Group #46832756 — subgroups

Subgroups — #2170×C₂₇₁ #46832756

Order 34688.

146 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 146 conjugacy classes of subgroups — too many to draw. Only its 92 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	#504×C₂₇₁	C₂	1	2	yes	maximal, complemented
2	#443×C₂₇₁	C₂	1	1	yes	maximal, characteristic
3	#364×C₂₇₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#350×C₂₇₁	C₂	1	2	yes	maximal, complemented
5	C₄×C₁₆×C₂₇₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#135×C₂₇₁	C₄	1	1	yes	characteristic
7	#131×C₂₇₁	C₂²	1	2	yes	—
8	M₃₂×C₂₇₁	—	2	2	no	—
9	M₃₂×C₂₇₁	C₂²	1	2	yes	—
10	C₂×C₁₆×C₂₇₁	—	2	2	no	—
11	C₂×C₁₆×C₂₇₁	C₂²	1	1	yes	characteristic
12	C₂×C₁₆×C₂₇₁	C₂²	1	1	yes	characteristic
13	#108×C₂₇₁	C₄	1	1	yes	characteristic
14	#104×C₂₇₁	C₄	1	2	yes	—
15	C₄×C₈×C₂₇₁	C₂²	1	1	yes	characteristic
16	#41×C₂₇₁	C₂×C₄	1	2	yes	—
17	Q₁₆×C₂₇₁	C₈	1	1	yes	characteristic
18	SD₁₆×C₂₇₁	C₈	1	2	yes	—
19	D₈×C₂₇₁	C₈	1	1	yes	characteristic
20	M₁₆×C₂₇₁	—	2	2	no	—
21	M₁₆×C₂₇₁	C₂×C₄	1	2	yes	—
22	C₂×C₈×C₂₇₁	—	2	2	no	—
23	C₂×C₈×C₂₇₁	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×C₂₇₁	C₂³	1	1	yes	characteristic
25	C₄²×C₂₇₁	C₂×C₄	1	1	yes	characteristic
26	C₁₆×C₂₇₁	—	2	2	no	—
27	C₁₆×C₂₇₁	—	2	1	no	—
28	C₁₆×C₂₇₁	D₄	1	1	yes	characteristic
29	C₁₆×C₂₇₁	D₄	1	1	yes	characteristic
30	Q₈×C₂₇₁	C₂×C₈	1	2	yes	—
31	D₄×C₂₇₁	—	2	2	no	—
32	D₄×C₂₇₁	C₂×C₈	1	2	yes	—
33	C₂×C₄×C₂₇₁	—	2	1	no	—
34	C₂×C₄×C₂₇₁	—	2	2	no	—
35	C₂×C₄×C₂₇₁	C₂²×C₄	1	1	yes	characteristic
36	C₈×C₂₇₁	—	2	2	no	—
37	C₈×C₂₇₁	C₂×C₈	1	1	yes	characteristic
38	C₈×C₂₇₁	C₂×C₈	1	1	yes	characteristic
39	C₈×C₂₇₁	C₂×D₄	1	1	yes	characteristic
40	C₈×C₂₇₁	#41	1	1	yes	characteristic
41	C₂²×C₂₇₁	—	2	2	no	—
42	C₂²×C₂₇₁	#131	1	1	yes	characteristic
43	C₄×C₂₇₁	—	2	2	no	—
44	C₄×C₂₇₁	—	2	2	no	—
45	C₄×C₂₇₁	C₄×D₄	1	1	yes	characteristic
46	C₄×C₂₇₁	C₂²×C₈	1	1	yes	characteristic
47	C₂×C₂₇₁	—	2	1	no	—
48	C₂×C₂₇₁	—	4	2	no	—
49	C₂×C₂₇₁	C₈×D₄	1	1	yes	characteristic
50	C₂₇₁	#2170	1	1	yes	characteristic, direct product factor, Sylow
51	#2170	C₂₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
52	#504	C₂×C₂₇₁	1	2	yes	complemented
53	#443	C₂×C₂₇₁	1	1	yes	characteristic
54	#364	C₂×C₂₇₁	1	1	yes	characteristic, complemented
55	#350	C₂×C₂₇₁	1	2	yes	complemented
56	C₄×C₁₆	C₂×C₂₇₁	1	1	yes	characteristic, complemented
57	#135	C₄×C₂₇₁	1	1	yes	characteristic
58	#131	C₂²×C₂₇₁	1	2	yes	—
59	M₃₂	—	2	2	no	—
60	M₃₂	C₂²×C₂₇₁	1	2	yes	—
61	C₂×C₁₆	—	2	2	no	—
62	C₂×C₁₆	C₂²×C₂₇₁	1	1	yes	characteristic
63	C₂×C₁₆	C₂²×C₂₇₁	1	1	yes	characteristic
64	#108	C₄×C₂₇₁	1	1	yes	characteristic
65	#104	C₄×C₂₇₁	1	2	yes	—
66	C₄×C₈	C₂²×C₂₇₁	1	1	yes	characteristic
67	#41	C₂×C₄×C₂₇₁	1	2	yes	—
68	Q₁₆	C₈×C₂₇₁	1	1	yes	characteristic
69	SD₁₆	C₈×C₂₇₁	1	2	yes	—
70	D₈	C₈×C₂₇₁	1	1	yes	characteristic
71	M₁₆	—	2	2	no	—
72	M₁₆	C₂×C₄×C₂₇₁	1	2	yes	—
73	C₂×C₈	—	2	2	no	—
74	C₂×C₈	C₂×C₄×C₂₇₁	1	1	yes	characteristic
75	C₂×C₈	C₂³×C₂₇₁	1	1	yes	characteristic
76	C₄²	C₂×C₄×C₂₇₁	1	1	yes	characteristic
77	C₁₆	—	2	2	no	—
78	C₁₆	—	2	1	no	—
79	C₁₆	D₄×C₂₇₁	1	1	yes	characteristic
80	C₁₆	D₄×C₂₇₁	1	1	yes	characteristic
81	Q₈	C₂×C₈×C₂₇₁	1	2	yes	—
82	D₄	—	2	2	no	—
83	D₄	C₂×C₈×C₂₇₁	1	2	yes	—
84	C₂×C₄	—	2	1	no	—
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	C₂²×C₄×C₂₇₁	1	1	yes	characteristic
87	C₈	—	2	2	no	—
88	C₈	C₂×C₈×C₂₇₁	1	1	yes	characteristic
89	C₈	C₂×C₈×C₂₇₁	1	1	yes	characteristic
90	C₈	C₂×D₄×C₂₇₁	1	1	yes	characteristic
91	C₈	#41×C₂₇₁	1	1	yes	characteristic
92	C₂²	—	2	2	no	—
93	C₂²	#131×C₂₇₁	1	1	yes	characteristic
94	C₄	—	2	2	no	—
95	C₄	—	2	2	no	—
96	C₄	C₄×D₄×C₂₇₁	1	1	yes	characteristic
97	C₄	C₂²×C₈×C₂₇₁	1	1	yes	characteristic
98	C₂	—	2	1	no	—
99	C₂	—	4	2	no	—
100	C₂	C₈×D₄×C₂₇₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#504×C₂₇₁

C₂

yes

maximal, complemented

#443×C₂₇₁

C₂

yes

maximal, characteristic

#364×C₂₇₁

C₂

yes

maximal, characteristic, complemented

#350×C₂₇₁

C₂

yes

maximal, complemented

C₄×C₁₆×C₂₇₁

C₂

yes

maximal, characteristic, complemented

#135×C₂₇₁

C₄

yes

characteristic

#131×C₂₇₁

C₂²

yes

—

M₃₂×C₂₇₁

—

M₃₂×C₂₇₁

C₂²

yes

—

C₂×C₁₆×C₂₇₁

—

C₂×C₁₆×C₂₇₁

C₂²

yes

characteristic

C₂×C₁₆×C₂₇₁

C₂²

yes

characteristic

#108×C₂₇₁

C₄

yes

characteristic

#104×C₂₇₁

C₄

yes

—

C₄×C₈×C₂₇₁

C₂²

yes

characteristic

#41×C₂₇₁

C₂×C₄

yes

—

Q₁₆×C₂₇₁

C₈

yes

characteristic

SD₁₆×C₂₇₁

C₈

yes

—

D₈×C₂₇₁

C₈

yes

characteristic

M₁₆×C₂₇₁

—

M₁₆×C₂₇₁

C₂×C₄

yes

—

C₂×C₈×C₂₇₁

—

C₂×C₈×C₂₇₁

C₂×C₄

yes

characteristic

C₂×C₈×C₂₇₁

C₂³

yes

characteristic

C₄²×C₂₇₁

C₂×C₄

yes

characteristic

C₁₆×C₂₇₁

—

C₁₆×C₂₇₁

—

C₁₆×C₂₇₁

D₄

yes

characteristic

C₁₆×C₂₇₁

D₄

yes

characteristic

Q₈×C₂₇₁

C₂×C₈

yes

—

D₄×C₂₇₁

—

D₄×C₂₇₁

C₂×C₈

yes

—

C₂×C₄×C₂₇₁

—

C₂×C₄×C₂₇₁

—

C₂×C₄×C₂₇₁

C₂²×C₄

yes

characteristic

C₈×C₂₇₁

—

C₈×C₂₇₁

C₂×C₈

yes

characteristic

C₈×C₂₇₁

C₂×C₈

yes

characteristic

C₈×C₂₇₁

C₂×D₄

yes

characteristic

C₈×C₂₇₁

#41

yes

characteristic

C₂²×C₂₇₁

—

C₂²×C₂₇₁

#131

yes

characteristic

C₄×C₂₇₁

—

C₄×C₂₇₁

—

C₄×C₂₇₁

C₄×D₄

yes

characteristic

C₄×C₂₇₁

C₂²×C₈

yes

characteristic

C₂×C₂₇₁

—

C₂×C₂₇₁

—

C₂×C₂₇₁

C₈×D₄

yes

characteristic

C₂₇₁

#2170

yes

characteristic, direct product factor, Sylow

#2170

C₂₇₁

yes

maximal, characteristic, direct product factor, Sylow

#504

C₂×C₂₇₁

yes

complemented

#443

C₂×C₂₇₁

yes

characteristic

#364

C₂×C₂₇₁

yes

characteristic, complemented

#350

C₂×C₂₇₁

yes

complemented

C₄×C₁₆

C₂×C₂₇₁

yes

characteristic, complemented

#135

C₄×C₂₇₁

yes

characteristic

#131

C₂²×C₂₇₁

yes

—

M₃₂

—

M₃₂

C₂²×C₂₇₁

yes

—

C₂×C₁₆

—

C₂×C₁₆

C₂²×C₂₇₁

yes

characteristic

C₂×C₁₆

C₂²×C₂₇₁

yes

characteristic

#108

C₄×C₂₇₁

yes

characteristic

#104

C₄×C₂₇₁

yes

—

C₄×C₈

C₂²×C₂₇₁

yes

characteristic

#41

C₂×C₄×C₂₇₁

yes

—

Q₁₆

C₈×C₂₇₁

yes

characteristic

SD₁₆

C₈×C₂₇₁

yes

—

D₈

C₈×C₂₇₁

yes

characteristic

M₁₆

—

M₁₆

C₂×C₄×C₂₇₁

yes

—

C₂×C₈

—

C₂×C₈

C₂×C₄×C₂₇₁

yes

characteristic

C₂×C₈

C₂³×C₂₇₁

yes

characteristic

C₄²

C₂×C₄×C₂₇₁

yes

characteristic

C₁₆

—

C₁₆

—

C₁₆

D₄×C₂₇₁

yes

characteristic

C₁₆

D₄×C₂₇₁

yes

characteristic

Q₈

C₂×C₈×C₂₇₁

yes

—

D₄

—

D₄

C₂×C₈×C₂₇₁

yes

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²×C₄×C₂₇₁

yes

characteristic

C₈

—

C₈

C₂×C₈×C₂₇₁

yes

characteristic

C₈

C₂×C₈×C₂₇₁

yes

characteristic

C₈

C₂×D₄×C₂₇₁

yes

characteristic

C₈

#41×C₂₇₁

yes

characteristic

C₂²

—

C₂²

#131×C₂₇₁

yes

characteristic

C₄

—

C₄

—

C₄

C₄×D₄×C₂₇₁

yes

characteristic

C₄

C₂²×C₈×C₂₇₁

yes

characteristic

C₂

—

C₂

—

100

C₂

C₈×D₄×C₂₇₁

yes

characteristic

	subgroup	G/H
center	C₁₆×C₂₇₁	D₄
commutator	C₄	C₂²×C₈×C₂₇₁
Frattini, cosocle	C₂×C₈	C₂³×C₂₇₁
socle	C₂×C₂₇₁	C₈×D₄
prime radical	#135×C₂₇₁	C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₆×C₂₇₁ ◁ C₂×C₁₆×C₂₇₁ ◁ G

subgroup

G/H

center

C₁₆×C₂₇₁

D₄

commutator

C₄

C₂²×C₈×C₂₇₁

Frattini, cosocle

C₂×C₈

C₂³×C₂₇₁

socle

C₂×C₂₇₁

C₈×D₄

prime radical

#135×C₂₇₁

C₄

LCS

G ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₁₆×C₂₇₁ ◁ C₂×C₁₆×C₂₇₁ ◁ G

N	H ≅ G/N	type
1. #504×C₂₇₁	C₂	semidirect
3. #364×C₂₇₁	C₂	semidirect
4. #350×C₂₇₁	C₂	semidirect
5. C₄×C₁₆×C₂₇₁	C₂	semidirect
50. C₂₇₁	#2170	direct (N × H)
51. #2170	C₂₇₁	direct (N × H)
52. #504	C₂×C₂₇₁	semidirect
54. #364	C₂×C₂₇₁	semidirect
55. #350	C₂×C₂₇₁	semidirect
56. C₄×C₁₆	C₂×C₂₇₁	semidirect

H ≅ G/N

type

1. #504×C₂₇₁

C₂

semidirect

3. #364×C₂₇₁

C₂

semidirect

4. #350×C₂₇₁

C₂

semidirect

5. C₄×C₁₆×C₂₇₁

C₂

semidirect

50. C₂₇₁

#2170

direct (N × H)

51. #2170

C₂₇₁

direct (N × H)

52. #504

C₂×C₂₇₁

semidirect

54. #364

C₂×C₂₇₁

semidirect

55. #350

C₂×C₂₇₁

semidirect

56. C₄×C₁₆

C₂×C₂₇₁

semidirect

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

subgroup	index
#504×C₂₇₁	2
#443×C₂₇₁	2
#364×C₂₇₁	2
#350×C₂₇₁	2
C₄×C₁₆×C₂₇₁	2
#2170	271

subgroup

index

#504×C₂₇₁

#443×C₂₇₁

#364×C₂₇₁

#350×C₂₇₁

C₄×C₁₆×C₂₇₁

#2170

271

Subgroups — #2170×C₂₇₁ #46832756

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2170×C271 #46832756

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2170×C₂₇₁ #46832756