Subgroups — Dic₁₃×M₆₄ #15783094

Order 3328.

250 conjugacy classes of subgroups.

The subgroup lattice has 150 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718091	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₆×Dic₁₃	C₂	1	1	yes	maximal, characteristic
3	#13112969	C₂	1	2	yes	maximal, complemented
4	C₃₂×Dic₁₃	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₃×M₆₄	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
7	C₂×C₈×Dic₁₃	C₄	1	1	yes	characteristic
8	C₂²×#6090	C₄	1	1	yes	characteristic
9	C₁₃×M₆₄	C₄	1	4	yes	complemented
10	C₂×C₁₃×C₃₂	C₂²	1	2	yes	—
11	C₁₆×Dic₁₃	C₄	1	1	yes	characteristic
12	C₁₆×Dic₁₃	C₄	1	1	yes	characteristic
13	C₁₆×Dic₁₃	C₂²	1	1	yes	characteristic
14	C₁₆×Dic₁₃	C₂²	1	1	yes	characteristic
15	#11718091	C₄	1	4	yes	complemented
16	C₂×#88034	C₂²	1	2	yes	—
17	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×Dic₁₃	C₈	1	1	yes	characteristic
19	C₂²×#1055	C₈	1	1	yes	characteristic
20	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
21	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
22	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
23	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
24	C₈×Dic₁₃	C₈	1	1	yes	characteristic
25	C₈×Dic₁₃	C₈	1	1	yes	characteristic
26	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
27	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
28	C₂×#6090	C₈	1	4	yes	—
29	C₂×#6090	C₂×C₄	1	1	yes	characteristic
30	C₂×#6090	C₂×C₄	1	1	yes	characteristic
31	C₁₃×C₃₂	C₂×C₄	1	4	yes	complemented
32	#88034	C₂×C₄	1	4	yes	complemented
33	C₄×M₆₄	—	13	1	no	maximal, Sylow
34	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
35	C₂²×Dic₁₃	C₁₆	1	2	yes	—
36	C₂×C₈×C₁₃	C₄²	1	2	yes	—
37	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
38	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
39	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
40	C₄×Dic₁₃	C₁₆	1	2	yes	—
41	C₄×Dic₁₃	C₂×C₈	1	1	yes	characteristic
42	C₄×Dic₁₃	C₂×C₈	1	1	yes	characteristic
43	C₂×#1055	C₁₆	1	4	yes	—
44	C₂×#1055	C₂×C₈	1	1	yes	characteristic
45	C₂×#1055	C₂×C₈	1	1	yes	characteristic
46	C₁₃×C₁₆	C₄²	1	2	yes	—
47	C₁₃×C₁₆	C₂²×C₄	1	1	yes	characteristic
48	C₁₃×C₁₆	C₂²×C₄	1	1	yes	characteristic
49	#6090	C₂×C₈	1	2	yes	—
50	#6090	C₂×C₈	1	2	yes	—
51	C₂×M₆₄	—	13	1	no	—
52	C₂×M₆₄	D₁₃	1	1	yes	characteristic
53	C₂×C₄×C₁₆	—	13	1	no	—
54	#1397	—	13	2	no	—
55	C₄×C₃₂	—	13	2	no	—
56	C₂³×C₁₃	C₂×C₁₆	1	1	yes	characteristic
57	C₂×C₄×C₁₃	C₄×C₈	1	2	yes	—
58	C₂×C₄×C₁₃	C₂×C₁₆	1	1	yes	characteristic
59	C₂×C₄×C₁₃	C₂×C₁₆	1	2	yes	—
60	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
61	C₂×Dic₁₃	—	2	4	no	—
62	C₂×Dic₁₃	C₂×C₁₆	1	2	yes	—
63	C₂×Dic₁₃	C₂×C₁₆	1	2	yes	—
64	C₈×C₁₃	C₄×C₈	1	2	yes	—
65	C₈×C₁₃	C₂×C₄²	1	1	yes	characteristic
66	C₈×C₁₃	C₂²×C₈	1	1	yes	characteristic
67	#1055	C₂×C₁₆	1	2	yes	—
68	#1055	C₂×C₁₆	1	2	yes	—
69	C₂²×C₁₆	—	13	1	no	—
70	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
71	C₂×C₄×C₈	—	13	1	no	—
72	M₆₄	—	13	4	no	—
73	M₆₄	Dic₁₃	1	4	yes	direct product factor
74	C₂×C₃₂	—	13	2	no	—
75	C₂×C₃₂	C₂×D₁₃	1	2	yes	—
76	C₄×C₁₆	—	13	1	no	—
77	C₄×C₁₆	—	13	1	no	—
78	C₄×C₁₆	—	13	1	no	—
79	C₄×C₁₆	—	13	1	no	—
80	C₂²×C₁₃	—	2	2	no	—
81	C₂²×C₁₃	C₄×C₁₆	1	2	yes	—
82	C₂²×C₁₃	C₂²×C₁₆	1	1	yes	characteristic
83	C₄×C₁₃	C₄×C₁₆	1	2	yes	—
84	C₄×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
85	C₄×C₁₃	C₂²×C₁₆	1	1	yes	characteristic
86	Dic₁₃	—	2	2	no	—
87	Dic₁₃	M₆₄	1	4	yes	direct product factor
88	C₂²×C₈	—	13	1	no	—
89	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
90	C₂×C₄²	—	13	1	no	—
91	C₂×C₁₆	—	13	4	no	—
92	C₂×C₁₆	—	13	1	no	—
93	C₂×C₁₆	—	13	1	no	—
94	C₂×C₁₆	C₄×D₁₃	1	1	yes	characteristic
95	C₂×C₁₆	C₄×D₁₃	1	2	yes	—
96	C₂×C₁₆	C₂×Dic₁₃	1	2	yes	—
97	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
98	C₄×C₈	—	13	1	no	—
99	C₄×C₈	—	13	1	no	—
100	C₄×C₈	—	13	1	no	—
101	C₄×C₈	—	13	1	no	—
102	C₃₂	—	13	4	no	—
103	C₃₂	C₂×Dic₁₃	1	4	yes	complemented
104	C₂×C₁₃	—	2	2	no	—
105	C₂×C₁₃	C₂×C₄×C₁₆	1	1	yes	characteristic
106	C₂×C₁₃	C₂×M₆₄	1	2	yes	—
107	C₂²×C₄	—	13	2	no	—
108	C₂²×C₄	C₈×D₁₃	1	1	yes	characteristic
109	C₂×C₈	—	13	1	no	—
110	C₂×C₈	—	13	1	no	—
111	C₂×C₈	—	13	4	no	—
112	C₂×C₈	C₈×D₁₃	1	2	yes	—
113	C₂×C₈	C₈×D₁₃	1	1	yes	characteristic
114	C₂×C₈	C₄×Dic₁₃	1	2	yes	—
115	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
116	C₄²	—	13	1	no	—
117	C₄²	—	13	1	no	—
118	C₄²	—	13	2	no	—
119	C₁₆	—	13	2	no	—
120	C₁₆	—	13	2	no	—
121	C₁₆	C₄×Dic₁₃	1	2	yes	—
122	C₁₆	C₂×C₄×D₁₃	1	1	yes	characteristic
123	C₁₆	C₂²×Dic₁₃	1	1	yes	characteristic
124	C₁₃	C₄×M₆₄	1	1	yes	characteristic, complemented, Sylow
125	C₂³	C₁₆×D₁₃	1	1	yes	characteristic
126	C₂×C₄	—	13	2	no	—
127	C₂×C₄	—	13	2	no	—
128	C₂×C₄	—	26	4	no	—
129	C₂×C₄	C₁₆×D₁₃	1	2	yes	—
130	C₂×C₄	C₁₆×D₁₃	1	1	yes	characteristic
131	C₂×C₄	C₈×Dic₁₃	1	2	yes	—
132	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
133	C₈	—	13	2	no	—
134	C₈	—	13	2	no	—
135	C₈	C₈×Dic₁₃	1	2	yes	—
136	C₈	C₂×C₈×D₁₃	1	1	yes	characteristic
137	C₈	C₂×C₄×Dic₁₃	1	1	yes	characteristic
138	C₂²	—	2	2	no	—
139	C₂²	C₁₆×Dic₁₃	1	2	yes	—
140	C₂²	C₂×C₁₆×D₁₃	1	1	yes	characteristic
141	C₄	—	13	4	no	—
142	C₄	—	26	2	no	—
143	C₄	C₁₆×Dic₁₃	1	2	yes	—
144	C₄	C₂×C₁₆×D₁₃	1	1	yes	characteristic
145	C₄	C₂×C₈×Dic₁₃	1	1	yes	characteristic
146	C₂	—	2	2	no	—
147	C₂	C₂×C₁₆×Dic₁₃	1	1	yes	characteristic
148	C₂	D₁₃×M₆₄	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂²×D₁₃
commutator	C₂×C₁₃	C₂×C₄×C₁₆
Fitting	C₂×C₁₃×M₆₄	C₂
socle	C₂²×C₁₃	C₂²×C₁₆
prime radical	C₂³×C₁₃	C₂×C₁₆
cosocle	C₂×C₁₃×C₁₆	C₂³
hypercenter	C₂×M₆₄	D₁₃
hypocenter	C₁₃	C₄×M₆₄
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂×M₆₄

Complements

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

N	H ≅ G/N	type
3. #13112969	C₂	semidirect
4. C₃₂×Dic₁₃	C₂	semidirect
9. C₁₃×M₆₄	C₄	semidirect
15. #11718091	C₄	semidirect
31. C₁₃×C₃₂	C₂×C₄	semidirect
32. #88034	C₂×C₄	semidirect
73. M₆₄	Dic₁₃	direct (N × H)
87. Dic₁₃	M₆₄	direct (N × H)
103. C₃₂	C₂×Dic₁₃	semidirect
124. C₁₃	C₄×M₆₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718091	2
C₂×C₁₆×Dic₁₃	2
#13112969	2
C₃₂×Dic₁₃	2
C₂×C₁₃×M₆₄	2
C₄×M₆₄	13

Subgroups — Dic13×M64 #15783094

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Dic₁₃×M₆₄ #15783094