Subgroups — C₂×C₃×#63308 #13132654

Order 1680.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×Dic₃₅	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₇×Dic₅	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₅×Dic₇	C₂	1	1	yes	maximal, characteristic
4	C₃×#63308	C₂	1	4	yes	maximal, direct product factor
5	C₂×#63308	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂²×C₃×C₅×C₇	C₂²	1	1	yes	characteristic
7	C₃×Dic₃₅	C₂²	1	2	yes	—
8	C₃×C₇×Dic₅	C₂²	1	2	yes	—
9	C₃×C₅×Dic₇	C₂²	1	2	yes	—
10	C₂×C₃×Dic₁₄	—	5	1	no	maximal, Hall
11	C₂×Dic₃₅	C₂×C₃	1	1	yes	characteristic
12	C₂×C₇×Dic₅	C₂×C₃	1	1	yes	characteristic
13	C₂×C₅×Dic₇	C₂×C₃	1	1	yes	characteristic
14	#63308	C₂×C₃	1	4	yes	direct product factor
15	C₂×C₃×Dic₁₀	—	7	1	no	maximal, Hall
16	C₂×C₃×C₅×C₇	Q₈	1	2	yes	complemented
17	C₂×C₃×C₅×C₇	C₂³	1	1	yes	characteristic
18	C₂×C₃×C₄×C₇	—	5	1	no	—
19	C₂×C₃×Dic₇	—	5	1	no	—
20	C₂×C₃×Dic₇	D₅	1	1	yes	characteristic
21	C₃×Dic₁₄	—	5	4	no	—
22	C₂²×C₅×C₇	C₂²×C₃	1	1	yes	characteristic
23	Dic₃₅	C₂²×C₃	1	2	yes	—
24	C₅×Dic₇	C₂²×C₃	1	2	yes	—
25	C₇×Dic₅	C₂²×C₃	1	2	yes	—
26	C₂×C₃×C₄×C₅	—	7	1	no	—
27	C₂×C₃×Dic₅	—	7	1	no	—
28	C₂×C₃×Dic₅	D₇	1	1	yes	characteristic
29	C₃×Dic₁₀	—	7	4	no	—
30	C₂×Dic₁₄	—	5	1	no	Hall
31	C₃×C₅×C₇	C₂×Q₈	1	1	yes	characteristic, complemented, Hall
32	C₂²×C₃×C₇	C₂×D₅	1	1	yes	characteristic
33	C₃×C₄×C₇	—	5	2	no	—
34	C₃×Dic₇	—	5	2	no	—
35	C₃×Dic₇	C₂×D₅	1	2	yes	—
36	C₂×Dic₁₀	—	7	1	no	Hall
37	C₂×C₅×C₇	C₃×Q₈	1	2	yes	complemented
38	C₂×C₅×C₇	C₂³×C₃	1	1	yes	characteristic
39	C₂²×C₃×C₅	C₂×D₇	1	1	yes	characteristic
40	C₃×C₄×C₅	—	7	2	no	—
41	C₃×Dic₅	—	7	2	no	—
42	C₃×Dic₅	C₂×D₇	1	2	yes	—
43	C₂×C₄×C₇	—	5	1	no	—
44	C₂×Dic₇	—	5	1	no	—
45	C₂×Dic₇	C₃×D₅	1	1	yes	characteristic
46	Dic₁₄	—	5	4	no	—
47	C₂×C₃×Q₈	—	35	1	no	Hall
48	C₂×C₃×C₇	Dic₁₀	1	2	yes	complemented
49	C₂×C₃×C₇	C₂²×D₅	1	1	yes	characteristic
50	C₂×C₄×C₅	—	7	1	no	—
51	C₂×Dic₅	—	7	1	no	—
52	C₂×Dic₅	C₃×D₇	1	1	yes	characteristic
53	Dic₁₀	—	7	4	no	—
54	C₅×C₇	C₂×C₃×Q₈	1	1	yes	characteristic, complemented, Hall
55	C₂×C₃×C₅	Dic₁₄	1	2	yes	complemented
56	C₂×C₃×C₅	C₂²×D₇	1	1	yes	characteristic
57	C₂²×C₇	C₂×C₃×D₅	1	1	yes	characteristic
58	C₄×C₇	—	5	2	no	—
59	Dic₇	—	5	2	no	—
60	Dic₇	C₂×C₃×D₅	1	2	yes	—
61	C₃×Q₈	—	35	4	no	—
62	C₂×C₃×C₄	—	5	1	no	—
63	C₂×C₃×C₄	—	7	1	no	—
64	C₂×C₃×C₄	—	35	1	no	—
65	C₃×C₇	C₂×Dic₁₀	1	1	yes	characteristic, complemented, Hall
66	C₂²×C₅	C₂×C₃×D₇	1	1	yes	characteristic
67	C₄×C₅	—	7	2	no	—
68	Dic₅	—	7	2	no	—
69	Dic₅	C₂×C₃×D₇	1	2	yes	—
70	C₂×Q₈	—	35	1	no	Sylow
71	C₃×C₅	C₂×Dic₁₄	1	1	yes	characteristic, complemented, Hall
72	C₂×C₇	C₃×Dic₁₀	1	2	yes	complemented
73	C₂×C₇	C₂²×C₃×D₅	1	1	yes	characteristic
74	C₂²×C₃	D₅×D₇	1	1	yes	characteristic
75	C₃×C₄	—	5	2	no	—
76	C₃×C₄	—	7	2	no	—
77	C₃×C₄	—	35	2	no	—
78	C₂×C₅	C₃×Dic₁₄	1	2	yes	complemented
79	C₂×C₅	C₂²×C₃×D₇	1	1	yes	characteristic
80	Q₈	—	35	4	no	—
81	C₂×C₄	—	5	1	no	—
82	C₂×C₄	—	7	1	no	—
83	C₂×C₄	—	35	1	no	—
84	C₇	C₂×C₃×Dic₁₀	1	1	yes	characteristic, complemented, Sylow
85	C₂×C₃	#63308	1	2	yes	direct product factor
86	C₂×C₃	C₂×D₅×D₇	1	1	yes	characteristic
87	C₅	C₂×C₃×Dic₁₄	1	1	yes	characteristic, complemented, Sylow
88	C₂²	C₃×D₅×D₇	1	1	yes	characteristic
89	C₄	—	5	2	no	—
90	C₄	—	7	2	no	—
91	C₄	—	35	2	no	—
92	C₃	C₂×#63308	1	1	yes	characteristic, direct product factor, Sylow
93	C₂	C₃×#63308	1	2	yes	direct product factor
94	C₂	C₂×C₃×D₅×D₇	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂²×C₃	D₅×D₇
commutator, cosocle	C₂×C₅×C₇	C₂³×C₃
Frattini	C₂	C₂×C₃×D₅×D₇
Fitting, socle, prime radical	C₂²×C₃×C₅×C₇	C₂²
hypocenter	C₅×C₇	C₂×C₃×Q₈
LCS	G ▷ C₂×C₅×C₇ ▷ C₅×C₇
UCS	C₁ ◁ C₂²×C₃

Complements

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

N	H ≅ G/N	type
4. C₃×#63308	C₂	direct (N × H)
5. C₂×#63308	C₃	direct (N × H)
14. #63308	C₂×C₃	direct (N × H)
16. C₂×C₃×C₅×C₇	Q₈	semidirect
31. C₃×C₅×C₇	C₂×Q₈	semidirect
37. C₂×C₅×C₇	C₃×Q₈	semidirect
48. C₂×C₃×C₇	Dic₁₀	semidirect
54. C₅×C₇	C₂×C₃×Q₈	semidirect
55. C₂×C₃×C₅	Dic₁₄	semidirect
65. C₃×C₇	C₂×Dic₁₀	semidirect
71. C₃×C₅	C₂×Dic₁₄	semidirect
72. C₂×C₇	C₃×Dic₁₀	semidirect
78. C₂×C₅	C₃×Dic₁₄	semidirect
84. C₇	C₂×C₃×Dic₁₀	semidirect
85. C₂×C₃	#63308	direct (N × H)
87. C₅	C₂×C₃×Dic₁₄	semidirect
92. C₃	C₂×#63308	direct (N × H)
93. C₂	C₃×#63308	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×Dic₃₅	2
C₂×C₃×C₇×Dic₅	2
C₂×C₃×C₅×Dic₇	2
C₃×#63308	2
C₂×#63308	3
C₂×C₃×Dic₁₄	5
C₂×C₃×Dic₁₀	7

Subgroups — C2×C3×#63308 #13132654

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#63308 #13132654