Subgroups — C₃×#89707 #13055367

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×C₃×C₄⋊Dic₇	C₂	1	6	yes	maximal
3	#89707	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₂²×C₃×C₄×C₇	C₂²	1	3	yes	—
5	C₂²×C₃×Dic₇	C₂²	1	4	yes	—
6	C₃×C₄²×C₇	C₄	1	4	yes	complemented
7	C₃×C₄⋊Dic₇	—	2	12	no	—
8	C₂×C₄²×C₇	C₂×C₃	1	1	yes	characteristic
9	C₂×C₄⋊Dic₇	C₂×C₃	1	6	yes	—
10	C₃×#384	—	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×C₃×Dic₇	—	2	12	no	—
16	C₂²×C₄×C₇	C₂²×C₃	1	3	yes	—
17	C₂²×Dic₇	C₂²×C₃	1	4	yes	—
18	C₄²×C₇	C₃×C₄	1	4	yes	complemented
19	C₄⋊Dic₇	—	2	12	no	—
20	C₂×C₃×C₄⋊C₄	—	7	6	no	—
21	C₂×C₃×C₄²	D₇	1	1	yes	characteristic
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	C₃×Dic₇	—	4	4	no	—
27	#384	—	7	1	no	Sylow
28	C₂³×C₇	C₂³×C₃	1	1	yes	characteristic
29	C₂×C₄×C₇	C₂×C₃×C₄	1	6	yes	—
30	C₂×C₄×C₇	C₃×D₄	1	6	yes	—
31	C₂×C₄×C₇	C₃×Q₈	1	6	yes	—
32	C₂×Dic₇	—	2	12	no	—
33	C₂²×C₃×C₄	—	7	4	no	—
34	C₂²×C₃×C₄	C₂×D₇	1	3	yes	—
35	C₃×C₄⋊C₄	—	14	12	no	—
36	C₃×C₄²	Dic₇	1	4	yes	complemented
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	—	7	6	no	—
41	C₂×C₄²	C₃×D₇	1	1	yes	characteristic
42	C₂²×C₇	C₂²×C₃×C₄	1	1	yes	characteristic
43	C₂²×C₇	C₂×C₃×D₄	1	3	yes	—
44	C₂²×C₇	C₂×C₃×Q₈	1	3	yes	—
45	C₄×C₇	C₃×C₄⋊C₄	1	12	yes	complemented
46	Dic₇	—	4	4	no	—
47	C₂³×C₃	C₂²×D₇	1	1	yes	characteristic
48	C₂×C₃×C₄	—	14	12	no	—
49	C₂×C₃×C₄	Dic₁₄	1	6	yes	—
50	C₂×C₃×C₄	D₂₈	1	6	yes	—
51	C₂×C₃×C₄	C₂×Dic₇	1	6	yes	—
52	C₃×C₇	#384	1	1	yes	characteristic, complemented, Hall
53	C₂²×C₄	—	7	4	no	—
54	C₂²×C₄	C₂×C₃×D₇	1	3	yes	—
55	C₄⋊C₄	—	14	12	no	—
56	C₄²	C₃×Dic₇	1	4	yes	complemented
57	C₂×C₇	C₂×C₃×C₄⋊C₄	1	3	yes	—
58	C₂×C₇	C₃×#127	1	1	yes	characteristic
59	C₂×C₇	C₃×C₄⋊Q₈	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	—	28	4	no	—
64	C₃×C₄	C₄⋊Dic₇	1	12	yes	complemented
65	C₂³	C₂²×C₃×D₇	1	1	yes	characteristic
66	C₂×C₄	—	14	12	no	—
67	C₂×C₄	C₃×Dic₁₄	1	6	yes	—
68	C₂×C₄	C₃×D₂₈	1	6	yes	—
69	C₂×C₄	C₂×C₃×Dic₇	1	6	yes	—
70	C₇	C₃×#384	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×C₃×Dic₁₄	1	3	yes	—
75	C₂²	C₂×C₃×D₂₈	1	3	yes	—
76	C₂²	C₂²×C₃×Dic₇	1	1	yes	characteristic
77	C₄	—	28	4	no	—
78	C₄	C₃×C₄⋊Dic₇	1	12	yes	complemented
79	C₃	#89707	1	1	yes	characteristic, direct product factor, Sylow
80	C₂	C₃×#6432	1	3	yes	—
81	C₂	C₃×#6437	1	1	yes	characteristic
82	C₂	C₂×C₃×C₄⋊Dic₇	1	3	yes	—

	subgroup	G/H
center	C₂³×C₃	C₂²×D₇
commutator	C₂²×C₇	C₂²×C₃×C₄
Frattini	C₂³	C₂²×C₃×D₇
Fitting	C₂×C₃×C₄²×C₇	C₂
socle, prime radical	C₂³×C₃×C₇	C₂³
cosocle	C₂³×C₇	C₂³×C₃
hypercenter	C₂×C₃×C₄²	D₇
hypocenter	C₇	C₃×#384
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄²

Complements

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

N	H ≅ G/N	type
3. #89707	C₃	direct (N × H)
6. C₃×C₄²×C₇	C₄	semidirect
18. C₄²×C₇	C₃×C₄	semidirect
25. C₃×C₄×C₇	C₄⋊C₄	semidirect
36. C₃×C₄²	Dic₇	semidirect
45. C₄×C₇	C₃×C₄⋊C₄	semidirect
52. C₃×C₇	#384	semidirect
56. C₄²	C₃×Dic₇	semidirect
64. C₃×C₄	C₄⋊Dic₇	semidirect
70. C₇	C₃×#384	semidirect
78. C₄	C₃×C₄⋊Dic₇	semidirect
79. C₃	#89707	direct (N × H)

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×C₃×C₄⋊Dic₇	2
#89707	3
C₃×#384	7

Subgroups — C3×#89707 #13055367

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#89707 #13055367