Subgroups — #41013570

Order 27008.

220 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20269913	C₂	1	4	yes	maximal
2	#27213170	C₂	1	2	yes	maximal
3	C₂×C₄×C₈×C₂₁₁	C₂	1	1	yes	maximal, characteristic
4	C₂×#16809946	C₂²	1	4	yes	—
5	#20269913	—	2	8	no	—
6	C₄⋊(C₂₁₁⋊C₈)	—	2	4	no	—
7	C₂²×C₈×C₂₁₁	C₂²	1	2	yes	—
8	C₂×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
9	C₄×C₈×C₂₁₁	C₄	1	2	yes	—
10	C₄×C₈×C₂₁₁	C₄	1	2	yes	—
11	#16809946	—	2	8	no	—
12	C₂×C₂₁₁⋊C₈	—	2	4	no	—
13	C₂²×C₄×C₂₁₁	Q₈	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	1	yes	characteristic
19	C₄²×C₂₁₁	C₂×C₄	1	1	yes	characteristic
20	C₄²×C₂₁₁	D₄	1	2	yes	—
21	C₂₁₁⋊C₈	—	4	4	no	—
22	C₂³×C₂₁₁	C₂×Q₈	1	1	yes	characteristic
23	C₂×C₄×C₂₁₁	—	2	4	no	—
24	C₂×C₄×C₂₁₁	C₄⋊C₄	1	2	yes	—
25	C₂×C₄×C₂₁₁	C₄⋊C₄	1	2	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×D₄	1	1	yes	characteristic
29	C₂×C₄×C₂₁₁	C₂×Q₈	1	2	yes	—
30	C₈×C₂₁₁	C₄⋊C₄	1	8	yes	—
31	C₂²×C₂₁₁	—	2	2	no	—
32	C₂²×C₂₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
33	C₂²×C₂₁₁	C₄⋊Q₈	1	2	yes	—
34	C₄×C₂₁₁	—	2	2	no	—
35	C₄×C₂₁₁	#108	1	4	yes	complemented
36	C₄×C₂₁₁	C₂×C₄⋊C₄	1	2	yes	—
37	C₄×C₂₁₁	#127	1	1	yes	characteristic
38	C₄×C₂₁₁	C₄⋊Q₈	1	1	yes	characteristic
39	C₂×C₂₁₁	—	2	2	no	—
40	C₂×C₂₁₁	#384	1	1	yes	characteristic
41	C₂×C₂₁₁	C₂×#108	1	2	yes	—
42	C₂₁₁	#1848	1	1	yes	characteristic, complemented, Sylow
43	#1848	—	211	1	no	maximal, Sylow
44	C₂×#108	—	211	4	no	—
45	#423	—	211	2	no	—
46	C₂×C₄×C₈	D₂₁₁	1	1	yes	characteristic
47	C₂×M₁₆	—	211	4	no	—
48	C₂²×C₈	C₂×D₂₁₁	1	2	yes	—
49	C₂×C₄²	C₂×D₂₁₁	1	1	yes	characteristic
50	#108	—	422	8	no	—
51	C₄⋊C₈	—	422	4	no	—
52	C₄×C₈	Dic₂₁₁	1	2	yes	—
53	C₄×C₈	Dic₂₁₁	1	2	yes	—
54	C₂²×C₄	Dic₄₂₂	1	2	yes	—
55	C₂²×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
56	M₁₆	—	422	8	no	—
57	C₂×C₈	—	422	4	no	—
58	C₂×C₈	C₂×Dic₂₁₁	1	4	yes	—
59	C₂×C₈	D₈₄₄	1	4	yes	—
60	C₂×C₈	Dic₄₂₂	1	4	yes	—
61	C₄²	C₂×Dic₂₁₁	1	1	yes	characteristic
62	C₄²	C₂×Dic₂₁₁	1	1	yes	characteristic
63	C₄²	D₈₄₄	1	2	yes	—
64	C₂³	C₂×Dic₄₂₂	1	1	yes	characteristic
65	C₂×C₄	—	2	4	no	—
66	C₂×C₄	C₄⋊Dic₂₁₁	1	2	yes	—
67	C₂×C₄	C₄⋊Dic₂₁₁	1	2	yes	—
68	C₂×C₄	C₂²×Dic₂₁₁	1	1	yes	characteristic
69	C₂×C₄	C₂×D₈₄₄	1	2	yes	—
70	C₂×C₄	C₂×D₈₄₄	1	1	yes	characteristic
71	C₂×C₄	C₂×Dic₄₂₂	1	2	yes	—
72	C₈	—	844	4	no	—
73	C₈	C₄⋊Dic₂₁₁	1	8	yes	—
74	C₂²	—	2	2	no	—
75	C₂²	C₂×C₄⋊Dic₂₁₁	1	1	yes	characteristic
76	C₂²	#20269977	1	2	yes	—
77	C₄	—	2	2	no	—
78	C₄	#20269913	1	4	yes	complemented
79	C₄	C₂×C₄⋊Dic₂₁₁	1	2	yes	—
80	C₄	#20269974	1	1	yes	characteristic
81	C₄	#20269977	1	1	yes	characteristic
82	C₂	—	2	2	no	—
83	C₂	#27213016	1	1	yes	characteristic
84	C₂	C₂×#20269913	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₈₄₄
commutator	C₂×C₄×C₂₁₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₁₁
Fitting	C₂×C₄×C₈×C₂₁₁	C₂
socle	C₂²×C₂₁₁	C₂×C₄⋊C₄
prime radical	C₂³×C₂₁₁	C₂×Q₈
cosocle	C₂²×C₄×C₂₁₁	C₂³
hypercenter	C₂×C₄×C₈	D₂₁₁
hypocenter	C₂₁₁	#1848
LCS	G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
35. C₄×C₂₁₁	#108	semidirect
42. C₂₁₁	#1848	semidirect
78. C₄	#20269913	semidirect

Maximal subgroups

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

subgroup	index
C₂×#20269913	2
#27213170	2
C₂×C₄×C₈×C₂₁₁	2
#1848	211