Subgroups — H(3,𝔽₄)×C₂₁₁ #27212839

Order 13504.

260 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#124×C₂₁₁	C₂	1	9	yes	maximal, complemented
2	#120×C₂₁₁	C₂	1	6	yes	maximal, complemented
3	C₂⁴×C₂₁₁	C₂²	1	2	yes	complemented
4	C₂×Q₈×C₂₁₁	C₂²	1	3	yes	complemented
5	C₂×D₄×C₂₁₁	C₂²	1	9	yes	—
6	C₂²⋊C₄×C₂₁₁	C₂²	1	18	yes	complemented
7	C₄²×C₂₁₁	C₂²	1	3	yes	complemented
8	C₂³×C₂₁₁	—	4	6	no	—
9	C₂³×C₂₁₁	C₂³	1	6	yes	—
10	Q₈×C₂₁₁	—	4	3	no	—
11	D₄×C₂₁₁	—	4	9	no	—
12	C₂×C₄×C₂₁₁	C₂³	1	9	yes	—
13	C₂²×C₂₁₁	—	2	18	no	—
14	C₂²×C₂₁₁	—	4	8	no	—
15	C₂²×C₂₁₁	C₂⁴	1	1	yes	characteristic
16	C₄×C₂₁₁	—	2	9	no	—
17	C₂×C₂₁₁	—	4	6	no	—
18	C₂×C₂₁₁	H(5,𝔽₂)	1	3	yes	—
19	C₂₁₁	H(3,𝔽₄)	1	1	yes	characteristic, direct product factor, Sylow
20	H(3,𝔽₄)	C₂₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	#124	C₂×C₂₁₁	1	9	yes	complemented
22	#120	C₂×C₂₁₁	1	6	yes	complemented
23	C₂⁴	C₂²×C₂₁₁	1	2	yes	complemented
24	C₂×Q₈	C₂²×C₂₁₁	1	3	yes	complemented
25	C₂×D₄	C₂²×C₂₁₁	1	9	yes	—
26	C₂²⋊C₄	C₂²×C₂₁₁	1	18	yes	complemented
27	C₄²	C₂²×C₂₁₁	1	3	yes	complemented
28	C₂³	—	4	6	no	—
29	C₂³	C₂³×C₂₁₁	1	6	yes	—
30	Q₈	—	4	3	no	—
31	D₄	—	4	9	no	—
32	C₂×C₄	C₂³×C₂₁₁	1	9	yes	—
33	C₂²	—	2	18	no	—
34	C₂²	—	4	8	no	—
35	C₂²	C₂⁴×C₂₁₁	1	1	yes	characteristic
36	C₄	—	2	9	no	—
37	C₂	—	4	6	no	—
38	C₂	H(5,𝔽₂)×C₂₁₁	1	3	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₁₁	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₂₁₁
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₁₁ ◁ G

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.