2	no	—
35	C₂²⋊D₂₆₉	—	4	2	no	—
36	C₂×Dic₂₆₉	—	2	1	no	—
37	C₄×D₂₆₉	—	4	1	no	—
38	C₄×D₂₆₉	—	4	1	no	—
39	#14524130	—	4	1	no	—
40	#14524130	—	4	2	no	—
41	C₂³×C₂₆₉	#41	1	1	yes	characteristic
42	C₂×C₄×C₂₆₉	—	2	4	no	—
43	C₂×C₄×C₂₆₉	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₂₆₉	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₂₆₉	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄×C₂₆₉	#41	1	2	yes	—
47	C₈×C₂₆₉	—	2	2	no	—
48	C₂²×C₂₆₉	—	4	1	no	—
49	C₂²×C₂₆₉	C₂²⋊Q₈	1	1	yes	characteristic
50	C₂²×C₂₆₉	#123	1	2	yes	—
51	C₂×D₂₆₉	—	4	2	no	—
52	C₂×D₂₆₉	—	4	1	no	—
53	C₄×C₂₆₉	—	4	4	no	—
54	C₄×C₂₆₉	C₂²⋊Q₈	1	2	yes	—
55	C₄×C₂₆₉	#123	1	1	yes	characteristic
56	C₄×C₂₆₉	#127	1	1	yes	characteristic
57	Dic₂₆₉	—	4	1	no	—
58	C₂×C₂₆₉	—	2	2	no	—
59	C₂×C₂₆₉	—	2	1	no	—
60	C₂×C₂₆₉	#399	1	1	yes	characteristic
61	D₂₆₉	—	8	1	no	—
62	C₂₆₉	#2098	1	1	yes	characteristic, complemented, Sylow
63	#2098	—	269	1	no	maximal, Sylow
64	#430	—	269	1	no	—
65	#424	—	269	2	no	—
66	#421	—	269	2	no	—
67	#408	—	269	1	no	—
68	H(3,ℤ/4)	D₂₆₉	1	1	yes	characteristic, complemented
69	C₂×#41	—	269	1	no	—
70	C₂×M₁₆	—	269	2	no	—
71	C₂×M₁₆	C₂×D₂₆₉	1	1	yes	characteristic
72	C₂²×C₈	—	269	1	no	—
73	#117	C₂×D₂₆₉	1	2	yes	—
74	#108	—	538	2	no	—
75	C₄⋊C₈	—	538	4	no	—
76	#104	—	538	4	no	—
77	C₂²⋊C₈	—	538	2	no	—
78	#41	—	538	2	no	—
79	#41	—	538	2	no	—
80	C₂×Q₈	—	269	1	no	—
81	C₂×D₄	—	269	1	no	—
82	C₂×D₄	—	538	1	no	—
83	C₂²×C₄	—	538	1	no	—
84	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
85	M₁₆	—	2	2	no	—
86	M₁₆	—	538	2	no	—
87	M₁₆	—	538	2	no	—
88	C₂×C₈	—	538	1	no	—
89	C₂×C₈	—	538	2	no	—
90	C₂×C₈	—	538	2	no	—
91	C₂×C₈	D₁₀₇₆	1	2	yes	—
92	C₄⋊C₄	—	2	2	no	—
93	C₂²⋊C₄	—	2	2	no	—
94	C₄²	D₁₀₇₆	1	4	yes	—
95	C₂³	—	538	1	no	—
96	C₂³	#16864401	1	1	yes	characteristic
97	Q₈	—	538	2	no	—
98	D₄	—	538	2	no	—
99	D₄	—	1076	2	no	—
100	C₂×C₄	—	2	4	no	—
101	C₂×C₄	—	538	1	no	—
102	C₂×C₄	—	1076	1	no	—
103	C₂×C₄	—	1076	1	no	—
104	C₂×C₄	C₂×D₁₀₇₆	1	2	yes	—
105	C₂×C₄	C₂×D₁₀₇₆	1	1	yes	characteristic
106	C₂×C₄	Q₈×D₂₆₉	1	1	yes	characteristic
107	C₂×C₄	#16864401	1	2	yes	—
108	C₈	—	2	2	no	—
109	C₈	—	1076	1	no	—
110	C₈	—	1076	2	no	—
111	C₂²	—	4	1	no	—
112	C₂²	—	1076	2	no	—
113	C₂²	—	1076	1	no	—
114	C₂²	#22575433	1	1	yes	characteristic
115	C₂²	#22575437	1	2	yes	—
116	C₄	—	4	4	no	—
117	C₄	—	1076	1	no	—
118	C₄	#22575433	1	2	yes	—
119	C₄	#22575437	1	1	yes	characteristic
120	C₄	#22575450	1	1	yes	characteristic
121	C₂	—	2	2	no	—
122	C₂	—	2	1	no	—
123	C₂	—	2152	1	no	—
124	C₂	#31770542	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#31770654

C₂

yes

maximal, characteristic, complemented

#31770632

C₂

yes

maximal, complemented

#31770623

C₂

yes

maximal

#31770573

C₂

yes

maximal, characteristic

H(3,ℤ/4)×C₂₆₉

C₂

yes

maximal, characteristic, complemented

C₂×#16864402

C₂²

yes

characteristic

C₂×#16864386

C₂²

yes

—

C₂²×#14524130

C₂²

yes

—

—

—

—

C₂×M₁₆×C₂₆₉

C₂²

yes

characteristic

#117×C₂₆₉

C₂²

yes

—

#16864402

—

#16864402

—

C₂×Dic₅₃₈

Q₈

yes

characteristic

C₂×D₁₀₇₆

Q₈

yes

characteristic

C₂×C₂²⋊D₂₆₉

—

C₂×C₄×D₂₆₉

—

—

—

—

—

—

C₂²×C₄×C₂₆₉

C₂³

yes

characteristic

M₁₆×C₂₆₉

—

C₂×C₈×C₂₆₉

D₄

yes

—

C₄⋊C₄×C₂₆₉

—

C₂²⋊C₄×C₂₆₉

—

C₄²×C₂₆₉

D₄

yes

—

C₂²×D₂₆₉

—

Dic₅₃₈

—

D₁₀₇₆

—

C₂²⋊D₂₆₉

—

C₂×Dic₂₆₉

—

C₄×D₂₆₉

—

C₄×D₂₆₉

—

#14524130

—

#14524130

—

C₂³×C₂₆₉

#41

yes

characteristic

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

C₂×D₄

yes

—

C₂×C₄×C₂₆₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₆₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₆₉

#41

yes

—

C₈×C₂₆₉

—

C₂²×C₂₆₉

—

C₂²×C₂₆₉

C₂²⋊Q₈

yes

characteristic

C₂²×C₂₆₉

#123

yes

—

C₂×D₂₆₉

—

C₂×D₂₆₉

—

C₄×C₂₆₉

—

C₄×C₂₆₉

C₂²⋊Q₈

yes

—

C₄×C₂₆₉

yes

C₄×C₂₆₉

yes

Dic₂₆₉

—

C₂×C₂₆₉

—

C₂×C₂₆₉

—

C₂×C₂₆₉

#399

yes

characteristic

D₂₆₉

—

C₂₆₉

#2098

yes

characteristic, complemented, Sylow

#2098

—

269

maximal, Sylow

#430

—

269

—

#424

—

269

—

#421

—

269

—

#408

—

269

—

H(3,ℤ/4)

D₂₆₉

yes

characteristic, complemented

C₂×#41

—

269

—

C₂×M₁₆

—

269

—

C₂×M₁₆

C₂×D₂₆₉

yes

characteristic

C₂²×C₈

—

269

—

#117

C₂×D₂₆₉

yes

—

#108

—

538

—

C₄⋊C₈

—

538

—

#104

—

538

—

C₂²⋊C₈

—

538

—

#41

—

538

—

#41

—

538

—

C₂×Q₈

—

269

—

C₂×D₄

—

269

—

C₂×D₄

—

538

—

C₂²×C₄

—

538

—

C₂²×C₄

C₂²×D₂₆₉

yes

characteristic

M₁₆

—

M₁₆

—

538

—

M₁₆

—

538

—

C₂×C₈

—

538

—

C₂×C₈

—

538

—

C₂×C₈

—

538

—

C₂×C₈

D₁₀₇₆

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

D₁₀₇₆

yes

—

C₂³

—

538

—

C₂³

#16864401

yes

characteristic

Q₈

—

538

—

D₄

—

538

—

D₄

—

1076

—

100

C₂×C₄

—

101

C₂×C₄

—

538

—

102

C₂×C₄

—

1076

—

103

C₂×C₄

—

1076

—

104

C₂×C₄

C₂×D₁₀₇₆

yes

—

105

C₂×C₄

C₂×D₁₀₇₆

yes

characteristic

106

C₂×C₄

Q₈×D₂₆₉

yes

characteristic

107

C₂×C₄

#16864401

yes

—

108

C₈

—

109

C₈

—

1076

—

110

C₈

—

1076

—

111

C₂²

—

112

C₂²

—

1076

—

113

C₂²

—

1076

—

114

C₂²

#22575433

yes

characteristic

115

C₂²

#22575437

yes

—

116

C₄

—

117

C₄

—

1076

—

118

C₄

#22575433

yes

—

119

C₄

#22575437

yes

characteristic

120

C₄

#22575450

yes

characteristic

121

C₂

—

122

C₂

—

123

C₂

—

2152

—

124

C₂

#31770542

yes

characteristic

	subgroup	G/H
center	C₄	#22575450
commutator, cosocle	C₂²×C₄×C₂₆₉	C₂³
Frattini	C₂²×C₄	C₂²×D₂₆₉
Fitting	H(3,ℤ/4)×C₂₆₉	C₂
socle	C₂×C₂₆₉	#399
prime radical	C₂×#16864402	C₂²
hypercenter	H(3,ℤ/4)	D₂₆₉
hypocenter	C₂₆₉	#2098
LCS	G ▷ C₂²×C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

#22575450

commutator, cosocle

C₂²×C₄×C₂₆₉

C₂³

Frattini

C₂²×C₄

C₂²×D₂₆₉

Fitting

H(3,ℤ/4)×C₂₆₉

C₂

C₂×C₂₆₉

C₂²

D₂₆₉

C₂₆₉

G ▷ C₂²×C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
1. #31770654	C₂	semidirect
2. #31770632	C₂	semidirect
5. H(3,ℤ/4)×C₂₆₉	C₂	semidirect
62. C₂₆₉	#2098	semidirect
68. H(3,ℤ/4)	D₂₆₉	semidirect

H ≅ G/N

type

1. #31770654

C₂

semidirect

2. #31770632

C₂

semidirect

5. H(3,ℤ/4)×C₂₆₉

C₂

semidirect

62. C₂₆₉

#2098

semidirect

68. H(3,ℤ/4)

D₂₆₉

semidirect

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

subgroup	index
#31770654	2
#31770632	2
#31770623	2
#31770573	2
H(3,ℤ/4)×C₂₆₉	2
#2098	269

subgroup

index

H(3,ℤ/4)×C₂₆₉

269

Subgroups — #46812439

Conjugacy classes of subgroups

Complements

Maximal subgroups