Group #46811047 — subgroups

Subgroups — #46811047

Order 34432.

248 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 117 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	C₂×#22575386	C₂	1	2	yes	maximal
2	C₂×#22575384	C₂	1	2	yes	maximal
3	C₂×C₄×#14524130	C₂	1	1	yes	maximal, characteristic
4	#31770471	C₂	1	1	yes	maximal, characteristic
5	#382×C₂₆₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×#14524130	C₂²	1	1	yes	characteristic
7	C₂²×#14524130	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₂₆₉	C₂²	1	2	yes	—
9	#22575386	—	2	4	no	—
10	#22575384	—	2	4	no	—
11	C₄×#14524130	C₄	1	4	yes	complemented
12	#22575360	—	2	2	no	—
13	C₂×C₄⋊C₄×C₂₆₉	C₂²	1	2	yes	—
14	C₂×C₄²×C₂₆₉	C₂²	1	1	yes	characteristic
15	#95×C₂₆₉	—	2	2	no	—
16	C₂²×Dic₂₆₉	—	2	2	no	—
17	C₂×#14524130	C₂×C₄	1	2	yes	—
18	C₂×#14524130	C₂×C₄	1	2	yes	—
19	C₂×#14524130	D₄	1	2	yes	—
20	C₂×#14524130	D₄	1	2	yes	—
21	C₂×#14524130	Q₈	1	2	yes	—
22	C₂×#14524130	Q₈	1	2	yes	—
23	C₄⋊Dic₂₆₉	—	2	2	no	—
24	C₄⋊Dic₂₆₉	—	2	2	no	—
25	C₂²×C₄×C₂₆₉	—	2	2	no	—
26	C₂²×C₄×C₂₆₉	D₄	1	2	yes	—
27	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
28	C₄⋊C₄×C₂₆₉	—	2	2	no	—
29	C₄⋊C₄×C₂₆₉	—	2	2	no	—
30	C₄²×C₂₆₉	C₂×C₄	1	2	yes	—
31	C₄²×C₂₆₉	Q₈	1	2	yes	—
32	C₂×Dic₂₆₉	—	4	2	no	—
33	C₂×Dic₂₆₉	—	4	4	no	—
34	#14524130	C₄⋊C₄	1	4	yes	—
35	#14524130	C₄⋊C₄	1	4	yes	—
36	C₂³×C₂₆₉	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₂₆₉	—	2	4	no	—
38	C₂×C₄×C₂₆₉	—	4	2	no	—
39	C₂×C₄×C₂₆₉	—	4	4	no	—
40	C₂×C₄×C₂₆₉	C₄⋊C₄	1	4	yes	—
41	C₂×C₄×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₂₆₉	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₂₆₉	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₂₆₉	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₄×C₂₆₉	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄×C₂₆₉	C₂×Q₈	1	1	yes	characteristic
47	C₂²×C₂₆₉	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₂₆₉	#127	1	1	yes	characteristic
49	C₂²×C₂₆₉	C₄⋊Q₈	1	1	yes	characteristic
50	C₂²×C₂₆₉	#135	1	4	yes	—
51	C₄×C₂₆₉	—	2	4	no	—
52	C₄×C₂₆₉	—	8	2	no	—
53	C₄×C₂₆₉	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₄×C₂₆₉	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₄×C₂₆₉	C₄⋊Q₈	1	1	yes	characteristic
56	C₄×C₂₆₉	C₄⋊Q₈	1	1	yes	characteristic
57	Dic₂₆₉	—	8	2	no	—
58	C₂×C₂₆₉	#384	1	1	yes	characteristic
59	C₂×C₂₆₉	#427	1	2	yes	—
60	C₂×C₂₆₉	#495	1	2	yes	—
61	C₂×C₂₆₉	#499	1	2	yes	—
62	C₂₆₉	#1846	1	1	yes	characteristic, complemented, Sylow
63	#1846	—	269	1	no	maximal, Sylow
64	C₂×#107	—	269	2	no	—
65	C₂×#106	—	269	2	no	—
66	C₂×C₄×C₈	—	269	1	no	—
67	#382	—	269	1	no	—
68	#382	D₂₆₉	1	1	yes	characteristic
69	C₂²×C₈	—	269	1	no	—
70	C₂²×C₈	—	269	1	no	—
71	C₂×C₄⋊C₄	—	269	2	no	—
72	C₂×C₄⋊C₄	C₂×D₂₆₉	1	2	yes	—
73	C₂×C₄²	C₂×D₂₆₉	1	1	yes	characteristic
74	#107	—	538	4	no	—
75	#106	—	538	4	no	—
76	C₄×C₈	—	269	4	no	—
77	#95	—	2	2	no	—
78	#95	—	538	2	no	—
79	C₂²×C₄	—	2	2	no	—
80	C₂²×C₄	—	538	2	no	—
81	C₂²×C₄	C₂²⋊D₂₆₉	1	2	yes	—
82	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
83	C₂×C₈	—	269	2	no	—
84	C₂×C₈	—	269	2	no	—
85	C₂×C₈	—	269	2	no	—
86	C₂×C₈	—	269	2	no	—
87	C₂×C₈	—	269	2	no	—
88	C₂×C₈	—	269	2	no	—
89	C₄⋊C₄	—	2	2	no	—
90	C₄⋊C₄	—	2	2	no	—
91	C₄⋊C₄	—	538	2	no	—
92	C₄⋊C₄	—	538	2	no	—
93	C₄²	C₄×D₂₆₉	1	2	yes	—
94	C₄²	Dic₅₃₈	1	2	yes	—
95	C₂³	C₂×C₂²⋊D₂₆₉	1	1	yes	characteristic
96	C₂×C₄	—	2	4	no	—
97	C₂×C₄	—	4	2	no	—
98	C₂×C₄	—	4	4	no	—
99	C₂×C₄	—	1076	2	no	—
100	C₂×C₄	—	1076	4	no	—
101	C₂×C₄	#16864381	1	4	yes	—
102	C₂×C₄	C₂×C₄×D₂₆₉	1	1	yes	characteristic
103	C₂×C₄	D₄×D₂₆₉	1	1	yes	characteristic
104	C₂×C₄	D₄×D₂₆₉	1	1	yes	characteristic
105	C₂×C₄	Q₈×D₂₆₉	1	1	yes	characteristic
106	C₂×C₄	Q₈×D₂₆₉	1	1	yes	characteristic
107	C₂×C₄	C₂×Dic₅₃₈	1	1	yes	characteristic
108	C₈	—	269	4	no	—
109	C₈	—	269	4	no	—
110	C₂²	C₂×#16864381	1	1	yes	characteristic
111	C₂²	#22575449	1	1	yes	characteristic
112	C₂²	#22575451	1	1	yes	characteristic
113	C₂²	#22575473	1	4	yes	—
114	C₄	—	2	4	no	—
115	C₄	—	8	2	no	—
116	C₄	—	2152	2	no	—
117	C₄	C₄⋊C₄×D₂₆₉	1	1	yes	characteristic
118	C₄	C₄⋊C₄×D₂₆₉	1	1	yes	characteristic
119	C₄	#22575452	1	1	yes	characteristic
120	C₄	#22575452	1	1	yes	characteristic
121	C₂	#31770474	1	1	yes	characteristic
122	C₂	#31770641	1	2	yes	—
123	C₂	#31770987	1	2	yes	—
124	C₂	#31771004	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22575386

C₂

yes

maximal

C₂×#22575384

C₂

yes

maximal

C₂×C₄×#14524130

C₂

yes

maximal, characteristic

#31770471

C₂

yes

maximal, characteristic

#382×C₂₆₉

C₂

yes

maximal, characteristic

C₂²×#14524130

C₂²

yes

characteristic

C₂²×#14524130

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₆₉

C₂²

yes

—

#22575386

—

#22575384

—

C₄×#14524130

C₄

yes

complemented

#22575360

—

C₂×C₄⋊C₄×C₂₆₉

C₂²

yes

—

C₂×C₄²×C₂₆₉

C₂²

yes

characteristic

#95×C₂₆₉

—

C₂²×Dic₂₆₉

—

C₂×#14524130

C₂×C₄

yes

—

C₂×#14524130

C₂×C₄

yes

—

C₂×#14524130

D₄

yes

—

C₂×#14524130

D₄

yes

—

C₂×#14524130

Q₈

yes

—

C₂×#14524130

Q₈

yes

—

C₄⋊Dic₂₆₉

—

C₄⋊Dic₂₆₉

—

C₂²×C₄×C₂₆₉

—

C₂²×C₄×C₂₆₉

D₄

yes

—

C₂²×C₄×C₂₆₉

C₂³

yes

characteristic

C₄⋊C₄×C₂₆₉

—

C₄⋊C₄×C₂₆₉

—

C₄²×C₂₆₉

C₂×C₄

yes

—

C₄²×C₂₆₉

Q₈

yes

—

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

—

#14524130

C₄⋊C₄

yes

—

#14524130

C₄⋊C₄

yes

—

C₂³×C₂₆₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

C₄⋊C₄

yes

—

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

C₂×Q₈

yes

characteristic

C₂²×C₂₆₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₆₉

#127

yes

characteristic

C₂²×C₂₆₉

C₄⋊Q₈

yes

characteristic

C₂²×C₂₆₉

#135

yes

—

C₄×C₂₆₉

—

C₄×C₂₆₉

—

C₄×C₂₆₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₂₆₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₂₆₉

C₄⋊Q₈

yes

characteristic

C₄×C₂₆₉

C₄⋊Q₈

yes

characteristic

Dic₂₆₉

—

C₂×C₂₆₉

#384

yes

characteristic

C₂×C₂₆₉

#427

yes

—

C₂×C₂₆₉

#495

yes

—

C₂×C₂₆₉

#499

yes

—

C₂₆₉

#1846

yes

characteristic, complemented, Sylow

#1846

—

269

maximal, Sylow

C₂×#107

—

269

—

C₂×#106

—

269

—

C₂×C₄×C₈

—

269

—

#382

—

269

—

#382

D₂₆₉

yes

characteristic

C₂²×C₈

—

269

—

C₂²×C₈

—

269

—

C₂×C₄⋊C₄

—

269

—

C₂×C₄⋊C₄

C₂×D₂₆₉

yes

—

C₂×C₄²

C₂×D₂₆₉

yes

characteristic

#107

—

538

—

#106

—

538

—

C₄×C₈

—

269

—

#95

—

#95

—

538

—

C₂²×C₄

—

C₂²×C₄

—

538

—

C₂²×C₄

C₂²⋊D₂₆₉

yes

—

C₂²×C₄

C₂²×D₂₆₉

yes

characteristic

C₂×C₈

—

269

—

C₂×C₈

—

269

—

C₂×C₈

—

269

—

C₂×C₈

—

269

—

C₂×C₈

—

269

—

C₂×C₈

—

269

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

538

—

C₄⋊C₄

—

538

—

C₄²

C₄×D₂₆₉

yes

—

C₄²

Dic₅₃₈

yes

—

C₂³

C₂×C₂²⋊D₂₆₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

1076

—

100

C₂×C₄

—

1076

—

101

C₂×C₄

#16864381

yes

—

102

C₂×C₄

C₂×C₄×D₂₆₉

yes

characteristic

103

C₂×C₄

D₄×D₂₆₉

yes

characteristic

104

C₂×C₄

D₄×D₂₆₉

yes

characteristic

105

C₂×C₄

Q₈×D₂₆₉

yes

characteristic

106

C₂×C₄

Q₈×D₂₆₉

yes

characteristic

107

C₂×C₄

C₂×Dic₅₃₈

yes

characteristic

108

C₈

—

269

—

109

C₈

—

269

—

110

C₂²

C₂×#16864381

yes

characteristic

111

C₂²

#22575449

yes

characteristic

112

C₂²

#22575451

yes

characteristic

113

C₂²

#22575473

yes

—

114

C₄

—

115

C₄

—

116

C₄

—

2152

—

117

C₄

C₄⋊C₄×D₂₆₉

yes

characteristic

118

C₄

C₄⋊C₄×D₂₆₉

yes

characteristic

119

C₄

#22575452

yes

characteristic

120

C₄

#22575452

yes

characteristic

121

C₂

#31770474

yes

characteristic

122

C₂

#31770641

yes

—

123

C₂

#31770987

yes

—

124

C₂

#31771004

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₂₆₉
commutator	C₂×C₄×C₂₆₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₆₉
Fitting	#382×C₂₆₉	C₂
socle, prime radical	C₂³×C₂₆₉	C₂×D₄
cosocle	C₂²×C₄×C₂₆₉	C₂³
hypercenter	#382	D₂₆₉
hypocenter	C₂₆₉	#1846
LCS	G ▷ C₂×C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₂₆₉

commutator

C₂×C₄×C₂₆₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₆₉

Fitting

#382×C₂₆₉

C₂

socle, prime radical

C₂³×C₂₆₉

C₂×D₄

cosocle

C₂²×C₄×C₂₆₉

C₂³

D₂₆₉

C₂₆₉

G ▷ C₂×C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

N	H ≅ G/N	type
11. C₄×#14524130	C₄	semidirect
62. C₂₆₉	#1846	semidirect

H ≅ G/N

type

11. C₄×#14524130

C₄

semidirect

62. C₂₆₉

#1846

semidirect

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

subgroup	index
C₂×#22575386	2
C₂×#22575384	2
C₂×C₄×#14524130	2
#31770471	2
#382×C₂₆₉	2
#1846	269

subgroup

index

#382×C₂₆₉

269

Subgroups — #46811047

Conjugacy classes of subgroups

Complements

Maximal subgroups