Group #46838119 — subgroups

Subgroups — #46838119

Order 34688.

172 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 97 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×#22576100	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×Dic₂₇₁	C₂	1	1	yes	maximal, characteristic
3	Dic₂₇₁⋊C₁₆	C₂	1	4	yes	maximal, complemented
4	C₂×M₃₂×C₂₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₂₇₁⋊C₈	C₄	1	1	yes	characteristic
6	C₂×C₄×Dic₂₇₁	C₄	1	1	yes	characteristic
7	#22576100	—	2	2	no	—
8	C₂×C₂₇₁⋊C₁₆	C₂²	1	2	yes	—
9	C₈×Dic₂₇₁	C₄	1	2	yes	—
10	C₈×Dic₂₇₁	C₂²	1	2	yes	—
11	C₂²×C₈×C₂₇₁	C₂²	1	1	yes	characteristic
12	M₃₂×C₂₇₁	—	2	2	no	—
13	C₂×C₁₆×C₂₇₁	C₂²	1	2	yes	—
14	C₂²×Dic₂₇₁	C₈	1	2	yes	—
15	C₂×C₂₇₁⋊C₈	—	2	2	no	—
16	C₂×C₂₇₁⋊C₈	C₂×C₄	1	2	yes	—
17	C₄×Dic₂₇₁	C₈	1	1	yes	characteristic
18	C₄×Dic₂₇₁	C₈	1	1	yes	characteristic
19	C₄×Dic₂₇₁	C₂×C₄	1	2	yes	—
20	C₂₇₁⋊C₁₆	—	2	2	no	—
21	C₂²×C₄×C₂₇₁	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×C₂₇₁	C₂×C₄	1	1	yes	characteristic
23	C₂×C₈×C₂₇₁	D₄	1	2	yes	—
24	C₂×C₈×C₂₇₁	Q₈	1	2	yes	—
25	C₂×C₈×C₂₇₁	C₂³	1	1	yes	characteristic
26	C₁₆×C₂₇₁	—	2	2	no	—
27	C₂×Dic₂₇₁	—	2	4	no	—
28	C₂×Dic₂₇₁	C₂×C₈	1	2	yes	—
29	C₂×Dic₂₇₁	C₂×C₈	1	2	yes	—
30	C₂₇₁⋊C₈	—	2	2	no	—
31	C₂³×C₂₇₁	C₂×C₈	1	1	yes	characteristic
32	C₂×C₄×C₂₇₁	C₄⋊C₄	1	2	yes	—
33	C₂×C₄×C₂₇₁	C₂×C₈	1	1	yes	characteristic
34	C₂×C₄×C₂₇₁	M₁₆	1	2	yes	—
35	C₂×C₄×C₂₇₁	C₂²×C₄	1	1	yes	characteristic
36	C₈×C₂₇₁	C₄⋊C₄	1	2	yes	—
37	C₈×C₂₇₁	C₂×D₄	1	1	yes	characteristic
38	C₈×C₂₇₁	C₂×Q₈	1	1	yes	characteristic
39	C₂²×C₂₇₁	—	2	2	no	—
40	C₂²×C₂₇₁	C₄⋊C₈	1	2	yes	—
41	C₂²×C₂₇₁	C₂²×C₈	1	1	yes	characteristic
42	C₄×C₂₇₁	C₄⋊C₈	1	2	yes	—
43	C₄×C₂₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₄×C₂₇₁	C₂×M₁₆	1	1	yes	characteristic
45	Dic₂₇₁	—	2	2	no	—
46	Dic₂₇₁	M₃₂	1	4	yes	complemented
47	C₂×C₂₇₁	—	2	2	no	—
48	C₂×C₂₇₁	C₂×C₄⋊C₈	1	1	yes	characteristic
49	C₂×C₂₇₁	C₂×M₃₂	1	2	yes	—
50	C₂₇₁	#2150	1	1	yes	characteristic, complemented, Sylow
51	#2150	—	271	1	no	maximal, Sylow
52	C₂×M₃₂	—	271	1	no	—
53	C₂×M₃₂	D₂₇₁	1	1	yes	characteristic
54	C₂×C₄×C₈	—	271	1	no	—
55	C₄⋊C₁₆	—	271	4	no	—
56	C₂²×C₈	—	271	1	no	—
57	C₂²×C₈	C₂×D₂₇₁	1	1	yes	characteristic
58	C₂×C₄²	—	271	1	no	—
59	M₃₂	—	2	2	no	—
60	M₃₂	—	542	2	no	—
61	C₂×C₁₆	—	271	2	no	—
62	C₂×C₁₆	C₂×D₂₇₁	1	2	yes	—
63	C₄×C₈	—	271	2	no	—
64	C₄×C₈	—	271	2	no	—
65	C₂²×C₄	—	271	2	no	—
66	C₂²×C₄	C₄×D₂₇₁	1	1	yes	characteristic
67	C₂×C₈	—	271	2	no	—
68	C₂×C₈	—	542	2	no	—
69	C₂×C₈	C₄×D₂₇₁	1	1	yes	characteristic
70	C₂×C₈	C₂²⋊D₂₇₁	1	2	yes	—
71	C₂×C₈	Dic₅₄₂	1	2	yes	—
72	C₂×C₈	C₂²×D₂₇₁	1	1	yes	characteristic
73	C₄²	—	271	1	no	—
74	C₄²	—	271	1	no	—
75	C₄²	—	271	2	no	—
76	C₁₆	—	2	2	no	—
77	C₁₆	—	542	2	no	—
78	C₂³	C₈×D₂₇₁	1	1	yes	characteristic
79	C₂×C₄	—	271	2	no	—
80	C₂×C₄	—	271	2	no	—
81	C₂×C₄	—	542	4	no	—
82	C₂×C₄	Dic₂₇₁⋊C₄	1	2	yes	—
83	C₂×C₄	C₈×D₂₇₁	1	1	yes	characteristic
84	C₂×C₄	#16864637	1	2	yes	—
85	C₂×C₄	C₂×C₄×D₂₇₁	1	1	yes	characteristic
86	C₈	—	542	2	no	—
87	C₈	Dic₂₇₁⋊C₄	1	2	yes	—
88	C₈	C₂×C₂²⋊D₂₇₁	1	1	yes	characteristic
89	C₈	C₂×Dic₅₄₂	1	1	yes	characteristic
90	C₂²	—	2	2	no	—
91	C₂²	Dic₂₇₁⋊C₈	1	2	yes	—
92	C₂²	C₂×C₈×D₂₇₁	1	1	yes	characteristic
93	C₄	—	271	4	no	—
94	C₄	—	542	2	no	—
95	C₄	Dic₂₇₁⋊C₈	1	2	yes	—
96	C₄	C₂×Dic₂₇₁⋊C₄	1	1	yes	characteristic
97	C₄	C₂×#16864637	1	1	yes	characteristic
98	C₂	—	2	2	no	—
99	C₂	C₂×Dic₂₇₁⋊C₈	1	1	yes	characteristic
100	C₂	M₃₂×D₂₇₁	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22576100

C₂

yes

maximal, characteristic

C₂×C₈×Dic₂₇₁

C₂

yes

maximal, characteristic

Dic₂₇₁⋊C₁₆

C₂

yes

maximal, complemented

C₂×M₃₂×C₂₇₁

C₂

yes

maximal, characteristic

C₂²×C₂₇₁⋊C₈

C₄

yes

characteristic

C₂×C₄×Dic₂₇₁

C₄

yes

characteristic

#22576100

—

C₂×C₂₇₁⋊C₁₆

C₂²

yes

—

C₈×Dic₂₇₁

C₄

yes

—

C₈×Dic₂₇₁

C₂²

yes

—

C₂²×C₈×C₂₇₁

C₂²

yes

characteristic

M₃₂×C₂₇₁

—

C₂×C₁₆×C₂₇₁

C₂²

yes

—

C₂²×Dic₂₇₁

C₈

yes

—

C₂×C₂₇₁⋊C₈

—

C₂×C₂₇₁⋊C₈

C₂×C₄

yes

—

C₄×Dic₂₇₁

C₈

yes

characteristic

C₄×Dic₂₇₁

C₈

yes

characteristic

C₄×Dic₂₇₁

C₂×C₄

yes

—

C₂₇₁⋊C₁₆

—

C₂²×C₄×C₂₇₁

C₂×C₄

yes

characteristic

C₂×C₈×C₂₇₁

C₂×C₄

yes

characteristic

C₂×C₈×C₂₇₁

D₄

yes

—

C₂×C₈×C₂₇₁

Q₈

yes

—

C₂×C₈×C₂₇₁

C₂³

yes

characteristic

C₁₆×C₂₇₁

—

C₂×Dic₂₇₁

—

C₂×Dic₂₇₁

C₂×C₈

yes

—

C₂×Dic₂₇₁

C₂×C₈

yes

—

C₂₇₁⋊C₈

—

C₂³×C₂₇₁

C₂×C₈

yes

characteristic

C₂×C₄×C₂₇₁

C₄⋊C₄

yes

—

C₂×C₄×C₂₇₁

C₂×C₈

yes

characteristic

C₂×C₄×C₂₇₁

M₁₆

yes

—

C₂×C₄×C₂₇₁

C₂²×C₄

yes

characteristic

C₈×C₂₇₁

C₄⋊C₄

yes

—

C₈×C₂₇₁

C₂×D₄

yes

characteristic

C₈×C₂₇₁

C₂×Q₈

yes

characteristic

C₂²×C₂₇₁

—

C₂²×C₂₇₁

C₄⋊C₈

yes

—

C₂²×C₂₇₁

C₂²×C₈

yes

characteristic

C₄×C₂₇₁

C₄⋊C₈

yes

—

C₄×C₂₇₁

C₂×C₄⋊C₄

yes

characteristic

C₄×C₂₇₁

C₂×M₁₆

yes

characteristic

Dic₂₇₁

—

Dic₂₇₁

M₃₂

yes

complemented

C₂×C₂₇₁

—

C₂×C₂₇₁

C₂×C₄⋊C₈

yes

characteristic

C₂×C₂₇₁

C₂×M₃₂

yes

—

C₂₇₁

#2150

yes

characteristic, complemented, Sylow

#2150

—

271

maximal, Sylow

C₂×M₃₂

—

271

—

C₂×M₃₂

D₂₇₁

yes

characteristic

C₂×C₄×C₈

—

271

—

C₄⋊C₁₆

—

271

—

C₂²×C₈

—

271

—

C₂²×C₈

C₂×D₂₇₁

yes

characteristic

C₂×C₄²

—

271

—

M₃₂

—

M₃₂

—

542

—

C₂×C₁₆

—

271

—

C₂×C₁₆

C₂×D₂₇₁

yes

—

C₄×C₈

—

271

—

C₄×C₈

—

271

—

C₂²×C₄

—

271

—

C₂²×C₄

C₄×D₂₇₁

yes

characteristic

C₂×C₈

—

271

—

C₂×C₈

—

542

—

C₂×C₈

C₄×D₂₇₁

yes

characteristic

C₂×C₈

C₂²⋊D₂₇₁

yes

—

C₂×C₈

Dic₅₄₂

yes

—

C₂×C₈

C₂²×D₂₇₁

yes

characteristic

C₄²

—

271

—

C₄²

—

271

—

C₄²

—

271

—

C₁₆

—

C₁₆

—

542

—

C₂³

C₈×D₂₇₁

yes

characteristic

C₂×C₄

—

271

—

C₂×C₄

—

271

—

C₂×C₄

—

542

—

C₂×C₄

Dic₂₇₁⋊C₄

yes

—

C₂×C₄

C₈×D₂₇₁

yes

characteristic

C₂×C₄

#16864637

yes

—

C₂×C₄

C₂×C₄×D₂₇₁

yes

characteristic

C₈

—

542

—

C₈

Dic₂₇₁⋊C₄

yes

—

C₈

C₂×C₂²⋊D₂₇₁

yes

characteristic

C₈

C₂×Dic₅₄₂

yes

characteristic

C₂²

—

C₂²

Dic₂₇₁⋊C₈

yes

—

C₂²

C₂×C₈×D₂₇₁

yes

characteristic

C₄

—

271

—

C₄

—

542

—

C₄

Dic₂₇₁⋊C₈

yes

—

C₄

C₂×Dic₂₇₁⋊C₄

yes

characteristic

C₄

C₂×#16864637

yes

characteristic

C₂

—

C₂

C₂×Dic₂₇₁⋊C₈

yes

characteristic

100

C₂

M₃₂×D₂₇₁

yes

—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₂₇₁
commutator, socle	C₂²×C₂₇₁	C₂²×C₈
Fitting	C₂×M₃₂×C₂₇₁	C₂
prime radical	C₂³×C₂₇₁	C₂×C₈
cosocle	C₂×C₈×C₂₇₁	C₂³
hypercenter	C₂×M₃₂	D₂₇₁
hypocenter	C₂₇₁	#2150
LCS	G ▷ C₂²×C₂₇₁ ▷ C₂₇₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₂₇₁

commutator, socle

C₂²×C₂₇₁

C₂²×C₈

Fitting

C₂×M₃₂×C₂₇₁

C₂

prime radical

C₂³×C₂₇₁

C₂×C₈

cosocle

C₂×C₈×C₂₇₁

C₂³

hypercenter

C₂×M₃₂

D₂₇₁

C₂₇₁

G ▷ C₂²×C₂₇₁ ▷ C₂₇₁

C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

N	H ≅ G/N	type
3. Dic₂₇₁⋊C₁₆	C₂	semidirect
46. Dic₂₇₁	M₃₂	semidirect
50. C₂₇₁	#2150	semidirect

H ≅ G/N

type

3. Dic₂₇₁⋊C₁₆

C₂

semidirect

46. Dic₂₇₁

M₃₂

semidirect

50. C₂₇₁

#2150

semidirect

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

subgroup	index
C₂×#22576100	2
C₂×C₈×Dic₂₇₁	2
Dic₂₇₁⋊C₁₆	2
C₂×M₃₂×C₂₇₁	2
#2150	271

subgroup

index

C₂×#22576100

C₂×C₈×Dic₂₇₁

Dic₂₇₁⋊C₁₆

C₂×M₃₂×C₂₇₁

#2150

271

Subgroups — #46838119

Conjugacy classes of subgroups

Complements

Maximal subgroups