Subgroups — C₂×M₆₄×D₅₃ #27542196

Order 13568.

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×D₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810457	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16810456	C₂	1	2	yes	maximal, complemented
4	M₆₄×D₅₃	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃₂×D₅₃	C₂	1	2	yes	maximal, complemented
6	C₂×C₅₃×M₆₄	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₈×D₅₃	C₄	1	1	yes	characteristic
8	C₂²×#11719908	C₄	1	1	yes	characteristic
9	C₂×C₁₆×D₅₃	C₄	1	4	yes	—
10	C₂×C₁₆×D₅₃	C₄	1	1	yes	characteristic
11	C₂×C₁₆×D₅₃	C₄	1	1	yes	characteristic
12	C₂×C₁₆×D₅₃	C₂²	1	1	yes	characteristic
13	C₂×C₁₆×D₅₃	C₂²	1	4	yes	—
14	C₂×C₁₆×D₅₃	C₂²	1	1	yes	characteristic
15	#16810457	C₂²	1	4	yes	complemented
16	#16810456	C₂²	1	8	yes	complemented
17	C₂×#13133123	C₂²	1	2	yes	complemented
18	C₃₂×D₅₃	C₂²	1	8	yes	complemented
19	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
20	C₅₃×M₆₄	C₂²	1	4	yes	complemented
21	C₂×C₃₂×C₅₃	C₂²	1	2	yes	complemented
22	C₂²×C₄×D₅₃	C₈	1	1	yes	characteristic
23	C₂²×#88322	C₈	1	1	yes	characteristic
24	C₂×C₈×D₅₃	C₈	1	1	yes	characteristic
25	C₂×C₈×D₅₃	C₈	1	4	yes	—
26	C₂×C₈×D₅₃	C₈	1	1	yes	characteristic
27	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈×D₅₃	C₂×C₄	1	4	yes	—
29	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂×#11719908	C₂×C₄	1	1	yes	characteristic
31	C₂×#11719908	C₂×C₄	1	4	yes	—
32	C₂×#11719908	C₂×C₄	1	1	yes	characteristic
33	C₁₆×D₅₃	C₂×C₄	1	2	yes	—
34	C₁₆×D₅₃	C₂×C₄	1	4	yes	—
35	C₁₆×D₅₃	C₂×C₄	1	4	yes	—
36	C₁₆×D₅₃	C₂×C₄	1	2	yes	—
37	C₁₆×D₅₃	C₂³	1	2	yes	—
38	C₁₆×D₅₃	C₂³	1	2	yes	—
39	#13133123	C₂³	1	4	yes	complemented
40	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
43	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
44	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
45	C₃₂×C₅₃	C₂³	1	4	yes	complemented
46	C₂³×D₅₃	C₁₆	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
48	C₂²×Dic₅₃	C₁₆	1	1	yes	characteristic
49	C₂×C₄×D₅₃	C₁₆	1	1	yes	characteristic
50	C₂×C₄×D₅₃	C₁₆	1	4	yes	—
51	C₂×C₄×D₅₃	C₁₆	1	1	yes	characteristic
52	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
53	C₂×C₄×D₅₃	C₂×C₈	1	4	yes	—
54	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
55	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
56	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
57	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
58	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
59	C₂×#88322	C₂×C₈	1	4	yes	—
60	C₂×#88322	C₂×C₈	1	1	yes	characteristic
61	C₂×#88322	C₂×C₈	1	1	yes	characteristic
62	C₈×D₅₃	C₂×C₈	1	4	yes	—
63	C₈×D₅₃	C₂×C₈	1	2	yes	—
64	C₈×D₅₃	C₂×C₈	1	2	yes	—
65	C₈×D₅₃	C₂×C₈	1	4	yes	—
66	C₈×D₅₃	C₂²×C₄	1	2	yes	—
67	C₈×D₅₃	C₂²×C₄	1	2	yes	—
68	C₁₆×C₅₃	C₂²×C₄	1	2	yes	—
69	C₁₆×C₅₃	C₂²×C₄	1	1	yes	characteristic
70	C₁₆×C₅₃	C₂⁴	1	1	yes	characteristic
71	#11719908	C₂²×C₄	1	2	yes	—
72	#11719908	C₂²×C₄	1	2	yes	—
73	C₂³×C₅₃	C₂×C₁₆	1	1	yes	characteristic
74	C₂²×D₅₃	—	2	4	no	—
75	C₂²×D₅₃	C₂×C₁₆	1	1	yes	characteristic
76	C₂²×D₅₃	C₂×C₁₆	1	4	yes	—
77	C₂²×D₅₃	C₂×C₁₆	1	1	yes	characteristic
78	C₂×C₄×C₅₃	C₂×C₁₆	1	2	yes	—
79	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
80	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
81	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
82	C₂×Dic₅₃	C₂×C₁₆	1	4	yes	—
83	C₂×Dic₅₃	C₂×C₁₆	1	1	yes	characteristic
84	C₂×Dic₅₃	C₂×C₁₆	1	1	yes	characteristic
85	C₄×D₅₃	C₂×C₁₆	1	4	yes	—
86	C₄×D₅₃	C₂×C₁₆	1	2	yes	—
87	C₄×D₅₃	C₂×C₁₆	1	2	yes	—
88	C₄×D₅₃	C₂×C₁₆	1	4	yes	—
89	C₄×D₅₃	C₂²×C₈	1	2	yes	—
90	C₄×D₅₃	C₂²×C₈	1	2	yes	—
91	C₈×C₅₃	C₂²×C₈	1	2	yes	—
92	C₈×C₅₃	C₂²×C₈	1	1	yes	characteristic
93	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
94	#88322	C₂²×C₈	1	2	yes	—
95	#88322	C₂²×C₈	1	2	yes	—
96	C₂²×M₆₄	—	53	1	no	maximal, Sylow
97	C₂²×C₅₃	—	2	2	no	—
98	C₂²×C₅₃	C₂²×C₁₆	1	2	yes	—
99	C₂²×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
100	C₂×D₅₃	—	2	2	no	—
101	C₂×D₅₃	—	2	8	no	—
102	C₂×D₅₃	M₆₄	1	4	yes	direct product factor
103	C₂×D₅₃	C₂²×C₁₆	1	2	yes	—
104	C₂×D₅₃	C₂²×C₁₆	1	2	yes	—
105	C₄×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
106	C₄×C₅₃	C₂²×C₁₆	1	2	yes	—
107	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
108	Dic₅₃	C₂²×C₁₆	1	2	yes	—
109	Dic₅₃	C₂²×C₁₆	1	2	yes	—
110	C₂³×C₁₆	—	53	1	no	—
111	C₂×M₆₄	—	53	1	no	—
112	C₂×M₆₄	—	53	8	no	—
113	C₂×M₆₄	—	53	2	no	—
114	C₂×M₆₄	D₅₃	1	1	yes	characteristic, direct product factor
115	C₂²×C₃₂	—	53	2	no	—
116	C₂×C₅₃	—	2	2	no	—
117	C₂×C₅₃	C₂×M₆₄	1	2	yes	complemented
118	C₂×C₅₃	C₂³×C₁₆	1	1	yes	characteristic
119	D₅₃	—	2	2	no	—
120	D₅₃	C₂×M₆₄	1	4	yes	direct product factor
121	C₂³×C₈	—	53	1	no	—
122	C₂²×C₁₆	—	53	1	no	—
123	C₂²×C₁₆	—	53	4	no	—
124	C₂²×C₁₆	—	53	1	no	—
125	C₂²×C₁₆	—	53	1	no	—
126	C₂²×C₁₆	—	53	4	no	—
127	C₂²×C₁₆	—	53	1	no	—
128	C₂²×C₁₆	—	53	1	no	—
129	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
130	M₆₄	—	53	4	no	—
131	M₆₄	—	53	8	no	—
132	M₆₄	C₂×D₅₃	1	4	yes	direct product factor
133	C₂×C₃₂	—	53	2	no	—
134	C₂×C₃₂	—	53	8	no	—
135	C₂×C₃₂	C₂×D₅₃	1	2	yes	complemented
136	C₅₃	C₂²×M₆₄	1	1	yes	characteristic, complemented, Sylow
137	C₂³×C₄	—	53	1	no	—
138	C₂²×C₈	—	53	1	no	—
139	C₂²×C₈	—	53	4	no	—
140	C₂²×C₈	—	53	1	no	—
141	C₂²×C₈	—	53	1	no	—
142	C₂²×C₈	—	53	1	no	—
143	C₂²×C₈	—	53	4	no	—
144	C₂²×C₈	—	53	1	no	—
145	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
146	C₂×C₁₆	—	53	4	no	—
147	C₂×C₁₆	—	53	1	no	—
148	C₂×C₁₆	—	53	1	no	—
149	C₂×C₁₆	—	53	4	no	—
150	C₂×C₁₆	—	53	4	no	—
151	C₂×C₁₆	—	53	2	no	—
152	C₂×C₁₆	—	53	2	no	—
153	C₂×C₁₆	—	53	2	no	—
154	C₂×C₁₆	—	53	2	no	—
155	C₂×C₁₆	C₄×D₅₃	1	1	yes	characteristic
156	C₂×C₁₆	C₄×D₅₃	1	2	yes	—
157	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
158	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
159	C₃₂	—	53	4	no	—
160	C₃₂	C₂²×D₅₃	1	4	yes	complemented
161	C₂⁴	—	53	1	no	—
162	C₂²×C₄	—	53	1	no	—
163	C₂²×C₄	—	53	1	no	—
164	C₂²×C₄	—	53	4	no	—
165	C₂²×C₄	—	53	1	no	—
166	C₂²×C₄	—	53	1	no	—
167	C₂²×C₄	—	53	4	no	—
168	C₂²×C₄	—	53	1	no	—
169	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
170	C₂×C₈	—	53	4	no	—
171	C₂×C₈	—	53	1	no	—
172	C₂×C₈	—	53	1	no	—
173	C₂×C₈	—	53	4	no	—
174	C₂×C₈	—	53	4	no	—
175	C₂×C₈	—	53	2	no	—
176	C₂×C₈	—	53	2	no	—
177	C₂×C₈	—	53	2	no	—
178	C₂×C₈	—	53	2	no	—
179	C₂×C₈	C₈×D₅₃	1	1	yes	characteristic
180	C₂×C₈	C₈×D₅₃	1	2	yes	—
181	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
182	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
183	C₁₆	—	53	2	no	—
184	C₁₆	—	53	2	no	—
185	C₁₆	C₂×C₄×D₅₃	1	2	yes	—
186	C₁₆	C₂×C₄×D₅₃	1	1	yes	characteristic
187	C₁₆	C₂³×D₅₃	1	1	yes	characteristic
188	C₂³	—	53	1	no	—
189	C₂³	—	53	4	no	—
190	C₂³	—	53	1	no	—
191	C₂³	—	106	4	no	—
192	C₂³	C₁₆×D₅₃	1	1	yes	characteristic
193	C₂×C₄	—	53	4	no	—
194	C₂×C₄	—	53	1	no	—
195	C₂×C₄	—	53	1	no	—
196	C₂×C₄	—	53	4	no	—
197	C₂×C₄	—	53	2	no	—
198	C₂×C₄	—	53	2	no	—
199	C₂×C₄	—	53	4	no	—
200	C₂×C₄	—	53	2	no	—
201	C₂×C₄	—	53	2	no	—
202	C₂×C₄	C₁₆×D₅₃	1	2	yes	—
203	C₂×C₄	C₁₆×D₅₃	1	1	yes	characteristic
204	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
205	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
206	C₈	—	53	2	no	—
207	C₈	—	53	2	no	—
208	C₈	C₂×C₈×D₅₃	1	2	yes	—
209	C₈	C₂×C₈×D₅₃	1	1	yes	characteristic
210	C₈	C₂²×C₄×D₅₃	1	1	yes	characteristic
211	C₂²	—	2	2	no	—
212	C₂²	—	53	4	no	—
213	C₂²	—	53	2	no	—
214	C₂²	—	53	2	no	—
215	C₂²	—	106	8	no	—
216	C₂²	—	106	2	no	—
217	C₂²	C₂×C₁₆×D₅₃	1	2	yes	—
218	C₂²	C₂×C₁₆×D₅₃	1	1	yes	characteristic
219	C₄	—	53	2	no	—
220	C₄	—	53	2	no	—
221	C₄	C₂×C₁₆×D₅₃	1	1	yes	characteristic
222	C₄	C₂×C₁₆×D₅₃	1	2	yes	—
223	C₄	C₂²×C₈×D₅₃	1	1	yes	characteristic
224	C₂	—	2	2	no	—
225	C₂	—	53	4	no	—
226	C₂	—	106	2	no	—
227	C₂	M₆₄×D₅₃	1	2	yes	direct product factor
228	C₂	C₂²×C₁₆×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₁₆	C₂²×D₅₃
commutator	C₂×C₅₃	C₂³×C₁₆
Frattini	C₁₆	C₂³×D₅₃
Fitting	C₂×C₅₃×M₆₄	C₂
socle	C₂²×C₅₃	C₂²×C₁₆
prime radical	C₂³×D₅₃	C₁₆
cosocle	C₁₆×C₅₃	C₂⁴
hypercenter	C₂×M₆₄	D₅₃
hypocenter	C₅₃	C₂²×M₆₄
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂×M₆₄

Complements

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

N	H ≅ G/N	type
2. C₂×#16810457	C₂	semidirect
3. C₂×#16810456	C₂	semidirect
4. M₆₄×D₅₃	C₂	direct (N × H)
5. C₂×C₃₂×D₅₃	C₂	semidirect
6. C₂×C₅₃×M₆₄	C₂	semidirect
15. #16810457	C₂²	semidirect
16. #16810456	C₂²	semidirect
17. C₂×#13133123	C₂²	semidirect
18. C₃₂×D₅₃	C₂²	semidirect
20. C₅₃×M₆₄	C₂²	semidirect
21. C₂×C₃₂×C₅₃	C₂²	semidirect
39. #13133123	C₂³	semidirect
45. C₃₂×C₅₃	C₂³	semidirect
102. C₂×D₅₃	M₆₄	direct (N × H)
114. C₂×M₆₄	D₅₃	direct (N × H)
117. C₂×C₅₃	C₂×M₆₄	semidirect
120. D₅₃	C₂×M₆₄	direct (N × H)
132. M₆₄	C₂×D₅₃	direct (N × H)
135. C₂×C₃₂	C₂×D₅₃	semidirect
136. C₅₃	C₂²×M₆₄	semidirect
160. C₃₂	C₂²×D₅₃	semidirect
227. C₂	M₆₄×D₅₃	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×D₅₃	2
C₂×#16810457	2
C₂×#16810456	2
M₆₄×D₅₃	2
C₂×C₃₂×D₅₃	2
C₂×C₅₃×M₆₄	2
C₂²×M₆₄	53

Subgroups — C2×M64×D53 #27542196

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×M₆₄×D₅₃ #27542196