Subgroups — #27213444

Order 13504.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20269968	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×Dic₂₁₁	C₂	1	1	yes	maximal, characteristic
3	#20269911	C₂	1	1	yes	maximal, characteristic, complemented
4	#20269908	C₂	1	1	yes	maximal, characteristic, complemented
5	Dic₂₁₁⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	#20269898	C₂	1	1	yes	maximal, characteristic
7	#102×C₂₁₁	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₂₁₁	—	2	1	no	—
9	C₂×C₂₁₁⋊C₈	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₂₁₁	—	2	1	no	—
11	C₄⋊Dic₂₁₁	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₂₁₁	C₂²	1	1	yes	characteristic
13	Dic₂₁₁⋊C₄	—	2	1	no	—
14	C₂²⋊Dic₂₁₁	—	2	1	no	—
15	C₄×Dic₂₁₁	C₂²	1	1	yes	characteristic
16	C₂×D₄×C₂₁₁	C₂²	1	1	yes	characteristic
17	C₂×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×C₂₁₁	C₂²	1	1	yes	characteristic
19	C₂×Dic₂₁₁	—	2	1	no	—
20	C₂×Dic₂₁₁	—	2	1	no	—
21	C₂×Dic₂₁₁	—	4	2	no	—
22	C₂×Dic₂₁₁	D₄	1	1	yes	characteristic
23	C₂×Dic₂₁₁	D₄	1	1	yes	characteristic
24	C₂₁₁⋊C₈	—	2	1	no	—
25	C₂³×C₂₁₁	—	2	1	no	—
26	D₄×C₂₁₁	—	2	1	no	—
27	D₄×C₂₁₁	Q₈	1	2	yes	—
28	C₂×C₄×C₂₁₁	—	2	1	no	—
29	C₂×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
30	C₈×C₂₁₁	—	2	1	no	—
31	C₂²×C₂₁₁	—	2	2	no	—
32	C₂²×C₂₁₁	—	4	1	no	—
33	C₂²×C₂₁₁	—	4	1	no	—
34	C₂²×C₂₁₁	C₂×D₄	1	1	yes	characteristic
35	C₄×C₂₁₁	—	4	1	no	—
36	C₄×C₂₁₁	C₂×Q₈	1	1	yes	characteristic
37	C₄×C₂₁₁	#41	1	1	yes	characteristic
38	Dic₂₁₁	—	2	1	no	—
39	Dic₂₁₁	—	2	1	no	—
40	Dic₂₁₁	—	4	1	no	—
41	Dic₂₁₁	—	4	1	no	—
42	C₂×C₂₁₁	—	4	2	no	—
43	C₂×C₂₁₁	C₂²⋊Q₈	1	1	yes	characteristic
44	C₂×C₂₁₁	#135	1	1	yes	characteristic
45	C₂×C₂₁₁	AGL(1,ℤ/8)	1	1	yes	characteristic
46	C₂₁₁	#478	1	1	yes	characteristic, complemented, Sylow
47	#478	—	211	1	no	maximal, Sylow
48	#125	—	211	1	no	—
49	C₄×D₄	—	211	1	no	—
50	#107	—	211	1	no	—
51	#106	—	211	1	no	—
52	C₄⋊C₈	—	211	1	no	—
53	#102	—	211	1	no	—
54	#102	D₂₁₁	1	1	yes	characteristic
55	C₂×D₄	C₂×D₂₁₁	1	1	yes	characteristic
56	C₂²×C₄	—	422	1	no	—
57	C₂×C₈	—	211	1	no	—
58	C₂×C₈	C₂×D₂₁₁	1	1	yes	characteristic
59	C₄⋊C₄	—	422	1	no	—
60	C₄⋊C₄	—	422	1	no	—
61	C₄⋊C₄	—	211	1	no	—
62	C₄⋊C₄	—	211	1	no	—
63	C₄⋊C₄	C₂×D₂₁₁	1	1	yes	characteristic
64	C₂²⋊C₄	—	422	1	no	—
65	C₄²	—	211	1	no	—
66	C₂³	—	2	1	no	—
67	D₄	—	2	1	no	—
68	D₄	Dic₄₂₂	1	2	yes	—
69	C₂×C₄	—	2	1	no	—
70	C₂×C₄	—	844	2	no	—
71	C₂×C₄	—	422	1	no	—
72	C₂×C₄	—	422	1	no	—
73	C₂×C₄	—	211	1	no	—
74	C₂×C₄	—	211	1	no	—
75	C₂×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
76	C₈	—	2	1	no	—
77	C₈	—	422	1	no	—
78	C₂²	—	2	2	no	—
79	C₂²	—	4	1	no	—
80	C₂²	—	4	1	no	—
81	C₂²	D₄×D₂₁₁	1	1	yes	characteristic
82	C₄	—	4	1	no	—
83	C₄	—	844	1	no	—
84	C₄	—	844	1	no	—
85	C₄	—	422	1	no	—
86	C₄	—	422	1	no	—
87	C₄	C₂×Dic₄₂₂	1	1	yes	characteristic
88	C₄	#16809961	1	1	yes	characteristic
89	C₂	—	4	2	no	—
90	C₂	#20269954	1	1	yes	characteristic
91	C₂	#20269996	1	1	yes	characteristic
92	C₂	#20270004	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₂₁₁
commutator, cosocle	C₂×C₄×C₂₁₁	C₂³
Frattini	C₂×C₄	C₂²×D₂₁₁
Fitting	#102×C₂₁₁	C₂
socle	C₂²×C₂₁₁	C₂×D₄
prime radical	C₂×D₄×C₂₁₁	C₂²
hypercenter	#102	D₂₁₁
hypocenter	C₂₁₁	#478
LCS	G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #102

Complements

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

N	H ≅ G/N	type
1. #20269968	C₂	semidirect
3. #20269911	C₂	semidirect
4. #20269908	C₂	semidirect
5. Dic₂₁₁⋊C₈	C₂	semidirect
46. C₂₁₁	#478	semidirect

Maximal subgroups

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

subgroup	index
#20269968	2
D₄×Dic₂₁₁	2
#20269911	2
#20269908	2
Dic₂₁₁⋊C₈	2
#20269898	2
#102×C₂₁₁	2
#478	211