Subgroups — #2104×C₂₁₁ #41009724

Order 27008.

178 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#342×C₂₁₁	C₂	1	7	yes	maximal
2	C₂×C₂²⋊C₄×C₂₁₁	C₂²	1	7	yes	—
3	C₂⁴×C₂₁₁	C₂³	1	1	yes	characteristic
4	C₂²×C₄×C₂₁₁	—	2	7	no	—
5	C₂²⋊C₄×C₂₁₁	—	2	14	no	—
6	C₂³×C₂₁₁	—	8	1	no	—
7	C₂³×C₂₁₁	#41	1	7	yes	—
8	C₂×C₄×C₂₁₁	—	4	7	no	—
9	C₂×C₄×C₂₁₁	—	8	7	no	—
10	C₂²×C₂₁₁	—	4	7	no	—
11	C₂²×C₂₁₁	#126	1	7	yes	—
12	C₄×C₂₁₁	—	8	7	no	—
13	C₂×C₂₁₁	—	2	7	no	—
14	C₂×C₂₁₁	#401	1	1	yes	characteristic
15	C₂₁₁	#2104	1	1	yes	characteristic, direct product factor, Sylow
16	#2104	C₂₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
17	#342	C₂×C₂₁₁	1	7	yes	—
18	C₂×C₂²⋊C₄	C₂²×C₂₁₁	1	7	yes	—
19	C₂⁴	C₂³×C₂₁₁	1	1	yes	characteristic
20	C₂²×C₄	—	2	7	no	—
21	C₂²⋊C₄	—	2	14	no	—
22	C₂³	—	8	1	no	—
23	C₂³	#41×C₂₁₁	1	7	yes	—
24	C₂×C₄	—	4	7	no	—
25	C₂×C₄	—	8	7	no	—
26	C₂²	—	4	7	no	—
27	C₂²	#126×C₂₁₁	1	7	yes	—
28	C₄	—	8	7	no	—
29	C₂	—	2	7	no	—
30	C₂	#401×C₂₁₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₂₁₁	#401
commutator, Frattini, cosocle	C₂⁴	C₂³×C₂₁₁
prime radical	C₂⁴×C₂₁₁	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₂₁₁ ◁ C₂⁴×C₂₁₁ ◁ G

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

N	H ≅ G/N	type
15. C₂₁₁	#2104	direct (N × H)
16. #2104	C₂₁₁	direct (N × H)

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

subgroup	index
#342×C₂₁₁	2
#2104	211