Subgroups — #33999979

Order 18688.

150 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#11924031	C₂	1	1	yes	maximal, characteristic
2	C₈×#11924031	C₂	1	1	yes	maximal, characteristic
3	C₁₆×#10596831	C₂	1	1	yes	maximal, characteristic
4	C₄×#11924031	C₄	1	2	yes	—
5	C₄×#11924031	C₄	1	2	yes	—
6	C₄×#11924003	C₄	1	1	yes	characteristic
7	C₈×#10596831	C₂²	1	1	yes	characteristic
8	C₄×C₁₆×C₇₃	C₄	1	1	yes	characteristic
9	C₂×#11924031	C₈	1	4	yes	—
10	C₂×#11924031	C₈	1	4	yes	—
11	C₂×#11924003	C₈	1	2	yes	—
12	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
13	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
14	C₂×C₁₆×C₇₃	C₈	1	2	yes	—
15	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
16	#11924031	—	2	4	no	—
17	#11924031	C₁₆	1	8	yes	complemented
18	C₂×#10596831	C₂×C₈	1	2	yes	—
19	C₂×#10596831	C₂×C₈	1	2	yes	—
20	#11924003	—	2	2	no	—
21	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
22	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
23	C₄²×C₇₃	C₄²	1	1	yes	characteristic
24	C₁₆×C₇₃	—	2	2	no	—
25	C₂×C₄×C₇₃	C₄×C₈	1	1	yes	characteristic
26	C₂×C₄×C₇₃	C₄×C₈	1	1	yes	characteristic
27	C₂×C₄×C₇₃	C₄×C₈	1	1	yes	characteristic
28	C₈×C₇₃	M₃₂	1	2	yes	—
29	C₈×C₇₃	M₃₂	1	2	yes	—
30	#10596831	C₂×C₁₆	1	4	yes	—
31	#10596831	M₃₂	1	4	yes	—
32	C₂²×C₇₃	C₈²	1	1	yes	characteristic
33	C₄×C₇₃	C₄×C₁₆	1	2	yes	—
34	C₄×C₇₃	#346	1	2	yes	—
35	C₄×C₇₃	#346	1	1	yes	characteristic
36	C₄×C₇₃	#346	1	1	yes	characteristic
37	#7052	—	73	1	no	maximal, Sylow
38	C₂×C₇₃	C₈×C₁₆	1	1	yes	characteristic
39	C₂×C₇₃	#1311	1	1	yes	characteristic
40	C₂×C₇₃	#1311	1	1	yes	characteristic
41	C₈×C₁₆	—	73	1	no	—
42	C₈×C₁₆	—	73	1	no	—
43	C₈×C₁₆	—	73	1	no	—
44	C₇₃	#7052	1	1	yes	characteristic, complemented, Sylow
45	C₄×C₁₆	—	73	2	no	—
46	C₄×C₁₆	—	73	2	no	—
47	C₄×C₁₆	—	73	1	no	—
48	C₄×C₁₆	#64408	1	1	yes	characteristic
49	C₈²	—	73	1	no	—
50	C₂×C₁₆	—	73	4	no	—
51	C₂×C₁₆	—	73	4	no	—
52	C₂×C₁₆	—	73	2	no	—
53	C₂×C₁₆	#10596833	1	2	yes	—
54	C₄×C₈	—	73	1	no	—
55	C₄×C₈	—	73	1	no	—
56	C₄×C₈	C₂×#64408	1	1	yes	characteristic
57	C₂×C₈	—	73	2	no	—
58	C₂×C₈	—	73	2	no	—
59	C₂×C₈	C₂×#10596833	1	1	yes	characteristic
60	C₂×C₈	#11924036	1	1	yes	characteristic
61	C₄²	C₄×#64408	1	1	yes	characteristic
62	C₁₆	—	2	2	no	—
63	C₁₆	—	73	8	no	—
64	C₁₆	—	146	4	no	—
65	C₁₆	—	146	2	no	—
66	C₂×C₄	C₈×#64408	1	1	yes	characteristic
67	C₂×C₄	C₈×#64408	1	1	yes	characteristic
68	C₂×C₄	C₄×#10596833	1	1	yes	characteristic
69	C₈	—	73	4	no	—
70	C₈	—	73	4	no	—
71	C₈	#14556665	1	2	yes	—
72	C₈	#14556666	1	2	yes	—
73	C₂²	C₈×#10596833	1	1	yes	characteristic
74	C₄	C₁₆×#64408	1	2	yes	—
75	C₄	#17989474	1	2	yes	—
76	C₄	#17989475	1	1	yes	characteristic
77	C₄	#17989476	1	1	yes	characteristic
78	C₂	C₁₆×#10596833	1	1	yes	characteristic
79	C₂	#22624842	1	1	yes	characteristic
80	C₂	#22624843	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄×C₈	C₂×#64408
commutator	C₂×C₇₃	C₈×C₁₆
Fitting	C₄×C₁₆×C₇₃	C₄
socle, prime radical	C₂²×C₇₃	C₈²
cosocle	C₈×#10596831	C₂²
hypercenter	C₄×C₁₆	#64408
hypocenter	C₇₃	#7052
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
17. #11924031	C₁₆	semidirect
44. C₇₃	#7052	semidirect

Maximal subgroups

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

subgroup	index
C₈×#11924031	2
C₁₆×#10596831	2
#7052	73