Subgroups — #29601068

Order 15616.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21448129	C₂	1	2	yes	maximal, complemented
2	#21444033	C₂	1	2	yes	maximal, complemented
3	D₃₉₀₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₄×D₆₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₁×Q₁₂₈	C₂	1	1	yes	maximal, characteristic, complemented
6	#16840195	—	2	2	no	—
7	#16839599	—	2	2	no	—
8	D₁₉₅₂	C₂²	1	2	yes	—
9	C₃₂×D₆₁	C₂²	1	1	yes	characteristic
10	#16839481	C₂²	1	1	yes	characteristic
11	C₆₁×Q₆₄	C₂²	1	2	yes	—
12	C₆₁×C₆₄	C₂²	1	1	yes	characteristic
13	#14519800	—	4	2	no	—
14	#14519720	—	4	2	no	—
15	D₉₇₆	—	2	2	no	—
16	C₁₆×D₆₁	D₄	1	1	yes	characteristic
17	#14519683	D₄	1	1	yes	characteristic
18	Q₃₂×C₆₁	—	2	2	no	—
19	C₃₂×C₆₁	C₂³	1	1	yes	characteristic
20	#11758226	—	8	2	no	—
21	Q₁₆×C₆₁	—	4	2	no	—
22	#11758201	—	8	2	no	—
23	D₄₈₈	—	4	2	no	—
24	C₈×D₆₁	D₈	1	1	yes	characteristic
25	C₁₆×C₆₁	C₂×D₄	1	1	yes	characteristic
26	#11758185	D₈	1	1	yes	characteristic
27	Q₈×C₆₁	—	8	2	no	—
28	D₂₄₄	—	8	2	no	—
29	D₂₄₄	—	16	2	no	—
30	C₄×D₆₁	—	16	2	no	—
31	C₄×D₆₁	D₁₆	1	1	yes	characteristic
32	C₈×C₆₁	C₂×D₈	1	1	yes	characteristic
33	#92418	D₁₆	1	1	yes	characteristic
34	#13741	—	61	1	no	maximal, Sylow
35	C₂×D₆₁	—	16	2	no	—
36	C₂×D₆₁	D₃₂	1	1	yes	characteristic
37	C₄×C₆₁	—	16	2	no	—
38	C₄×C₆₁	C₂×D₁₆	1	1	yes	characteristic
39	Dic₆₁	D₃₂	1	1	yes	characteristic
40	#2262	—	61	2	no	—
41	Q₁₂₈	D₆₁	1	1	yes	characteristic, complemented
42	SD₁₂₈	—	61	2	no	—
43	D₆₄	—	61	1	no	—
44	C₂×C₆₄	—	61	1	no	—
45	C₂×C₆₁	C₂×D₃₂	1	1	yes	characteristic
46	D₆₁	—	2	1	no	—
47	D₆₁	—	32	2	no	—
48	#508	—	122	2	no	—
49	Q₆₄	C₂×D₆₁	1	2	yes	—
50	SD₆₄	—	122	2	no	—
51	D₃₂	—	61	2	no	—
52	C₂×C₃₂	—	61	1	no	—
53	C₆₄	—	61	1	no	—
54	C₆₄	C₂×D₆₁	1	1	yes	characteristic
55	C₆₁	#13741	1	1	yes	characteristic, complemented, Sylow
56	#135	—	244	2	no	—
57	Q₃₂	—	2	2	no	—
58	SD₃₂	—	244	2	no	—
59	D₁₆	—	122	2	no	—
60	C₂×C₁₆	—	61	1	no	—
61	C₃₂	—	61	1	no	—
62	C₃₂	C₂²×D₆₁	1	1	yes	characteristic
63	#41	—	488	2	no	—
64	Q₁₆	—	4	2	no	—
65	SD₁₆	—	488	2	no	—
66	D₈	—	244	2	no	—
67	C₂×C₈	—	61	1	no	—
68	C₁₆	—	61	1	no	—
69	C₁₆	D₄×D₆₁	1	1	yes	characteristic
70	Q₈	—	8	2	no	—
71	D₄	—	976	2	no	—
72	D₄	—	488	2	no	—
73	C₂×C₄	—	61	1	no	—
74	C₂×C₄	—	976	2	no	—
75	C₈	—	61	1	no	—
76	C₈	D₈×D₆₁	1	1	yes	characteristic
77	C₂²	—	61	1	no	—
78	C₂²	—	976	2	no	—
79	C₄	—	16	2	no	—
80	C₄	—	61	1	no	—
81	C₄	D₁₆×D₆₁	1	1	yes	characteristic
82	C₂	—	122	1	no	—
83	C₂	—	1952	2	no	—
84	C₂	D₃₂×D₆₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₃₂×D₆₁
commutator, cosocle	C₃₂×C₆₁	C₂³
Frattini	C₃₂	C₂²×D₆₁
Fitting	C₆₁×Q₁₂₈	C₂
socle	C₂×C₆₁	C₂×D₃₂
hypercenter	Q₁₂₈	D₆₁
hypocenter	C₆₁	#13741
LCS	G ▷ C₃₂×C₆₁ ▷ C₁₆×C₆₁ ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂ ◁ Q₁₂₈

Complements

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

N	H ≅ G/N	type
1. #21448129	C₂	semidirect
2. #21444033	C₂	semidirect
3. D₃₉₀₄	C₂	semidirect
4. C₆₄×D₆₁	C₂	semidirect
5. C₆₁×Q₁₂₈	C₂	semidirect
41. Q₁₂₈	D₆₁	semidirect
55. C₆₁	#13741	semidirect

Maximal subgroups

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

subgroup	index
#21448129	2
#21444033	2
D₃₉₀₄	2
C₆₄×D₆₁	2
C₆₁×Q₁₂₈	2
#13741	61