Subgroups — #34001212

Order 18688.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22625100	C₂	1	2	yes	maximal, complemented
2	C₂×#17988421	C₂	1	1	yes	maximal, characteristic
3	#17989426	—	2	2	no	—
4	#17988421	—	2	2	no	—
5	C₂×#14556497	C₄	1	1	yes	characteristic
6	C₂×#14556497	C₂²	1	1	yes	characteristic
7	C₂²×C₁₆×C₇₃	C₄	1	1	yes	characteristic
8	#14556497	—	2	1	no	—
9	#14556497	D₄	1	2	yes	—
10	C₂²×C₈×C₇₃	C₈	1	1	yes	characteristic
11	C₂×C₁₆×C₇₃	—	2	1	no	—
12	C₂×C₁₆×C₇₃	—	2	1	no	—
13	C₂×C₁₆×C₇₃	C₈	1	1	yes	characteristic
14	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄×C₇₃	C₁₆	1	1	yes	characteristic
16	C₂×C₈×C₇₃	—	2	1	no	—
17	C₂×C₈×C₇₃	—	2	1	no	—
18	C₂×C₈×C₇₃	C₁₆	1	1	yes	characteristic
19	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
20	C₁₆×C₇₃	—	2	1	no	—
21	C₁₆×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
22	C₁₆×C₇₃	M₁₆	1	1	yes	characteristic
23	C₂³×C₇₃	C₃₂	1	1	yes	characteristic
24	C₂×C₄×C₇₃	—	2	1	no	—
25	C₂×C₄×C₇₃	—	2	1	no	—
26	C₂×C₄×C₇₃	C₃₂	1	1	yes	characteristic
27	C₂×C₄×C₇₃	C₂×C₁₆	1	1	yes	characteristic
28	C₈×C₇₃	—	2	1	no	—
29	C₈×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
30	C₈×C₇₃	M₃₂	1	1	yes	characteristic
31	C₂²×C₇₃	—	2	1	no	—
32	C₂²×C₇₃	—	4	1	no	—
33	C₂²×C₇₃	C₂×C₃₂	1	1	yes	characteristic
34	C₄×C₇₃	—	2	1	no	—
35	C₄×C₇₃	C₂²⋊C₁₆	1	1	yes	characteristic
36	C₄×C₇₃	M₆₄	1	1	yes	characteristic
37	#7513	—	73	1	no	maximal, Sylow
38	C₂×C₇₃	—	2	1	no	—
39	C₂×C₇₃	—	4	1	no	—
40	C₂×C₇₃	C₂²⋊C₃₂	1	1	yes	characteristic
41	C₂×M₆₄	—	73	1	no	—
42	M₁₂₈	—	73	2	no	—
43	C₇₃	#7513	1	1	yes	characteristic, complemented, Sylow
44	C₂²×C₁₆	#64408	1	1	yes	characteristic
45	M₆₄	—	146	2	no	—
46	C₂×C₃₂	—	73	1	no	—
47	C₂×C₃₂	—	73	1	no	—
48	C₆₄	—	146	2	no	—
49	C₂²×C₈	#10596833	1	1	yes	characteristic
50	C₂×C₁₆	—	2	1	no	—
51	C₂×C₁₆	—	2	1	no	—
52	C₂×C₁₆	#10596833	1	1	yes	characteristic
53	C₂×C₁₆	C₂×#64408	1	1	yes	characteristic
54	C₃₂	—	73	2	no	—
55	C₃₂	—	146	1	no	—
56	C₂²×C₄	#11924031	1	1	yes	characteristic
57	C₂×C₈	—	2	1	no	—
58	C₂×C₈	—	2	1	no	—
59	C₂×C₈	#11924031	1	1	yes	characteristic
60	C₂×C₈	C₂×#10596833	1	1	yes	characteristic
61	C₁₆	—	2	1	no	—
62	C₁₆	C₂²⋊#64408	1	1	yes	characteristic
63	C₁₆	#11924037	1	1	yes	characteristic
64	C₂³	#14556641	1	1	yes	characteristic
65	C₂×C₄	—	2	1	no	—
66	C₂×C₄	—	2	1	no	—
67	C₂×C₄	#14556641	1	1	yes	characteristic
68	C₂×C₄	C₂×#11924031	1	1	yes	characteristic
69	C₈	—	2	1	no	—
70	C₈	C₂²⋊#10596833	1	1	yes	characteristic
71	C₈	#14556665	1	1	yes	characteristic
72	C₂²	—	2	1	no	—
73	C₂²	—	4	1	no	—
74	C₂²	C₂×#14556641	1	1	yes	characteristic
75	C₄	—	2	1	no	—
76	C₄	C₂²⋊#11924031	1	1	yes	characteristic
77	C₄	#17989519	1	1	yes	characteristic
78	C₂	—	2	1	no	—
79	C₂	—	4	1	no	—
80	C₂	C₂²⋊#14556641	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	C₂²⋊#64408
commutator	C₂²×C₇₃	C₂×C₃₂
Frattini	C₂×C₁₆	C₂×#64408
Fitting	C₂²×C₁₆×C₇₃	C₄
socle	C₂×C₇₃	C₂²⋊C₃₂
prime radical	C₂³×C₇₃	C₃₂
cosocle	C₂×#14556497	C₂²
hypercenter	C₂²×C₁₆	#64408
hypocenter	C₇₃	#7513
LCS	G ▷ C₂²×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₁₆ ◁ C₂×C₁₆ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
1. #22625100	C₂	semidirect
43. C₇₃	#7513	semidirect

Maximal subgroups

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

subgroup	index
#22625100	2
C₂×#17988421	2
#7513	73