Group #37529317 — subgroups

Subgroups — #1374×C₁₈₁ #37529317

Order 23168.

160 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 112 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	C₂²×C₁₆×C₁₈₁	C₂	1	2	yes	maximal
2	C₂×C₄×C₈×C₁₈₁	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₈×C₁₈₁	C₄	1	1	yes	characteristic
4	C₂²×C₈×C₁₈₁	C₂²	1	1	yes	characteristic
5	C₂×C₄²×C₁₈₁	C₄	1	1	yes	characteristic
6	C₂×C₁₆×C₁₈₁	—	2	2	no	—
7	C₂×C₁₆×C₁₈₁	—	2	2	no	—
8	C₂×C₁₆×C₁₈₁	C₄	1	4	yes	complemented
9	C₄×C₈×C₁₈₁	—	2	2	no	—
10	C₂²×C₄×C₁₈₁	C₈	1	2	yes	—
11	C₂²×C₄×C₁₈₁	C₂×C₄	1	1	yes	characteristic
12	C₂×C₈×C₁₈₁	—	2	1	no	—
13	C₂×C₈×C₁₈₁	—	2	1	no	—
14	C₂×C₈×C₁₈₁	C₈	1	2	yes	—
15	C₂×C₈×C₁₈₁	C₂×C₄	1	2	yes	—
16	C₂×C₈×C₁₈₁	D₄	1	1	yes	characteristic
17	C₂×C₈×C₁₈₁	D₄	1	2	yes	—
18	C₂×C₈×C₁₈₁	Q₈	1	1	yes	characteristic
19	C₄²×C₁₈₁	—	2	1	no	—
20	C₄²×C₁₈₁	—	2	1	no	—
21	C₁₆×C₁₈₁	—	2	4	no	—
22	C₂³×C₁₈₁	C₂×C₈	1	1	yes	characteristic
23	C₂×C₄×C₁₈₁	—	2	2	no	—
24	C₂×C₄×C₁₈₁	—	2	2	no	—
25	C₂×C₄×C₁₈₁	C₁₆	1	4	yes	complemented
26	C₂×C₄×C₁₈₁	C₄²	1	1	yes	characteristic
27	C₂×C₄×C₁₈₁	C₂²⋊C₄	1	1	yes	characteristic
28	C₂×C₄×C₁₈₁	C₄⋊C₄	1	1	yes	characteristic
29	C₂×C₄×C₁₈₁	C₂×C₈	1	1	yes	characteristic
30	C₂×C₄×C₁₈₁	M₁₆	1	1	yes	characteristic
31	C₂×C₄×C₁₈₁	M₁₆	1	1	yes	characteristic
32	C₈×C₁₈₁	—	2	2	no	—
33	C₈×C₁₈₁	C₂²⋊C₄	1	2	yes	—
34	C₈×C₁₈₁	C₄⋊C₄	1	2	yes	—
35	C₂²×C₁₈₁	C₄×C₈	1	1	yes	characteristic
36	C₂²×C₁₈₁	C₂²⋊C₈	1	1	yes	characteristic
37	C₂²×C₁₈₁	C₄⋊C₈	1	1	yes	characteristic
38	C₂²×C₁₈₁	C₂×C₁₆	1	2	yes	—
39	C₂²×C₁₈₁	M₃₂	1	2	yes	—
40	C₄×C₁₈₁	—	2	4	no	—
41	C₄×C₁₈₁	#95	1	1	yes	characteristic
42	C₄×C₁₈₁	#97	1	1	yes	characteristic
43	C₄×C₁₈₁	C₂²⋊C₈	1	1	yes	characteristic
44	C₄×C₁₈₁	C₄⋊C₈	1	1	yes	characteristic
45	C₂×C₁₈₁	#336	1	1	yes	characteristic
46	C₂×C₁₈₁	C₄×C₁₆	1	1	yes	characteristic
47	C₂×C₁₈₁	#346	1	1	yes	characteristic
48	C₂×C₁₈₁	C₂²⋊C₁₆	1	2	yes	—
49	C₂×C₁₈₁	C₄⋊C₁₆	1	2	yes	—
50	C₁₈₁	#1374	1	1	yes	characteristic, direct product factor, Sylow
51	#1374	C₁₈₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
52	C₂²×C₁₆	C₂×C₁₈₁	1	2	yes	—
53	C₂×C₄×C₈	C₂×C₁₈₁	1	1	yes	characteristic
54	C₂²×C₈	C₄×C₁₈₁	1	1	yes	characteristic
55	C₂²×C₈	C₂²×C₁₈₁	1	1	yes	characteristic
56	C₂×C₄²	C₄×C₁₈₁	1	1	yes	characteristic
57	C₂×C₁₆	—	2	2	no	—
58	C₂×C₁₆	—	2	2	no	—
59	C₂×C₁₆	C₄×C₁₈₁	1	4	yes	complemented
60	C₄×C₈	—	2	2	no	—
61	C₂²×C₄	C₈×C₁₈₁	1	2	yes	—
62	C₂²×C₄	C₂×C₄×C₁₈₁	1	1	yes	characteristic
63	C₂×C₈	—	2	1	no	—
64	C₂×C₈	—	2	1	no	—
65	C₂×C₈	C₈×C₁₈₁	1	2	yes	—
66	C₂×C₈	C₂×C₄×C₁₈₁	1	2	yes	—
67	C₂×C₈	D₄×C₁₈₁	1	1	yes	characteristic
68	C₂×C₈	D₄×C₁₈₁	1	2	yes	—
69	C₂×C₈	Q₈×C₁₈₁	1	1	yes	characteristic
70	C₄²	—	2	1	no	—
71	C₄²	—	2	1	no	—
72	C₁₆	—	2	4	no	—
73	C₂³	C₂×C₈×C₁₈₁	1	1	yes	characteristic
74	C₂×C₄	—	2	2	no	—
75	C₂×C₄	—	2	2	no	—
76	C₂×C₄	C₁₆×C₁₈₁	1	4	yes	complemented
77	C₂×C₄	C₄²×C₁₈₁	1	1	yes	characteristic
78	C₂×C₄	C₂²⋊C₄×C₁₈₁	1	1	yes	characteristic
79	C₂×C₄	C₄⋊C₄×C₁₈₁	1	1	yes	characteristic
80	C₂×C₄	C₂×C₈×C₁₈₁	1	1	yes	characteristic
81	C₂×C₄	M₁₆×C₁₈₁	1	1	yes	characteristic
82	C₂×C₄	M₁₆×C₁₈₁	1	1	yes	characteristic
83	C₈	—	2	2	no	—
84	C₈	C₂²⋊C₄×C₁₈₁	1	2	yes	—
85	C₈	C₄⋊C₄×C₁₈₁	1	2	yes	—
86	C₂²	C₄×C₈×C₁₈₁	1	1	yes	characteristic
87	C₂²	C₂²⋊C₈×C₁₈₁	1	1	yes	characteristic
88	C₂²	C₄⋊C₈×C₁₈₁	1	1	yes	characteristic
89	C₂²	C₂×C₁₆×C₁₈₁	1	2	yes	—
90	C₂²	M₃₂×C₁₈₁	1	2	yes	—
91	C₄	—	2	4	no	—
92	C₄	#95×C₁₈₁	1	1	yes	characteristic
93	C₄	#97×C₁₈₁	1	1	yes	characteristic
94	C₄	C₂²⋊C₈×C₁₈₁	1	1	yes	characteristic
95	C₄	C₄⋊C₈×C₁₈₁	1	1	yes	characteristic
96	C₂	#336×C₁₈₁	1	1	yes	characteristic
97	C₂	C₄×C₁₆×C₁₈₁	1	1	yes	characteristic
98	C₂	#346×C₁₈₁	1	1	yes	characteristic
99	C₂	C₂²⋊C₁₆×C₁₈₁	1	2	yes	—
100	C₂	C₄⋊C₁₆×C₁₈₁	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₁₆×C₁₈₁

C₂

yes

maximal

C₂×C₄×C₈×C₁₈₁

C₂

yes

maximal, characteristic

C₂²×C₈×C₁₈₁

C₄

yes

characteristic

C₂²×C₈×C₁₈₁

C₂²

yes

characteristic

C₂×C₄²×C₁₈₁

C₄

yes

characteristic

C₂×C₁₆×C₁₈₁

—

C₂×C₁₆×C₁₈₁

—

C₂×C₁₆×C₁₈₁

C₄

yes

complemented

C₄×C₈×C₁₈₁

—

C₂²×C₄×C₁₈₁

C₈

yes

—

C₂²×C₄×C₁₈₁

C₂×C₄

yes

characteristic

C₂×C₈×C₁₈₁

—

C₂×C₈×C₁₈₁

—

C₂×C₈×C₁₈₁

C₈

yes

—

C₂×C₈×C₁₈₁

C₂×C₄

yes

—

C₂×C₈×C₁₈₁

D₄

yes

characteristic

C₂×C₈×C₁₈₁

D₄

yes

—

C₂×C₈×C₁₈₁

Q₈

yes

characteristic

C₄²×C₁₈₁

—

C₄²×C₁₈₁

—

C₁₆×C₁₈₁

—

C₂³×C₁₈₁

C₂×C₈

yes

characteristic

C₂×C₄×C₁₈₁

—

C₂×C₄×C₁₈₁

—

C₂×C₄×C₁₈₁

C₁₆

yes

complemented

C₂×C₄×C₁₈₁

C₄²

yes

characteristic

C₂×C₄×C₁₈₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₈₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₈₁

C₂×C₈

yes

characteristic

C₂×C₄×C₁₈₁

M₁₆

yes

characteristic

C₂×C₄×C₁₈₁

M₁₆

yes

characteristic

C₈×C₁₈₁

—

C₈×C₁₈₁

C₂²⋊C₄

yes

—

C₈×C₁₈₁

C₄⋊C₄

yes

—

C₂²×C₁₈₁

C₄×C₈

yes

characteristic

C₂²×C₁₈₁

C₂²⋊C₈

yes

characteristic

C₂²×C₁₈₁

C₄⋊C₈

yes

characteristic

C₂²×C₁₈₁

C₂×C₁₆

yes

—

C₂²×C₁₈₁

M₃₂

yes

—

C₄×C₁₈₁

—

C₄×C₁₈₁

yes

C₄×C₁₈₁

yes

C₄×C₁₈₁

C₂²⋊C₈

yes

characteristic

C₄×C₁₈₁

C₄⋊C₈

yes

characteristic

C₂×C₁₈₁

#336

yes

characteristic

C₂×C₁₈₁

C₄×C₁₆

yes

characteristic

C₂×C₁₈₁

#346

yes

characteristic

C₂×C₁₈₁

C₂²⋊C₁₆

yes

—

C₂×C₁₈₁

C₄⋊C₁₆

yes

—

C₁₈₁

#1374

yes

characteristic, direct product factor, Sylow

#1374

C₁₈₁

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₁₆

C₂×C₁₈₁

yes

—

C₂×C₄×C₈

C₂×C₁₈₁

yes

characteristic

C₂²×C₈

C₄×C₁₈₁

yes

characteristic

C₂²×C₈

C₂²×C₁₈₁

yes

characteristic

C₂×C₄²

C₄×C₁₈₁

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₄×C₁₈₁

yes

complemented

C₄×C₈

—

C₂²×C₄

C₈×C₁₈₁

yes

—

C₂²×C₄

C₂×C₄×C₁₈₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₈×C₁₈₁

yes

—

C₂×C₈

C₂×C₄×C₁₈₁

yes

—

C₂×C₈

D₄×C₁₈₁

yes

characteristic

C₂×C₈

D₄×C₁₈₁

yes

—

C₂×C₈

Q₈×C₁₈₁

yes

characteristic

C₄²

—

C₄²

—

C₁₆

—

C₂³

C₂×C₈×C₁₈₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₁₆×C₁₈₁

yes

complemented

C₂×C₄

C₄²×C₁₈₁

yes

characteristic

C₂×C₄

C₂²⋊C₄×C₁₈₁

yes

characteristic

C₂×C₄

C₄⋊C₄×C₁₈₁

yes

characteristic

C₂×C₄

C₂×C₈×C₁₈₁

yes

characteristic

C₂×C₄

M₁₆×C₁₈₁

yes

characteristic

C₂×C₄

M₁₆×C₁₈₁

yes

characteristic

C₈

—

C₈

C₂²⋊C₄×C₁₈₁

yes

—

C₈

C₄⋊C₄×C₁₈₁

yes

—

C₂²

C₄×C₈×C₁₈₁

yes

characteristic

C₂²

C₂²⋊C₈×C₁₈₁

yes

characteristic

C₂²

C₄⋊C₈×C₁₈₁

yes

characteristic

C₂²

C₂×C₁₆×C₁₈₁

yes

—

C₂²

M₃₂×C₁₈₁

yes

—

C₄

—

C₄

#95×C₁₈₁

yes

characteristic

C₄

#97×C₁₈₁

yes

characteristic

C₄

C₂²⋊C₈×C₁₈₁

yes

characteristic

C₄

C₄⋊C₈×C₁₈₁

yes

characteristic

C₂

#336×C₁₈₁

yes

characteristic

C₂

C₄×C₁₆×C₁₈₁

yes

characteristic

C₂

#346×C₁₈₁

yes

characteristic

C₂

C₂²⋊C₁₆×C₁₈₁

yes

—

100

C₂

C₄⋊C₁₆×C₁₈₁

yes

—

	subgroup	G/H
center	C₂²×C₈×C₁₈₁	C₂²
commutator	C₂	C₄×C₁₆×C₁₈₁
Frattini, cosocle	C₂²×C₈	C₂²×C₁₈₁
socle, prime radical	C₂³×C₁₈₁	C₂×C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈×C₁₈₁ ◁ G

subgroup

G/H

center

C₂²×C₈×C₁₈₁

C₂²

commutator

C₂

C₄×C₁₆×C₁₈₁

Frattini, cosocle

C₂²×C₈

C₂²×C₁₈₁

socle, prime radical

C₂³×C₁₈₁

C₂×C₈

LCS

G ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₈×C₁₈₁ ◁ G

N	H ≅ G/N	type
8. C₂×C₁₆×C₁₈₁	C₄	semidirect
25. C₂×C₄×C₁₈₁	C₁₆	semidirect
50. C₁₈₁	#1374	direct (N × H)
51. #1374	C₁₈₁	direct (N × H)
59. C₂×C₁₆	C₄×C₁₈₁	semidirect
76. C₂×C₄	C₁₆×C₁₈₁	semidirect

H ≅ G/N

type

8. C₂×C₁₆×C₁₈₁

C₄

semidirect

25. C₂×C₄×C₁₈₁

C₁₆

semidirect

50. C₁₈₁

#1374

direct (N × H)

51. #1374

C₁₈₁

direct (N × H)

59. C₂×C₁₆

C₄×C₁₈₁

semidirect

76. C₂×C₄

C₁₆×C₁₈₁

semidirect

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

subgroup	index
C₂²×C₁₆×C₁₈₁	2
C₂×C₄×C₈×C₁₈₁	2
#1374	181

subgroup

index

C₂²×C₁₆×C₁₈₁

C₂×C₄×C₈×C₁₈₁

#1374

181

Subgroups — #1374×C₁₈₁ #37529317

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1374×C181 #37529317

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1374×C₁₈₁ #37529317