Subgroups — #29570446

Order 15616.

414 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21445565	C₂	1	2	yes	maximal
2	#21443646	C₂	1	4	yes	maximal
3	C₄×#97×C₆₁	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×#92418	C₂²	1	4	yes	—
5	#16839517	C₄	1	8	yes	—
6	C₂×#97×C₆₁	C₄	1	4	yes	—
7	C₂×C₄×C₈×C₆₁	C₂²	1	2	yes	—
8	C₄³×C₆₁	C₂²	1	1	yes	characteristic
9	C₂²×#92418	C₂×C₄	1	8	yes	—
10	C₄×#92418	—	2	8	no	—
11	C₂²×C₈×C₆₁	C₂×C₄	1	4	yes	—
12	C₂×C₄²×C₆₁	C₂×C₄	1	4	yes	—
13	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
14	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
15	#97×C₆₁	—	2	8	no	—
16	C₄×C₈×C₆₁	—	2	4	no	—
17	C₂²×C₄×C₆₁	C₄²	1	4	yes	—
18	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
20	C₂×C₈×C₆₁	—	2	4	no	—
21	C₂×C₈×C₆₁	—	2	4	no	—
22	C₂×C₈×C₆₁	M₁₆	1	8	yes	—
23	C₄²×C₆₁	—	2	4	no	—
24	C₄²×C₆₁	—	2	8	no	—
25	C₄²×C₆₁	#41	1	4	yes	—
26	C₂×#92418	—	2	8	no	—
27	C₂×#92418	—	2	16	no	—
28	C₂³×C₆₁	C₂×C₄²	1	1	yes	characteristic
29	C₂×C₄×C₆₁	—	2	4	no	—
30	C₂×C₄×C₆₁	—	2	8	no	—
31	C₂×C₄×C₆₁	#117	1	2	yes	—
32	C₂×C₄×C₆₁	#117	1	4	yes	—
33	C₂×C₄×C₆₁	C₂×M₁₆	1	4	yes	—
34	C₂×C₄×C₆₁	#131	1	8	yes	—
35	C₈×C₆₁	—	2	8	no	—
36	#92418	—	4	8	no	—
37	#8134	—	61	1	no	maximal, Sylow
38	C₂²×C₆₁	#376	1	1	yes	characteristic
39	C₂²×C₆₁	C₄×M₁₆	1	2	yes	—
40	C₂²×C₆₁	#405	1	4	yes	—
41	C₄×C₆₁	—	4	4	no	—
42	C₄×C₆₁	#432	1	4	yes	—
43	C₄×C₆₁	#433	1	8	yes	—
44	#1744	—	61	2	no	—
45	C₄×#97	D₆₁	1	1	yes	characteristic
46	#1273	—	61	4	no	—
47	C₂×C₆₁	#1453	1	4	yes	—
48	C₂×C₆₁	#1745	1	1	yes	characteristic
49	C₂×C₆₁	#1746	1	2	yes	—
50	C₂×#97	Dic₆₁	1	4	yes	—
51	C₂×C₄×C₈	—	61	4	no	—
52	C₂×C₄×C₈	C₂×D₆₁	1	2	yes	—
53	C₄³	C₂×D₆₁	1	1	yes	characteristic
54	#336	—	61	8	no	—
55	C₆₁	#8134	1	1	yes	characteristic, complemented, Sylow
56	C₂²×C₈	—	61	8	no	—
57	C₂²×C₈	C₂×Dic₆₁	1	4	yes	—
58	C₂×C₄²	C₄×D₆₁	1	4	yes	—
59	C₂×C₄²	C₂×Dic₆₁	1	2	yes	—
60	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
61	#97	—	2	8	no	—
62	C₄×C₈	—	2	4	no	—
63	C₄×C₈	—	122	8	no	—
64	C₂²×C₄	C₄×Dic₆₁	1	4	yes	—
65	C₂²×C₄	C₂×C₄×D₆₁	1	2	yes	—
66	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
67	C₂×C₈	—	2	4	no	—
68	C₂×C₈	—	2	4	no	—
69	C₂×C₈	—	122	8	no	—
70	C₂×C₈	—	122	16	no	—
71	C₂×C₈	#11758194	1	8	yes	—
72	C₄²	—	2	4	no	—
73	C₄²	—	2	8	no	—
74	C₄²	#11758222	1	4	yes	—
75	C₂³	C₂×C₄×Dic₆₁	1	1	yes	characteristic
76	C₂×C₄	—	2	4	no	—
77	C₂×C₄	—	2	8	no	—
78	C₂×C₄	#14519734	1	4	yes	—
79	C₂×C₄	#14519736	1	2	yes	—
80	C₂×C₄	C₂×#11758194	1	4	yes	—
81	C₂×C₄	#14519785	1	8	yes	—
82	C₈	—	2	8	no	—
83	C₈	—	244	8	no	—
84	C₂²	#16839608	1	1	yes	characteristic
85	C₂²	C₄×#11758194	1	2	yes	—
86	C₂²	#16839720	1	4	yes	—
87	C₄	—	4	4	no	—
88	C₄	#16839819	1	4	yes	—
89	C₄	#16839824	1	8	yes	—
90	C₂	#21444116	1	4	yes	—
91	C₂	#21445568	1	1	yes	characteristic
92	C₂	#21445572	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₆₁
commutator, socle, prime radical	C₂³×C₆₁	C₂×C₄²
Fitting	C₄×#97×C₆₁	C₂
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	C₄×#97	D₆₁
hypocenter	C₆₁	#8134
LCS	G ▷ C₂³×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄² ◁ C₄×#97

Complements

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

N	H ≅ G/N	type
55. C₆₁	#8134	semidirect

Maximal subgroups

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

subgroup	index
#21445565	2
#21443646	2
C₄×#97×C₆₁	2
#8134	61