Subgroups — C₂×#20272804 #27298961

Order 13568.

606 conjugacy classes of subgroups.

The subgroup lattice has 322 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	C₂³×#11719908	C₂	1	1	yes	maximal, characteristic
3	#20272804	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₅₃×#405	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133157	—	2	1	no	—
6	C₂×#13133157	—	2	2	no	—
7	C₂×#13133157	C₄	1	4	yes	complemented
8	C₂×#13133157	C₂²	1	2	yes	—
9	C₂²×#11719908	—	2	1	no	—
10	C₂²×#11719908	—	2	1	no	—
11	C₂²×#11719908	—	2	2	no	—
12	C₂²×#11719908	—	2	1	no	—
13	C₂²×#11719908	C₄	1	4	yes	complemented
14	C₂²×#11719908	C₂²	1	2	yes	—
15	C₂²×M₁₆×C₅₃	C₄	1	1	yes	characteristic
16	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×#117×C₅₃	C₄	1	1	yes	characteristic
18	C₅₃×#405	—	2	1	no	—
19	C₅₃×#405	—	2	1	no	—
20	C₅₃×#405	C₄	1	2	yes	—
21	C₅₃×#405	C₂²	1	2	yes	—
22	C₂×#97×C₅₃	—	2	1	no	—
23	C₂×C₄×C₈×C₅₃	—	2	1	no	—
24	#13133157	—	2	4	no	—
25	#13133157	—	2	2	no	—
26	#13133157	C₂×C₄	1	4	yes	complemented
27	C₂×#11719908	—	2	4	no	—
28	C₂×#11719908	—	2	4	no	—
29	C₂×#11719908	—	2	4	no	—
30	C₂×#11719908	—	2	2	no	—
31	C₂×#11719908	—	2	2	no	—
32	C₂×#11719908	—	2	2	no	—
33	C₂×#11719908	C₂×C₄	1	4	yes	complemented
34	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂×M₁₆×C₅₃	—	2	1	no	—
36	C₂×M₁₆×C₅₃	—	2	2	no	—
37	C₂×M₁₆×C₅₃	C₈	1	4	yes	—
38	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
39	C₂²×C₈×C₅₃	—	2	1	no	—
40	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
41	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
42	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₅₃	D₄	1	2	yes	—
48	C₂²×C₈×C₅₃	Q₈	1	2	yes	—
49	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
50	#117×C₅₃	—	2	1	no	—
51	#117×C₅₃	—	2	1	no	—
52	#117×C₅₃	C₈	1	2	yes	—
53	#117×C₅₃	C₂×C₄	1	2	yes	—
54	C₂×C₄⋊C₄×C₅₃	C₈	1	2	yes	—
55	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
56	C₂×C₄²×C₅₃	—	2	1	no	—
57	#97×C₅₃	—	2	2	no	—
58	#97×C₅₃	—	2	2	no	—
59	C₄×C₈×C₅₃	—	2	2	no	—
60	C₄×C₈×C₅₃	—	2	2	no	—
61	C₂⁴×C₅₃	M₁₆	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	—	2	2	no	—
63	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
64	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
67	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
68	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
69	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
70	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	M₁₆	1	2	yes	—
72	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
73	M₁₆×C₅₃	—	2	4	no	—
74	M₁₆×C₅₃	—	2	2	no	—
75	M₁₆×C₅₃	C₂×C₈	1	4	yes	—
76	C₂×C₈×C₅₃	—	2	4	no	—
77	C₂×C₈×C₅₃	—	2	2	no	—
78	C₂×C₈×C₅₃	C₄²	1	2	yes	—
79	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
82	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
83	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
84	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
85	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
86	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
87	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
88	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
89	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
90	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
96	C₄⋊C₄×C₅₃	—	2	2	no	—
97	C₄⋊C₄×C₅₃	C₂×C₈	1	4	yes	—
98	C₂²⋊C₄×C₅₃	—	2	2	no	—
99	C₂²⋊C₄×C₅₃	—	2	2	no	—
100	C₄²×C₅₃	—	2	2	no	—
101	C₄²×C₅₃	—	2	2	no	—
102	#11719908	—	2	4	no	—
103	#11719908	—	2	4	no	—
104	C₂³×C₅₃	—	2	4	no	—
105	C₂³×C₅₃	#97	1	2	yes	—
106	C₂³×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
107	C₂³×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
108	C₂³×C₅₃	C₄⋊C₈	1	2	yes	—
109	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
110	C₂×C₄×C₅₃	—	2	4	no	—
111	C₂×C₄×C₅₃	—	4	4	no	—
112	C₂×C₄×C₅₃	#95	1	2	yes	—
113	C₂×C₄×C₅₃	#95	1	2	yes	—
114	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
115	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
116	C₂×C₄×C₅₃	#97	1	2	yes	—
117	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
118	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
119	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
120	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
121	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
122	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
123	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
124	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
125	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
128	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
129	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
130	C₈×C₅₃	—	2	4	no	—
131	C₈×C₅₃	#95	1	2	yes	—
132	C₈×C₅₃	#95	1	2	yes	—
133	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
134	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
135	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
136	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
137	C₂×#1379	—	53	1	no	maximal, Sylow
138	C₂²×C₅₃	—	2	2	no	—
139	C₂²×C₅₃	—	2	2	no	—
140	C₂²×C₅₃	—	2	4	no	—
141	C₂²×C₅₃	—	2	4	no	—
142	C₂²×C₅₃	#336	1	2	yes	—
143	C₂²×C₅₃	#336	1	2	yes	—
144	C₂²×C₅₃	#350	1	4	yes	—
145	C₂²×C₅₃	C₂×#97	1	1	yes	characteristic
146	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
147	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
148	C₄×C₅₃	—	4	4	no	—
149	C₄×C₅₃	#336	1	2	yes	—
150	C₄×C₅₃	#336	1	2	yes	—
151	C₄×C₅₃	C₂×#95	1	1	yes	characteristic
152	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
153	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
154	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
155	C₂²×M₃₂	—	53	1	no	—
156	C₂³×C₁₆	—	53	1	no	—
157	C₂×#405	D₅₃	1	1	yes	characteristic
158	#1379	—	53	4	no	—
159	C₂×C₅₃	—	2	2	no	—
160	C₂×C₅₃	—	2	1	no	—
161	C₂×C₅₃	—	2	1	no	—
162	C₂×C₅₃	#1379	1	4	yes	complemented
163	C₂×C₅₃	C₂×#336	1	1	yes	characteristic
164	C₂×C₅₃	C₂×#350	1	2	yes	—
165	C₂²×M₁₆	Dic₅₃	1	1	yes	characteristic
166	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
167	C₂×#117	Dic₅₃	1	1	yes	characteristic
168	C₂×M₃₂	—	53	4	no	—
169	C₂×M₃₂	—	53	2	no	—
170	C₂×M₃₂	—	106	2	no	—
171	C₂×M₃₂	—	106	1	no	—
172	C₂²×C₁₆	—	53	4	no	—
173	C₂²×C₁₆	—	53	2	no	—
174	C₂²×C₁₆	—	106	2	no	—
175	C₂²×C₁₆	—	106	1	no	—
176	C₂²×C₁₆	—	106	1	no	—
177	C₂²×C₁₆	—	106	1	no	—
178	#405	—	2	1	no	—
179	#405	—	2	1	no	—
180	#405	Dic₅₃	1	2	yes	—
181	#405	C₂×D₅₃	1	2	yes	—
182	C₂×#97	—	2	1	no	—
183	C₂×C₄×C₈	—	2	1	no	—
184	C₅₃	C₂×#1379	1	1	yes	characteristic, complemented, Sylow
185	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
186	C₂×M₁₆	—	2	2	no	—
187	C₂×M₁₆	—	2	1	no	—
188	C₂×M₁₆	#88322	1	4	yes	—
189	C₂×M₁₆	C₂×Dic₅₃	1	2	yes	—
190	C₂²×C₈	—	2	1	no	—
191	C₂²×C₈	Dic₁₀₆	1	2	yes	—
192	C₂²×C₈	C₄×D₅₃	1	2	yes	—
193	C₂²×C₈	C₄×D₅₃	1	2	yes	—
194	C₂²×C₈	D₂₁₂	1	2	yes	—
195	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
196	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
197	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
198	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
199	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
200	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
201	#117	—	2	1	no	—
202	#117	—	2	1	no	—
203	#117	#88322	1	2	yes	—
204	#117	C₂×Dic₅₃	1	2	yes	—
205	C₂×C₄⋊C₄	#88322	1	2	yes	—
206	C₂×C₂²⋊C₄	—	2	1	no	—
207	C₂×C₄²	—	2	1	no	—
208	M₃₂	—	53	4	no	—
209	M₃₂	—	106	2	no	—
210	M₃₂	—	106	4	no	—
211	C₂×C₁₆	—	53	4	no	—
212	C₂×C₁₆	—	106	2	no	—
213	C₂×C₁₆	—	106	2	no	—
214	C₂×C₁₆	—	106	2	no	—
215	C₂×C₁₆	—	106	4	no	—
216	C₂×C₁₆	—	106	4	no	—
217	C₂×C₁₆	—	106	4	no	—
218	#97	—	2	2	no	—
219	#97	—	2	2	no	—
220	C₄×C₈	—	2	2	no	—
221	C₄×C₈	—	2	2	no	—
222	C₂⁴	#11719917	1	1	yes	characteristic
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	C₂×#88322	1	2	yes	—
225	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
226	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
227	C₂²×C₄	#11719917	1	1	yes	characteristic
228	C₂²×C₄	#11719917	1	2	yes	—
229	C₂²×C₄	C₄×Dic₅₃	1	2	yes	—
230	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
231	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
232	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
233	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
234	M₁₆	—	2	4	no	—
235	M₁₆	—	2	2	no	—
236	M₁₆	C₂×#88322	1	4	yes	—
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	C₄×Dic₅₃	1	2	yes	—
240	C₂×C₈	#11719919	1	2	yes	—
241	C₂×C₈	#11719919	1	2	yes	—
242	C₂×C₈	#11719919	1	2	yes	—
243	C₂×C₈	#11719919	1	2	yes	—
244	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
245	C₂×C₈	#11719921	1	2	yes	—
246	C₂×C₈	#11719921	1	2	yes	—
247	C₂×C₈	#11719921	1	2	yes	—
248	C₂×C₈	#11719921	1	2	yes	—
249	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
250	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
251	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
252	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
253	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
254	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
255	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
256	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	C₂×#88322	1	4	yes	—
259	C₂²⋊C₄	—	2	2	no	—
260	C₂²⋊C₄	—	2	2	no	—
261	C₄²	—	2	2	no	—
262	C₄²	—	2	2	no	—
263	C₁₆	—	106	4	no	—
264	C₁₆	—	106	4	no	—
265	C₂³	—	2	4	no	—
266	C₂³	#13133128	1	2	yes	—
267	C₂³	C₂²⋊#88322	1	1	yes	characteristic
268	C₂³	C₂²⋊#88322	1	1	yes	characteristic
269	C₂³	C₄⋊#88322	1	2	yes	—
270	C₂³	C₂×#11719917	1	1	yes	characteristic
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	4	4	no	—
273	C₂×C₄	#13133124	1	2	yes	—
274	C₂×C₄	#13133124	1	2	yes	—
275	C₂×C₄	C₄×#88322	1	2	yes	—
276	C₂×C₄	C₄×#88322	1	2	yes	—
277	C₂×C₄	#13133128	1	2	yes	—
278	C₂×C₄	C₂²⋊#88322	1	2	yes	—
279	C₂×C₄	C₂²⋊#88322	1	2	yes	—
280	C₂×C₄	C₂²⋊#88322	1	1	yes	characteristic
281	C₂×C₄	C₂²⋊#88322	1	1	yes	characteristic
282	C₂×C₄	C₄⋊#88322	1	2	yes	—
283	C₂×C₄	C₄⋊#88322	1	2	yes	—
284	C₂×C₄	C₄⋊#88322	1	2	yes	—
285	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
286	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
287	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
288	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
289	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
290	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
291	C₈	—	2	4	no	—
292	C₈	#13133124	1	2	yes	—
293	C₈	#13133124	1	2	yes	—
294	C₈	C₂×#11719921	1	1	yes	characteristic
295	C₈	C₂×#11719921	1	1	yes	characteristic
296	C₈	C₂×#11719919	1	1	yes	characteristic
297	C₈	C₂×#11719919	1	1	yes	characteristic
298	C₂²	—	2	2	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	4	no	—
301	C₂²	—	2	4	no	—
302	C₂²	#16810378	1	2	yes	—
303	C₂²	#16810378	1	2	yes	—
304	C₂²	#16810407	1	4	yes	—
305	C₂²	C₂×#13133128	1	1	yes	characteristic
306	C₂²	C₂×C₂²⋊#88322	1	1	yes	characteristic
307	C₂²	C₂×C₄⋊#88322	1	1	yes	characteristic
308	C₄	—	4	4	no	—
309	C₄	#16810378	1	2	yes	—
310	C₄	#16810378	1	2	yes	—
311	C₄	C₂×#13133124	1	1	yes	characteristic
312	C₄	C₂×C₄×#88322	1	1	yes	characteristic
313	C₄	C₂×C₂²⋊#88322	1	1	yes	characteristic
314	C₄	C₂×C₄⋊#88322	1	1	yes	characteristic
315	C₂	—	2	1	no	—
316	C₂	—	2	2	no	—
317	C₂	—	2	1	no	—
318	C₂	#20272804	1	4	yes	direct product factor
319	C₂	C₂×#16810378	1	1	yes	characteristic
320	C₂	C₂×#16810407	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈	C₂²⋊D₅₃
commutator	C₄×C₅₃	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#405	C₂
socle	C₂³×C₅₃	C₂²⋊C₈
prime radical	C₂⁴×C₅₃	M₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#405	D₅₃
hypocenter	C₅₃	C₂×#1379
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂³×C₈ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
3. #20272804	C₂	direct (N × H)
7. C₂×#13133157	C₄	semidirect
13. C₂²×#11719908	C₄	semidirect
26. #13133157	C₂×C₄	semidirect
33. C₂×#11719908	C₂×C₄	semidirect
162. C₂×C₅₃	#1379	semidirect
184. C₅₃	C₂×#1379	semidirect
318. C₂	#20272804	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
C₂³×#11719908	2
#20272804	2
C₂×C₅₃×#405	2
C₂×#1379	53

Subgroups — C2×#20272804 #27298961

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20272804 #27298961