Subgroups — #72216364

Order 1024.

399 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#151726	C₂	1	4	yes	maximal, complemented
2	#123404	C₂	1	1	yes	maximal, characteristic, complemented
3	#123403	C₂	1	1	yes	maximal, characteristic
4	C₁₆×C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
5	#13727	—	2	8	no	—
6	#12521	—	2	2	no	—
7	#12516	C₂²	1	2	yes	—
8	#12515	—	2	2	no	—
9	#12515	C₂²	1	2	yes	—
10	C₈×C₃₂	C₂²	1	2	yes	—
11	C₁₆²	C₂²	1	1	yes	characteristic
12	C₂×SD₆₄	—	4	16	no	—
13	#2252	—	4	4	no	—
14	#2247	—	2	4	no	—
15	#2246	—	2	4	no	—
16	#2246	—	4	4	no	—
17	#1713	—	2	2	no	—
18	#1712	—	2	2	no	—
19	C₄×C₃₂	D₄	1	4	yes	—
20	C₈×C₁₆	D₄	1	2	yes	—
21	C₈×C₁₆	C₂³	1	1	yes	characteristic
22	C₂×Q₃₂	—	4	8	no	—
23	C₂×D₁₆	—	4	8	no	—
24	C₂×D₁₆	—	8	8	no	—
25	#500	—	4	4	no	—
26	#494	—	4	2	no	—
27	#493	—	4	2	no	—
28	#493	—	4	4	no	—
29	SD₆₄	—	8	32	no	—
30	C₂×C₃₂	D₈	1	8	yes	—
31	#366	—	8	8	no	—
32	C₄×C₁₆	D₈	1	4	yes	—
33	C₄×C₁₆	C₂×D₄	1	2	yes	—
34	C₈²	C₂×D₄	1	1	yes	characteristic
35	C₂×Q₁₆	—	8	4	no	—
36	C₂×D₈	—	8	4	no	—
37	C₂×D₈	—	8	8	no	—
38	C₄⋊Q₈	—	8	2	no	—
39	#127	—	8	2	no	—
40	Q₃₂	—	8	16	no	—
41	D₁₆	—	8	16	no	—
42	D₁₆	—	16	16	no	—
43	C₂×C₁₆	D₁₆	1	8	yes	—
44	C₂×C₁₆	C₂×D₈	1	4	yes	—
45	#107	—	8	8	no	—
46	C₄×C₈	#127	1	1	yes	characteristic
47	C₄×C₈	C₂×D₈	1	2	yes	—
48	C₃₂	D₁₆	1	16	yes	complemented
49	C₂×Q₈	—	16	2	no	—
50	C₂×D₄	—	16	2	no	—
51	C₂×D₄	—	16	4	no	—
52	Q₁₆	—	16	8	no	—
53	D₈	—	16	8	no	—
54	D₈	—	16	16	no	—
55	C₂×C₈	#493	1	2	yes	—
56	C₂×C₈	C₂×D₁₆	1	4	yes	—
57	C₄⋊C₄	—	16	4	no	—
58	C₄²	#493	1	1	yes	characteristic
59	C₁₆	SD₆₄	1	16	yes	complemented
60	C₁₆	C₂×D₁₆	1	8	yes	—
61	C₂³	—	32	2	no	—
62	Q₈	—	32	4	no	—
63	D₄	—	32	8	no	—
64	D₄	—	32	4	no	—
65	C₂×C₄	—	32	2	no	—
66	C₂×C₄	#1712	1	1	yes	characteristic
67	C₂×C₄	#2246	1	2	yes	—
68	C₈	#2246	1	4	yes	—
69	C₈	C₂×SD₆₄	1	8	yes	—
70	C₂²	—	64	2	no	—
71	C₂²	—	64	4	no	—
72	C₂²	#12515	1	1	yes	characteristic
73	C₄	—	64	2	no	—
74	C₄	#12515	1	2	yes	—
75	C₄	#13727	1	4	yes	—
76	C₂	—	128	2	no	—
77	C₂	#123403	1	1	yes	characteristic
78	C₂	#151726	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	#12515
commutator, Frattini, cosocle	C₈×C₁₆	C₂³
prime radical	#123403	C₂
LCS	G ▷ C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₁₆² ◁ G
LECS	G ▷ C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #151726	C₂	semidirect
2. #123404	C₂	semidirect
4. C₁₆×C₃₂	C₂	semidirect
48. C₃₂	D₁₆	semidirect
59. C₁₆	SD₆₄	semidirect

Maximal subgroups

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

subgroup	index
#151726	2
#123404	2
#123403	2
C₁₆×C₃₂	2