Subgroups — #29567667

Order 15616.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#14519683	C₂	1	1	yes	maximal, characteristic
2	#21443891	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#2252	C₂	1	1	yes	maximal, characteristic
4	C₂×#14519683	—	2	1	no	—
5	#16839515	—	2	1	no	—
6	C₆₁×#500	—	2	1	no	—
7	C₆₁×#366	—	2	1	no	—
8	C₆₁×#366	C₄	1	2	yes	—
9	C₄×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
10	C₄⋊#92418	—	4	1	no	—
11	#14519683	—	4	1	no	—
12	C₄⋊Q₈×C₆₁	—	4	1	no	—
13	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
14	C₂×C₁₆×C₆₁	D₄	1	1	yes	characteristic
15	#107×C₆₁	—	2	2	no	—
16	#107×C₆₁	—	4	1	no	—
17	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
18	C₂×Q₈×C₆₁	—	8	1	no	—
19	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
20	C₂×C₈×C₆₁	SD₁₆	1	1	yes	characteristic
21	C₄⋊C₄×C₆₁	—	4	2	no	—
22	C₄⋊C₄×C₆₁	—	8	1	no	—
23	C₄²×C₆₁	D₈	1	1	yes	characteristic
24	C₂×#92418	—	8	1	no	—
25	C₁₆×C₆₁	—	2	1	no	—
26	C₁₆×C₆₁	SD₁₆	1	1	yes	characteristic
27	C₁₆×C₆₁	Q₁₆	1	1	yes	characteristic
28	Q₈×C₆₁	—	16	2	no	—
29	C₂×C₄×C₆₁	—	8	2	no	—
30	C₂×C₄×C₆₁	#102	1	1	yes	characteristic
31	C₂×C₄×C₆₁	D₁₆	1	1	yes	characteristic
32	C₂×C₄×C₆₁	SD₃₂	1	1	yes	characteristic
33	C₈×C₆₁	—	2	1	no	—
34	C₈×C₆₁	#101	1	1	yes	characteristic
35	C₈×C₆₁	#103	1	1	yes	characteristic
36	#92418	—	16	1	no	—
37	#7451	—	61	1	no	maximal, Sylow
38	C₂²×C₆₁	#357	1	1	yes	characteristic
39	C₄×C₆₁	—	2	1	no	—
40	C₄×C₆₁	—	16	2	no	—
41	C₄×C₆₁	#332	1	1	yes	characteristic
42	C₄×C₆₁	#362	1	1	yes	characteristic
43	C₄×C₆₁	SD₆₄	1	1	yes	characteristic
44	C₄×C₆₁	Q₆₄	1	1	yes	characteristic
45	#2252	D₆₁	1	1	yes	characteristic
46	C₄⋊C₃₂	—	61	1	no	—
47	#1371	—	61	1	no	—
48	C₂×C₆₁	#1364	1	1	yes	characteristic
49	C₂×C₆₁	#1416	1	1	yes	characteristic
50	C₂×C₆₁	#1419	1	1	yes	characteristic
51	#500	—	2	1	no	—
52	C₂×C₃₂	—	122	1	no	—
53	#366	—	2	1	no	—
54	#366	Dic₆₁	1	2	yes	—
55	C₄×C₁₆	C₂×D₆₁	1	1	yes	characteristic
56	#335	—	122	1	no	—
57	C₆₁	#7451	1	1	yes	characteristic, complemented, Sylow
58	C₄⋊Q₈	—	4	1	no	—
59	C₂×C₁₆	C₂×Dic₆₁	1	1	yes	characteristic
60	C₂×C₁₆	C₂²⋊D₆₁	1	1	yes	characteristic
61	#107	—	2	2	no	—
62	#107	—	4	1	no	—
63	C₄⋊C₈	—	244	1	no	—
64	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
65	C₃₂	—	244	1	no	—
66	C₂×Q₈	—	8	1	no	—
67	C₂×C₈	—	488	1	no	—
68	C₂×C₈	#11758200	1	1	yes	characteristic
69	C₂×C₈	C₂²⋊Dic₆₁	1	1	yes	characteristic
70	C₄⋊C₄	—	4	2	no	—
71	C₄⋊C₄	—	8	1	no	—
72	C₄²	#11758199	1	1	yes	characteristic
73	C₁₆	—	2	1	no	—
74	C₁₆	#11758201	1	1	yes	characteristic
75	C₁₆	#11758202	1	1	yes	characteristic
76	Q₈	—	16	2	no	—
77	C₂×C₄	—	8	2	no	—
78	C₂×C₄	#14519698	1	1	yes	characteristic
79	C₂×C₄	#14519718	1	1	yes	characteristic
80	C₂×C₄	#14519721	1	1	yes	characteristic
81	C₈	—	2	1	no	—
82	C₈	—	976	1	no	—
83	C₈	#14519696	1	1	yes	characteristic
84	C₈	#14519701	1	1	yes	characteristic
85	C₂²	#16839563	1	1	yes	characteristic
86	C₄	—	2	1	no	—
87	C₄	—	16	2	no	—
88	C₄	#16839508	1	1	yes	characteristic
89	C₄	#16839578	1	1	yes	characteristic
90	C₄	#16839599	1	1	yes	characteristic
91	C₄	#16839602	1	1	yes	characteristic
92	C₂	#21443874	1	1	yes	characteristic
93	C₂	#21444000	1	1	yes	characteristic
94	C₂	#21444009	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16839563
commutator	C₂×C₁₆×C₆₁	C₂×C₄
Frattini	C₄×C₁₆	C₂×D₆₁
Fitting	C₆₁×#2252	C₂
socle, prime radical	C₂²×C₆₁	#357
cosocle	C₄×C₁₆×C₆₁	C₂²
hypercenter	#2252	D₆₁
hypocenter	C₆₁	#7451
LCS	G ▷ C₂×C₁₆×C₆₁ ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #2252

Complements

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

N	H ≅ G/N	type
57. C₆₁	#7451	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊#14519683	2
#21443891	2
C₆₁×#2252	2
#7451	61