Subgroups — #41013069

Order 27008.

198 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27213551	C₂	1	2	yes	maximal, complemented
2	#27213526	C₂	1	1	yes	maximal, characteristic, complemented
3	#27213519	C₂	1	1	yes	maximal, characteristic
4	#27213493	C₂	1	2	yes	maximal
5	#322×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₈₄₄	—	2	2	no	—
7	C₂×#16809951	—	2	2	no	—
8	#20269977	C₂²	1	2	yes	—
9	#20269977	C₂²	1	1	yes	characteristic
10	#20269974	C₂²	1	1	yes	characteristic
11	#20269911	—	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	1	no	—
16	C₂×Dic₄₂₂	—	2	2	no	—
17	C₂×D₈₄₄	—	2	2	no	—
18	C₂×D₈₄₄	—	2	1	no	—
19	Dic₈₄₄	—	4	4	no	—
20	#16809951	—	4	4	no	—
21	C₄⋊Dic₂₁₁	—	2	2	no	—
22	C₄⋊Dic₂₁₁	—	2	2	no	—
23	C₄⋊Dic₂₁₁	—	2	2	no	—
24	C₂×C₈×C₂₁₁	D₄	1	4	yes	—
25	C₂×C₈×C₂₁₁	D₄	1	2	yes	—
26	C₄²×C₂₁₁	C₂³	1	1	yes	characteristic
27	C₂²×D₂₁₁	—	4	1	no	—
28	Dic₄₂₂	—	4	2	no	—
29	Dic₄₂₂	—	4	4	no	—
30	D₈₄₄	—	4	2	no	—
31	D₈₄₄	—	4	4	no	—
32	C₂×Dic₂₁₁	—	4	2	no	—
33	C₂×Dic₂₁₁	—	4	1	no	—
34	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₂₁₁	C₂×D₄	1	2	yes	—
36	C₈×C₂₁₁	—	2	4	no	—
37	C₈×C₂₁₁	D₈	1	4	yes	—
38	C₂²×C₂₁₁	#127	1	1	yes	characteristic
39	C₂×D₂₁₁	—	8	1	no	—
40	C₂×D₂₁₁	—	8	2	no	—
41	C₄×C₂₁₁	C₂×D₈	1	2	yes	—
42	C₄×C₂₁₁	#137	1	4	yes	—
43	Dic₂₁₁	—	8	1	no	—
44	Dic₂₁₁	—	8	2	no	—
45	C₂×C₂₁₁	#493	1	1	yes	characteristic
46	C₂×C₂₁₁	#497	1	2	yes	—
47	D₂₁₁	—	16	1	no	—
48	C₂₁₁	#1722	1	1	yes	characteristic, complemented, Sylow
49	#1722	—	211	1	no	maximal, Sylow
50	#500	—	211	2	no	—
51	#494	—	211	1	no	—
52	#492	—	211	1	no	—
53	#486	—	211	2	no	—
54	#322	D₂₁₁	1	1	yes	characteristic, complemented
55	C₂×Q₁₆	—	422	2	no	—
56	C₂×SD₁₆	—	422	2	no	—
57	C₄⋊Q₈	—	211	2	no	—
58	C₄⋊Q₈	—	211	1	no	—
59	#127	—	211	1	no	—
60	#107	—	422	4	no	—
61	#102	—	422	4	no	—
62	C₄×C₈	C₂×D₂₁₁	1	2	yes	—
63	C₄×C₈	C₂×D₂₁₁	1	1	yes	characteristic
64	C₂×Q₈	—	422	1	no	—
65	C₂×Q₈	—	422	2	no	—
66	C₂×D₄	—	422	2	no	—
67	C₂×D₄	—	422	1	no	—
68	Q₁₆	—	844	4	no	—
69	SD₁₆	—	844	4	no	—
70	C₂×C₈	D₈₄₄	1	4	yes	—
71	C₂×C₈	D₈₄₄	1	2	yes	—
72	C₄⋊C₄	—	422	2	no	—
73	C₄⋊C₄	—	422	2	no	—
74	C₄⋊C₄	—	422	2	no	—
75	C₄²	C₂²×D₂₁₁	1	1	yes	characteristic
76	C₂³	—	844	1	no	—
77	Q₈	—	844	2	no	—
78	Q₈	—	844	4	no	—
79	D₄	—	844	2	no	—
80	D₄	—	844	4	no	—
81	C₂×C₄	—	844	2	no	—
82	C₂×C₄	—	844	1	no	—
83	C₂×C₄	C₂×D₈₄₄	1	1	yes	characteristic
84	C₂×C₄	C₂×D₈₄₄	1	2	yes	—
85	C₈	—	2	4	no	—
86	C₈	D₁₆₈₈	1	4	yes	—
87	C₂²	—	1688	1	no	—
88	C₂²	—	1688	2	no	—
89	C₂²	#20269974	1	1	yes	characteristic
90	C₄	—	1688	1	no	—
91	C₄	—	1688	2	no	—
92	C₄	C₂×D₁₆₈₈	1	2	yes	—
93	C₄	#20270010	1	4	yes	—
94	C₂	—	3376	1	no	—
95	C₂	#27213523	1	1	yes	characteristic
96	C₂	#27213540	1	2	yes	—

	subgroup	G/H
center	C₂²	#20269974
commutator, cosocle	C₄²×C₂₁₁	C₂³
Frattini	C₄²	C₂²×D₂₁₁
Fitting	#322×C₂₁₁	C₂
socle	C₂²×C₂₁₁	#127
prime radical	#20269974	C₂²
hypercenter	#322	D₂₁₁
hypocenter	C₂₁₁	#1722
LCS	G ▷ C₄²×C₂₁₁ ▷ C₂²×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322

Complements

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

N	H ≅ G/N	type
1. #27213551	C₂	semidirect
2. #27213526	C₂	semidirect
5. #322×C₂₁₁	C₂	semidirect
48. C₂₁₁	#1722	semidirect
54. #322	D₂₁₁	semidirect

Maximal subgroups

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

subgroup	index
#27213551	2
#27213526	2
#27213519	2
#27213493	2
#322×C₂₁₁	2
#1722	211