Group #46811054 — subgroups

Subgroups — #46811054

Order 34432.

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

normal subgroups only

This group has 256 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×#22575364	C₂	1	1	yes	maximal, characteristic
4	#31770475	C₂	1	1	yes	maximal, characteristic
5	#382×C₂₆₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×#14524130	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₂₆₉	—	2	2	no	—
8	C₂×C₄⋊Dic₂₆₉	C₂²	1	2	yes	—
9	#22575386	—	2	4	no	—
10	#22575384	—	2	4	no	—
11	#22575364	C₄	1	4	yes	complemented
12	C₂×C₄⋊C₄×C₂₆₉	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₂₆₉	C₂²	1	1	yes	characteristic
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	4	yes	—
18	C₂×#14524130	D₄	1	4	yes	—
19	C₂×#14524130	Q₈	1	4	yes	—
20	C₄⋊Dic₂₆₉	—	2	2	no	—
21	C₄⋊Dic₂₆₉	—	2	2	no	—
22	C₄⋊Dic₂₆₉	—	4	4	no	—
23	C₂²×C₄×C₂₆₉	—	2	1	no	—
24	C₂²×C₄×C₂₆₉	—	2	1	no	—
25	C₂²×C₄×C₂₆₉	D₄	1	2	yes	—
26	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
27	C₄⋊C₄×C₂₆₉	—	2	2	no	—
28	C₄⋊C₄×C₂₆₉	—	2	2	no	—
29	C₄²×C₂₆₉	C₂×C₄	1	2	yes	—
30	C₄²×C₂₆₉	Q₈	1	2	yes	—
31	C₂×Dic₂₆₉	—	4	2	no	—
32	C₂×Dic₂₆₉	—	4	4	no	—
33	#14524130	C₄⋊C₄	1	8	yes	complemented
34	C₂³×C₂₆₉	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₂₆₉	—	2	2	no	—
36	C₂×C₄×C₂₆₉	—	2	2	no	—
37	C₂×C₄×C₂₆₉	—	4	1	no	—
38	C₂×C₄×C₂₆₉	—	4	1	no	—
39	C₂×C₄×C₂₆₉	—	4	2	no	—
40	C₂×C₄×C₂₆₉	—	4	2	no	—
41	C₂×C₄×C₂₆₉	C₄⋊C₄	1	4	yes	—
42	C₂×C₄×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₂₆₉	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₂₆₉	C₂×Q₈	1	2	yes	—
45	C₂×C₄×C₂₆₉	C₂×Q₈	1	1	yes	characteristic
46	C₂²×C₂₆₉	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂²×C₂₆₉	#127	1	1	yes	characteristic
48	C₂²×C₂₆₉	C₄⋊Q₈	1	1	yes	characteristic
49	C₂²×C₂₆₉	AGL(1,ℤ/8)	1	2	yes	—
50	C₂²×C₂₆₉	#137	1	2	yes	—
51	C₄×C₂₆₉	—	2	4	no	—
52	C₄×C₂₆₉	—	8	1	no	—
53	C₄×C₂₆₉	—	8	1	no	—
54	C₄×C₂₆₉	C₂×C₄⋊C₄	1	2	yes	—
55	C₄×C₂₆₉	C₄⋊Q₈	1	2	yes	—
56	Dic₂₆₉	—	8	2	no	—
57	C₂×C₂₆₉	#384	1	1	yes	characteristic
58	C₂×C₂₆₉	#428	1	2	yes	—
59	C₂×C₂₆₉	#496	1	1	yes	characteristic
60	C₂×C₂₆₉	#497	1	1	yes	characteristic
61	C₂×C₂₆₉	#501	1	2	yes	—
62	C₂₆₉	#1847	1	1	yes	characteristic, complemented, Sylow
63	#1847	—	269	1	no	maximal, Sylow
64	C₂×#107	—	269	2	no	—
65	C₂×#106	—	269	2	no	—
66	C₂×#97	—	269	1	no	—
67	#384	—	269	1	no	—
68	#382	D₂₆₉	1	1	yes	characteristic
69	C₂²×C₈	—	269	2	no	—
70	C₂×C₄⋊C₄	—	269	2	no	—
71	C₂×C₄⋊C₄	—	538	2	no	—
72	C₂×C₄⋊C₄	C₂×D₂₆₉	1	1	yes	characteristic
73	C₂×C₄⋊C₄	C₂×D₂₆₉	1	1	yes	characteristic
74	C₂×C₄²	C₂×D₂₆₉	1	1	yes	characteristic
75	#107	—	538	4	no	—
76	#106	—	538	4	no	—
77	#97	—	269	4	no	—
78	#95	—	2	2	no	—
79	C₂²×C₄	—	2	1	no	—
80	C₂²×C₄	—	2	1	no	—
81	C₂²×C₄	—	538	2	no	—
82	C₂²×C₄	C₂²⋊D₂₆₉	1	2	yes	—
83	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
84	C₂×C₈	—	269	4	no	—
85	C₂×C₈	—	269	4	no	—
86	C₂×C₈	—	269	4	no	—
87	C₄⋊C₄	—	2	2	no	—
88	C₄⋊C₄	—	2	2	no	—
89	C₄⋊C₄	—	538	2	no	—
90	C₄⋊C₄	—	538	2	no	—
91	C₄⋊C₄	—	1076	4	no	—
92	C₄²	C₄×D₂₆₉	1	2	yes	—
93	C₄²	Dic₅₃₈	1	2	yes	—
94	C₂³	C₂×C₂²⋊D₂₆₉	1	1	yes	characteristic
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	2	2	no	—
97	C₂×C₄	—	4	1	no	—
98	C₂×C₄	—	4	1	no	—
99	C₂×C₄	—	4	2	no	—
100	C₂×C₄	—	4	2	no	—
101	C₂×C₄	—	1076	2	no	—
102	C₂×C₄	—	1076	4	no	—
103	C₂×C₄	#16864381	1	4	yes	—
104	C₂×C₄	C₂×C₄×D₂₆₉	1	1	yes	characteristic
105	C₂×C₄	D₄×D₂₆₉	1	2	yes	—
106	C₂×C₄	Q₈×D₂₆₉	1	2	yes	—
107	C₂×C₄	C₂×Dic₅₃₈	1	1	yes	characteristic
108	C₈	—	269	8	no	—
109	C₂²	C₂×#16864381	1	1	yes	characteristic
110	C₂²	#22575449	1	1	yes	characteristic
111	C₂²	#22575451	1	1	yes	characteristic
112	C₂²	#22575478	1	2	yes	—
113	C₂²	#22575483	1	2	yes	—
114	C₄	—	2	4	no	—
115	C₄	—	8	1	no	—
116	C₄	—	8	1	no	—
117	C₄	—	2152	2	no	—
118	C₄	C₄⋊C₄×D₂₆₉	1	2	yes	—
119	C₄	#22575452	1	2	yes	—
120	C₂	#31770474	1	1	yes	characteristic
121	C₂	#31770648	1	2	yes	—
122	C₂	#31770995	1	1	yes	characteristic
123	C₂	#31771000	1	1	yes	characteristic
124	C₂	#31771015	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22575386

C₂

yes

maximal

C₂×#22575384

C₂

yes

maximal

C₂×#22575364

C₂

yes

maximal, characteristic

#31770475

C₂

yes

maximal, characteristic

#382×C₂₆₉

C₂

yes

maximal, characteristic

C₂²×#14524130

C₂²

yes

—

C₂×C₄⋊Dic₂₆₉

—

C₂×C₄⋊Dic₂₆₉

C₂²

yes

—

#22575386

—

#22575384

—

#22575364

C₄

yes

complemented

C₂×C₄⋊C₄×C₂₆₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₂₆₉

C₂²

yes

characteristic

C₂×C₄²×C₂₆₉

C₂²

yes

characteristic

#95×C₂₆₉

—

C₂²×Dic₂₆₉

—

C₂×#14524130

C₂×C₄

yes

—

C₂×#14524130

D₄

yes

—

C₂×#14524130

Q₈

yes

—

C₄⋊Dic₂₆₉

—

C₄⋊Dic₂₆₉

—

C₄⋊Dic₂₆₉

—

C₂²×C₄×C₂₆₉

—

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

complemented

C₂³×C₂₆₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

—

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

—

C₂×C₄×C₂₆₉

C₂×Q₈

yes

—

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

AGL(1,ℤ/8)

yes

—

C₂²×C₂₆₉

#137

yes

—

C₄×C₂₆₉

—

C₄×C₂₆₉

—

C₄×C₂₆₉

—

C₄×C₂₆₉

C₂×C₄⋊C₄

yes

—

C₄×C₂₆₉

C₄⋊Q₈

yes

—

Dic₂₆₉

—

C₂×C₂₆₉

#384

yes

characteristic

C₂×C₂₆₉

#428

yes

—

C₂×C₂₆₉

yes

C₂×C₂₆₉

yes

C₂×C₂₆₉

yes

—

C₂₆₉

#1847

yes

characteristic, complemented, Sylow

#1847

—

269

maximal, Sylow

C₂×#107

—

269

—

C₂×#106

—

269

—

C₂×#97

—

269

—

#384

—

269

—

#382

D₂₆₉

yes

characteristic

C₂²×C₈

—

269

—

C₂×C₄⋊C₄

—

269

—

C₂×C₄⋊C₄

—

538

—

C₂×C₄⋊C₄

C₂×D₂₆₉

yes

characteristic

C₂×C₄⋊C₄

C₂×D₂₆₉

yes

characteristic

C₂×C₄²

C₂×D₂₆₉

yes

characteristic

#107

—

538

—

#106

—

538

—

#97

—

269

—

#95

—

C₂²×C₄

—

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

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

538

—

C₄⋊C₄

—

538

—

C₄⋊C₄

—

1076

—

C₄²

C₄×D₂₆₉

yes

—

C₄²

Dic₅₃₈

yes

—

C₂³

C₂×C₂²⋊D₂₆₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

1076

—

102

C₂×C₄

—

1076

—

103

C₂×C₄

#16864381

yes

—

104

C₂×C₄

C₂×C₄×D₂₆₉

yes

characteristic

105

C₂×C₄

D₄×D₂₆₉

yes

—

106

C₂×C₄

Q₈×D₂₆₉

yes

—

107

C₂×C₄

C₂×Dic₅₃₈

yes

characteristic

108

C₈

—

269

—

109

C₂²

C₂×#16864381

yes

characteristic

110

C₂²

#22575449

yes

characteristic

111

C₂²

#22575451

yes

characteristic

112

C₂²

#22575478

yes

—

113

C₂²

#22575483

yes

—

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

—

2152

—

118

C₄

C₄⋊C₄×D₂₆₉

yes

—

119

C₄

#22575452

yes

—

120

C₂

#31770474

yes

characteristic

121

C₂

#31770648

yes

—

122

C₂

#31770995

yes

characteristic

123

C₂

#31771000

yes

characteristic

124

C₂

#31771015

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	#1847
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. #22575364	C₄	semidirect
33. #14524130	C₄⋊C₄	semidirect
62. C₂₆₉	#1847	semidirect

H ≅ G/N

type

11. #22575364

C₄

semidirect

33. #14524130

C₄⋊C₄

semidirect

62. C₂₆₉

#1847

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×#22575364	2
#31770475	2
#382×C₂₆₉	2
#1847	269

subgroup

index

#382×C₂₆₉

269

Subgroups — #46811054

Conjugacy classes of subgroups

Complements

Maximal subgroups