Subgroups — #13056571

Order 1344.

372 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄²×C₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊Dic₂₁	C₂	1	6	yes	maximal
3	#89707	—	3	1	no	maximal, Hall
4	C₂²×C₃×C₄×C₇	C₂²	1	3	yes	—
5	C₂²×Dic₂₁	C₂²	1	4	yes	—
6	C₃×C₄²×C₇	C₄	1	4	yes	complemented
7	C₄⋊Dic₂₁	—	2	12	no	—
8	C₂×C₄²×C₇	D₃	1	1	yes	characteristic
9	C₂×C₄⋊Dic₇	—	3	6	no	—
10	#4935	—	7	1	no	maximal, Hall
11	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
12	C₂×C₃×C₄×C₇	C₂×C₄	1	6	yes	—
13	C₂×C₃×C₄×C₇	D₄	1	6	yes	—
14	C₂×C₃×C₄×C₇	Q₈	1	6	yes	—
15	C₂×Dic₂₁	—	2	12	no	—
16	C₂²×C₄×C₇	C₂×D₃	1	3	yes	—
17	C₂²×Dic₇	—	3	4	no	—
18	C₄²×C₇	Dic₃	1	4	yes	complemented
19	C₄⋊Dic₇	—	6	12	no	—
20	C₂×C₃×C₄²	D₇	1	1	yes	characteristic
21	C₂×C₄⋊Dic₃	—	7	6	no	—
22	C₂²×C₃×C₇	C₂²×C₄	1	1	yes	characteristic
23	C₂²×C₃×C₇	C₂×D₄	1	3	yes	—
24	C₂²×C₃×C₇	C₂×Q₈	1	3	yes	—
25	C₃×C₄×C₇	C₄⋊C₄	1	12	yes	complemented
26	Dic₂₁	—	4	4	no	—
27	#384	—	21	1	no	Sylow
28	C₂³×C₇	C₂²×D₃	1	1	yes	characteristic
29	C₂×C₄×C₇	Dic₆	1	6	yes	—
30	C₂×C₄×C₇	D₁₂	1	6	yes	—
31	C₂×C₄×C₇	C₂×Dic₃	1	6	yes	—
32	C₂×Dic₇	—	6	12	no	—
33	C₂²×C₃×C₄	C₂×D₇	1	3	yes	—
34	C₂²×Dic₃	—	7	4	no	—
35	C₃×C₄²	Dic₇	1	4	yes	complemented
36	C₄⋊Dic₃	—	14	12	no	—
37	C₂×C₃×C₇	C₂×C₄⋊C₄	1	3	yes	—
38	C₂×C₃×C₇	#127	1	1	yes	characteristic
39	C₂×C₃×C₇	C₄⋊Q₈	1	3	yes	—
40	C₂×C₄⋊C₄	—	21	6	no	—
41	C₂×C₄²	D₂₁	1	1	yes	characteristic
42	C₂²×C₇	C₂×Dic₆	1	3	yes	—
43	C₂²×C₇	C₂×D₁₂	1	3	yes	—
44	C₂²×C₇	C₂²×Dic₃	1	1	yes	characteristic
45	C₄×C₇	C₄⋊Dic₃	1	12	yes	complemented
46	Dic₇	—	12	4	no	—
47	C₂³×C₃	C₂²×D₇	1	1	yes	characteristic
48	C₂×C₃×C₄	Dic₁₄	1	6	yes	—
49	C₂×C₃×C₄	D₂₈	1	6	yes	—
50	C₂×C₃×C₄	C₂×Dic₇	1	6	yes	—
51	C₂×Dic₃	—	14	12	no	—
52	C₃×C₇	#384	1	1	yes	characteristic, complemented, Hall
53	C₂²×C₄	—	21	4	no	—
54	C₂²×C₄	C₂×D₂₁	1	3	yes	—
55	C₄⋊C₄	—	42	12	no	—
56	C₄²	Dic₂₁	1	4	yes	complemented
57	C₂×C₇	#869	1	3	yes	—
58	C₂×C₇	#874	1	1	yes	characteristic
59	C₂×C₇	C₂×C₄⋊Dic₃	1	3	yes	—
60	C₂²×C₃	C₂×Dic₁₄	1	3	yes	—
61	C₂²×C₃	C₂×D₂₈	1	3	yes	—
62	C₂²×C₃	C₂²×Dic₇	1	1	yes	characteristic
63	C₃×C₄	C₄⋊Dic₇	1	12	yes	complemented
64	Dic₃	—	28	4	no	—
65	C₂³	C₂²×D₂₁	1	1	yes	characteristic
66	C₂×C₄	—	42	12	no	—
67	C₂×C₄	Dic₄₂	1	6	yes	—
68	C₂×C₄	D₈₄	1	6	yes	—
69	C₂×C₄	C₂×Dic₂₁	1	6	yes	—
70	C₇	#4935	1	1	yes	characteristic, complemented, Sylow
71	C₂×C₃	#6432	1	3	yes	—
72	C₂×C₃	#6437	1	1	yes	characteristic
73	C₂×C₃	C₂×C₄⋊Dic₇	1	3	yes	—
74	C₂²	C₂×Dic₄₂	1	3	yes	—
75	C₂²	C₂×D₈₄	1	3	yes	—
76	C₂²	C₂²×Dic₂₁	1	1	yes	characteristic
77	C₄	—	84	4	no	—
78	C₄	C₄⋊Dic₂₁	1	12	yes	complemented
79	C₃	#89707	1	1	yes	characteristic, complemented, Sylow
80	C₂	#10621352	1	3	yes	—
81	C₂	#10621357	1	1	yes	characteristic
82	C₂	C₂×C₄⋊Dic₂₁	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₂₁
commutator	C₂²×C₃×C₇	C₂²×C₄
Fitting	C₂×C₃×C₄²×C₇	C₂
socle, prime radical, cosocle	C₂³×C₃×C₇	C₂³
hypercenter	C₂×C₄²	D₂₁
hypocenter	C₃×C₇	#384
LCS	G ▷ C₂²×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
6. C₃×C₄²×C₇	C₄	semidirect
18. C₄²×C₇	Dic₃	semidirect
25. C₃×C₄×C₇	C₄⋊C₄	semidirect
35. C₃×C₄²	Dic₇	semidirect
45. C₄×C₇	C₄⋊Dic₃	semidirect
52. C₃×C₇	#384	semidirect
56. C₄²	Dic₂₁	semidirect
63. C₃×C₄	C₄⋊Dic₇	semidirect
70. C₇	#4935	semidirect
78. C₄	C₄⋊Dic₂₁	semidirect
79. C₃	#89707	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄²×C₇	2
C₂×C₄⋊Dic₂₁	2
#89707	3
#4935	7