Subgroups — #350×C₁₆₇ #24922751

Order 10688.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#131×C₁₆₇	C₂	1	1	yes	maximal, characteristic
2	M₃₂×C₁₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₆×C₁₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#41×C₁₆₇	C₄	1	1	yes	characteristic
5	M₁₆×C₁₆₇	—	2	1	no	—
6	M₁₆×C₁₆₇	C₄	1	1	yes	characteristic
7	C₂×C₈×C₁₆₇	—	2	1	no	—
8	C₂×C₈×C₁₆₇	C₂²	1	1	yes	characteristic
9	C₁₆×C₁₆₇	—	2	1	no	—
10	C₁₆×C₁₆₇	—	2	1	no	—
11	Q₈×C₁₆₇	C₈	1	1	yes	characteristic
12	D₄×C₁₆₇	—	2	1	no	—
13	D₄×C₁₆₇	C₈	1	1	yes	characteristic
14	C₂×C₄×C₁₆₇	—	2	1	no	—
15	C₂×C₄×C₁₆₇	C₂×C₄	1	1	yes	characteristic
16	C₈×C₁₆₇	—	2	1	no	—
17	C₈×C₁₆₇	D₄	1	1	yes	characteristic
18	C₈×C₁₆₇	D₄	1	1	yes	characteristic
19	C₂²×C₁₆₇	—	2	1	no	—
20	C₂²×C₁₆₇	M₁₆	1	1	yes	characteristic
21	C₄×C₁₆₇	—	2	1	no	—
22	C₄×C₁₆₇	C₂²⋊C₄	1	1	yes	characteristic
23	C₄×C₁₆₇	C₂×C₈	1	1	yes	characteristic
24	C₂×C₁₆₇	—	2	1	no	—
25	C₂×C₁₆₇	—	4	1	no	—
26	C₂×C₁₆₇	C₂²⋊C₈	1	1	yes	characteristic
27	C₁₆₇	#350	1	1	yes	characteristic, direct product factor, Sylow
28	#350	C₁₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
29	#131	C₂×C₁₆₇	1	1	yes	characteristic
30	M₃₂	C₂×C₁₆₇	1	1	yes	characteristic, complemented
31	C₂×C₁₆	C₂×C₁₆₇	1	1	yes	characteristic, complemented
32	#41	C₄×C₁₆₇	1	1	yes	characteristic
33	M₁₆	—	2	1	no	—
34	M₁₆	C₄×C₁₆₇	1	1	yes	characteristic
35	C₂×C₈	—	2	1	no	—
36	C₂×C₈	C₂²×C₁₆₇	1	1	yes	characteristic
37	C₁₆	—	2	1	no	—
38	C₁₆	—	2	1	no	—
39	Q₈	C₈×C₁₆₇	1	1	yes	characteristic
40	D₄	—	2	1	no	—
41	D₄	C₈×C₁₆₇	1	1	yes	characteristic
42	C₂×C₄	—	2	1	no	—
43	C₂×C₄	C₂×C₄×C₁₆₇	1	1	yes	characteristic
44	C₈	—	2	1	no	—
45	C₈	D₄×C₁₆₇	1	1	yes	characteristic
46	C₈	D₄×C₁₆₇	1	1	yes	characteristic
47	C₂²	—	2	1	no	—
48	C₂²	M₁₆×C₁₆₇	1	1	yes	characteristic
49	C₄	—	2	1	no	—
50	C₄	C₂²⋊C₄×C₁₆₇	1	1	yes	characteristic
51	C₄	C₂×C₈×C₁₆₇	1	1	yes	characteristic
52	C₂	—	2	1	no	—
53	C₂	—	4	1	no	—
54	C₂	C₂²⋊C₈×C₁₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₁₆₇	D₄
commutator	C₄	C₂×C₈×C₁₆₇
Frattini, cosocle	C₂×C₈	C₂²×C₁₆₇
socle	C₂×C₁₆₇	C₂²⋊C₈
prime radical	#41×C₁₆₇	C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₁₆₇ ◁ C₂×C₈×C₁₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. M₃₂×C₁₆₇	C₂	semidirect
3. C₂×C₁₆×C₁₆₇	C₂	semidirect
27. C₁₆₇	#350	direct (N × H)
28. #350	C₁₆₇	direct (N × H)
30. M₃₂	C₂×C₁₆₇	semidirect
31. C₂×C₁₆	C₂×C₁₆₇	semidirect

Maximal subgroups

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

subgroup	index
#131×C₁₆₇	2
M₃₂×C₁₆₇	2
C₂×C₁₆×C₁₆₇	2
#350	167

Subgroups — #350×C167 #24922751

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #350×C₁₆₇ #24922751