Subgroups — #34012101

Order 18688.

350 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22626154	C₂	1	3	yes	maximal
2	#22626142	C₂	1	3	yes	maximal
3	#22609687	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556665	C₂²	1	3	yes	—
5	C₂²×#11924031	C₂²	1	1	yes	characteristic
6	#17989480	C₄	1	12	yes	complemented
7	#17989476	C₄	1	6	yes	complemented
8	#17989473	C₄	1	6	yes	complemented
9	#17988545	C₂²	1	3	yes	—
10	#17988344	C₄	1	3	yes	—
11	#338×C₇₃	C₄	1	1	yes	characteristic
12	#14556665	C₂×C₄	1	12	yes	—
13	C₂×#11924031	C₂×C₄	1	3	yes	—
14	C₂×#11924031	C₂×C₄	1	3	yes	—
15	C₂×#11924031	C₂×C₄	1	6	yes	—
16	C₂²×#10596831	C₂³	1	1	yes	characteristic
17	#14556550	C₂×C₄	1	3	yes	—
18	#14556501	C₂×C₄	1	6	yes	—
19	C₈×Dic₇₃	C₂×C₄	1	6	yes	—
20	C₂×M₁₆×C₇₃	C₂×C₄	1	3	yes	—
21	#11924031	C₄²	1	1	yes	characteristic
22	#11924031	C₄²	1	3	yes	—
23	#11924031	C₄²	1	12	yes	—
24	C₂×#10596831	C₂²×C₄	1	3	yes	—
25	C₂×#10596831	C₂²×C₄	1	3	yes	—
26	C₄⋊Dic₇₃	—	2	3	no	—
27	C₂²⋊Dic₇₃	—	2	3	no	—
28	C₄×Dic₇₃	C₄²	1	6	yes	—
29	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
30	M₁₆×C₇₃	—	2	6	no	—
31	C₂×C₈×C₇₃	C₄²	1	6	yes	—
32	C₂³×C₇₃	#131	1	1	yes	characteristic
33	C₂×C₄×C₇₃	C₂×C₄²	1	3	yes	—
34	C₂×C₄×C₇₃	#131	1	3	yes	—
35	C₂×Dic₇₃	—	2	6	no	—
36	C₈×C₇₃	—	2	6	no	—
37	#10596831	C₂×C₄²	1	1	yes	characteristic
38	#10596831	C₂×C₄²	1	3	yes	—
39	C₂²×C₇₃	—	4	1	no	—
40	C₂²×C₇₃	#405	1	3	yes	—
41	C₄×C₇₃	C₄³	1	1	yes	characteristic
42	C₄×C₇₃	#405	1	3	yes	—
43	Dic₇₃	—	4	6	no	—
44	#12542	—	73	1	no	maximal, Sylow
45	C₂×C₇₃	—	2	3	no	—
46	C₂×C₇₃	#1726	1	1	yes	characteristic
47	#2110	—	73	3	no	—
48	#2108	—	73	3	no	—
49	#1747	—	73	1	no	—
50	C₇₃	#12542	1	1	yes	characteristic, complemented, Sylow
51	C₂×M₃₂	—	73	3	no	—
52	C₂²×C₁₆	—	73	1	no	—
53	#405	—	73	3	no	—
54	#347	—	73	12	no	—
55	#346	—	73	6	no	—
56	C₄×C₁₆	—	73	6	no	—
57	#338	#64408	1	1	yes	characteristic
58	H(3,ℤ/4)	—	73	3	no	—
59	C₂×M₁₆	C₂×#64408	1	3	yes	—
60	C₂²×C₈	—	73	1	no	—
61	#117	—	73	3	no	—
62	M₃₂	—	73	12	no	—
63	C₂×C₁₆	—	73	3	no	—
64	C₂×C₁₆	—	73	6	no	—
65	C₂×C₁₆	—	73	3	no	—
66	#97	—	73	6	no	—
67	C₄×C₈	—	73	6	no	—
68	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
69	M₁₆	—	2	6	no	—
70	C₂×C₈	—	73	3	no	—
71	C₂×C₈	—	73	3	no	—
72	C₂×C₈	C₄×#64408	1	6	yes	—
73	C₄⋊C₄	—	146	3	no	—
74	C₂²⋊C₄	—	146	3	no	—
75	C₄²	—	73	6	no	—
76	C₁₆	—	73	1	no	—
77	C₁₆	—	73	3	no	—
78	C₁₆	—	73	12	no	—
79	C₂³	#14556679	1	1	yes	characteristic
80	C₂×C₄	—	146	6	no	—
81	C₂×C₄	C₂×C₄×#64408	1	3	yes	—
82	C₂×C₄	#14556678	1	3	yes	—
83	C₈	—	2	6	no	—
84	C₈	—	73	1	no	—
85	C₈	—	73	3	no	—
86	C₂²	—	4	1	no	—
87	C₂²	#17989552	1	3	yes	—
88	C₄	—	292	6	no	—
89	C₄	C₄²×#64408	1	1	yes	characteristic
90	C₄	#17989551	1	3	yes	—
91	C₂	—	2	3	no	—
92	C₂	#22625588	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₄²×#64408
commutator	C₄×C₇₃	C₄³
Frattini	C₂²×C₄	C₂²×#64408
Fitting	#338×C₇₃	C₄
socle	C₂×C₇₃	#1726
prime radical	C₂³×C₇₃	#131
cosocle	C₂²×#10596831	C₂³
hypercenter	#338	#64408
hypocenter	C₇₃	#12542
LCS	G ▷ C₄×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ #338

Complements

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

N	H ≅ G/N	type
6. #17989480	C₄	semidirect
7. #17989476	C₄	semidirect
8. #17989473	C₄	semidirect
50. C₇₃	#12542	semidirect

Maximal subgroups

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

subgroup	index
#22626154	2
#22626142	2
#22609687	2
#12542	73