Subgroups — #41013541

Order 27008.

240 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27212901	C₂	1	6	yes	maximal
2	C₄³×C₂₁₁	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₂₁₁⋊C₈	C₂²	1	4	yes	—
4	C₂×C₄²×C₂₁₁	C₄	1	3	yes	—
5	C₂×C₄²×C₂₁₁	C₄	1	1	yes	characteristic
6	C₂×C₄²×C₂₁₁	C₂²	1	3	yes	—
7	C₂×C₂₁₁⋊C₈	—	2	12	no	—
8	C₂²×C₄×C₂₁₁	C₂×C₄	1	3	yes	—
9	C₂²×C₄×C₂₁₁	C₂×C₄	1	3	yes	—
10	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
11	C₄²×C₂₁₁	—	2	6	no	—
12	C₄²×C₂₁₁	—	2	6	no	—
13	C₄²×C₂₁₁	C₈	1	4	yes	complemented
14	C₂₁₁⋊C₈	—	4	4	no	—
15	C₂³×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
16	C₂×C₄×C₂₁₁	—	2	3	no	—
17	C₂×C₄×C₂₁₁	—	2	3	no	—
18	C₂×C₄×C₂₁₁	—	2	3	no	—
19	C₂×C₄×C₂₁₁	—	2	3	no	—
20	C₂×C₄×C₂₁₁	C₂×C₈	1	6	yes	—
21	C₂×C₄×C₂₁₁	M₁₆	1	6	yes	—
22	C₂×C₄×C₂₁₁	#41	1	6	yes	—
23	C₂²×C₂₁₁	#117	1	3	yes	—
24	C₂²×C₂₁₁	C₂²×C₈	1	1	yes	characteristic
25	C₂²×C₂₁₁	C₂×M₁₆	1	3	yes	—
26	C₄×C₂₁₁	—	2	6	no	—
27	C₄×C₂₁₁	—	2	6	no	—
28	C₄×C₂₁₁	#126	1	4	yes	—
29	C₂×C₂₁₁	#383	1	1	yes	characteristic
30	C₂×C₂₁₁	#431	1	3	yes	—
31	C₂×C₂₁₁	#432	1	3	yes	—
32	C₂₁₁	#1839	1	1	yes	characteristic, complemented, Sylow
33	#1839	—	211	1	no	maximal, Sylow
34	C₄³	D₂₁₁	1	1	yes	characteristic
35	#336	—	211	6	no	—
36	C₂²×C₈	—	211	4	no	—
37	C₂×C₄²	Dic₂₁₁	1	3	yes	—
38	C₂×C₄²	Dic₂₁₁	1	1	yes	characteristic
39	C₂×C₄²	C₂×D₂₁₁	1	3	yes	—
40	C₂²×C₄	C₂×Dic₂₁₁	1	3	yes	—
41	C₂²×C₄	C₂×Dic₂₁₁	1	3	yes	—
42	C₂²×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
43	C₂×C₈	—	422	12	no	—
44	C₄²	—	2	6	no	—
45	C₄²	—	2	6	no	—
46	C₄²	C₂₁₁⋊C₈	1	4	yes	complemented
47	C₂³	C₂²×Dic₂₁₁	1	1	yes	characteristic
48	C₂×C₄	—	2	3	no	—
49	C₂×C₄	—	2	3	no	—
50	C₂×C₄	—	2	3	no	—
51	C₂×C₄	—	2	3	no	—
52	C₂×C₄	C₂×C₂₁₁⋊C₈	1	6	yes	—
53	C₂×C₄	#16809946	1	6	yes	—
54	C₂×C₄	#16809962	1	6	yes	—
55	C₈	—	844	4	no	—
56	C₂²	#20269936	1	3	yes	—
57	C₂²	C₂²×C₂₁₁⋊C₈	1	1	yes	characteristic
58	C₂²	C₂×#16809946	1	3	yes	—
59	C₄	—	2	6	no	—
60	C₄	—	2	6	no	—
61	C₄	#20269971	1	4	yes	—
62	C₂	#27213014	1	1	yes	characteristic
63	C₂	#27213199	1	3	yes	—
64	C₂	#27213203	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₂₁₁
commutator	C₂²×C₂₁₁	C₂²×C₈
Fitting	C₄³×C₂₁₁	C₂
socle, prime radical	C₂³×C₂₁₁	C₂²×C₄
cosocle	C₂²×C₄×C₂₁₁	C₂³
hypercenter	C₄³	D₂₁₁
hypocenter	C₂₁₁	#1839
LCS	G ▷ C₂²×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³

Complements

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

N	H ≅ G/N	type
13. C₄²×C₂₁₁	C₈	semidirect
32. C₂₁₁	#1839	semidirect
46. C₄²	C₂₁₁⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#27212901	2
C₄³×C₂₁₁	2
#1839	211