Subgroups — #27213083

Order 13504.

186 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊Dic₂₁₁	C₂	1	3	yes	maximal, complemented
2	C₂×(C₂×D₂₁₁)⋊C₄	C₂	1	3	yes	maximal
3	#95×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×D₂₁₁	C₂²	1	1	yes	characteristic
5	C₂²×Dic₂₁₁	C₂²	1	3	yes	—
6	C₄⋊Dic₂₁₁	—	2	6	no	—
7	(C₂×D₂₁₁)⋊C₄	—	2	6	no	—
8	C₂²×C₄×C₂₁₁	C₂²	1	3	yes	—
9	C₂²×D₂₁₁	—	2	3	no	—
10	C₂²×D₂₁₁	—	2	3	no	—
11	C₂²×D₂₁₁	Q₈	1	2	yes	—
12	C₂×Dic₂₁₁	—	2	3	no	—
13	C₂×Dic₂₁₁	—	2	6	no	—
14	C₂³×C₂₁₁	C₂³	1	1	yes	characteristic
15	C₂×C₄×C₂₁₁	—	2	6	no	—
16	C₂×C₄×C₂₁₁	D₄	1	6	yes	—
17	C₂²×C₂₁₁	C₂×D₄	1	3	yes	—
18	C₂²×C₂₁₁	C₂×Q₈	1	1	yes	characteristic
19	C₂²×C₂₁₁	#41	1	3	yes	—
20	C₂×D₂₁₁	—	2	6	no	—
21	C₂×D₂₁₁	—	4	3	no	—
22	C₂×D₂₁₁	—	4	1	no	—
23	C₄×C₂₁₁	—	2	6	no	—
24	Dic₂₁₁	—	4	3	no	—
25	C₂×C₂₁₁	C₂²⋊Q₈	1	3	yes	—
26	C₂×C₂₁₁	#123	1	3	yes	—
27	C₂×C₂₁₁	#127	1	1	yes	characteristic
28	D₂₁₁	—	4	2	no	—
29	C₂₁₁	#399	1	1	yes	characteristic, complemented, Sylow
30	#399	—	211	1	no	maximal, Sylow
31	C₂×C₄⋊C₄	—	211	3	no	—
32	C₂×C₂²⋊C₄	—	211	3	no	—
33	#95	D₂₁₁	1	1	yes	characteristic, complemented
34	C₂⁴	—	211	1	no	—
35	C₂²×C₄	—	211	3	no	—
36	C₂²×C₄	C₂×D₂₁₁	1	3	yes	—
37	C₄⋊C₄	—	422	6	no	—
38	C₂²⋊C₄	—	422	6	no	—
39	C₂³	—	422	3	no	—
40	C₂³	—	422	3	no	—
41	C₂³	—	211	2	no	—
42	C₂³	C₂²×D₂₁₁	1	1	yes	characteristic
43	C₂×C₄	—	2	6	no	—
44	C₂×C₄	—	422	3	no	—
45	C₂×C₄	—	422	6	no	—
46	C₂×C₄	D₈₄₄	1	6	yes	—
47	C₂²	—	844	3	no	—
48	C₂²	—	844	1	no	—
49	C₂²	—	422	6	no	—
50	C₂²	C₂×D₈₄₄	1	3	yes	—
51	C₂²	Q₈×D₂₁₁	1	1	yes	characteristic
52	C₂²	#16809961	1	3	yes	—
53	C₄	—	2	6	no	—
54	C₄	—	844	3	no	—
55	C₂	—	844	2	no	—
56	C₂	#20269957	1	3	yes	—
57	C₂	#20269961	1	3	yes	—
58	C₂	#20269974	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₂₁₁
commutator, socle, cosocle	C₂³×C₂₁₁	C₂³
Fitting	#95×C₂₁₁	C₂
prime radical	C₂³×D₂₁₁	C₂²
hypercenter	#95	D₂₁₁
hypocenter	C₂₁₁	#399
LCS	G ▷ C₂³×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂³ ◁ #95

Complements

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

N	H ≅ G/N	type
1. C₂×C₄⋊Dic₂₁₁	C₂	semidirect
3. #95×C₂₁₁	C₂	semidirect
29. C₂₁₁	#399	semidirect
33. #95	D₂₁₁	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
C₂×(C₂×D₂₁₁)⋊C₄	2
#95×C₂₁₁	2
#399	211