Subgroups — C₂×C₇×C₉×C₁₃×C₁₆ #72203346

Order 26208.

168 conjugacy classes of subgroups.

The subgroup lattice has 144 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×C₈×C₉×C₁₃	C₂	1	1	yes	maximal, characteristic
2	C₇×C₉×C₁₃×C₁₆	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₃×C₇×C₁₃×C₁₆	C₃	1	1	yes	maximal, characteristic
4	C₂×C₄×C₇×C₉×C₁₃	C₄	1	1	yes	characteristic
5	C₇×C₈×C₉×C₁₃	C₄	1	1	yes	characteristic
6	C₇×C₈×C₉×C₁₃	C₂²	1	1	yes	characteristic
7	C₂×C₃×C₇×C₈×C₁₃	C₂×C₃	1	1	yes	characteristic
8	C₃×C₇×C₁₃×C₁₆	C₂×C₃	1	2	yes	—
9	C₂×C₉×C₁₃×C₁₆	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₂×C₇×C₁₃×C₁₆	C₉	1	1	yes	characteristic, direct product factor, Hall
11	C₂×C₃×C₄×C₇×C₁₃	C₃×C₄	1	1	yes	characteristic
12	C₃×C₇×C₈×C₁₃	C₃×C₄	1	1	yes	characteristic
13	C₃×C₇×C₈×C₁₃	C₂²×C₃	1	1	yes	characteristic
14	C₂×C₇×C₉×C₁₆	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Hall
15	C₂²×C₇×C₉×C₁₃	C₈	1	1	yes	characteristic
16	C₄×C₇×C₉×C₁₃	C₈	1	1	yes	characteristic
17	C₄×C₇×C₉×C₁₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₈×C₉×C₁₃	C₂×C₇	1	1	yes	characteristic
19	C₉×C₁₃×C₁₆	C₂×C₇	1	2	yes	direct product factor
20	C₂×C₇×C₉×C₁₃	C₁₆	1	2	yes	direct product factor
21	C₂×C₇×C₉×C₁₃	C₂×C₈	1	1	yes	characteristic
22	C₂×C₇×C₈×C₁₃	C₂×C₉	1	1	yes	characteristic
23	C₇×C₁₃×C₁₆	C₂×C₉	1	2	yes	direct product factor
24	C₂×C₃×C₁₃×C₁₆	C₃×C₇	1	1	yes	characteristic
25	C₂²×C₃×C₇×C₁₃	C₃×C₈	1	1	yes	characteristic
26	C₃×C₄×C₇×C₁₃	C₃×C₈	1	1	yes	characteristic
27	C₃×C₄×C₇×C₁₃	C₂×C₃×C₄	1	1	yes	characteristic
28	C₂×C₇×C₈×C₉	C₂×C₁₃	1	1	yes	characteristic
29	C₇×C₉×C₁₆	C₂×C₁₃	1	2	yes	direct product factor
30	C₂×C₄×C₉×C₁₃	C₄×C₇	1	1	yes	characteristic
31	C₈×C₉×C₁₃	C₄×C₇	1	1	yes	characteristic
32	C₈×C₉×C₁₃	C₂²×C₇	1	1	yes	characteristic
33	C₇×C₉×C₁₃	C₂×C₁₆	1	1	yes	characteristic, direct product factor, Hall
34	C₂×C₄×C₇×C₁₃	C₄×C₉	1	1	yes	characteristic
35	C₇×C₈×C₁₃	C₄×C₉	1	1	yes	characteristic
36	C₇×C₈×C₁₃	C₂²×C₉	1	1	yes	characteristic
37	C₂×C₃×C₇×C₁₆	C₃×C₁₃	1	1	yes	characteristic
38	C₂×C₃×C₈×C₁₃	C₂×C₃×C₇	1	1	yes	characteristic
39	C₃×C₁₃×C₁₆	C₂×C₃×C₇	1	2	yes	—
40	C₂×C₃×C₇×C₁₃	C₃×C₁₆	1	2	yes	—
41	C₂×C₃×C₇×C₁₃	C₂×C₃×C₈	1	1	yes	characteristic
42	C₂×C₄×C₇×C₉	C₄×C₁₃	1	1	yes	characteristic
43	C₇×C₈×C₉	C₄×C₁₃	1	1	yes	characteristic
44	C₇×C₈×C₉	C₂²×C₁₃	1	1	yes	characteristic
45	C₂²×C₉×C₁₃	C₇×C₈	1	1	yes	characteristic
46	C₄×C₉×C₁₃	C₇×C₈	1	1	yes	characteristic
47	C₄×C₉×C₁₃	C₂×C₄×C₇	1	1	yes	characteristic
48	C₂×C₁₃×C₁₆	C₇×C₉	1	1	yes	characteristic, direct product factor, Hall
49	C₂²×C₇×C₁₃	C₈×C₉	1	1	yes	characteristic
50	C₄×C₇×C₁₃	C₈×C₉	1	1	yes	characteristic
51	C₄×C₇×C₁₃	C₂×C₄×C₉	1	1	yes	characteristic
52	C₂×C₃×C₇×C₈	C₂×C₃×C₁₃	1	1	yes	characteristic
53	C₃×C₇×C₁₆	C₂×C₃×C₁₃	1	2	yes	—
54	C₂×C₃×C₄×C₁₃	C₃×C₄×C₇	1	1	yes	characteristic
55	C₃×C₈×C₁₃	C₃×C₄×C₇	1	1	yes	characteristic
56	C₃×C₈×C₁₃	C₂²×C₃×C₇	1	1	yes	characteristic
57	C₂×C₉×C₁₆	C₇×C₁₃	1	1	yes	characteristic, direct product factor, Hall
58	C₃×C₇×C₁₃	C₂×C₃×C₁₆	1	1	yes	characteristic
59	C₂²×C₇×C₉	C₈×C₁₃	1	1	yes	characteristic
60	C₄×C₇×C₉	C₈×C₁₃	1	1	yes	characteristic
61	C₄×C₇×C₉	C₂×C₄×C₁₃	1	1	yes	characteristic
62	C₂×C₉×C₁₃	C₇×C₁₆	1	2	yes	direct product factor
63	C₂×C₉×C₁₃	C₂×C₇×C₈	1	1	yes	characteristic
64	C₂×C₇×C₁₆	C₉×C₁₃	1	1	yes	characteristic, direct product factor, Hall
65	C₂×C₈×C₁₃	C₂×C₇×C₉	1	1	yes	characteristic
66	C₁₃×C₁₆	C₂×C₇×C₉	1	2	yes	direct product factor
67	C₂×C₇×C₁₃	C₉×C₁₆	1	2	yes	direct product factor
68	C₂×C₇×C₁₃	C₂×C₈×C₉	1	1	yes	characteristic
69	C₂×C₃×C₄×C₇	C₃×C₄×C₁₃	1	1	yes	characteristic
70	C₃×C₇×C₈	C₃×C₄×C₁₃	1	1	yes	characteristic
71	C₃×C₇×C₈	C₂²×C₃×C₁₃	1	1	yes	characteristic
72	C₂²×C₃×C₁₃	C₃×C₇×C₈	1	1	yes	characteristic
73	C₃×C₄×C₁₃	C₃×C₇×C₈	1	1	yes	characteristic
74	C₃×C₄×C₁₃	C₂×C₃×C₄×C₇	1	1	yes	characteristic
75	C₂×C₈×C₉	C₂×C₇×C₁₃	1	1	yes	characteristic
76	C₉×C₁₆	C₂×C₇×C₁₃	1	2	yes	direct product factor
77	C₂×C₇×C₉	C₁₃×C₁₆	1	2	yes	direct product factor
78	C₂×C₇×C₉	C₂×C₈×C₁₃	1	1	yes	characteristic
79	C₉×C₁₃	C₂×C₇×C₁₆	1	1	yes	characteristic, direct product factor, Hall
80	C₂×C₇×C₈	C₂×C₉×C₁₃	1	1	yes	characteristic
81	C₇×C₁₆	C₂×C₉×C₁₃	1	2	yes	direct product factor
82	C₂×C₄×C₁₃	C₄×C₇×C₉	1	1	yes	characteristic
83	C₈×C₁₃	C₄×C₇×C₉	1	1	yes	characteristic
84	C₈×C₁₃	C₂²×C₇×C₉	1	1	yes	characteristic
85	C₂×C₃×C₁₆	C₃×C₇×C₁₃	1	1	yes	characteristic
86	C₇×C₁₃	C₂×C₉×C₁₆	1	1	yes	characteristic, direct product factor, Hall
87	C₂²×C₃×C₇	C₃×C₈×C₁₃	1	1	yes	characteristic
88	C₃×C₄×C₇	C₃×C₈×C₁₃	1	1	yes	characteristic
89	C₃×C₄×C₇	C₂×C₃×C₄×C₁₃	1	1	yes	characteristic
90	C₂×C₃×C₁₃	C₃×C₇×C₁₆	1	2	yes	—
91	C₂×C₃×C₁₃	C₂×C₃×C₇×C₈	1	1	yes	characteristic
92	C₂×C₄×C₉	C₄×C₇×C₁₃	1	1	yes	characteristic
93	C₈×C₉	C₄×C₇×C₁₃	1	1	yes	characteristic
94	C₈×C₉	C₂²×C₇×C₁₃	1	1	yes	characteristic
95	C₇×C₉	C₂×C₁₃×C₁₆	1	1	yes	characteristic, direct product factor, Hall
96	C₂×C₄×C₇	C₄×C₉×C₁₃	1	1	yes	characteristic
97	C₇×C₈	C₄×C₉×C₁₃	1	1	yes	characteristic
98	C₇×C₈	C₂²×C₉×C₁₃	1	1	yes	characteristic
99	C₂²×C₁₃	C₇×C₈×C₉	1	1	yes	characteristic
100	C₄×C₁₃	C₇×C₈×C₉	1	1	yes	characteristic
101	C₄×C₁₃	C₂×C₄×C₇×C₉	1	1	yes	characteristic
102	C₂×C₃×C₈	C₂×C₃×C₇×C₁₃	1	1	yes	characteristic
103	C₃×C₁₆	C₂×C₃×C₇×C₁₃	1	2	yes	—
104	C₂×C₃×C₇	C₃×C₁₃×C₁₆	1	2	yes	—
105	C₂×C₃×C₇	C₂×C₃×C₈×C₁₃	1	1	yes	characteristic
106	C₃×C₁₃	C₂×C₃×C₇×C₁₆	1	1	yes	characteristic
107	C₂²×C₉	C₇×C₈×C₁₃	1	1	yes	characteristic
108	C₄×C₉	C₇×C₈×C₁₃	1	1	yes	characteristic
109	C₄×C₉	C₂×C₄×C₇×C₁₃	1	1	yes	characteristic
110	C₂×C₁₆	C₇×C₉×C₁₃	1	1	yes	characteristic, direct product factor, Sylow
111	C₂²×C₇	C₈×C₉×C₁₃	1	1	yes	characteristic
112	C₄×C₇	C₈×C₉×C₁₃	1	1	yes	characteristic
113	C₄×C₇	C₂×C₄×C₉×C₁₃	1	1	yes	characteristic
114	C₂×C₁₃	C₇×C₉×C₁₆	1	2	yes	direct product factor
115	C₂×C₁₃	C₂×C₇×C₈×C₉	1	1	yes	characteristic
116	C₂×C₃×C₄	C₃×C₄×C₇×C₁₃	1	1	yes	characteristic
117	C₃×C₈	C₃×C₄×C₇×C₁₃	1	1	yes	characteristic
118	C₃×C₈	C₂²×C₃×C₇×C₁₃	1	1	yes	characteristic
119	C₃×C₇	C₂×C₃×C₁₃×C₁₆	1	1	yes	characteristic
120	C₂×C₉	C₇×C₁₃×C₁₆	1	2	yes	direct product factor
121	C₂×C₉	C₂×C₇×C₈×C₁₃	1	1	yes	characteristic
122	C₂×C₈	C₂×C₇×C₉×C₁₃	1	1	yes	characteristic
123	C₁₆	C₂×C₇×C₉×C₁₃	1	2	yes	direct product factor
124	C₂×C₇	C₉×C₁₃×C₁₆	1	2	yes	direct product factor
125	C₂×C₇	C₂×C₈×C₉×C₁₃	1	1	yes	characteristic
126	C₁₃	C₂×C₇×C₉×C₁₆	1	1	yes	characteristic, direct product factor, Sylow
127	C₂²×C₃	C₃×C₇×C₈×C₁₃	1	1	yes	characteristic
128	C₃×C₄	C₃×C₇×C₈×C₁₃	1	1	yes	characteristic
129	C₃×C₄	C₂×C₃×C₄×C₇×C₁₃	1	1	yes	characteristic
130	C₉	C₂×C₇×C₁₃×C₁₆	1	1	yes	characteristic, direct product factor, Sylow
131	C₂×C₄	C₄×C₇×C₉×C₁₃	1	1	yes	characteristic
132	C₈	C₄×C₇×C₉×C₁₃	1	1	yes	characteristic
133	C₈	C₂²×C₇×C₉×C₁₃	1	1	yes	characteristic
134	C₇	C₂×C₉×C₁₃×C₁₆	1	1	yes	characteristic, direct product factor, Sylow
135	C₂×C₃	C₃×C₇×C₁₃×C₁₆	1	2	yes	—
136	C₂×C₃	C₂×C₃×C₇×C₈×C₁₃	1	1	yes	characteristic
137	C₂²	C₇×C₈×C₉×C₁₃	1	1	yes	characteristic
138	C₄	C₇×C₈×C₉×C₁₃	1	1	yes	characteristic
139	C₄	C₂×C₄×C₇×C₉×C₁₃	1	1	yes	characteristic
140	C₃	C₂×C₃×C₇×C₁₃×C₁₆	1	1	yes	characteristic
141	C₂	C₇×C₉×C₁₃×C₁₆	1	2	yes	direct product factor
142	C₂	C₂×C₇×C₈×C₉×C₁₃	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₃×C₈	C₂²×C₃×C₇×C₁₃
socle, prime radical	C₂²×C₃×C₇×C₁₃	C₃×C₈
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇×C₉×C₁₃×C₁₆	C₂	direct (N × H)
9. C₂×C₉×C₁₃×C₁₆	C₇	direct (N × H)
10. C₂×C₇×C₁₃×C₁₆	C₉	direct (N × H)
14. C₂×C₇×C₉×C₁₆	C₁₃	direct (N × H)
19. C₉×C₁₃×C₁₆	C₂×C₇	direct (N × H)
20. C₂×C₇×C₉×C₁₃	C₁₆	direct (N × H)
23. C₇×C₁₃×C₁₆	C₂×C₉	direct (N × H)
29. C₇×C₉×C₁₆	C₂×C₁₃	direct (N × H)
33. C₇×C₉×C₁₃	C₂×C₁₆	direct (N × H)
48. C₂×C₁₃×C₁₆	C₇×C₉	direct (N × H)
57. C₂×C₉×C₁₆	C₇×C₁₃	direct (N × H)
62. C₂×C₉×C₁₃	C₇×C₁₆	direct (N × H)
64. C₂×C₇×C₁₆	C₉×C₁₃	direct (N × H)
66. C₁₃×C₁₆	C₂×C₇×C₉	direct (N × H)
67. C₂×C₇×C₁₃	C₉×C₁₆	direct (N × H)
76. C₉×C₁₆	C₂×C₇×C₁₃	direct (N × H)
77. C₂×C₇×C₉	C₁₃×C₁₆	direct (N × H)
79. C₉×C₁₃	C₂×C₇×C₁₆	direct (N × H)
81. C₇×C₁₆	C₂×C₉×C₁₃	direct (N × H)
86. C₇×C₁₃	C₂×C₉×C₁₆	direct (N × H)
95. C₇×C₉	C₂×C₁₃×C₁₆	direct (N × H)
110. C₂×C₁₆	C₇×C₉×C₁₃	direct (N × H)
114. C₂×C₁₃	C₇×C₉×C₁₆	direct (N × H)
120. C₂×C₉	C₇×C₁₃×C₁₆	direct (N × H)
123. C₁₆	C₂×C₇×C₉×C₁₃	direct (N × H)
124. C₂×C₇	C₉×C₁₃×C₁₆	direct (N × H)
126. C₁₃	C₂×C₇×C₉×C₁₆	direct (N × H)
130. C₉	C₂×C₇×C₁₃×C₁₆	direct (N × H)
134. C₇	C₂×C₉×C₁₃×C₁₆	direct (N × H)
141. C₂	C₇×C₉×C₁₃×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
C₂×C₇×C₈×C₉×C₁₃	2
C₇×C₉×C₁₃×C₁₆	2
C₂×C₃×C₇×C₁₃×C₁₆	3
C₂×C₉×C₁₃×C₁₆	7
C₂×C₇×C₉×C₁₆	13

Subgroups — C2×C7×C9×C13×C16 #72203346

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×C₉×C₁₃×C₁₆ #72203346