Subgroups — C₂×D₁₃×M₆₄ #16021023

Order 3328.

502 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₆×D₁₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718091	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#11718039	C₂	1	2	yes	maximal, complemented
4	D₁₃×M₆₄	C₂	1	8	yes	maximal, direct product factor
5	C₂×D₁₃×C₃₂	C₂	1	2	yes	maximal, complemented
6	C₂×C₁₃×M₆₄	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₈×D₁₃	C₄	1	1	yes	characteristic
8	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
9	C₂²×#6090	C₄	1	1	yes	characteristic
10	C₂×C₁₆×D₁₃	C₄	1	1	yes	characteristic
11	C₂×C₁₆×D₁₃	C₄	1	4	yes	—
12	C₂×C₁₆×D₁₃	C₄	1	1	yes	characteristic
13	C₂×C₁₆×D₁₃	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×D₁₃	C₂²	1	4	yes	—
15	C₂×C₁₆×D₁₃	C₂²	1	1	yes	characteristic
16	C₁₃×M₆₄	C₂²	1	4	yes	complemented
17	C₂×C₁₃×C₃₂	C₂²	1	2	yes	complemented
18	#11718091	C₂²	1	4	yes	complemented
19	C₂×#88034	C₂²	1	2	yes	complemented
20	#11718039	C₂²	1	8	yes	complemented
21	D₁₃×C₃₂	C₂²	1	8	yes	complemented
22	C₂²×C₄×D₁₃	C₈	1	1	yes	characteristic
23	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×#1055	C₈	1	1	yes	characteristic
25	C₂×C₈×D₁₃	C₈	1	1	yes	characteristic
26	C₂×C₈×D₁₃	C₈	1	4	yes	—
27	C₂×C₈×D₁₃	C₈	1	1	yes	characteristic
28	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₈×D₁₃	C₂×C₄	1	4	yes	—
30	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
32	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
33	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
34	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
35	C₂×#6090	C₂×C₄	1	1	yes	characteristic
36	C₂×#6090	C₂×C₄	1	4	yes	—
37	C₂×#6090	C₂×C₄	1	1	yes	characteristic
38	C₁₆×D₁₃	C₂×C₄	1	2	yes	—
39	C₁₆×D₁₃	C₂×C₄	1	4	yes	—
40	C₁₆×D₁₃	C₂×C₄	1	4	yes	—
41	C₁₆×D₁₃	C₂×C₄	1	2	yes	—
42	C₁₆×D₁₃	C₂³	1	2	yes	—
43	C₁₆×D₁₃	C₂³	1	2	yes	—
44	C₁₃×C₃₂	C₂³	1	4	yes	complemented
45	#88034	C₂³	1	4	yes	complemented
46	C₂²×M₆₄	—	13	1	no	maximal, Sylow
47	C₂³×D₁₃	C₁₆	1	1	yes	characteristic
48	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
49	C₂²×Dic₁₃	C₁₆	1	1	yes	characteristic
50	C₂×C₄×D₁₃	C₁₆	1	1	yes	characteristic
51	C₂×C₄×D₁₃	C₁₆	1	4	yes	—
52	C₂×C₄×D₁₃	C₁₆	1	1	yes	characteristic
53	C₂×C₄×D₁₃	C₂×C₈	1	1	yes	characteristic
54	C₂×C₄×D₁₃	C₂×C₈	1	4	yes	—
55	C₂×C₄×D₁₃	C₂×C₈	1	1	yes	characteristic
56	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
57	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
58	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
59	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
60	C₂×#1055	C₂×C₈	1	4	yes	—
61	C₂×#1055	C₂×C₈	1	1	yes	characteristic
62	C₂×#1055	C₂×C₈	1	1	yes	characteristic
63	C₈×D₁₃	C₂×C₈	1	4	yes	—
64	C₈×D₁₃	C₂×C₈	1	2	yes	—
65	C₈×D₁₃	C₂×C₈	1	2	yes	—
66	C₈×D₁₃	C₂×C₈	1	4	yes	—
67	C₈×D₁₃	C₂²×C₄	1	2	yes	—
68	C₈×D₁₃	C₂²×C₄	1	2	yes	—
69	C₁₃×C₁₆	C₂²×C₄	1	2	yes	—
70	C₁₃×C₁₆	C₂²×C₄	1	1	yes	characteristic
71	C₁₃×C₁₆	C₂⁴	1	1	yes	characteristic
72	#6090	C₂²×C₄	1	2	yes	—
73	#6090	C₂²×C₄	1	2	yes	—
74	C₂³×C₁₆	—	13	1	no	—
75	C₂×M₆₄	—	13	1	no	—
76	C₂×M₆₄	—	13	8	no	—
77	C₂×M₆₄	—	13	2	no	—
78	C₂×M₆₄	D₁₃	1	1	yes	characteristic, direct product factor
79	C₂²×C₃₂	—	13	2	no	—
80	C₂³×C₁₃	C₂×C₁₆	1	1	yes	characteristic
81	C₂²×D₁₃	—	2	4	no	—
82	C₂²×D₁₃	C₂×C₁₆	1	1	yes	characteristic
83	C₂²×D₁₃	C₂×C₁₆	1	4	yes	—
84	C₂²×D₁₃	C₂×C₁₆	1	1	yes	characteristic
85	C₂×C₄×C₁₃	C₂×C₁₆	1	2	yes	—
86	C₂×C₄×C₁₃	C₂×C₁₆	1	1	yes	characteristic
87	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
88	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
89	C₂×Dic₁₃	C₂×C₁₆	1	4	yes	—
90	C₂×Dic₁₃	C₂×C₁₆	1	1	yes	characteristic
91	C₂×Dic₁₃	C₂×C₁₆	1	1	yes	characteristic
92	C₄×D₁₃	C₂×C₁₆	1	4	yes	—
93	C₄×D₁₃	C₂×C₁₆	1	2	yes	—
94	C₄×D₁₃	C₂×C₁₆	1	2	yes	—
95	C₄×D₁₃	C₂×C₁₆	1	4	yes	—
96	C₄×D₁₃	C₂²×C₈	1	2	yes	—
97	C₄×D₁₃	C₂²×C₈	1	2	yes	—
98	C₈×C₁₃	C₂²×C₈	1	2	yes	—
99	C₈×C₁₃	C₂²×C₈	1	1	yes	characteristic
100	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
101	#1055	C₂²×C₈	1	2	yes	—
102	#1055	C₂²×C₈	1	2	yes	—
103	C₂³×C₈	—	13	1	no	—
104	C₂²×C₁₆	—	13	1	no	—
105	C₂²×C₁₆	—	13	4	no	—
106	C₂²×C₁₆	—	13	1	no	—
107	C₂²×C₁₆	—	13	1	no	—
108	C₂²×C₁₆	—	13	4	no	—
109	C₂²×C₁₆	—	13	1	no	—
110	C₂²×C₁₆	—	13	1	no	—
111	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
112	M₆₄	—	13	4	no	—
113	M₆₄	—	13	8	no	—
114	M₆₄	C₂×D₁₃	1	4	yes	direct product factor
115	C₂×C₃₂	—	13	2	no	—
116	C₂×C₃₂	—	13	8	no	—
117	C₂×C₃₂	C₂×D₁₃	1	2	yes	complemented
118	C₂²×C₁₃	—	2	2	no	—
119	C₂²×C₁₃	C₂²×C₁₆	1	2	yes	—
120	C₂²×C₁₃	C₂²×C₁₆	1	1	yes	characteristic
121	C₂×D₁₃	—	2	2	no	—
122	C₂×D₁₃	—	2	8	no	—
123	C₂×D₁₃	M₆₄	1	4	yes	direct product factor
124	C₂×D₁₃	C₂²×C₁₆	1	2	yes	—
125	C₂×D₁₃	C₂²×C₁₆	1	2	yes	—
126	C₄×C₁₃	C₂²×C₁₆	1	1	yes	characteristic
127	C₄×C₁₃	C₂²×C₁₆	1	2	yes	—
128	C₄×C₁₃	C₂³×C₈	1	1	yes	characteristic
129	Dic₁₃	C₂²×C₁₆	1	2	yes	—
130	Dic₁₃	C₂²×C₁₆	1	2	yes	—
131	C₂³×C₄	—	13	1	no	—
132	C₂²×C₈	—	13	1	no	—
133	C₂²×C₈	—	13	1	no	—
134	C₂²×C₈	—	13	4	no	—
135	C₂²×C₈	—	13	1	no	—
136	C₂²×C₈	—	13	4	no	—
137	C₂²×C₈	—	13	1	no	—
138	C₂²×C₈	—	13	1	no	—
139	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
140	C₂×C₁₆	—	13	4	no	—
141	C₂×C₁₆	—	13	1	no	—
142	C₂×C₁₆	—	13	1	no	—
143	C₂×C₁₆	—	13	4	no	—
144	C₂×C₁₆	—	13	4	no	—
145	C₂×C₁₆	—	13	2	no	—
146	C₂×C₁₆	—	13	2	no	—
147	C₂×C₁₆	—	13	2	no	—
148	C₂×C₁₆	—	13	2	no	—
149	C₂×C₁₆	C₄×D₁₃	1	1	yes	characteristic
150	C₂×C₁₆	C₄×D₁₃	1	2	yes	—
151	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
152	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
153	C₃₂	—	13	4	no	—
154	C₃₂	C₂²×D₁₃	1	4	yes	complemented
155	C₂×C₁₃	—	2	2	no	—
156	C₂×C₁₃	C₂×M₆₄	1	2	yes	complemented
157	C₂×C₁₃	C₂³×C₁₆	1	1	yes	characteristic
158	D₁₃	—	2	2	no	—
159	D₁₃	C₂×M₆₄	1	4	yes	direct product factor
160	C₂⁴	—	13	1	no	—
161	C₂²×C₄	—	13	1	no	—
162	C₂²×C₄	—	13	1	no	—
163	C₂²×C₄	—	13	4	no	—
164	C₂²×C₄	—	13	1	no	—
165	C₂²×C₄	—	13	1	no	—
166	C₂²×C₄	—	13	4	no	—
167	C₂²×C₄	—	13	1	no	—
168	C₂²×C₄	C₈×D₁₃	1	1	yes	characteristic
169	C₂×C₈	—	13	4	no	—
170	C₂×C₈	—	13	1	no	—
171	C₂×C₈	—	13	1	no	—
172	C₂×C₈	—	13	4	no	—
173	C₂×C₈	—	13	4	no	—
174	C₂×C₈	—	13	2	no	—
175	C₂×C₈	—	13	2	no	—
176	C₂×C₈	—	13	2	no	—
177	C₂×C₈	—	13	2	no	—
178	C₂×C₈	C₈×D₁₃	1	1	yes	characteristic
179	C₂×C₈	C₈×D₁₃	1	2	yes	—
180	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
181	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
182	C₁₆	—	13	2	no	—
183	C₁₆	—	13	2	no	—
184	C₁₆	C₂×C₄×D₁₃	1	2	yes	—
185	C₁₆	C₂×C₄×D₁₃	1	1	yes	characteristic
186	C₁₆	C₂³×D₁₃	1	1	yes	characteristic
187	C₁₃	C₂²×M₆₄	1	1	yes	characteristic, complemented, Sylow
188	C₂³	—	13	1	no	—
189	C₂³	—	13	4	no	—
190	C₂³	—	13	1	no	—
191	C₂³	—	26	4	no	—
192	C₂³	C₁₆×D₁₃	1	1	yes	characteristic
193	C₂×C₄	—	13	4	no	—
194	C₂×C₄	—	13	1	no	—
195	C₂×C₄	—	13	1	no	—
196	C₂×C₄	—	13	2	no	—
197	C₂×C₄	—	13	2	no	—
198	C₂×C₄	—	13	2	no	—
199	C₂×C₄	—	13	4	no	—
200	C₂×C₄	—	13	4	no	—
201	C₂×C₄	—	13	2	no	—
202	C₂×C₄	C₁₆×D₁₃	1	2	yes	—
203	C₂×C₄	C₁₆×D₁₃	1	1	yes	characteristic
204	C₂×C₄	C₂×C₈×D₁₃	1	2	yes	—
205	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
206	C₈	—	13	2	no	—
207	C₈	—	13	2	no	—
208	C₈	C₂×C₈×D₁₃	1	2	yes	—
209	C₈	C₂×C₈×D₁₃	1	1	yes	characteristic
210	C₈	C₂²×C₄×D₁₃	1	1	yes	characteristic
211	C₂²	—	2	2	no	—
212	C₂²	—	13	4	no	—
213	C₂²	—	13	2	no	—
214	C₂²	—	13	2	no	—
215	C₂²	—	26	8	no	—
216	C₂²	—	26	2	no	—
217	C₂²	C₂×C₁₆×D₁₃	1	2	yes	—
218	C₂²	C₂×C₁₆×D₁₃	1	1	yes	characteristic
219	C₄	—	13	2	no	—
220	C₄	—	13	2	no	—
221	C₄	C₂×C₁₆×D₁₃	1	2	yes	—
222	C₄	C₂×C₁₆×D₁₃	1	1	yes	characteristic
223	C₄	C₂²×C₈×D₁₃	1	1	yes	characteristic
224	C₂	—	2	2	no	—
225	C₂	—	13	4	no	—
226	C₂	—	26	2	no	—
227	C₂	D₁₃×M₆₄	1	2	yes	direct product factor
228	C₂	C₂²×C₁₆×D₁₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₁₆	C₂²×D₁₃
commutator	C₂×C₁₃	C₂³×C₁₆
Frattini	C₁₆	C₂³×D₁₃
Fitting	C₂×C₁₃×M₆₄	C₂
socle	C₂²×C₁₃	C₂²×C₁₆
prime radical	C₂³×D₁₃	C₁₆
cosocle	C₁₃×C₁₆	C₂⁴
hypercenter	C₂×M₆₄	D₁₃
hypocenter	C₁₃	C₂²×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
2. C₂×#11718091	C₂	semidirect
3. C₂×#11718039	C₂	semidirect
4. D₁₃×M₆₄	C₂	direct (N × H)
5. C₂×D₁₃×C₃₂	C₂	semidirect
6. C₂×C₁₃×M₆₄	C₂	semidirect
16. C₁₃×M₆₄	C₂²	semidirect
17. C₂×C₁₃×C₃₂	C₂²	semidirect
18. #11718091	C₂²	semidirect
19. C₂×#88034	C₂²	semidirect
20. #11718039	C₂²	semidirect
21. D₁₃×C₃₂	C₂²	semidirect
44. C₁₃×C₃₂	C₂³	semidirect
45. #88034	C₂³	semidirect
78. C₂×M₆₄	D₁₃	direct (N × H)
114. M₆₄	C₂×D₁₃	direct (N × H)
117. C₂×C₃₂	C₂×D₁₃	semidirect
123. C₂×D₁₃	M₆₄	direct (N × H)
154. C₃₂	C₂²×D₁₃	semidirect
156. C₂×C₁₃	C₂×M₆₄	semidirect
159. D₁₃	C₂×M₆₄	direct (N × H)
187. C₁₃	C₂²×M₆₄	semidirect
227. C₂	D₁₃×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
C₂²×C₁₆×D₁₃	2
C₂×#11718091	2
C₂×#11718039	2
D₁₃×M₆₄	2
C₂×D₁₃×C₃₂	2
C₂×C₁₃×M₆₄	2
C₂²×M₆₄	13

Subgroups — C2×D13×M64 #16021023

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₁₃×M₆₄ #16021023