Subgroups — C₂⁴×#3862 #72246052

Order 2352.

670 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#3862	C₂	1	15	yes	maximal, direct product factor
2	C₂⁴×C₇²	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₂²×#3862	C₂²	1	35	yes	direct product factor
4	C₂³×C₇²	C₂×C₃	1	15	yes	complemented
5	C₂⁴×ΓL(1,𝔽₈)	—	7	2	no	maximal
6	C₂×#3862	C₂³	1	15	yes	direct product factor
7	C₂²×C₇²	C₂²×C₃	1	35	yes	complemented
8	C₂³×ΓL(1,𝔽₈)	—	7	30	no	—
9	#3862	C₂⁴	1	1	yes	characteristic, direct product factor, Hall
10	C₂⁴×C₇	—	3	2	no	—
11	C₂⁴×C₇	ΓL(1,𝔽₈)	1	2	yes	complemented
12	C₂×C₇²	C₂³×C₃	1	15	yes	complemented
13	C₂²×ΓL(1,𝔽₈)	—	7	70	no	—
14	C₂³×C₇	—	3	30	no	—
15	C₂³×C₇	C₂×ΓL(1,𝔽₈)	1	30	yes	complemented
16	C₇²	C₂⁴×C₃	1	1	yes	characteristic, complemented, Sylow
17	C₂⁴×C₃	—	49	1	no	Hall
18	C₂×ΓL(1,𝔽₈)	—	7	30	no	—
19	C₂²×C₇	—	3	70	no	—
20	C₂²×C₇	C₂²×ΓL(1,𝔽₈)	1	70	yes	complemented
21	C₂³×C₃	—	49	15	no	—
22	ΓL(1,𝔽₈)	—	7	2	no	—
23	C₂⁴	#3862	1	1	yes	characteristic, direct product factor, Sylow
24	C₂×C₇	—	3	30	no	—
25	C₂×C₇	C₂³×ΓL(1,𝔽₈)	1	30	yes	complemented
26	C₂²×C₃	—	49	35	no	—
27	C₂³	C₂×#3862	1	15	yes	direct product factor
28	C₇	—	3	2	no	—
29	C₇	C₂⁴×ΓL(1,𝔽₈)	1	2	yes	complemented
30	C₂×C₃	—	49	15	no	—
31	C₂²	C₂²×#3862	1	35	yes	direct product factor
32	C₃	—	49	1	no	Sylow
33	C₂	C₂³×#3862	1	15	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂⁴	#3862
commutator, cosocle, hypocenter	C₇²	C₂⁴×C₃
Frattini	C₁	G
Fitting, socle	C₂⁴×C₇²	C₃
LCS	G ▷ C₇²
UCS	C₁ ◁ C₂⁴

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

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