Subgroups — #34033159

Order 18688.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22627035	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#17989669	C₂	1	1	yes	maximal, characteristic, complemented
3	#22626354	C₂	1	1	yes	maximal, characteristic
4	#17989669	—	2	1	no	—
5	#17989669	—	2	1	no	—
6	#17989519	—	2	1	no	—
7	#17989519	C₄	1	1	yes	characteristic
8	C₂×#14556641	—	2	1	no	—
9	C₂×#14556641	C₂²	1	1	yes	characteristic
10	#17989004	C₄	1	1	yes	characteristic
11	#14556641	—	2	1	no	—
12	#14556641	D₄	1	1	yes	characteristic
13	#14556641	D₄	1	1	yes	characteristic
14	#14556531	—	2	1	no	—
15	#14556531	C₈	1	1	yes	characteristic
16	C₂×#11924003	—	2	1	no	—
17	C₂×#11924003	C₂×C₄	1	1	yes	characteristic
18	#131×C₇₃	C₈	1	1	yes	characteristic
19	#11924003	—	2	1	no	—
20	#11924003	C₂²⋊C₄	1	1	yes	characteristic
21	#11924003	M₁₆	1	1	yes	characteristic
22	#41×C₇₃	C₁₆	1	1	yes	characteristic
23	M₁₆×C₇₃	—	2	1	no	—
24	M₁₆×C₇₃	C₁₆	1	1	yes	characteristic
25	C₂×C₈×C₇₃	—	2	1	no	—
26	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
27	Q₈×C₇₃	C₃₂	1	1	yes	characteristic
28	D₄×C₇₃	—	2	1	no	—
29	D₄×C₇₃	C₃₂	1	1	yes	characteristic
30	C₂×C₄×C₇₃	—	2	1	no	—
31	C₂×C₄×C₇₃	C₂×C₁₆	1	1	yes	characteristic
32	C₈×C₇₃	—	2	1	no	—
33	C₈×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
34	C₈×C₇₃	M₃₂	1	1	yes	characteristic
35	C₂²×C₇₃	—	2	1	no	—
36	C₂²×C₇₃	M₆₄	1	1	yes	characteristic
37	C₄×C₇₃	—	2	1	no	—
38	C₄×C₇₃	C₂²⋊C₁₆	1	1	yes	characteristic
39	C₄×C₇₃	C₂×C₃₂	1	1	yes	characteristic
40	#7514	—	73	1	no	maximal, Sylow
41	C₂×C₇₃	—	2	1	no	—
42	C₂×C₇₃	—	4	1	no	—
43	C₂×C₇₃	C₂²⋊C₃₂	1	1	yes	characteristic
44	#2258	—	73	1	no	—
45	M₁₂₈	—	73	1	no	—
46	C₂×C₆₄	—	73	1	no	—
47	C₇₃	#7514	1	1	yes	characteristic, complemented, Sylow
48	#504	—	73	1	no	—
49	M₆₄	—	73	1	no	—
50	M₆₄	—	146	1	no	—
51	C₂×C₃₂	—	73	1	no	—
52	C₂×C₃₂	—	146	1	no	—
53	C₆₄	—	146	1	no	—
54	C₆₄	—	146	1	no	—
55	#131	#10596842	1	1	yes	characteristic
56	M₃₂	—	73	1	no	—
57	M₃₂	—	146	1	no	—
58	C₂×C₁₆	—	73	1	no	—
59	C₂×C₁₆	—	146	1	no	—
60	C₃₂	—	73	1	no	—
61	C₃₂	—	73	1	no	—
62	C₃₂	—	146	1	no	—
63	#41	#11924040	1	1	yes	characteristic
64	M₁₆	—	2	1	no	—
65	M₁₆	#11924040	1	1	yes	characteristic
66	C₂×C₈	—	2	1	no	—
67	C₂×C₈	C₂×#10596842	1	1	yes	characteristic
68	C₁₆	—	73	1	no	—
69	C₁₆	—	73	1	no	—
70	C₁₆	—	146	1	no	—
71	Q₈	#14556681	1	1	yes	characteristic
72	D₄	—	2	1	no	—
73	D₄	#14556681	1	1	yes	characteristic
74	C₂×C₄	—	2	1	no	—
75	C₂×C₄	C₂×#11924040	1	1	yes	characteristic
76	C₈	—	2	1	no	—
77	C₈	C₂²⋊#10596842	1	1	yes	characteristic
78	C₈	#14556687	1	1	yes	characteristic
79	C₂²	—	2	1	no	—
80	C₂²	#17989695	1	1	yes	characteristic
81	C₄	—	2	1	no	—
82	C₄	C₂²⋊#11924040	1	1	yes	characteristic
83	C₄	C₂×#14556681	1	1	yes	characteristic
84	C₂	—	2	1	no	—
85	C₂	—	4	1	no	—
86	C₂	C₂²⋊#14556681	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂²⋊#10596842
commutator	C₄×C₇₃	C₂×C₃₂
Frattini	C₂×C₈	C₂×#10596842
Fitting	#131×C₇₃	C₈
socle	C₂×C₇₃	C₂²⋊C₃₂
prime radical	#41×C₇₃	C₁₆
cosocle	C₂×#14556641	C₂²
hypercenter	#131	#10596842
hypocenter	C₇₃	#7514
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ #131

Complements

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

N	H ≅ G/N	type
1. #22627035	C₂	semidirect
2. C₂×#17989669	C₂	semidirect
47. C₇₃	#7514	semidirect

Maximal subgroups

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

subgroup	index
#22627035	2
C₂×#17989669	2
#22626354	2
#7514	73