Subgroups — C₂×C₇×#204 #10621299

Order 672.

136 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×D₁₂	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×C₇×C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#204	C₂	1	4	yes	maximal, direct product factor
5	C₂×C₇×SD₁₆	—	3	1	no	maximal, Hall
6	C₂²×D₃×C₇	—	2	1	no	—
7	C₂×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
8	C₂×C₇×Dic₃	—	2	1	no	—
9	C₇×D₁₂	—	2	1	no	—
10	C₇×D₁₂	C₂²	1	2	yes	—
11	C₇×Dic₆	—	2	1	no	—
12	C₇×Dic₆	C₂²	1	2	yes	complemented
13	C₃×C₇×C₈	C₂²	1	2	yes	complemented
14	C₂×C₇×Q₈	—	3	1	no	—
15	C₂×C₇×D₄	—	3	1	no	—
16	C₇×SD₁₆	—	3	4	no	—
17	C₂×C₇×C₈	D₃	1	1	yes	characteristic, complemented
18	C₂×#204	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₂²×C₃×C₇	D₄	1	1	yes	characteristic
20	C₂×D₃×C₇	—	2	2	no	—
21	C₂×D₃×C₇	—	4	2	no	—
22	C₃×C₄×C₇	D₄	1	1	yes	characteristic
23	C₃×C₄×C₇	C₂³	1	1	yes	characteristic
24	C₇×Dic₃	—	2	2	no	—
25	C₂³×C₇	—	6	1	no	—
26	C₇×Q₈	—	3	2	no	—
27	C₇×Q₈	—	6	1	no	—
28	C₇×D₄	—	3	2	no	—
29	C₇×D₄	—	6	1	no	—
30	C₂×C₄×C₇	—	6	1	no	—
31	C₂×C₄×C₇	C₂×D₃	1	1	yes	characteristic
32	C₇×C₈	C₂×D₃	1	2	yes	complemented
33	C₂×D₁₂	C₂×C₇	1	1	yes	characteristic
34	C₂×Dic₆	C₂×C₇	1	1	yes	characteristic, complemented
35	C₂×C₃×C₈	C₂×C₇	1	1	yes	characteristic, complemented
36	#204	C₂×C₇	1	4	yes	direct product factor
37	C₂×C₃×C₇	SD₁₆	1	2	yes	complemented
38	C₂×C₃×C₇	C₂×D₄	1	1	yes	characteristic
39	D₃×C₇	—	4	2	no	—
40	C₂×SD₁₆	—	3	1	no	Sylow
41	C₂²×C₇	—	6	2	no	—
42	C₂²×C₇	—	12	2	no	—
43	C₂²×C₇	D₁₂	1	1	yes	characteristic
44	C₄×C₇	—	6	2	no	—
45	C₄×C₇	D₁₂	1	1	yes	characteristic
46	C₄×C₇	C₂²×D₃	1	1	yes	characteristic
47	C₂²×D₃	—	2	1	no	—
48	C₂×C₃×C₄	C₂²×C₇	1	1	yes	characteristic
49	C₂×Dic₃	—	2	1	no	—
50	D₁₂	—	2	1	no	—
51	D₁₂	C₂²×C₇	1	2	yes	—
52	Dic₆	—	2	1	no	—
53	Dic₆	C₂²×C₇	1	2	yes	complemented
54	C₃×C₈	C₂²×C₇	1	2	yes	complemented
55	C₃×C₇	C₂×SD₁₆	1	1	yes	characteristic, complemented, Hall
56	C₂×Q₈	—	3	1	no	—
57	C₂×D₄	—	3	1	no	—
58	SD₁₆	—	3	4	no	—
59	C₂×C₈	D₃×C₇	1	1	yes	characteristic, complemented
60	C₂×C₇	—	12	2	no	—
61	C₂×C₇	#204	1	2	yes	direct product factor
62	C₂×C₇	C₂×D₁₂	1	1	yes	characteristic
63	C₂²×C₃	C₇×D₄	1	1	yes	characteristic
64	C₂×D₃	—	2	2	no	—
65	C₂×D₃	—	4	2	no	—
66	C₃×C₄	C₇×D₄	1	1	yes	characteristic
67	C₃×C₄	C₂³×C₇	1	1	yes	characteristic
68	Dic₃	—	2	2	no	—
69	C₂³	—	6	1	no	—
70	Q₈	—	3	2	no	—
71	Q₈	—	6	1	no	—
72	D₄	—	3	2	no	—
73	D₄	—	6	1	no	—
74	C₂×C₄	—	6	1	no	—
75	C₂×C₄	C₂×D₃×C₇	1	1	yes	characteristic
76	C₈	C₂×D₃×C₇	1	2	yes	complemented
77	C₇	C₂×#204	1	1	yes	characteristic, direct product factor, Sylow
78	C₂×C₃	C₇×SD₁₆	1	2	yes	complemented
79	C₂×C₃	C₂×C₇×D₄	1	1	yes	characteristic
80	D₃	—	4	2	no	—
81	C₂²	—	6	2	no	—
82	C₂²	—	12	2	no	—
83	C₂²	C₇×D₁₂	1	1	yes	characteristic
84	C₄	—	6	2	no	—
85	C₄	C₇×D₁₂	1	1	yes	characteristic
86	C₄	C₂²×D₃×C₇	1	1	yes	characteristic
87	C₃	C₂×C₇×SD₁₆	1	1	yes	characteristic, complemented, Sylow
88	C₂	—	12	2	no	—
89	C₂	C₇×#204	1	2	yes	direct product factor
90	C₂	C₂×C₇×D₁₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₇	D₁₂
commutator, cosocle	C₃×C₄	C₂³×C₇
Frattini	C₄	C₂²×D₃×C₇
Fitting	C₂×C₃×C₇×C₈	C₂
socle	C₂²×C₃×C₇	D₄
prime radical	C₂×C₇×D₁₂	C₂
hypercenter	C₂×C₇×C₈	D₃
hypocenter	C₃	C₂×C₇×SD₁₆
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄×C₇ ◁ C₂×C₇×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×C₇×Dic₆	C₂	semidirect
3. C₂×C₃×C₇×C₈	C₂	semidirect
4. C₇×#204	C₂	direct (N × H)
12. C₇×Dic₆	C₂²	semidirect
13. C₃×C₇×C₈	C₂²	semidirect
17. C₂×C₇×C₈	D₃	semidirect
18. C₂×#204	C₇	direct (N × H)
32. C₇×C₈	C₂×D₃	semidirect
34. C₂×Dic₆	C₂×C₇	semidirect
35. C₂×C₃×C₈	C₂×C₇	semidirect
36. #204	C₂×C₇	direct (N × H)
37. C₂×C₃×C₇	SD₁₆	semidirect
53. Dic₆	C₂²×C₇	semidirect
54. C₃×C₈	C₂²×C₇	semidirect
55. C₃×C₇	C₂×SD₁₆	semidirect
59. C₂×C₈	D₃×C₇	semidirect
61. C₂×C₇	#204	direct (N × H)
76. C₈	C₂×D₃×C₇	semidirect
77. C₇	C₂×#204	direct (N × H)
78. C₂×C₃	C₇×SD₁₆	semidirect
87. C₃	C₂×C₇×SD₁₆	semidirect
89. C₂	C₇×#204	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×C₇×D₁₂	2
C₂×C₇×Dic₆	2
C₂×C₃×C₇×C₈	2
C₇×#204	2
C₂×C₇×SD₁₆	3
C₂×#204	7

Subgroups — C2×C7×#204 #10621299

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#204 #10621299