Subgroups — #34034212

Order 18688.

342 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22627001	C₂	1	6	yes	maximal
2	C₄²×#10596833	C₂	1	1	yes	maximal, characteristic
3	C₂²×#11924040	C₂²	1	4	yes	—
4	C₂×C₄×#10596833	C₄	1	3	yes	—
5	C₂×C₄×#10596833	C₂²	1	3	yes	—
6	C₄²×Dic₇₃	C₄	1	1	yes	characteristic
7	C₂×#11924040	—	2	12	no	—
8	C₂²×#10596833	C₂×C₄	1	3	yes	—
9	C₂²×#10596833	C₂³	1	1	yes	characteristic
10	C₄×#10596833	—	2	6	no	—
11	C₄×#10596833	—	2	6	no	—
12	C₂×C₄×Dic₇₃	C₈	1	3	yes	—
13	C₂×C₄×Dic₇₃	C₂×C₄	1	3	yes	—
14	C₂×C₄²×C₇₃	C₈	1	1	yes	characteristic
15	#11924040	—	4	4	no	—
16	C₂×#10596833	—	2	3	no	—
17	C₂×#10596833	—	2	3	no	—
18	C₂×#10596833	M₁₆	1	6	yes	—
19	C₂×#10596833	#41	1	6	yes	—
20	C₂²×Dic₇₃	C₂×C₈	1	3	yes	—
21	C₂²×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
22	C₄×Dic₇₃	—	2	6	no	—
23	C₄×Dic₇₃	—	2	3	no	—
24	C₄×Dic₇₃	—	2	3	no	—
25	C₂²×C₄×C₇₃	C₂×C₈	1	3	yes	—
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	C₂×C₁₆	1	6	yes	—
31	C₂×Dic₇₃	—	2	3	no	—
32	C₂×Dic₇₃	—	2	3	no	—
33	C₂×Dic₇₃	M₃₂	1	6	yes	—
34	C₂×Dic₇₃	#117	1	3	yes	—
35	C₂×Dic₇₃	C₂×M₁₆	1	3	yes	—
36	#10596833	—	2	6	no	—
37	#10596833	#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	Dic₇₃	—	2	6	no	—
43	Dic₇₃	#383	1	1	yes	characteristic
44	Dic₇₃	#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	—	73	1	no	—
56	C₂²×C₈	—	73	1	no	—
57	C₂²×C₈	—	73	3	no	—
58	C₂×C₄²	—	73	3	no	—
59	C₂×C₄²	—	73	3	no	—
60	C₂×C₄²	#10596842	1	1	yes	characteristic
61	C₂×C₁₆	—	146	12	no	—
62	C₄×C₈	—	146	6	no	—
63	C₄×C₈	—	146	6	no	—
64	C₂²×C₄	—	73	1	no	—
65	C₂²×C₄	—	73	3	no	—
66	C₂²×C₄	C₂×#10596842	1	3	yes	—
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	—	146	3	no	—
72	C₄²	—	146	3	no	—
73	C₄²	—	146	6	no	—
74	C₄²	#11924040	1	4	yes	complemented
75	C₁₆	—	292	4	no	—
76	C₂³	C₂²×#10596842	1	1	yes	characteristic
77	C₂×C₄	—	2	3	no	—
78	C₂×C₄	—	2	3	no	—
79	C₂×C₄	—	73	3	no	—
80	C₂×C₄	—	73	3	no	—
81	C₂×C₄	—	73	6	no	—
82	C₂×C₄	—	146	3	no	—
83	C₂×C₄	—	146	3	no	—
84	C₂×C₄	C₂×#11924040	1	6	yes	—
85	C₈	—	73	4	no	—
86	C₈	—	146	6	no	—
87	C₂²	#17989700	1	3	yes	—
88	C₂²	C₂²×#11924040	1	1	yes	characteristic
89	C₂²	#17989706	1	3	yes	—
90	C₄	—	2	6	no	—
91	C₄	—	73	1	no	—
92	C₄	—	73	3	no	—
93	C₄	—	146	6	no	—
94	C₂	#22627074	1	1	yes	characteristic
95	C₂	#22627128	1	3	yes	—
96	C₂	#22627135	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂²×#10596842
commutator	C₂²×C₇₃	C₂²×C₁₆
Fitting	C₂×C₄²×C₇₃	C₈
socle, prime radical	C₂³×C₇₃	C₂²×C₈
cosocle	C₂²×#10596833	C₂³
hypercenter	C₂×C₄²	#10596842
hypocenter	C₇₃	#12894
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

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	#11924040	semidirect

Maximal subgroups

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

subgroup	index
#22627001	2
C₄²×#10596833	2
#12894	73