Subgroups — C₂³×C₂²⋊Dic₁₆₇ #36337296

Order 21376.

4632 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×Dic₁₆₇	C₂	1	2	yes	maximal, complemented
2	C₂²×C₂²⋊Dic₁₆₇	C₂	1	28	yes	maximal, direct product factor
3	C₂⁶×C₁₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂³×Dic₁₆₇	—	2	16	no	—
5	C₂³×Dic₁₆₇	C₂²	1	28	yes	complemented
6	C₂×C₂²⋊Dic₁₆₇	C₂²	1	112	yes	direct product factor
7	C₂⁵×C₁₆₇	—	2	16	no	—
8	C₂⁵×C₁₆₇	C₄	1	16	yes	complemented
9	C₂⁵×C₁₆₇	C₂²	1	14	yes	—
10	C₂⁵×C₁₆₇	C₂²	1	1	yes	characteristic
11	C₂²×Dic₁₆₇	—	2	112	no	—
12	C₂²×Dic₁₆₇	C₂³	1	56	yes	complemented
13	C₂²⋊Dic₁₆₇	C₂³	1	64	yes	direct product factor
14	C₂⁴×C₁₆₇	—	2	128	no	—
15	C₂⁴×C₁₆₇	—	2	112	no	—
16	C₂⁴×C₁₆₇	C₂×C₄	1	8	yes	—
17	C₂⁴×C₁₆₇	C₂×C₄	1	112	yes	complemented
18	C₂⁴×C₁₆₇	D₄	1	16	yes	—
19	C₂⁴×C₁₆₇	C₂³	1	28	yes	—
20	C₂⁴×C₁₆₇	C₂³	1	7	yes	—
21	C₂×Dic₁₆₇	—	2	112	no	—
22	C₂×Dic₁₆₇	C₂⁴	1	16	yes	complemented
23	C₂³×C₁₆₇	—	2	448	no	—
24	C₂³×C₁₆₇	—	2	112	no	—
25	C₂³×C₁₆₇	C₂²⋊C₄	1	64	yes	complemented
26	C₂³×C₁₆₇	C₂²×C₄	1	28	yes	—
27	C₂³×C₁₆₇	C₂²×C₄	1	112	yes	complemented
28	C₂³×C₁₆₇	C₂×D₄	1	56	yes	—
29	C₂³×C₁₆₇	C₂⁴	1	8	yes	—
30	C₂³×C₁₆₇	C₂⁴	1	7	yes	—
31	C₂²×C₁₆₇	—	2	224	no	—
32	C₂²×C₁₆₇	—	2	16	no	—
33	C₂²×C₁₆₇	C₂×C₂²⋊C₄	1	112	yes	complemented
34	C₂²×C₁₆₇	C₂³×C₄	1	14	yes	—
35	C₂²×C₁₆₇	C₂³×C₄	1	16	yes	complemented
36	C₂²×C₁₆₇	C₂²×D₄	1	28	yes	—
37	C₂²×C₁₆₇	C₂⁵	1	1	yes	characteristic
38	Dic₁₆₇	—	2	16	no	—
39	C₂×C₁₆₇	—	2	16	no	—
40	C₂×C₁₆₇	C₂²×C₂²⋊C₄	1	28	yes	complemented
41	C₂×C₁₆₇	C₂⁴×C₄	1	1	yes	characteristic
42	C₂×C₁₆₇	C₂³×D₄	1	2	yes	—
43	C₁₆₇	C₂³×C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
44	C₂³×C₂²⋊C₄	—	167	1	no	maximal, Sylow
45	C₂⁶	D₁₆₇	1	1	yes	characteristic
46	C₂⁴×C₄	—	167	2	no	—
47	C₂²×C₂²⋊C₄	—	167	28	no	—
48	C₂⁵	—	2	16	no	—
49	C₂⁵	Dic₁₆₇	1	16	yes	complemented
50	C₂⁵	C₂×D₁₆₇	1	14	yes	—
51	C₂⁵	C₂×D₁₆₇	1	1	yes	characteristic
52	C₂³×C₄	—	334	16	no	—
53	C₂³×C₄	—	167	28	no	—
54	C₂×C₂²⋊C₄	—	167	112	no	—
55	C₂⁴	—	2	128	no	—
56	C₂⁴	—	2	112	no	—
57	C₂⁴	C₂×Dic₁₆₇	1	8	yes	—
58	C₂⁴	C₂×Dic₁₆₇	1	112	yes	complemented
59	C₂⁴	C₂²⋊D₁₆₇	1	16	yes	—
60	C₂⁴	C₂²×D₁₆₇	1	28	yes	—
61	C₂⁴	C₂²×D₁₆₇	1	7	yes	—
62	C₂²×C₄	—	334	112	no	—
63	C₂²×C₄	—	167	56	no	—
64	C₂²⋊C₄	—	167	64	no	—
65	C₂³	—	2	112	no	—
66	C₂³	—	2	448	no	—
67	C₂³	C₂²⋊Dic₁₆₇	1	64	yes	direct product factor
68	C₂³	C₂²×Dic₁₆₇	1	28	yes	—
69	C₂³	C₂²×Dic₁₆₇	1	112	yes	complemented
70	C₂³	C₂×C₂²⋊D₁₆₇	1	56	yes	—
71	C₂³	C₂³×D₁₆₇	1	8	yes	—
72	C₂³	C₂³×D₁₆₇	1	7	yes	—
73	C₂×C₄	—	334	112	no	—
74	C₂×C₄	—	167	16	no	—
75	C₂²	—	2	224	no	—
76	C₂²	—	2	16	no	—
77	C₂²	C₂×C₂²⋊Dic₁₆₇	1	112	yes	direct product factor
78	C₂²	C₂³×Dic₁₆₇	1	14	yes	—
79	C₂²	C₂³×Dic₁₆₇	1	16	yes	complemented
80	C₂²	C₂²×C₂²⋊D₁₆₇	1	28	yes	—
81	C₂²	C₂⁴×D₁₆₇	1	1	yes	characteristic
82	C₄	—	334	16	no	—
83	C₂	—	2	16	no	—
84	C₂	C₂²×C₂²⋊Dic₁₆₇	1	28	yes	direct product factor
85	C₂	C₂⁴×Dic₁₆₇	1	1	yes	characteristic
86	C₂	C₂³×C₂²⋊D₁₆₇	1	2	yes	—

	subgroup	G/H
center	C₂⁵	C₂×D₁₆₇
commutator	C₂×C₁₆₇	C₂⁴×C₄
Frattini	C₂²	C₂⁴×D₁₆₇
Fitting, prime radical	C₂⁶×C₁₆₇	C₂
socle	C₂⁵×C₁₆₇	C₂²
cosocle	C₂²×C₁₆₇	C₂⁵
hypercenter	C₂⁶	D₁₆₇
hypocenter	C₁₆₇	C₂³×C₂²⋊C₄
LCS	G ▷ C₂×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₂⁵ ◁ C₂⁶

Complements

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

N	H ≅ G/N	type
1. C₂⁴×Dic₁₆₇	C₂	semidirect
2. C₂²×C₂²⋊Dic₁₆₇	C₂	direct (N × H)
5. C₂³×Dic₁₆₇	C₂²	semidirect
6. C₂×C₂²⋊Dic₁₆₇	C₂²	direct (N × H)
8. C₂⁵×C₁₆₇	C₄	semidirect
12. C₂²×Dic₁₆₇	C₂³	semidirect
13. C₂²⋊Dic₁₆₇	C₂³	direct (N × H)
17. C₂⁴×C₁₆₇	C₂×C₄	semidirect
22. C₂×Dic₁₆₇	C₂⁴	semidirect
25. C₂³×C₁₆₇	C₂²⋊C₄	semidirect
27. C₂³×C₁₆₇	C₂²×C₄	semidirect
33. C₂²×C₁₆₇	C₂×C₂²⋊C₄	semidirect
35. C₂²×C₁₆₇	C₂³×C₄	semidirect
40. C₂×C₁₆₇	C₂²×C₂²⋊C₄	semidirect
43. C₁₆₇	C₂³×C₂²⋊C₄	semidirect
49. C₂⁵	Dic₁₆₇	semidirect
58. C₂⁴	C₂×Dic₁₆₇	semidirect
67. C₂³	C₂²⋊Dic₁₆₇	direct (N × H)
69. C₂³	C₂²×Dic₁₆₇	semidirect
77. C₂²	C₂×C₂²⋊Dic₁₆₇	direct (N × H)
79. C₂²	C₂³×Dic₁₆₇	semidirect
84. C₂	C₂²×C₂²⋊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×C₂²⋊Dic₁₆₇	2
C₂⁶×C₁₆₇	2
C₂³×C₂²⋊C₄	167

Subgroups — C23×C22⋊Dic167 #36337296

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₂²⋊Dic₁₆₇ #36337296