Subgroups — #41011438

Order 27008.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27213519	C₂	1	1	yes	maximal, characteristic
2	#27212967	C₂	1	1	yes	maximal, characteristic, complemented
3	#364×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#16809951	—	2	1	no	—
5	C₂×#16809951	—	2	1	no	—
6	#20269977	C₄	1	1	yes	characteristic
7	#20269974	C₄	1	1	yes	characteristic
8	#20269917	—	2	1	no	—
9	M₃₂×C₂₁₁	—	2	1	no	—
10	C₄×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
11	C₂×Dic₄₂₂	—	2	1	no	—
12	C₂×D₈₄₄	—	2	1	no	—
13	C₂×D₈₄₄	—	4	1	no	—
14	#16809951	—	4	1	no	—
15	#16809951	—	4	1	no	—
16	#16809951	—	4	1	no	—
17	#16809951	—	4	1	no	—
18	C₄⋊Dic₂₁₁	—	4	1	no	—
19	C₂₁₁⋊C₁₆	—	4	1	no	—
20	C₂×C₈×C₂₁₁	D₄	1	1	yes	characteristic
21	C₂×C₈×C₂₁₁	D₄	1	1	yes	characteristic
22	C₄²×C₂₁₁	C₂×C₄	1	1	yes	characteristic
23	C₁₆×C₂₁₁	—	4	1	no	—
24	C₂²×D₂₁₁	—	4	1	no	—
25	Dic₄₂₂	—	4	1	no	—
26	Dic₄₂₂	—	4	1	no	—
27	D₈₄₄	—	4	1	no	—
28	D₈₄₄	—	4	1	no	—
29	D₈₄₄	—	8	2	no	—
30	C₂×Dic₂₁₁	—	4	1	no	—
31	C₂×C₄×C₂₁₁	—	2	1	no	—
32	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
33	C₈×C₂₁₁	D₈	1	1	yes	characteristic
34	C₈×C₂₁₁	D₈	1	1	yes	characteristic
35	C₈×C₂₁₁	SD₁₆	1	1	yes	characteristic
36	C₈×C₂₁₁	SD₁₆	1	1	yes	characteristic
37	C₂²×C₂₁₁	#100	1	1	yes	characteristic
38	C₂×D₂₁₁	—	8	2	no	—
39	C₂×D₂₁₁	—	8	1	no	—
40	C₄×C₂₁₁	—	2	2	no	—
41	C₄×C₂₁₁	#102	1	1	yes	characteristic
42	C₄×C₂₁₁	#102	1	1	yes	characteristic
43	Dic₂₁₁	—	8	1	no	—
44	C₂×C₂₁₁	—	2	1	no	—
45	C₂×C₂₁₁	#331	1	1	yes	characteristic
46	D₂₁₁	—	16	1	no	—
47	C₂₁₁	#1359	1	1	yes	characteristic, complemented, Sylow
48	#1359	—	211	1	no	maximal, Sylow
49	#492	—	211	1	no	—
50	#364	—	211	1	no	—
51	#364	D₂₁₁	1	1	yes	characteristic, complemented
52	C₂×SD₁₆	—	422	1	no	—
53	C₂×SD₁₆	—	422	1	no	—
54	C₄⋊Q₈	—	211	1	no	—
55	#127	—	211	1	no	—
56	M₃₂	—	2	1	no	—
57	M₃₂	—	422	1	no	—
58	C₄×C₈	C₂×D₂₁₁	1	1	yes	characteristic
59	C₂×Q₈	—	422	1	no	—
60	C₂×D₄	—	844	1	no	—
61	C₂×D₄	—	422	1	no	—
62	SD₁₆	—	844	1	no	—
63	SD₁₆	—	844	1	no	—
64	SD₁₆	—	844	1	no	—
65	SD₁₆	—	844	1	no	—
66	C₂×C₈	C₂²⋊D₂₁₁	1	1	yes	characteristic
67	C₂×C₈	D₈₄₄	1	1	yes	characteristic
68	C₄⋊C₄	—	844	1	no	—
69	C₄²	C₄×D₂₁₁	1	1	yes	characteristic
70	C₁₆	—	4	1	no	—
71	C₁₆	—	844	1	no	—
72	C₂³	—	844	1	no	—
73	Q₈	—	844	1	no	—
74	Q₈	—	844	1	no	—
75	D₄	—	844	1	no	—
76	D₄	—	844	1	no	—
77	D₄	—	1688	2	no	—
78	C₂×C₄	—	2	1	no	—
79	C₂×C₄	—	844	1	no	—
80	C₂×C₄	(C₂×D₂₁₁)⋊C₄	1	1	yes	characteristic
81	C₈	#16809947	1	1	yes	characteristic
82	C₈	D₁₆₈₈	1	1	yes	characteristic
83	C₈	#16809949	1	1	yes	characteristic
84	C₈	#16809951	1	1	yes	characteristic
85	C₂²	—	1688	2	no	—
86	C₂²	—	1688	1	no	—
87	C₂²	#20269893	1	1	yes	characteristic
88	C₄	—	2	2	no	—
89	C₄	—	1688	1	no	—
90	C₄	#20269897	1	1	yes	characteristic
91	C₄	#20269899	1	1	yes	characteristic
92	C₂	—	2	1	no	—
93	C₂	—	3376	1	no	—
94	C₂	#27212889	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#27212889
commutator	C₄²×C₂₁₁	C₂×C₄
Frattini	C₄×C₈	C₂×D₂₁₁
Fitting	#364×C₂₁₁	C₂
socle	C₂×C₂₁₁	#331
prime radical	#20269974	C₄
cosocle	C₄×C₈×C₂₁₁	C₂²
hypercenter	#364	D₂₁₁
hypocenter	C₂₁₁	#1359
LCS	G ▷ C₄²×C₂₁₁ ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #364

Complements

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

N	H ≅ G/N	type
2. #27212967	C₂	semidirect
3. #364×C₂₁₁	C₂	semidirect
47. C₂₁₁	#1359	semidirect
51. #364	D₂₁₁	semidirect

Maximal subgroups

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

subgroup	index
#27213519	2
#27212967	2
#364×C₂₁₁	2
#1359	211