Subgroups — C₅²×C₇×C₈×C₁₆ #72199813

Order 22400.

832 conjugacy classes of subgroups.

The subgroup lattice has 120 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅²×C₇×C₈²	C₂	1	1	yes	maximal, characteristic
2	C₄×C₅²×C₇×C₁₆	C₂	1	2	yes	maximal
3	C₄×C₅²×C₇×C₈	C₄	1	2	yes	—
4	C₄×C₅²×C₇×C₈	C₂²	1	1	yes	characteristic
5	C₂×C₅²×C₇×C₁₆	C₄	1	4	yes	—
6	C₅×C₇×C₈×C₁₆	C₅	1	6	yes	maximal, direct product factor
7	C₅²×C₈×C₁₆	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₄²×C₅²×C₇	C₂×C₄	1	1	yes	characteristic
9	C₂×C₅²×C₇×C₈	C₈	1	4	yes	—
10	C₂×C₅²×C₇×C₈	C₂×C₄	1	2	yes	—
11	C₅²×C₇×C₁₆	C₈	1	8	yes	direct product factor
12	C₅×C₇×C₈²	C₂×C₅	1	6	yes	—
13	C₄×C₅×C₇×C₁₆	C₂×C₅	1	12	yes	—
14	C₄×C₅²×C₁₆	C₂×C₇	1	2	yes	—
15	C₅²×C₈²	C₂×C₇	1	1	yes	characteristic
16	C₂×C₄×C₅²×C₇	C₄²	1	1	yes	characteristic
17	C₂×C₄×C₅²×C₇	C₂×C₈	1	2	yes	—
18	C₅²×C₇×C₈	C₁₆	1	8	yes	direct product factor
19	C₅²×C₇×C₈	C₂×C₈	1	4	yes	—
20	C₂×C₅×C₇×C₁₆	C₄×C₅	1	24	yes	—
21	C₄×C₅×C₇×C₈	C₄×C₅	1	12	yes	—
22	C₄×C₅×C₇×C₈	C₂²×C₅	1	6	yes	—
23	C₇×C₈×C₁₆	C₅²	1	1	yes	characteristic, direct product factor, Hall
24	C₂×C₅²×C₁₆	C₄×C₇	1	4	yes	—
25	C₄×C₅²×C₈	C₄×C₇	1	2	yes	—
26	C₄×C₅²×C₈	C₂²×C₇	1	1	yes	characteristic
27	C₂²×C₅²×C₇	C₄×C₈	1	1	yes	characteristic
28	C₄×C₅²×C₇	C₄×C₈	1	2	yes	—
29	C₄×C₅²×C₇	C₂×C₁₆	1	4	yes	—
30	C₅×C₈×C₁₆	C₅×C₇	1	6	yes	direct product factor
31	C₂×C₅×C₇×C₈	C₅×C₈	1	24	yes	—
32	C₂×C₅×C₇×C₈	C₂×C₄×C₅	1	12	yes	—
33	C₄²×C₅×C₇	C₂×C₄×C₅	1	6	yes	—
34	C₅×C₇×C₁₆	C₅×C₈	1	48	yes	direct product factor
35	C₄×C₇×C₁₆	C₂×C₅²	1	2	yes	—
36	C₇×C₈²	C₂×C₅²	1	1	yes	characteristic
37	C₂×C₅²×C₈	C₇×C₈	1	4	yes	—
38	C₂×C₅²×C₈	C₂×C₄×C₇	1	2	yes	—
39	C₄²×C₅²	C₂×C₄×C₇	1	1	yes	characteristic
40	C₅²×C₁₆	C₇×C₈	1	8	yes	direct product factor
41	C₂×C₅²×C₇	C₈²	1	1	yes	characteristic
42	C₂×C₅²×C₇	C₄×C₁₆	1	2	yes	—
43	C₄×C₅×C₁₆	C₂×C₅×C₇	1	12	yes	—
44	C₅×C₈²	C₂×C₅×C₇	1	6	yes	—
45	C₂×C₄×C₅×C₇	C₄²×C₅	1	6	yes	—
46	C₂×C₄×C₅×C₇	C₂×C₅×C₈	1	12	yes	—
47	C₅×C₇×C₈	C₅×C₁₆	1	48	yes	direct product factor
48	C₅×C₇×C₈	C₂×C₅×C₈	1	24	yes	—
49	C₂×C₇×C₁₆	C₄×C₅²	1	4	yes	—
50	C₄×C₇×C₈	C₄×C₅²	1	2	yes	—
51	C₄×C₇×C₈	C₂²×C₅²	1	1	yes	characteristic
52	C₂×C₄×C₅²	C₄²×C₇	1	1	yes	characteristic
53	C₂×C₄×C₅²	C₂×C₇×C₈	1	2	yes	—
54	C₅²×C₈	C₇×C₁₆	1	8	yes	direct product factor
55	C₅²×C₈	C₂×C₇×C₈	1	4	yes	—
56	C₅²×C₇	C₈×C₁₆	1	1	yes	characteristic, direct product factor, Hall
57	C₂×C₅×C₁₆	C₄×C₅×C₇	1	24	yes	—
58	C₄×C₅×C₈	C₄×C₅×C₇	1	12	yes	—
59	C₄×C₅×C₈	C₂²×C₅×C₇	1	6	yes	—
60	C₂²×C₅×C₇	C₄×C₅×C₈	1	6	yes	—
61	C₄×C₅×C₇	C₄×C₅×C₈	1	12	yes	—
62	C₄×C₅×C₇	C₂×C₅×C₁₆	1	24	yes	—
63	C₈×C₁₆	C₅²×C₇	1	1	yes	characteristic, direct product factor, Sylow
64	C₂×C₇×C₈	C₅²×C₈	1	4	yes	—
65	C₂×C₇×C₈	C₂×C₄×C₅²	1	2	yes	—
66	C₄²×C₇	C₂×C₄×C₅²	1	1	yes	characteristic
67	C₇×C₁₆	C₅²×C₈	1	8	yes	direct product factor
68	C₂²×C₅²	C₄×C₇×C₈	1	1	yes	characteristic
69	C₄×C₅²	C₄×C₇×C₈	1	2	yes	—
70	C₄×C₅²	C₂×C₇×C₁₆	1	4	yes	—
71	C₂×C₅×C₈	C₅×C₇×C₈	1	24	yes	—
72	C₂×C₅×C₈	C₂×C₄×C₅×C₇	1	12	yes	—
73	C₄²×C₅	C₂×C₄×C₅×C₇	1	6	yes	—
74	C₅×C₁₆	C₅×C₇×C₈	1	48	yes	direct product factor
75	C₂×C₅×C₇	C₅×C₈²	1	6	yes	—
76	C₂×C₅×C₇	C₄×C₅×C₁₆	1	12	yes	—
77	C₄×C₁₆	C₂×C₅²×C₇	1	2	yes	—
78	C₈²	C₂×C₅²×C₇	1	1	yes	characteristic
79	C₂×C₄×C₇	C₄²×C₅²	1	1	yes	characteristic
80	C₂×C₄×C₇	C₂×C₅²×C₈	1	2	yes	—
81	C₇×C₈	C₅²×C₁₆	1	8	yes	direct product factor
82	C₇×C₈	C₂×C₅²×C₈	1	4	yes	—
83	C₂×C₅²	C₇×C₈²	1	1	yes	characteristic
84	C₂×C₅²	C₄×C₇×C₁₆	1	2	yes	—
85	C₂×C₄×C₅	C₄²×C₅×C₇	1	6	yes	—
86	C₂×C₄×C₅	C₂×C₅×C₇×C₈	1	12	yes	—
87	C₅×C₈	C₅×C₇×C₁₆	1	48	yes	direct product factor
88	C₅×C₈	C₂×C₅×C₇×C₈	1	24	yes	—
89	C₅×C₇	C₅×C₈×C₁₆	1	6	yes	direct product factor
90	C₂×C₁₆	C₄×C₅²×C₇	1	4	yes	—
91	C₄×C₈	C₄×C₅²×C₇	1	2	yes	—
92	C₄×C₈	C₂²×C₅²×C₇	1	1	yes	characteristic
93	C₂²×C₇	C₄×C₅²×C₈	1	1	yes	characteristic
94	C₄×C₇	C₄×C₅²×C₈	1	2	yes	—
95	C₄×C₇	C₂×C₅²×C₁₆	1	4	yes	—
96	C₅²	C₇×C₈×C₁₆	1	1	yes	characteristic, direct product factor, Sylow
97	C₂²×C₅	C₄×C₅×C₇×C₈	1	6	yes	—
98	C₄×C₅	C₄×C₅×C₇×C₈	1	12	yes	—
99	C₄×C₅	C₂×C₅×C₇×C₁₆	1	24	yes	—
100	C₂×C₈	C₅²×C₇×C₈	1	4	yes	—
101	C₂×C₈	C₂×C₄×C₅²×C₇	1	2	yes	—
102	C₄²	C₂×C₄×C₅²×C₇	1	1	yes	characteristic
103	C₁₆	C₅²×C₇×C₈	1	8	yes	direct product factor
104	C₂×C₇	C₅²×C₈²	1	1	yes	characteristic
105	C₂×C₇	C₄×C₅²×C₁₆	1	2	yes	—
106	C₂×C₅	C₄×C₅×C₇×C₁₆	1	12	yes	—
107	C₂×C₅	C₅×C₇×C₈²	1	6	yes	—
108	C₂×C₄	C₂×C₅²×C₇×C₈	1	2	yes	—
109	C₂×C₄	C₄²×C₅²×C₇	1	1	yes	characteristic
110	C₈	C₅²×C₇×C₁₆	1	8	yes	direct product factor
111	C₈	C₂×C₅²×C₇×C₈	1	4	yes	—
112	C₇	C₅²×C₈×C₁₆	1	1	yes	characteristic, direct product factor, Sylow
113	C₅	C₅×C₇×C₈×C₁₆	1	6	yes	direct product factor
114	C₂²	C₄×C₅²×C₇×C₈	1	1	yes	characteristic
115	C₄	C₂×C₅²×C₇×C₁₆	1	4	yes	—
116	C₄	C₄×C₅²×C₇×C₈	1	2	yes	—
117	C₂	C₄×C₅²×C₇×C₁₆	1	2	yes	—
118	C₂	C₅²×C₇×C₈²	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₄×C₈	C₂²×C₅²×C₇
socle, prime radical	C₂²×C₅²×C₇	C₄×C₈
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
6. C₅×C₇×C₈×C₁₆	C₅	direct (N × H)
7. C₅²×C₈×C₁₆	C₇	direct (N × H)
11. C₅²×C₇×C₁₆	C₈	direct (N × H)
18. C₅²×C₇×C₈	C₁₆	direct (N × H)
23. C₇×C₈×C₁₆	C₅²	direct (N × H)
30. C₅×C₈×C₁₆	C₅×C₇	direct (N × H)
34. C₅×C₇×C₁₆	C₅×C₈	direct (N × H)
40. C₅²×C₁₆	C₇×C₈	direct (N × H)
47. C₅×C₇×C₈	C₅×C₁₆	direct (N × H)
54. C₅²×C₈	C₇×C₁₆	direct (N × H)
56. C₅²×C₇	C₈×C₁₆	direct (N × H)
63. C₈×C₁₆	C₅²×C₇	direct (N × H)
67. C₇×C₁₆	C₅²×C₈	direct (N × H)
74. C₅×C₁₆	C₅×C₇×C₈	direct (N × H)
81. C₇×C₈	C₅²×C₁₆	direct (N × H)
87. C₅×C₈	C₅×C₇×C₁₆	direct (N × H)
89. C₅×C₇	C₅×C₈×C₁₆	direct (N × H)
96. C₅²	C₇×C₈×C₁₆	direct (N × H)
103. C₁₆	C₅²×C₇×C₈	direct (N × H)
110. C₈	C₅²×C₇×C₁₆	direct (N × H)
112. C₇	C₅²×C₈×C₁₆	direct (N × H)
113. C₅	C₅×C₇×C₈×C₁₆	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅²×C₇×C₈²	2
C₄×C₅²×C₇×C₁₆	2
C₅×C₇×C₈×C₁₆	5
C₅²×C₈×C₁₆	7

Subgroups — C52×C7×C8×C16 #72199813

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅²×C₇×C₈×C₁₆ #72199813