Subgroups — C₂⁴×#63216 #17981015

Order 4352.

2590 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#63216	C₂	1	30	yes	maximal, direct product factor
2	C₂⁴×#3622	C₂	1	1	yes	maximal, characteristic
3	C₂²×#63216	C₂²	1	140	yes	direct product factor
4	C₂³×#3622	C₄	1	15	yes	—
5	C₂³×#3622	C₂²	1	15	yes	—
6	C₂⁴×C₄×C₁₇	C₄	1	1	yes	characteristic
7	C₂⁵×C₁₇	C₈	1	1	yes	characteristic
8	C₂³×C₄×C₁₇	C₈	1	15	yes	—
9	C₂³×C₄×C₁₇	C₂×C₄	1	15	yes	—
10	C₂²×#3622	C₂×C₄	1	105	yes	—
11	C₂²×#3622	C₂³	1	35	yes	—
12	C₂×#63216	C₂³	1	120	yes	direct product factor
13	C₂⁴×C₁₇	C₁₆	1	16	yes	complemented
14	C₂⁴×C₁₇	C₂×C₈	1	15	yes	—
15	C₂²×C₄×C₁₇	C₂×C₈	1	105	yes	—
16	C₂²×C₄×C₁₇	C₂²×C₄	1	35	yes	—
17	C₂×#3622	C₂²×C₄	1	105	yes	—
18	C₂×#3622	C₂⁴	1	15	yes	—
19	#63216	C₂⁴	1	16	yes	direct product factor
20	C₂⁴×C₁₆	—	17	1	no	maximal, Sylow
21	C₂³×C₁₇	C₂×C₁₆	1	120	yes	complemented
22	C₂³×C₁₇	C₂²×C₈	1	35	yes	—
23	C₂×C₄×C₁₇	C₂²×C₈	1	105	yes	—
24	C₂×C₄×C₁₇	C₂³×C₄	1	15	yes	—
25	#3622	C₂³×C₄	1	15	yes	—
26	#3622	C₂⁵	1	1	yes	characteristic
27	C₂⁴×C₈	—	17	1	no	—
28	C₂³×C₁₆	—	17	30	no	—
29	C₂²×C₁₇	C₂²×C₁₆	1	140	yes	complemented
30	C₂²×C₁₇	C₂³×C₈	1	15	yes	—
31	C₄×C₁₇	C₂³×C₈	1	15	yes	—
32	C₄×C₁₇	C₂⁴×C₄	1	1	yes	characteristic
33	C₂⁴×C₄	#595	1	1	yes	characteristic
34	C₂³×C₈	—	17	15	no	—
35	C₂³×C₈	—	17	15	no	—
36	C₂²×C₁₆	—	17	140	no	—
37	C₂×C₁₇	C₂³×C₁₆	1	30	yes	complemented
38	C₂×C₁₇	C₂⁴×C₈	1	1	yes	characteristic
39	C₂⁵	#3624	1	1	yes	characteristic
40	C₂³×C₄	#3624	1	15	yes	—
41	C₂³×C₄	C₂×#595	1	15	yes	—
42	C₂²×C₈	—	17	105	no	—
43	C₂²×C₈	—	17	35	no	—
44	C₂×C₁₆	—	17	120	no	—
45	C₁₇	C₂⁴×C₁₆	1	1	yes	characteristic, complemented, Sylow
46	C₂⁴	#63216	1	16	yes	direct product factor
47	C₂⁴	C₂×#3624	1	15	yes	—
48	C₂²×C₄	C₂×#3624	1	105	yes	—
49	C₂²×C₄	C₂²×#595	1	35	yes	—
50	C₂×C₈	—	17	15	no	—
51	C₂×C₈	—	17	105	no	—
52	C₁₆	—	17	16	no	—
53	C₂³	C₂×#63216	1	120	yes	direct product factor
54	C₂³	C₂²×#3624	1	35	yes	—
55	C₂×C₄	C₂²×#3624	1	105	yes	—
56	C₂×C₄	C₂³×#595	1	15	yes	—
57	C₈	—	17	1	no	—
58	C₈	—	17	15	no	—
59	C₂²	C₂²×#63216	1	140	yes	direct product factor
60	C₂²	C₂³×#3624	1	15	yes	—
61	C₄	C₂³×#3624	1	15	yes	—
62	C₄	C₂⁴×#595	1	1	yes	characteristic
63	C₂	C₂³×#63216	1	30	yes	direct product factor
64	C₂	C₂⁴×#3624	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂⁴×C₄	#595
commutator, hypocenter	C₁₇	C₂⁴×C₁₆
Frattini	C₄	C₂⁴×#595
Fitting	C₂⁴×C₄×C₁₇	C₄
socle, prime radical	C₂⁵×C₁₇	C₈
cosocle	#3622	C₂⁵
LCS	G ▷ C₁₇
UCS	C₁ ◁ C₂⁴×C₄

Complements

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

N	H ≅ G/N	type
1. C₂³×#63216	C₂	direct (N × H)
3. C₂²×#63216	C₂²	direct (N × H)
12. C₂×#63216	C₂³	direct (N × H)
13. C₂⁴×C₁₇	C₁₆	semidirect
19. #63216	C₂⁴	direct (N × H)
21. C₂³×C₁₇	C₂×C₁₆	semidirect
29. C₂²×C₁₇	C₂²×C₁₆	semidirect
37. C₂×C₁₇	C₂³×C₁₆	semidirect
45. C₁₇	C₂⁴×C₁₆	semidirect
46. C₂⁴	#63216	direct (N × H)
53. C₂³	C₂×#63216	direct (N × H)
59. C₂²	C₂²×#63216	direct (N × H)
63. C₂	C₂³×#63216	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#63216	2
C₂⁴×#3622	2
C₂⁴×C₁₆	17

Subgroups — C24×#63216 #17981015

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×#63216 #17981015