Subgroups — #36322209

Order 21376.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₁₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	#24923022	C₂	1	1	yes	maximal, characteristic
3	#335×C₁₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₆₇⋊C₁₆	C₄	1	1	yes	characteristic
5	C₂×C₁₆₇⋊C₁₆	C₄	1	1	yes	characteristic
6	C₄⋊(C₁₆₇⋊C₈)	—	2	1	no	—
7	C₄⋊C₈×C₁₆₇	—	2	1	no	—
8	C₄×C₈×C₁₆₇	C₂²	1	1	yes	characteristic
9	C₂×C₁₆₇⋊C₈	—	4	1	no	—
10	C₁₆₇⋊C₁₆	—	2	1	no	—
11	C₁₆₇⋊C₁₆	C₈	1	2	yes	complemented
12	C₂×C₈×C₁₆₇	—	4	1	no	—
13	C₂×C₈×C₁₆₇	C₂×C₄	1	1	yes	characteristic
14	C₂×C₈×C₁₆₇	Q₈	1	1	yes	characteristic
15	C₄²×C₁₆₇	D₄	1	1	yes	characteristic
16	C₁₆₇⋊C₈	—	8	1	no	—
17	C₂×C₄×C₁₆₇	C₄⋊C₄	1	1	yes	characteristic
18	C₂×C₄×C₁₆₇	D₈	1	1	yes	characteristic
19	C₂×C₄×C₁₆₇	Q₁₆	1	1	yes	characteristic
20	C₈×C₁₆₇	—	2	1	no	—
21	C₈×C₁₆₇	—	8	1	no	—
22	C₈×C₁₆₇	C₂×C₈	1	1	yes	characteristic
23	C₈×C₁₆₇	M₁₆	1	1	yes	characteristic
24	C₂²×C₁₆₇	#107	1	1	yes	characteristic
25	C₄×C₁₆₇	—	2	1	no	—
26	C₄×C₁₆₇	C₄⋊C₈	1	1	yes	characteristic
27	C₄×C₁₆₇	#108	1	1	yes	characteristic
28	C₄×C₁₆₇	D₁₆	1	1	yes	characteristic
29	C₄×C₁₆₇	Q₃₂	1	1	yes	characteristic
30	C₂×C₁₆₇	#335	1	1	yes	characteristic
31	C₂×C₁₆₇	#366	1	1	yes	characteristic
32	C₂×C₁₆₇	#368	1	1	yes	characteristic
33	C₁₆₇	#1371	1	1	yes	characteristic, complemented, Sylow
34	#1371	—	167	1	no	maximal, Sylow
35	C₄×C₁₆	—	167	1	no	—
36	#335	—	167	1	no	—
37	#335	D₁₆₇	1	1	yes	characteristic
38	C₂×C₁₆	—	167	1	no	—
39	C₂×C₁₆	—	167	1	no	—
40	C₄⋊C₈	—	2	1	no	—
41	C₄⋊C₈	—	334	1	no	—
42	C₄×C₈	C₂×D₁₆₇	1	1	yes	characteristic
43	C₂×C₈	—	4	1	no	—
44	C₂×C₈	—	668	1	no	—
45	C₂×C₈	C₄×D₁₆₇	1	1	yes	characteristic
46	C₂×C₈	Dic₃₃₄	1	1	yes	characteristic
47	C₄²	C₂²⋊D₁₆₇	1	1	yes	characteristic
48	C₁₆	—	334	1	no	—
49	C₁₆	—	167	2	no	—
50	C₂×C₄	Dic₁₆₇⋊C₄	1	1	yes	characteristic
51	C₂×C₄	#14581935	1	1	yes	characteristic
52	C₂×C₄	#14581940	1	1	yes	characteristic
53	C₈	—	2	1	no	—
54	C₈	—	8	1	no	—
55	C₈	—	1336	1	no	—
56	C₈	C₈×D₁₆₇	1	1	yes	characteristic
57	C₈	#14581933	1	1	yes	characteristic
58	C₂²	#19128765	1	1	yes	characteristic
59	C₄	—	2	1	no	—
60	C₄	Dic₁₆₇⋊C₈	1	1	yes	characteristic
61	C₄	#19128767	1	1	yes	characteristic
62	C₄	#19128773	1	1	yes	characteristic
63	C₄	#19128778	1	1	yes	characteristic
64	C₂	#24923021	1	1	yes	characteristic
65	C₂	#24923094	1	1	yes	characteristic
66	C₂	#24923098	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14581935
commutator	C₈×C₁₆₇	C₂×C₈
Frattini	C₄×C₈	C₂×D₁₆₇
Fitting	#335×C₁₆₇	C₂
socle, prime radical	C₂²×C₁₆₇	#107
cosocle	C₄×C₈×C₁₆₇	C₂²
hypercenter	#335	D₁₆₇
hypocenter	C₁₆₇	#1371
LCS	G ▷ C₈×C₁₆₇ ▷ C₄×C₁₆₇ ▷ C₂×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ #335

Complements

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

N	H ≅ G/N	type
11. C₁₆₇⋊C₁₆	C₈	semidirect
33. C₁₆₇	#1371	semidirect

Maximal subgroups

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

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