Subgroups — #41011370

Order 27008.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27213264	C₂	1	1	yes	maximal, characteristic
2	C₈×C₂₁₁⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#364×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#20269999	C₄	1	1	yes	characteristic
5	#20269983	—	2	1	no	—
6	#20269913	C₄	1	1	yes	characteristic
7	#20269905	—	2	1	no	—
8	C₄×C₂₁₁⋊C₈	—	2	1	no	—
9	M₃₂×C₂₁₁	—	2	1	no	—
10	C₄×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
11	#16809962	—	2	1	no	—
12	Dic₈₄₄	C₈	1	1	yes	characteristic, complemented
13	#16809951	—	2	1	no	—
14	D₁₆₈₈	C₈	1	1	yes	characteristic, complemented
15	#16809946	—	2	1	no	—
16	#16809945	—	4	1	no	—
17	C₂×C₂₁₁⋊C₈	—	2	2	no	—
18	C₈×D₂₁₁	—	4	1	no	—
19	C₂×C₈×C₂₁₁	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈×C₂₁₁	D₄	1	1	yes	characteristic
21	C₄²×C₂₁₁	D₄	1	1	yes	characteristic
22	C₁₆×C₂₁₁	—	4	1	no	—
23	Dic₄₂₂	—	2	1	no	—
24	D₈₄₄	—	2	1	no	—
25	C₂²⋊D₂₁₁	—	4	1	no	—
26	C₄×D₂₁₁	—	4	1	no	—
27	C₂₁₁⋊C₈	—	2	4	no	—
28	C₂₁₁⋊C₈	—	4	1	no	—
29	C₂×C₄×C₂₁₁	—	2	1	no	—
30	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
31	C₈×C₂₁₁	C₂×C₈	1	1	yes	characteristic
32	C₈×C₂₁₁	M₁₆	1	1	yes	characteristic
33	C₈×C₂₁₁	D₈	1	1	yes	characteristic
34	C₈×C₂₁₁	SD₁₆	1	1	yes	characteristic
35	C₂²×C₂₁₁	#104	1	1	yes	characteristic
36	C₂×D₂₁₁	—	4	1	no	—
37	C₄×C₂₁₁	—	2	2	no	—
38	C₄×C₂₁₁	C₂²⋊C₈	1	1	yes	characteristic
39	C₄×C₂₁₁	#102	1	1	yes	characteristic
40	Dic₂₁₁	—	4	1	no	—
41	C₂×C₂₁₁	—	2	1	no	—
42	C₂×C₂₁₁	D₄⋊C₈	1	1	yes	characteristic
43	D₂₁₁	—	8	1	no	—
44	C₂₁₁	#1335	1	1	yes	characteristic, complemented, Sylow
45	#1335	—	211	1	no	maximal, Sylow
46	#443	—	211	1	no	—
47	#364	D₂₁₁	1	1	yes	characteristic, complemented
48	C₈²	—	211	1	no	—
49	#135	—	211	1	no	—
50	#131	—	422	1	no	—
51	M₃₂	—	2	1	no	—
52	#108	—	211	1	no	—
53	#104	—	422	1	no	—
54	C₄×C₈	—	422	1	no	—
55	C₄×C₈	C₂×D₂₁₁	1	1	yes	characteristic
56	#41	—	422	1	no	—
57	Q₁₆	—	211	1	no	—
58	SD₁₆	—	422	1	no	—
59	D₈	—	211	1	no	—
60	M₁₆	—	844	1	no	—
61	M₁₆	—	422	1	no	—
62	C₂×C₈	—	844	1	no	—
63	C₂×C₈	—	422	2	no	—
64	C₂×C₈	C₄×D₂₁₁	1	1	yes	characteristic
65	C₂×C₈	C₂²⋊D₂₁₁	1	1	yes	characteristic
66	C₄²	D₈₄₄	1	1	yes	characteristic
67	C₁₆	—	4	1	no	—
68	Q₈	—	422	1	no	—
69	D₄	—	844	1	no	—
70	D₄	—	422	1	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	844	1	no	—
73	C₂×C₄	(C₂×D₂₁₁)⋊C₄	1	1	yes	characteristic
74	C₈	—	844	1	no	—
75	C₈	—	422	4	no	—
76	C₈	C₈×D₂₁₁	1	1	yes	characteristic
77	C₈	#16809945	1	1	yes	characteristic
78	C₈	#16809947	1	1	yes	characteristic
79	C₈	#16809949	1	1	yes	characteristic
80	C₂²	—	844	1	no	—
81	C₂²	#20269903	1	1	yes	characteristic
82	C₄	—	2	2	no	—
83	C₄	—	844	1	no	—
84	C₄	(C₂×D₂₁₁)⋊C₈	1	1	yes	characteristic
85	C₄	#20269897	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	1688	1	no	—
88	C₂	#27212873	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#16809947
commutator	C₈×C₂₁₁	C₂×C₈
Frattini	C₄×C₈	C₂×D₂₁₁
Fitting	#364×C₂₁₁	C₂
socle	C₂×C₂₁₁	D₄⋊C₈
prime radical	#20269999	C₄
cosocle	C₄×C₈×C₂₁₁	C₂²
hypercenter	#364	D₂₁₁
hypocenter	C₂₁₁	#1335
LCS	G ▷ C₈×C₂₁₁ ▷ C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #364

Complements

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

N	H ≅ G/N	type
2. C₈×C₂₁₁⋊C₈	C₂	semidirect
3. #364×C₂₁₁	C₂	semidirect
12. Dic₈₄₄	C₈	semidirect
14. D₁₆₈₈	C₈	semidirect
44. C₂₁₁	#1335	semidirect
47. #364	D₂₁₁	semidirect

Maximal subgroups

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

subgroup	index
#27213264	2
C₈×C₂₁₁⋊C₈	2
#364×C₂₁₁	2
#1335	211