Group #46821399 — subgroups

Subgroups — #46821399

Order 34432.

356 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 173 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	#31771162	C₂	1	2	yes	maximal
2	#31771010	C₂	1	2	yes	maximal, complemented
3	#31771007	C₂	1	4	yes	maximal, complemented
4	#31771003	C₂	1	2	yes	maximal, complemented
5	#31770643	C₂	1	4	yes	maximal
6	C₂×C₄×C₈×C₂₆₉	C₂	1	1	yes	maximal, characteristic
7	#22575453	C₂²	1	4	yes	—
8	#22575445	C₂²	1	4	yes	—
9	#22575432	—	2	4	no	—
10	C₄×Dic₅₃₈	—	2	4	no	—
11	#22575412	C₂²	1	4	yes	—
12	#22575387	C₂²	1	8	yes	—
13	#22575385	C₂²	1	8	yes	—
14	C₂²×C₈×C₂₆₉	C₂²	1	2	yes	—
15	C₂×C₄²×C₂₆₉	C₂²	1	1	yes	characteristic
16	C₄×C₈×C₂₆₉	C₂²	1	1	yes	characteristic
17	C₄×C₈×C₂₆₉	C₂²	1	2	yes	—
18	C₄×C₈×C₂₆₉	C₂²	1	1	yes	characteristic
19	C₂×Dic₅₃₈	—	2	4	no	—
20	C₄⋊Dic₂₆₉	—	2	4	no	—
21	C₄⋊Dic₂₆₉	C₂³	1	8	yes	—
22	#16864381	—	2	8	no	—
23	C₂²⋊Dic₂₆₉	—	2	4	no	—
24	C₄×Dic₂₆₉	—	2	4	no	—
25	C₂²×C₄×C₂₆₉	D₄	1	2	yes	—
26	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
27	C₂×C₈×C₂₆₉	Q₈	1	8	yes	complemented
28	C₂×C₈×C₂₆₉	C₂³	1	2	yes	—
29	C₂×C₈×C₂₆₉	C₂³	1	2	yes	—
30	C₄²×C₂₆₉	D₄	1	2	yes	—
31	C₄²×C₂₆₉	C₂³	1	1	yes	characteristic
32	C₄²×C₂₆₉	C₂³	1	1	yes	characteristic
33	Dic₅₃₈	—	4	8	no	—
34	C₂×Dic₂₆₉	—	2	8	no	—
35	C₂³×C₂₆₉	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₂₆₉	—	2	4	no	—
37	C₂×C₄×C₂₆₉	C₂×D₄	1	2	yes	—
38	C₂×C₄×C₂₆₉	C₂×D₄	1	2	yes	—
39	C₂×C₄×C₂₆₉	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₂₆₉	C₂×Q₈	1	4	yes	—
41	C₂×C₄×C₂₆₉	C₂⁴	1	1	yes	characteristic
42	C₈×C₂₆₉	C₂×Q₈	1	4	yes	—
43	C₈×C₂₆₉	C₂×Q₈	1	4	yes	—
44	C₂²×C₂₆₉	—	2	2	no	—
45	C₂²×C₂₆₉	C₄⋊Q₈	1	2	yes	—
46	C₂²×C₂₆₉	C₂²×D₄	1	1	yes	characteristic
47	C₄×C₂₆₉	—	2	2	no	—
48	C₄×C₂₆₉	C₄⋊Q₈	1	2	yes	—
49	C₄×C₂₆₉	#135	1	4	yes	—
50	C₄×C₂₆₉	C₂²×Q₈	1	2	yes	—
51	Dic₂₆₉	—	4	8	no	—
52	C₂×C₂₆₉	—	2	2	no	—
53	C₂×C₂₆₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
54	C₂×C₂₆₉	C₂×#135	1	2	yes	—
55	C₂₆₉	#3160	1	1	yes	characteristic, complemented, Sylow
56	#3160	—	269	1	no	maximal, Sylow
57	#533	—	269	2	no	—
58	#500	—	269	2	no	—
59	#499	—	269	4	no	—
60	#498	—	269	2	no	—
61	#427	—	269	4	no	—
62	C₂×C₄×C₈	D₂₆₉	1	1	yes	characteristic
63	C₂²×C₈	C₂×D₂₆₉	1	2	yes	—
64	C₄⋊Q₈	—	269	4	no	—
65	#125	—	269	4	no	—
66	C₂²⋊Q₈	—	538	4	no	—
67	C₄×Q₈	—	538	4	no	—
68	#117	—	269	4	no	—
69	C₂×C₄²	C₂×D₂₆₉	1	1	yes	characteristic
70	#107	—	269	8	no	—
71	#106	—	269	8	no	—
72	C₄×C₈	C₂×D₂₆₉	1	1	yes	characteristic
73	C₄×C₈	C₂×D₂₆₉	1	2	yes	—
74	C₄×C₈	C₂×D₂₆₉	1	1	yes	characteristic
75	C₂×Q₈	—	538	4	no	—
76	C₂²×C₄	D₁₀₇₆	1	2	yes	—
77	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
78	C₂×C₈	Dic₅₃₈	1	8	yes	complemented
79	C₂×C₈	C₂²×D₂₆₉	1	2	yes	—
80	C₂×C₈	C₂²×D₂₆₉	1	2	yes	—
81	C₄⋊C₄	—	269	8	no	—
82	C₄⋊C₄	—	538	8	no	—
83	C₄⋊C₄	—	538	4	no	—
84	C₂²⋊C₄	—	538	4	no	—
85	C₄²	—	538	4	no	—
86	C₄²	D₁₀₇₆	1	2	yes	—
87	C₄²	C₂²×D₂₆₉	1	1	yes	characteristic
88	C₄²	C₂²×D₂₆₉	1	1	yes	characteristic
89	C₂³	C₂×D₁₀₇₆	1	1	yes	characteristic
90	Q₈	—	1076	8	no	—
91	C₂×C₄	—	2	4	no	—
92	C₂×C₄	—	538	8	no	—
93	C₂×C₄	C₂×D₁₀₇₆	1	2	yes	—
94	C₂×C₄	C₂×D₁₀₇₆	1	2	yes	—
95	C₂×C₄	C₂×D₁₀₇₆	1	1	yes	characteristic
96	C₂×C₄	C₂×Dic₅₃₈	1	4	yes	—
97	C₂×C₄	C₂³×D₂₆₉	1	1	yes	characteristic
98	C₈	C₂×Dic₅₃₈	1	4	yes	—
99	C₈	C₂×Dic₅₃₈	1	4	yes	—
100	C₂²	—	2	2	no	—
101	C₂²	#22575453	1	2	yes	—
102	C₂²	C₂²×D₁₀₇₆	1	1	yes	characteristic
103	C₄	—	2	2	no	—
104	C₄	—	1076	8	no	—
105	C₄	#22575453	1	2	yes	—
106	C₄	#22575475	1	4	yes	—
107	C₄	C₂²×Dic₅₃₈	1	2	yes	—
108	C₂	—	2	2	no	—
109	C₂	C₂×#22575453	1	1	yes	characteristic
110	C₂	C₂×#22575475	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#31771162

C₂

yes

maximal

#31771010

C₂

yes

maximal, complemented

#31771007

C₂

yes

maximal, complemented

#31771003

C₂

yes

maximal, complemented

#31770643

C₂

yes

maximal

C₂×C₄×C₈×C₂₆₉

C₂

yes

maximal, characteristic

#22575453

C₂²

yes

—

#22575445

C₂²

yes

—

#22575432

—

C₄×Dic₅₃₈

—

#22575412

C₂²

yes

—

#22575387

C₂²

yes

—

#22575385

C₂²

yes

—

C₂²×C₈×C₂₆₉

C₂²

yes

—

C₂×C₄²×C₂₆₉

C₂²

yes

characteristic

C₄×C₈×C₂₆₉

C₂²

yes

characteristic

C₄×C₈×C₂₆₉

C₂²

yes

—

C₄×C₈×C₂₆₉

C₂²

yes

characteristic

C₂×Dic₅₃₈

—

C₄⋊Dic₂₆₉

—

C₄⋊Dic₂₆₉

C₂³

yes

—

#16864381

—

C₂²⋊Dic₂₆₉

—

C₄×Dic₂₆₉

—

C₂²×C₄×C₂₆₉

D₄

yes

—

C₂²×C₄×C₂₆₉

C₂³

yes

characteristic

C₂×C₈×C₂₆₉

Q₈

yes

complemented

C₂×C₈×C₂₆₉

C₂³

yes

—

C₂×C₈×C₂₆₉

C₂³

yes

—

C₄²×C₂₆₉

D₄

yes

—

C₄²×C₂₆₉

C₂³

yes

characteristic

C₄²×C₂₆₉

C₂³

yes

characteristic

Dic₅₃₈

—

C₂×Dic₂₆₉

—

C₂³×C₂₆₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

C₂×D₄

yes

—

C₂×C₄×C₂₆₉

C₂×D₄

yes

—

C₂×C₄×C₂₆₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₆₉

C₂×Q₈

yes

—

C₂×C₄×C₂₆₉

C₂⁴

yes

characteristic

C₈×C₂₆₉

C₂×Q₈

yes

—

C₈×C₂₆₉

C₂×Q₈

yes

—

C₂²×C₂₆₉

—

C₂²×C₂₆₉

C₄⋊Q₈

yes

—

C₂²×C₂₆₉

C₂²×D₄

yes

characteristic

C₄×C₂₆₉

—

C₄×C₂₆₉

C₄⋊Q₈

yes

—

C₄×C₂₆₉

#135

yes

—

C₄×C₂₆₉

C₂²×Q₈

yes

—

Dic₂₆₉

—

C₂×C₂₆₉

—

C₂×C₂₆₉

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₂₆₉

C₂×#135

yes

—

C₂₆₉

#3160

yes

characteristic, complemented, Sylow

#3160

—

269

maximal, Sylow

#533

—

269

—

#500

—

269

—

#499

—

269

—

#498

—

269

—

#427

—

269

—

C₂×C₄×C₈

D₂₆₉

yes

characteristic

C₂²×C₈

C₂×D₂₆₉

yes

—

C₄⋊Q₈

—

269

—

#125

—

269

—

C₂²⋊Q₈

—

538

—

C₄×Q₈

—

538

—

#117

—

269

—

C₂×C₄²

C₂×D₂₆₉

yes

characteristic

#107

—

269

—

#106

—

269

—

C₄×C₈

C₂×D₂₆₉

yes

characteristic

C₄×C₈

C₂×D₂₆₉

yes

—

C₄×C₈

C₂×D₂₆₉

yes

characteristic

C₂×Q₈

—

538

—

C₂²×C₄

D₁₀₇₆

yes

—

C₂²×C₄

C₂²×D₂₆₉

yes

characteristic

C₂×C₈

Dic₅₃₈

yes

complemented

C₂×C₈

C₂²×D₂₆₉

yes

—

C₂×C₈

C₂²×D₂₆₉

yes

—

C₄⋊C₄

—

269

—

C₄⋊C₄

—

538

—

C₄⋊C₄

—

538

—

C₂²⋊C₄

—

538

—

C₄²

—

538

—

C₄²

D₁₀₇₆

yes

—

C₄²

C₂²×D₂₆₉

yes

characteristic

C₄²

C₂²×D₂₆₉

yes

characteristic

C₂³

C₂×D₁₀₇₆

yes

characteristic

Q₈

—

1076

—

C₂×C₄

—

C₂×C₄

—

538

—

C₂×C₄

C₂×D₁₀₇₆

yes

—

C₂×C₄

C₂×D₁₀₇₆

yes

—

C₂×C₄

C₂×D₁₀₇₆

yes

characteristic

C₂×C₄

C₂×Dic₅₃₈

yes

—

C₂×C₄

C₂³×D₂₆₉

yes

characteristic

C₈

C₂×Dic₅₃₈

yes

—

C₈

C₂×Dic₅₃₈

yes

—

100

C₂²

—

101

C₂²

#22575453

yes

—

102

C₂²

C₂²×D₁₀₇₆

yes

characteristic

103

C₄

—

104

C₄

—

1076

—

105

C₄

#22575453

yes

—

106

C₄

#22575475

yes

—

107

C₄

C₂²×Dic₅₃₈

yes

—

108

C₂

—

109

C₂

C₂×#22575453

yes

characteristic

110

C₂

C₂×#22575475

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₁₀₇₆
commutator, cosocle	C₂×C₄×C₂₆₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₆₉
Fitting	C₂×C₄×C₈×C₂₆₉	C₂
socle	C₂²×C₂₆₉	C₂²×D₄
prime radical	C₂³×C₂₆₉	C₂×D₄
hypercenter	C₂×C₄×C₈	D₂₆₉
hypocenter	C₂₆₉	#3160
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, cosocle

C₂×C₄×C₂₆₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₂₆₉

Fitting

C₂×C₄×C₈×C₂₆₉

C₂

socle

C₂²×C₂₆₉

C₂²×D₄

prime radical

C₂³×C₂₆₉

C₂×D₄

hypercenter

C₂×C₄×C₈

D₂₆₉

hypocenter

C₂₆₉

#3160

LCS

G ▷ C₂×C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
2. #31771010	C₂	semidirect
3. #31771007	C₂	semidirect
4. #31771003	C₂	semidirect
27. C₂×C₈×C₂₆₉	Q₈	semidirect
55. C₂₆₉	#3160	semidirect
78. C₂×C₈	Dic₅₃₈	semidirect

H ≅ G/N

type

2. #31771010

C₂

semidirect

3. #31771007

C₂

semidirect

4. #31771003

C₂

semidirect

27. C₂×C₈×C₂₆₉

Q₈

semidirect

55. C₂₆₉

#3160

semidirect

78. C₂×C₈

Dic₅₃₈

semidirect

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

subgroup	index
#31771162	2
#31771010	2
#31771007	2
#31771003	2
#31770643	2
C₂×C₄×C₈×C₂₆₉	2
#3160	269

subgroup

index

C₂×C₄×C₈×C₂₆₉

269

Subgroups — #46821399

Conjugacy classes of subgroups

Complements

Maximal subgroups