Subgroups — #30606130

Order 15616.

350 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21462042	C₂	1	3	yes	maximal
2	#21462030	C₂	1	3	yes	maximal
3	#21445575	C₂	1	1	yes	maximal, characteristic
4	C₂×#14519851	C₂²	1	3	yes	—
5	C₂²×#11758187	C₂²	1	1	yes	characteristic
6	#16840655	C₄	1	12	yes	complemented
7	#16840651	C₄	1	6	yes	complemented
8	#16840648	C₄	1	6	yes	complemented
9	#16839720	C₂²	1	3	yes	—
10	#16839519	C₄	1	3	yes	—
11	C₆₁×#338	C₄	1	1	yes	characteristic
12	#14519851	C₂×C₄	1	12	yes	—
13	C₂×#11758187	C₂×C₄	1	3	yes	—
14	C₂×#11758187	C₂×C₄	1	3	yes	—
15	C₂×#11758187	C₂×C₄	1	6	yes	—
16	C₂²×#92418	C₂³	1	1	yes	characteristic
17	#14519736	C₂×C₄	1	3	yes	—
18	#14519687	C₂×C₄	1	6	yes	—
19	C₈×Dic₆₁	C₂×C₄	1	6	yes	—
20	C₂×M₁₆×C₆₁	C₂×C₄	1	3	yes	—
21	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
22	M₁₆×C₆₁	—	2	6	no	—
23	C₂×C₈×C₆₁	C₄²	1	6	yes	—
24	C₂²⋊Dic₆₁	—	2	3	no	—
25	C₄⋊Dic₆₁	—	2	3	no	—
26	C₄×Dic₆₁	C₄²	1	6	yes	—
27	C₂×#92418	C₂²×C₄	1	3	yes	—
28	C₂×#92418	C₂²×C₄	1	3	yes	—
29	#11758187	C₄²	1	1	yes	characteristic
30	#11758187	C₄²	1	3	yes	—
31	#11758187	C₄²	1	12	yes	—
32	C₂³×C₆₁	#131	1	1	yes	characteristic
33	C₂×C₄×C₆₁	C₂×C₄²	1	3	yes	—
34	C₂×C₄×C₆₁	#131	1	3	yes	—
35	C₂×Dic₆₁	—	2	6	no	—
36	C₈×C₆₁	—	2	6	no	—
37	#92418	C₂×C₄²	1	1	yes	characteristic
38	#92418	C₂×C₄²	1	3	yes	—
39	#12542	—	61	1	no	maximal, Sylow
40	C₂²×C₆₁	—	4	1	no	—
41	C₂²×C₆₁	#405	1	3	yes	—
42	C₄×C₆₁	C₄³	1	1	yes	characteristic
43	C₄×C₆₁	#405	1	3	yes	—
44	Dic₆₁	—	4	6	no	—
45	#2110	—	61	3	no	—
46	#2108	—	61	3	no	—
47	#1747	—	61	1	no	—
48	C₂×C₆₁	—	2	3	no	—
49	C₂×C₆₁	#1726	1	1	yes	characteristic
50	C₂×M₃₂	—	61	3	no	—
51	C₂²×C₁₆	—	61	1	no	—
52	#405	—	61	3	no	—
53	#347	—	61	12	no	—
54	#346	—	61	6	no	—
55	C₄×C₁₆	—	61	6	no	—
56	#338	#6923	1	1	yes	characteristic
57	H(3,ℤ/4)	—	61	3	no	—
58	C₆₁	#12542	1	1	yes	characteristic, complemented, Sylow
59	C₂×M₁₆	C₂×#6923	1	3	yes	—
60	C₂²×C₈	—	61	1	no	—
61	#117	—	61	3	no	—
62	M₃₂	—	61	12	no	—
63	C₂×C₁₆	—	61	3	no	—
64	C₂×C₁₆	—	61	6	no	—
65	C₂×C₁₆	—	61	3	no	—
66	#97	—	61	6	no	—
67	C₄×C₈	—	61	6	no	—
68	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
69	M₁₆	—	2	6	no	—
70	C₂×C₈	—	61	3	no	—
71	C₂×C₈	—	61	3	no	—
72	C₂×C₈	C₄×#6923	1	6	yes	—
73	C₄⋊C₄	—	122	3	no	—
74	C₂²⋊C₄	—	122	3	no	—
75	C₄²	—	61	6	no	—
76	C₁₆	—	61	1	no	—
77	C₁₆	—	61	3	no	—
78	C₁₆	—	61	12	no	—
79	C₂³	#14519865	1	1	yes	characteristic
80	C₂×C₄	—	122	6	no	—
81	C₂×C₄	C₂×C₄×#6923	1	3	yes	—
82	C₂×C₄	#14519864	1	3	yes	—
83	C₈	—	2	6	no	—
84	C₈	—	61	1	no	—
85	C₈	—	61	3	no	—
86	C₂²	—	4	1	no	—
87	C₂²	#16840727	1	3	yes	—
88	C₄	—	244	6	no	—
89	C₄	C₄²×#6923	1	1	yes	characteristic
90	C₄	#16840726	1	3	yes	—
91	C₂	—	2	3	no	—
92	C₂	#21461476	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₄²×#6923
commutator	C₄×C₆₁	C₄³
Frattini	C₂²×C₄	C₂²×#6923
Fitting	C₆₁×#338	C₄
socle	C₂×C₆₁	#1726
prime radical	C₂³×C₆₁	#131
cosocle	C₂²×#92418	C₂³
hypercenter	#338	#6923
hypocenter	C₆₁	#12542
LCS	G ▷ C₄×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄ ◁ #338

Complements

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

N	H ≅ G/N	type
6. #16840655	C₄	semidirect
7. #16840651	C₄	semidirect
8. #16840648	C₄	semidirect
58. C₆₁	#12542	semidirect

Maximal subgroups

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

subgroup	index
#21462042	2
#21462030	2
#21445575	2
#12542	61