Subgroups — C₂²×#16810522 #27542440

Order 13568.

4894 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄⋊Dic₅₃	C₂	1	2	yes	maximal
2	C₂³×C₄⋊Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂³×C₄×Dic₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#16810522	C₂	1	24	yes	maximal, direct product factor
5	C₂²×#13133124	C₂	1	2	yes	maximal
6	C₂³×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂⁴×Dic₅₃	C₂²	1	2	yes	—
8	C₂⁴×Dic₅₃	C₂²	1	1	yes	characteristic
9	C₂⁴×Dic₅₃	C₂²	1	1	yes	characteristic
10	C₂²×C₄⋊Dic₅₃	—	2	16	no	—
11	C₂²×C₄⋊Dic₅₃	—	2	8	no	—
12	C₂²×C₄⋊Dic₅₃	C₂²	1	24	yes	—
13	C₂²×C₄⋊Dic₅₃	C₂²	1	12	yes	—
14	C₂²×C₄×Dic₅₃	—	2	8	no	—
15	C₂²×C₄×Dic₅₃	C₂²	1	12	yes	—
16	#16810522	C₂²	1	64	yes	direct product factor
17	C₂×#13133124	C₂²	1	24	yes	—
18	C₂⁴×C₄×C₅₃	C₂²	1	2	yes	—
19	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
20	C₂²×C₄⋊C₄×C₅₃	C₄	1	16	yes	complemented
21	C₂²×C₄⋊C₄×C₅₃	C₂²	1	12	yes	—
22	C₂³×Dic₅₃	—	2	4	no	—
23	C₂³×Dic₅₃	—	2	4	no	—
24	C₂³×Dic₅₃	—	2	4	no	—
25	C₂³×Dic₅₃	—	2	4	no	—
26	C₂³×Dic₅₃	—	2	8	no	—
27	C₂³×Dic₅₃	—	2	8	no	—
28	C₂³×Dic₅₃	—	2	8	no	—
29	C₂³×Dic₅₃	D₄	1	8	yes	—
30	C₂³×Dic₅₃	Q₈	1	8	yes	—
31	C₂³×Dic₅₃	C₂³	1	12	yes	—
32	C₂³×Dic₅₃	C₂³	1	6	yes	—
33	C₂³×Dic₅₃	C₂³	1	6	yes	—
34	C₂×C₄⋊Dic₅₃	—	2	48	no	—
35	C₂×C₄⋊Dic₅₃	—	2	96	no	—
36	C₂×C₄⋊Dic₅₃	C₂³	1	16	yes	—
37	C₂×C₄⋊Dic₅₃	C₂³	1	32	yes	—
38	C₂×C₄×Dic₅₃	—	2	48	no	—
39	C₂×C₄×Dic₅₃	C₂³	1	16	yes	—
40	#13133124	C₂³	1	32	yes	—
41	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
42	C₂³×C₄×C₅₃	—	2	8	no	—
43	C₂³×C₄×C₅₃	—	2	8	no	—
44	C₂³×C₄×C₅₃	C₂×C₄	1	16	yes	—
45	C₂³×C₄×C₅₃	C₂×C₄	1	8	yes	—
46	C₂³×C₄×C₅₃	D₄	1	8	yes	—
47	C₂³×C₄×C₅₃	Q₈	1	8	yes	—
48	C₂³×C₄×C₅₃	C₂³	1	12	yes	—
49	C₂³×C₄×C₅₃	C₂³	1	6	yes	—
50	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	96	yes	complemented
51	C₂×C₄⋊C₄×C₅₃	C₂³	1	16	yes	—
52	C₂⁴×C₅₃	C₂²×C₄	1	4	yes	—
53	C₂⁴×C₅₃	C₂×D₄	1	4	yes	—
54	C₂⁴×C₅₃	C₂×D₄	1	4	yes	—
55	C₂⁴×C₅₃	C₂×Q₈	1	4	yes	—
56	C₂⁴×C₅₃	C₂×Q₈	1	4	yes	—
57	C₂⁴×C₅₃	#41	1	4	yes	—
58	C₂⁴×C₅₃	#41	1	4	yes	—
59	C₂⁴×C₅₃	C₂⁴	1	3	yes	—
60	C₂²×C₄×C₅₃	—	2	64	no	—
61	C₂²×C₄×C₅₃	—	2	24	no	—
62	C₂²×C₄×C₅₃	—	2	24	no	—
63	C₂²×C₄×C₅₃	C₄⋊C₄	1	64	yes	complemented
64	C₂²×C₄×C₅₃	C₂²×C₄	1	48	yes	—
65	C₂²×C₄×C₅₃	C₂²×C₄	1	24	yes	—
66	C₂²×C₄×C₅₃	C₂×D₄	1	24	yes	—
67	C₂²×C₄×C₅₃	C₂×Q₈	1	24	yes	—
68	C₂²×C₄×C₅₃	C₂⁴	1	8	yes	—
69	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
70	C₂²×Dic₅₃	—	2	32	no	—
71	C₂²×Dic₅₃	—	2	32	no	—
72	C₂²×Dic₅₃	—	2	12	no	—
73	C₂²×Dic₅₃	—	2	12	no	—
74	C₂²×Dic₅₃	—	2	12	no	—
75	C₂²×Dic₅₃	—	2	12	no	—
76	C₂²×Dic₅₃	—	2	24	no	—
77	C₂²×Dic₅₃	—	2	24	no	—
78	C₂²×Dic₅₃	—	2	24	no	—
79	C₂²×Dic₅₃	—	4	32	no	—
80	C₂²×Dic₅₃	C₂×D₄	1	24	yes	—
81	C₂²×Dic₅₃	C₂×Q₈	1	24	yes	—
82	C₂²×Dic₅₃	C₂⁴	1	8	yes	—
83	C₂²×Dic₅₃	C₂⁴	1	4	yes	—
84	C₂²×Dic₅₃	C₂⁴	1	4	yes	—
85	C₄⋊C₄×C₅₃	C₂²×C₄	1	64	yes	complemented
86	C₄⋊Dic₅₃	—	2	32	no	—
87	C₄⋊Dic₅₃	—	2	64	no	—
88	C₄×Dic₅₃	—	2	32	no	—
89	C₂³×C₅₃	C₂×C₄⋊C₄	1	16	yes	—
90	C₂³×C₅₃	C₄×D₄	1	16	yes	—
91	C₂³×C₅₃	C₄×Q₈	1	16	yes	—
92	C₂³×C₅₃	#121	1	16	yes	—
93	C₂³×C₅₃	C₂²⋊Q₈	1	16	yes	—
94	C₂³×C₅₃	#125	1	16	yes	—
95	C₂³×C₅₃	C₄⋊Q₈	1	16	yes	—
96	C₂³×C₅₃	C₂³×C₄	1	6	yes	—
97	C₂³×C₅₃	C₂²×D₄	1	6	yes	—
98	C₂³×C₅₃	C₂²×D₄	1	6	yes	—
99	C₂³×C₅₃	C₂²×Q₈	1	6	yes	—
100	C₂³×C₅₃	C₂²×Q₈	1	6	yes	—
101	C₂³×C₅₃	C₂×#41	1	6	yes	—
102	C₂³×C₅₃	C₂×#41	1	6	yes	—
103	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
104	C₂×C₄×C₅₃	—	2	96	no	—
105	C₂×C₄×C₅₃	—	2	8	no	—
106	C₂×C₄×C₅₃	—	2	8	no	—
107	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	96	yes	complemented
108	C₂×C₄×C₅₃	C₂³×C₄	1	16	yes	—
109	C₂×C₄×C₅₃	C₂³×C₄	1	8	yes	—
110	C₂×C₄×C₅₃	C₂²×D₄	1	8	yes	—
111	C₂×C₄×C₅₃	C₂²×Q₈	1	8	yes	—
112	C₂×Dic₅₃	—	2	48	no	—
113	C₂×Dic₅₃	—	2	48	no	—
114	C₂×Dic₅₃	—	2	4	no	—
115	C₂×Dic₅₃	—	2	4	no	—
116	C₂×Dic₅₃	—	2	4	no	—
117	C₂×Dic₅₃	—	2	4	no	—
118	C₂×Dic₅₃	—	2	8	no	—
119	C₂×Dic₅₃	—	2	8	no	—
120	C₂×Dic₅₃	—	2	8	no	—
121	C₂×Dic₅₃	—	4	48	no	—
122	C₂×Dic₅₃	C₂²×D₄	1	8	yes	—
123	C₂×Dic₅₃	C₂²×Q₈	1	8	yes	—
124	C₂²×#389	—	53	1	no	maximal, Sylow
125	C₂²×C₅₃	#389	1	64	yes	complemented
126	C₂²×C₅₃	C₂²×C₄⋊C₄	1	12	yes	—
127	C₂²×C₅₃	C₂×C₄×D₄	1	12	yes	—
128	C₂²×C₅₃	C₂×C₄×Q₈	1	12	yes	—
129	C₂²×C₅₃	C₂×#121	1	12	yes	—
130	C₂²×C₅₃	C₂×C₂²⋊Q₈	1	12	yes	—
131	C₂²×C₅₃	C₂×#125	1	12	yes	—
132	C₂²×C₅₃	C₂×C₄⋊Q₈	1	12	yes	—
133	C₂²×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
134	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
135	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
136	C₂²×C₅₃	C₂³×Q₈	1	1	yes	characteristic
137	C₂²×C₅₃	C₂³×Q₈	1	1	yes	characteristic
138	C₂²×C₅₃	C₂²×#41	1	1	yes	characteristic
139	C₂²×C₅₃	C₂²×#41	1	1	yes	characteristic
140	C₄×C₅₃	—	2	16	no	—
141	C₄×C₅₃	C₂²×C₄⋊C₄	1	16	yes	complemented
142	Dic₅₃	—	2	8	no	—
143	Dic₅₃	—	2	8	no	—
144	Dic₅₃	—	4	8	no	—
145	C₂³×C₄⋊C₄	—	53	2	no	—
146	C₂³×C₄⋊C₄	—	53	1	no	—
147	C₂³×C₄⋊C₄	D₅₃	1	1	yes	characteristic
148	C₂³×C₄²	—	53	1	no	—
149	C₂×#389	—	53	24	no	—
150	C₂²×#95	—	53	2	no	—
151	C₂×C₅₃	C₂×#389	1	24	yes	complemented
152	C₂×C₅₃	C₂³×C₄⋊C₄	1	1	yes	characteristic
153	C₂×C₅₃	C₂²×C₄×D₄	1	1	yes	characteristic
154	C₂×C₅₃	C₂²×C₄×Q₈	1	1	yes	characteristic
155	C₂×C₅₃	C₂²×#121	1	1	yes	characteristic
156	C₂×C₅₃	C₂²×C₂²⋊Q₈	1	1	yes	characteristic
157	C₂×C₅₃	C₂²×#125	1	1	yes	characteristic
158	C₂×C₅₃	C₂²×C₄⋊Q₈	1	1	yes	characteristic
159	C₂⁴×C₄	—	53	1	no	—
160	C₂⁴×C₄	—	53	1	no	—
161	C₂⁴×C₄	—	53	2	no	—
162	C₂⁴×C₄	C₂×D₅₃	1	2	yes	—
163	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic
164	C₂²×C₄⋊C₄	—	53	24	no	—
165	C₂²×C₄⋊C₄	—	53	12	no	—
166	C₂²×C₄⋊C₄	—	106	16	no	—
167	C₂²×C₄⋊C₄	—	106	8	no	—
168	C₂²×C₄⋊C₄	Dic₅₃	1	16	yes	complemented
169	C₂²×C₄⋊C₄	C₂×D₅₃	1	12	yes	—
170	C₂²×C₄²	—	53	12	no	—
171	C₂²×C₄²	—	106	8	no	—
172	#389	—	53	64	no	—
173	C₂×#95	—	53	24	no	—
174	C₅₃	C₂²×#389	1	1	yes	characteristic, complemented, Sylow
175	C₂⁵	C₂²×D₅₃	1	1	yes	characteristic
176	C₂³×C₄	—	2	8	no	—
177	C₂³×C₄	—	2	8	no	—
178	C₂³×C₄	—	53	8	no	—
179	C₂³×C₄	—	53	8	no	—
180	C₂³×C₄	—	53	6	no	—
181	C₂³×C₄	—	53	6	no	—
182	C₂³×C₄	—	53	12	no	—
183	C₂³×C₄	—	106	4	no	—
184	C₂³×C₄	—	106	4	no	—
185	C₂³×C₄	—	106	4	no	—
186	C₂³×C₄	—	106	4	no	—
187	C₂³×C₄	—	106	8	no	—
188	C₂³×C₄	—	106	8	no	—
189	C₂³×C₄	—	106	8	no	—
190	C₂³×C₄	Dic₁₀₆	1	8	yes	—
191	C₂³×C₄	D₂₁₂	1	8	yes	—
192	C₂³×C₄	C₂×Dic₅₃	1	16	yes	—
193	C₂³×C₄	C₂×Dic₅₃	1	8	yes	—
194	C₂³×C₄	C₂²×D₅₃	1	12	yes	—
195	C₂³×C₄	C₂²×D₅₃	1	6	yes	—
196	C₂×C₄⋊C₄	—	53	32	no	—
197	C₂×C₄⋊C₄	—	53	16	no	—
198	C₂×C₄⋊C₄	—	106	96	no	—
199	C₂×C₄⋊C₄	—	106	48	no	—
200	C₂×C₄⋊C₄	C₂×Dic₅₃	1	96	yes	complemented
201	C₂×C₄⋊C₄	C₂²×D₅₃	1	16	yes	—
202	C₂×C₄²	—	53	16	no	—
203	C₂×C₄²	—	106	48	no	—
204	#95	—	53	32	no	—
205	C₂⁴	C₂×Dic₁₀₆	1	4	yes	—
206	C₂⁴	C₂×D₂₁₂	1	4	yes	—
207	C₂⁴	D₄×D₅₃	1	4	yes	—
208	C₂⁴	#11719947	1	4	yes	—
209	C₂⁴	Q₈×D₅₃	1	4	yes	—
210	C₂⁴	#11719949	1	4	yes	—
211	C₂⁴	C₂²×Dic₅₃	1	4	yes	—
212	C₂⁴	C₂³×D₅₃	1	3	yes	—
213	C₂²×C₄	—	2	64	no	—
214	C₂²×C₄	—	2	24	no	—
215	C₂²×C₄	—	2	24	no	—
216	C₂²×C₄	—	53	24	no	—
217	C₂²×C₄	—	53	24	no	—
218	C₂²×C₄	—	53	4	no	—
219	C₂²×C₄	—	53	4	no	—
220	C₂²×C₄	—	53	8	no	—
221	C₂²×C₄	—	106	32	no	—
222	C₂²×C₄	—	106	32	no	—
223	C₂²×C₄	—	106	12	no	—
224	C₂²×C₄	—	106	12	no	—
225	C₂²×C₄	—	106	12	no	—
226	C₂²×C₄	—	106	12	no	—
227	C₂²×C₄	—	106	24	no	—
228	C₂²×C₄	—	106	24	no	—
229	C₂²×C₄	—	106	24	no	—
230	C₂²×C₄	—	212	32	no	—
231	C₂²×C₄	C₄⋊Dic₅₃	1	64	yes	complemented
232	C₂²×C₄	C₂×Dic₁₀₆	1	24	yes	—
233	C₂²×C₄	C₂×D₂₁₂	1	24	yes	—
234	C₂²×C₄	C₂²×Dic₅₃	1	48	yes	—
235	C₂²×C₄	C₂²×Dic₅₃	1	24	yes	—
236	C₂²×C₄	C₂³×D₅₃	1	8	yes	—
237	C₂²×C₄	C₂³×D₅₃	1	4	yes	—
238	C₄⋊C₄	—	106	64	no	—
239	C₄⋊C₄	—	106	32	no	—
240	C₄⋊C₄	C₂²×Dic₅₃	1	64	yes	complemented
241	C₄²	—	106	32	no	—
242	C₂³	C₂×C₄⋊Dic₅₃	1	16	yes	—
243	C₂³	D₄×Dic₅₃	1	16	yes	—
244	C₂³	Q₈×Dic₅₃	1	16	yes	—
245	C₂³	#13133192	1	16	yes	—
246	C₂³	#13133197	1	16	yes	—
247	C₂³	#13133208	1	16	yes	—
248	C₂³	#13133216	1	16	yes	—
249	C₂³	C₂³×Dic₅₃	1	6	yes	—
250	C₂³	C₂×D₄×D₅₃	1	6	yes	—
251	C₂³	C₂²×D₂₁₂	1	6	yes	—
252	C₂³	C₂×Q₈×D₅₃	1	6	yes	—
253	C₂³	C₂²×Dic₁₀₆	1	6	yes	—
254	C₂³	C₂×#11719947	1	6	yes	—
255	C₂³	C₂×#11719949	1	6	yes	—
256	C₂³	C₂⁴×D₅₃	1	1	yes	characteristic
257	C₂×C₄	—	2	96	no	—
258	C₂×C₄	—	2	8	no	—
259	C₂×C₄	—	2	8	no	—
260	C₂×C₄	—	53	8	no	—
261	C₂×C₄	—	53	8	no	—
262	C₂×C₄	—	106	48	no	—
263	C₂×C₄	—	106	48	no	—
264	C₂×C₄	—	106	4	no	—
265	C₂×C₄	—	106	4	no	—
266	C₂×C₄	—	106	4	no	—
267	C₂×C₄	—	106	4	no	—
268	C₂×C₄	—	106	8	no	—
269	C₂×C₄	—	106	8	no	—
270	C₂×C₄	—	106	8	no	—
271	C₂×C₄	—	212	48	no	—
272	C₂×C₄	C₂×C₄⋊Dic₅₃	1	96	yes	complemented
273	C₂×C₄	C₂³×Dic₅₃	1	16	yes	—
274	C₂×C₄	C₂³×Dic₅₃	1	8	yes	—
275	C₂×C₄	C₂²×D₂₁₂	1	8	yes	—
276	C₂×C₄	C₂²×Dic₁₀₆	1	8	yes	—
277	C₂²	#16810522	1	64	yes	direct product factor
278	C₂²	C₂²×C₄⋊Dic₅₃	1	12	yes	—
279	C₂²	C₂×D₄×Dic₅₃	1	12	yes	—
280	C₂²	C₂×Q₈×Dic₅₃	1	12	yes	—
281	C₂²	C₂×#13133192	1	12	yes	—
282	C₂²	C₂×#13133197	1	12	yes	—
283	C₂²	C₂×#13133208	1	12	yes	—
284	C₂²	C₂×#13133216	1	12	yes	—
285	C₂²	C₂⁴×Dic₅₃	1	1	yes	characteristic
286	C₂²	C₂²×D₄×D₅₃	1	1	yes	characteristic
287	C₂²	C₂³×D₂₁₂	1	1	yes	characteristic
288	C₂²	C₂²×Q₈×D₅₃	1	1	yes	characteristic
289	C₂²	C₂³×Dic₁₀₆	1	1	yes	characteristic
290	C₂²	C₂²×#11719947	1	1	yes	characteristic
291	C₂²	C₂²×#11719949	1	1	yes	characteristic
292	C₄	—	2	16	no	—
293	C₄	—	106	8	no	—
294	C₄	—	106	8	no	—
295	C₄	—	212	8	no	—
296	C₄	C₂²×C₄⋊Dic₅₃	1	16	yes	complemented
297	C₂	C₂×#16810522	1	24	yes	direct product factor
298	C₂	C₂³×C₄⋊Dic₅₃	1	1	yes	characteristic
299	C₂	C₂²×D₄×Dic₅₃	1	1	yes	characteristic
300	C₂	C₂²×Q₈×Dic₅₃	1	1	yes	characteristic
301	C₂	C₂²×#13133192	1	1	yes	characteristic
302	C₂	C₂²×#13133197	1	1	yes	characteristic
303	C₂	C₂²×#13133208	1	1	yes	characteristic
304	C₂	C₂²×#13133216	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂⁴×C₄
Frattini	C₂³	C₂⁴×D₅₃
Fitting	C₂³×C₄⋊C₄×C₅₃	C₂
socle, prime radical	C₂⁵×C₅₃	C₂³
cosocle	C₂³×C₅₃	C₂⁵
hypercenter	C₂³×C₄⋊C₄	D₅₃
hypocenter	C₅₃	C₂²×#389
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁵ ◁ C₂³×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
4. C₂×#16810522	C₂	direct (N × H)
16. #16810522	C₂²	direct (N × H)
20. C₂²×C₄⋊C₄×C₅₃	C₄	semidirect
50. C₂×C₄⋊C₄×C₅₃	C₂×C₄	semidirect
63. C₂²×C₄×C₅₃	C₄⋊C₄	semidirect
85. C₄⋊C₄×C₅₃	C₂²×C₄	semidirect
107. C₂×C₄×C₅₃	C₂×C₄⋊C₄	semidirect
125. C₂²×C₅₃	#389	semidirect
141. C₄×C₅₃	C₂²×C₄⋊C₄	semidirect
151. C₂×C₅₃	C₂×#389	semidirect
168. C₂²×C₄⋊C₄	Dic₅₃	semidirect
174. C₅₃	C₂²×#389	semidirect
200. C₂×C₄⋊C₄	C₂×Dic₅₃	semidirect
231. C₂²×C₄	C₄⋊Dic₅₃	semidirect
240. C₄⋊C₄	C₂²×Dic₅₃	semidirect
272. C₂×C₄	C₂×C₄⋊Dic₅₃	semidirect
277. C₂²	#16810522	direct (N × H)
296. C₄	C₂²×C₄⋊Dic₅₃	semidirect
297. C₂	C₂×#16810522	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×C₄⋊Dic₅₃	2
C₂³×C₄×Dic₅₃	2
C₂×#16810522	2
C₂²×#13133124	2
C₂³×C₄⋊C₄×C₅₃	2
C₂²×#389	53

Subgroups — C22×#16810522 #27542440

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16810522 #27542440