Subgroups — C₂²×C₇×C₈×C₁₁² #72204166

Order 27104.

1064 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₇×C₁₁²	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×C₈×C₁₁²	C₂	1	6	yes	maximal, direct product factor
3	C₂³×C₇×C₁₁²	C₄	1	1	yes	characteristic
4	C₂×C₄×C₇×C₁₁²	C₄	1	3	yes	—
5	C₂×C₄×C₇×C₁₁²	C₂²	1	3	yes	—
6	C₇×C₈×C₁₁²	C₂²	1	4	yes	direct product factor
7	C₂²×C₈×C₁₁²	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₂²×C₇×C₈×C₁₁	C₁₁	1	12	yes	maximal, direct product factor
9	C₂²×C₇×C₁₁²	C₈	1	4	yes	direct product factor
10	C₂²×C₇×C₁₁²	C₂×C₄	1	3	yes	—
11	C₄×C₇×C₁₁²	C₂×C₄	1	3	yes	—
12	C₄×C₇×C₁₁²	C₂³	1	1	yes	characteristic
13	C₂²×C₄×C₁₁²	C₂×C₇	1	1	yes	characteristic
14	C₂×C₈×C₁₁²	C₂×C₇	1	6	yes	direct product factor
15	C₂×C₇×C₁₁²	C₂×C₈	1	6	yes	direct product factor
16	C₂×C₇×C₁₁²	C₂²×C₄	1	1	yes	characteristic
17	C₂²×C₄×C₇×C₁₁	C₂×C₁₁	1	12	yes	—
18	C₂×C₇×C₈×C₁₁	C₂×C₁₁	1	72	yes	direct product factor
19	C₂³×C₁₁²	C₄×C₇	1	1	yes	characteristic
20	C₂×C₄×C₁₁²	C₄×C₇	1	3	yes	—
21	C₂×C₄×C₁₁²	C₂²×C₇	1	3	yes	—
22	C₈×C₁₁²	C₂²×C₇	1	4	yes	direct product factor
23	C₇×C₁₁²	C₂²×C₈	1	1	yes	characteristic, direct product factor, Hall
24	C₂³×C₇×C₁₁	C₄×C₁₁	1	12	yes	—
25	C₂×C₄×C₇×C₁₁	C₄×C₁₁	1	36	yes	—
26	C₂×C₄×C₇×C₁₁	C₂²×C₁₁	1	36	yes	—
27	C₇×C₈×C₁₁	C₂²×C₁₁	1	48	yes	direct product factor
28	C₂²×C₁₁²	C₇×C₈	1	4	yes	direct product factor
29	C₂²×C₁₁²	C₂×C₄×C₇	1	3	yes	—
30	C₄×C₁₁²	C₂×C₄×C₇	1	3	yes	—
31	C₄×C₁₁²	C₂³×C₇	1	1	yes	characteristic
32	C₂²×C₈×C₁₁	C₇×C₁₁	1	12	yes	direct product factor
33	C₂²×C₇×C₁₁	C₈×C₁₁	1	48	yes	direct product factor
34	C₂²×C₇×C₁₁	C₂×C₄×C₁₁	1	36	yes	—
35	C₄×C₇×C₁₁	C₂×C₄×C₁₁	1	36	yes	—
36	C₄×C₇×C₁₁	C₂³×C₁₁	1	12	yes	—
37	C₂×C₁₁²	C₂×C₇×C₈	1	6	yes	direct product factor
38	C₂×C₁₁²	C₂²×C₄×C₇	1	1	yes	characteristic
39	C₂²×C₇×C₈	C₁₁²	1	1	yes	characteristic, direct product factor, Hall
40	C₂²×C₄×C₁₁	C₂×C₇×C₁₁	1	12	yes	—
41	C₂×C₈×C₁₁	C₂×C₇×C₁₁	1	72	yes	direct product factor
42	C₂×C₇×C₁₁	C₂×C₈×C₁₁	1	72	yes	direct product factor
43	C₂×C₇×C₁₁	C₂²×C₄×C₁₁	1	12	yes	—
44	C₁₁²	C₂²×C₇×C₈	1	1	yes	characteristic, direct product factor, Sylow
45	C₂²×C₄×C₇	C₂×C₁₁²	1	1	yes	characteristic
46	C₂×C₇×C₈	C₂×C₁₁²	1	6	yes	direct product factor
47	C₂³×C₁₁	C₄×C₇×C₁₁	1	12	yes	—
48	C₂×C₄×C₁₁	C₄×C₇×C₁₁	1	36	yes	—
49	C₂×C₄×C₁₁	C₂²×C₇×C₁₁	1	36	yes	—
50	C₈×C₁₁	C₂²×C₇×C₁₁	1	48	yes	direct product factor
51	C₇×C₁₁	C₂²×C₈×C₁₁	1	12	yes	direct product factor
52	C₂³×C₇	C₄×C₁₁²	1	1	yes	characteristic
53	C₂×C₄×C₇	C₄×C₁₁²	1	3	yes	—
54	C₂×C₄×C₇	C₂²×C₁₁²	1	3	yes	—
55	C₇×C₈	C₂²×C₁₁²	1	4	yes	direct product factor
56	C₂²×C₁₁	C₇×C₈×C₁₁	1	48	yes	direct product factor
57	C₂²×C₁₁	C₂×C₄×C₇×C₁₁	1	36	yes	—
58	C₄×C₁₁	C₂×C₄×C₇×C₁₁	1	36	yes	—
59	C₄×C₁₁	C₂³×C₇×C₁₁	1	12	yes	—
60	C₂²×C₈	C₇×C₁₁²	1	1	yes	characteristic, direct product factor, Sylow
61	C₂²×C₇	C₈×C₁₁²	1	4	yes	direct product factor
62	C₂²×C₇	C₂×C₄×C₁₁²	1	3	yes	—
63	C₄×C₇	C₂×C₄×C₁₁²	1	3	yes	—
64	C₄×C₇	C₂³×C₁₁²	1	1	yes	characteristic
65	C₂×C₁₁	C₂×C₇×C₈×C₁₁	1	72	yes	direct product factor
66	C₂×C₁₁	C₂²×C₄×C₇×C₁₁	1	12	yes	—
67	C₂²×C₄	C₂×C₇×C₁₁²	1	1	yes	characteristic
68	C₂×C₈	C₂×C₇×C₁₁²	1	6	yes	direct product factor
69	C₂×C₇	C₂×C₈×C₁₁²	1	6	yes	direct product factor
70	C₂×C₇	C₂²×C₄×C₁₁²	1	1	yes	characteristic
71	C₁₁	C₂²×C₇×C₈×C₁₁	1	12	yes	direct product factor
72	C₂³	C₄×C₇×C₁₁²	1	1	yes	characteristic
73	C₂×C₄	C₄×C₇×C₁₁²	1	3	yes	—
74	C₂×C₄	C₂²×C₇×C₁₁²	1	3	yes	—
75	C₈	C₂²×C₇×C₁₁²	1	4	yes	direct product factor
76	C₇	C₂²×C₈×C₁₁²	1	1	yes	characteristic, direct product factor, Sylow
77	C₂²	C₇×C₈×C₁₁²	1	4	yes	direct product factor
78	C₂²	C₂×C₄×C₇×C₁₁²	1	3	yes	—
79	C₄	C₂×C₄×C₇×C₁₁²	1	3	yes	—
80	C₄	C₂³×C₇×C₁₁²	1	1	yes	characteristic
81	C₂	C₂×C₇×C₈×C₁₁²	1	6	yes	direct product factor
82	C₂	C₂²×C₄×C₇×C₁₁²	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₄	C₂³×C₇×C₁₁²
socle, prime radical	C₂³×C₇×C₁₁²	C₄
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₇×C₈×C₁₁²	C₂	direct (N × H)
6. C₇×C₈×C₁₁²	C₂²	direct (N × H)
7. C₂²×C₈×C₁₁²	C₇	direct (N × H)
8. C₂²×C₇×C₈×C₁₁	C₁₁	direct (N × H)
9. C₂²×C₇×C₁₁²	C₈	direct (N × H)
14. C₂×C₈×C₁₁²	C₂×C₇	direct (N × H)
15. C₂×C₇×C₁₁²	C₂×C₈	direct (N × H)
18. C₂×C₇×C₈×C₁₁	C₂×C₁₁	direct (N × H)
22. C₈×C₁₁²	C₂²×C₇	direct (N × H)
23. C₇×C₁₁²	C₂²×C₈	direct (N × H)
27. C₇×C₈×C₁₁	C₂²×C₁₁	direct (N × H)
28. C₂²×C₁₁²	C₇×C₈	direct (N × H)
32. C₂²×C₈×C₁₁	C₇×C₁₁	direct (N × H)
33. C₂²×C₇×C₁₁	C₈×C₁₁	direct (N × H)
37. C₂×C₁₁²	C₂×C₇×C₈	direct (N × H)
39. C₂²×C₇×C₈	C₁₁²	direct (N × H)
41. C₂×C₈×C₁₁	C₂×C₇×C₁₁	direct (N × H)
42. C₂×C₇×C₁₁	C₂×C₈×C₁₁	direct (N × H)
44. C₁₁²	C₂²×C₇×C₈	direct (N × H)
46. C₂×C₇×C₈	C₂×C₁₁²	direct (N × H)
50. C₈×C₁₁	C₂²×C₇×C₁₁	direct (N × H)
51. C₇×C₁₁	C₂²×C₈×C₁₁	direct (N × H)
55. C₇×C₈	C₂²×C₁₁²	direct (N × H)
56. C₂²×C₁₁	C₇×C₈×C₁₁	direct (N × H)
60. C₂²×C₈	C₇×C₁₁²	direct (N × H)
61. C₂²×C₇	C₈×C₁₁²	direct (N × H)
65. C₂×C₁₁	C₂×C₇×C₈×C₁₁	direct (N × H)
68. C₂×C₈	C₂×C₇×C₁₁²	direct (N × H)
69. C₂×C₇	C₂×C₈×C₁₁²	direct (N × H)
71. C₁₁	C₂²×C₇×C₈×C₁₁	direct (N × H)
75. C₈	C₂²×C₇×C₁₁²	direct (N × H)
76. C₇	C₂²×C₈×C₁₁²	direct (N × H)
77. C₂²	C₇×C₈×C₁₁²	direct (N × H)
81. C₂	C₂×C₇×C₈×C₁₁²	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₇×C₁₁²	2
C₂×C₇×C₈×C₁₁²	2
C₂²×C₈×C₁₁²	7
C₂²×C₇×C₈×C₁₁	11

Subgroups — C22×C7×C8×C112 #72204166

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₇×C₈×C₁₁² #72204166