Subgroups — #34001259

Order 18688.

136 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22625069	C₂	1	2	yes	maximal
2	#22609549	C₂	1	1	yes	maximal, characteristic
3	#17989498	—	2	2	no	—
4	#17988992	—	2	2	no	—
5	#17988967	C₄	1	1	yes	characteristic
6	#17988967	C₂²	1	1	yes	characteristic
7	C₈²×C₇₃	C₄	1	1	yes	characteristic
8	#14556652	—	4	2	no	—
9	C₂×Dic₂₉₂	—	4	1	no	—
10	C₂×Dic₂₉₂	—	4	1	no	—
11	#14556591	—	2	1	no	—
12	#14556591	D₄	1	2	yes	—
13	#14556525	—	4	4	no	—
14	C₄×C₈×C₇₃	—	2	1	no	—
15	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
16	#11924037	—	8	2	no	—
17	C₂×Dic₁₄₆	—	2	2	no	—
18	C₂×Dic₁₄₆	—	4	1	no	—
19	Dic₂₉₂	—	8	2	no	—
20	Dic₂₉₂	—	8	2	no	—
21	C₄⋊Dic₇₃	—	4	2	no	—
22	C₄⋊Dic₇₃	—	4	2	no	—
23	C₂×C₈×C₇₃	—	2	2	no	—
24	C₂×C₈×C₇₃	—	2	1	no	—
25	C₄²×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
26	C₂×C₄×C₇₃	—	2	1	no	—
27	C₂×C₄×C₇₃	#99	1	1	yes	characteristic
28	C₂×Dic₇₃	—	4	2	no	—
29	C₂×Dic₇₃	—	8	1	no	—
30	Dic₁₄₆	—	8	2	no	—
31	Dic₁₄₆	—	8	2	no	—
32	#10596833	—	16	2	no	—
33	C₈×C₇₃	—	2	4	no	—
34	C₈×C₇₃	—	2	2	no	—
35	C₂²×C₇₃	#353	1	1	yes	characteristic
36	C₄×C₇₃	—	2	2	no	—
37	C₄×C₇₃	—	2	1	no	—
38	Dic₇₃	—	8	2	no	—
39	Dic₇₃	—	16	1	no	—
40	#7532	—	73	1	no	maximal, Sylow
41	C₂×C₇₃	—	2	1	no	—
42	C₂×C₇₃	#1408	1	1	yes	characteristic
43	#1713	—	73	1	no	—
44	#1411	—	73	2	no	—
45	C₇₃	#7532	1	1	yes	characteristic, complemented, Sylow
46	#500	—	146	2	no	—
47	#494	—	73	1	no	—
48	#494	—	73	1	no	—
49	#356	—	146	2	no	—
50	C₈²	#64408	1	1	yes	characteristic
51	C₂×Q₁₆	—	292	1	no	—
52	C₂×Q₁₆	—	292	1	no	—
53	C₄⋊Q₈	—	73	2	no	—
54	C₄⋊Q₈	—	146	1	no	—
55	#107	—	292	4	no	—
56	#101	—	292	2	no	—
57	C₄×C₈	—	2	1	no	—
58	C₄×C₈	C₂×#64408	1	1	yes	characteristic
59	C₂×Q₈	—	292	1	no	—
60	C₂×Q₈	—	146	2	no	—
61	Q₁₆	—	584	2	no	—
62	Q₁₆	—	584	2	no	—
63	M₁₆	—	584	2	no	—
64	C₂×C₈	—	2	2	no	—
65	C₂×C₈	—	2	1	no	—
66	C₄⋊C₄	—	292	2	no	—
67	C₄⋊C₄	—	292	2	no	—
68	C₄²	C₂²⋊#64408	1	1	yes	characteristic
69	Q₈	—	584	2	no	—
70	Q₈	—	584	2	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	292	2	no	—
73	C₂×C₄	—	584	1	no	—
74	C₂×C₄	#14556650	1	1	yes	characteristic
75	C₈	—	2	2	no	—
76	C₈	—	2	4	no	—
77	C₈	—	1168	2	no	—
78	C₂²	#17989492	1	1	yes	characteristic
79	C₄	—	2	2	no	—
80	C₄	—	2	1	no	—
81	C₄	—	584	2	no	—
82	C₄	—	1168	1	no	—
83	C₂	—	2	1	no	—
84	C₂	#22625063	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22625063
commutator	C₄×C₈×C₇₃	C₂×C₄
Frattini	C₄×C₈	C₂×#64408
Fitting	C₈²×C₇₃	C₄
socle	C₂×C₇₃	#1408
prime radical	C₂²×C₇₃	#353
cosocle	#17988967	C₂²
hypercenter	C₈²	#64408
hypocenter	C₇₃	#7532
LCS	G ▷ C₄×C₈×C₇₃ ▷ C₄²×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₈²

Complements

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

N	H ≅ G/N	type
45. C₇₃	#7532	semidirect

Maximal subgroups

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

subgroup	index
#22625069	2
#22609549	2
#7532	73