Subgroups — #473×C₂₁₁ #27212751

Order 13504.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

	subgroup	G/H
center, socle	C₂×C₂₁₁	#121
commutator, Frattini, cosocle	C₂×C₄	C₂³×C₂₁₁
prime radical	#41×C₂₁₁	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₂₁₁ ◁ C₂×C₄×C₂₁₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×Q₁₆×C₂₁₁	C₂	semidirect
4. #108×C₂₁₁	C₂	semidirect
5. #101×C₂₁₁	C₂	semidirect
38. C₂₁₁	#473	direct (N × H)
39. #473	C₂₁₁	direct (N × H)
41. C₂×Q₁₆	C₂×C₂₁₁	semidirect
43. #108	C₂×C₂₁₁	semidirect
44. #101	C₂×C₂₁₁	semidirect

Maximal subgroups

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

subgroup	index
#137×C₂₁₁	2
C₂×Q₁₆×C₂₁₁	2
#131×C₂₁₁	2
#108×C₂₁₁	2
#101×C₂₁₁	2
#473	211

Subgroups — #473×C211 #27212751

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #473×C₂₁₁ #27212751