Group #37529520 — subgroups

Subgroups — C₈×Q₁₆×C₁₈₁ #37529520

Order 23168.

164 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 164 conjugacy classes of subgroups — too many to draw. Only its 104 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	C₈×Q₈×C₁₈₁	C₂	1	2	yes	maximal
2	C₄×Q₁₆×C₁₈₁	C₂	1	1	yes	maximal, characteristic
3	#335×C₁₈₁	C₂	1	1	yes	maximal, characteristic
4	Q₈⋊C₈×C₁₈₁	C₂	1	2	yes	maximal
5	C₈²×C₁₈₁	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₁₆×C₁₈₁	C₄	1	1	yes	characteristic
7	C₄×Q₈×C₁₈₁	C₂²	1	2	yes	—
8	#107×C₁₈₁	C₄	1	1	yes	characteristic
9	C₄⋊C₈×C₁₈₁	—	2	2	no	—
10	C₄⋊C₈×C₁₈₁	C₂²	1	2	yes	—
11	#103×C₁₈₁	C₄	1	2	yes	—
12	C₄×C₈×C₁₈₁	—	2	2	no	—
13	C₄×C₈×C₁₈₁	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₁₈₁	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₁₈₁	C₂²	1	1	yes	characteristic
16	C₂×Q₈×C₁₈₁	C₂×C₄	1	2	yes	—
17	Q₁₆×C₁₈₁	C₈	1	4	yes	direct product factor
18	C₂×C₈×C₁₈₁	—	2	1	no	—
19	C₂×C₈×C₁₈₁	—	2	2	no	—
20	C₂×C₈×C₁₈₁	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×C₁₈₁	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×C₁₈₁	D₄	1	1	yes	characteristic
23	C₂×C₈×C₁₈₁	D₄	1	1	yes	characteristic
24	C₄⋊C₄×C₁₈₁	—	2	2	no	—
25	C₄⋊C₄×C₁₈₁	C₂×C₄	1	2	yes	—
26	C₄²×C₁₈₁	—	2	2	no	—
27	C₄²×C₁₈₁	C₂³	1	1	yes	characteristic
28	Q₈×C₁₈₁	—	2	2	no	—
29	Q₈×C₁₈₁	C₂×C₈	1	4	yes	—
30	C₂×C₄×C₁₈₁	—	2	2	no	—
31	C₂×C₄×C₁₈₁	—	2	2	no	—
32	C₂×C₄×C₁₈₁	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₁₈₁	#41	1	1	yes	characteristic
35	C₈×C₁₈₁	—	2	2	no	—
36	C₈×C₁₈₁	—	2	1	no	—
37	C₈×C₁₈₁	—	2	1	no	—
38	C₈×C₁₈₁	—	4	2	no	—
39	C₈×C₁₈₁	C₂×C₈	1	2	yes	—
40	C₈×C₁₈₁	Q₁₆	1	2	yes	direct product factor
41	C₂²×C₁₈₁	C₄×D₄	1	1	yes	characteristic
42	C₄×C₁₈₁	—	2	4	no	—
43	C₄×C₁₈₁	—	2	1	no	—
44	C₄×C₁₈₁	—	4	2	no	—
45	C₄×C₁₈₁	C₂²×C₈	1	1	yes	characteristic
46	C₄×C₁₈₁	#131	1	1	yes	characteristic
47	C₄×C₁₈₁	C₂×Q₁₆	1	1	yes	characteristic
48	C₄×C₁₈₁	#135	1	1	yes	characteristic
49	C₂×C₁₈₁	C₈×D₄	1	1	yes	characteristic
50	C₂×C₁₈₁	C₄×Q₁₆	1	1	yes	characteristic
51	C₂×C₁₈₁	#443	1	1	yes	characteristic
52	C₁₈₁	C₈×Q₁₆	1	1	yes	characteristic, direct product factor, Sylow
53	C₈×Q₁₆	C₁₈₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
54	C₈×Q₈	C₂×C₁₈₁	1	2	yes	—
55	C₄×Q₁₆	C₂×C₁₈₁	1	1	yes	characteristic
56	#335	C₂×C₁₈₁	1	1	yes	characteristic
57	Q₈⋊C₈	C₂×C₁₈₁	1	2	yes	—
58	C₈²	C₂×C₁₈₁	1	1	yes	characteristic
59	C₂×Q₁₆	C₄×C₁₈₁	1	1	yes	characteristic
60	C₄×Q₈	C₂²×C₁₈₁	1	2	yes	—
61	#107	C₄×C₁₈₁	1	1	yes	characteristic
62	C₄⋊C₈	—	2	2	no	—
63	C₄⋊C₈	C₂²×C₁₈₁	1	2	yes	—
64	#103	C₄×C₁₈₁	1	2	yes	—
65	C₄×C₈	—	2	2	no	—
66	C₄×C₈	C₂²×C₁₈₁	1	1	yes	characteristic
67	C₄×C₈	C₂²×C₁₈₁	1	1	yes	characteristic
68	C₄×C₈	C₂²×C₁₈₁	1	1	yes	characteristic
69	C₂×Q₈	C₂×C₄×C₁₈₁	1	2	yes	—
70	Q₁₆	C₈×C₁₈₁	1	4	yes	direct product factor
71	C₂×C₈	—	2	1	no	—
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	C₂×C₄×C₁₈₁	1	1	yes	characteristic
74	C₂×C₈	C₂×C₄×C₁₈₁	1	1	yes	characteristic
75	C₂×C₈	D₄×C₁₈₁	1	1	yes	characteristic
76	C₂×C₈	D₄×C₁₈₁	1	1	yes	characteristic
77	C₄⋊C₄	—	2	2	no	—
78	C₄⋊C₄	C₂×C₄×C₁₈₁	1	2	yes	—
79	C₄²	—	2	2	no	—
80	C₄²	C₂³×C₁₈₁	1	1	yes	characteristic
81	Q₈	—	2	2	no	—
82	Q₈	C₂×C₈×C₁₈₁	1	4	yes	—
83	C₂×C₄	—	2	2	no	—
84	C₂×C₄	—	2	2	no	—
85	C₂×C₄	C₂²×C₄×C₁₈₁	1	1	yes	characteristic
86	C₂×C₄	C₂×D₄×C₁₈₁	1	1	yes	characteristic
87	C₂×C₄	#41×C₁₈₁	1	1	yes	characteristic
88	C₈	—	2	2	no	—
89	C₈	—	2	1	no	—
90	C₈	—	2	1	no	—
91	C₈	—	4	2	no	—
92	C₈	C₂×C₈×C₁₈₁	1	2	yes	—
93	C₈	Q₁₆×C₁₈₁	1	2	yes	direct product factor
94	C₂²	C₄×D₄×C₁₈₁	1	1	yes	characteristic
95	C₄	—	2	4	no	—
96	C₄	—	2	1	no	—
97	C₄	—	4	2	no	—
98	C₄	C₂²×C₈×C₁₈₁	1	1	yes	characteristic
99	C₄	#131×C₁₈₁	1	1	yes	characteristic
100	C₄	C₂×Q₁₆×C₁₈₁	1	1	yes	characteristic
101	C₄	#135×C₁₈₁	1	1	yes	characteristic
102	C₂	C₈×D₄×C₁₈₁	1	1	yes	characteristic
103	C₂	C₄×Q₁₆×C₁₈₁	1	1	yes	characteristic
104	C₂	#443×C₁₈₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₈×Q₈×C₁₈₁

C₂

yes

maximal

C₄×Q₁₆×C₁₈₁

C₂

yes

maximal, characteristic

#335×C₁₈₁

C₂

yes

maximal, characteristic

Q₈⋊C₈×C₁₈₁

C₂

yes

maximal

C₈²×C₁₈₁

C₂

yes

maximal, characteristic

C₂×Q₁₆×C₁₈₁

C₄

yes

characteristic

C₄×Q₈×C₁₈₁

C₂²

yes

—

#107×C₁₈₁

C₄

yes

characteristic

C₄⋊C₈×C₁₈₁

—

C₄⋊C₈×C₁₈₁

C₂²

yes

—

#103×C₁₈₁

C₄

yes

—

C₄×C₈×C₁₈₁

—

C₄×C₈×C₁₈₁

C₂²

yes

characteristic

C₄×C₈×C₁₈₁

C₂²

yes

characteristic

C₄×C₈×C₁₈₁

C₂²

yes

characteristic

C₂×Q₈×C₁₈₁

C₂×C₄

yes

—

Q₁₆×C₁₈₁

C₈

yes

direct product factor

C₂×C₈×C₁₈₁

—

C₂×C₈×C₁₈₁

—

C₂×C₈×C₁₈₁

C₂×C₄

yes

characteristic

C₂×C₈×C₁₈₁

C₂×C₄

yes

characteristic

C₂×C₈×C₁₈₁

D₄

yes

characteristic

C₂×C₈×C₁₈₁

D₄

yes

characteristic

C₄⋊C₄×C₁₈₁

—

C₄⋊C₄×C₁₈₁

C₂×C₄

yes

—

C₄²×C₁₈₁

—

C₄²×C₁₈₁

C₂³

yes

characteristic

Q₈×C₁₈₁

—

Q₈×C₁₈₁

C₂×C₈

yes

—

C₂×C₄×C₁₈₁

—

C₂×C₄×C₁₈₁

—

C₂×C₄×C₁₈₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₈₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₈₁

#41

yes

characteristic

C₈×C₁₈₁

—

C₈×C₁₈₁

—

C₈×C₁₈₁

—

C₈×C₁₈₁

—

C₈×C₁₈₁

C₂×C₈

yes

—

C₈×C₁₈₁

Q₁₆

yes

direct product factor

C₂²×C₁₈₁

C₄×D₄

yes

characteristic

C₄×C₁₈₁

—

C₄×C₁₈₁

—

C₄×C₁₈₁

—

C₄×C₁₈₁

C₂²×C₈

yes

characteristic

C₄×C₁₈₁

#131

yes

characteristic

C₄×C₁₈₁

C₂×Q₁₆

yes

characteristic

C₄×C₁₈₁

#135

yes

characteristic

C₂×C₁₈₁

C₈×D₄

yes

characteristic

C₂×C₁₈₁

C₄×Q₁₆

yes

characteristic

C₂×C₁₈₁

#443

yes

characteristic

C₁₈₁

C₈×Q₁₆

yes

characteristic, direct product factor, Sylow

C₈×Q₁₆

C₁₈₁

yes

maximal, characteristic, direct product factor, Sylow

C₈×Q₈

C₂×C₁₈₁

yes

—

C₄×Q₁₆

C₂×C₁₈₁

yes

characteristic

#335

C₂×C₁₈₁

yes

characteristic

Q₈⋊C₈

C₂×C₁₈₁

yes

—

C₈²

C₂×C₁₈₁

yes

characteristic

C₂×Q₁₆

C₄×C₁₈₁

yes

characteristic

C₄×Q₈

C₂²×C₁₈₁

yes

—

#107

C₄×C₁₈₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

C₂²×C₁₈₁

yes

—

#103

C₄×C₁₈₁

yes

—

C₄×C₈

—

C₄×C₈

C₂²×C₁₈₁

yes

characteristic

C₄×C₈

C₂²×C₁₈₁

yes

characteristic

C₄×C₈

C₂²×C₁₈₁

yes

characteristic

C₂×Q₈

C₂×C₄×C₁₈₁

yes

—

Q₁₆

C₈×C₁₈₁

yes

direct product factor

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×C₄×C₁₈₁

yes

characteristic

C₂×C₈

C₂×C₄×C₁₈₁

yes

characteristic

C₂×C₈

D₄×C₁₈₁

yes

characteristic

C₂×C₈

D₄×C₁₈₁

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

C₂×C₄×C₁₈₁

yes

—

C₄²

—

C₄²

C₂³×C₁₈₁

yes

characteristic

Q₈

—

Q₈

C₂×C₈×C₁₈₁

yes

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²×C₄×C₁₈₁

yes

characteristic

C₂×C₄

C₂×D₄×C₁₈₁

yes

characteristic

C₂×C₄

#41×C₁₈₁

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

—

C₈

C₂×C₈×C₁₈₁

yes

—

C₈

Q₁₆×C₁₈₁

yes

direct product factor

C₂²

C₄×D₄×C₁₈₁

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

C₂²×C₈×C₁₈₁

yes

characteristic

C₄

#131×C₁₈₁

yes

characteristic

100

C₄

C₂×Q₁₆×C₁₈₁

yes

characteristic

101

C₄

#135×C₁₈₁

yes

characteristic

102

C₂

C₈×D₄×C₁₈₁

yes

characteristic

103

C₂

C₄×Q₁₆×C₁₈₁

yes

characteristic

104

C₂

#443×C₁₈₁

yes

characteristic

	subgroup	G/H
center	C₂×C₈×C₁₈₁	D₄
commutator	C₄	C₂²×C₈×C₁₈₁
Frattini, cosocle	C₄²	C₂³×C₁₈₁
socle, prime radical	C₂²×C₁₈₁	C₄×D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₁₈₁ ◁ C₄×C₈×C₁₈₁ ◁ G

subgroup

G/H

center

C₂×C₈×C₁₈₁

D₄

commutator

C₄

C₂²×C₈×C₁₈₁

Frattini, cosocle

C₄²

C₂³×C₁₈₁

socle, prime radical

C₂²×C₁₈₁

C₄×D₄

LCS

G ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₈×C₁₈₁ ◁ C₄×C₈×C₁₈₁ ◁ G

N	H ≅ G/N	type
17. Q₁₆×C₁₈₁	C₈	direct (N × H)
40. C₈×C₁₈₁	Q₁₆	direct (N × H)
52. C₁₈₁	C₈×Q₁₆	direct (N × H)
53. C₈×Q₁₆	C₁₈₁	direct (N × H)
70. Q₁₆	C₈×C₁₈₁	direct (N × H)
93. C₈	Q₁₆×C₁₈₁	direct (N × H)

H ≅ G/N

type

17. Q₁₆×C₁₈₁

C₈

direct (N × H)

40. C₈×C₁₈₁

Q₁₆

direct (N × H)

52. C₁₈₁

C₈×Q₁₆

direct (N × H)

53. C₈×Q₁₆

C₁₈₁

direct (N × H)

70. Q₁₆

C₈×C₁₈₁

direct (N × H)

93. C₈

Q₁₆×C₁₈₁

direct (N × H)

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

subgroup	index
C₈×Q₈×C₁₈₁	2
C₄×Q₁₆×C₁₈₁	2
#335×C₁₈₁	2
Q₈⋊C₈×C₁₈₁	2
C₈²×C₁₈₁	2
C₈×Q₁₆	181

subgroup

index

C₈×Q₈×C₁₈₁

C₄×Q₁₆×C₁₈₁

#335×C₁₈₁

Q₈⋊C₈×C₁₈₁

C₈²×C₁₈₁

C₈×Q₁₆

181

Subgroups — C₈×Q₁₆×C₁₈₁ #37529520

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C8×Q16×C181 #37529520

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×Q₁₆×C₁₈₁ #37529520