Subgroups — #41014797

Order 27008.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27213194	C₂	1	3	yes	maximal, complemented
2	C₂×#20269893	C₂	1	3	yes	maximal
3	#338×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×D₈₄₄	C₂²	1	1	yes	characteristic
5	C₂×#16809946	C₂²	1	3	yes	—
6	#20269912	—	2	6	no	—
7	#20269893	—	2	6	no	—
8	C₂×M₁₆×C₂₁₁	C₂²	1	3	yes	—
9	C₂³×D₂₁₁	Q₈	1	2	yes	—
10	C₂×D₈₄₄	—	2	3	no	—
11	C₂×D₈₄₄	—	2	3	no	—
12	#16809946	—	2	3	no	—
13	#16809946	D₄	1	6	yes	—
14	C₂×C₂₁₁⋊C₈	—	2	3	no	—
15	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
16	M₁₆×C₂₁₁	—	2	3	no	—
17	M₁₆×C₂₁₁	—	2	3	no	—
18	C₂×C₈×C₂₁₁	—	2	3	no	—
19	C₂²×D₂₁₁	—	2	6	no	—
20	C₂²×D₂₁₁	—	8	2	no	—
21	D₈₄₄	—	4	3	no	—
22	D₈₄₄	—	4	1	no	—
23	C₂₁₁⋊C₈	—	2	6	no	—
24	C₂³×C₂₁₁	C₂×Q₈	1	1	yes	characteristic
25	C₂×C₄×C₂₁₁	C₂×D₄	1	3	yes	—
26	C₂×C₄×C₂₁₁	#41	1	3	yes	—
27	C₈×C₂₁₁	—	4	3	no	—
28	C₂²×C₂₁₁	—	4	1	no	—
29	C₂²×C₂₁₁	C₂²⋊Q₈	1	3	yes	—
30	C₂×D₂₁₁	—	4	2	no	—
31	C₂×D₂₁₁	—	8	6	no	—
32	C₄×C₂₁₁	#121	1	3	yes	—
33	C₄×C₂₁₁	#126	1	1	yes	characteristic
34	C₂×C₂₁₁	—	2	3	no	—
35	C₂×C₂₁₁	#396	1	1	yes	characteristic
36	D₂₁₁	—	8	2	no	—
37	C₂₁₁	#2069	1	1	yes	characteristic, complemented, Sylow
38	#2069	—	211	1	no	maximal, Sylow
39	#430	—	211	3	no	—
40	C₂×#100	—	211	3	no	—
41	#338	D₂₁₁	1	1	yes	characteristic, complemented
42	C₂²×D₄	—	211	1	no	—
43	C₂×M₁₆	—	211	3	no	—
44	C₂×M₁₆	C₂×D₂₁₁	1	3	yes	—
45	#108	—	422	6	no	—
46	#100	—	422	6	no	—
47	C₂⁴	—	211	2	no	—
48	C₂×D₄	—	422	3	no	—
49	C₂×D₄	—	422	3	no	—
50	C₂²×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
51	M₁₆	—	2	3	no	—
52	M₁₆	—	2	3	no	—
53	M₁₆	—	422	3	no	—
54	M₁₆	—	211	6	no	—
55	C₂×C₈	—	2	3	no	—
56	C₂×C₈	—	422	3	no	—
57	C₂³	—	1688	2	no	—
58	C₂³	—	422	6	no	—
59	C₂³	Q₈×D₂₁₁	1	1	yes	characteristic
60	D₄	—	844	3	no	—
61	D₄	—	844	1	no	—
62	C₂×C₄	D₄×D₂₁₁	1	3	yes	—
63	C₂×C₄	#16809962	1	3	yes	—
64	C₈	—	4	3	no	—
65	C₈	—	422	6	no	—
66	C₂²	—	4	1	no	—
67	C₂²	—	844	2	no	—
68	C₂²	—	1688	6	no	—
69	C₂²	#20269955	1	3	yes	—
70	C₄	#20269950	1	3	yes	—
71	C₄	#20269971	1	1	yes	characteristic
72	C₂	—	2	3	no	—
73	C₂	—	1688	2	no	—
74	C₂	#27213070	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#27213070
commutator, cosocle	C₂²×C₄×C₂₁₁	C₂³
Frattini	C₂²×C₄	C₂²×D₂₁₁
Fitting	#338×C₂₁₁	C₂
socle	C₂×C₂₁₁	#396
prime radical	C₂²×D₈₄₄	C₂²
hypercenter	#338	D₂₁₁
hypocenter	C₂₁₁	#2069
LCS	G ▷ C₂²×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #338

Complements

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

N	H ≅ G/N	type
1. #27213194	C₂	semidirect
3. #338×C₂₁₁	C₂	semidirect
37. C₂₁₁	#2069	semidirect
41. #338	D₂₁₁	semidirect

Maximal subgroups

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

subgroup	index
#27213194	2
C₂×#20269893	2
#338×C₂₁₁	2
#2069	211