Subgroups — C₃²×C₄×C₇×C₈×C₁₃ #72203349

Order 26208.

528 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	C₃²×C₄²×C₇×C₁₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃²×C₇×C₈×C₁₃	C₂	1	2	yes	maximal
3	C₃×C₄×C₇×C₈×C₁₃	C₃	1	4	yes	maximal, direct product factor
4	C₂×C₃²×C₄×C₇×C₁₃	C₄	1	2	yes	—
5	C₂×C₃²×C₄×C₇×C₁₃	C₂²	1	1	yes	characteristic
6	C₃²×C₇×C₈×C₁₃	C₄	1	4	yes	direct product factor
7	C₃×C₄²×C₇×C₁₃	C₂×C₃	1	4	yes	—
8	C₂×C₃×C₇×C₈×C₁₃	C₂×C₃	1	8	yes	—
9	C₃²×C₄×C₈×C₁₃	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₄×C₇×C₈×C₁₃	C₃²	1	1	yes	characteristic, direct product factor, Hall
11	C₂×C₃×C₄×C₇×C₁₃	C₃×C₄	1	8	yes	—
12	C₂×C₃×C₄×C₇×C₁₃	C₂²×C₃	1	4	yes	—
13	C₃×C₇×C₈×C₁₃	C₃×C₄	1	16	yes	direct product factor
14	C₃²×C₄×C₇×C₈	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Hall
15	C₂²×C₃²×C₇×C₁₃	C₂×C₄	1	1	yes	characteristic
16	C₃²×C₄×C₇×C₁₃	C₈	1	4	yes	direct product factor
17	C₃²×C₄×C₇×C₁₃	C₂×C₄	1	2	yes	—
18	C₂×C₃²×C₈×C₁₃	C₂×C₇	1	2	yes	—
19	C₃²×C₄²×C₁₃	C₂×C₇	1	1	yes	characteristic
20	C₂×C₃²×C₇×C₁₃	C₄²	1	1	yes	characteristic
21	C₂×C₃²×C₇×C₁₃	C₂×C₈	1	2	yes	—
22	C₂×C₇×C₈×C₁₃	C₂×C₃²	1	2	yes	—
23	C₄²×C₇×C₁₃	C₂×C₃²	1	1	yes	characteristic
24	C₃×C₄×C₈×C₁₃	C₃×C₇	1	4	yes	direct product factor
25	C₂²×C₃×C₇×C₁₃	C₂×C₃×C₄	1	4	yes	—
26	C₃×C₄×C₇×C₁₃	C₃×C₈	1	16	yes	direct product factor
27	C₃×C₄×C₇×C₁₃	C₂×C₃×C₄	1	8	yes	—
28	C₂×C₃²×C₇×C₈	C₂×C₁₃	1	2	yes	—
29	C₃²×C₄²×C₇	C₂×C₁₃	1	1	yes	characteristic
30	C₂×C₃²×C₄×C₁₃	C₄×C₇	1	2	yes	—
31	C₂×C₃²×C₄×C₁₃	C₂²×C₇	1	1	yes	characteristic
32	C₃²×C₈×C₁₃	C₄×C₇	1	4	yes	direct product factor
33	C₃²×C₇×C₁₃	C₄×C₈	1	1	yes	characteristic, direct product factor, Hall
34	C₂×C₄×C₇×C₁₃	C₃²×C₄	1	2	yes	—
35	C₂×C₄×C₇×C₁₃	C₂²×C₃²	1	1	yes	characteristic
36	C₇×C₈×C₁₃	C₃²×C₄	1	4	yes	direct product factor
37	C₃×C₄×C₇×C₈	C₃×C₁₃	1	4	yes	direct product factor
38	C₂×C₃×C₈×C₁₃	C₂×C₃×C₇	1	8	yes	—
39	C₃×C₄²×C₁₃	C₂×C₃×C₇	1	4	yes	—
40	C₂×C₃×C₇×C₁₃	C₃×C₄²	1	4	yes	—
41	C₂×C₃×C₇×C₁₃	C₂×C₃×C₈	1	8	yes	—
42	C₂×C₃²×C₄×C₇	C₄×C₁₃	1	2	yes	—
43	C₂×C₃²×C₄×C₇	C₂²×C₁₃	1	1	yes	characteristic
44	C₃²×C₇×C₈	C₄×C₁₃	1	4	yes	direct product factor
45	C₂²×C₃²×C₁₃	C₂×C₄×C₇	1	1	yes	characteristic
46	C₃²×C₄×C₁₃	C₇×C₈	1	4	yes	direct product factor
47	C₃²×C₄×C₁₃	C₂×C₄×C₇	1	2	yes	—
48	C₄×C₈×C₁₃	C₃²×C₇	1	1	yes	characteristic, direct product factor, Hall
49	C₂²×C₇×C₁₃	C₂×C₃²×C₄	1	1	yes	characteristic
50	C₄×C₇×C₁₃	C₃²×C₈	1	4	yes	direct product factor
51	C₄×C₇×C₁₃	C₂×C₃²×C₄	1	2	yes	—
52	C₂×C₃×C₇×C₈	C₂×C₃×C₁₃	1	8	yes	—
53	C₃×C₄²×C₇	C₂×C₃×C₁₃	1	4	yes	—
54	C₂×C₃×C₄×C₁₃	C₃×C₄×C₇	1	8	yes	—
55	C₂×C₃×C₄×C₁₃	C₂²×C₃×C₇	1	4	yes	—
56	C₃×C₈×C₁₃	C₃×C₄×C₇	1	16	yes	direct product factor
57	C₃²×C₄×C₈	C₇×C₁₃	1	1	yes	characteristic, direct product factor, Hall
58	C₃×C₇×C₁₃	C₃×C₄×C₈	1	4	yes	direct product factor
59	C₂²×C₃²×C₇	C₂×C₄×C₁₃	1	1	yes	characteristic
60	C₃²×C₄×C₇	C₈×C₁₃	1	4	yes	direct product factor
61	C₃²×C₄×C₇	C₂×C₄×C₁₃	1	2	yes	—
62	C₂×C₃²×C₁₃	C₄²×C₇	1	1	yes	characteristic
63	C₂×C₃²×C₁₃	C₂×C₇×C₈	1	2	yes	—
64	C₄×C₇×C₈	C₃²×C₁₃	1	1	yes	characteristic, direct product factor, Hall
65	C₂×C₈×C₁₃	C₂×C₃²×C₇	1	2	yes	—
66	C₄²×C₁₃	C₂×C₃²×C₇	1	1	yes	characteristic
67	C₂×C₇×C₁₃	C₃²×C₄²	1	1	yes	characteristic
68	C₂×C₇×C₁₃	C₂×C₃²×C₈	1	2	yes	—
69	C₂×C₃×C₄×C₇	C₃×C₄×C₁₃	1	8	yes	—
70	C₂×C₃×C₄×C₇	C₂²×C₃×C₁₃	1	4	yes	—
71	C₃×C₇×C₈	C₃×C₄×C₁₃	1	16	yes	direct product factor
72	C₂²×C₃×C₁₃	C₂×C₃×C₄×C₇	1	4	yes	—
73	C₃×C₄×C₁₃	C₃×C₇×C₈	1	16	yes	direct product factor
74	C₃×C₄×C₁₃	C₂×C₃×C₄×C₇	1	8	yes	—
75	C₂×C₃²×C₈	C₂×C₇×C₁₃	1	2	yes	—
76	C₃²×C₄²	C₂×C₇×C₁₃	1	1	yes	characteristic
77	C₂×C₃²×C₇	C₄²×C₁₃	1	1	yes	characteristic
78	C₂×C₃²×C₇	C₂×C₈×C₁₃	1	2	yes	—
79	C₃²×C₁₃	C₄×C₇×C₈	1	1	yes	characteristic, direct product factor, Hall
80	C₂×C₇×C₈	C₂×C₃²×C₁₃	1	2	yes	—
81	C₄²×C₇	C₂×C₃²×C₁₃	1	1	yes	characteristic
82	C₂×C₄×C₁₃	C₃²×C₄×C₇	1	2	yes	—
83	C₂×C₄×C₁₃	C₂²×C₃²×C₇	1	1	yes	characteristic
84	C₈×C₁₃	C₃²×C₄×C₇	1	4	yes	direct product factor
85	C₃×C₄×C₈	C₃×C₇×C₁₃	1	4	yes	direct product factor
86	C₇×C₁₃	C₃²×C₄×C₈	1	1	yes	characteristic, direct product factor, Hall
87	C₂²×C₃×C₇	C₂×C₃×C₄×C₁₃	1	4	yes	—
88	C₃×C₄×C₇	C₃×C₈×C₁₃	1	16	yes	direct product factor
89	C₃×C₄×C₇	C₂×C₃×C₄×C₁₃	1	8	yes	—
90	C₂×C₃×C₁₃	C₃×C₄²×C₇	1	4	yes	—
91	C₂×C₃×C₁₃	C₂×C₃×C₇×C₈	1	8	yes	—
92	C₂×C₃²×C₄	C₄×C₇×C₁₃	1	2	yes	—
93	C₂×C₃²×C₄	C₂²×C₇×C₁₃	1	1	yes	characteristic
94	C₃²×C₈	C₄×C₇×C₁₃	1	4	yes	direct product factor
95	C₃²×C₇	C₄×C₈×C₁₃	1	1	yes	characteristic, direct product factor, Hall
96	C₂×C₄×C₇	C₃²×C₄×C₁₃	1	2	yes	—
97	C₂×C₄×C₇	C₂²×C₃²×C₁₃	1	1	yes	characteristic
98	C₇×C₈	C₃²×C₄×C₁₃	1	4	yes	direct product factor
99	C₂²×C₁₃	C₂×C₃²×C₄×C₇	1	1	yes	characteristic
100	C₄×C₁₃	C₃²×C₇×C₈	1	4	yes	direct product factor
101	C₄×C₁₃	C₂×C₃²×C₄×C₇	1	2	yes	—
102	C₂×C₃×C₈	C₂×C₃×C₇×C₁₃	1	8	yes	—
103	C₃×C₄²	C₂×C₃×C₇×C₁₃	1	4	yes	—
104	C₂×C₃×C₇	C₃×C₄²×C₁₃	1	4	yes	—
105	C₂×C₃×C₇	C₂×C₃×C₈×C₁₃	1	8	yes	—
106	C₃×C₁₃	C₃×C₄×C₇×C₈	1	4	yes	direct product factor
107	C₂²×C₃²	C₂×C₄×C₇×C₁₃	1	1	yes	characteristic
108	C₃²×C₄	C₇×C₈×C₁₃	1	4	yes	direct product factor
109	C₃²×C₄	C₂×C₄×C₇×C₁₃	1	2	yes	—
110	C₄×C₈	C₃²×C₇×C₁₃	1	1	yes	characteristic, direct product factor, Sylow
111	C₂²×C₇	C₂×C₃²×C₄×C₁₃	1	1	yes	characteristic
112	C₄×C₇	C₃²×C₈×C₁₃	1	4	yes	direct product factor
113	C₄×C₇	C₂×C₃²×C₄×C₁₃	1	2	yes	—
114	C₂×C₁₃	C₃²×C₄²×C₇	1	1	yes	characteristic
115	C₂×C₁₃	C₂×C₃²×C₇×C₈	1	2	yes	—
116	C₂×C₃×C₄	C₃×C₄×C₇×C₁₃	1	8	yes	—
117	C₂×C₃×C₄	C₂²×C₃×C₇×C₁₃	1	4	yes	—
118	C₃×C₈	C₃×C₄×C₇×C₁₃	1	16	yes	direct product factor
119	C₃×C₇	C₃×C₄×C₈×C₁₃	1	4	yes	direct product factor
120	C₂×C₃²	C₄²×C₇×C₁₃	1	1	yes	characteristic
121	C₂×C₃²	C₂×C₇×C₈×C₁₃	1	2	yes	—
122	C₂×C₈	C₂×C₃²×C₇×C₁₃	1	2	yes	—
123	C₄²	C₂×C₃²×C₇×C₁₃	1	1	yes	characteristic
124	C₂×C₇	C₃²×C₄²×C₁₃	1	1	yes	characteristic
125	C₂×C₇	C₂×C₃²×C₈×C₁₃	1	2	yes	—
126	C₁₃	C₃²×C₄×C₇×C₈	1	1	yes	characteristic, direct product factor, Sylow
127	C₂²×C₃	C₂×C₃×C₄×C₇×C₁₃	1	4	yes	—
128	C₃×C₄	C₃×C₇×C₈×C₁₃	1	16	yes	direct product factor
129	C₃×C₄	C₂×C₃×C₄×C₇×C₁₃	1	8	yes	—
130	C₃²	C₄×C₇×C₈×C₁₃	1	1	yes	characteristic, direct product factor, Sylow
131	C₂×C₄	C₃²×C₄×C₇×C₁₃	1	2	yes	—
132	C₂×C₄	C₂²×C₃²×C₇×C₁₃	1	1	yes	characteristic
133	C₈	C₃²×C₄×C₇×C₁₃	1	4	yes	direct product factor
134	C₇	C₃²×C₄×C₈×C₁₃	1	1	yes	characteristic, direct product factor, Sylow
135	C₂×C₃	C₂×C₃×C₇×C₈×C₁₃	1	8	yes	—
136	C₂×C₃	C₃×C₄²×C₇×C₁₃	1	4	yes	—
137	C₂²	C₂×C₃²×C₄×C₇×C₁₃	1	1	yes	characteristic
138	C₄	C₃²×C₇×C₈×C₁₃	1	4	yes	direct product factor
139	C₄	C₂×C₃²×C₄×C₇×C₁₃	1	2	yes	—
140	C₃	C₃×C₄×C₇×C₈×C₁₃	1	4	yes	direct product factor
141	C₂	C₂×C₃²×C₇×C₈×C₁₃	1	2	yes	—
142	C₂	C₃²×C₄²×C₇×C₁₃	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₄	C₂²×C₃²×C₇×C₁₃
socle, prime radical	C₂²×C₃²×C₇×C₁₃	C₂×C₄
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃×C₄×C₇×C₈×C₁₃	C₃	direct (N × H)
6. C₃²×C₇×C₈×C₁₃	C₄	direct (N × H)
9. C₃²×C₄×C₈×C₁₃	C₇	direct (N × H)
10. C₄×C₇×C₈×C₁₃	C₃²	direct (N × H)
13. C₃×C₇×C₈×C₁₃	C₃×C₄	direct (N × H)
14. C₃²×C₄×C₇×C₈	C₁₃	direct (N × H)
16. C₃²×C₄×C₇×C₁₃	C₈	direct (N × H)
24. C₃×C₄×C₈×C₁₃	C₃×C₇	direct (N × H)
26. C₃×C₄×C₇×C₁₃	C₃×C₈	direct (N × H)
32. C₃²×C₈×C₁₃	C₄×C₇	direct (N × H)
33. C₃²×C₇×C₁₃	C₄×C₈	direct (N × H)
36. C₇×C₈×C₁₃	C₃²×C₄	direct (N × H)
37. C₃×C₄×C₇×C₈	C₃×C₁₃	direct (N × H)
44. C₃²×C₇×C₈	C₄×C₁₃	direct (N × H)
46. C₃²×C₄×C₁₃	C₇×C₈	direct (N × H)
48. C₄×C₈×C₁₃	C₃²×C₇	direct (N × H)
50. C₄×C₇×C₁₃	C₃²×C₈	direct (N × H)
56. C₃×C₈×C₁₃	C₃×C₄×C₇	direct (N × H)
57. C₃²×C₄×C₈	C₇×C₁₃	direct (N × H)
58. C₃×C₇×C₁₃	C₃×C₄×C₈	direct (N × H)
60. C₃²×C₄×C₇	C₈×C₁₃	direct (N × H)
64. C₄×C₇×C₈	C₃²×C₁₃	direct (N × H)
71. C₃×C₇×C₈	C₃×C₄×C₁₃	direct (N × H)
73. C₃×C₄×C₁₃	C₃×C₇×C₈	direct (N × H)
79. C₃²×C₁₃	C₄×C₇×C₈	direct (N × H)
84. C₈×C₁₃	C₃²×C₄×C₇	direct (N × H)
85. C₃×C₄×C₈	C₃×C₇×C₁₃	direct (N × H)
86. C₇×C₁₃	C₃²×C₄×C₈	direct (N × H)
88. C₃×C₄×C₇	C₃×C₈×C₁₃	direct (N × H)
94. C₃²×C₈	C₄×C₇×C₁₃	direct (N × H)
95. C₃²×C₇	C₄×C₈×C₁₃	direct (N × H)
98. C₇×C₈	C₃²×C₄×C₁₃	direct (N × H)
100. C₄×C₁₃	C₃²×C₇×C₈	direct (N × H)
106. C₃×C₁₃	C₃×C₄×C₇×C₈	direct (N × H)
108. C₃²×C₄	C₇×C₈×C₁₃	direct (N × H)
110. C₄×C₈	C₃²×C₇×C₁₃	direct (N × H)
112. C₄×C₇	C₃²×C₈×C₁₃	direct (N × H)
118. C₃×C₈	C₃×C₄×C₇×C₁₃	direct (N × H)
119. C₃×C₇	C₃×C₄×C₈×C₁₃	direct (N × H)
126. C₁₃	C₃²×C₄×C₇×C₈	direct (N × H)
128. C₃×C₄	C₃×C₇×C₈×C₁₃	direct (N × H)
130. C₃²	C₄×C₇×C₈×C₁₃	direct (N × H)
133. C₈	C₃²×C₄×C₇×C₁₃	direct (N × H)
134. C₇	C₃²×C₄×C₈×C₁₃	direct (N × H)
138. C₄	C₃²×C₇×C₈×C₁₃	direct (N × H)
140. C₃	C₃×C₄×C₇×C₈×C₁₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃²×C₄²×C₇×C₁₃	2
C₂×C₃²×C₇×C₈×C₁₃	2
C₃×C₄×C₇×C₈×C₁₃	3
C₃²×C₄×C₈×C₁₃	7
C₃²×C₄×C₇×C₈	13

Subgroups — C32×C4×C7×C8×C13 #72203349

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₄×C₇×C₈×C₁₃ #72203349