Subgroups — #27212992

Order 13504.

178 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₂₁₁	C₂	1	2	yes	maximal
2	C₂₁₁⋊#95	C₂	1	4	yes	maximal
3	C₂×C₄²×C₂₁₁	C₂	1	1	yes	maximal, characteristic
4	C₂²×Dic₂₁₁	C₂²	1	4	yes	—
5	C₄×Dic₂₁₁	C₄	1	8	yes	complemented
6	C₂²×C₄×C₂₁₁	C₂²	1	2	yes	—
7	C₂²×C₄×C₂₁₁	C₂²	1	1	yes	characteristic
8	C₄²×C₂₁₁	C₄	1	4	yes	complemented
9	C₂×Dic₂₁₁	—	2	8	no	—
10	C₂×Dic₂₁₁	C₂×C₄	1	8	yes	—
11	C₂³×C₂₁₁	C₂³	1	1	yes	characteristic
12	C₂×C₄×C₂₁₁	—	2	4	no	—
13	C₂×C₄×C₂₁₁	C₂×C₄	1	4	yes	—
14	C₂×C₄×C₂₁₁	C₂×C₄	1	4	yes	—
15	C₂×C₄×C₂₁₁	C₂×C₄	1	2	yes	—
16	C₂²×C₂₁₁	C₂²×C₄	1	2	yes	—
17	C₂²×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
18	C₂²×C₂₁₁	#41	1	4	yes	—
19	C₄×C₂₁₁	—	2	4	no	—
20	C₄×C₂₁₁	C₄²	1	4	yes	—
21	Dic₂₁₁	—	2	8	no	—
22	C₂×C₂₁₁	C₂×C₄²	1	1	yes	characteristic
23	C₂×C₂₁₁	#117	1	4	yes	—
24	C₂×C₂₁₁	#117	1	2	yes	—
25	C₂₁₁	#376	1	1	yes	characteristic, complemented, Sylow
26	#376	—	211	1	no	maximal, Sylow
27	C₂×C₄²	—	211	2	no	—
28	C₂×C₄²	D₂₁₁	1	1	yes	characteristic
29	#95	—	211	4	no	—
30	C₂²×C₄	—	211	4	no	—
31	C₂²×C₄	C₂×D₂₁₁	1	2	yes	—
32	C₂²×C₄	C₂×D₂₁₁	1	1	yes	characteristic
33	C₄²	—	211	8	no	—
34	C₄²	Dic₂₁₁	1	4	yes	complemented
35	C₂³	C₂²×D₂₁₁	1	1	yes	characteristic
36	C₂×C₄	—	2	4	no	—
37	C₂×C₄	—	422	8	no	—
38	C₂×C₄	—	211	8	no	—
39	C₂×C₄	C₄×D₂₁₁	1	4	yes	—
40	C₂×C₄	C₂×Dic₂₁₁	1	4	yes	—
41	C₂×C₄	C₂×Dic₂₁₁	1	2	yes	—
42	C₂²	C₂×C₄×D₂₁₁	1	2	yes	—
43	C₂²	C₂²×Dic₂₁₁	1	1	yes	characteristic
44	C₂²	#16809962	1	4	yes	—
45	C₄	—	2	4	no	—
46	C₄	—	422	8	no	—
47	C₄	C₄×Dic₂₁₁	1	4	yes	—
48	C₂	C₂×C₄×Dic₂₁₁	1	1	yes	characteristic
49	C₂	#20269934	1	4	yes	—
50	C₂	#20269936	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₁₁
commutator	C₂×C₂₁₁	C₂×C₄²
Frattini	C₂³	C₂²×D₂₁₁
Fitting	C₂×C₄²×C₂₁₁	C₂
socle, prime radical, cosocle	C₂³×C₂₁₁	C₂³
hypercenter	C₂×C₄²	D₂₁₁
hypocenter	C₂₁₁	#376
LCS	G ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
5. C₄×Dic₂₁₁	C₄	semidirect
8. C₄²×C₂₁₁	C₄	semidirect
25. C₂₁₁	#376	semidirect
34. C₄²	Dic₂₁₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₂₁₁	2
C₂₁₁⋊#95	2
C₂×C₄²×C₂₁₁	2
#376	211