Subgroups — C₂³×#14556641 #34030494

Order 18688.

542 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14556641	C₂	1	14	yes	maximal, direct product factor
2	C₂³×#11924003	C₂	1	1	yes	maximal, characteristic
3	C₂×#14556641	C₂²	1	28	yes	direct product factor
4	C₂²×#11924003	C₄	1	7	yes	—
5	C₂²×#11924003	C₂²	1	7	yes	—
6	C₂³×C₈×C₇₃	C₄	1	1	yes	characteristic
7	#14556641	C₂³	1	8	yes	direct product factor
8	C₂×#11924003	C₂×C₄	1	21	yes	—
9	C₂×#11924003	C₂³	1	7	yes	—
10	C₂³×C₄×C₇₃	C₈	1	1	yes	characteristic
11	C₂²×C₈×C₇₃	C₈	1	7	yes	—
12	C₂²×C₈×C₇₃	C₂×C₄	1	7	yes	—
13	#11924003	C₂²×C₄	1	7	yes	—
14	#11924003	C₂⁴	1	1	yes	characteristic
15	C₂⁴×C₇₃	C₁₆	1	1	yes	characteristic
16	C₂²×C₄×C₇₃	C₁₆	1	7	yes	—
17	C₂²×C₄×C₇₃	C₂×C₈	1	7	yes	—
18	C₂×C₈×C₇₃	C₂×C₈	1	21	yes	—
19	C₂×C₈×C₇₃	C₂²×C₄	1	7	yes	—
20	C₂³×C₇₃	C₃₂	1	8	yes	complemented
21	C₂³×C₇₃	C₂×C₁₆	1	7	yes	—
22	C₂×C₄×C₇₃	C₂×C₁₆	1	21	yes	—
23	C₂×C₄×C₇₃	C₂²×C₈	1	7	yes	—
24	C₈×C₇₃	C₂²×C₈	1	7	yes	—
25	C₈×C₇₃	C₂³×C₄	1	1	yes	characteristic
26	C₂²×C₇₃	C₂×C₃₂	1	28	yes	complemented
27	C₂²×C₇₃	C₂²×C₁₆	1	7	yes	—
28	C₄×C₇₃	C₂²×C₁₆	1	7	yes	—
29	C₄×C₇₃	C₂³×C₈	1	1	yes	characteristic
30	C₂³×C₃₂	—	73	1	no	maximal, Sylow
31	C₂×C₇₃	C₂²×C₃₂	1	14	yes	complemented
32	C₂×C₇₃	C₂³×C₁₆	1	1	yes	characteristic
33	C₂³×C₁₆	—	73	1	no	—
34	C₂²×C₃₂	—	73	14	no	—
35	C₇₃	C₂³×C₃₂	1	1	yes	characteristic, complemented, Sylow
36	C₂³×C₈	#64408	1	1	yes	characteristic
37	C₂²×C₁₆	—	73	7	no	—
38	C₂²×C₁₆	—	73	7	no	—
39	C₂×C₃₂	—	73	28	no	—
40	C₂³×C₄	#10596833	1	1	yes	characteristic
41	C₂²×C₈	#10596833	1	7	yes	—
42	C₂²×C₈	C₂×#64408	1	7	yes	—
43	C₂×C₁₆	—	73	7	no	—
44	C₂×C₁₆	—	73	21	no	—
45	C₃₂	—	73	8	no	—
46	C₂⁴	#11924031	1	1	yes	characteristic
47	C₂²×C₄	#11924031	1	7	yes	—
48	C₂²×C₄	C₂×#10596833	1	7	yes	—
49	C₂×C₈	C₂×#10596833	1	21	yes	—
50	C₂×C₈	C₂²×#64408	1	7	yes	—
51	C₁₆	—	73	7	no	—
52	C₁₆	—	73	1	no	—
53	C₂³	#14556641	1	8	yes	direct product factor
54	C₂³	C₂×#11924031	1	7	yes	—
55	C₂×C₄	C₂×#11924031	1	21	yes	—
56	C₂×C₄	C₂²×#10596833	1	7	yes	—
57	C₈	C₂²×#10596833	1	7	yes	—
58	C₈	C₂³×#64408	1	1	yes	characteristic
59	C₂²	C₂×#14556641	1	28	yes	direct product factor
60	C₂²	C₂²×#11924031	1	7	yes	—
61	C₄	C₂²×#11924031	1	7	yes	—
62	C₄	C₂³×#10596833	1	1	yes	characteristic
63	C₂	C₂²×#14556641	1	14	yes	direct product factor
64	C₂	C₂³×#11924031	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂³×C₈	#64408
commutator, hypocenter	C₇₃	C₂³×C₃₂
Frattini	C₈	C₂³×#64408
Fitting	C₂³×C₈×C₇₃	C₄
socle, prime radical	C₂⁴×C₇₃	C₁₆
cosocle	#11924003	C₂⁴
LCS	G ▷ C₇₃
UCS	C₁ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
1. C₂²×#14556641	C₂	direct (N × H)
3. C₂×#14556641	C₂²	direct (N × H)
7. #14556641	C₂³	direct (N × H)
20. C₂³×C₇₃	C₃₂	semidirect
26. C₂²×C₇₃	C₂×C₃₂	semidirect
31. C₂×C₇₃	C₂²×C₃₂	semidirect
35. C₇₃	C₂³×C₃₂	semidirect
53. C₂³	#14556641	direct (N × H)
59. C₂²	C₂×#14556641	direct (N × H)
63. C₂	C₂²×#14556641	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#14556641	2
C₂³×#11924003	2
C₂³×C₃₂	73

Subgroups — C23×#14556641 #34030494

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#14556641 #34030494