Subgroups — #16839701

Order 3904.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊Dic₆₁	C₂	1	1	yes	maximal, characteristic
2	#14519684	C₂	1	1	yes	maximal, characteristic
3	#14519684	C₂	1	2	yes	maximal
4	#14519684	C₂	1	2	yes	maximal
5	C₂×C₄⋊C₄×C₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄×C₆₁	C₂²	1	1	yes	characteristic
7	C₂²×C₄×C₆₁	C₂²	1	1	yes	characteristic
8	C₂²×C₄×C₆₁	C₂²	1	1	yes	characteristic
9	C₂²×Dic₆₁	C₂²	1	2	yes	—
10	C₂²×Dic₆₁	C₂²	1	2	yes	—
11	C₄⋊C₄×C₆₁	—	2	2	no	—
12	C₄⋊Dic₆₁	—	2	2	no	—
13	C₂³×C₆₁	C₂³	1	1	yes	characteristic
14	C₂×C₄×C₆₁	—	2	1	no	—
15	C₂×C₄×C₆₁	—	2	1	no	—
16	C₂×C₄×C₆₁	—	2	2	no	—
17	C₂×C₄×C₆₁	—	2	1	no	—
18	C₂×C₄×C₆₁	—	2	2	no	—
19	C₂×C₄×C₆₁	D₄	1	2	yes	—
20	C₂×C₄×C₆₁	Q₈	1	2	yes	—
21	C₂×Dic₆₁	—	2	2	no	—
22	C₂×Dic₆₁	—	2	4	no	—
23	C₂×Dic₆₁	—	2	2	no	—
24	C₂×Dic₆₁	—	2	2	no	—
25	C₂×Dic₆₁	—	2	2	no	—
26	C₂²×C₆₁	C₂×D₄	1	1	yes	characteristic
27	C₂²×C₆₁	C₂×Q₈	1	1	yes	characteristic
28	C₂²×C₆₁	#41	1	1	yes	characteristic
29	C₂²×C₆₁	#41	1	2	yes	—
30	C₂²×C₆₁	#41	1	2	yes	—
31	C₄×C₆₁	—	2	2	no	—
32	C₄×C₆₁	—	2	2	no	—
33	C₄×C₆₁	—	4	1	no	—
34	Dic₆₁	—	4	2	no	—
35	Dic₆₁	—	4	2	no	—
36	C₂×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
37	C₂×C₆₁	#123	1	1	yes	characteristic
38	C₂×C₆₁	#124	1	1	yes	characteristic
39	C₂×C₆₁	#125	1	2	yes	—
40	C₂×C₆₁	#126	1	2	yes	—
41	#400	—	61	1	no	maximal, Sylow
42	C₆₁	#400	1	1	yes	characteristic, complemented, Sylow
43	C₂×C₄⋊C₄	—	61	1	no	—
44	C₂×C₄⋊C₄	D₆₁	1	1	yes	characteristic
45	#95	—	61	1	no	—
46	#95	—	61	2	no	—
47	#95	—	61	2	no	—
48	C₂²×C₄	—	61	2	no	—
49	C₂²×C₄	—	61	2	no	—
50	C₂²×C₄	C₂×D₆₁	1	1	yes	characteristic
51	C₂²×C₄	C₂×D₆₁	1	1	yes	characteristic
52	C₂²×C₄	C₂×D₆₁	1	1	yes	characteristic
53	C₄⋊C₄	—	2	2	no	—
54	C₄⋊C₄	—	122	2	no	—
55	C₂³	C₂²×D₆₁	1	1	yes	characteristic
56	C₂×C₄	—	2	1	no	—
57	C₂×C₄	—	2	2	no	—
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	—	2	2	no	—
60	C₂×C₄	—	2	1	no	—
61	C₂×C₄	—	122	2	no	—
62	C₂×C₄	—	122	2	no	—
63	C₂×C₄	—	122	4	no	—
64	C₂×C₄	—	122	2	no	—
65	C₂×C₄	—	122	2	no	—
66	C₂×C₄	Dic₁₂₂	1	2	yes	—
67	C₂×C₄	C₂²⋊D₆₁	1	2	yes	—
68	C₂²	C₂×Dic₁₂₂	1	1	yes	characteristic
69	C₂²	#11758222	1	1	yes	characteristic
70	C₂²	#11758224	1	2	yes	—
71	C₂²	#11758226	1	2	yes	—
72	C₂²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
73	C₄	—	2	2	no	—
74	C₄	—	2	2	no	—
75	C₄	—	4	1	no	—
76	C₄	—	244	2	no	—
77	C₄	—	244	2	no	—
78	C₂	#14519756	1	1	yes	characteristic
79	C₂	#14519758	1	1	yes	characteristic
80	C₂	#14519766	1	1	yes	characteristic
81	C₂	#14519768	1	2	yes	—
82	C₂	#14519772	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₆₁
commutator, socle, prime radical, cosocle	C₂³×C₆₁	C₂³
Fitting	C₂×C₄⋊C₄×C₆₁	C₂
hypercenter	C₂×C₄⋊C₄	D₆₁
hypocenter	C₆₁	#400
LCS	G ▷ C₂³×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
42. C₆₁	#400	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊Dic₆₁	2
#14519684	2
C₂×C₄⋊C₄×C₆₁	2
#400	61