Subgroups — #498×C₃₆₇ #37553816

Order 23488.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊Q₈×C₃₆₇	C₂	1	2	yes	maximal
2	#106×C₃₆₇	C₂	1	4	yes	maximal
3	C₄×C₈×C₃₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₈×C₃₆₇	—	2	2	no	—
5	C₂×C₈×C₃₆₇	C₂²	1	2	yes	—
6	C₄⋊C₄×C₃₆₇	—	2	2	no	—
7	C₄⋊C₄×C₃₆₇	C₂²	1	4	yes	—
8	C₄²×C₃₆₇	C₂²	1	1	yes	characteristic
9	Q₈×C₃₆₇	—	4	4	no	—
10	C₂×C₄×C₃₆₇	—	2	4	no	—
11	C₂×C₄×C₃₆₇	D₄	1	2	yes	—
12	C₂×C₄×C₃₆₇	C₂³	1	1	yes	characteristic
13	C₈×C₃₆₇	Q₈	1	4	yes	complemented
14	C₂²×C₃₆₇	C₂×D₄	1	1	yes	characteristic
15	C₄×C₃₆₇	—	4	4	no	—
16	C₄×C₃₆₇	SD₁₆	1	4	yes	—
17	C₄×C₃₆₇	C₂×Q₈	1	2	yes	—
18	C₂×C₃₆₇	C₄⋊Q₈	1	1	yes	characteristic
19	C₂×C₃₆₇	C₂×SD₁₆	1	2	yes	—
20	C₃₆₇	#498	1	1	yes	characteristic, direct product factor, Sylow
21	#498	C₃₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
22	C₄⋊Q₈	C₂×C₃₆₇	1	2	yes	—
23	#106	C₂×C₃₆₇	1	4	yes	—
24	C₄×C₈	C₂×C₃₆₇	1	1	yes	characteristic
25	C₂×Q₈	—	2	2	no	—
26	C₂×C₈	C₂²×C₃₆₇	1	2	yes	—
27	C₄⋊C₄	—	2	2	no	—
28	C₄⋊C₄	C₂²×C₃₆₇	1	4	yes	—
29	C₄²	C₂²×C₃₆₇	1	1	yes	characteristic
30	Q₈	—	4	4	no	—
31	C₂×C₄	—	2	4	no	—
32	C₂×C₄	D₄×C₃₆₇	1	2	yes	—
33	C₂×C₄	C₂³×C₃₆₇	1	1	yes	characteristic
34	C₈	Q₈×C₃₆₇	1	4	yes	complemented
35	C₂²	C₂×D₄×C₃₆₇	1	1	yes	characteristic
36	C₄	—	4	4	no	—
37	C₄	SD₁₆×C₃₆₇	1	4	yes	—
38	C₄	C₂×Q₈×C₃₆₇	1	2	yes	—
39	C₂	C₄⋊Q₈×C₃₆₇	1	1	yes	characteristic
40	C₂	C₂×SD₁₆×C₃₆₇	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂²×C₃₆₇	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂³×C₃₆₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₆₇ ◁ C₄²×C₃₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
13. C₈×C₃₆₇	Q₈	semidirect
20. C₃₆₇	#498	direct (N × H)
21. #498	C₃₆₇	direct (N × H)
34. C₈	Q₈×C₃₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊Q₈×C₃₆₇	2
#106×C₃₆₇	2
C₄×C₈×C₃₆₇	2
#498	367

Subgroups — #498×C367 #37553816

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #498×C₃₆₇ #37553816