Subgroups — #27213418

Order 13504.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20270008	C₂	1	2	yes	maximal
2	#20269983	C₂	1	1	yes	maximal, characteristic
3	#20269913	C₂	1	1	yes	maximal, characteristic, complemented
4	#20269896	C₂	1	2	yes	maximal, complemented
5	C₂×Q₁₆×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#16809962	C₂²	1	1	yes	characteristic
7	#16809952	—	2	2	no	—
8	#16809949	—	2	2	no	—
9	#16809946	C₂²	1	1	yes	characteristic
10	#16809946	C₂²	1	2	yes	—
11	#16809945	—	2	1	no	—
12	C₈×D₂₁₁	—	2	1	no	—
13	C₂×Q₈×C₂₁₁	C₂²	1	2	yes	—
14	Q₁₆×C₂₁₁	—	2	1	no	—
15	Q₁₆×C₂₁₁	—	2	1	no	—
16	C₂×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
17	Dic₄₂₂	D₄	1	1	yes	characteristic
18	D₈₄₄	D₄	1	1	yes	characteristic
19	C₂²⋊D₂₁₁	—	2	1	no	—
20	C₄×D₂₁₁	—	2	1	no	—
21	C₂₁₁⋊C₈	—	2	1	no	—
22	C₂₁₁⋊C₈	—	2	2	no	—
23	Q₈×C₂₁₁	—	2	2	no	—
24	Q₈×C₂₁₁	—	2	2	no	—
25	C₂×C₄×C₂₁₁	—	2	2	no	—
26	C₂×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
27	C₈×C₂₁₁	D₄	1	1	yes	characteristic
28	C₈×C₂₁₁	D₄	1	1	yes	characteristic
29	C₂²×C₂₁₁	#41	1	1	yes	characteristic
30	C₂×D₂₁₁	—	2	1	no	—
31	C₄×C₂₁₁	—	4	2	no	—
32	C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
33	C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
34	Dic₂₁₁	—	2	1	no	—
35	C₂×C₂₁₁	—	2	1	no	—
36	C₂×C₂₁₁	#121	1	1	yes	characteristic
37	D₂₁₁	—	4	1	no	—
38	C₂₁₁	#473	1	1	yes	characteristic, complemented, Sylow
39	#473	—	211	1	no	maximal, Sylow
40	#137	—	211	2	no	—
41	C₂×Q₁₆	D₂₁₁	1	1	yes	characteristic, complemented
42	#131	—	211	1	no	—
43	#108	—	211	1	no	—
44	#101	—	211	2	no	—
45	#41	—	211	1	no	—
46	C₂×Q₈	C₂×D₂₁₁	1	2	yes	—
47	Q₁₆	—	2	1	no	—
48	Q₁₆	—	2	1	no	—
49	Q₁₆	—	422	2	no	—
50	SD₁₆	—	422	2	no	—
51	M₁₆	—	422	1	no	—
52	M₁₆	—	211	1	no	—
53	M₁₆	—	211	2	no	—
54	C₂×C₈	—	422	1	no	—
55	C₂×C₈	C₂×D₂₁₁	1	1	yes	characteristic
56	Q₈	—	2	2	no	—
57	Q₈	—	2	2	no	—
58	Q₈	—	211	1	no	—
59	D₄	—	422	1	no	—
60	D₄	—	211	1	no	—
61	C₂×C₄	—	2	2	no	—
62	C₂×C₄	—	422	1	no	—
63	C₂×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
64	C₈	—	422	2	no	—
65	C₈	—	422	1	no	—
66	C₈	C₂²⋊D₂₁₁	1	1	yes	characteristic
67	C₈	C₂²⋊D₂₁₁	1	1	yes	characteristic
68	C₂²	—	422	1	no	—
69	C₂²	#16809961	1	1	yes	characteristic
70	C₄	—	4	2	no	—
71	C₄	—	422	1	no	—
72	C₄	D₄×D₂₁₁	1	1	yes	characteristic
73	C₄	C₂×C₂²⋊D₂₁₁	1	1	yes	characteristic
74	C₂	—	2	1	no	—
75	C₂	—	844	1	no	—
76	C₂	#20269948	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20269948
commutator, cosocle	C₂×C₄×C₂₁₁	C₂³
Frattini	C₂×C₄	C₂²×D₂₁₁
Fitting	C₂×Q₁₆×C₂₁₁	C₂
socle	C₂×C₂₁₁	#121
prime radical	#16809962	C₂²
hypercenter	C₂×Q₁₆	D₂₁₁
hypocenter	C₂₁₁	#473
LCS	G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×Q₁₆

Complements

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

N	H ≅ G/N	type
3. #20269913	C₂	semidirect
4. #20269896	C₂	semidirect
5. C₂×Q₁₆×C₂₁₁	C₂	semidirect
38. C₂₁₁	#473	semidirect
41. C₂×Q₁₆	D₂₁₁	semidirect

Maximal subgroups

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

subgroup	index
#20270008	2
#20269983	2
#20269913	2
#20269896	2
C₂×Q₁₆×C₂₁₁	2
#473	211