Subgroups — #33007022

Order 18688.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22612224	C₂	1	1	yes	maximal, characteristic
2	#22608142	C₂	1	1	yes	maximal, characteristic, complemented
3	#22608141	C₂	1	2	yes	maximal, complemented
4	C₆₄×D₇₃	C₂	1	2	yes	maximal, complemented
5	C₂×C₆₄×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#17989003	C₄	1	1	yes	characteristic
7	#17988421	C₄	1	1	yes	characteristic
8	#17988420	C₄	1	2	yes	—
9	C₃₂×D₇₃	C₂²	1	2	yes	—
10	#17988306	C₂²	1	2	yes	—
11	C₂×C₃₂×C₇₃	C₂²	1	1	yes	characteristic
12	C₆₄×C₇₃	C₂²	1	1	yes	characteristic
13	C₆₄×C₇₃	C₂²	1	1	yes	characteristic
14	#14556597	C₈	1	1	yes	characteristic
15	#14556531	C₈	1	1	yes	characteristic
16	#14556530	C₈	1	2	yes	—
17	C₁₆×D₇₃	C₂×C₄	1	2	yes	—
18	#14556497	C₂×C₄	1	2	yes	—
19	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
20	C₃₂×C₇₃	C₂×C₄	1	1	yes	characteristic
21	C₃₂×C₇₃	C₂³	1	1	yes	characteristic
22	#11924028	C₁₆	1	1	yes	characteristic
23	#11924012	C₁₆	1	1	yes	characteristic
24	#11924011	C₁₆	1	2	yes	—
25	C₈×D₇₃	C₂×C₈	1	2	yes	—
26	#11924003	C₂×C₈	1	2	yes	—
27	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
28	C₁₆×C₇₃	C₂×C₈	1	1	yes	characteristic
29	C₁₆×C₇₃	C₂²×C₄	1	1	yes	characteristic
30	C₂×C₄×C₇₃	C₂×C₁₆	1	1	yes	characteristic
31	C₂²⋊D₇₃	C₃₂	1	2	yes	—
32	D₂₉₂	C₃₂	1	1	yes	characteristic
33	C₄×D₇₃	C₂×C₁₆	1	2	yes	—
34	Dic₁₄₆	C₃₂	1	1	yes	characteristic
35	C₈×C₇₃	C₂×C₁₆	1	1	yes	characteristic
36	C₈×C₇₃	C₂²×C₈	1	1	yes	characteristic
37	#10596831	C₂×C₁₆	1	2	yes	—
38	C₂²×C₇₃	C₂×C₃₂	1	1	yes	characteristic
39	C₂×D₇₃	C₂×C₃₂	1	2	yes	—
40	C₄×C₇₃	C₂×C₃₂	1	1	yes	characteristic
41	C₄×C₇₃	C₂²×C₁₆	1	1	yes	characteristic
42	Dic₇₃	C₂×C₃₂	1	2	yes	—
43	#13737	—	73	1	no	maximal, Sylow
44	C₂×C₇₃	—	2	1	no	—
45	C₂×C₇₃	C₂²×C₃₂	1	1	yes	characteristic
46	D₇₃	—	2	2	no	—
47	#2258	—	73	1	no	—
48	M₁₂₈	—	73	1	no	—
49	M₁₂₈	—	73	2	no	—
50	C₂×C₆₄	—	73	2	no	—
51	C₂×C₆₄	D₇₃	1	1	yes	characteristic, complemented
52	C₇₃	#13737	1	1	yes	characteristic, complemented, Sylow
53	#504	—	73	1	no	—
54	M₆₄	—	73	1	no	—
55	M₆₄	—	73	2	no	—
56	C₂×C₃₂	—	73	2	no	—
57	C₂×C₃₂	C₂×D₇₃	1	1	yes	characteristic
58	C₆₄	—	73	2	no	—
59	C₆₄	C₂×D₇₃	1	1	yes	characteristic
60	C₆₄	C₂×D₇₃	1	1	yes	characteristic
61	#131	—	73	1	no	—
62	M₃₂	—	73	1	no	—
63	M₃₂	—	73	2	no	—
64	C₂×C₁₆	—	73	2	no	—
65	C₂×C₁₆	C₄×D₇₃	1	1	yes	characteristic
66	C₃₂	—	73	2	no	—
67	C₃₂	C₄×D₇₃	1	1	yes	characteristic
68	C₃₂	C₂²×D₇₃	1	1	yes	characteristic
69	#41	—	73	1	no	—
70	M₁₆	—	73	1	no	—
71	M₁₆	—	73	2	no	—
72	C₂×C₈	—	73	2	no	—
73	C₂×C₈	C₈×D₇₃	1	1	yes	characteristic
74	C₁₆	—	73	2	no	—
75	C₁₆	C₈×D₇₃	1	1	yes	characteristic
76	C₁₆	C₂×C₄×D₇₃	1	1	yes	characteristic
77	Q₈	—	73	1	no	—
78	D₄	—	73	2	no	—
79	D₄	—	73	1	no	—
80	C₂×C₄	—	73	2	no	—
81	C₂×C₄	C₁₆×D₇₃	1	1	yes	characteristic
82	C₈	—	73	2	no	—
83	C₈	C₁₆×D₇₃	1	1	yes	characteristic
84	C₈	C₂×C₈×D₇₃	1	1	yes	characteristic
85	C₂²	—	73	2	no	—
86	C₂²	C₃₂×D₇₃	1	1	yes	characteristic
87	C₄	—	73	2	no	—
88	C₄	C₃₂×D₇₃	1	1	yes	characteristic
89	C₄	C₂×C₁₆×D₇₃	1	1	yes	characteristic
90	C₂	—	2	1	no	—
91	C₂	—	146	2	no	—
92	C₂	C₂×C₃₂×D₇₃	1	1	yes	characteristic

	subgroup	G/H
center	C₆₄	C₂×D₇₃
commutator, socle	C₂×C₇₃	C₂²×C₃₂
Frattini	C₃₂	C₂²×D₇₃
Fitting	C₂×C₆₄×C₇₃	C₂
prime radical	#11924028	C₁₆
cosocle	C₃₂×C₇₃	C₂³
hypercenter	C₂×C₆₄	D₇₃
hypocenter	C₇₃	#13737
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₆₄ ◁ C₂×C₆₄

Complements

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

N	H ≅ G/N	type
2. #22608142	C₂	semidirect
3. #22608141	C₂	semidirect
4. C₆₄×D₇₃	C₂	semidirect
5. C₂×C₆₄×C₇₃	C₂	semidirect
51. C₂×C₆₄	D₇₃	semidirect
52. C₇₃	#13737	semidirect

Maximal subgroups

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

subgroup	index
#22612224	2
#22608142	2
#22608141	2
C₆₄×D₇₃	2
C₂×C₆₄×C₇₃	2
#13737	73