Subgroups — #16839867

Order 3904.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11758202	C₂	1	1	yes	maximal, characteristic
2	C₄×Dic₁₂₂	C₂	1	1	yes	maximal, characteristic
3	#14519708	C₂	1	1	yes	maximal, characteristic
4	#14519701	C₂	1	1	yes	maximal, characteristic
5	#14519700	C₂	1	1	yes	maximal, characteristic
6	#14519688	C₂	1	1	yes	maximal, characteristic
7	C₄×Q₈×C₆₁	C₂	1	1	yes	maximal, characteristic
8	C₂×Q₈×C₆₁	C₂²	1	1	yes	characteristic
9	C₂×Dic₁₂₂	C₂²	1	1	yes	characteristic
10	C₄⋊C₄×C₆₁	—	2	1	no	—
11	C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
12	C₄²×C₆₁	—	2	1	no	—
13	C₄²×C₆₁	C₂²	1	1	yes	characteristic
14	#11758202	C₄	1	4	yes	complemented
15	C₄⋊Dic₆₁	C₂²	1	1	yes	characteristic
16	#11758196	—	2	1	no	—
17	C₄×Dic₆₁	—	2	1	no	—
18	C₂×#92418	C₂²	1	1	yes	characteristic
19	C₂×#92418	C₂²	1	1	yes	characteristic
20	Q₈×C₆₁	—	2	1	no	—
21	Q₈×C₆₁	C₂×C₄	1	2	yes	—
22	C₂×C₄×C₆₁	—	2	1	no	—
23	C₂×C₄×C₆₁	—	2	1	no	—
24	C₂×C₄×C₆₁	D₄	1	1	yes	characteristic
25	C₂×C₄×C₆₁	D₄	1	1	yes	characteristic
26	C₂×C₄×C₆₁	C₂³	1	1	yes	characteristic
27	C₂×Dic₆₁	—	2	1	no	—
28	C₂×Dic₆₁	—	2	1	no	—
29	Dic₁₂₂	—	2	1	no	—
30	Dic₁₂₂	C₂×C₄	1	2	yes	—
31	#92418	—	2	1	no	—
32	#92418	C₂×C₄	1	2	yes	—
33	C₂²×C₆₁	C₂×D₄	1	1	yes	characteristic
34	C₄×C₆₁	—	2	2	no	—
35	C₄×C₆₁	—	2	1	no	—
36	C₄×C₆₁	—	2	1	no	—
37	C₄×C₆₁	—	4	1	no	—
38	C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
39	C₄×C₆₁	#41	1	1	yes	characteristic
40	Dic₆₁	—	2	2	no	—
41	Dic₆₁	—	4	1	no	—
42	C₂×C₆₁	C₄×D₄	1	1	yes	characteristic
43	C₂×C₆₁	#137	1	1	yes	characteristic
44	C₂×C₆₁	#137	1	1	yes	characteristic
45	#441	—	61	1	no	maximal, Sylow
46	C₆₁	#441	1	1	yes	characteristic, complemented, Sylow
47	C₂×Q₁₆	—	61	1	no	—
48	C₄×Q₈	—	61	1	no	—
49	C₄×Q₈	D₆₁	1	1	yes	characteristic
50	#106	—	61	1	no	—
51	#103	—	61	1	no	—
52	#103	—	61	1	no	—
53	#97	—	61	1	no	—
54	C₂×Q₈	—	61	1	no	—
55	C₂×Q₈	C₂×D₆₁	1	1	yes	characteristic
56	Q₁₆	—	61	4	no	—
57	C₂×C₈	—	61	1	no	—
58	C₂×C₈	—	61	1	no	—
59	C₄⋊C₄	—	2	1	no	—
60	C₄⋊C₄	—	61	1	no	—
61	C₄⋊C₄	—	122	1	no	—
62	C₄⋊C₄	C₂×D₆₁	1	1	yes	characteristic
63	C₄²	—	2	1	no	—
64	C₄²	—	122	1	no	—
65	C₄²	C₂×D₆₁	1	1	yes	characteristic
66	Q₈	—	2	1	no	—
67	Q₈	—	61	2	no	—
68	Q₈	—	122	1	no	—
69	Q₈	C₄×D₆₁	1	2	yes	—
70	C₂×C₄	—	2	1	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	122	1	no	—
73	C₂×C₄	—	122	1	no	—
74	C₂×C₄	C₂²⋊D₆₁	1	1	yes	characteristic
75	C₂×C₄	C₂²⋊D₆₁	1	1	yes	characteristic
76	C₂×C₄	C₂²×D₆₁	1	1	yes	characteristic
77	C₈	—	61	2	no	—
78	C₈	—	122	1	no	—
79	C₂²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
80	C₄	—	2	1	no	—
81	C₄	—	2	2	no	—
82	C₄	—	2	1	no	—
83	C₄	—	4	1	no	—
84	C₄	—	122	2	no	—
85	C₄	—	244	1	no	—
86	C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
87	C₄	#11758222	1	1	yes	characteristic
88	C₂	C₄×C₂²⋊D₆₁	1	1	yes	characteristic
89	C₂	#14519807	1	1	yes	characteristic
90	C₂	#14519808	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₆₁
commutator	C₄×C₆₁	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₆₁
Fitting	C₄×Q₈×C₆₁	C₂
socle, prime radical	C₂²×C₆₁	C₂×D₄
cosocle	C₂×C₄×C₆₁	C₂³
hypercenter	C₄×Q₈	D₆₁
hypocenter	C₆₁	#441
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
14. #11758202	C₄	semidirect
46. C₆₁	#441	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11758202	2
C₄×Dic₁₂₂	2
#14519708	2
#14519701	2
#14519700	2
#14519688	2
C₄×Q₈×C₆₁	2
#441	61