Subgroups — C₂²×C₁₃×#504 #15748961

Order 3328.

2118 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	C₂²×C₁₃×#131	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×#504	C₂	1	24	yes	maximal, direct product factor
3	C₂²×C₁₃×M₃₂	C₂	1	3	yes	maximal, complemented
4	C₂³×C₁₃×C₁₆	C₂	1	3	yes	maximal, complemented
5	C₂²×C₁₃×#41	C₄	1	1	yes	characteristic
6	C₂×C₁₃×#131	C₄	1	12	yes	—
7	C₂×C₁₃×#131	C₂²	1	12	yes	—
8	C₂²×C₁₃×M₁₆	C₄	1	3	yes	—
9	C₂³×C₈×C₁₃	C₂²	1	3	yes	—
10	C₁₃×#504	C₂²	1	64	yes	direct product factor
11	C₂×C₁₃×M₃₂	C₂²	1	36	yes	complemented
12	C₂²×C₁₃×C₁₆	C₂²	1	3	yes	—
13	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
14	C₂²×C₁₃×C₁₆	C₂²	1	36	yes	complemented
15	C₂×C₁₃×#41	C₈	1	12	yes	—
16	C₂×C₁₃×#41	C₂×C₄	1	12	yes	—
17	C₂²×Q₈×C₁₃	C₈	1	1	yes	characteristic
18	C₂²×D₄×C₁₃	C₈	1	3	yes	—
19	C₂³×C₄×C₁₃	C₂×C₄	1	3	yes	—
20	C₁₃×#131	C₂×C₄	1	48	yes	—
21	C₁₃×#131	C₂³	1	16	yes	—
22	C₂×C₁₃×M₁₆	C₂×C₄	1	36	yes	—
23	C₂²×C₈×C₁₃	C₂×C₄	1	3	yes	—
24	C₂²×C₈×C₁₃	C₂×C₄	1	18	yes	—
25	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
26	C₂²×C₈×C₁₃	C₂³	1	18	yes	—
27	C₁₃×M₃₂	C₂³	1	48	yes	complemented
28	C₂×C₁₃×C₁₆	C₂³	1	18	yes	—
29	C₂×C₁₃×C₁₆	C₂³	1	6	yes	—
30	C₂×C₁₃×C₁₆	C₂³	1	48	yes	complemented
31	C₂²×#504	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
32	C₂⁴×C₁₃	C₂×C₈	1	3	yes	—
33	C₁₃×#41	C₂×C₈	1	48	yes	—
34	C₁₃×#41	C₂²×C₄	1	16	yes	—
35	C₂×Q₈×C₁₃	C₂×C₈	1	12	yes	—
36	C₂×D₄×C₁₃	C₂×C₈	1	36	yes	—
37	C₂²×C₄×C₁₃	C₂×C₈	1	3	yes	—
38	C₂²×C₄×C₁₃	C₂×C₈	1	18	yes	—
39	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₃	C₂²×C₄	1	18	yes	—
41	C₁₃×M₁₆	C₂²×C₄	1	48	yes	—
42	C₂×C₈×C₁₃	C₂²×C₄	1	3	yes	—
43	C₂×C₈×C₁₃	C₂²×C₄	1	18	yes	—
44	C₂×C₈×C₁₃	C₂²×C₄	1	36	yes	—
45	C₂×C₈×C₁₃	C₂⁴	1	3	yes	—
46	C₂×C₈×C₁₃	C₂⁴	1	12	yes	—
47	C₁₃×C₁₆	C₂⁴	1	4	yes	—
48	C₁₃×C₁₆	C₂⁴	1	12	yes	—
49	C₂²×#131	C₂×C₁₃	1	1	yes	characteristic
50	C₂×#504	C₂×C₁₃	1	24	yes	direct product factor
51	C₂²×M₃₂	C₂×C₁₃	1	3	yes	complemented
52	C₂³×C₁₆	C₂×C₁₃	1	3	yes	complemented
53	C₂³×C₁₃	—	2	12	no	—
54	C₂³×C₁₃	C₂²×C₈	1	18	yes	—
55	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
56	Q₈×C₁₃	C₂²×C₈	1	16	yes	—
57	D₄×C₁₃	C₂²×C₈	1	48	yes	—
58	C₂×C₄×C₁₃	C₂²×C₈	1	3	yes	—
59	C₂×C₄×C₁₃	C₂²×C₈	1	18	yes	—
60	C₂×C₄×C₁₃	C₂²×C₈	1	36	yes	—
61	C₂×C₄×C₁₃	C₂³×C₄	1	3	yes	—
62	C₂×C₄×C₁₃	C₂³×C₄	1	12	yes	—
63	C₈×C₁₃	C₂³×C₄	1	3	yes	—
64	C₈×C₁₃	C₂³×C₄	1	12	yes	—
65	C₈×C₁₃	C₂⁵	1	1	yes	characteristic
66	C₂²×#41	C₄×C₁₃	1	1	yes	characteristic
67	C₂×#131	C₄×C₁₃	1	12	yes	—
68	C₂×#131	C₂²×C₁₃	1	12	yes	—
69	C₂²×M₁₆	C₄×C₁₃	1	3	yes	—
70	C₂³×C₈	C₂²×C₁₃	1	3	yes	—
71	#504	C₂²×C₁₃	1	64	yes	direct product factor
72	C₂×M₃₂	C₂²×C₁₃	1	36	yes	complemented
73	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
74	C₂²×C₁₆	C₂²×C₁₃	1	3	yes	—
75	C₂²×C₁₆	C₂²×C₁₃	1	36	yes	complemented
76	C₂²×C₁₃	—	2	36	no	—
77	C₂²×C₁₃	#504	1	4	yes	direct product factor
78	C₂²×C₁₃	C₂³×C₈	1	12	yes	—
79	C₂²×C₁₃	C₂³×C₈	1	3	yes	—
80	C₄×C₁₃	C₂³×C₈	1	3	yes	—
81	C₄×C₁₃	C₂³×C₈	1	12	yes	—
82	C₄×C₁₃	C₂⁴×C₄	1	1	yes	characteristic
83	C₂×#41	C₈×C₁₃	1	12	yes	—
84	C₂×#41	C₂×C₄×C₁₃	1	12	yes	—
85	C₂²×Q₈	C₈×C₁₃	1	1	yes	characteristic
86	C₂²×D₄	C₈×C₁₃	1	3	yes	—
87	C₂³×C₄	C₂×C₄×C₁₃	1	3	yes	—
88	#131	C₂×C₄×C₁₃	1	48	yes	—
89	#131	C₂³×C₁₃	1	16	yes	—
90	C₂×M₁₆	C₂×C₄×C₁₃	1	36	yes	—
91	C₂²×C₈	C₂×C₄×C₁₃	1	3	yes	—
92	C₂²×C₈	C₂×C₄×C₁₃	1	18	yes	—
93	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
94	C₂²×C₈	C₂³×C₁₃	1	18	yes	—
95	M₃₂	C₂³×C₁₃	1	48	yes	complemented
96	C₂×C₁₆	C₂³×C₁₃	1	6	yes	—
97	C₂×C₁₆	C₂³×C₁₃	1	18	yes	—
98	C₂×C₁₆	C₂³×C₁₃	1	48	yes	complemented
99	C₂×C₁₃	—	2	12	no	—
100	C₂×C₁₃	C₂×#504	1	6	yes	direct product factor
101	C₂×C₁₃	C₂⁴×C₈	1	1	yes	characteristic
102	C₂⁴	C₂×C₈×C₁₃	1	3	yes	—
103	#41	C₂×C₈×C₁₃	1	48	yes	—
104	#41	C₂²×C₄×C₁₃	1	16	yes	—
105	C₂×Q₈	C₂×C₈×C₁₃	1	12	yes	—
106	C₂×D₄	C₂×C₈×C₁₃	1	36	yes	—
107	C₂²×C₄	C₂×C₈×C₁₃	1	3	yes	—
108	C₂²×C₄	C₂×C₈×C₁₃	1	18	yes	—
109	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
110	C₂²×C₄	C₂²×C₄×C₁₃	1	18	yes	—
111	M₁₆	C₂²×C₄×C₁₃	1	48	yes	—
112	C₂×C₈	C₂²×C₄×C₁₃	1	3	yes	—
113	C₂×C₈	C₂²×C₄×C₁₃	1	18	yes	—
114	C₂×C₈	C₂²×C₄×C₁₃	1	36	yes	—
115	C₂×C₈	C₂⁴×C₁₃	1	3	yes	—
116	C₂×C₈	C₂⁴×C₁₃	1	12	yes	—
117	C₁₆	C₂⁴×C₁₃	1	4	yes	—
118	C₁₆	C₂⁴×C₁₃	1	12	yes	—
119	C₁₃	C₂²×#504	1	1	yes	characteristic, direct product factor, Sylow
120	C₂³	—	2	12	no	—
121	C₂³	C₂²×C₈×C₁₃	1	18	yes	—
122	C₂³	C₂²×C₈×C₁₃	1	1	yes	characteristic
123	Q₈	C₂²×C₈×C₁₃	1	16	yes	—
124	D₄	C₂²×C₈×C₁₃	1	48	yes	—
125	C₂×C₄	C₂²×C₈×C₁₃	1	3	yes	—
126	C₂×C₄	C₂²×C₈×C₁₃	1	18	yes	—
127	C₂×C₄	C₂²×C₈×C₁₃	1	36	yes	—
128	C₂×C₄	C₂³×C₄×C₁₃	1	3	yes	—
129	C₂×C₄	C₂³×C₄×C₁₃	1	12	yes	—
130	C₈	C₂³×C₄×C₁₃	1	3	yes	—
131	C₈	C₂³×C₄×C₁₃	1	12	yes	—
132	C₈	C₂⁵×C₁₃	1	1	yes	characteristic
133	C₂²	—	2	36	no	—
134	C₂²	C₁₃×#504	1	4	yes	direct product factor
135	C₂²	C₂³×C₈×C₁₃	1	12	yes	—
136	C₂²	C₂³×C₈×C₁₃	1	3	yes	—
137	C₄	C₂³×C₈×C₁₃	1	3	yes	—
138	C₄	C₂³×C₈×C₁₃	1	12	yes	—
139	C₄	C₂⁴×C₄×C₁₃	1	1	yes	characteristic
140	C₂	—	2	12	no	—
141	C₂	C₂×C₁₃×#504	1	6	yes	direct product factor
142	C₂	C₂⁴×C₈×C₁₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₁₃×C₁₆	C₂²
commutator	C₂	C₂⁴×C₈×C₁₃
Frattini, cosocle	C₈	C₂⁵×C₁₃
socle	C₂³×C₁₃	C₂²×C₈
prime radical	C₂²×C₁₃×#41	C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃×C₁₆ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₃×#504	C₂	direct (N × H)
3. C₂²×C₁₃×M₃₂	C₂	semidirect
4. C₂³×C₁₃×C₁₆	C₂	semidirect
10. C₁₃×#504	C₂²	direct (N × H)
11. C₂×C₁₃×M₃₂	C₂²	semidirect
14. C₂²×C₁₃×C₁₆	C₂²	semidirect
27. C₁₃×M₃₂	C₂³	semidirect
30. C₂×C₁₃×C₁₆	C₂³	semidirect
31. C₂²×#504	C₁₃	direct (N × H)
50. C₂×#504	C₂×C₁₃	direct (N × H)
51. C₂²×M₃₂	C₂×C₁₃	semidirect
52. C₂³×C₁₆	C₂×C₁₃	semidirect
71. #504	C₂²×C₁₃	direct (N × H)
72. C₂×M₃₂	C₂²×C₁₃	semidirect
75. C₂²×C₁₆	C₂²×C₁₃	semidirect
77. C₂²×C₁₃	#504	direct (N × H)
95. M₃₂	C₂³×C₁₃	semidirect
98. C₂×C₁₆	C₂³×C₁₃	semidirect
100. C₂×C₁₃	C₂×#504	direct (N × H)
119. C₁₃	C₂²×#504	direct (N × H)
134. C₂²	C₁₃×#504	direct (N × H)
141. C₂	C₂×C₁₃×#504	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₁₃×#131	2
C₂×C₁₃×#504	2
C₂²×C₁₃×M₃₂	2
C₂³×C₁₃×C₁₆	2
C₂²×#504	13

Subgroups — C22×C13×#504 #15748961

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₁₃×#504 #15748961