Subgroups — #34033003

Order 18688.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22627014	C₂	1	2	yes	maximal
2	C₁₆×#10596833	C₂	1	1	yes	maximal, characteristic
3	C₂×#14556681	C₄	1	4	yes	—
4	#17989473	C₂²	1	1	yes	characteristic
5	C₈×#10596833	C₄	1	1	yes	characteristic
6	C₁₆×Dic₇₃	C₄	1	1	yes	characteristic
7	#14556681	—	2	4	no	—
8	C₂×#11924031	C₂×C₄	1	2	yes	—
9	C₄×#10596833	C₈	1	1	yes	characteristic
10	C₄×#10596833	C₈	1	1	yes	characteristic
11	C₂×#11924003	C₈	1	1	yes	characteristic
12	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
13	C₂×C₁₆×C₇₃	C₈	1	1	yes	characteristic
14	C₂×#10596833	C₁₆	1	2	yes	—
15	C₂×#10596833	C₁₆	1	2	yes	—
16	#11924031	M₁₆	1	4	yes	—
17	C₂×#10596831	C₄²	1	1	yes	characteristic
18	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
19	#11924003	—	2	1	no	—
20	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
21	C₁₆×C₇₃	—	2	1	no	—
22	C₂×C₄×C₇₃	C₄×C₈	1	1	yes	characteristic
23	C₂×Dic₇₃	C₂×C₁₆	1	1	yes	characteristic
24	C₂×Dic₇₃	C₂×C₁₆	1	1	yes	characteristic
25	#10596833	—	2	2	no	—
26	#10596833	C₃₂	1	4	yes	complemented
27	C₈×C₇₃	M₃₂	1	1	yes	characteristic
28	C₈×C₇₃	M₃₂	1	1	yes	characteristic
29	#10596831	#97	1	2	yes	—
30	C₂²×C₇₃	C₄×C₁₆	1	1	yes	characteristic
31	C₄×C₇₃	#322	1	1	yes	characteristic
32	C₄×C₇₃	#346	1	1	yes	characteristic
33	Dic₇₃	C₂×C₃₂	1	2	yes	—
34	Dic₇₃	M₆₄	1	2	yes	—
35	#7331	—	73	1	no	maximal, Sylow
36	C₂×C₇₃	#1312	1	1	yes	characteristic
37	C₂×C₇₃	C₄×C₃₂	1	1	yes	characteristic
38	C₂×C₇₃	#1397	1	1	yes	characteristic
39	C₄×C₃₂	—	73	2	no	—
40	C₈×C₁₆	—	73	1	no	—
41	C₇₃	#7331	1	1	yes	characteristic, complemented, Sylow
42	C₂×C₃₂	—	73	4	no	—
43	C₄×C₁₆	—	73	1	no	—
44	C₄×C₁₆	—	73	1	no	—
45	C₈²	—	73	1	no	—
46	C₂×C₁₆	—	73	2	no	—
47	C₂×C₁₆	—	73	1	no	—
48	C₂×C₁₆	#10596842	1	1	yes	characteristic
49	C₄×C₈	—	73	1	no	—
50	C₄×C₈	—	73	1	no	—
51	C₄×C₈	—	73	1	no	—
52	C₃₂	—	146	4	no	—
53	C₂×C₈	—	73	2	no	—
54	C₂×C₈	—	73	1	no	—
55	C₂×C₈	—	73	2	no	—
56	C₂×C₈	C₂×#10596842	1	1	yes	characteristic
57	C₄²	—	73	1	no	—
58	C₁₆	—	2	1	no	—
59	C₁₆	—	73	4	no	—
60	C₁₆	—	146	1	no	—
61	C₂×C₄	—	73	1	no	—
62	C₂×C₄	—	73	1	no	—
63	C₂×C₄	C₄×#10596842	1	1	yes	characteristic
64	C₈	—	73	4	no	—
65	C₈	—	73	2	no	—
66	C₈	—	146	2	no	—
67	C₈	#14556687	1	1	yes	characteristic
68	C₈	#14556688	1	1	yes	characteristic
69	C₂²	#64408⋊C₁₆	1	1	yes	characteristic
70	C₄	—	73	2	no	—
71	C₄	—	73	2	no	—
72	C₄	#17989671	1	1	yes	characteristic
73	C₄	#17989681	1	1	yes	characteristic
74	C₂	#22626949	1	1	yes	characteristic
75	C₂	#64408⋊C₃₂	1	1	yes	characteristic
76	C₂	#22627018	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂×#10596842
commutator	C₂×C₇₃	C₄×C₃₂
Fitting	C₂×C₁₆×C₇₃	C₈
socle, prime radical	C₂²×C₇₃	C₄×C₁₆
cosocle	#17989473	C₂²
hypercenter	C₂×C₁₆	#10596842
hypocenter	C₇₃	#7331
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
26. #10596833	C₃₂	semidirect
41. C₇₃	#7331	semidirect

Maximal subgroups

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

subgroup	index
#22627014	2
C₁₆×#10596833	2
#7331	73