Subgroups — #41012992

Order 27008.

180 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27213551	C₂	1	1	yes	maximal, characteristic
2	#27213542	C₂	1	2	yes	maximal
3	#27213498	C₂	1	2	yes	maximal
4	C₈×C₂₁₁⋊C₈	C₂	1	1	yes	maximal, characteristic
5	#494×C₂₁₁	C₂	1	1	yes	maximal, characteristic
6	#20269977	C₂²	1	2	yes	—
7	#20269911	—	2	2	no	—
8	#20269908	—	2	4	no	—
9	#20269900	—	2	4	no	—
10	C₄×C₂₁₁⋊C₈	C₂²	1	1	yes	characteristic
11	C₄×C₂₁₁⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×Q₁₆×C₂₁₁	—	2	2	no	—
13	C₄⋊Q₈×C₂₁₁	C₂²	1	2	yes	—
14	C₄×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
15	C₂×Dic₄₂₂	—	2	2	no	—
16	C₂×C₂₁₁⋊C₈	D₄	1	2	yes	—
17	C₂×C₂₁₁⋊C₈	D₄	1	2	yes	—
18	C₄⋊Dic₂₁₁	—	2	2	no	—
19	C₄⋊Dic₂₁₁	—	2	2	no	—
20	C₂×Q₈×C₂₁₁	—	2	2	no	—
21	Q₁₆×C₂₁₁	—	4	4	no	—
22	C₂×C₈×C₂₁₁	D₄	1	2	yes	—
23	C₄⋊C₄×C₂₁₁	—	2	2	no	—
24	C₄⋊C₄×C₂₁₁	—	2	2	no	—
25	C₄²×C₂₁₁	C₂³	1	1	yes	characteristic
26	Dic₄₂₂	—	4	4	no	—
27	C₂×Dic₂₁₁	—	4	2	no	—
28	C₂₁₁⋊C₈	—	2	2	no	—
29	C₂₁₁⋊C₈	Q₁₆	1	4	yes	complemented
30	Q₈×C₂₁₁	—	4	4	no	—
31	C₂×C₄×C₂₁₁	—	4	2	no	—
32	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
35	C₈×C₂₁₁	SD₁₆	1	4	yes	—
36	C₂²×C₂₁₁	#127	1	1	yes	characteristic
37	C₄×C₂₁₁	—	8	2	no	—
38	C₄×C₂₁₁	C₂×SD₁₆	1	2	yes	—
39	C₄×C₂₁₁	C₂×Q₁₆	1	2	yes	—
40	C₄×C₂₁₁	#135	1	2	yes	—
41	Dic₂₁₁	—	8	2	no	—
42	C₂×C₂₁₁	#492	1	1	yes	characteristic
43	C₂×C₂₁₁	#494	1	1	yes	characteristic
44	C₂×C₂₁₁	#495	1	1	yes	characteristic
45	C₂₁₁	#1707	1	1	yes	characteristic, complemented, Sylow
46	#1707	—	211	1	no	maximal, Sylow
47	#500	—	211	1	no	—
48	#498	—	211	2	no	—
49	#494	D₂₁₁	1	1	yes	characteristic
50	#487	—	211	2	no	—
51	C₈²	—	211	1	no	—
52	C₂×Q₁₆	—	2	2	no	—
53	C₄⋊Q₈	—	211	2	no	—
54	C₄⋊Q₈	C₂×D₂₁₁	1	2	yes	—
55	#107	—	422	2	no	—
56	#106	—	422	4	no	—
57	#103	—	422	4	no	—
58	C₄×C₈	—	211	1	no	—
59	C₄×C₈	—	211	1	no	—
60	C₄×C₈	C₂×D₂₁₁	1	1	yes	characteristic
61	C₂×Q₈	—	2	2	no	—
62	C₂×Q₈	—	422	2	no	—
63	Q₁₆	—	4	4	no	—
64	C₂×C₈	—	211	2	no	—
65	C₂×C₈	—	211	2	no	—
66	C₂×C₈	C₂²⋊D₂₁₁	1	2	yes	—
67	C₄⋊C₄	—	2	2	no	—
68	C₄⋊C₄	—	2	2	no	—
69	C₄⋊C₄	—	422	2	no	—
70	C₄⋊C₄	—	422	2	no	—
71	C₄²	C₂²×D₂₁₁	1	1	yes	characteristic
72	Q₈	—	4	4	no	—
73	Q₈	—	844	4	no	—
74	C₂×C₄	—	4	2	no	—
75	C₂×C₄	—	844	2	no	—
76	C₂×C₄	D₄×D₂₁₁	1	1	yes	characteristic
77	C₂×C₄	D₄×D₂₁₁	1	1	yes	characteristic
78	C₂×C₄	C₂×C₂²⋊D₂₁₁	1	1	yes	characteristic
79	C₈	—	422	2	no	—
80	C₈	—	211	4	no	—
81	C₈	#16809950	1	4	yes	—
82	C₂²	#20269973	1	1	yes	characteristic
83	C₄	—	8	2	no	—
84	C₄	—	1688	2	no	—
85	C₄	C₂×#16809950	1	2	yes	—
86	C₄	Q₁₆×D₂₁₁	1	2	yes	—
87	C₄	#20270000	1	2	yes	—
88	C₂	#27213518	1	1	yes	characteristic
89	C₂	#27213524	1	1	yes	characteristic
90	C₂	#27213531	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#20269973
commutator, cosocle	C₄²×C₂₁₁	C₂³
Frattini	C₄²	C₂²×D₂₁₁
Fitting	#494×C₂₁₁	C₂
socle, prime radical	C₂²×C₂₁₁	#127
hypercenter	#494	D₂₁₁
hypocenter	C₂₁₁	#1707
LCS	G ▷ C₄²×C₂₁₁ ▷ C₂²×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

Complements

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

N	H ≅ G/N	type
29. C₂₁₁⋊C₈	Q₁₆	semidirect
45. C₂₁₁	#1707	semidirect

Maximal subgroups

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

subgroup	index
#27213551	2
#27213542	2
#27213498	2
C₈×C₂₁₁⋊C₈	2
#494×C₂₁₁	2
#1707	211