Subgroups — C₂⁴×#11924031 #34032675

Order 18688.

2590 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#11924031	C₂	1	30	yes	maximal, direct product factor
2	C₂⁴×#10596831	C₂	1	1	yes	maximal, characteristic
3	C₂²×#11924031	C₂²	1	140	yes	direct product factor
4	C₂³×#10596831	C₄	1	15	yes	—
5	C₂³×#10596831	C₂²	1	15	yes	—
6	C₂⁴×C₄×C₇₃	C₄	1	1	yes	characteristic
7	C₂×#11924031	C₂³	1	120	yes	direct product factor
8	C₂²×#10596831	C₂×C₄	1	105	yes	—
9	C₂²×#10596831	C₂³	1	35	yes	—
10	C₂⁵×C₇₃	C₈	1	1	yes	characteristic
11	C₂³×C₄×C₇₃	C₈	1	15	yes	—
12	C₂³×C₄×C₇₃	C₂×C₄	1	15	yes	—
13	#11924031	C₂⁴	1	16	yes	direct product factor
14	C₂×#10596831	C₂²×C₄	1	105	yes	—
15	C₂×#10596831	C₂⁴	1	15	yes	—
16	C₂⁴×C₇₃	C₁₆	1	16	yes	complemented
17	C₂⁴×C₇₃	C₂×C₈	1	15	yes	—
18	C₂²×C₄×C₇₃	C₂×C₈	1	105	yes	—
19	C₂²×C₄×C₇₃	C₂²×C₄	1	35	yes	—
20	C₂³×C₇₃	C₂×C₁₆	1	120	yes	complemented
21	C₂³×C₇₃	C₂²×C₈	1	35	yes	—
22	C₂×C₄×C₇₃	C₂²×C₈	1	105	yes	—
23	C₂×C₄×C₇₃	C₂³×C₄	1	15	yes	—
24	#10596831	C₂³×C₄	1	15	yes	—
25	#10596831	C₂⁵	1	1	yes	characteristic
26	C₂²×C₇₃	C₂²×C₁₆	1	140	yes	complemented
27	C₂²×C₇₃	C₂³×C₈	1	15	yes	—
28	C₄×C₇₃	C₂³×C₈	1	15	yes	—
29	C₄×C₇₃	C₂⁴×C₄	1	1	yes	characteristic
30	C₂⁴×C₁₆	—	73	1	no	maximal, Sylow
31	C₂×C₇₃	C₂³×C₁₆	1	30	yes	complemented
32	C₂×C₇₃	C₂⁴×C₈	1	1	yes	characteristic
33	C₂⁴×C₈	—	73	1	no	—
34	C₂³×C₁₆	—	73	30	no	—
35	C₇₃	C₂⁴×C₁₆	1	1	yes	characteristic, complemented, Sylow
36	C₂⁴×C₄	#64408	1	1	yes	characteristic
37	C₂³×C₈	—	73	15	no	—
38	C₂³×C₈	—	73	15	no	—
39	C₂²×C₁₆	—	73	140	no	—
40	C₂⁵	#10596833	1	1	yes	characteristic
41	C₂³×C₄	#10596833	1	15	yes	—
42	C₂³×C₄	C₂×#64408	1	15	yes	—
43	C₂²×C₈	—	73	105	no	—
44	C₂²×C₈	—	73	35	no	—
45	C₂×C₁₆	—	73	120	no	—
46	C₂⁴	#11924031	1	16	yes	direct product factor
47	C₂⁴	C₂×#10596833	1	15	yes	—
48	C₂²×C₄	C₂×#10596833	1	105	yes	—
49	C₂²×C₄	C₂²×#64408	1	35	yes	—
50	C₂×C₈	—	73	105	no	—
51	C₂×C₈	—	73	15	no	—
52	C₁₆	—	73	16	no	—
53	C₂³	C₂×#11924031	1	120	yes	direct product factor
54	C₂³	C₂²×#10596833	1	35	yes	—
55	C₂×C₄	C₂²×#10596833	1	105	yes	—
56	C₂×C₄	C₂³×#64408	1	15	yes	—
57	C₈	—	73	1	no	—
58	C₈	—	73	15	no	—
59	C₂²	C₂²×#11924031	1	140	yes	direct product factor
60	C₂²	C₂³×#10596833	1	15	yes	—
61	C₄	C₂³×#10596833	1	15	yes	—
62	C₄	C₂⁴×#64408	1	1	yes	characteristic
63	C₂	C₂³×#11924031	1	30	yes	direct product factor
64	C₂	C₂⁴×#10596833	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂⁴×C₄	#64408
commutator, hypocenter	C₇₃	C₂⁴×C₁₆
Frattini	C₄	C₂⁴×#64408
Fitting	C₂⁴×C₄×C₇₃	C₄
socle, prime radical	C₂⁵×C₇₃	C₈
cosocle	#10596831	C₂⁵
LCS	G ▷ C₇₃
UCS	C₁ ◁ C₂⁴×C₄

Complements

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

N	H ≅ G/N	type
1. C₂³×#11924031	C₂	direct (N × H)
3. C₂²×#11924031	C₂²	direct (N × H)
7. C₂×#11924031	C₂³	direct (N × H)
13. #11924031	C₂⁴	direct (N × H)
16. C₂⁴×C₇₃	C₁₆	semidirect
20. C₂³×C₇₃	C₂×C₁₆	semidirect
26. C₂²×C₇₃	C₂²×C₁₆	semidirect
31. C₂×C₇₃	C₂³×C₁₆	semidirect
35. C₇₃	C₂⁴×C₁₆	semidirect
46. C₂⁴	#11924031	direct (N × H)
53. C₂³	C₂×#11924031	direct (N × H)
59. C₂²	C₂²×#11924031	direct (N × H)
63. C₂	C₂³×#11924031	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×#11924031	2
C₂⁴×#10596831	2
C₂⁴×C₁₆	73

Subgroups — C24×#11924031 #34032675

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×#11924031 #34032675