Subgroups — #36322210

Order 21376.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24923022	C₂	1	2	yes	maximal
2	C₄×C₁₆×C₁₆₇	C₂	1	1	yes	maximal, characteristic
3	C₄⋊(C₁₆₇⋊C₈)	—	2	2	no	—
4	C₂×C₁₆×C₁₆₇	C₄	1	1	yes	characteristic
5	C₂×C₁₆×C₁₆₇	C₄	1	1	yes	characteristic
6	C₄×C₈×C₁₆₇	C₂²	1	1	yes	characteristic
7	C₂×C₁₆₇⋊C₈	—	4	2	no	—
8	C₂×C₈×C₁₆₇	C₂×C₄	1	1	yes	characteristic
9	C₂×C₈×C₁₆₇	Q₈	1	1	yes	characteristic
10	C₄²×C₁₆₇	D₄	1	1	yes	characteristic
11	C₁₆×C₁₆₇	—	2	1	no	—
12	C₁₆×C₁₆₇	C₈	1	2	yes	complemented
13	C₁₆₇⋊C₈	—	8	2	no	—
14	C₂×C₄×C₁₆₇	C₄⋊C₄	1	1	yes	characteristic
15	C₂×C₄×C₁₆₇	D₈	1	1	yes	characteristic
16	C₂×C₄×C₁₆₇	Q₁₆	1	1	yes	characteristic
17	C₈×C₁₆₇	—	2	1	no	—
18	C₈×C₁₆₇	C₂×C₈	1	1	yes	characteristic
19	C₈×C₁₆₇	M₁₆	1	1	yes	characteristic
20	C₂²×C₁₆₇	#107	1	1	yes	characteristic
21	C₄×C₁₆₇	—	2	1	no	—
22	C₄×C₁₆₇	C₄⋊C₈	1	1	yes	characteristic
23	C₄×C₁₆₇	#108	1	1	yes	characteristic
24	C₄×C₁₆₇	D₁₆	1	1	yes	characteristic
25	C₄×C₁₆₇	Q₃₂	1	1	yes	characteristic
26	C₂×C₁₆₇	#335	1	1	yes	characteristic
27	C₂×C₁₆₇	#366	1	1	yes	characteristic
28	C₂×C₁₆₇	#368	1	1	yes	characteristic
29	C₁₆₇	#1371	1	1	yes	characteristic, complemented, Sylow
30	#1371	—	167	1	no	maximal, Sylow
31	C₄×C₁₆	D₁₆₇	1	1	yes	characteristic
32	#335	—	167	2	no	—
33	C₂×C₁₆	Dic₁₆₇	1	1	yes	characteristic
34	C₂×C₁₆	Dic₁₆₇	1	1	yes	characteristic
35	C₄⋊C₈	—	334	2	no	—
36	C₄×C₈	C₂×D₁₆₇	1	1	yes	characteristic
37	C₂×C₈	—	668	2	no	—
38	C₂×C₈	C₂×Dic₁₆₇	1	1	yes	characteristic
39	C₂×C₈	Dic₃₃₄	1	1	yes	characteristic
40	C₄²	D₆₆₈	1	1	yes	characteristic
41	C₁₆	—	2	1	no	—
42	C₁₆	C₁₆₇⋊C₈	1	2	yes	complemented
43	C₂×C₄	C₄⋊Dic₁₆₇	1	1	yes	characteristic
44	C₂×C₄	D₁₃₃₆	1	1	yes	characteristic
45	C₂×C₄	Dic₆₆₈	1	1	yes	characteristic
46	C₈	—	2	1	no	—
47	C₈	—	1336	2	no	—
48	C₈	C₂×C₁₆₇⋊C₈	1	1	yes	characteristic
49	C₈	#14581934	1	1	yes	characteristic
50	C₂²	#19128766	1	1	yes	characteristic
51	C₄	—	2	1	no	—
52	C₄	C₄⋊(C₁₆₇⋊C₈)	1	1	yes	characteristic
53	C₄	#19128768	1	1	yes	characteristic
54	C₄	D₂₆₇₂	1	1	yes	characteristic
55	C₄	Dic₁₃₃₆	1	1	yes	characteristic
56	C₂	#24923022	1	1	yes	characteristic
57	C₂	#24923095	1	1	yes	characteristic
58	C₂	#24923099	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₁₃₃₆
commutator	C₈×C₁₆₇	C₂×C₈
Frattini	C₄×C₈	C₂×D₁₆₇
Fitting	C₄×C₁₆×C₁₆₇	C₂
socle, prime radical	C₂²×C₁₆₇	#107
cosocle	C₄×C₈×C₁₆₇	C₂²
hypercenter	C₄×C₁₆	D₁₆₇
hypocenter	C₁₆₇	#1371
LCS	G ▷ C₈×C₁₆₇ ▷ C₄×C₁₆₇ ▷ C₂×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
12. C₁₆×C₁₆₇	C₈	semidirect
29. C₁₆₇	#1371	semidirect
42. C₁₆	C₁₆₇⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#24923022	2
C₄×C₁₆×C₁₆₇	2
#1371	167