Subgroups — C₄×C₁₃×M₆₄ #15699887

Order 3328.

250 conjugacy classes of subgroups.

The subgroup lattice has 138 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×M₆₄	C₂	1	2	yes	maximal
2	C₂×C₄×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#1397	C₂	1	2	yes	maximal, complemented
4	C₄×C₁₃×C₃₂	C₂	1	2	yes	maximal, complemented
5	C₂²×C₁₃×C₁₆	C₄	1	1	yes	characteristic
6	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₈×C₁₃	C₄	1	1	yes	characteristic
8	C₁₃×M₆₄	C₄	1	8	yes	direct product factor
9	C₂×C₁₃×C₃₂	C₂²	1	4	yes	—
10	C₄×C₁₃×C₁₆	C₄	1	1	yes	characteristic
11	C₄×C₁₃×C₁₆	C₄	1	1	yes	characteristic
12	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
13	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₁₃	C₈	1	1	yes	characteristic
15	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄²×C₁₃	C₈	1	1	yes	characteristic
17	C₂×C₁₃×C₁₆	C₈	1	4	yes	—
18	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
19	C₂×C₁₃×C₁₆	C₂×C₄	1	4	yes	—
20	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
21	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
22	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
23	C₄×C₈×C₁₃	C₈	1	1	yes	characteristic
24	C₄×C₈×C₁₃	C₈	1	1	yes	characteristic
25	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₁₃×C₃₂	C₂×C₄	1	8	yes	complemented
28	C₄×M₆₄	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
29	C₂²×C₄×C₁₃	C₁₆	1	2	yes	—
30	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
31	C₂×C₈×C₁₃	C₁₆	1	4	yes	—
32	C₂×C₈×C₁₃	C₄²	1	2	yes	—
33	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
34	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
35	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
36	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
37	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
38	C₄²×C₁₃	C₁₆	1	2	yes	—
39	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
40	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
41	C₁₃×C₁₆	C₄²	1	2	yes	—
42	C₁₃×C₁₆	C₂×C₈	1	2	yes	—
43	C₁₃×C₁₆	C₂×C₈	1	2	yes	—
44	C₁₃×C₁₆	C₂²×C₄	1	2	yes	—
45	C₂×M₆₄	C₂×C₁₃	1	2	yes	—
46	C₂×C₄×C₁₆	C₂×C₁₃	1	1	yes	characteristic
47	#1397	C₂×C₁₃	1	2	yes	complemented
48	C₄×C₃₂	C₂×C₁₃	1	2	yes	complemented
49	C₂³×C₁₃	C₂×C₁₆	1	1	yes	characteristic
50	C₂×C₄×C₁₃	—	2	4	no	—
51	C₂×C₄×C₁₃	C₄×C₈	1	2	yes	—
52	C₂×C₄×C₁₃	C₂×C₁₆	1	2	yes	—
53	C₂×C₄×C₁₃	C₂×C₁₆	1	1	yes	characteristic
54	C₂×C₄×C₁₃	C₂×C₁₆	1	2	yes	—
55	C₂×C₄×C₁₃	C₂×C₁₆	1	2	yes	—
56	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
57	C₈×C₁₃	C₄×C₈	1	2	yes	—
58	C₈×C₁₃	C₂×C₁₆	1	2	yes	—
59	C₈×C₁₃	C₂×C₁₆	1	2	yes	—
60	C₈×C₁₃	C₂×C₄²	1	1	yes	characteristic
61	C₈×C₁₃	C₂²×C₈	1	1	yes	characteristic
62	C₂²×C₁₆	C₄×C₁₃	1	1	yes	characteristic
63	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
64	C₂×C₄×C₈	C₄×C₁₃	1	1	yes	characteristic
65	M₆₄	C₄×C₁₃	1	8	yes	direct product factor
66	C₂×C₃₂	C₂²×C₁₃	1	4	yes	—
67	C₄×C₁₆	C₄×C₁₃	1	1	yes	characteristic
68	C₄×C₁₆	C₄×C₁₃	1	1	yes	characteristic
69	C₄×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
70	C₄×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
71	C₂²×C₁₃	—	2	2	no	—
72	C₂²×C₁₃	C₄×C₁₆	1	2	yes	—
73	C₂²×C₁₃	C₂²×C₁₆	1	1	yes	characteristic
74	C₄×C₁₃	—	2	2	no	—
75	C₄×C₁₃	C₄×C₁₆	1	2	yes	—
76	C₄×C₁₃	M₆₄	1	4	yes	direct product factor
77	C₄×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
78	C₄×C₁₃	C₂²×C₁₆	1	1	yes	characteristic
79	C₂²×C₈	C₈×C₁₃	1	1	yes	characteristic
80	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
81	C₂×C₄²	C₈×C₁₃	1	1	yes	characteristic
82	C₂×C₁₆	C₈×C₁₃	1	4	yes	—
83	C₂×C₁₆	C₂×C₄×C₁₃	1	1	yes	characteristic
84	C₂×C₁₆	C₂×C₄×C₁₃	1	4	yes	—
85	C₂×C₁₆	C₂×C₄×C₁₃	1	1	yes	characteristic
86	C₂×C₁₆	C₂×C₄×C₁₃	1	1	yes	characteristic
87	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
88	C₄×C₈	C₈×C₁₃	1	1	yes	characteristic
89	C₄×C₈	C₈×C₁₃	1	1	yes	characteristic
90	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
91	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
92	C₃₂	C₂×C₄×C₁₃	1	8	yes	complemented
93	C₂×C₁₃	—	2	2	no	—
94	C₂×C₁₃	C₂×C₄×C₁₆	1	1	yes	characteristic
95	C₂×C₁₃	C₂×M₆₄	1	2	yes	—
96	C₂²×C₄	C₁₃×C₁₆	1	2	yes	—
97	C₂²×C₄	C₂×C₈×C₁₃	1	1	yes	characteristic
98	C₂×C₈	C₁₃×C₁₆	1	4	yes	—
99	C₂×C₈	C₄²×C₁₃	1	2	yes	—
100	C₂×C₈	C₂×C₈×C₁₃	1	1	yes	characteristic
101	C₂×C₈	C₂×C₈×C₁₃	1	2	yes	—
102	C₂×C₈	C₂×C₈×C₁₃	1	1	yes	characteristic
103	C₂×C₈	C₂×C₈×C₁₃	1	1	yes	characteristic
104	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
105	C₄²	C₁₃×C₁₆	1	2	yes	—
106	C₄²	C₂×C₈×C₁₃	1	1	yes	characteristic
107	C₄²	C₂×C₈×C₁₃	1	1	yes	characteristic
108	C₁₆	C₄²×C₁₃	1	2	yes	—
109	C₁₆	C₂×C₈×C₁₃	1	2	yes	—
110	C₁₆	C₂×C₈×C₁₃	1	2	yes	—
111	C₁₆	C₂²×C₄×C₁₃	1	2	yes	—
112	C₁₃	C₄×M₆₄	1	1	yes	characteristic, direct product factor, Sylow
113	C₂³	C₂×C₁₃×C₁₆	1	1	yes	characteristic
114	C₂×C₄	—	2	4	no	—
115	C₂×C₄	C₄×C₈×C₁₃	1	2	yes	—
116	C₂×C₄	C₂×C₁₃×C₁₆	1	2	yes	—
117	C₂×C₄	C₂×C₁₃×C₁₆	1	1	yes	characteristic
118	C₂×C₄	C₂×C₁₃×C₁₆	1	2	yes	—
119	C₂×C₄	C₂×C₁₃×C₁₆	1	2	yes	—
120	C₂×C₄	C₂²×C₈×C₁₃	1	1	yes	characteristic
121	C₈	C₄×C₈×C₁₃	1	2	yes	—
122	C₈	C₂×C₁₃×C₁₆	1	2	yes	—
123	C₈	C₂×C₁₃×C₁₆	1	2	yes	—
124	C₈	C₂×C₄²×C₁₃	1	1	yes	characteristic
125	C₈	C₂²×C₈×C₁₃	1	1	yes	characteristic
126	C₂²	—	2	2	no	—
127	C₂²	C₄×C₁₃×C₁₆	1	2	yes	—
128	C₂²	C₂²×C₁₃×C₁₆	1	1	yes	characteristic
129	C₄	—	2	2	no	—
130	C₄	C₄×C₁₃×C₁₆	1	2	yes	—
131	C₄	C₁₃×M₆₄	1	4	yes	direct product factor
132	C₄	C₂×C₄×C₈×C₁₃	1	1	yes	characteristic
133	C₄	C₂²×C₁₃×C₁₆	1	1	yes	characteristic
134	C₂	—	2	2	no	—
135	C₂	C₂×C₄×C₁₃×C₁₆	1	1	yes	characteristic
136	C₂	C₂×C₁₃×M₆₄	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₁₃×#1397	C₂	semidirect
4. C₄×C₁₃×C₃₂	C₂	semidirect
8. C₁₃×M₆₄	C₄	direct (N × H)
27. C₁₃×C₃₂	C₂×C₄	semidirect
28. C₄×M₆₄	C₁₃	direct (N × H)
47. #1397	C₂×C₁₃	semidirect
48. C₄×C₃₂	C₂×C₁₃	semidirect
65. M₆₄	C₄×C₁₃	direct (N × H)
76. C₄×C₁₃	M₆₄	direct (N × H)
92. C₃₂	C₂×C₄×C₁₃	semidirect
112. C₁₃	C₄×M₆₄	direct (N × H)
131. C₄	C₁₃×M₆₄	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×C₁₃×M₆₄	2
C₂×C₄×C₁₃×C₁₆	2
C₁₃×#1397	2
C₄×C₁₃×C₃₂	2
C₄×M₆₄	13

Subgroups — C4×C13×M64 #15699887

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₁₃×M₆₄ #15699887