Subgroups — #34000749

Order 18688.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14556641	C₂	1	2	yes	maximal
2	C₈×#11924003	C₂	1	1	yes	maximal, characteristic
3	C₂×#14556641	C₄	1	4	yes	—
4	C₄×#11924003	C₄	1	1	yes	characteristic
5	C₄×#11924003	C₂²	1	1	yes	characteristic
6	C₈²×C₇₃	C₄	1	1	yes	characteristic
7	#14556641	C₈	1	8	yes	complemented
8	C₂×#11924003	C₈	1	2	yes	—
9	C₂×#11924003	C₂×C₄	1	2	yes	—
10	C₄×C₈×C₇₃	C₈	1	2	yes	—
11	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
12	#11924003	C₁₆	1	4	yes	—
13	#11924003	C₂×C₈	1	4	yes	—
14	C₂×C₈×C₇₃	C₁₆	1	4	yes	—
15	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
16	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
17	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
18	C₂×C₄×C₇₃	C₄×C₈	1	1	yes	characteristic
19	C₂×C₄×C₇₃	C₂×C₁₆	1	2	yes	—
20	C₈×C₇₃	—	2	4	no	—
21	C₈×C₇₃	C₄×C₈	1	2	yes	—
22	C₈×C₇₃	C₂×C₁₆	1	2	yes	—
23	C₂²×C₇₃	C₄×C₁₆	1	1	yes	characteristic
24	C₄×C₇₃	C₈²	1	1	yes	characteristic
25	C₄×C₇₃	C₄×C₁₆	1	1	yes	characteristic
26	C₄×C₇₃	M₆₄	1	4	yes	—
27	#7329	—	73	1	no	maximal, Sylow
28	C₂×C₇₃	C₈×C₁₆	1	1	yes	characteristic
29	C₂×C₇₃	#1397	1	2	yes	—
30	C₄×C₃₂	—	73	2	no	—
31	C₈×C₁₆	—	73	1	no	—
32	C₇₃	#7329	1	1	yes	characteristic, complemented, Sylow
33	C₂×C₃₂	—	73	4	no	—
34	C₄×C₁₆	—	73	1	no	—
35	C₄×C₁₆	—	73	1	no	—
36	C₈²	#64408	1	1	yes	characteristic
37	C₂×C₁₆	—	73	2	no	—
38	C₂×C₁₆	—	73	2	no	—
39	C₄×C₈	#10596833	1	2	yes	—
40	C₄×C₈	C₂×#64408	1	1	yes	characteristic
41	C₃₂	—	73	8	no	—
42	C₂×C₈	#11924031	1	4	yes	—
43	C₂×C₈	C₄×#64408	1	1	yes	characteristic
44	C₂×C₈	#11924036	1	1	yes	characteristic
45	C₄²	C₂×#10596833	1	1	yes	characteristic
46	C₁₆	—	73	4	no	—
47	C₁₆	—	73	4	no	—
48	C₂×C₄	C₄×#10596833	1	1	yes	characteristic
49	C₂×C₄	C₂×#11924031	1	2	yes	—
50	C₈	—	2	4	no	—
51	C₈	C₈×#64408	1	2	yes	—
52	C₈	#14556664	1	2	yes	—
53	C₂²	C₄×#11924031	1	1	yes	characteristic
54	C₄	C₈×#10596833	1	1	yes	characteristic
55	C₄	#17989473	1	1	yes	characteristic
56	C₄	#17989519	1	4	yes	—
57	C₂	C₈×#11924031	1	1	yes	characteristic
58	C₂	#22625038	1	2	yes	—

	subgroup	G/H
center, Frattini	C₄×C₈	C₂×#64408
commutator	C₂×C₇₃	C₈×C₁₆
Fitting	C₈²×C₇₃	C₄
socle, prime radical	C₂²×C₇₃	C₄×C₁₆
cosocle	C₄×#11924003	C₂²
hypercenter	C₈²	#64408
hypocenter	C₇₃	#7329
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄×C₈ ◁ C₈²

Complements

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

N	H ≅ G/N	type
7. #14556641	C₈	semidirect
32. C₇₃	#7329	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14556641	2
C₈×#11924003	2
#7329	73