Subgroups — C₂×M₃₂×Dic₅₃ #27536721

Order 13568.

918 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133157	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	M₃₂×Dic₅₃	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16810398	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₆×Dic₅₃	C₂	1	2	yes	maximal, complemented
6	C₂²×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂³×#88322	C₄	1	1	yes	characteristic
8	C₂²×C₄×Dic₅₃	C₄	1	1	yes	characteristic
9	C₂×#13133157	C₄	1	8	yes	complemented
10	C₂×#13133157	C₂²	1	4	yes	—
11	C₂²×#11719908	C₂²	1	2	yes	—
12	C₂×C₈×Dic₅₃	C₄	1	4	yes	—
13	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
14	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
15	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
17	C₂×C₈×Dic₅₃	C₂²	1	4	yes	—
18	#16810398	C₂²	1	8	yes	complemented
19	C₁₆×Dic₅₃	C₂²	1	8	yes	complemented
20	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×M₃₂×C₅₃	C₄	1	8	yes	complemented
22	C₂×M₃₂×C₅₃	C₂²	1	4	yes	—
23	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
24	C₂³×Dic₅₃	C₈	1	2	yes	—
25	C₂²×#88322	C₈	1	8	yes	—
26	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
27	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
28	C₂²×#88322	C₂×C₄	1	4	yes	—
29	C₂×C₄×Dic₅₃	C₈	1	4	yes	—
30	C₂×C₄×Dic₅₃	C₈	1	2	yes	—
31	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×Dic₅₃	C₂×C₄	1	4	yes	—
33	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
34	#13133157	C₂×C₄	1	16	yes	complemented
35	C₂×#11719908	C₂×C₄	1	8	yes	complemented
36	C₂×#11719908	C₂³	1	4	yes	—
37	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
38	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
39	C₈×Dic₅₃	C₂×C₄	1	4	yes	—
40	C₈×Dic₅₃	C₂×C₄	1	4	yes	—
41	C₈×Dic₅₃	C₂³	1	2	yes	—
42	C₈×Dic₅₃	C₂³	1	2	yes	—
43	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
45	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
46	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
47	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
49	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
50	M₃₂×C₅₃	C₂×C₄	1	16	yes	complemented
51	C₂×C₁₆×C₅₃	C₂×C₄	1	8	yes	complemented
52	C₂×C₁₆×C₅₃	C₂³	1	4	yes	—
53	C₂⁴×C₅₃	C₂×C₈	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	C₄²	1	4	yes	—
55	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
56	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	C₂×C₈	1	4	yes	—
58	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
60	C₂²×Dic₅₃	—	2	8	no	—
61	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
62	C₂²×Dic₅₃	C₂×C₈	1	8	yes	—
63	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
64	C₂×C₈×C₅₃	C₄²	1	8	yes	—
65	C₂×C₈×C₅₃	C₄²	1	4	yes	—
66	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
67	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
69	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
70	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
71	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
72	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
73	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
74	C₄×Dic₅₃	C₂×C₈	1	8	yes	—
75	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
76	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
77	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
78	C₂×#88322	C₂×C₈	1	16	yes	—
79	C₂×#88322	C₂×C₈	1	4	yes	—
80	C₂×#88322	C₂×C₈	1	4	yes	—
81	C₂×#88322	C₂²×C₄	1	2	yes	—
82	C₂×#88322	C₂²×C₄	1	2	yes	—
83	C₁₆×C₅₃	C₂²×C₄	1	8	yes	complemented
84	#11719908	C₂²×C₄	1	8	yes	complemented
85	C₂³×C₅₃	—	2	4	no	—
86	C₂³×C₅₃	C₄×C₈	1	4	yes	—
87	C₂³×C₅₃	C₂²×C₈	1	2	yes	—
88	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
89	C₂×C₄×C₅₃	C₄×C₈	1	8	yes	—
90	C₂×C₄×C₅₃	C₄×C₈	1	4	yes	—
91	C₂×C₄×C₅₃	C₂×C₄²	1	4	yes	—
92	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
93	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
94	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
95	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
96	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
97	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
98	C₂×Dic₅₃	—	2	4	no	—
99	C₂×Dic₅₃	—	2	16	no	—
100	C₂×Dic₅₃	M₃₂	1	8	yes	direct product factor
101	C₂×Dic₅₃	C₂²×C₈	1	4	yes	—
102	C₂×Dic₅₃	C₂²×C₈	1	4	yes	—
103	C₈×C₅₃	C₂×C₄²	1	2	yes	—
104	C₈×C₅₃	C₂×C₄²	1	4	yes	—
105	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
106	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
107	#88322	C₂²×C₈	1	4	yes	—
108	#88322	C₂²×C₈	1	4	yes	—
109	C₂×C₄×M₃₂	—	53	1	no	maximal, Sylow
110	C₂²×C₅₃	—	2	8	no	—
111	C₂²×C₅₃	—	2	4	no	—
112	C₂²×C₅₃	C₂×C₄×C₈	1	4	yes	—
113	C₂²×C₅₃	C₂×C₄×C₈	1	2	yes	—
114	C₂²×C₅₃	C₂×M₃₂	1	4	yes	—
115	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
116	C₄×C₅₃	C₂×C₄×C₈	1	2	yes	—
117	C₄×C₅₃	C₂×C₄×C₈	1	4	yes	—
118	C₄×C₅₃	C₂²×C₄²	1	1	yes	characteristic
119	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
120	Dic₅₃	—	2	4	no	—
121	Dic₅₃	C₂×M₃₂	1	8	yes	direct product factor
122	C₂²×M₃₂	—	53	1	no	—
123	C₂²×M₃₂	D₅₃	1	1	yes	characteristic
124	C₂²×C₄×C₈	—	53	1	no	—
125	C₄×M₃₂	—	53	8	no	—
126	C₂×#346	—	53	2	no	—
127	C₂×C₄×C₁₆	—	53	2	no	—
128	C₂×C₅₃	—	2	4	no	—
129	C₂×C₅₃	C₄×M₃₂	1	4	yes	complemented
130	C₂×C₅₃	C₂²×C₄×C₈	1	1	yes	characteristic
131	C₂×C₅₃	C₂²×M₃₂	1	2	yes	—
132	C₂³×C₈	—	53	1	no	—
133	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
134	C₂²×C₄²	—	53	1	no	—
135	C₂×M₃₂	—	53	8	no	—
136	C₂×M₃₂	—	53	4	no	—
137	C₂×M₃₂	Dic₅₃	1	8	yes	direct product factor
138	C₂×M₃₂	C₂×D₅₃	1	4	yes	—
139	C₂²×C₁₆	—	53	2	no	—
140	C₂²×C₁₆	C₂×D₅₃	1	2	yes	—
141	C₂×C₄×C₈	—	53	1	no	—
142	C₂×C₄×C₈	—	53	1	no	—
143	C₂×C₄×C₈	—	53	4	no	—
144	C₂×C₄×C₈	—	53	4	no	—
145	C₂×C₄×C₈	—	53	1	no	—
146	C₂×C₄×C₈	—	53	1	no	—
147	#346	—	53	8	no	—
148	C₄×C₁₆	—	53	8	no	—
149	C₅₃	C₂×C₄×M₃₂	1	1	yes	characteristic, complemented, Sylow
150	C₂³×C₄	—	53	2	no	—
151	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
152	C₂²×C₈	—	53	4	no	—
153	C₂²×C₈	—	53	1	no	—
154	C₂²×C₈	—	53	1	no	—
155	C₂²×C₈	—	53	8	no	—
156	C₂²×C₈	C₄×D₅₃	1	4	yes	—
157	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
158	C₂²×C₈	C₄×D₅₃	1	2	yes	—
159	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
160	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
161	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
162	C₂×C₄²	—	53	1	no	—
163	C₂×C₄²	—	53	4	no	—
164	C₂×C₄²	—	53	1	no	—
165	C₂×C₄²	—	53	2	no	—
166	C₂×C₄²	—	53	4	no	—
167	M₃₂	—	53	16	no	—
168	M₃₂	C₂×Dic₅₃	1	16	yes	direct product factor
169	C₂×C₁₆	—	53	8	no	—
170	C₂×C₁₆	—	53	4	no	—
171	C₂×C₁₆	C₂×Dic₅₃	1	8	yes	complemented
172	C₂×C₁₆	C₂²×D₅₃	1	4	yes	—
173	C₄×C₈	—	53	4	no	—
174	C₄×C₈	—	53	4	no	—
175	C₄×C₈	—	53	2	no	—
176	C₄×C₈	—	53	2	no	—
177	C₄×C₈	—	53	2	no	—
178	C₄×C₈	—	53	2	no	—
179	C₂⁴	C₈×D₅₃	1	1	yes	characteristic
180	C₂²×C₄	—	53	2	no	—
181	C₂²×C₄	—	53	8	no	—
182	C₂²×C₄	—	53	2	no	—
183	C₂²×C₄	—	106	8	no	—
184	C₂²×C₄	C₈×D₅₃	1	2	yes	—
185	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
186	C₂²×C₄	C₈×D₅₃	1	4	yes	—
187	C₂²×C₄	C₄×Dic₅₃	1	4	yes	—
188	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
189	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
190	C₂×C₈	—	53	16	no	—
191	C₂×C₈	—	53	4	no	—
192	C₂×C₈	—	53	4	no	—
193	C₂×C₈	—	53	2	no	—
194	C₂×C₈	—	53	2	no	—
195	C₂×C₈	C₄×Dic₅₃	1	4	yes	—
196	C₂×C₈	C₄×Dic₅₃	1	8	yes	—
197	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
198	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
199	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
200	C₂×C₈	C₂×C₄×D₅₃	1	4	yes	—
201	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
202	C₂×C₈	C₂²×Dic₅₃	1	4	yes	—
203	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
204	C₄²	—	53	2	no	—
205	C₄²	—	53	8	no	—
206	C₄²	—	53	2	no	—
207	C₄²	—	53	2	no	—
208	C₄²	—	53	2	no	—
209	C₁₆	—	53	8	no	—
210	C₁₆	C₂²×Dic₅₃	1	8	yes	complemented
211	C₂³	—	2	4	no	—
212	C₂³	C₈×Dic₅₃	1	4	yes	—
213	C₂³	C₂×C₈×D₅₃	1	2	yes	—
214	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
215	C₂×C₄	—	53	8	no	—
216	C₂×C₄	—	53	4	no	—
217	C₂×C₄	—	53	4	no	—
218	C₂×C₄	—	106	16	no	—
219	C₂×C₄	—	106	4	no	—
220	C₂×C₄	C₈×Dic₅₃	1	8	yes	—
221	C₂×C₄	C₈×Dic₅₃	1	4	yes	—
222	C₂×C₄	C₂×C₄×Dic₅₃	1	4	yes	—
223	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
224	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
225	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
226	C₂×C₄	C₂×C₈×D₅₃	1	4	yes	—
227	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
228	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
229	C₈	—	53	4	no	—
230	C₈	—	53	4	no	—
231	C₈	C₂×C₄×Dic₅₃	1	4	yes	—
232	C₈	C₂×C₄×Dic₅₃	1	2	yes	—
233	C₈	C₂²×C₄×D₅₃	1	1	yes	characteristic
234	C₈	C₂³×Dic₅₃	1	1	yes	characteristic
235	C₂²	—	2	4	no	—
236	C₂²	—	2	8	no	—
237	C₂²	C₂×C₈×Dic₅₃	1	4	yes	—
238	C₂²	C₂×C₈×Dic₅₃	1	2	yes	—
239	C₂²	M₃₂×D₅₃	1	4	yes	—
240	C₂²	C₂²×C₈×D₅₃	1	1	yes	characteristic
241	C₄	—	53	8	no	—
242	C₄	—	106	4	no	—
243	C₄	C₂×C₈×Dic₅₃	1	2	yes	—
244	C₄	C₂×C₈×Dic₅₃	1	4	yes	—
245	C₄	C₂²×C₄×Dic₅₃	1	1	yes	characteristic
246	C₄	C₂²×C₈×D₅₃	1	1	yes	characteristic
247	C₂	—	2	4	no	—
248	C₂	M₃₂×Dic₅₃	1	4	yes	direct product factor
249	C₂	C₂²×C₈×Dic₅₃	1	1	yes	characteristic
250	C₂	C₂×M₃₂×D₅₃	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈	C₂²×D₅₃
commutator	C₂×C₅₃	C₂²×C₄×C₈
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂²×M₃₂×C₅₃	C₂
socle	C₂³×C₅₃	C₂²×C₈
prime radical	C₂⁴×C₅₃	C₂×C₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂²×M₃₂	D₅₃
hypocenter	C₅₃	C₂×C₄×M₃₂
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
3. M₃₂×Dic₅₃	C₂	direct (N × H)
4. C₂×#16810398	C₂	semidirect
5. C₂×C₁₆×Dic₅₃	C₂	semidirect
9. C₂×#13133157	C₄	semidirect
18. #16810398	C₂²	semidirect
19. C₁₆×Dic₅₃	C₂²	semidirect
21. C₂×M₃₂×C₅₃	C₄	semidirect
34. #13133157	C₂×C₄	semidirect
35. C₂×#11719908	C₂×C₄	semidirect
50. M₃₂×C₅₃	C₂×C₄	semidirect
51. C₂×C₁₆×C₅₃	C₂×C₄	semidirect
83. C₁₆×C₅₃	C₂²×C₄	semidirect
84. #11719908	C₂²×C₄	semidirect
100. C₂×Dic₅₃	M₃₂	direct (N × H)
121. Dic₅₃	C₂×M₃₂	direct (N × H)
129. C₂×C₅₃	C₄×M₃₂	semidirect
137. C₂×M₃₂	Dic₅₃	direct (N × H)
149. C₅₃	C₂×C₄×M₃₂	semidirect
168. M₃₂	C₂×Dic₅₃	direct (N × H)
171. C₂×C₁₆	C₂×Dic₅₃	semidirect
210. C₁₆	C₂²×Dic₅₃	semidirect
248. C₂	M₃₂×Dic₅₃	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×#13133157	2
C₂²×C₈×Dic₅₃	2
M₃₂×Dic₅₃	2
C₂×#16810398	2
C₂×C₁₆×Dic₅₃	2
C₂²×M₃₂×C₅₃	2
C₂×C₄×M₃₂	53

Subgroups — C2×M32×Dic53 #27536721

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×M₃₂×Dic₅₃ #27536721