Subgroups — #41012999

Order 27008.

190 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27213544	C₂	1	1	yes	maximal, characteristic, complemented
2	#27213519	C₂	1	1	yes	maximal, characteristic
3	#27213497	C₂	1	2	yes	maximal, complemented
4	#27213493	C₂	1	2	yes	maximal
5	C₈²×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16809951	—	2	2	no	—
7	#20269977	C₂²	1	2	yes	—
8	#20269977	C₂²	1	1	yes	characteristic
9	#20269974	C₂²	1	1	yes	characteristic
10	#20269909	—	2	2	no	—
11	#20269902	—	2	4	no	—
12	#20269899	—	2	4	no	—
13	C₄×C₈×C₂₁₁	C₂²	1	2	yes	—
14	C₄×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
15	C₂×Dic₄₂₂	—	2	2	no	—
16	C₂×Dic₄₂₂	—	2	1	no	—
17	C₂×D₈₄₄	—	2	2	no	—
18	C₂×D₈₄₄	—	2	1	no	—
19	#16809951	—	4	4	no	—
20	C₄⋊Dic₂₁₁	—	2	2	no	—
21	C₄⋊Dic₂₁₁	—	2	2	no	—
22	C₄⋊Dic₂₁₁	—	2	2	no	—
23	C₂×C₈×C₂₁₁	D₄	1	4	yes	—
24	C₂×C₈×C₂₁₁	D₄	1	2	yes	—
25	C₄²×C₂₁₁	C₂³	1	1	yes	characteristic
26	C₂²×D₂₁₁	—	4	1	no	—
27	Dic₄₂₂	—	4	2	no	—
28	Dic₄₂₂	—	4	4	no	—
29	D₈₄₄	—	4	2	no	—
30	D₈₄₄	—	4	4	no	—
31	C₂×Dic₂₁₁	—	4	1	no	—
32	C₂×Dic₂₁₁	—	4	2	no	—
33	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₂₁₁	C₂×D₄	1	2	yes	—
35	C₈×C₂₁₁	—	2	4	no	—
36	C₈×C₂₁₁	SD₁₆	1	4	yes	—
37	C₂²×C₂₁₁	#127	1	1	yes	characteristic
38	C₂×D₂₁₁	—	8	1	no	—
39	C₂×D₂₁₁	—	8	2	no	—
40	C₄×C₂₁₁	C₂×SD₁₆	1	2	yes	—
41	C₄×C₂₁₁	#135	1	4	yes	—
42	Dic₂₁₁	—	8	1	no	—
43	Dic₂₁₁	—	8	2	no	—
44	C₂×C₂₁₁	#492	1	1	yes	characteristic
45	C₂×C₂₁₁	#495	1	2	yes	—
46	D₂₁₁	—	16	1	no	—
47	C₂₁₁	#1708	1	1	yes	characteristic, complemented, Sylow
48	#1708	—	211	1	no	maximal, Sylow
49	#498	—	211	1	no	—
50	#492	—	211	1	no	—
51	#487	—	211	2	no	—
52	#486	—	211	2	no	—
53	C₈²	D₂₁₁	1	1	yes	characteristic, complemented
54	C₂×SD₁₆	—	422	2	no	—
55	C₄⋊Q₈	—	211	2	no	—
56	C₄⋊Q₈	—	211	1	no	—
57	#127	—	211	1	no	—
58	#106	—	422	2	no	—
59	#103	—	422	4	no	—
60	#102	—	422	4	no	—
61	C₄×C₈	C₂×D₂₁₁	1	2	yes	—
62	C₄×C₈	C₂×D₂₁₁	1	1	yes	characteristic
63	C₂×Q₈	—	422	2	no	—
64	C₂×Q₈	—	422	1	no	—
65	C₂×D₄	—	422	2	no	—
66	C₂×D₄	—	422	1	no	—
67	SD₁₆	—	844	4	no	—
68	C₂×C₈	D₈₄₄	1	4	yes	—
69	C₂×C₈	D₈₄₄	1	2	yes	—
70	C₄⋊C₄	—	422	2	no	—
71	C₄⋊C₄	—	422	2	no	—
72	C₄⋊C₄	—	422	2	no	—
73	C₄²	C₂²×D₂₁₁	1	1	yes	characteristic
74	C₂³	—	844	1	no	—
75	Q₈	—	844	2	no	—
76	Q₈	—	844	4	no	—
77	D₄	—	844	2	no	—
78	D₄	—	844	4	no	—
79	C₂×C₄	—	844	1	no	—
80	C₂×C₄	—	844	2	no	—
81	C₂×C₄	C₂×D₈₄₄	1	1	yes	characteristic
82	C₂×C₄	C₂×D₈₄₄	1	2	yes	—
83	C₈	—	2	4	no	—
84	C₈	#16809951	1	4	yes	—
85	C₂²	—	1688	1	no	—
86	C₂²	—	1688	2	no	—
87	C₂²	#20269974	1	1	yes	characteristic
88	C₄	—	1688	1	no	—
89	C₄	—	1688	2	no	—
90	C₄	C₂×#16809951	1	2	yes	—
91	C₄	#20269999	1	4	yes	—
92	C₂	—	3376	1	no	—
93	C₂	#27213519	1	1	yes	characteristic
94	C₂	#27213530	1	2	yes	—

	subgroup	G/H
center	C₂²	#20269974
commutator, cosocle	C₄²×C₂₁₁	C₂³
Frattini	C₄²	C₂²×D₂₁₁
Fitting	C₈²×C₂₁₁	C₂
socle	C₂²×C₂₁₁	#127
prime radical	#20269974	C₂²
hypercenter	C₈²	D₂₁₁
hypocenter	C₂₁₁	#1708
LCS	G ▷ C₄²×C₂₁₁ ▷ C₂²×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈²

Complements

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

N	H ≅ G/N	type
1. #27213544	C₂	semidirect
3. #27213497	C₂	semidirect
5. C₈²×C₂₁₁	C₂	semidirect
47. C₂₁₁	#1708	semidirect
53. C₈²	D₂₁₁	semidirect

Maximal subgroups

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

subgroup	index
#27213544	2
#27213519	2
#27213497	2
#27213493	2
C₈²×C₂₁₁	2
#1708	211