Subgroups — #72225067

Order 2048.

555 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72225038	C₂	1	3	yes	maximal, complemented
2	C₁₆×M₆₄	C₂	1	3	yes	maximal
3	C₃₂²	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈×M₆₄	C₄	1	6	yes	—
5	C₂×C₁₆²	C₂²	1	1	yes	characteristic
6	#93664	C₄	1	6	yes	—
7	C₁₆×C₃₂	C₂²	1	3	yes	—
8	C₁₆×C₃₂	C₂²	1	3	yes	—
9	C₄×M₆₄	C₈	1	12	yes	—
10	C₂×C₈×C₁₆	C₂×C₄	1	3	yes	—
11	#7329	C₈	1	12	yes	—
12	C₈×C₃₂	C₂×C₄	1	6	yes	—
13	C₈×C₃₂	C₂×C₄	1	6	yes	—
14	C₁₆²	C₂×C₄	1	3	yes	—
15	C₁₆²	C₂³	1	1	yes	characteristic
16	C₂×M₆₄	C₁₆	1	24	yes	—
17	C₂×C₄×C₁₆	C₂×C₈	1	6	yes	—
18	C₂×C₈²	C₄²	1	1	yes	characteristic
19	#1397	C₁₆	1	24	yes	—
20	C₄×C₃₂	C₂×C₈	1	12	yes	—
21	C₄×C₃₂	C₂×C₈	1	12	yes	—
22	C₈×C₁₆	C₄²	1	3	yes	—
23	C₈×C₁₆	C₂×C₈	1	6	yes	—
24	C₈×C₁₆	C₂²×C₄	1	3	yes	—
25	C₂²×C₁₆	C₂×C₁₆	1	12	yes	—
26	C₂×C₄×C₈	C₄×C₈	1	3	yes	—
27	M₆₄	—	2	48	no	—
28	C₂×C₃₂	C₂×C₁₆	1	24	yes	—
29	C₂×C₃₂	C₂×C₁₆	1	24	yes	—
30	C₄×C₁₆	C₄×C₈	1	6	yes	—
31	C₄×C₁₆	C₂×C₁₆	1	12	yes	—
32	C₄×C₁₆	C₂²×C₈	1	6	yes	—
33	C₈²	C₄×C₈	1	3	yes	—
34	C₈²	C₂×C₄²	1	1	yes	characteristic
35	C₂²×C₈	C₄×C₁₆	1	6	yes	—
36	C₂×C₄²	C₈²	1	1	yes	characteristic
37	C₂×C₁₆	—	2	24	no	—
38	C₂×C₁₆	C₄×C₁₆	1	12	yes	—
39	C₂×C₁₆	C₂²×C₁₆	1	12	yes	—
40	C₄×C₈	C₈²	1	3	yes	—
41	C₄×C₈	C₄×C₁₆	1	6	yes	—
42	C₄×C₈	C₂×C₄×C₈	1	3	yes	—
43	C₃₂	—	2	24	no	—
44	C₃₂	M₆₄	1	48	yes	complemented
45	C₂²×C₄	C₈×C₁₆	1	3	yes	—
46	C₂×C₈	—	2	12	no	—
47	C₂×C₈	C₈×C₁₆	1	6	yes	—
48	C₂×C₈	C₂×C₄×C₁₆	1	6	yes	—
49	C₄²	C₈×C₁₆	1	3	yes	—
50	C₄²	C₂×C₈²	1	1	yes	characteristic
51	C₁₆	—	2	12	no	—
52	C₁₆	C₂×M₆₄	1	24	yes	—
53	C₂³	C₁₆²	1	1	yes	characteristic
54	C₂×C₄	—	2	6	no	—
55	C₂×C₄	C₁₆²	1	3	yes	—
56	C₂×C₄	C₂×C₈×C₁₆	1	3	yes	—
57	C₈	—	2	6	no	—
58	C₈	C₄×M₆₄	1	12	yes	—
59	C₂²	—	2	3	no	—
60	C₂²	C₂×C₁₆²	1	1	yes	characteristic
61	C₄	—	2	3	no	—
62	C₄	C₈×M₆₄	1	6	yes	—
63	C₂	—	4	1	no	—
64	C₂	C₁₆×M₆₄	1	3	yes	—

	subgroup	G/H
center, Frattini, cosocle	C₁₆²	C₂³
commutator, socle	C₂²	C₂×C₁₆²
prime radical	C₂³	C₁₆²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₁₆² ◁ G
LECS	G ▷ C₁₆² ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #72225038	C₂	semidirect
3. C₃₂²	C₂	semidirect
44. C₃₂	M₆₄	semidirect

Maximal subgroups

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

subgroup	index
#72225038	2
C₁₆×M₆₄	2
C₃₂²	2