Subgroups — #34001214

Order 18688.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22625100	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#17989426	C₂	1	1	yes	maximal, characteristic, complemented
3	#22612225	C₂	1	1	yes	maximal, characteristic
4	#17989426	—	2	1	no	—
5	#17989426	—	2	1	no	—
6	#17988421	—	2	1	no	—
7	#17988421	C₄	1	1	yes	characteristic
8	C₂×#14556497	—	2	1	no	—
9	C₂×#14556497	C₂²	1	1	yes	characteristic
10	#504×C₇₃	C₄	1	1	yes	characteristic
11	#14556497	—	2	1	no	—
12	#14556497	D₄	1	1	yes	characteristic
13	#14556497	D₄	1	1	yes	characteristic
14	#131×C₇₃	C₈	1	1	yes	characteristic
15	M₃₂×C₇₃	—	2	1	no	—
16	M₃₂×C₇₃	C₈	1	1	yes	characteristic
17	C₂×C₁₆×C₇₃	—	2	1	no	—
18	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
19	#41×C₇₃	C₁₆	1	1	yes	characteristic
20	M₁₆×C₇₃	—	2	1	no	—
21	M₁₆×C₇₃	C₁₆	1	1	yes	characteristic
22	C₂×C₈×C₇₃	—	2	1	no	—
23	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
24	C₁₆×C₇₃	—	2	1	no	—
25	C₁₆×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
26	C₁₆×C₇₃	M₁₆	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	#64408	1	1	yes	characteristic
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	#10596833	1	1	yes	characteristic
56	M₃₂	—	2	1	no	—
57	M₃₂	#10596833	1	1	yes	characteristic
58	C₂×C₁₆	—	2	1	no	—
59	C₂×C₁₆	C₂×#64408	1	1	yes	characteristic
60	C₃₂	—	73	1	no	—
61	C₃₂	—	73	1	no	—
62	C₃₂	—	146	1	no	—
63	#41	#11924031	1	1	yes	characteristic
64	M₁₆	—	2	1	no	—
65	M₁₆	#11924031	1	1	yes	characteristic
66	C₂×C₈	—	2	1	no	—
67	C₂×C₈	C₂×#10596833	1	1	yes	characteristic
68	C₁₆	—	2	1	no	—
69	C₁₆	C₂²⋊#64408	1	1	yes	characteristic
70	C₁₆	#11924037	1	1	yes	characteristic
71	Q₈	#14556641	1	1	yes	characteristic
72	D₄	—	2	1	no	—
73	D₄	#14556641	1	1	yes	characteristic
74	C₂×C₄	—	2	1	no	—
75	C₂×C₄	C₂×#11924031	1	1	yes	characteristic
76	C₈	—	2	1	no	—
77	C₈	C₂²⋊#10596833	1	1	yes	characteristic
78	C₈	#14556665	1	1	yes	characteristic
79	C₂²	—	2	1	no	—
80	C₂²	#17989519	1	1	yes	characteristic
81	C₄	—	2	1	no	—
82	C₄	C₂²⋊#11924031	1	1	yes	characteristic
83	C₄	C₂×#14556641	1	1	yes	characteristic
84	C₂	—	2	1	no	—
85	C₂	—	4	1	no	—
86	C₂	C₂²⋊#14556641	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	C₂²⋊#64408
commutator	C₄×C₇₃	C₂×C₃₂
Frattini	C₂×C₁₆	C₂×#64408
Fitting	#504×C₇₃	C₄
socle	C₂×C₇₃	C₂²⋊C₃₂
prime radical	#41×C₇₃	C₁₆
cosocle	C₂×#14556497	C₂²
hypercenter	#504	#64408
hypocenter	C₇₃	#7514
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₁₆ ◁ C₂×C₁₆ ◁ #504

Complements

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

N	H ≅ G/N	type
1. #22625100	C₂	semidirect
2. C₂×#17989426	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
#22625100	2
C₂×#17989426	2
#22612225	2
#7514	73