Subgroups — #41013558

Order 27008.

280 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20269911	C₂	1	4	yes	maximal
2	#27213016	C₂	1	2	yes	maximal
3	C₂×C₄×C₈×C₂₁₁	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊Dic₂₁₁	—	2	4	no	—
5	C₂×C₄⋊Dic₂₁₁	C₂²	1	4	yes	—
6	#20269911	—	2	8	no	—
7	C₂²×C₈×C₂₁₁	C₂²	1	2	yes	—
8	C₂×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
9	C₄×C₈×C₂₁₁	C₄	1	4	yes	complemented
10	C₂²×Dic₂₁₁	—	2	4	no	—
11	C₄⋊Dic₂₁₁	—	2	8	no	—
12	C₄⋊Dic₂₁₁	—	4	8	no	—
13	C₂²×C₄×C₂₁₁	D₄	1	2	yes	—
14	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
15	C₂×C₈×C₂₁₁	C₂×C₄	1	4	yes	—
16	C₂×C₈×C₂₁₁	D₄	1	4	yes	—
17	C₂×C₈×C₂₁₁	Q₈	1	4	yes	—
18	C₄²×C₂₁₁	C₂×C₄	1	2	yes	—
19	C₄²×C₂₁₁	Q₈	1	2	yes	—
20	C₂×Dic₂₁₁	—	4	4	no	—
21	C₂×Dic₂₁₁	—	4	8	no	—
22	C₂³×C₂₁₁	C₂×D₄	1	1	yes	characteristic
23	C₂×C₄×C₂₁₁	C₄⋊C₄	1	4	yes	—
24	C₂×C₄×C₂₁₁	D₈	1	4	yes	—
25	C₂×C₄×C₂₁₁	Q₁₆	1	4	yes	—
26	C₂×C₄×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
27	C₂×C₄×C₂₁₁	C₂×D₄	1	2	yes	—
28	C₂×C₄×C₂₁₁	C₂×Q₈	1	2	yes	—
29	C₂×C₄×C₂₁₁	C₂×Q₈	1	1	yes	characteristic
30	C₈×C₂₁₁	C₄⋊C₄	1	8	yes	complemented
31	C₂²×C₂₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
32	C₂²×C₂₁₁	#127	1	1	yes	characteristic
33	C₂²×C₂₁₁	C₄⋊Q₈	1	1	yes	characteristic
34	C₂²×C₂₁₁	C₂×D₈	1	2	yes	—
35	C₂²×C₂₁₁	C₂×Q₁₆	1	2	yes	—
36	C₄×C₂₁₁	#107	1	8	yes	complemented
37	C₄×C₂₁₁	C₂×C₄⋊C₄	1	2	yes	—
38	C₄×C₂₁₁	C₄⋊Q₈	1	2	yes	—
39	Dic₂₁₁	—	8	4	no	—
40	C₂×C₂₁₁	#384	1	1	yes	characteristic
41	C₂×C₂₁₁	C₂×#107	1	2	yes	—
42	C₂×C₂₁₁	#493	1	1	yes	characteristic
43	C₂×C₂₁₁	#494	1	1	yes	characteristic
44	C₂×C₂₁₁	#500	1	2	yes	—
45	C₂₁₁	#1845	1	1	yes	characteristic, complemented, Sylow
46	#1845	—	211	1	no	maximal, Sylow
47	C₂×#107	—	211	4	no	—
48	C₂×C₄×C₈	D₂₁₁	1	1	yes	characteristic
49	#384	—	211	2	no	—
50	C₂²×C₈	C₂×D₂₁₁	1	2	yes	—
51	C₂×C₄⋊C₄	—	422	4	no	—
52	C₂×C₄⋊C₄	—	211	4	no	—
53	C₂×C₄²	C₂×D₂₁₁	1	1	yes	characteristic
54	#107	—	422	8	no	—
55	C₄×C₈	Dic₂₁₁	1	4	yes	complemented
56	C₂²×C₄	—	422	4	no	—
57	C₂²×C₄	D₈₄₄	1	2	yes	—
58	C₂²×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
59	C₂×C₈	C₂×Dic₂₁₁	1	4	yes	—
60	C₂×C₈	D₈₄₄	1	4	yes	—
61	C₂×C₈	Dic₄₂₂	1	4	yes	—
62	C₄⋊C₄	—	844	8	no	—
63	C₄⋊C₄	—	422	8	no	—
64	C₄²	C₂×Dic₂₁₁	1	2	yes	—
65	C₄²	Dic₄₂₂	1	2	yes	—
66	C₂³	C₂×D₈₄₄	1	1	yes	characteristic
67	C₂×C₄	—	844	4	no	—
68	C₂×C₄	—	844	8	no	—
69	C₂×C₄	C₄⋊Dic₂₁₁	1	4	yes	—
70	C₂×C₄	D₁₆₈₈	1	4	yes	—
71	C₂×C₄	Dic₈₄₄	1	4	yes	—
72	C₂×C₄	C₂²×Dic₂₁₁	1	1	yes	characteristic
73	C₂×C₄	C₂×D₈₄₄	1	2	yes	—
74	C₂×C₄	C₂×Dic₄₂₂	1	2	yes	—
75	C₂×C₄	C₂×Dic₄₂₂	1	1	yes	characteristic
76	C₈	C₄⋊Dic₂₁₁	1	8	yes	complemented
77	C₂²	C₂×C₄⋊Dic₂₁₁	1	1	yes	characteristic
78	C₂²	#20269974	1	1	yes	characteristic
79	C₂²	#20269977	1	1	yes	characteristic
80	C₂²	C₂×D₁₆₈₈	1	2	yes	—
81	C₂²	C₂×Dic₈₄₄	1	2	yes	—
82	C₄	—	1688	4	no	—
83	C₄	#20269911	1	8	yes	complemented
84	C₄	C₂×C₄⋊Dic₂₁₁	1	2	yes	—
85	C₄	#20269977	1	2	yes	—
86	C₂	#27213016	1	1	yes	characteristic
87	C₂	C₂×#20269911	1	2	yes	—
88	C₂	#27213523	1	1	yes	characteristic
89	C₂	#27213526	1	1	yes	characteristic
90	C₂	#27213551	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₈₄₄
commutator	C₂×C₄×C₂₁₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₁₁
Fitting	C₂×C₄×C₈×C₂₁₁	C₂
socle, prime radical	C₂³×C₂₁₁	C₂×D₄
cosocle	C₂²×C₄×C₂₁₁	C₂³
hypercenter	C₂×C₄×C₈	D₂₁₁
hypocenter	C₂₁₁	#1845
LCS	G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
9. C₄×C₈×C₂₁₁	C₄	semidirect
30. C₈×C₂₁₁	C₄⋊C₄	semidirect
36. C₄×C₂₁₁	#107	semidirect
45. C₂₁₁	#1845	semidirect
55. C₄×C₈	Dic₂₁₁	semidirect
76. C₈	C₄⋊Dic₂₁₁	semidirect
83. C₄	#20269911	semidirect

Maximal subgroups

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

subgroup	index
C₂×#20269911	2
#27213016	2
C₂×C₄×C₈×C₂₁₁	2
#1845	211