Subgroups — C₂³×D₁₃₈₄ #36368279

Order 22144.

1976 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₆₉₂	C₂	1	2	yes	maximal, complemented
2	C₂²×D₁₃₈₄	C₂	1	28	yes	maximal, direct product factor
3	C₂³×C₈×C₁₇₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂⁴×D₁₇₃	—	2	2	no	—
5	C₂²×D₆₉₂	—	2	14	no	—
6	C₂²×D₆₉₂	C₂²	1	28	yes	complemented
7	C₂×D₁₃₈₄	C₂²	1	112	yes	direct product factor
8	C₂³×C₄×C₁₇₃	C₂²	1	1	yes	characteristic
9	C₂²×C₈×C₁₇₃	C₂²	1	14	yes	complemented
10	C₂³×D₁₇₃	—	2	28	no	—
11	C₂³×D₁₇₃	—	4	16	no	—
12	C₂×D₆₉₂	—	2	84	no	—
13	C₂×D₆₉₂	C₂³	1	56	yes	complemented
14	D₁₃₈₄	C₂³	1	64	yes	direct product factor
15	C₂⁴×C₁₇₃	D₄	1	1	yes	characteristic
16	C₂²×C₄×C₁₇₃	D₄	1	7	yes	—
17	C₂²×C₄×C₁₇₃	C₂³	1	7	yes	—
18	C₂×C₈×C₁₇₃	C₂³	1	28	yes	complemented
19	C₂²×D₁₇₃	—	2	56	no	—
20	C₂²×D₁₇₃	—	4	112	no	—
21	D₆₉₂	—	2	56	no	—
22	D₆₉₂	C₂⁴	1	16	yes	complemented
23	C₂³×C₁₇₃	D₈	1	8	yes	complemented
24	C₂³×C₁₇₃	C₂×D₄	1	7	yes	—
25	C₂×C₄×C₁₇₃	C₂×D₄	1	21	yes	—
26	C₂×C₄×C₁₇₃	C₂⁴	1	7	yes	—
27	C₈×C₁₇₃	C₂⁴	1	8	yes	complemented
28	C₂²×C₁₇₃	C₂×D₈	1	28	yes	complemented
29	C₂²×C₁₇₃	C₂²×D₄	1	7	yes	—
30	C₂×D₁₇₃	—	2	16	no	—
31	C₂×D₁₇₃	—	4	112	no	—
32	C₄×C₁₇₃	C₂²×D₄	1	7	yes	—
33	C₄×C₁₇₃	C₂⁵	1	1	yes	characteristic
34	C₂×C₁₇₃	C₂²×D₈	1	14	yes	complemented
35	C₂×C₁₇₃	C₂³×D₄	1	1	yes	characteristic
36	D₁₇₃	—	4	16	no	—
37	C₁₇₃	C₂³×D₈	1	1	yes	characteristic, complemented, Sylow
38	C₂³×D₈	—	173	1	no	maximal, Sylow
39	C₂³×D₄	—	173	2	no	—
40	C₂²×D₈	—	173	28	no	—
41	C₂³×C₈	D₁₇₃	1	1	yes	characteristic, complemented
42	C₂⁵	—	346	2	no	—
43	C₂²×D₄	—	346	14	no	—
44	C₂²×D₄	—	173	28	no	—
45	C₂³×C₄	C₂×D₁₇₃	1	1	yes	characteristic
46	C₂×D₈	—	173	112	no	—
47	C₂²×C₈	C₂×D₁₇₃	1	14	yes	complemented
48	C₂⁴	—	346	28	no	—
49	C₂⁴	—	692	16	no	—
50	C₂⁴	D₆₉₂	1	1	yes	characteristic
51	C₂×D₄	—	346	84	no	—
52	C₂×D₄	—	173	56	no	—
53	C₂²×C₄	D₆₉₂	1	7	yes	—
54	C₂²×C₄	C₂²×D₁₇₃	1	7	yes	—
55	D₈	—	173	64	no	—
56	C₂×C₈	C₂²×D₁₇₃	1	28	yes	complemented
57	C₂³	—	346	56	no	—
58	C₂³	—	692	112	no	—
59	C₂³	D₁₃₈₄	1	8	yes	direct product factor
60	C₂³	C₂×D₆₉₂	1	7	yes	—
61	D₄	—	346	56	no	—
62	D₄	—	173	16	no	—
63	C₂×C₄	C₂×D₆₉₂	1	21	yes	—
64	C₂×C₄	C₂³×D₁₇₃	1	7	yes	—
65	C₈	C₂³×D₁₇₃	1	8	yes	complemented
66	C₂²	—	346	16	no	—
67	C₂²	—	692	112	no	—
68	C₂²	C₂×D₁₃₈₄	1	28	yes	direct product factor
69	C₂²	C₂²×D₆₉₂	1	7	yes	—
70	C₄	C₂²×D₆₉₂	1	7	yes	—
71	C₄	C₂⁴×D₁₇₃	1	1	yes	characteristic
72	C₂	—	692	16	no	—
73	C₂	C₂²×D₁₃₈₄	1	14	yes	direct product factor
74	C₂	C₂³×D₆₉₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	D₆₉₂
commutator, cosocle	C₄×C₁₇₃	C₂⁵
Frattini	C₄	C₂⁴×D₁₇₃
Fitting	C₂³×C₈×C₁₇₃	C₂
socle	C₂⁴×C₁₇₃	D₄
hypercenter	C₂³×C₈	D₁₇₃
hypocenter	C₁₇₃	C₂³×D₈
LCS	G ▷ C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
1. C₂³×D₆₉₂	C₂	semidirect
2. C₂²×D₁₃₈₄	C₂	direct (N × H)
3. C₂³×C₈×C₁₇₃	C₂	semidirect
6. C₂²×D₆₉₂	C₂²	semidirect
7. C₂×D₁₃₈₄	C₂²	direct (N × H)
9. C₂²×C₈×C₁₇₃	C₂²	semidirect
13. C₂×D₆₉₂	C₂³	semidirect
14. D₁₃₈₄	C₂³	direct (N × H)
18. C₂×C₈×C₁₇₃	C₂³	semidirect
22. D₆₉₂	C₂⁴	semidirect
23. C₂³×C₁₇₃	D₈	semidirect
27. C₈×C₁₇₃	C₂⁴	semidirect
28. C₂²×C₁₇₃	C₂×D₈	semidirect
34. C₂×C₁₇₃	C₂²×D₈	semidirect
37. C₁₇₃	C₂³×D₈	semidirect
41. C₂³×C₈	D₁₇₃	semidirect
47. C₂²×C₈	C₂×D₁₇₃	semidirect
56. C₂×C₈	C₂²×D₁₇₃	semidirect
59. C₂³	D₁₃₈₄	direct (N × H)
65. C₈	C₂³×D₁₇₃	semidirect
68. C₂²	C₂×D₁₃₈₄	direct (N × H)
73. C₂	C₂²×D₁₃₈₄	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×D₆₉₂	2
C₂²×D₁₃₈₄	2
C₂³×C₈×C₁₇₃	2
C₂³×D₈	173

Subgroups — C23×D1384 #36368279

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×D₁₃₈₄ #36368279