Subgroups — C₃×C₇²×C₈×C₁₆ #72196552

Order 18816.

1040 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	C₃×C₇²×C₈²	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×C₇²×C₁₆	C₂	1	2	yes	maximal
3	C₇²×C₈×C₁₆	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₃×C₄×C₇²×C₈	C₄	1	2	yes	—
5	C₃×C₄×C₇²×C₈	C₂²	1	1	yes	characteristic
6	C₂×C₃×C₇²×C₁₆	C₄	1	4	yes	—
7	C₇²×C₈²	C₂×C₃	1	1	yes	characteristic
8	C₄×C₇²×C₁₆	C₂×C₃	1	2	yes	—
9	C₃×C₇×C₈×C₁₆	C₇	1	8	yes	maximal, direct product factor
10	C₃×C₄²×C₇²	C₂×C₄	1	1	yes	characteristic
11	C₂×C₃×C₇²×C₈	C₈	1	4	yes	—
12	C₂×C₃×C₇²×C₈	C₂×C₄	1	2	yes	—
13	C₃×C₇²×C₁₆	C₈	1	8	yes	direct product factor
14	C₂×C₇²×C₁₆	C₃×C₄	1	4	yes	—
15	C₄×C₇²×C₈	C₃×C₄	1	2	yes	—
16	C₄×C₇²×C₈	C₂²×C₃	1	1	yes	characteristic
17	C₃×C₄×C₇×C₁₆	C₂×C₇	1	16	yes	—
18	C₃×C₇×C₈²	C₂×C₇	1	8	yes	—
19	C₂×C₃×C₄×C₇²	C₄²	1	1	yes	characteristic
20	C₂×C₃×C₄×C₇²	C₂×C₈	1	2	yes	—
21	C₃×C₇²×C₈	C₁₆	1	8	yes	direct product factor
22	C₃×C₇²×C₈	C₂×C₈	1	4	yes	—
23	C₇×C₈×C₁₆	C₃×C₇	1	8	yes	direct product factor
24	C₂×C₇²×C₈	C₃×C₈	1	4	yes	—
25	C₂×C₇²×C₈	C₂×C₃×C₄	1	2	yes	—
26	C₄²×C₇²	C₂×C₃×C₄	1	1	yes	characteristic
27	C₇²×C₁₆	C₃×C₈	1	8	yes	direct product factor
28	C₂×C₃×C₇×C₁₆	C₄×C₇	1	32	yes	—
29	C₃×C₄×C₇×C₈	C₄×C₇	1	16	yes	—
30	C₃×C₄×C₇×C₈	C₂²×C₇	1	8	yes	—
31	C₂²×C₃×C₇²	C₄×C₈	1	1	yes	characteristic
32	C₃×C₄×C₇²	C₄×C₈	1	2	yes	—
33	C₃×C₄×C₇²	C₂×C₁₆	1	4	yes	—
34	C₄×C₇×C₁₆	C₂×C₃×C₇	1	16	yes	—
35	C₇×C₈²	C₂×C₃×C₇	1	8	yes	—
36	C₂×C₄×C₇²	C₃×C₄²	1	1	yes	characteristic
37	C₂×C₄×C₇²	C₂×C₃×C₈	1	2	yes	—
38	C₇²×C₈	C₃×C₁₆	1	8	yes	direct product factor
39	C₇²×C₈	C₂×C₃×C₈	1	4	yes	—
40	C₃×C₈×C₁₆	C₇²	1	1	yes	characteristic, direct product factor, Hall
41	C₂×C₃×C₇×C₈	C₇×C₈	1	32	yes	—
42	C₂×C₃×C₇×C₈	C₂×C₄×C₇	1	16	yes	—
43	C₃×C₄²×C₇	C₂×C₄×C₇	1	8	yes	—
44	C₃×C₇×C₁₆	C₇×C₈	1	64	yes	direct product factor
45	C₂×C₃×C₇²	C₈²	1	1	yes	characteristic
46	C₂×C₃×C₇²	C₄×C₁₆	1	2	yes	—
47	C₂×C₇×C₁₆	C₃×C₄×C₇	1	32	yes	—
48	C₄×C₇×C₈	C₃×C₄×C₇	1	16	yes	—
49	C₄×C₇×C₈	C₂²×C₃×C₇	1	8	yes	—
50	C₂²×C₇²	C₃×C₄×C₈	1	1	yes	characteristic
51	C₄×C₇²	C₃×C₄×C₈	1	2	yes	—
52	C₄×C₇²	C₂×C₃×C₁₆	1	4	yes	—
53	C₃×C₄×C₁₆	C₂×C₇²	1	2	yes	—
54	C₃×C₈²	C₂×C₇²	1	1	yes	characteristic
55	C₂×C₃×C₄×C₇	C₄²×C₇	1	8	yes	—
56	C₂×C₃×C₄×C₇	C₂×C₇×C₈	1	16	yes	—
57	C₃×C₇×C₈	C₇×C₁₆	1	64	yes	direct product factor
58	C₃×C₇×C₈	C₂×C₇×C₈	1	32	yes	—
59	C₃×C₇²	C₈×C₁₆	1	1	yes	characteristic, direct product factor, Hall
60	C₈×C₁₆	C₃×C₇²	1	1	yes	characteristic, direct product factor, Sylow
61	C₂×C₇×C₈	C₃×C₇×C₈	1	32	yes	—
62	C₂×C₇×C₈	C₂×C₃×C₄×C₇	1	16	yes	—
63	C₄²×C₇	C₂×C₃×C₄×C₇	1	8	yes	—
64	C₇×C₁₆	C₃×C₇×C₈	1	64	yes	direct product factor
65	C₂×C₇²	C₃×C₈²	1	1	yes	characteristic
66	C₂×C₇²	C₃×C₄×C₁₆	1	2	yes	—
67	C₂×C₃×C₁₆	C₄×C₇²	1	4	yes	—
68	C₃×C₄×C₈	C₄×C₇²	1	2	yes	—
69	C₃×C₄×C₈	C₂²×C₇²	1	1	yes	characteristic
70	C₂²×C₃×C₇	C₄×C₇×C₈	1	8	yes	—
71	C₃×C₄×C₇	C₄×C₇×C₈	1	16	yes	—
72	C₃×C₄×C₇	C₂×C₇×C₁₆	1	32	yes	—
73	C₄×C₁₆	C₂×C₃×C₇²	1	2	yes	—
74	C₈²	C₂×C₃×C₇²	1	1	yes	characteristic
75	C₂×C₄×C₇	C₃×C₄²×C₇	1	8	yes	—
76	C₂×C₄×C₇	C₂×C₃×C₇×C₈	1	16	yes	—
77	C₇×C₈	C₃×C₇×C₁₆	1	64	yes	direct product factor
78	C₇×C₈	C₂×C₃×C₇×C₈	1	32	yes	—
79	C₇²	C₃×C₈×C₁₆	1	1	yes	characteristic, direct product factor, Sylow
80	C₂×C₃×C₈	C₇²×C₈	1	4	yes	—
81	C₂×C₃×C₈	C₂×C₄×C₇²	1	2	yes	—
82	C₃×C₄²	C₂×C₄×C₇²	1	1	yes	characteristic
83	C₃×C₁₆	C₇²×C₈	1	8	yes	direct product factor
84	C₂×C₃×C₇	C₇×C₈²	1	8	yes	—
85	C₂×C₃×C₇	C₄×C₇×C₁₆	1	16	yes	—
86	C₂×C₁₆	C₃×C₄×C₇²	1	4	yes	—
87	C₄×C₈	C₃×C₄×C₇²	1	2	yes	—
88	C₄×C₈	C₂²×C₃×C₇²	1	1	yes	characteristic
89	C₂²×C₇	C₃×C₄×C₇×C₈	1	8	yes	—
90	C₄×C₇	C₃×C₄×C₇×C₈	1	16	yes	—
91	C₄×C₇	C₂×C₃×C₇×C₁₆	1	32	yes	—
92	C₂×C₃×C₄	C₄²×C₇²	1	1	yes	characteristic
93	C₂×C₃×C₄	C₂×C₇²×C₈	1	2	yes	—
94	C₃×C₈	C₇²×C₁₆	1	8	yes	direct product factor
95	C₃×C₈	C₂×C₇²×C₈	1	4	yes	—
96	C₃×C₇	C₇×C₈×C₁₆	1	8	yes	direct product factor
97	C₂×C₈	C₃×C₇²×C₈	1	4	yes	—
98	C₂×C₈	C₂×C₃×C₄×C₇²	1	2	yes	—
99	C₄²	C₂×C₃×C₄×C₇²	1	1	yes	characteristic
100	C₁₆	C₃×C₇²×C₈	1	8	yes	direct product factor
101	C₂×C₇	C₃×C₇×C₈²	1	8	yes	—
102	C₂×C₇	C₃×C₄×C₇×C₁₆	1	16	yes	—
103	C₂²×C₃	C₄×C₇²×C₈	1	1	yes	characteristic
104	C₃×C₄	C₄×C₇²×C₈	1	2	yes	—
105	C₃×C₄	C₂×C₇²×C₁₆	1	4	yes	—
106	C₂×C₄	C₂×C₃×C₇²×C₈	1	2	yes	—
107	C₂×C₄	C₃×C₄²×C₇²	1	1	yes	characteristic
108	C₈	C₃×C₇²×C₁₆	1	8	yes	direct product factor
109	C₈	C₂×C₃×C₇²×C₈	1	4	yes	—
110	C₇	C₃×C₇×C₈×C₁₆	1	8	yes	direct product factor
111	C₂×C₃	C₄×C₇²×C₁₆	1	2	yes	—
112	C₂×C₃	C₇²×C₈²	1	1	yes	characteristic
113	C₂²	C₃×C₄×C₇²×C₈	1	1	yes	characteristic
114	C₄	C₂×C₃×C₇²×C₁₆	1	4	yes	—
115	C₄	C₃×C₄×C₇²×C₈	1	2	yes	—
116	C₃	C₇²×C₈×C₁₆	1	1	yes	characteristic, direct product factor, Sylow
117	C₂	C₃×C₄×C₇²×C₁₆	1	2	yes	—
118	C₂	C₃×C₇²×C₈²	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₄×C₈	C₂²×C₃×C₇²
socle, prime radical	C₂²×C₃×C₇²	C₄×C₈
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₇²×C₈×C₁₆	C₃	direct (N × H)
9. C₃×C₇×C₈×C₁₆	C₇	direct (N × H)
13. C₃×C₇²×C₁₆	C₈	direct (N × H)
21. C₃×C₇²×C₈	C₁₆	direct (N × H)
23. C₇×C₈×C₁₆	C₃×C₇	direct (N × H)
27. C₇²×C₁₆	C₃×C₈	direct (N × H)
38. C₇²×C₈	C₃×C₁₆	direct (N × H)
40. C₃×C₈×C₁₆	C₇²	direct (N × H)
44. C₃×C₇×C₁₆	C₇×C₈	direct (N × H)
57. C₃×C₇×C₈	C₇×C₁₆	direct (N × H)
59. C₃×C₇²	C₈×C₁₆	direct (N × H)
60. C₈×C₁₆	C₃×C₇²	direct (N × H)
64. C₇×C₁₆	C₃×C₇×C₈	direct (N × H)
77. C₇×C₈	C₃×C₇×C₁₆	direct (N × H)
79. C₇²	C₃×C₈×C₁₆	direct (N × H)
83. C₃×C₁₆	C₇²×C₈	direct (N × H)
94. C₃×C₈	C₇²×C₁₆	direct (N × H)
96. C₃×C₇	C₇×C₈×C₁₆	direct (N × H)
100. C₁₆	C₃×C₇²×C₈	direct (N × H)
108. C₈	C₃×C₇²×C₁₆	direct (N × H)
110. C₇	C₃×C₇×C₈×C₁₆	direct (N × H)
116. C₃	C₇²×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
C₃×C₇²×C₈²	2
C₃×C₄×C₇²×C₁₆	2
C₇²×C₈×C₁₆	3
C₃×C₇×C₈×C₁₆	7

Subgroups — C3×C72×C8×C16 #72196552

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₇²×C₈×C₁₆ #72196552