Subgroups — #34013302

Order 18688.

342 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22624969	C₂	1	6	yes	maximal
2	C₄²×#10596831	C₂	1	1	yes	maximal, characteristic
3	C₂²×#11924031	C₂²	1	4	yes	—
4	C₂×C₄×#10596831	C₄	1	3	yes	—
5	C₂×C₄×#10596831	C₂²	1	3	yes	—
6	C₄³×C₇₃	C₄	1	1	yes	characteristic
7	C₂×#11924031	—	2	12	no	—
8	C₂²×#10596831	C₂×C₄	1	3	yes	—
9	C₂²×#10596831	C₂³	1	1	yes	characteristic
10	C₄×#10596831	—	2	6	no	—
11	C₄×#10596831	—	2	6	no	—
12	C₂×C₄²×C₇₃	C₈	1	3	yes	—
13	C₂×C₄²×C₇₃	C₈	1	1	yes	characteristic
14	C₂×C₄²×C₇₃	C₂×C₄	1	3	yes	—
15	#11924031	—	4	4	no	—
16	C₂×#10596831	—	2	3	no	—
17	C₂×#10596831	—	2	3	no	—
18	C₂×#10596831	M₁₆	1	6	yes	—
19	C₂×#10596831	#41	1	6	yes	—
20	C₂²×C₄×C₇₃	C₂×C₈	1	3	yes	—
21	C₂²×C₄×C₇₃	C₂×C₈	1	3	yes	—
22	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
23	C₄²×C₇₃	—	2	6	no	—
24	C₄²×C₇₃	—	2	3	no	—
25	C₄²×C₇₃	—	2	3	no	—
26	C₄²×C₇₃	C₁₆	1	4	yes	complemented
27	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
28	C₂×C₄×C₇₃	—	2	3	no	—
29	C₂×C₄×C₇₃	—	2	3	no	—
30	C₂×C₄×C₇₃	—	2	3	no	—
31	C₂×C₄×C₇₃	—	2	3	no	—
32	C₂×C₄×C₇₃	C₂×C₁₆	1	6	yes	—
33	C₂×C₄×C₇₃	M₃₂	1	6	yes	—
34	C₂×C₄×C₇₃	#117	1	3	yes	—
35	C₂×C₄×C₇₃	C₂×M₁₆	1	3	yes	—
36	#10596831	—	2	6	no	—
37	#10596831	#126	1	4	yes	—
38	C₂²×C₇₃	#431	1	3	yes	—
39	C₂²×C₇₃	C₂²×C₁₆	1	1	yes	characteristic
40	C₂²×C₇₃	C₂×M₃₂	1	3	yes	—
41	C₄×C₇₃	—	2	6	no	—
42	C₄×C₇₃	—	2	6	no	—
43	C₄×C₇₃	#383	1	1	yes	characteristic
44	C₄×C₇₃	#432	1	3	yes	—
45	#12894	—	73	1	no	maximal, Sylow
46	C₂×C₇₃	#1839	1	1	yes	characteristic
47	C₂×C₇₃	#2162	1	3	yes	—
48	C₂×C₇₃	#2163	1	3	yes	—
49	C₄²×C₈	—	73	1	no	—
50	#1374	—	73	6	no	—
51	C₇₃	#12894	1	1	yes	characteristic, complemented, Sylow
52	C₂²×C₁₆	—	73	4	no	—
53	C₂×C₄×C₈	—	73	3	no	—
54	C₂×C₄×C₈	—	73	3	no	—
55	C₄³	#64408	1	1	yes	characteristic
56	C₂²×C₈	—	73	1	no	—
57	C₂²×C₈	—	73	3	no	—
58	C₂×C₄²	#10596833	1	3	yes	—
59	C₂×C₄²	#10596833	1	1	yes	characteristic
60	C₂×C₄²	C₂×#64408	1	3	yes	—
61	C₂×C₁₆	—	146	12	no	—
62	C₄×C₈	—	146	6	no	—
63	C₄×C₈	—	146	6	no	—
64	C₂²×C₄	C₂×#10596833	1	3	yes	—
65	C₂²×C₄	C₂×#10596833	1	3	yes	—
66	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
67	C₂×C₈	—	73	6	no	—
68	C₂×C₈	—	73	6	no	—
69	C₂×C₈	—	146	3	no	—
70	C₂×C₈	—	146	3	no	—
71	C₄²	—	2	3	no	—
72	C₄²	—	2	6	no	—
73	C₄²	—	2	3	no	—
74	C₄²	#11924031	1	4	yes	complemented
75	C₁₆	—	292	4	no	—
76	C₂³	C₂²×#10596833	1	1	yes	characteristic
77	C₂×C₄	—	2	3	no	—
78	C₂×C₄	—	2	3	no	—
79	C₂×C₄	—	2	3	no	—
80	C₂×C₄	—	2	3	no	—
81	C₂×C₄	C₂×#11924031	1	6	yes	—
82	C₂×C₄	#14556665	1	6	yes	—
83	C₂×C₄	#14556670	1	3	yes	—
84	C₂×C₄	D₇₃⋊M₁₆	1	3	yes	—
85	C₈	—	73	4	no	—
86	C₈	—	146	6	no	—
87	C₂²	#17989603	1	3	yes	—
88	C₂²	C₂²×#11924031	1	1	yes	characteristic
89	C₂²	C₂×#14556665	1	3	yes	—
90	C₄	—	2	6	no	—
91	C₄	—	2	6	no	—
92	C₄	#17989532	1	1	yes	characteristic
93	C₄	#17989610	1	3	yes	—
94	C₂	#22625858	1	1	yes	characteristic
95	C₂	#22626288	1	3	yes	—
96	C₂	#22626292	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#64408
commutator	C₂²×C₇₃	C₂²×C₁₆
Fitting	C₄³×C₇₃	C₄
socle, prime radical	C₂³×C₇₃	C₂²×C₈
cosocle	C₂²×#10596831	C₂³
hypercenter	C₄³	#64408
hypocenter	C₇₃	#12894
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³

Complements

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

N	H ≅ G/N	type
26. C₄²×C₇₃	C₁₆	semidirect
51. C₇₃	#12894	semidirect
74. C₄²	#11924031	semidirect

Maximal subgroups

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

subgroup	index
#22624969	2
C₄²×#10596831	2
#12894	73