Subgroups — C₂×C₇×#214 #10621330

Order 672.

136 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₇×D₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#214	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₇×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₇×SD₁₆	—	3	1	no	maximal, Hall
6	C₂³×C₃×C₇	—	2	1	no	—
7	C₃×C₇×D₄	—	2	1	no	—
8	C₃×C₇×D₄	C₂²	1	2	yes	—
9	C₂×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
10	C₂×C₇×Dic₃	—	2	1	no	—
11	C₇×Dic₆	—	2	1	no	—
12	C₇×Dic₆	C₂²	1	2	yes	complemented
13	C₇×C₃⋊C₈	C₂²	1	2	yes	complemented
14	C₂×C₇×Q₈	—	3	1	no	—
15	C₂×C₇×D₄	D₃	1	1	yes	characteristic
16	C₇×SD₁₆	—	3	4	no	—
17	C₂×C₇×C₈	—	3	1	no	—
18	C₂×#214	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₂²×C₃×C₇	—	2	2	no	—
20	C₂²×C₃×C₇	—	4	2	no	—
21	C₂²×C₃×C₇	D₄	1	1	yes	characteristic
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	—	2	1	no	—
26	C₇×Q₈	—	3	2	no	—
27	C₇×Q₈	—	6	1	no	—
28	C₇×D₄	—	2	1	no	—
29	C₇×D₄	C₂×D₃	1	2	yes	—
30	C₂×C₄×C₇	—	6	1	no	—
31	C₂×C₄×C₇	C₂×D₃	1	1	yes	characteristic
32	C₇×C₈	—	3	2	no	—
33	C₂×C₃×D₄	C₂×C₇	1	1	yes	characteristic
34	C₂×Dic₆	C₂×C₇	1	1	yes	characteristic, complemented
35	#214	C₂×C₇	1	4	yes	direct product factor
36	C₂×C₃⋊C₈	C₂×C₇	1	1	yes	characteristic, complemented
37	C₂×C₃×C₇	—	4	2	no	—
38	C₂×C₃×C₇	SD₁₆	1	2	yes	complemented
39	C₂×C₃×C₇	C₂×D₄	1	1	yes	characteristic
40	C₂×SD₁₆	—	3	1	no	Sylow
41	C₂²×C₇	—	2	2	no	—
42	C₂²×C₇	—	4	2	no	—
43	C₂²×C₇	#67	1	1	yes	characteristic
44	C₄×C₇	—	6	2	no	—
45	C₄×C₇	#67	1	1	yes	characteristic
46	C₄×C₇	C₂²×D₃	1	1	yes	characteristic
47	C₂³×C₃	—	2	1	no	—
48	C₃×D₄	—	2	1	no	—
49	C₃×D₄	C₂²×C₇	1	2	yes	—
50	C₂×C₃×C₄	C₂²×C₇	1	1	yes	characteristic
51	C₂×Dic₃	—	2	1	no	—
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	D₃×C₇	1	1	yes	characteristic
58	SD₁₆	—	3	4	no	—
59	C₂×C₈	—	3	1	no	—
60	C₂×C₇	—	4	2	no	—
61	C₂×C₇	#214	1	2	yes	direct product factor
62	C₂×C₇	C₂×#67	1	1	yes	characteristic
63	C₂²×C₃	—	2	2	no	—
64	C₂²×C₃	—	4	2	no	—
65	C₂²×C₃	C₇×D₄	1	1	yes	characteristic
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	—	2	1	no	—
70	Q₈	—	3	2	no	—
71	Q₈	—	6	1	no	—
72	D₄	—	2	1	no	—
73	D₄	C₂×D₃×C₇	1	2	yes	—
74	C₂×C₄	—	6	1	no	—
75	C₂×C₄	C₂×D₃×C₇	1	1	yes	characteristic
76	C₈	—	3	2	no	—
77	C₇	C₂×#214	1	1	yes	characteristic, direct product factor, Sylow
78	C₂×C₃	—	4	2	no	—
79	C₂×C₃	C₇×SD₁₆	1	2	yes	complemented
80	C₂×C₃	C₂×C₇×D₄	1	1	yes	characteristic
81	C₂²	—	2	2	no	—
82	C₂²	—	4	2	no	—
83	C₂²	C₇×#67	1	1	yes	characteristic
84	C₄	—	6	2	no	—
85	C₄	C₇×#67	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	—	4	2	no	—
89	C₂	C₇×#214	1	2	yes	direct product factor
90	C₂	C₂×C₇×#67	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₇	#67
commutator, cosocle	C₃×C₄	C₂³×C₇
Frattini	C₄	C₂²×D₃×C₇
Fitting, prime radical	C₂×C₃×C₇×D₄	C₂
socle	C₂²×C₃×C₇	D₄
hypercenter	C₂×C₇×D₄	D₃
hypocenter	C₃	C₂×C₇×SD₁₆
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄×C₇ ◁ C₂×C₇×D₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₇×Dic₆	C₂	semidirect
3. C₇×#214	C₂	direct (N × H)
4. C₂×C₇×C₃⋊C₈	C₂	semidirect
12. C₇×Dic₆	C₂²	semidirect
13. C₇×C₃⋊C₈	C₂²	semidirect
18. C₂×#214	C₇	direct (N × H)
34. C₂×Dic₆	C₂×C₇	semidirect
35. #214	C₂×C₇	direct (N × H)
36. C₂×C₃⋊C₈	C₂×C₇	semidirect
38. C₂×C₃×C₇	SD₁₆	semidirect
53. Dic₆	C₂²×C₇	semidirect
54. C₃⋊C₈	C₂²×C₇	semidirect
55. C₃×C₇	C₂×SD₁₆	semidirect
61. C₂×C₇	#214	direct (N × H)
77. C₇	C₂×#214	direct (N × H)
79. C₂×C₃	C₇×SD₁₆	semidirect
87. C₃	C₂×C₇×SD₁₆	semidirect
89. C₂	C₇×#214	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×C₇×D₄	2
C₂×C₇×Dic₆	2
C₇×#214	2
C₂×C₇×C₃⋊C₈	2
C₂×C₇×SD₁₆	3
C₂×#214	7

Subgroups — C2×C7×#214 #10621330

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#214 #10621330