Subgroups — #34001215

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	#22612226	C₂	1	1	yes	maximal, characteristic
4	#17989426	—	2	1	no	—
5	#17989426	—	2	1	no	—
6	#17989003	C₄	1	1	yes	characteristic
7	#17988420	—	2	1	no	—
8	C₂×#14556497	C₂²	1	1	yes	characteristic
9	C₃₂×D₇₃	—	2	1	no	—
10	M₆₄×C₇₃	C₄	1	1	yes	characteristic
11	#14556597	C₈	1	1	yes	characteristic
12	#14556531	C₈	1	1	yes	characteristic
13	#14556530	—	2	1	no	—
14	C₁₆×D₇₃	—	2	1	no	—
15	#14556497	D₄	1	1	yes	characteristic
16	#14556497	D₄	1	1	yes	characteristic
17	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
18	C₃₂×C₇₃	—	2	1	no	—
19	#11924028	C₁₆	1	1	yes	characteristic
20	#11924012	C₁₆	1	1	yes	characteristic
21	#11924011	—	2	1	no	—
22	C₈×D₇₃	—	2	1	no	—
23	#11924003	—	2	1	no	—
24	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
25	C₁₆×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
26	C₁₆×C₇₃	M₁₆	1	1	yes	characteristic
27	C₂×C₄×C₇₃	C₂×C₁₆	1	1	yes	characteristic
28	C₂²⋊D₇₃	—	2	1	no	—
29	D₂₉₂	C₃₂	1	1	yes	characteristic
30	C₄×D₇₃	—	2	1	no	—
31	Dic₁₄₆	C₃₂	1	1	yes	characteristic
32	C₈×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
33	C₈×C₇₃	M₃₂	1	1	yes	characteristic
34	#10596831	—	2	1	no	—
35	C₂²×C₇₃	M₆₄	1	1	yes	characteristic
36	C₂×D₇₃	—	2	1	no	—
37	C₄×C₇₃	C₂²⋊C₁₆	1	1	yes	characteristic
38	C₄×C₇₃	C₂×C₃₂	1	1	yes	characteristic
39	Dic₇₃	—	2	1	no	—
40	#7514	—	73	1	no	maximal, Sylow
41	C₂×C₇₃	—	2	1	no	—
42	C₂×C₇₃	C₂²⋊C₃₂	1	1	yes	characteristic
43	D₇₃	—	4	1	no	—
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	—	146	1	no	—
50	M₆₄	#64408	1	1	yes	characteristic
51	C₂×C₃₂	—	73	1	no	—
52	C₂×C₃₂	—	146	1	no	—
53	C₆₄	—	146	1	no	—
54	C₆₄	—	146	1	no	—
55	#131	—	73	1	no	—
56	M₃₂	—	73	1	no	—
57	M₃₂	—	146	1	no	—
58	C₂×C₁₆	—	146	1	no	—
59	C₂×C₁₆	C₂×#64408	1	1	yes	characteristic
60	C₃₂	—	2	1	no	—
61	C₃₂	—	73	1	no	—
62	C₃₂	—	73	1	no	—
63	#41	—	73	1	no	—
64	M₁₆	—	73	1	no	—
65	M₁₆	—	146	1	no	—
66	C₂×C₈	—	146	1	no	—
67	C₂×C₈	#11924036	1	1	yes	characteristic
68	C₁₆	—	146	1	no	—
69	C₁₆	C₂²⋊#64408	1	1	yes	characteristic
70	C₁₆	#11924038	1	1	yes	characteristic
71	Q₈	—	73	1	no	—
72	D₄	—	73	1	no	—
73	D₄	—	146	1	no	—
74	C₂×C₄	—	146	1	no	—
75	C₂×C₄	#14556664	1	1	yes	characteristic
76	C₈	—	146	1	no	—
77	C₈	#14556648	1	1	yes	characteristic
78	C₈	#14556666	1	1	yes	characteristic
79	C₂²	—	146	1	no	—
80	C₂²	#17989520	1	1	yes	characteristic
81	C₄	—	146	1	no	—
82	C₄	#17989482	1	1	yes	characteristic
83	C₄	#17989518	1	1	yes	characteristic
84	C₂	—	2	1	no	—
85	C₂	—	292	1	no	—
86	C₂	#22625045	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	C₂²⋊#64408
commutator	C₄×C₇₃	C₂×C₃₂
Frattini	C₂×C₁₆	C₂×#64408
Fitting	M₆₄×C₇₃	C₄
socle	C₂×C₇₃	C₂²⋊C₃₂
prime radical	#11924028	C₁₆
cosocle	C₂×#14556497	C₂²
hypercenter	M₆₄	#64408
hypocenter	C₇₃	#7514
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₁₆ ◁ C₂×C₁₆ ◁ M₆₄

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
#22612226	2
#7514	73