Subgroups — C₃×#11716101 #72246602

Order 2352.

200 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×C₇⋊D₇	C₂	1	1	yes	maximal, characteristic
2	C₃×C₇×Dic₁₄	C₂	1	2	yes	maximal, complemented
3	C₃×#87642	C₂	1	2	yes	maximal, complemented
4	C₃×#87639	C₂	1	2	yes	maximal
5	#11716101	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂×C₃×C₇⋊D₇	C₂²	1	1	yes	characteristic
7	C₃×C₄×C₇²	C₂²	1	1	yes	characteristic
8	C₃×C₇⋊Dic₇	C₂²	1	1	yes	characteristic
9	C₃×C₇×Dic₇	C₂²	1	4	yes	—
10	C₄×C₇⋊D₇	C₂×C₃	1	1	yes	characteristic
11	C₇×Dic₁₄	C₂×C₃	1	2	yes	complemented
12	#87642	C₂×C₃	1	2	yes	complemented
13	#87639	C₂×C₃	1	2	yes	—
14	C₃×Q₈×D₇	—	7	2	no	maximal
15	C₂×C₃×C₇²	C₂³	1	1	yes	characteristic
16	C₃×C₇⋊D₇	Q₈	1	2	yes	complemented
17	C₂×C₇⋊D₇	C₂²×C₃	1	1	yes	characteristic
18	C₄×C₇²	C₂²×C₃	1	1	yes	characteristic
19	C₇⋊Dic₇	C₂²×C₃	1	1	yes	characteristic
20	C₇×Dic₇	C₂²×C₃	1	4	yes	—
21	C₃×C₇×Q₈	—	7	2	no	—
22	C₃×C₄×D₇	—	7	2	no	—
23	C₃×C₄×D₇	—	7	4	no	—
24	C₃×C₄×D₇	—	14	3	no	—
25	C₃×Dic₁₄	—	7	4	no	—
26	C₃×Dic₁₄	D₇	1	2	yes	complemented
27	C₃×C₇²	C₂×Q₈	1	1	yes	characteristic, complemented, Hall
28	Q₈×D₇	—	7	2	no	—
29	C₂×C₇²	C₂³×C₃	1	1	yes	characteristic
30	C₇⋊D₇	C₃×Q₈	1	2	yes	complemented
31	C₂×C₃×D₇	—	7	2	no	—
32	C₂×C₃×D₇	—	14	3	no	—
33	C₃×C₄×C₇	—	2	3	no	—
34	C₃×C₄×C₇	—	7	4	no	—
35	C₃×C₄×C₇	C₂×D₇	1	2	yes	—
36	C₃×Dic₇	—	7	2	no	—
37	C₃×Dic₇	—	14	3	no	—
38	C₃×Dic₇	C₂×D₇	1	4	yes	—
39	C₇×Q₈	—	7	2	no	—
40	C₄×D₇	—	7	2	no	—
41	C₄×D₇	—	7	4	no	—
42	C₄×D₇	—	14	3	no	—
43	Dic₁₄	—	7	4	no	—
44	Dic₁₄	C₃×D₇	1	2	yes	complemented
45	C₇²	C₂×C₃×Q₈	1	1	yes	characteristic, complemented, Sylow
46	C₂×C₃×Q₈	—	49	1	no	Hall
47	C₂×C₃×C₇	—	2	3	no	—
48	C₂×C₃×C₇	C₂²×D₇	1	2	yes	—
49	C₃×D₇	—	7	4	no	—
50	C₃×D₇	—	14	6	no	—
51	C₂×D₇	—	7	2	no	—
52	C₂×D₇	—	14	3	no	—
53	C₄×C₇	—	2	3	no	—
54	C₄×C₇	—	7	4	no	—
55	C₄×C₇	C₂×C₃×D₇	1	2	yes	—
56	Dic₇	—	7	2	no	—
57	Dic₇	—	14	3	no	—
58	Dic₇	C₂×C₃×D₇	1	4	yes	—
59	C₃×Q₈	—	7	2	no	—
60	C₃×Q₈	—	49	2	no	—
61	C₂×C₃×C₄	—	49	2	no	—
62	C₂×C₃×C₄	—	49	1	no	—
63	C₃×C₇	—	2	3	no	—
64	C₃×C₇	Q₈×D₇	1	2	yes	complemented
65	C₂×Q₈	—	49	1	no	Sylow
66	C₂×C₇	—	2	3	no	—
67	C₂×C₇	C₂²×C₃×D₇	1	2	yes	—
68	D₇	—	7	4	no	—
69	D₇	—	14	6	no	—
70	C₂²×C₃	—	49	1	no	—
71	C₃×C₄	—	7	4	no	—
72	C₃×C₄	—	49	1	no	—
73	C₃×C₄	D₇²	1	1	yes	characteristic
74	Q₈	—	7	2	no	—
75	Q₈	—	49	2	no	—
76	C₂×C₄	—	49	2	no	—
77	C₂×C₄	—	49	1	no	—
78	C₇	—	2	3	no	—
79	C₇	C₃×Q₈×D₇	1	2	yes	complemented
80	C₂×C₃	—	49	2	no	—
81	C₂×C₃	C₂×D₇²	1	1	yes	characteristic
82	C₂²	—	49	1	no	—
83	C₄	—	7	4	no	—
84	C₄	—	49	1	no	—
85	C₄	C₃×D₇²	1	1	yes	characteristic
86	C₃	#11716101	1	1	yes	characteristic, direct product factor, Sylow
87	C₂	—	49	2	no	—
88	C₂	C₂×C₃×D₇²	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃	C₂×D₇²
commutator, cosocle	C₂×C₇²	C₂³×C₃
Frattini	C₂	C₂×C₃×D₇²
Fitting	C₃×C₄×C₇²	C₂²
socle	C₂×C₃×C₇²	C₂³
prime radical	C₂×C₃×C₇⋊D₇	C₂²
hypercenter	C₃×C₄	D₇²
hypocenter	C₇²	C₂×C₃×Q₈
LCS	G ▷ C₂×C₇² ▷ C₇²
UCS	C₁ ◁ C₂×C₃ ◁ C₃×C₄

Complements

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

N	H ≅ G/N	type
2. C₃×C₇×Dic₁₄	C₂	semidirect
3. C₃×#87642	C₂	semidirect
5. #11716101	C₃	direct (N × H)
11. C₇×Dic₁₄	C₂×C₃	semidirect
12. #87642	C₂×C₃	semidirect
16. C₃×C₇⋊D₇	Q₈	semidirect
26. C₃×Dic₁₄	D₇	semidirect
27. C₃×C₇²	C₂×Q₈	semidirect
30. C₇⋊D₇	C₃×Q₈	semidirect
44. Dic₁₄	C₃×D₇	semidirect
45. C₇²	C₂×C₃×Q₈	semidirect
64. C₃×C₇	Q₈×D₇	semidirect
79. C₇	C₃×Q₈×D₇	semidirect
86. C₃	#11716101	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₄×C₇⋊D₇	2
C₃×C₇×Dic₁₄	2
C₃×#87642	2
C₃×#87639	2
#11716101	3
C₃×Q₈×D₇	7

Subgroups — C3×#11716101 #72246602

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#11716101 #72246602