Subgroups — C₂²×C₇×SD₁₆ #90591

Order 448.

298 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₇×D₄	C₂	1	1	yes	maximal, characteristic
3	C₂×C₇×SD₁₆	C₂	1	12	yes	maximal, direct product factor
4	C₂²×C₇×C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂⁴×C₇	—	2	1	no	—
6	C₂×C₇×Q₈	—	2	3	no	—
7	C₂×C₇×Q₈	C₂²	1	6	yes	complemented
8	C₂×C₇×D₄	—	2	3	no	—
9	C₂×C₇×D₄	C₂²	1	6	yes	—
10	C₂²×C₄×C₇	—	2	1	no	—
11	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
12	C₇×SD₁₆	C₂²	1	16	yes	direct product factor
13	C₂×C₇×C₈	C₂²	1	6	yes	complemented
14	C₂²×SD₁₆	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
15	C₂³×C₇	—	2	6	no	—
16	C₂³×C₇	—	4	4	no	—
17	C₂³×C₇	D₄	1	1	yes	characteristic
18	C₇×Q₈	—	2	6	no	—
19	C₇×Q₈	C₂³	1	4	yes	complemented
20	C₇×D₄	—	2	6	no	—
21	C₇×D₄	C₂³	1	4	yes	—
22	C₂×C₄×C₇	—	2	6	no	—
23	C₂×C₄×C₇	D₄	1	3	yes	—
24	C₂×C₄×C₇	C₂³	1	3	yes	—
25	C₇×C₈	C₂³	1	4	yes	complemented
26	C₂²×Q₈	C₂×C₇	1	1	yes	characteristic, complemented
27	C₂²×D₄	C₂×C₇	1	1	yes	characteristic
28	C₂×SD₁₆	C₂×C₇	1	12	yes	direct product factor
29	C₂²×C₈	C₂×C₇	1	1	yes	characteristic, complemented
30	C₂²×C₇	—	2	4	no	—
31	C₂²×C₇	—	4	12	no	—
32	C₂²×C₇	SD₁₆	1	4	yes	direct product factor
33	C₂²×C₇	C₂×D₄	1	3	yes	—
34	C₄×C₇	—	2	4	no	—
35	C₄×C₇	C₂×D₄	1	3	yes	—
36	C₄×C₇	C₂⁴	1	1	yes	characteristic
37	C₂⁴	—	2	1	no	—
38	C₂×Q₈	—	2	3	no	—
39	C₂×Q₈	C₂²×C₇	1	6	yes	complemented
40	C₂×D₄	—	2	3	no	—
41	C₂×D₄	C₂²×C₇	1	6	yes	—
42	C₂²×C₄	—	2	1	no	—
43	C₂²×C₄	C₂²×C₇	1	1	yes	characteristic
44	SD₁₆	C₂²×C₇	1	16	yes	direct product factor
45	C₂×C₈	C₂²×C₇	1	6	yes	complemented
46	C₂×C₇	—	4	4	no	—
47	C₂×C₇	C₂×SD₁₆	1	6	yes	direct product factor
48	C₂×C₇	C₂²×D₄	1	1	yes	characteristic
49	C₂³	—	2	6	no	—
50	C₂³	—	4	4	no	—
51	C₂³	C₇×D₄	1	1	yes	characteristic
52	Q₈	—	2	6	no	—
53	Q₈	C₂³×C₇	1	4	yes	complemented
54	D₄	—	2	6	no	—
55	D₄	C₂³×C₇	1	4	yes	—
56	C₂×C₄	—	2	6	no	—
57	C₂×C₄	C₇×D₄	1	3	yes	—
58	C₂×C₄	C₂³×C₇	1	3	yes	—
59	C₈	C₂³×C₇	1	4	yes	complemented
60	C₇	C₂²×SD₁₆	1	1	yes	characteristic, direct product factor, Sylow
61	C₂²	—	2	4	no	—
62	C₂²	—	4	12	no	—
63	C₂²	C₇×SD₁₆	1	4	yes	direct product factor
64	C₂²	C₂×C₇×D₄	1	3	yes	—
65	C₄	—	2	4	no	—
66	C₄	C₂×C₇×D₄	1	3	yes	—
67	C₄	C₂⁴×C₇	1	1	yes	characteristic
68	C₂	—	4	4	no	—
69	C₂	C₂×C₇×SD₁₆	1	6	yes	direct product factor
70	C₂	C₂²×C₇×D₄	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₇	D₄
commutator, Frattini, cosocle	C₄	C₂⁴×C₇
prime radical	C₂²×C₇×D₄	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂²×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₇×Q₈	C₂	semidirect
3. C₂×C₇×SD₁₆	C₂	direct (N × H)
4. C₂²×C₇×C₈	C₂	semidirect
7. C₂×C₇×Q₈	C₂²	semidirect
12. C₇×SD₁₆	C₂²	direct (N × H)
13. C₂×C₇×C₈	C₂²	semidirect
14. C₂²×SD₁₆	C₇	direct (N × H)
19. C₇×Q₈	C₂³	semidirect
25. C₇×C₈	C₂³	semidirect
26. C₂²×Q₈	C₂×C₇	semidirect
28. C₂×SD₁₆	C₂×C₇	direct (N × H)
29. C₂²×C₈	C₂×C₇	semidirect
32. C₂²×C₇	SD₁₆	direct (N × H)
39. C₂×Q₈	C₂²×C₇	semidirect
44. SD₁₆	C₂²×C₇	direct (N × H)
45. C₂×C₈	C₂²×C₇	semidirect
47. C₂×C₇	C₂×SD₁₆	direct (N × H)
53. Q₈	C₂³×C₇	semidirect
59. C₈	C₂³×C₇	semidirect
60. C₇	C₂²×SD₁₆	direct (N × H)
63. C₂²	C₇×SD₁₆	direct (N × H)
69. C₂	C₂×C₇×SD₁₆	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₇×Q₈	2
C₂²×C₇×D₄	2
C₂×C₇×SD₁₆	2
C₂²×C₇×C₈	2
C₂²×SD₁₆	7

Subgroups — C22×C7×SD16 #90591

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₇×SD₁₆ #90591