Subgroups — C₂³×C₄⋊Dic₁₆₇ #36337299

Order 21376.

2488 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×Dic₁₆₇	C₂	1	2	yes	maximal
2	C₂²×C₄⋊Dic₁₆₇	C₂	1	28	yes	maximal, direct product factor
3	C₂⁴×C₄×C₁₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂³×Dic₁₆₇	—	2	16	no	—
5	C₂³×Dic₁₆₇	C₂²	1	28	yes	—
6	C₂×C₄⋊Dic₁₆₇	C₂²	1	112	yes	direct product factor
7	C₂⁵×C₁₆₇	C₂²	1	1	yes	characteristic
8	C₂³×C₄×C₁₆₇	C₄	1	16	yes	complemented
9	C₂³×C₄×C₁₆₇	C₂²	1	14	yes	—
10	C₂²×Dic₁₆₇	—	2	112	no	—
11	C₂²×Dic₁₆₇	C₂³	1	56	yes	—
12	C₄⋊Dic₁₆₇	C₂³	1	64	yes	direct product factor
13	C₂⁴×C₁₆₇	C₂×C₄	1	8	yes	—
14	C₂⁴×C₁₆₇	D₄	1	8	yes	—
15	C₂⁴×C₁₆₇	Q₈	1	8	yes	—
16	C₂⁴×C₁₆₇	C₂³	1	7	yes	—
17	C₂²×C₄×C₁₆₇	C₂×C₄	1	112	yes	complemented
18	C₂²×C₄×C₁₆₇	C₂³	1	28	yes	—
19	C₂×Dic₁₆₇	—	2	112	no	—
20	C₂×Dic₁₆₇	C₂⁴	1	16	yes	—
21	C₂³×C₁₆₇	C₄⋊C₄	1	64	yes	complemented
22	C₂³×C₁₆₇	C₂²×C₄	1	28	yes	—
23	C₂³×C₁₆₇	C₂×D₄	1	28	yes	—
24	C₂³×C₁₆₇	C₂×Q₈	1	28	yes	—
25	C₂³×C₁₆₇	C₂⁴	1	7	yes	—
26	C₂×C₄×C₁₆₇	C₂²×C₄	1	112	yes	complemented
27	C₂×C₄×C₁₆₇	C₂⁴	1	8	yes	—
28	C₂²×C₁₆₇	C₂×C₄⋊C₄	1	112	yes	complemented
29	C₂²×C₁₆₇	C₂³×C₄	1	14	yes	—
30	C₂²×C₁₆₇	C₂²×D₄	1	14	yes	—
31	C₂²×C₁₆₇	C₂²×Q₈	1	14	yes	—
32	C₂²×C₁₆₇	C₂⁵	1	1	yes	characteristic
33	C₄×C₁₆₇	C₂³×C₄	1	16	yes	complemented
34	Dic₁₆₇	—	2	16	no	—
35	C₂×C₁₆₇	C₂²×C₄⋊C₄	1	28	yes	complemented
36	C₂×C₁₆₇	C₂⁴×C₄	1	1	yes	characteristic
37	C₂×C₁₆₇	C₂³×D₄	1	1	yes	characteristic
38	C₂×C₁₆₇	C₂³×Q₈	1	1	yes	characteristic
39	C₁₆₇	C₂³×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
40	C₂³×C₄⋊C₄	—	167	1	no	maximal, Sylow
41	C₂⁴×C₄	—	167	2	no	—
42	C₂⁴×C₄	D₁₆₇	1	1	yes	characteristic
43	C₂²×C₄⋊C₄	—	167	28	no	—
44	C₂⁵	C₂×D₁₆₇	1	1	yes	characteristic
45	C₂³×C₄	—	334	16	no	—
46	C₂³×C₄	—	167	28	no	—
47	C₂³×C₄	Dic₁₆₇	1	16	yes	complemented
48	C₂³×C₄	C₂×D₁₆₇	1	14	yes	—
49	C₂×C₄⋊C₄	—	167	112	no	—
50	C₂⁴	C₂×Dic₁₆₇	1	8	yes	—
51	C₂⁴	D₆₆₈	1	8	yes	—
52	C₂⁴	Dic₃₃₄	1	8	yes	—
53	C₂⁴	C₂²×D₁₆₇	1	7	yes	—
54	C₂²×C₄	—	334	112	no	—
55	C₂²×C₄	—	167	56	no	—
56	C₂²×C₄	C₂×Dic₁₆₇	1	112	yes	complemented
57	C₂²×C₄	C₂²×D₁₆₇	1	28	yes	—
58	C₄⋊C₄	—	167	64	no	—
59	C₂³	C₄⋊Dic₁₆₇	1	64	yes	direct product factor
60	C₂³	C₂²×Dic₁₆₇	1	28	yes	—
61	C₂³	C₂×D₆₆₈	1	28	yes	—
62	C₂³	C₂×Dic₃₃₄	1	28	yes	—
63	C₂³	C₂³×D₁₆₇	1	7	yes	—
64	C₂×C₄	—	334	112	no	—
65	C₂×C₄	—	167	16	no	—
66	C₂×C₄	C₂²×Dic₁₆₇	1	112	yes	complemented
67	C₂×C₄	C₂³×D₁₆₇	1	8	yes	—
68	C₂²	C₂×C₄⋊Dic₁₆₇	1	112	yes	direct product factor
69	C₂²	C₂³×Dic₁₆₇	1	14	yes	—
70	C₂²	C₂²×D₆₆₈	1	14	yes	—
71	C₂²	C₂²×Dic₃₃₄	1	14	yes	—
72	C₂²	C₂⁴×D₁₆₇	1	1	yes	characteristic
73	C₄	—	334	16	no	—
74	C₄	C₂³×Dic₁₆₇	1	16	yes	complemented
75	C₂	C₂²×C₄⋊Dic₁₆₇	1	28	yes	direct product factor
76	C₂	C₂⁴×Dic₁₆₇	1	1	yes	characteristic
77	C₂	C₂³×D₆₆₈	1	1	yes	characteristic
78	C₂	C₂³×Dic₃₃₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂×D₁₆₇
commutator	C₂×C₁₆₇	C₂⁴×C₄
Frattini	C₂²	C₂⁴×D₁₆₇
Fitting	C₂⁴×C₄×C₁₆₇	C₂
socle, prime radical	C₂⁵×C₁₆₇	C₂²
cosocle	C₂²×C₁₆₇	C₂⁵
hypercenter	C₂⁴×C₄	D₁₆₇
hypocenter	C₁₆₇	C₂³×C₄⋊C₄
LCS	G ▷ C₂×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₂⁵ ◁ C₂⁴×C₄

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄⋊Dic₁₆₇	C₂	direct (N × H)
6. C₂×C₄⋊Dic₁₆₇	C₂²	direct (N × H)
8. C₂³×C₄×C₁₆₇	C₄	semidirect
12. C₄⋊Dic₁₆₇	C₂³	direct (N × H)
17. C₂²×C₄×C₁₆₇	C₂×C₄	semidirect
21. C₂³×C₁₆₇	C₄⋊C₄	semidirect
26. C₂×C₄×C₁₆₇	C₂²×C₄	semidirect
28. C₂²×C₁₆₇	C₂×C₄⋊C₄	semidirect
33. C₄×C₁₆₇	C₂³×C₄	semidirect
35. C₂×C₁₆₇	C₂²×C₄⋊C₄	semidirect
39. C₁₆₇	C₂³×C₄⋊C₄	semidirect
47. C₂³×C₄	Dic₁₆₇	semidirect
56. C₂²×C₄	C₂×Dic₁₆₇	semidirect
59. C₂³	C₄⋊Dic₁₆₇	direct (N × H)
66. C₂×C₄	C₂²×Dic₁₆₇	semidirect
68. C₂²	C₂×C₄⋊Dic₁₆₇	direct (N × H)
74. C₄	C₂³×Dic₁₆₇	semidirect
75. C₂	C₂²×C₄⋊Dic₁₆₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×Dic₁₆₇	2
C₂²×C₄⋊Dic₁₆₇	2
C₂⁴×C₄×C₁₆₇	2
C₂³×C₄⋊C₄	167

Subgroups — C23×C4⋊Dic167 #36337299

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₄⋊Dic₁₆₇ #36337299