Subgroups — C₄²×C₇²×C₁₆ #72190899

Order 12544.

2790 conjugacy classes of subgroups.

The subgroup lattice has 117 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×C₇²×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₇²×C₁₆	C₂	1	6	yes	maximal
3	C₄³×C₇²	C₄	1	1	yes	characteristic
4	C₂×C₄×C₇²×C₈	C₄	1	3	yes	—
5	C₂×C₄×C₇²×C₈	C₂²	1	3	yes	—
6	C₂²×C₇²×C₁₆	C₂²	1	4	yes	—
7	C₄×C₇²×C₁₆	C₄	1	24	yes	direct product factor
8	C₄²×C₇×C₁₆	C₇	1	8	yes	maximal, direct product factor
9	C₂²×C₇²×C₈	C₂×C₄	1	3	yes	—
10	C₂²×C₇²×C₈	C₂³	1	1	yes	characteristic
11	C₂×C₄²×C₇²	C₈	1	4	yes	—
12	C₂×C₄²×C₇²	C₂×C₄	1	3	yes	—
13	C₂×C₇²×C₁₆	C₂×C₄	1	24	yes	—
14	C₄×C₇²×C₈	C₈	1	12	yes	—
15	C₄×C₇²×C₈	C₂×C₄	1	12	yes	—
16	C₂×C₄×C₇×C₁₆	C₂×C₇	1	48	yes	—
17	C₄²×C₇×C₈	C₂×C₇	1	8	yes	—
18	C₂²×C₄×C₇²	C₂×C₈	1	6	yes	—
19	C₂²×C₄×C₇²	C₂²×C₄	1	1	yes	characteristic
20	C₂×C₇²×C₈	C₄²	1	6	yes	—
21	C₂×C₇²×C₈	C₂×C₈	1	12	yes	—
22	C₂×C₇²×C₈	C₂²×C₄	1	6	yes	—
23	C₄²×C₇²	C₁₆	1	16	yes	direct product factor
24	C₄²×C₇²	C₄²	1	6	yes	—
25	C₄²×C₇²	C₂×C₈	1	6	yes	—
26	C₇²×C₁₆	C₄²	1	16	yes	direct product factor
27	C₂²×C₇×C₁₆	C₂²×C₇	1	32	yes	—
28	C₂×C₄×C₇×C₈	C₄×C₇	1	24	yes	—
29	C₂×C₄×C₇×C₈	C₂²×C₇	1	24	yes	—
30	C₄³×C₇	C₄×C₇	1	8	yes	—
31	C₄×C₇×C₁₆	C₄×C₇	1	192	yes	direct product factor
32	C₂³×C₇²	C₂²×C₈	1	1	yes	characteristic
33	C₂×C₄×C₇²	C₄×C₈	1	12	yes	—
34	C₂×C₄×C₇²	C₂×C₁₆	1	24	yes	—
35	C₂×C₄×C₇²	C₂×C₄²	1	3	yes	—
36	C₂×C₄×C₇²	C₂²×C₈	1	3	yes	—
37	C₇²×C₈	C₄×C₈	1	12	yes	—
38	C₇²×C₈	C₂×C₄²	1	4	yes	—
39	C₄²×C₁₆	C₇²	1	1	yes	characteristic, direct product factor, Sylow
40	C₂²×C₇×C₈	C₂×C₄×C₇	1	24	yes	—
41	C₂²×C₇×C₈	C₂³×C₇	1	8	yes	—
42	C₂×C₄²×C₇	C₇×C₈	1	32	yes	—
43	C₂×C₄²×C₇	C₂×C₄×C₇	1	24	yes	—
44	C₂×C₇×C₁₆	C₂×C₄×C₇	1	192	yes	—
45	C₄×C₇×C₈	C₇×C₈	1	96	yes	—
46	C₄×C₇×C₈	C₂×C₄×C₇	1	96	yes	—
47	C₂²×C₇²	C₂×C₄×C₈	1	3	yes	—
48	C₂²×C₇²	C₂²×C₁₆	1	4	yes	—
49	C₄×C₇²	C₄×C₁₆	1	24	yes	direct product factor
50	C₄×C₇²	C₄³	1	1	yes	characteristic
51	C₄×C₇²	C₂×C₄×C₈	1	3	yes	—
52	C₂×C₄×C₁₆	C₂×C₇²	1	6	yes	—
53	C₄²×C₈	C₂×C₇²	1	1	yes	characteristic
54	C₂²×C₄×C₇	C₂×C₇×C₈	1	48	yes	—
55	C₂²×C₄×C₇	C₂²×C₄×C₇	1	8	yes	—
56	C₂×C₇×C₈	C₄²×C₇	1	48	yes	—
57	C₂×C₇×C₈	C₂×C₇×C₈	1	96	yes	—
58	C₂×C₇×C₈	C₂²×C₄×C₇	1	48	yes	—
59	C₄²×C₇	C₇×C₁₆	1	128	yes	direct product factor
60	C₄²×C₇	C₄²×C₇	1	48	yes	—
61	C₄²×C₇	C₂×C₇×C₈	1	48	yes	—
62	C₇×C₁₆	C₄²×C₇	1	128	yes	direct product factor
63	C₂×C₇²	C₄²×C₈	1	1	yes	characteristic
64	C₂×C₇²	C₂×C₄×C₁₆	1	6	yes	—
65	C₂²×C₁₆	C₂²×C₇²	1	4	yes	—
66	C₂×C₄×C₈	C₄×C₇²	1	3	yes	—
67	C₂×C₄×C₈	C₂²×C₇²	1	3	yes	—
68	C₄³	C₄×C₇²	1	1	yes	characteristic
69	C₄×C₁₆	C₄×C₇²	1	24	yes	direct product factor
70	C₂³×C₇	C₂²×C₇×C₈	1	8	yes	—
71	C₂×C₄×C₇	C₄×C₇×C₈	1	96	yes	—
72	C₂×C₄×C₇	C₂×C₇×C₁₆	1	192	yes	—
73	C₂×C₄×C₇	C₂×C₄²×C₇	1	24	yes	—
74	C₂×C₄×C₇	C₂²×C₇×C₈	1	24	yes	—
75	C₇×C₈	C₄×C₇×C₈	1	96	yes	—
76	C₇×C₈	C₂×C₄²×C₇	1	32	yes	—
77	C₇²	C₄²×C₁₆	1	1	yes	characteristic, direct product factor, Sylow
78	C₂²×C₈	C₂×C₄×C₇²	1	3	yes	—
79	C₂²×C₈	C₂³×C₇²	1	1	yes	characteristic
80	C₂×C₄²	C₇²×C₈	1	4	yes	—
81	C₂×C₄²	C₂×C₄×C₇²	1	3	yes	—
82	C₂×C₁₆	C₂×C₄×C₇²	1	24	yes	—
83	C₄×C₈	C₇²×C₈	1	12	yes	—
84	C₄×C₈	C₂×C₄×C₇²	1	12	yes	—
85	C₂²×C₇	C₂×C₄×C₇×C₈	1	24	yes	—
86	C₂²×C₇	C₂²×C₇×C₁₆	1	32	yes	—
87	C₄×C₇	C₄×C₇×C₁₆	1	192	yes	direct product factor
88	C₄×C₇	C₄³×C₇	1	8	yes	—
89	C₄×C₇	C₂×C₄×C₇×C₈	1	24	yes	—
90	C₂²×C₄	C₂×C₇²×C₈	1	6	yes	—
91	C₂²×C₄	C₂²×C₄×C₇²	1	1	yes	characteristic
92	C₂×C₈	C₄²×C₇²	1	6	yes	—
93	C₂×C₈	C₂×C₇²×C₈	1	12	yes	—
94	C₂×C₈	C₂²×C₄×C₇²	1	6	yes	—
95	C₄²	C₇²×C₁₆	1	16	yes	direct product factor
96	C₄²	C₄²×C₇²	1	6	yes	—
97	C₄²	C₂×C₇²×C₈	1	6	yes	—
98	C₁₆	C₄²×C₇²	1	16	yes	direct product factor
99	C₂×C₇	C₄²×C₇×C₈	1	8	yes	—
100	C₂×C₇	C₂×C₄×C₇×C₁₆	1	48	yes	—
101	C₂³	C₂²×C₇²×C₈	1	1	yes	characteristic
102	C₂×C₄	C₄×C₇²×C₈	1	12	yes	—
103	C₂×C₄	C₂×C₇²×C₁₆	1	24	yes	—
104	C₂×C₄	C₂×C₄²×C₇²	1	3	yes	—
105	C₂×C₄	C₂²×C₇²×C₈	1	3	yes	—
106	C₈	C₄×C₇²×C₈	1	12	yes	—
107	C₈	C₂×C₄²×C₇²	1	4	yes	—
108	C₇	C₄²×C₇×C₁₆	1	8	yes	direct product factor
109	C₂²	C₂²×C₇²×C₁₆	1	4	yes	—
110	C₂²	C₂×C₄×C₇²×C₈	1	3	yes	—
111	C₄	C₄×C₇²×C₁₆	1	24	yes	direct product factor
112	C₄	C₂×C₄×C₇²×C₈	1	3	yes	—
113	C₄	C₄³×C₇²	1	1	yes	characteristic
114	C₂	C₂×C₄×C₇²×C₁₆	1	6	yes	—
115	C₂	C₄²×C₇²×C₈	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂²×C₈	C₂³×C₇²
socle, prime radical	C₂³×C₇²	C₂²×C₈
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
7. C₄×C₇²×C₁₆	C₄	direct (N × H)
8. C₄²×C₇×C₁₆	C₇	direct (N × H)
23. C₄²×C₇²	C₁₆	direct (N × H)
26. C₇²×C₁₆	C₄²	direct (N × H)
31. C₄×C₇×C₁₆	C₄×C₇	direct (N × H)
39. C₄²×C₁₆	C₇²	direct (N × H)
49. C₄×C₇²	C₄×C₁₆	direct (N × H)
59. C₄²×C₇	C₇×C₁₆	direct (N × H)
62. C₇×C₁₆	C₄²×C₇	direct (N × H)
69. C₄×C₁₆	C₄×C₇²	direct (N × H)
77. C₇²	C₄²×C₁₆	direct (N × H)
87. C₄×C₇	C₄×C₇×C₁₆	direct (N × H)
95. C₄²	C₇²×C₁₆	direct (N × H)
98. C₁₆	C₄²×C₇²	direct (N × H)
108. C₇	C₄²×C₇×C₁₆	direct (N × H)
111. C₄	C₄×C₇²×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
C₄²×C₇²×C₈	2
C₂×C₄×C₇²×C₁₆	2
C₄²×C₇×C₁₆	7

Subgroups — C42×C72×C16 #72190899

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄²×C₇²×C₁₆ #72190899