Group #46825706 — subgroups

Subgroups — #46825706

Order 34432.

166 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 54 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	#31771471	C₂	1	2	yes	maximal
2	#31770510	C₂	1	1	yes	maximal, characteristic
3	C₂²×#14524137	—	2	2	no	—
4	C₂²×Dic₅₃₈	C₄	1	1	yes	characteristic
5	C₂×C₄×Dic₂₆₉	C₂²	1	1	yes	characteristic
6	#22575360	—	2	2	no	—
7	C₂×C₄⋊C₄×C₂₆₉	C₄	1	1	yes	characteristic
8	C₂×#14524137	—	2	4	no	—
9	C₂×#14524137	—	4	2	no	—
10	C₂×#14524137	—	4	2	no	—
11	C₂×Dic₅₃₈	—	2	1	no	—
12	C₂×Dic₅₃₈	—	4	1	no	—
13	C₂×Dic₅₃₈	—	4	1	no	—
14	C₂×Dic₅₃₈	C₈	1	2	yes	complemented
15	C₂²×Dic₂₆₉	—	2	1	no	—
16	C₂²×Dic₂₆₉	D₄	1	2	yes	—
17	C₄×Dic₂₆₉	—	4	1	no	—
18	C₂²×C₄×C₂₆₉	—	2	1	no	—
19	C₂²×C₄×C₂₆₉	C₂×C₄	1	1	yes	characteristic
20	C₄⋊C₄×C₂₆₉	—	4	1	no	—
21	#14524137	—	4	4	no	—
22	Dic₅₃₈	—	4	2	no	—
23	Dic₅₃₈	—	4	2	no	—
24	C₂×Dic₂₆₉	—	2	2	no	—
25	C₂×Dic₂₆₉	—	2	2	no	—
26	C₂×Dic₂₆₉	—	2	2	no	—
27	C₂×Dic₂₆₉	—	4	1	no	—
28	C₂×Dic₂₆₉	—	4	1	no	—
29	C₂×Dic₂₆₉	SD₁₆	1	2	yes	—
30	C₂×Dic₂₆₉	Q₁₆	1	2	yes	—
31	C₂³×C₂₆₉	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×C₄×C₂₆₉	—	2	1	no	—
33	C₂×C₄×C₂₆₉	—	2	1	no	—
34	C₂×C₄×C₂₆₉	—	4	1	no	—
35	C₂×C₄×C₂₆₉	—	4	2	no	—
36	C₂×C₄×C₂₆₉	C₂×C₈	1	1	yes	characteristic
37	C₂×C₄×C₂₆₉	M₁₆	1	1	yes	characteristic
38	C₂²×C₂₆₉	C₂²⋊C₈	1	1	yes	characteristic
39	C₂²×C₂₆₉	#99	1	1	yes	characteristic
40	C₂²×C₂₆₉	#100	1	1	yes	characteristic
41	C₂²×C₂₆₉	#103	1	2	yes	—
42	C₂²×C₂₆₉	#104	1	2	yes	—
43	C₄×C₂₆₉	—	2	1	no	—
44	C₄×C₂₆₉	—	2	1	no	—
45	C₄×C₂₆₉	—	8	1	no	—
46	Dic₂₆₉	—	2	2	no	—
47	Dic₂₆₉	—	2	2	no	—
48	Dic₂₆₉	—	4	2	no	—
49	C₂×C₂₆₉	#323	1	1	yes	characteristic
50	C₂×C₂₆₉	Q₈⋊C₈	1	2	yes	—
51	C₂×C₂₆₉	#328	1	2	yes	—
52	C₂×C₂₆₉	#329	1	1	yes	characteristic
53	C₂×C₂₆₉	#333	1	1	yes	characteristic
54	C₂₆₉	#1272	1	1	yes	characteristic, complemented, Sylow
55	#1272	—	269	1	no	maximal, Sylow
56	#391	—	269	1	no	—
57	#336	—	269	2	no	—
58	C₂²×Q₈	—	269	1	no	—
59	C₂²×C₈	—	538	2	no	—
60	C₂×C₄⋊C₄	#11761640	1	1	yes	characteristic
61	C₂×C₄²	—	269	1	no	—
62	#95	—	538	2	no	—
63	C₂×Q₈	—	269	2	no	—
64	C₂×Q₈	—	538	1	no	—
65	C₂×Q₈	—	1076	1	no	—
66	C₂×Q₈	—	1076	1	no	—
67	C₂²×C₄	—	2	1	no	—
68	C₂²×C₄	—	269	2	no	—
69	C₂²×C₄	—	538	1	no	—
70	C₂²×C₄	C₂×#11761640	1	1	yes	characteristic
71	C₂×C₈	—	538	4	no	—
72	C₂×C₈	—	1076	2	no	—
73	C₂×C₈	—	1076	2	no	—
74	C₄⋊C₄	—	4	1	no	—
75	C₄²	—	1076	1	no	—
76	C₂³	C₂²⋊#11761640	1	1	yes	characteristic
77	Q₈	—	1076	2	no	—
78	Q₈	—	1076	2	no	—
79	C₂×C₄	—	2	1	no	—
80	C₂×C₄	—	2	1	no	—
81	C₂×C₄	—	4	1	no	—
82	C₂×C₄	—	4	2	no	—
83	C₂×C₄	—	269	2	no	—
84	C₂×C₄	—	269	2	no	—
85	C₂×C₄	—	538	2	no	—
86	C₂×C₄	—	538	2	no	—
87	C₂×C₄	—	538	2	no	—
88	C₂×C₄	—	1076	1	no	—
89	C₂×C₄	—	1076	1	no	—
90	C₂×C₄	D₂₆₉⋊C₈	1	1	yes	characteristic
91	C₂×C₄	#16864412	1	1	yes	characteristic
92	C₈	—	1076	4	no	—
93	C₂²	#22575510	1	1	yes	characteristic
94	C₂²	#22575512	1	1	yes	characteristic
95	C₂²	#22575513	1	1	yes	characteristic
96	C₂²	#22575516	1	2	yes	—
97	C₂²	#22575517	1	2	yes	—
98	C₄	—	2	1	no	—
99	C₄	—	2	1	no	—
100	C₄	—	8	1	no	—
101	C₄	—	538	2	no	—
102	C₄	—	538	2	no	—
103	C₄	—	1076	2	no	—
104	C₂	#31771449	1	1	yes	characteristic
105	C₂	#31771455	1	2	yes	—
106	C₂	#31771458	1	2	yes	—
107	C₂	#31771460	1	1	yes	characteristic
108	C₂	#31771465	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#31771471

C₂

yes

maximal

#31770510

C₂

yes

maximal, characteristic

C₂²×#14524137

—

C₂²×Dic₅₃₈

C₄

yes

characteristic

C₂×C₄×Dic₂₆₉

C₂²

yes

characteristic

#22575360

—

C₂×C₄⋊C₄×C₂₆₉

C₄

yes

—

—

—

C₂×Dic₅₃₈

—

C₂×Dic₅₃₈

—

C₂×Dic₅₃₈

—

C₂×Dic₅₃₈

C₈

yes

complemented

C₂²×Dic₂₆₉

—

C₂²×Dic₂₆₉

D₄

yes

—

C₄×Dic₂₆₉

—

C₂²×C₄×C₂₆₉

—

C₂²×C₄×C₂₆₉

C₂×C₄

yes

characteristic

C₄⋊C₄×C₂₆₉

—

#14524137

—

Dic₅₃₈

—

Dic₅₃₈

—

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

SD₁₆

yes

—

C₂×Dic₂₆₉

Q₁₆

yes

—

C₂³×C₂₆₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

C₂×C₈

yes

characteristic

C₂×C₄×C₂₆₉

M₁₆

yes

characteristic

C₂²×C₂₆₉

C₂²⋊C₈

yes

C₂²×C₂₆₉

yes

C₂²×C₂₆₉

yes

C₂²×C₂₆₉

yes

—

C₂²×C₂₆₉

#104

yes

—

C₄×C₂₆₉

—

C₄×C₂₆₉

—

C₄×C₂₆₉

—

Dic₂₆₉

—

Dic₂₆₉

—

Dic₂₆₉

—

C₂×C₂₆₉

#323

yes

characteristic

C₂×C₂₆₉

Q₈⋊C₈

yes

—

C₂×C₂₆₉

#328

yes

—

C₂×C₂₆₉

yes

C₂×C₂₆₉

yes

C₂₆₉

yes

characteristic, complemented, Sylow

#1272

—

269

maximal, Sylow

#391

—

269

—

#336

—

269

—

C₂²×Q₈

—

269

—

C₂²×C₈

—

538

—

C₂×C₄⋊C₄

#11761640

yes

characteristic

C₂×C₄²

—

269

—

#95

—

538

—

C₂×Q₈

—

269

—

C₂×Q₈

—

538

—

C₂×Q₈

—

1076

—

C₂×Q₈

—

1076

—

C₂²×C₄

—

C₂²×C₄

—

269

—

C₂²×C₄

—

538

—

C₂²×C₄

C₂×#11761640

yes

characteristic

C₂×C₈

—

538

—

C₂×C₈

—

1076

—

C₂×C₈

—

1076

—

C₄⋊C₄

—

C₄²

—

1076

—

C₂³

C₂²⋊#11761640

yes

characteristic

Q₈

—

1076

—

Q₈

—

1076

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

269

—

C₂×C₄

—

269

—

C₂×C₄

—

538

—

C₂×C₄

—

538

—

C₂×C₄

—

538

—

C₂×C₄

—

1076

—

C₂×C₄

—

1076

—

C₂×C₄

D₂₆₉⋊C₈

yes

characteristic

C₂×C₄

#16864412

yes

characteristic

C₈

—

1076

—

C₂²

yes

C₂²

yes

C₂²

yes

C₂²

yes

—

C₂²

#22575517

yes

—

C₄

—

C₄

—

100

C₄

—

101

C₄

—

538

—

102

C₄

—

538

—

103

C₄

—

1076

—

104

C₂

#31771449

yes

characteristic

105

C₂

#31771455

yes

—

106

C₂

#31771458

yes

—

107

C₂

#31771460

yes

characteristic

108

C₂

#31771465

yes

characteristic

	subgroup	G/H
center	C₂³	C₂²⋊#11761640
commutator	C₂×C₄×C₂₆₉	C₂×C₈
Frattini	C₂²×C₄	C₂×#11761640
Fitting	C₂×C₄⋊C₄×C₂₆₉	C₄
socle, prime radical	C₂³×C₂₆₉	C₂²⋊C₄
cosocle	C₂×C₄×Dic₂₆₉	C₂²
hypercenter	C₂×C₄⋊C₄	#11761640
hypocenter	C₂₆₉	#1272
LCS	G ▷ C₂×C₄×C₂₆₉ ▷ C₂²×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

subgroup

G/H

center

C₂³

C₂²⋊#11761640

commutator

C₂×C₄×C₂₆₉

C₂×C₈

Frattini

C₂²×C₄

C₂×#11761640

Fitting

C₂×C₄⋊C₄×C₂₆₉

C₄

socle, prime radical

C₂³×C₂₆₉

C₂²⋊C₄

cosocle

C₂×C₄×Dic₂₆₉

C₂²

C₂×C₄⋊C₄

C₂₆₉

G ▷ C₂×C₄×C₂₆₉ ▷ C₂²×C₂₆₉ ▷ C₂₆₉

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

N	H ≅ G/N	type
14. C₂×Dic₅₃₈	C₈	semidirect
54. C₂₆₉	#1272	semidirect

H ≅ G/N

type

14. C₂×Dic₅₃₈

C₈

semidirect

54. C₂₆₉

#1272

semidirect

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

subgroup	index
#31771471	2
#31770510	2
#1272	269

subgroup

index

#31771471

#31770510

#1272

269

Subgroups — #46825706

Conjugacy classes of subgroups

Complements

Maximal subgroups