Subgroups — #30595287

Order 15616.

136 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21460957	C₂	1	2	yes	maximal
2	#21445435	C₂	1	1	yes	maximal, characteristic
3	#16840673	—	2	2	no	—
4	#16840165	—	2	2	no	—
5	#16840142	C₂²	1	1	yes	characteristic
6	C₈×#92418	C₄	1	1	yes	characteristic
7	C₆₁×#494	C₄	1	1	yes	characteristic
8	#14519838	—	4	2	no	—
9	C₂×Dic₂₄₄	—	4	1	no	—
10	#14519777	D₄	1	2	yes	—
11	#14519710	—	4	4	no	—
12	C₄×#92418	—	2	1	no	—
13	C₂×Q₁₆×C₆₁	—	4	1	no	—
14	C₄⋊Q₈×C₆₁	—	2	1	no	—
15	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
16	C₂×Q₈×C₆₁	—	4	1	no	—
17	C₂×Dic₁₂₂	—	2	2	no	—
18	#11758217	—	8	2	no	—
19	Q₁₆×C₆₁	—	8	2	no	—
20	C₂×C₈×C₆₁	—	2	1	no	—
21	C₄⋊C₄×C₆₁	—	4	2	no	—
22	C₄²×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
23	C₄⋊Dic₆₁	—	4	2	no	—
24	C₂×#92418	—	2	2	no	—
25	Dic₂₄₄	—	8	2	no	—
26	Q₈×C₆₁	—	8	2	no	—
27	C₂×C₄×C₆₁	—	2	1	no	—
28	C₂×C₄×C₆₁	—	8	1	no	—
29	C₂×C₄×C₆₁	#99	1	1	yes	characteristic
30	C₂×Dic₆₁	—	4	2	no	—
31	Dic₁₂₂	—	8	2	no	—
32	#92420	—	16	2	no	—
33	C₈×C₆₁	—	2	2	no	—
34	#92418	—	2	4	no	—
35	#7532	—	61	1	no	maximal, Sylow
36	C₂²×C₆₁	#353	1	1	yes	characteristic
37	C₄×C₆₁	—	2	2	no	—
38	C₄×C₆₁	—	2	1	no	—
39	C₄×C₆₁	—	16	1	no	—
40	Dic₆₁	—	8	2	no	—
41	#1713	—	61	1	no	—
42	#1411	—	61	2	no	—
43	C₂×C₆₁	—	2	1	no	—
44	C₂×C₆₁	#1408	1	1	yes	characteristic
45	#500	—	122	2	no	—
46	#494	—	61	1	no	—
47	#494	#6923	1	1	yes	characteristic
48	#356	—	122	2	no	—
49	C₈²	—	61	1	no	—
50	C₆₁	#7532	1	1	yes	characteristic, complemented, Sylow
51	C₂×Q₁₆	—	4	1	no	—
52	C₂×Q₁₆	—	244	1	no	—
53	C₄⋊Q₈	—	2	1	no	—
54	C₄⋊Q₈	—	61	2	no	—
55	#107	—	244	4	no	—
56	#101	—	244	2	no	—
57	C₄×C₈	—	122	1	no	—
58	C₄×C₈	C₂×#6923	1	1	yes	characteristic
59	C₂×Q₈	—	4	1	no	—
60	C₂×Q₈	—	122	2	no	—
61	Q₁₆	—	8	2	no	—
62	Q₁₆	—	488	2	no	—
63	M₁₆	—	488	2	no	—
64	C₂×C₈	—	2	1	no	—
65	C₂×C₈	—	122	2	no	—
66	C₄⋊C₄	—	4	2	no	—
67	C₄⋊C₄	—	244	2	no	—
68	C₄²	C₂²⋊#6923	1	1	yes	characteristic
69	Q₈	—	8	2	no	—
70	Q₈	—	488	2	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	8	1	no	—
73	C₂×C₄	—	244	2	no	—
74	C₂×C₄	#14519835	1	1	yes	characteristic
75	C₈	—	2	2	no	—
76	C₈	—	122	4	no	—
77	C₈	—	976	2	no	—
78	C₂²	#16840666	1	1	yes	characteristic
79	C₄	—	2	2	no	—
80	C₄	—	2	1	no	—
81	C₄	—	16	1	no	—
82	C₄	—	488	2	no	—
83	C₂	—	2	1	no	—
84	C₂	#21460950	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#21460950
commutator	C₄×C₈×C₆₁	C₂×C₄
Frattini	C₄×C₈	C₂×#6923
Fitting	C₆₁×#494	C₄
socle	C₂×C₆₁	#1408
prime radical	C₂²×C₆₁	#353
cosocle	#16840142	C₂²
hypercenter	#494	#6923
hypocenter	C₆₁	#7532
LCS	G ▷ C₄×C₈×C₆₁ ▷ C₄²×C₆₁ ▷ C₂×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ #494

Complements

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

N	H ≅ G/N	type
50. C₆₁	#7532	semidirect

Maximal subgroups

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

subgroup	index
#21460957	2
#21445435	2
#7532	61