Subgroups — #13056573

Order 1344.

276 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	#10620670	C₂	1	6	yes	maximal
3	#89709	—	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	C₂×C₄²×C₇	D₃	1	1	yes	characteristic
8	C₇⋊#95	—	3	6	no	—
9	#4937	—	7	1	no	maximal, Hall
10	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
11	C₂×C₃×C₄×C₇	—	2	6	no	—
12	C₂×C₃×C₄×C₇	C₂×C₄	1	6	yes	—
13	C₂×Dic₂₁	—	2	12	no	—
14	C₂²×C₄×C₇	C₂×D₃	1	3	yes	—
15	C₂²×Dic₇	—	3	4	no	—
16	C₄²×C₇	Dic₃	1	4	yes	complemented
17	C₂×C₃×C₄²	D₇	1	1	yes	characteristic
18	C₃⋊#95	—	7	6	no	—
19	C₂²×C₃×C₇	C₂²×C₄	1	1	yes	characteristic
20	C₂²×C₃×C₇	#41	1	6	yes	—
21	C₃×C₄×C₇	—	2	6	no	—
22	Dic₂₁	—	4	4	no	—
23	#383	—	21	1	no	Sylow
24	C₂³×C₇	C₂²×D₃	1	1	yes	characteristic
25	C₂×C₄×C₇	—	2	6	no	—
26	C₂×C₄×C₇	C₂×Dic₃	1	6	yes	—
27	C₂×Dic₇	—	6	12	no	—
28	C₂²×C₃×C₄	C₂×D₇	1	3	yes	—
29	C₂²×Dic₃	—	7	4	no	—
30	C₃×C₄²	Dic₇	1	4	yes	complemented
31	C₂×C₃×C₇	#117	1	3	yes	—
32	C₂×C₃×C₇	#126	1	4	yes	—
33	C₂×C₄²	D₂₁	1	1	yes	characteristic
34	#95	—	21	6	no	—
35	C₂²×C₇	#235	1	6	yes	—
36	C₂²×C₇	C₂²×Dic₃	1	1	yes	characteristic
37	C₄×C₇	—	2	6	no	—
38	Dic₇	—	12	4	no	—
39	C₂³×C₃	C₂²×D₇	1	1	yes	characteristic
40	C₂×C₃×C₄	—	2	6	no	—
41	C₂×C₃×C₄	C₂×Dic₇	1	6	yes	—
42	C₂×Dic₃	—	14	12	no	—
43	C₃×C₇	#383	1	1	yes	characteristic, complemented, Hall
44	C₂²×C₄	—	21	4	no	—
45	C₂²×C₄	C₂×D₂₁	1	3	yes	—
46	C₄²	Dic₂₁	1	4	yes	complemented
47	C₂×C₇	#876	1	4	yes	—
48	C₂×C₇	#926	1	3	yes	—
49	C₂²×C₃	#1157	1	6	yes	—
50	C₂²×C₃	C₂²×Dic₇	1	1	yes	characteristic
51	C₃×C₄	—	2	6	no	—
52	Dic₃	—	28	4	no	—
53	C₂³	C₂²×D₂₁	1	1	yes	characteristic
54	C₂×C₄	—	2	6	no	—
55	C₂×C₄	—	42	12	no	—
56	C₂×C₄	C₂×Dic₂₁	1	6	yes	—
57	C₇	#4937	1	1	yes	characteristic, complemented, Sylow
58	C₂×C₃	#6439	1	4	yes	—
59	C₂×C₃	#6489	1	3	yes	—
60	C₂²	#66728	1	6	yes	—
61	C₂²	C₂²×Dic₂₁	1	1	yes	characteristic
62	C₄	—	2	6	no	—
63	C₄	—	84	4	no	—
64	C₃	#89709	1	1	yes	characteristic, complemented, Sylow
65	C₂	#10621359	1	4	yes	—
66	C₂	#10621409	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	#383
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
16. C₄²×C₇	Dic₃	semidirect
30. C₃×C₄²	Dic₇	semidirect
43. C₃×C₇	#383	semidirect
46. C₄²	Dic₂₁	semidirect
57. C₇	#4937	semidirect
64. C₃	#89709	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
#10620670	2
#89709	3
#4937	7