Group #46809648 — subgroups

Subgroups — #46809648

Order 34432.

168 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 93 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	#31770662	C₂	1	2	yes	maximal
2	#31770358	C₂	1	2	yes	maximal, complemented
3	#31770356	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×C₈×C₂₆₉	C₂	1	1	yes	maximal, characteristic
5	C₄⋊#14524130	C₂²	1	4	yes	—
6	C₂²⋊#14524130	—	2	2	no	—
7	#22575364	—	2	2	no	—
8	C₂²×C₈×C₂₆₉	C₄	1	1	yes	characteristic
9	C₂²×C₈×C₂₆₉	C₄	1	1	yes	characteristic
10	C₂×C₄²×C₂₆₉	C₂²	1	1	yes	characteristic
11	C₄×C₈×C₂₆₉	C₄	1	1	yes	characteristic
12	C₄×C₈×C₂₆₉	C₄	1	1	yes	characteristic
13	C₄×C₈×C₂₆₉	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₂₆₉	C₂²	1	1	yes	characteristic
15	C₂×#14524130	—	2	4	no	—
16	C₂²×C₄×C₂₆₉	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₄×C₂₆₉	D₄	1	1	yes	characteristic
18	C₂²×C₄×C₂₆₉	Q₈	1	1	yes	characteristic
19	C₂×C₈×C₂₆₉	—	2	2	no	—
20	C₂×C₈×C₂₆₉	—	2	2	no	—
21	C₂×C₈×C₂₆₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×C₂₆₉	C₂×C₄	1	1	yes	characteristic
23	C₂×C₈×C₂₆₉	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×C₂₆₉	C₂×C₄	1	1	yes	characteristic
25	C₄²×C₂₆₉	C₂×C₄	1	1	yes	characteristic
26	C₄²×C₂₆₉	D₄	1	1	yes	characteristic
27	C₄²×C₂₆₉	Q₈	1	1	yes	characteristic
28	C₄²×C₂₆₉	C₂³	1	1	yes	characteristic
29	#14524130	—	4	4	no	—
30	C₂³×C₂₆₉	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₂₆₉	—	2	2	no	—
32	C₂×C₄×C₂₆₉	—	2	1	no	—
33	C₂×C₄×C₂₆₉	—	2	1	no	—
34	C₂×C₄×C₂₆₉	—	2	2	no	—
35	C₂×C₄×C₂₆₉	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₄×C₂₆₉	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₄×C₂₆₉	C₄⋊C₄	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×Q₈	1	1	yes	characteristic
41	C₈×C₂₆₉	—	2	1	no	—
42	C₈×C₂₆₉	—	2	1	no	—
43	C₈×C₂₆₉	M₁₆	1	2	yes	—
44	C₈×C₂₆₉	M₁₆	1	2	yes	—
45	C₂²×C₂₆₉	—	2	1	no	—
46	C₂²×C₂₆₉	—	2	1	no	—
47	C₂²×C₂₆₉	#108	1	2	yes	—
48	C₂²×C₂₆₉	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₄×C₂₆₉	—	2	1	no	—
50	C₄×C₂₆₉	—	2	1	no	—
51	C₄×C₂₆₉	—	2	1	no	—
52	C₄×C₂₆₉	—	2	1	no	—
53	C₄×C₂₆₉	C₂×M₁₆	1	1	yes	characteristic
54	C₄×C₂₆₉	C₂×M₁₆	1	1	yes	characteristic
55	C₄×C₂₆₉	#135	1	2	yes	—
56	C₂×C₂₆₉	—	2	2	no	—
57	C₂×C₂₆₉	#423	1	1	yes	characteristic
58	C₂×C₂₆₉	#427	1	1	yes	characteristic
59	C₂×C₂₆₉	C₂×#108	1	1	yes	characteristic
60	C₂₆₉	#1568	1	1	yes	characteristic, complemented, Sylow
61	#1568	—	269	1	no	maximal, Sylow
62	#432	—	269	2	no	—
63	C₂×C₄×C₈	D₂₆₉	1	1	yes	characteristic
64	#335	—	269	2	no	—
65	#334	—	269	2	no	—
66	C₂²×C₈	Dic₂₆₉	1	1	yes	characteristic
67	C₂²×C₈	Dic₂₆₉	1	1	yes	characteristic
68	C₂×C₄²	C₂×D₂₆₉	1	1	yes	characteristic
69	C₄⋊C₈	—	269	4	no	—
70	C₂²⋊C₈	—	538	2	no	—
71	#97	—	538	2	no	—
72	C₄×C₈	Dic₂₆₉	1	1	yes	characteristic
73	C₄×C₈	Dic₂₆₉	1	1	yes	characteristic
74	C₄×C₈	C₂×D₂₆₉	1	1	yes	characteristic
75	C₄×C₈	C₂×D₂₆₉	1	1	yes	characteristic
76	C₂²×C₄	C₂×Dic₂₆₉	1	1	yes	characteristic
77	C₂²×C₄	D₁₀₇₆	1	1	yes	characteristic
78	C₂²×C₄	Dic₅₃₈	1	1	yes	characteristic
79	C₂×C₈	—	2	2	no	—
80	C₂×C₈	—	2	2	no	—
81	C₂×C₈	—	538	4	no	—
82	C₂×C₈	C₂×Dic₂₆₉	1	1	yes	characteristic
83	C₂×C₈	C₂×Dic₂₆₉	1	1	yes	characteristic
84	C₂×C₈	C₂×Dic₂₆₉	1	1	yes	characteristic
85	C₂×C₈	C₂×Dic₂₆₉	1	1	yes	characteristic
86	C₄²	C₂×Dic₂₆₉	1	1	yes	characteristic
87	C₄²	D₁₀₇₆	1	1	yes	characteristic
88	C₄²	Dic₅₃₈	1	1	yes	characteristic
89	C₄²	C₂²×D₂₆₉	1	1	yes	characteristic
90	C₂³	C₄⋊Dic₂₆₉	1	1	yes	characteristic
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	—	2	1	no	—
93	C₂×C₄	—	2	1	no	—
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	C₄⋊Dic₂₆₉	1	1	yes	characteristic
96	C₂×C₄	C₄⋊Dic₂₆₉	1	1	yes	characteristic
97	C₂×C₄	C₄⋊Dic₂₆₉	1	1	yes	characteristic
98	C₂×C₄	C₂²×Dic₂₆₉	1	1	yes	characteristic
99	C₂×C₄	C₂×D₁₀₇₆	1	1	yes	characteristic
100	C₂×C₄	C₂×Dic₅₃₈	1	1	yes	characteristic
101	C₈	—	2	1	no	—
102	C₈	—	2	1	no	—
103	C₈	—	1076	4	no	—
104	C₈	#16864386	1	2	yes	—
105	C₈	#16864386	1	2	yes	—
106	C₂²	—	2	1	no	—
107	C₂²	—	2	1	no	—
108	C₂²	#22575389	1	2	yes	—
109	C₂²	C₂×C₄⋊Dic₂₆₉	1	1	yes	characteristic
110	C₄	—	2	1	no	—
111	C₄	—	2	1	no	—
112	C₄	—	2	1	no	—
113	C₄	—	2	1	no	—
114	C₄	C₂×#16864386	1	1	yes	characteristic
115	C₄	C₂×#16864386	1	1	yes	characteristic
116	C₄	#22575475	1	2	yes	—
117	C₂	—	2	2	no	—
118	C₂	#31770629	1	1	yes	characteristic
119	C₂	#31770643	1	1	yes	characteristic
120	C₂	C₂×#22575389	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#31770662

C₂

yes

maximal

#31770358

C₂

yes

maximal, complemented

#31770356

C₂

yes

maximal, complemented

C₂×C₄×C₈×C₂₆₉

C₂

yes

maximal, characteristic

C₄⋊#14524130

C₂²

yes

—

C₂²⋊#14524130

—

#22575364

—

C₂²×C₈×C₂₆₉

C₄

yes

characteristic

C₂²×C₈×C₂₆₉

C₄

yes

characteristic

C₂×C₄²×C₂₆₉

C₂²

yes

characteristic

C₄×C₈×C₂₆₉

C₄

yes

characteristic

C₄×C₈×C₂₆₉

C₄

yes

characteristic

C₄×C₈×C₂₆₉

C₂²

yes

characteristic

C₄×C₈×C₂₆₉

C₂²

yes

characteristic

C₂×#14524130

—

C₂²×C₄×C₂₆₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₆₉

D₄

yes

characteristic

C₂²×C₄×C₂₆₉

Q₈

yes

characteristic

C₂×C₈×C₂₆₉

—

C₂×C₈×C₂₆₉

—

C₂×C₈×C₂₆₉

C₂×C₄

yes

characteristic

C₂×C₈×C₂₆₉

C₂×C₄

yes

characteristic

C₂×C₈×C₂₆₉

C₂×C₄

yes

characteristic

C₂×C₈×C₂₆₉

C₂×C₄

yes

characteristic

C₄²×C₂₆₉

C₂×C₄

yes

characteristic

C₄²×C₂₆₉

D₄

yes

characteristic

C₄²×C₂₆₉

Q₈

yes

characteristic

C₄²×C₂₆₉

C₂³

yes

characteristic

#14524130

—

C₂³×C₂₆₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₂₆₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₂₆₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₂₆₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₆₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₆₉

C₂×Q₈

yes

characteristic

C₈×C₂₆₉

—

C₈×C₂₆₉

—

C₈×C₂₆₉

M₁₆

yes

—

C₈×C₂₆₉

M₁₆

yes

—

C₂²×C₂₆₉

—

C₂²×C₂₆₉

—

C₂²×C₂₆₉

#108

yes

—

C₂²×C₂₆₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₂₆₉

—

C₄×C₂₆₉

—

C₄×C₂₆₉

—

C₄×C₂₆₉

—

C₄×C₂₆₉

C₂×M₁₆

yes

characteristic

C₄×C₂₆₉

C₂×M₁₆

yes

characteristic

C₄×C₂₆₉

#135

yes

—

C₂×C₂₆₉

—

C₂×C₂₆₉

yes

C₂×C₂₆₉

yes

C₂×C₂₆₉

yes

C₂₆₉

yes

characteristic, complemented, Sylow

#1568

—

269

maximal, Sylow

#432

—

269

—

C₂×C₄×C₈

D₂₆₉

yes

characteristic

#335

—

269

—

#334

—

269

—

C₂²×C₈

Dic₂₆₉

yes

characteristic

C₂²×C₈

Dic₂₆₉

yes

characteristic

C₂×C₄²

C₂×D₂₆₉

yes

characteristic

C₄⋊C₈

—

269

—

C₂²⋊C₈

—

538

—

#97

—

538

—

C₄×C₈

Dic₂₆₉

yes

characteristic

C₄×C₈

Dic₂₆₉

yes

characteristic

C₄×C₈

C₂×D₂₆₉

yes

characteristic

C₄×C₈

C₂×D₂₆₉

yes

characteristic

C₂²×C₄

C₂×Dic₂₆₉

yes

characteristic

C₂²×C₄

D₁₀₇₆

yes

characteristic

C₂²×C₄

Dic₅₃₈

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

538

—

C₂×C₈

C₂×Dic₂₆₉

yes

characteristic

C₂×C₈

C₂×Dic₂₆₉

yes

characteristic

C₂×C₈

C₂×Dic₂₆₉

yes

characteristic

C₂×C₈

C₂×Dic₂₆₉

yes

characteristic

C₄²

C₂×Dic₂₆₉

yes

characteristic

C₄²

D₁₀₇₆

yes

characteristic

C₄²

Dic₅₃₈

yes

characteristic

C₄²

C₂²×D₂₆₉

yes

characteristic

C₂³

C₄⋊Dic₂₆₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄⋊Dic₂₆₉

yes

characteristic

C₂×C₄

C₄⋊Dic₂₆₉

yes

characteristic

C₂×C₄

C₄⋊Dic₂₆₉

yes

characteristic

C₂×C₄

C₂²×Dic₂₆₉

yes

characteristic

C₂×C₄

C₂×D₁₀₇₆

yes

characteristic

100

C₂×C₄

C₂×Dic₅₃₈

yes

characteristic

101

C₈

—

102

C₈

—

103

C₈

—

1076

—

104

C₈

#16864386

yes

—

105

C₈

#16864386

yes

—

106

C₂²

—

107

C₂²

—

108

C₂²

#22575389

yes

—

109

C₂²

C₂×C₄⋊Dic₂₆₉

yes

characteristic

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

—

114

C₄

C₂×#16864386

yes

characteristic

115

C₄

C₂×#16864386

yes

characteristic

116

C₄

#22575475

yes

—

117

C₂

—

118

C₂

#31770629

yes

characteristic

119

C₂

#31770643

yes

characteristic

120

C₂

C₂×#22575389

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₁₀₇₆
commutator	C₂×C₄×C₂₆₉	C₂²×C₄
Frattini	C₄²	C₂²×D₂₆₉
Fitting	C₂×C₄×C₈×C₂₆₉	C₂
socle	C₂²×C₂₆₉	C₂×C₄⋊C₄
prime radical	C₂³×C₂₆₉	C₄⋊C₄
cosocle	C₄²×C₂₆₉	C₂³
hypercenter	C₂×C₄×C₈	D₂₆₉
hypocenter	C₂₆₉	#1568
LCS	G ▷ C₂×C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂×C₄

C₂×D₁₀₇₆

commutator

C₂×C₄×C₂₆₉

C₂²×C₄

Frattini

C₄²

C₂²×D₂₆₉

Fitting

C₂×C₄×C₈×C₂₆₉

C₂

socle

C₂²×C₂₆₉

C₂×C₄⋊C₄

prime radical

C₂³×C₂₆₉

C₄⋊C₄

cosocle

C₄²×C₂₆₉

C₂³

hypercenter

C₂×C₄×C₈

D₂₆₉

hypocenter

C₂₆₉

#1568

LCS

G ▷ C₂×C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
2. #31770358	C₂	semidirect
3. #31770356	C₂	semidirect
60. C₂₆₉	#1568	semidirect

H ≅ G/N

type

2. #31770358

C₂

semidirect

3. #31770356

C₂

semidirect

60. C₂₆₉

#1568

semidirect

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

subgroup	index
#31770662	2
#31770358	2
#31770356	2
C₂×C₄×C₈×C₂₆₉	2
#1568	269

subgroup

index

C₂×C₄×C₈×C₂₆₉

269

Subgroups — #46809648

Conjugacy classes of subgroups

Complements

Maximal subgroups