Subgroups — C₂²×M₁₆×Dic₅₃ #28247478

Order 13568.

3950 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#11719917	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₄×Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×M₁₆×Dic₅₃	C₂	1	24	yes	maximal, direct product factor
4	C₂²×#13133127	C₂	1	2	yes	maximal, complemented
5	C₂²×C₈×Dic₅₃	C₂	1	2	yes	maximal, complemented
6	C₂³×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂⁴×Dic₅₃	C₄	1	2	yes	—
8	C₂²×#11719917	C₄	1	16	yes	complemented
9	C₂²×#11719917	C₂²	1	12	yes	—
10	C₂³×#88322	C₂²	1	2	yes	—
11	C₂²×C₄×Dic₅₃	C₄	1	12	yes	—
12	C₂²×C₄×Dic₅₃	C₄	1	2	yes	—
13	C₂²×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
14	C₂²×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
15	C₂²×C₄×Dic₅₃	C₂²	1	12	yes	—
16	M₁₆×Dic₅₃	C₂²	1	64	yes	direct product factor
17	C₂×#13133127	C₂²	1	24	yes	complemented
18	C₂×C₈×Dic₅₃	C₂²	1	24	yes	complemented
19	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
20	C₂²×M₁₆×C₅₃	C₄	1	16	yes	complemented
21	C₂²×M₁₆×C₅₃	C₂²	1	12	yes	—
22	C₂³×C₈×C₅₃	C₂²	1	2	yes	—
23	C₂³×Dic₅₃	—	2	16	no	—
24	C₂³×Dic₅₃	C₂×C₄	1	2	yes	—
25	C₂³×Dic₅₃	C₂×C₄	1	24	yes	—
26	C₂³×Dic₅₃	C₂×C₄	1	2	yes	—
27	C₂×#11719917	C₂×C₄	1	96	yes	complemented
28	C₂×#11719917	C₂³	1	16	yes	—
29	C₂²×#88322	C₂×C₄	1	16	yes	complemented
30	C₂²×#88322	C₂³	1	12	yes	—
31	C₂×C₄×Dic₅₃	C₂×C₄	1	6	yes	—
32	C₂×C₄×Dic₅₃	C₂×C₄	1	48	yes	—
33	C₂×C₄×Dic₅₃	C₂×C₄	1	24	yes	—
34	C₂×C₄×Dic₅₃	C₂×C₄	1	6	yes	—
35	C₂×C₄×Dic₅₃	C₂³	1	6	yes	—
36	C₂×C₄×Dic₅₃	C₂³	1	6	yes	—
37	C₂×C₄×Dic₅₃	C₂³	1	16	yes	—
38	#13133127	C₂³	1	32	yes	complemented
39	C₈×Dic₅₃	C₂³	1	32	yes	complemented
40	C₂⁵×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₂³×C₄×C₅₃	C₂×C₄	1	6	yes	—
42	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₂³×C₄×C₅₃	C₂×C₄	1	8	yes	—
44	C₂³×C₄×C₅₃	C₂×C₄	1	8	yes	—
45	C₂³×C₄×C₅₃	C₂³	1	6	yes	—
46	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
47	C₂×M₁₆×C₅₃	C₂×C₄	1	96	yes	complemented
48	C₂×M₁₆×C₅₃	C₂³	1	16	yes	—
49	C₂²×C₈×C₅₃	C₂×C₄	1	16	yes	complemented
50	C₂²×C₈×C₅₃	C₂³	1	12	yes	—
51	C₂⁴×C₅₃	—	2	8	no	—
52	C₂⁴×C₅₃	C₄²	1	8	yes	—
53	C₂⁴×C₅₃	C₂²×C₄	1	6	yes	—
54	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	C₄²	1	48	yes	—
56	C₂²×C₄×C₅₃	C₄²	1	8	yes	—
57	C₂²×C₄×C₅₃	C₂²×C₄	1	12	yes	—
58	C₂²×C₄×C₅₃	C₂²×C₄	1	6	yes	—
59	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
60	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
61	C₂²×C₄×C₅₃	C₂²×C₄	1	24	yes	—
62	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
63	C₂²×C₄×C₅₃	C₂²×C₄	1	24	yes	—
64	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
65	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
66	C₂²×Dic₅₃	—	2	8	no	—
67	C₂²×Dic₅₃	—	2	96	no	—
68	C₂²×Dic₅₃	M₁₆	1	16	yes	direct product factor
69	C₂²×Dic₅₃	C₂²×C₄	1	12	yes	—
70	C₂²×Dic₅₃	C₂²×C₄	1	32	yes	—
71	C₂²×Dic₅₃	C₂²×C₄	1	12	yes	—
72	M₁₆×C₅₃	C₂²×C₄	1	64	yes	complemented
73	C₂×C₈×C₅₃	C₂²×C₄	1	48	yes	complemented
74	C₂×C₈×C₅₃	C₂⁴	1	8	yes	—
75	C₄×Dic₅₃	C₂²×C₄	1	12	yes	—
76	C₄×Dic₅₃	C₂²×C₄	1	32	yes	—
77	C₄×Dic₅₃	C₂²×C₄	1	12	yes	—
78	C₄×Dic₅₃	C₂⁴	1	4	yes	—
79	C₄×Dic₅₃	C₂⁴	1	4	yes	—
80	#11719917	C₂²×C₄	1	64	yes	complemented
81	C₂×#88322	C₂²×C₄	1	48	yes	complemented
82	C₂×#88322	C₂⁴	1	8	yes	—
83	C₂³×C₅₃	—	2	24	no	—
84	C₂³×C₅₃	—	2	32	no	—
85	C₂³×C₅₃	C₂×C₄²	1	24	yes	—
86	C₂³×C₅₃	C₂×C₄²	1	4	yes	—
87	C₂³×C₅₃	C₂×M₁₆	1	8	yes	—
88	C₂³×C₅₃	C₂³×C₄	1	4	yes	—
89	C₂³×C₅₃	C₂³×C₄	1	3	yes	—
90	C₂×C₄×C₅₃	C₂×C₄²	1	48	yes	—
91	C₂×C₄×C₅₃	C₂×C₄²	1	12	yes	—
92	C₂×C₄×C₅₃	C₂×C₄²	1	24	yes	—
93	C₂×C₄×C₅₃	C₂³×C₄	1	6	yes	—
94	C₂×C₄×C₅₃	C₂³×C₄	1	8	yes	—
95	C₂×C₄×C₅₃	C₂³×C₄	1	3	yes	—
96	C₂×C₄×C₅₃	C₂³×C₄	1	4	yes	—
97	C₂×C₄×C₅₃	C₂³×C₄	1	6	yes	—
98	C₂×C₄×C₅₃	C₂³×C₄	1	8	yes	—
99	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
100	C₂×Dic₅₃	—	2	24	no	—
101	C₂×Dic₅₃	—	2	64	no	—
102	C₂×Dic₅₃	C₂×M₁₆	1	48	yes	direct product factor
103	C₂×Dic₅₃	C₂³×C₄	1	8	yes	—
104	C₂×Dic₅₃	C₂³×C₄	1	8	yes	—
105	C₈×C₅₃	C₂³×C₄	1	16	yes	complemented
106	#88322	C₂³×C₄	1	16	yes	complemented
107	C₂²×C₄×M₁₆	—	53	1	no	maximal, Sylow
108	C₂²×C₅₃	—	2	48	no	—
109	C₂²×C₅₃	—	2	8	no	—
110	C₂²×C₅₃	C₄×M₁₆	1	16	yes	complemented
111	C₂²×C₅₃	C₂²×C₄²	1	8	yes	—
112	C₂²×C₅₃	C₂²×C₄²	1	6	yes	—
113	C₂²×C₅₃	C₂²×M₁₆	1	12	yes	—
114	C₂²×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
115	C₄×C₅₃	C₂²×C₄²	1	6	yes	—
116	C₄×C₅₃	C₂²×C₄²	1	8	yes	—
117	C₄×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
118	C₄×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
119	Dic₅₃	—	2	8	no	—
120	Dic₅₃	C₂²×M₁₆	1	16	yes	direct product factor
121	C₂³×M₁₆	—	53	1	no	—
122	C₂³×M₁₆	D₅₃	1	1	yes	characteristic
123	C₂³×C₄²	—	53	1	no	—
124	C₂×C₄×M₁₆	—	53	24	no	—
125	C₂²×#97	—	53	2	no	—
126	C₂²×C₄×C₈	—	53	2	no	—
127	C₂×C₅₃	—	2	8	no	—
128	C₂×C₅₃	C₂×C₄×M₁₆	1	12	yes	complemented
129	C₂×C₅₃	C₂³×C₄²	1	1	yes	characteristic
130	C₂×C₅₃	C₂³×M₁₆	1	2	yes	—
131	C₂⁴×C₄	—	53	2	no	—
132	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic
133	C₂²×M₁₆	—	53	16	no	—
134	C₂²×M₁₆	—	53	12	no	—
135	C₂²×M₁₆	Dic₅₃	1	16	yes	direct product factor
136	C₂²×M₁₆	C₂×D₅₃	1	12	yes	—
137	C₂³×C₈	—	53	2	no	—
138	C₂³×C₈	C₂×D₅₃	1	2	yes	—
139	C₂²×C₄²	—	53	2	no	—
140	C₂²×C₄²	—	53	12	no	—
141	C₂²×C₄²	—	53	12	no	—
142	C₂²×C₄²	—	53	1	no	—
143	C₂²×C₄²	—	53	1	no	—
144	C₄×M₁₆	—	53	64	no	—
145	C₂×#97	—	53	24	no	—
146	C₂×C₄×C₈	—	53	24	no	—
147	C₅₃	C₂²×C₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
148	C₂⁵	C₄×D₅₃	1	1	yes	characteristic
149	C₂³×C₄	—	53	2	no	—
150	C₂³×C₄	—	53	24	no	—
151	C₂³×C₄	—	53	2	no	—
152	C₂³×C₄	—	106	16	no	—
153	C₂³×C₄	C₄×D₅₃	1	8	yes	—
154	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
155	C₂³×C₄	C₄×D₅₃	1	6	yes	—
156	C₂³×C₄	C₂×Dic₅₃	1	8	yes	—
157	C₂³×C₄	C₂²×D₅₃	1	6	yes	—
158	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
159	C₂×M₁₆	—	53	96	no	—
160	C₂×M₁₆	—	53	16	no	—
161	C₂×M₁₆	C₂×Dic₅₃	1	96	yes	direct product factor
162	C₂×M₁₆	C₂²×D₅₃	1	16	yes	—
163	C₂²×C₈	—	53	16	no	—
164	C₂²×C₈	—	53	12	no	—
165	C₂²×C₈	C₂×Dic₅₃	1	16	yes	complemented
166	C₂²×C₈	C₂²×D₅₃	1	12	yes	—
167	C₂×C₄²	—	53	6	no	—
168	C₂×C₄²	—	53	24	no	—
169	C₂×C₄²	—	53	48	no	—
170	C₂×C₄²	—	53	6	no	—
171	C₂×C₄²	—	53	6	no	—
172	C₂×C₄²	—	53	6	no	—
173	C₂×C₄²	—	53	16	no	—
174	#97	—	53	32	no	—
175	C₄×C₈	—	53	32	no	—
176	C₂⁴	—	2	8	no	—
177	C₂⁴	C₄×Dic₅₃	1	8	yes	—
178	C₂⁴	C₂×C₄×D₅₃	1	6	yes	—
179	C₂⁴	C₂×C₄×D₅₃	1	1	yes	characteristic
180	C₂²×C₄	—	53	16	no	—
181	C₂²×C₄	—	53	12	no	—
182	C₂²×C₄	—	53	32	no	—
183	C₂²×C₄	—	53	12	no	—
184	C₂²×C₄	—	106	96	no	—
185	C₂²×C₄	—	106	8	no	—
186	C₂²×C₄	C₄×Dic₅₃	1	48	yes	—
187	C₂²×C₄	C₄×Dic₅₃	1	8	yes	—
188	C₂²×C₄	C₂×C₄×D₅₃	1	12	yes	—
189	C₂²×C₄	C₂×C₄×D₅₃	1	6	yes	—
190	C₂²×C₄	C₂×C₄×D₅₃	1	3	yes	—
191	C₂²×C₄	C₂×C₄×D₅₃	1	4	yes	—
192	C₂²×C₄	C₂×C₄×D₅₃	1	24	yes	—
193	C₂²×C₄	C₂²×Dic₅₃	1	4	yes	—
194	C₂²×C₄	C₂²×Dic₅₃	1	24	yes	—
195	C₂²×C₄	C₂³×D₅₃	1	3	yes	—
196	C₂²×C₄	C₂³×D₅₃	1	4	yes	—
197	M₁₆	—	53	64	no	—
198	M₁₆	C₂²×Dic₅₃	1	64	yes	direct product factor
199	C₂×C₈	—	53	48	no	—
200	C₂×C₈	—	53	8	no	—
201	C₂×C₈	C₂²×Dic₅₃	1	48	yes	complemented
202	C₂×C₈	C₂³×D₅₃	1	8	yes	—
203	C₄²	—	53	12	no	—
204	C₄²	—	53	32	no	—
205	C₄²	—	53	12	no	—
206	C₄²	—	53	4	no	—
207	C₄²	—	53	4	no	—
208	C₂³	—	2	32	no	—
209	C₂³	—	2	24	no	—
210	C₂³	C₂×C₄×Dic₅₃	1	24	yes	—
211	C₂³	C₂×C₄×Dic₅₃	1	4	yes	—
212	C₂³	M₁₆×D₅₃	1	8	yes	—
213	C₂³	C₂²×C₄×D₅₃	1	4	yes	—
214	C₂³	C₂²×C₄×D₅₃	1	3	yes	—
215	C₂×C₄	—	53	48	no	—
216	C₂×C₄	—	53	8	no	—
217	C₂×C₄	—	53	8	no	—
218	C₂×C₄	—	106	64	no	—
219	C₂×C₄	—	106	24	no	—
220	C₂×C₄	C₂×C₄×Dic₅₃	1	48	yes	—
221	C₂×C₄	C₂×C₄×Dic₅₃	1	24	yes	—
222	C₂×C₄	C₂×C₄×Dic₅₃	1	12	yes	—
223	C₂×C₄	C₂²×C₄×D₅₃	1	4	yes	—
224	C₂×C₄	C₂²×C₄×D₅₃	1	3	yes	—
225	C₂×C₄	C₂²×C₄×D₅₃	1	6	yes	—
226	C₂×C₄	C₂²×C₄×D₅₃	1	8	yes	—
227	C₂×C₄	C₂³×Dic₅₃	1	6	yes	—
228	C₂×C₄	C₂³×Dic₅₃	1	8	yes	—
229	C₂×C₄	C₂⁴×D₅₃	1	1	yes	characteristic
230	C₈	—	53	16	no	—
231	C₈	C₂³×Dic₅₃	1	16	yes	complemented
232	C₂²	—	2	8	no	—
233	C₂²	—	2	48	no	—
234	C₂²	M₁₆×Dic₅₃	1	16	yes	direct product factor
235	C₂²	C₂²×C₄×Dic₅₃	1	8	yes	—
236	C₂²	C₂²×C₄×Dic₅₃	1	6	yes	—
237	C₂²	C₂×M₁₆×D₅₃	1	12	yes	—
238	C₂²	C₂³×C₄×D₅₃	1	1	yes	characteristic
239	C₄	—	53	16	no	—
240	C₄	—	106	8	no	—
241	C₄	C₂²×C₄×Dic₅₃	1	6	yes	—
242	C₄	C₂²×C₄×Dic₅₃	1	8	yes	—
243	C₄	C₂³×C₄×D₅₃	1	1	yes	characteristic
244	C₄	C₂⁴×Dic₅₃	1	1	yes	characteristic
245	C₂	—	2	8	no	—
246	C₂	C₂×M₁₆×Dic₅₃	1	12	yes	direct product factor
247	C₂	C₂³×C₄×Dic₅₃	1	1	yes	characteristic
248	C₂	C₂²×M₁₆×D₅₃	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₅₃
commutator	C₂×C₅₃	C₂³×C₄²
Frattini	C₂×C₄	C₂⁴×D₅₃
Fitting	C₂³×M₁₆×C₅₃	C₂
socle	C₂⁴×C₅₃	C₂²×C₄
prime radical	C₂⁵×C₅₃	C₂×C₄
cosocle	C₂×C₄×C₅₃	C₂⁵
hypercenter	C₂³×M₁₆	D₅₃
hypocenter	C₅₃	C₂²×C₄×M₁₆
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂³×M₁₆

Complements

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

N	H ≅ G/N	type
3. C₂×M₁₆×Dic₅₃	C₂	direct (N × H)
4. C₂²×#13133127	C₂	semidirect
5. C₂²×C₈×Dic₅₃	C₂	semidirect
8. C₂²×#11719917	C₄	semidirect
16. M₁₆×Dic₅₃	C₂²	direct (N × H)
17. C₂×#13133127	C₂²	semidirect
18. C₂×C₈×Dic₅₃	C₂²	semidirect
20. C₂²×M₁₆×C₅₃	C₄	semidirect
27. C₂×#11719917	C₂×C₄	semidirect
29. C₂²×#88322	C₂×C₄	semidirect
38. #13133127	C₂³	semidirect
39. C₈×Dic₅₃	C₂³	semidirect
47. C₂×M₁₆×C₅₃	C₂×C₄	semidirect
49. C₂²×C₈×C₅₃	C₂×C₄	semidirect
68. C₂²×Dic₅₃	M₁₆	direct (N × H)
72. M₁₆×C₅₃	C₂²×C₄	semidirect
73. C₂×C₈×C₅₃	C₂²×C₄	semidirect
80. #11719917	C₂²×C₄	semidirect
81. C₂×#88322	C₂²×C₄	semidirect
102. C₂×Dic₅₃	C₂×M₁₆	direct (N × H)
105. C₈×C₅₃	C₂³×C₄	semidirect
106. #88322	C₂³×C₄	semidirect
110. C₂²×C₅₃	C₄×M₁₆	semidirect
120. Dic₅₃	C₂²×M₁₆	direct (N × H)
128. C₂×C₅₃	C₂×C₄×M₁₆	semidirect
135. C₂²×M₁₆	Dic₅₃	direct (N × H)
147. C₅₃	C₂²×C₄×M₁₆	semidirect
161. C₂×M₁₆	C₂×Dic₅₃	direct (N × H)
165. C₂²×C₈	C₂×Dic₅₃	semidirect
198. M₁₆	C₂²×Dic₅₃	direct (N × H)
201. C₂×C₈	C₂²×Dic₅₃	semidirect
231. C₈	C₂³×Dic₅₃	semidirect
234. C₂²	M₁₆×Dic₅₃	direct (N × H)
246. C₂	C₂×M₁₆×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
C₂³×#11719917	2
C₂³×C₄×Dic₅₃	2
C₂×M₁₆×Dic₅₃	2
C₂²×#13133127	2
C₂²×C₈×Dic₅₃	2
C₂³×M₁₆×C₅₃	2
C₂²×C₄×M₁₆	53

Subgroups — C22×M16×Dic53 #28247478

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×M₁₆×Dic₅₃ #28247478