Subgroups — C₃×C₇×C₈×Q₈ #13056956

Order 1344.

204 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×C₇×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₃×C₇×C₄⋊C₈	C₂	1	3	yes	maximal
3	C₃×C₄×C₇×C₈	C₂	1	3	yes	maximal
4	C₇×C₈×Q₈	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂×C₃×C₇×Q₈	C₄	1	1	yes	characteristic
6	C₂×C₃×C₇×C₈	C₂²	1	3	yes	—
7	C₂×C₃×C₇×C₈	C₂²	1	1	yes	characteristic
8	C₃×C₇×C₄⋊C₄	C₄	1	3	yes	—
9	C₃×C₄²×C₇	C₂²	1	3	yes	—
10	C₄×C₇×Q₈	C₂×C₃	1	1	yes	characteristic
11	C₇×C₄⋊C₈	C₂×C₃	1	3	yes	—
12	C₄×C₇×C₈	C₂×C₃	1	3	yes	—
13	C₃×C₈×Q₈	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
14	C₃×C₇×Q₈	C₈	1	4	yes	direct product factor
15	C₂×C₃×C₄×C₇	C₂×C₄	1	3	yes	—
16	C₂×C₃×C₄×C₇	C₂×C₄	1	3	yes	—
17	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
18	C₃×C₇×C₈	—	2	3	no	—
19	C₃×C₇×C₈	Q₈	1	2	yes	direct product factor
20	C₂×C₇×Q₈	C₃×C₄	1	1	yes	characteristic
21	C₂×C₇×C₈	C₂²×C₃	1	3	yes	—
22	C₂×C₇×C₈	C₂²×C₃	1	1	yes	characteristic
23	C₇×C₄⋊C₄	C₃×C₄	1	3	yes	—
24	C₄²×C₇	C₂²×C₃	1	3	yes	—
25	C₃×C₄×Q₈	C₂×C₇	1	1	yes	characteristic
26	C₃×C₄⋊C₈	C₂×C₇	1	3	yes	—
27	C₃×C₄×C₈	C₂×C₇	1	3	yes	—
28	C₂²×C₃×C₇	C₂²×C₄	1	1	yes	characteristic
29	C₃×C₄×C₇	—	2	3	no	—
30	C₃×C₄×C₇	C₂×C₈	1	6	yes	—
31	C₃×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
32	C₃×C₄×C₇	#41	1	1	yes	characteristic
33	C₈×Q₈	C₃×C₇	1	1	yes	characteristic, direct product factor, Sylow
34	C₇×Q₈	C₃×C₈	1	4	yes	direct product factor
35	C₂×C₄×C₇	C₂×C₃×C₄	1	3	yes	—
36	C₂×C₄×C₇	C₂×C₃×C₄	1	3	yes	—
37	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
38	C₇×C₈	—	2	3	no	—
39	C₇×C₈	C₃×Q₈	1	2	yes	direct product factor
40	C₂×C₃×Q₈	C₄×C₇	1	1	yes	characteristic
41	C₂×C₃×C₈	C₂²×C₇	1	3	yes	—
42	C₂×C₃×C₈	C₂²×C₇	1	1	yes	characteristic
43	C₃×C₄⋊C₄	C₄×C₇	1	3	yes	—
44	C₃×C₄²	C₂²×C₇	1	3	yes	—
45	C₂×C₃×C₇	C₄×Q₈	1	1	yes	characteristic
46	C₂×C₃×C₇	C₂²×C₈	1	1	yes	characteristic
47	C₂×C₃×C₇	#131	1	1	yes	characteristic
48	C₄×Q₈	C₂×C₃×C₇	1	1	yes	characteristic
49	C₄⋊C₈	C₂×C₃×C₇	1	3	yes	—
50	C₄×C₈	C₂×C₃×C₇	1	3	yes	—
51	C₂²×C₇	C₂²×C₃×C₄	1	1	yes	characteristic
52	C₄×C₇	—	2	3	no	—
53	C₄×C₇	C₂×C₃×C₈	1	6	yes	—
54	C₄×C₇	C₂×C₃×Q₈	1	1	yes	characteristic
55	C₄×C₇	C₃×#41	1	1	yes	characteristic
56	C₃×Q₈	C₇×C₈	1	4	yes	direct product factor
57	C₂×C₃×C₄	C₂×C₄×C₇	1	3	yes	—
58	C₂×C₃×C₄	C₂×C₄×C₇	1	3	yes	—
59	C₂×C₃×C₄	C₂³×C₇	1	1	yes	characteristic
60	C₃×C₈	—	2	3	no	—
61	C₃×C₈	C₇×Q₈	1	2	yes	direct product factor
62	C₃×C₇	C₈×Q₈	1	1	yes	characteristic, direct product factor, Hall
63	C₂×Q₈	C₃×C₄×C₇	1	1	yes	characteristic
64	C₂×C₈	C₂²×C₃×C₇	1	3	yes	—
65	C₂×C₈	C₂²×C₃×C₇	1	1	yes	characteristic
66	C₄⋊C₄	C₃×C₄×C₇	1	3	yes	—
67	C₄²	C₂²×C₃×C₇	1	3	yes	—
68	C₂×C₇	C₃×C₄×Q₈	1	1	yes	characteristic
69	C₂×C₇	C₂²×C₃×C₈	1	1	yes	characteristic
70	C₂×C₇	C₃×#131	1	1	yes	characteristic
71	C₂²×C₃	C₂²×C₄×C₇	1	1	yes	characteristic
72	C₃×C₄	—	2	3	no	—
73	C₃×C₄	C₂×C₇×C₈	1	6	yes	—
74	C₃×C₄	C₂×C₇×Q₈	1	1	yes	characteristic
75	C₃×C₄	C₇×#41	1	1	yes	characteristic
76	Q₈	C₃×C₇×C₈	1	4	yes	direct product factor
77	C₂×C₄	C₂×C₃×C₄×C₇	1	3	yes	—
78	C₂×C₄	C₂×C₃×C₄×C₇	1	3	yes	—
79	C₂×C₄	C₂³×C₃×C₇	1	1	yes	characteristic
80	C₈	—	2	3	no	—
81	C₈	C₃×C₇×Q₈	1	2	yes	direct product factor
82	C₇	C₃×C₈×Q₈	1	1	yes	characteristic, direct product factor, Sylow
83	C₂×C₃	C₄×C₇×Q₈	1	1	yes	characteristic
84	C₂×C₃	C₂²×C₇×C₈	1	1	yes	characteristic
85	C₂×C₃	C₇×#131	1	1	yes	characteristic
86	C₂²	C₂²×C₃×C₄×C₇	1	1	yes	characteristic
87	C₄	—	2	3	no	—
88	C₄	C₂×C₃×C₇×C₈	1	6	yes	—
89	C₄	C₂×C₃×C₇×Q₈	1	1	yes	characteristic
90	C₄	C₃×C₇×#41	1	1	yes	characteristic
91	C₃	C₇×C₈×Q₈	1	1	yes	characteristic, direct product factor, Sylow
92	C₂	C₃×C₄×C₇×Q₈	1	1	yes	characteristic
93	C₂	C₂²×C₃×C₇×C₈	1	1	yes	characteristic
94	C₂	C₃×C₇×#131	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₇×C₈	C₂²
commutator	C₂	C₂²×C₃×C₇×C₈
Frattini, cosocle	C₂×C₄	C₂³×C₃×C₇
socle, prime radical	C₂²×C₃×C₇	C₂²×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₇×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₇×C₈×Q₈	C₃	direct (N × H)
13. C₃×C₈×Q₈	C₇	direct (N × H)
14. C₃×C₇×Q₈	C₈	direct (N × H)
19. C₃×C₇×C₈	Q₈	direct (N × H)
33. C₈×Q₈	C₃×C₇	direct (N × H)
34. C₇×Q₈	C₃×C₈	direct (N × H)
39. C₇×C₈	C₃×Q₈	direct (N × H)
56. C₃×Q₈	C₇×C₈	direct (N × H)
61. C₃×C₈	C₇×Q₈	direct (N × H)
62. C₃×C₇	C₈×Q₈	direct (N × H)
76. Q₈	C₃×C₇×C₈	direct (N × H)
81. C₈	C₃×C₇×Q₈	direct (N × H)
82. C₇	C₃×C₈×Q₈	direct (N × H)
91. C₃	C₇×C₈×Q₈	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×Q₈	2
C₃×C₇×C₄⋊C₈	2
C₃×C₄×C₇×C₈	2
C₇×C₈×Q₈	3
C₃×C₈×Q₈	7

Subgroups — C3×C7×C8×Q8 #13056956

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₇×C₈×Q₈ #13056956