Subgroups — #13132245

Order 1680.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×Dic₁₅	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×Dic₂₁	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×Dic₃₅	C₂	1	1	yes	maximal, characteristic
4	Dic₇⋊Dic₅	—	3	1	no	maximal, Hall
5	C₂²×C₃×C₅×C₇	C₂²	1	1	yes	characteristic
6	C₇×Dic₁₅	—	2	1	no	—
7	C₅×Dic₂₁	C₄	1	2	yes	complemented
8	C₃×Dic₃₅	—	2	1	no	—
9	#66577	—	5	1	no	maximal, Hall
10	C₂×Dic₃₅	D₃	1	1	yes	characteristic
11	C₂×C₇×Dic₅	—	3	1	no	—
12	C₂×C₅×Dic₇	—	3	1	no	—
13	Dic₃⋊Dic₅	—	7	1	no	maximal, Hall
14	C₂×C₃×C₅×C₇	C₂×C₄	1	1	yes	characteristic
15	C₂×C₃×C₅×C₇	D₄	1	1	yes	characteristic
16	C₂×C₃×C₅×C₇	Q₈	1	1	yes	characteristic
17	C₂×Dic₂₁	D₅	1	1	yes	characteristic
18	C₂×C₇×Dic₃	—	5	1	no	—
19	C₂×C₃×Dic₇	—	5	1	no	—
20	C₂²×C₅×C₇	C₂×D₃	1	1	yes	characteristic
21	Dic₃₅	—	2	1	no	—
22	C₅×Dic₇	—	3	2	no	—
23	C₇×Dic₅	—	6	1	no	—
24	C₂×Dic₁₅	D₇	1	1	yes	characteristic
25	C₂×C₅×Dic₃	—	7	1	no	—
26	C₂×C₃×Dic₅	—	7	1	no	—
27	Dic₇⋊C₄	—	15	1	no	Hall
28	C₃×C₅×C₇	C₄⋊C₄	1	1	yes	characteristic, complemented, Hall
29	C₂²×C₃×C₇	C₂×D₅	1	1	yes	characteristic
30	Dic₂₁	Dic₅	1	2	yes	complemented
31	C₃×Dic₇	—	10	1	no	—
32	C₇×Dic₃	—	10	1	no	—
33	C₄⋊Dic₅	—	21	1	no	Hall
34	C₂×C₅×C₇	Dic₆	1	1	yes	characteristic
35	C₂×C₅×C₇	C₄×D₃	1	1	yes	characteristic
36	C₂×C₅×C₇	#67	1	1	yes	characteristic
37	C₂²×C₃×C₅	C₂×D₇	1	1	yes	characteristic
38	Dic₁₅	—	2	1	no	—
39	C₃×Dic₅	—	14	1	no	—
40	C₅×Dic₃	—	7	2	no	—
41	C₂×C₄×C₇	—	15	1	no	—
42	C₂×Dic₇	—	3	1	no	—
43	C₂×Dic₇	—	5	1	no	—
44	Dic₃⋊C₄	—	35	1	no	Hall
45	C₂×C₃×C₇	Dic₁₀	1	1	yes	characteristic
46	C₂×C₃×C₇	D₂₀	1	1	yes	characteristic
47	C₂×C₃×C₇	C₂×Dic₅	1	1	yes	characteristic
48	C₂×C₄×C₅	—	21	1	no	—
49	C₂×Dic₅	—	3	1	no	—
50	C₂×Dic₅	—	7	1	no	—
51	C₅×C₇	Dic₃⋊C₄	1	1	yes	characteristic, complemented, Hall
52	C₂×C₃×C₅	Dic₁₄	1	1	yes	characteristic
53	C₂×C₃×C₅	C₄×D₇	1	1	yes	characteristic
54	C₂×C₃×C₅	C₂²⋊D₇	1	1	yes	characteristic
55	C₂²×C₇	D₃×D₅	1	1	yes	characteristic
56	C₄×C₇	—	30	1	no	—
57	Dic₇	—	3	2	no	—
58	Dic₇	—	10	1	no	—
59	C₂×C₃×C₄	—	35	1	no	—
60	C₂×Dic₃	—	5	1	no	—
61	C₂×Dic₃	—	7	1	no	—
62	C₃×C₇	C₄⋊Dic₅	1	1	yes	characteristic, complemented, Hall
63	C₂²×C₅	D₃×D₇	1	1	yes	characteristic
64	C₄×C₅	—	21	2	no	—
65	Dic₅	—	6	1	no	—
66	Dic₅	—	14	1	no	—
67	C₄⋊C₄	—	105	1	no	Sylow
68	C₃×C₅	Dic₇⋊C₄	1	1	yes	characteristic, complemented, Hall
69	C₂×C₇	D₃×Dic₅	1	1	yes	characteristic
70	C₂×C₇	#1202	1	1	yes	characteristic
71	C₂×C₇	#1204	1	1	yes	characteristic
72	C₂²×C₃	D₅×D₇	1	1	yes	characteristic
73	C₃×C₄	—	70	1	no	—
74	Dic₃	—	7	2	no	—
75	Dic₃	—	10	1	no	—
76	C₂×C₅	D₂₁⋊C₄	1	1	yes	characteristic
77	C₂×C₅	#4245	1	1	yes	characteristic
78	C₂×C₅	#4248	1	1	yes	characteristic
79	C₂×C₄	—	15	1	no	—
80	C₂×C₄	—	21	1	no	—
81	C₂×C₄	—	35	1	no	—
82	C₇	Dic₃⋊Dic₅	1	1	yes	characteristic, complemented, Sylow
83	C₂×C₃	D₇×Dic₅	1	1	yes	characteristic
84	C₂×C₃	#63307	1	1	yes	characteristic
85	C₂×C₃	#63308	1	1	yes	characteristic
86	C₅	#66577	1	1	yes	characteristic, complemented, Sylow
87	C₂²	D₁₅⋊D₇	1	1	yes	characteristic
88	C₄	—	21	2	no	—
89	C₄	—	30	1	no	—
90	C₄	—	70	1	no	—
91	C₃	Dic₇⋊Dic₅	1	1	yes	characteristic, complemented, Sylow
92	C₂	D₂₁⋊Dic₅	1	1	yes	characteristic
93	C₂	#11719787	1	1	yes	characteristic
94	C₂	#11719788	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂²	D₁₅⋊D₇
commutator	C₂×C₃×C₅×C₇	C₂×C₄
Fitting, socle, prime radical, cosocle	C₂²×C₃×C₅×C₇	C₂²
hypocenter	C₃×C₅×C₇	C₄⋊C₄
LCS	G ▷ C₂×C₃×C₅×C₇ ▷ C₃×C₅×C₇
UCS	C₁ ◁ C₂²

Complements

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

N	H ≅ G/N	type
7. C₅×Dic₂₁	C₄	semidirect
28. C₃×C₅×C₇	C₄⋊C₄	semidirect
30. Dic₂₁	Dic₅	semidirect
51. C₅×C₇	Dic₃⋊C₄	semidirect
62. C₃×C₇	C₄⋊Dic₅	semidirect
68. C₃×C₅	Dic₇⋊C₄	semidirect
82. C₇	Dic₃⋊Dic₅	semidirect
86. C₅	#66577	semidirect
91. C₃	Dic₇⋊Dic₅	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×Dic₁₅	2
C₂×C₅×Dic₂₁	2
C₂×C₃×Dic₃₅	2
Dic₇⋊Dic₅	3
#66577	5
Dic₃⋊Dic₅	7