Subgroups — #543×C₃₆₇ #37553861

Order 23488.

242 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²⋊Q₈×C₃₆₇	C₂	1	12	yes	maximal, complemented
2	C₂×C₂²⋊C₄×C₃₆₇	C₂	1	3	yes	maximal
3	C₂⁴×C₃₆₇	C₂²	1	1	yes	characteristic
4	C₂×Q₈×C₃₆₇	C₂²	1	4	yes	complemented
5	C₂²×C₄×C₃₆₇	C₂²	1	6	yes	—
6	C₄⋊C₄×C₃₆₇	C₂²	1	12	yes	complemented
7	C₂²⋊C₄×C₃₆₇	C₂²	1	12	yes	—
8	C₂³×C₃₆₇	—	4	2	no	—
9	C₂³×C₃₆₇	Q₈	1	4	yes	complemented
10	C₂³×C₃₆₇	C₂³	1	3	yes	—
11	Q₈×C₃₆₇	—	4	4	no	—
12	C₂×C₄×C₃₆₇	—	4	6	no	—
13	C₂×C₄×C₃₆₇	C₂³	1	12	yes	—
14	C₂²×C₃₆₇	—	2	6	no	—
15	C₂²×C₃₆₇	—	4	4	no	—
16	C₂²×C₃₆₇	C₂×Q₈	1	6	yes	—
17	C₂²×C₃₆₇	C₂⁴	1	1	yes	characteristic
18	C₄×C₃₆₇	—	2	12	no	—
19	C₂×C₃₆₇	—	2	6	no	—
20	C₂×C₃₆₇	C₂²×Q₈	1	1	yes	characteristic
21	C₂×C₃₆₇	H(5,𝔽₂)	1	2	yes	—
22	C₃₆₇	#543	1	1	yes	characteristic, direct product factor, Sylow
23	#543	C₃₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂²⋊Q₈	C₂×C₃₆₇	1	12	yes	complemented
25	C₂×C₂²⋊C₄	C₂×C₃₆₇	1	3	yes	—
26	C₂⁴	C₂²×C₃₆₇	1	1	yes	characteristic
27	C₂×Q₈	C₂²×C₃₆₇	1	4	yes	complemented
28	C₂²×C₄	C₂²×C₃₆₇	1	6	yes	—
29	C₄⋊C₄	C₂²×C₃₆₇	1	12	yes	complemented
30	C₂²⋊C₄	C₂²×C₃₆₇	1	12	yes	—
31	C₂³	—	4	2	no	—
32	C₂³	Q₈×C₃₆₇	1	4	yes	complemented
33	C₂³	C₂³×C₃₆₇	1	3	yes	—
34	Q₈	—	4	4	no	—
35	C₂×C₄	—	4	6	no	—
36	C₂×C₄	C₂³×C₃₆₇	1	12	yes	—
37	C₂²	—	2	6	no	—
38	C₂²	—	4	4	no	—
39	C₂²	C₂×Q₈×C₃₆₇	1	6	yes	—
40	C₂²	C₂⁴×C₃₆₇	1	1	yes	characteristic
41	C₄	—	2	12	no	—
42	C₂	—	2	6	no	—
43	C₂	C₂²×Q₈×C₃₆₇	1	1	yes	characteristic
44	C₂	H(5,𝔽₂)×C₃₆₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₆₇	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₃₆₇
prime radical	C₂⁴×C₃₆₇	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²⋊Q₈×C₃₆₇	C₂	semidirect
4. C₂×Q₈×C₃₆₇	C₂²	semidirect
6. C₄⋊C₄×C₃₆₇	C₂²	semidirect
9. C₂³×C₃₆₇	Q₈	semidirect
22. C₃₆₇	#543	direct (N × H)
23. #543	C₃₆₇	direct (N × H)
24. C₂²⋊Q₈	C₂×C₃₆₇	semidirect
27. C₂×Q₈	C₂²×C₃₆₇	semidirect
29. C₄⋊C₄	C₂²×C₃₆₇	semidirect
32. C₂³	Q₈×C₃₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²⋊Q₈×C₃₆₇	2
C₂×C₂²⋊C₄×C₃₆₇	2
#543	367

Subgroups — #543×C367 #37553861

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #543×C₃₆₇ #37553861