Subgroups — C₂²×#16810586 #28247522

Order 13568.

7214 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₄×D₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂³×#11719917	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16810586	C₂	1	24	yes	maximal, direct product factor
4	C₂²×#13133129	C₂	1	4	yes	maximal, complemented
5	C₂³×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂⁵×D₅₃	C₄	1	1	yes	characteristic
7	C₂⁴×Dic₅₃	C₄	1	1	yes	characteristic
8	C₂³×C₄×D₅₃	—	2	8	no	—
9	C₂³×C₄×D₅₃	—	2	8	no	—
10	C₂³×C₄×D₅₃	C₄	1	12	yes	—
11	C₂³×C₄×D₅₃	C₄	1	2	yes	—
12	C₂³×C₄×D₅₃	C₂²	1	2	yes	—
13	C₂³×C₄×D₅₃	C₂²	1	12	yes	—
14	C₂²×#11719917	—	2	8	no	—
15	C₂²×#11719917	C₂²	1	12	yes	complemented
16	C₂³×#88322	C₂²	1	2	yes	complemented
17	#16810586	C₂²	1	64	yes	direct product factor
18	C₂×#13133129	C₂²	1	48	yes	complemented
19	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
20	C₂²×M₁₆×C₅₃	—	2	8	no	—
21	C₂²×M₁₆×C₅₃	C₂²	1	12	yes	complemented
22	C₂³×C₈×C₅₃	C₂²	1	2	yes	complemented
23	C₂⁴×D₅₃	—	2	8	no	—
24	C₂⁴×D₅₃	—	2	16	no	—
25	C₂⁴×D₅₃	C₂×C₄	1	12	yes	—
26	C₂⁴×D₅₃	C₂×C₄	1	2	yes	—
27	C₂³×Dic₅₃	—	2	8	no	—
28	C₂³×Dic₅₃	C₂×C₄	1	12	yes	—
29	C₂³×Dic₅₃	C₂×C₄	1	2	yes	—
30	C₂²×C₄×D₅₃	—	2	96	no	—
31	C₂²×C₄×D₅₃	—	2	16	no	—
32	C₂²×C₄×D₅₃	—	2	8	no	—
33	C₂²×C₄×D₅₃	—	2	48	no	—
34	C₂²×C₄×D₅₃	—	2	8	no	—
35	C₂²×C₄×D₅₃	—	2	48	no	—
36	C₂²×C₄×D₅₃	C₂×C₄	1	48	yes	—
37	C₂²×C₄×D₅₃	C₂×C₄	1	12	yes	—
38	C₂²×C₄×D₅₃	C₂×C₄	1	24	yes	—
39	C₂²×C₄×D₅₃	C₂³	1	12	yes	—
40	C₂²×C₄×D₅₃	C₂³	1	16	yes	—
41	C₂×#11719917	—	2	48	no	—
42	C₂×#11719917	C₂³	1	16	yes	complemented
43	C₂²×#88322	—	2	8	no	—
44	C₂²×#88322	C₂³	1	12	yes	complemented
45	#13133129	C₂³	1	64	yes	complemented
46	C₂⁵×C₅₃	C₂×C₄	1	1	yes	characteristic
47	C₂³×C₄×C₅₃	C₂×C₄	1	6	yes	—
48	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
49	C₂³×C₄×C₅₃	D₄	1	8	yes	—
50	C₂³×C₄×C₅₃	D₄	1	8	yes	—
51	C₂³×C₄×C₅₃	C₂³	1	6	yes	—
52	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
53	C₂×M₁₆×C₅₃	—	2	48	no	—
54	C₂×M₁₆×C₅₃	C₂³	1	16	yes	complemented
55	C₂²×C₈×C₅₃	—	2	8	no	—
56	C₂²×C₈×C₅₃	C₂³	1	12	yes	complemented
57	C₂⁴×C₅₃	—	2	8	no	—
58	C₂⁴×C₅₃	C₂²⋊C₄	1	8	yes	—
59	C₂⁴×C₅₃	C₂²×C₄	1	6	yes	—
60	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
61	C₂³×D₅₃	—	2	128	no	—
62	C₂³×D₅₃	—	2	48	no	—
63	C₂³×D₅₃	—	2	8	no	—
64	C₂³×D₅₃	—	2	8	no	—
65	C₂³×D₅₃	—	2	96	no	—
66	C₂³×D₅₃	M₁₆	1	16	yes	complemented
67	C₂³×D₅₃	C₂²×C₄	1	16	yes	—
68	C₂³×D₅₃	C₂²×C₄	1	12	yes	—
69	C₂²×C₄×C₅₃	C₂²⋊C₄	1	48	yes	—
70	C₂²×C₄×C₅₃	C₂²⋊C₄	1	8	yes	—
71	C₂²×C₄×C₅₃	C₂²×C₄	1	12	yes	—
72	C₂²×C₄×C₅₃	C₂²×C₄	1	6	yes	—
73	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
74	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
75	C₂²×C₄×C₅₃	C₂×D₄	1	24	yes	—
76	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
77	C₂²×C₄×C₅₃	C₂×D₄	1	24	yes	—
78	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
79	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
80	C₂²×Dic₅₃	—	2	48	no	—
81	C₂²×Dic₅₃	—	2	8	no	—
82	C₂²×Dic₅₃	C₂²×C₄	1	16	yes	—
83	C₂²×Dic₅₃	C₂²×C₄	1	12	yes	—
84	C₂×C₄×D₅₃	—	2	192	no	—
85	C₂×C₄×D₅₃	—	2	48	no	—
86	C₂×C₄×D₅₃	—	2	96	no	—
87	C₂×C₄×D₅₃	—	2	24	no	—
88	C₂×C₄×D₅₃	—	2	32	no	—
89	C₂×C₄×D₅₃	—	2	24	no	—
90	C₂×C₄×D₅₃	—	2	32	no	—
91	C₂×C₄×D₅₃	C₂²×C₄	1	24	yes	—
92	C₂×C₄×D₅₃	C₂²×C₄	1	32	yes	—
93	C₂×C₄×D₅₃	C₂⁴	1	8	yes	—
94	M₁₆×C₅₃	—	2	32	no	—
95	C₂×C₈×C₅₃	—	2	24	no	—
96	C₂×C₈×C₅₃	C₂⁴	1	8	yes	complemented
97	#11719917	—	2	32	no	—
98	C₂×#88322	—	2	24	no	—
99	C₂×#88322	C₂⁴	1	8	yes	complemented
100	C₂³×C₅₃	—	2	24	no	—
101	C₂³×C₅₃	—	2	32	no	—
102	C₂³×C₅₃	C₂×C₂²⋊C₄	1	24	yes	—
103	C₂³×C₅₃	C₂×C₂²⋊C₄	1	4	yes	—
104	C₂³×C₅₃	C₂×M₁₆	1	8	yes	—
105	C₂³×C₅₃	C₂³×C₄	1	4	yes	—
106	C₂³×C₅₃	C₂³×C₄	1	3	yes	—
107	C₂²×D₅₃	—	2	384	no	—
108	C₂²×D₅₃	—	2	32	no	—
109	C₂²×D₅₃	—	2	24	no	—
110	C₂²×D₅₃	—	2	24	no	—
111	C₂²×D₅₃	—	2	64	no	—
112	C₂²×D₅₃	—	2	64	no	—
113	C₂²×D₅₃	C₂×M₁₆	1	48	yes	complemented
114	C₂²×D₅₃	C₂³×C₄	1	8	yes	—
115	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	48	yes	—
116	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	12	yes	—
117	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	24	yes	—
118	C₂×C₄×C₅₃	C₂³×C₄	1	3	yes	—
119	C₂×C₄×C₅₃	C₂³×C₄	1	4	yes	—
120	C₂×C₄×C₅₃	C₂²×D₄	1	6	yes	—
121	C₂×C₄×C₅₃	C₂²×D₄	1	8	yes	—
122	C₂×C₄×C₅₃	C₂²×D₄	1	6	yes	—
123	C₂×C₄×C₅₃	C₂²×D₄	1	8	yes	—
124	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
125	C₂×Dic₅₃	—	2	32	no	—
126	C₂×Dic₅₃	—	2	24	no	—
127	C₂×Dic₅₃	C₂³×C₄	1	8	yes	—
128	C₄×D₅₃	—	2	48	no	—
129	C₄×D₅₃	—	2	64	no	—
130	C₄×D₅₃	—	2	8	no	—
131	C₄×D₅₃	—	2	8	no	—
132	C₈×C₅₃	—	2	8	no	—
133	#88322	—	2	8	no	—
134	C₂²×#407	—	53	1	no	maximal, Sylow
135	C₂²×C₅₃	—	2	48	no	—
136	C₂²×C₅₃	—	2	8	no	—
137	C₂²×C₅₃	#407	1	16	yes	complemented
138	C₂²×C₅₃	C₂²×C₂²⋊C₄	1	8	yes	—
139	C₂²×C₅₃	C₂²×C₂²⋊C₄	1	6	yes	—
140	C₂²×C₅₃	C₂²×M₁₆	1	12	yes	—
141	C₂²×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
142	C₂×D₅₃	—	2	128	no	—
143	C₂×D₅₃	—	2	96	no	—
144	C₂×D₅₃	—	2	8	no	—
145	C₂×D₅₃	—	2	8	no	—
146	C₂×D₅₃	C₂²×M₁₆	1	16	yes	complemented
147	C₄×C₅₃	C₂²×C₂²⋊C₄	1	6	yes	—
148	C₄×C₅₃	C₂²×C₂²⋊C₄	1	8	yes	—
149	C₄×C₅₃	C₂³×D₄	1	1	yes	characteristic
150	C₄×C₅₃	C₂³×D₄	1	1	yes	characteristic
151	Dic₅₃	—	2	8	no	—
152	C₂⁵×C₄	—	53	1	no	—
153	C₂³×M₁₆	—	53	1	no	—
154	C₂³×M₁₆	D₅₃	1	1	yes	characteristic, complemented
155	C₂×#407	—	53	24	no	—
156	C₂²×C₂²⋊C₈	—	53	4	no	—
157	C₂×C₅₃	—	2	8	no	—
158	C₂×C₅₃	C₂×#407	1	12	yes	complemented
159	C₂×C₅₃	C₂³×C₂²⋊C₄	1	1	yes	characteristic
160	C₂×C₅₃	C₂³×M₁₆	1	2	yes	—
161	D₅₃	—	2	16	no	—
162	C₂⁶	—	53	1	no	—
163	C₂⁴×C₄	—	53	1	no	—
164	C₂⁴×C₄	—	53	2	no	—
165	C₂⁴×C₄	—	53	12	no	—
166	C₂⁴×C₄	—	53	12	no	—
167	C₂⁴×C₄	—	53	2	no	—
168	C₂⁴×C₄	—	106	8	no	—
169	C₂⁴×C₄	—	106	8	no	—
170	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic
171	C₂²×M₁₆	—	2	8	no	—
172	C₂²×M₁₆	—	53	12	no	—
173	C₂²×M₁₆	—	106	8	no	—
174	C₂²×M₁₆	C₂×D₅₃	1	12	yes	complemented
175	C₂³×C₈	—	53	2	no	—
176	C₂³×C₈	C₂×D₅₃	1	2	yes	complemented
177	#407	—	53	64	no	—
178	C₂×C₂²⋊C₈	—	53	48	no	—
179	C₅₃	C₂²×#407	1	1	yes	characteristic, complemented, Sylow
180	C₂⁵	—	53	2	no	—
181	C₂⁵	—	53	12	no	—
182	C₂⁵	—	106	16	no	—
183	C₂⁵	—	106	8	no	—
184	C₂⁵	C₄×D₅₃	1	1	yes	characteristic
185	C₂³×C₄	—	53	12	no	—
186	C₂³×C₄	—	53	2	no	—
187	C₂³×C₄	—	53	24	no	—
188	C₂³×C₄	—	53	48	no	—
189	C₂³×C₄	—	53	12	no	—
190	C₂³×C₄	—	53	12	no	—
191	C₂³×C₄	—	53	16	no	—
192	C₂³×C₄	—	106	8	no	—
193	C₂³×C₄	—	106	8	no	—
194	C₂³×C₄	—	106	48	no	—
195	C₂³×C₄	—	106	8	no	—
196	C₂³×C₄	—	106	48	no	—
197	C₂³×C₄	—	106	96	no	—
198	C₂³×C₄	—	106	16	no	—
199	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
200	C₂³×C₄	C₄×D₅₃	1	6	yes	—
201	C₂³×C₄	D₂₁₂	1	8	yes	—
202	C₂³×C₄	C₂²⋊D₅₃	1	8	yes	—
203	C₂³×C₄	C₂²×D₅₃	1	6	yes	—
204	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
205	C₂×M₁₆	—	2	48	no	—
206	C₂×M₁₆	—	53	16	no	—
207	C₂×M₁₆	—	106	48	no	—
208	C₂×M₁₆	C₂²×D₅₃	1	16	yes	complemented
209	C₂²×C₈	—	2	8	no	—
210	C₂²×C₈	—	53	12	no	—
211	C₂²×C₈	—	106	8	no	—
212	C₂²×C₈	C₂²×D₅₃	1	12	yes	complemented
213	C₂²⋊C₈	—	53	64	no	—
214	C₂⁴	—	2	8	no	—
215	C₂⁴	—	53	16	no	—
216	C₂⁴	—	53	12	no	—
217	C₂⁴	—	53	16	no	—
218	C₂⁴	—	106	96	no	—
219	C₂⁴	—	106	128	no	—
220	C₂⁴	—	106	48	no	—
221	C₂⁴	—	106	8	no	—
222	C₂⁴	—	106	8	no	—
223	C₂⁴	#11719921	1	8	yes	—
224	C₂⁴	C₂×C₄×D₅₃	1	6	yes	—
225	C₂⁴	C₂×C₄×D₅₃	1	1	yes	characteristic
226	C₂²×C₄	—	53	16	no	—
227	C₂²×C₄	—	53	12	no	—
228	C₂²×C₄	—	53	24	no	—
229	C₂²×C₄	—	53	32	no	—
230	C₂²×C₄	—	53	8	no	—
231	C₂²×C₄	—	106	48	no	—
232	C₂²×C₄	—	106	8	no	—
233	C₂²×C₄	—	106	192	no	—
234	C₂²×C₄	—	106	96	no	—
235	C₂²×C₄	—	106	48	no	—
236	C₂²×C₄	—	106	24	no	—
237	C₂²×C₄	—	106	32	no	—
238	C₂²×C₄	—	106	24	no	—
239	C₂²×C₄	—	106	32	no	—
240	C₂²×C₄	#11719921	1	48	yes	—
241	C₂²×C₄	#11719921	1	8	yes	—
242	C₂²×C₄	C₂×C₄×D₅₃	1	12	yes	—
243	C₂²×C₄	C₂×C₄×D₅₃	1	6	yes	—
244	C₂²×C₄	C₂×C₄×D₅₃	1	3	yes	—
245	C₂²×C₄	C₂×D₂₁₂	1	4	yes	—
246	C₂²×C₄	C₂×D₂₁₂	1	24	yes	—
247	C₂²×C₄	C₂×C₂²⋊D₅₃	1	4	yes	—
248	C₂²×C₄	C₂×C₂²⋊D₅₃	1	24	yes	—
249	C₂²×C₄	C₂³×D₅₃	1	3	yes	—
250	C₂²×C₄	C₂³×D₅₃	1	4	yes	—
251	M₁₆	—	2	32	no	—
252	M₁₆	—	106	32	no	—
253	C₂×C₈	—	2	24	no	—
254	C₂×C₈	—	53	8	no	—
255	C₂×C₈	—	106	24	no	—
256	C₂×C₈	C₂³×D₅₃	1	8	yes	complemented
257	C₂³	—	2	32	no	—
258	C₂³	—	2	24	no	—
259	C₂³	—	53	48	no	—
260	C₂³	—	53	8	no	—
261	C₂³	—	106	64	no	—
262	C₂³	—	106	384	no	—
263	C₂³	—	106	32	no	—
264	C₂³	—	106	24	no	—
265	C₂³	—	106	64	no	—
266	C₂³	—	106	24	no	—
267	C₂³	C₂×#11719921	1	24	yes	—
268	C₂³	C₂×#11719921	1	4	yes	—
269	C₂³	M₁₆×D₅₃	1	8	yes	—
270	C₂³	C₂²×C₄×D₅₃	1	4	yes	—
271	C₂³	C₂²×C₄×D₅₃	1	3	yes	—
272	C₂×C₄	—	53	8	no	—
273	C₂×C₄	—	106	32	no	—
274	C₂×C₄	—	106	24	no	—
275	C₂×C₄	—	106	48	no	—
276	C₂×C₄	—	106	64	no	—
277	C₂×C₄	—	106	8	no	—
278	C₂×C₄	—	106	8	no	—
279	C₂×C₄	C₂×#11719921	1	48	yes	—
280	C₂×C₄	C₂×#11719921	1	24	yes	—
281	C₂×C₄	C₂×#11719921	1	12	yes	—
282	C₂×C₄	C₂²×C₄×D₅₃	1	4	yes	—
283	C₂×C₄	C₂²×C₄×D₅₃	1	3	yes	—
284	C₂×C₄	C₂²×C₂²⋊D₅₃	1	6	yes	—
285	C₂×C₄	C₂²×C₂²⋊D₅₃	1	8	yes	—
286	C₂×C₄	C₂²×D₂₁₂	1	6	yes	—
287	C₂×C₄	C₂²×D₂₁₂	1	8	yes	—
288	C₂×C₄	C₂⁴×D₅₃	1	1	yes	characteristic
289	C₈	—	2	8	no	—
290	C₈	—	106	8	no	—
291	C₂²	—	2	48	no	—
292	C₂²	—	2	8	no	—
293	C₂²	—	53	16	no	—
294	C₂²	—	106	128	no	—
295	C₂²	—	106	96	no	—
296	C₂²	—	106	8	no	—
297	C₂²	—	106	8	no	—
298	C₂²	#16810586	1	16	yes	direct product factor
299	C₂²	C₂²×#11719921	1	8	yes	—
300	C₂²	C₂²×#11719921	1	6	yes	—
301	C₂²	C₂×M₁₆×D₅₃	1	12	yes	—
302	C₂²	C₂³×C₄×D₅₃	1	1	yes	characteristic
303	C₄	—	106	8	no	—
304	C₄	C₂²×#11719921	1	6	yes	—
305	C₄	C₂²×#11719921	1	8	yes	—
306	C₄	C₂³×C₂²⋊D₅₃	1	1	yes	characteristic
307	C₄	C₂³×D₂₁₂	1	1	yes	characteristic
308	C₂	—	2	8	no	—
309	C₂	—	106	16	no	—
310	C₂	C₂×#16810586	1	12	yes	direct product factor
311	C₂	C₂³×#11719921	1	1	yes	characteristic
312	C₂	C₂²×M₁₆×D₅₃	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₅₃
Fitting	C₂³×M₁₆×C₅₃	C₂
socle	C₂⁴×C₅₃	C₂²×C₄
prime radical	C₂⁵×D₅₃	C₄
cosocle	C₂×C₄×C₅₃	C₂⁵
hypercenter	C₂³×M₁₆	D₅₃
hypocenter	C₅₃	C₂²×#407
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂³×M₁₆

Complements

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

N	H ≅ G/N	type
2. C₂³×#11719917	C₂	semidirect
3. C₂×#16810586	C₂	direct (N × H)
4. C₂²×#13133129	C₂	semidirect
5. C₂³×M₁₆×C₅₃	C₂	semidirect
15. C₂²×#11719917	C₂²	semidirect
16. C₂³×#88322	C₂²	semidirect
17. #16810586	C₂²	direct (N × H)
18. C₂×#13133129	C₂²	semidirect
21. C₂²×M₁₆×C₅₃	C₂²	semidirect
22. C₂³×C₈×C₅₃	C₂²	semidirect
42. C₂×#11719917	C₂³	semidirect
44. C₂²×#88322	C₂³	semidirect
45. #13133129	C₂³	semidirect
54. C₂×M₁₆×C₅₃	C₂³	semidirect
56. C₂²×C₈×C₅₃	C₂³	semidirect
66. C₂³×D₅₃	M₁₆	semidirect
96. C₂×C₈×C₅₃	C₂⁴	semidirect
99. C₂×#88322	C₂⁴	semidirect
113. C₂²×D₅₃	C₂×M₁₆	semidirect
137. C₂²×C₅₃	#407	semidirect
146. C₂×D₅₃	C₂²×M₁₆	semidirect
154. C₂³×M₁₆	D₅₃	semidirect
158. C₂×C₅₃	C₂×#407	semidirect
174. C₂²×M₁₆	C₂×D₅₃	semidirect
176. C₂³×C₈	C₂×D₅₃	semidirect
179. C₅₃	C₂²×#407	semidirect
208. C₂×M₁₆	C₂²×D₅₃	semidirect
212. C₂²×C₈	C₂²×D₅₃	semidirect
256. C₂×C₈	C₂³×D₅₃	semidirect
298. C₂²	#16810586	direct (N × H)
310. C₂	C₂×#16810586	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂⁴×C₄×D₅₃	2
C₂³×#11719917	2
C₂×#16810586	2
C₂²×#13133129	2
C₂³×M₁₆×C₅₃	2
C₂²×#407	53

Subgroups — C22×#16810586 #28247522

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16810586 #28247522