Subgroups — C₂×#20272796 #27298944

Order 13568.

670 conjugacy classes of subgroups.

The subgroup lattice has 312 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	#20272796	C₂	1	4	yes	maximal, direct product factor
4	C₂²×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂³×#88322	C₄	1	1	yes	characteristic
6	C₂²×C₄×Dic₅₃	C₄	1	1	yes	characteristic
7	C₂×#13133157	—	2	2	no	—
8	C₂×#13133157	—	2	1	no	—
9	C₂×#13133157	C₄	1	4	yes	complemented
10	C₂×#13133157	C₂²	1	2	yes	—
11	C₂²×#11719908	—	2	1	no	—
12	C₂×C₈×Dic₅₃	—	2	2	no	—
13	C₂×C₈×Dic₅₃	—	2	2	no	—
14	C₂×C₈×Dic₅₃	C₄	1	2	yes	—
15	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
16	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×M₃₂×C₅₃	—	2	2	no	—
18	C₂×M₃₂×C₅₃	—	2	1	no	—
19	C₂×M₃₂×C₅₃	C₄	1	4	yes	complemented
20	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
21	C₂²×C₁₆×C₅₃	—	2	1	no	—
22	C₂³×Dic₅₃	—	2	1	no	—
23	C₂²×#88322	—	2	2	no	—
24	C₂²×#88322	—	2	2	no	—
25	C₂²×#88322	C₈	1	2	yes	—
26	C₂²×#88322	C₈	1	2	yes	—
27	C₂²×#88322	C₂×C₄	1	2	yes	—
28	C₂×C₄×Dic₅₃	—	2	2	no	—
29	C₂×C₄×Dic₅₃	—	2	1	no	—
30	C₂×C₄×Dic₅₃	C₈	1	4	yes	—
31	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
32	#13133157	—	2	4	no	—
33	#13133157	—	2	2	no	—
34	#13133157	C₂×C₄	1	4	yes	complemented
35	C₂×#11719908	—	2	4	no	—
36	C₂×#11719908	—	2	2	no	—
37	C₈×Dic₅₃	—	2	4	no	—
38	C₈×Dic₅₃	—	2	2	no	—
39	C₈×Dic₅₃	—	2	2	no	—
40	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
42	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
43	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₅₃	D₄	1	2	yes	—
45	C₂²×C₈×C₅₃	D₄	1	2	yes	—
46	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
48	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
49	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
50	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
51	M₃₂×C₅₃	—	2	4	no	—
52	M₃₂×C₅₃	—	2	2	no	—
53	M₃₂×C₅₃	C₂×C₄	1	4	yes	complemented
54	C₂×C₁₆×C₅₃	—	2	4	no	—
55	C₂×C₁₆×C₅₃	—	2	2	no	—
56	C₂⁴×C₅₃	M₁₆	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
58	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
59	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
64	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	M₁₆	1	2	yes	—
66	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×Dic₅₃	—	2	8	no	—
68	C₂²×Dic₅₃	—	2	4	no	—
69	C₂²×Dic₅₃	—	2	2	no	—
70	C₂×C₈×C₅₃	C₄²	1	2	yes	—
71	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
72	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
73	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
74	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
75	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
76	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
79	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
80	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
81	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
82	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
88	C₄×Dic₅₃	—	2	4	no	—
89	C₄×Dic₅₃	—	2	2	no	—
90	C₄×Dic₅₃	C₂×C₈	1	4	yes	—
91	C₂×#88322	—	2	4	no	—
92	C₂×#88322	—	2	4	no	—
93	C₂×#88322	—	2	2	no	—
94	C₂×#88322	—	2	2	no	—
95	C₂×#88322	C₂×C₈	1	4	yes	—
96	C₁₆×C₅₃	—	2	4	no	—
97	#11719908	—	2	4	no	—
98	C₂³×C₅₃	—	2	4	no	—
99	C₂³×C₅₃	#97	1	2	yes	—
100	C₂³×C₅₃	C₂²⋊C₈	1	2	yes	—
101	C₂³×C₅₃	C₄⋊C₈	1	1	yes	characteristic
102	C₂³×C₅₃	C₄⋊C₈	1	1	yes	characteristic
103	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
104	C₂×C₄×C₅₃	#95	1	2	yes	—
105	C₂×C₄×C₅₃	#95	1	2	yes	—
106	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
107	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
108	C₂×C₄×C₅₃	#97	1	2	yes	—
109	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
110	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
111	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
112	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
113	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
114	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
115	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
116	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
117	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
120	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
121	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
122	C₂×Dic₅₃	—	2	16	no	—
123	C₂×Dic₅₃	—	2	8	no	—
124	C₂×Dic₅₃	—	2	4	no	—
125	C₈×C₅₃	#95	1	2	yes	—
126	C₈×C₅₃	#95	1	2	yes	—
127	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
128	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
129	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
130	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
131	#88322	—	2	4	no	—
132	C₂×#1376	—	53	1	no	maximal, Sylow
133	C₂²×C₅₃	—	2	4	no	—
134	C₂²×C₅₃	—	2	4	no	—
135	C₂²×C₅₃	—	2	2	no	—
136	C₂²×C₅₃	—	2	2	no	—
137	C₂²×C₅₃	#336	1	2	yes	—
138	C₂²×C₅₃	#336	1	2	yes	—
139	C₂²×C₅₃	#364	1	4	yes	—
140	C₂²×C₅₃	C₂×#97	1	1	yes	characteristic
141	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
142	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
143	C₄×C₅₃	#336	1	2	yes	—
144	C₄×C₅₃	#336	1	2	yes	—
145	C₄×C₅₃	C₂×#95	1	1	yes	characteristic
146	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
147	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
148	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
149	Dic₅₃	—	2	8	no	—
150	C₂²×M₃₂	—	53	1	no	—
151	C₂²×M₃₂	D₅₃	1	1	yes	characteristic
152	C₂²×C₄×C₈	—	53	1	no	—
153	#1376	—	53	4	no	—
154	C₂×C₅₃	—	2	1	no	—
155	C₂×C₅₃	—	2	2	no	—
156	C₂×C₅₃	—	2	1	no	—
157	C₂×C₅₃	#1376	1	4	yes	complemented
158	C₂×C₅₃	C₂×#336	1	1	yes	characteristic
159	C₂×C₅₃	C₂×#364	1	2	yes	—
160	C₂³×C₈	—	53	1	no	—
161	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
162	C₂²×C₄²	—	53	1	no	—
163	C₂×M₃₂	—	2	1	no	—
164	C₂×M₃₂	—	2	2	no	—
165	C₂×M₃₂	—	53	4	no	—
166	C₂×M₃₂	—	53	2	no	—
167	C₂×M₃₂	—	106	2	no	—
168	C₂×M₃₂	—	106	1	no	—
169	C₂×M₃₂	Dic₅₃	1	4	yes	complemented
170	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
171	C₂²×C₁₆	—	2	1	no	—
172	C₂²×C₁₆	—	106	1	no	—
173	C₂×C₄×C₈	—	53	2	no	—
174	C₂×C₄×C₈	—	53	2	no	—
175	C₂×C₄×C₈	—	106	2	no	—
176	C₂×C₄×C₈	—	106	2	no	—
177	C₅₃	C₂×#1376	1	1	yes	characteristic, complemented, Sylow
178	C₂³×C₄	—	106	1	no	—
179	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
180	C₂²×C₈	—	53	2	no	—
181	C₂²×C₈	—	53	2	no	—
182	C₂²×C₈	—	53	2	no	—
183	C₂²×C₈	—	106	2	no	—
184	C₂²×C₈	—	106	2	no	—
185	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
186	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
187	C₂²×C₈	C₄×D₅₃	1	2	yes	—
188	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
189	C₂²×C₈	D₂₁₂	1	2	yes	—
190	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
191	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
192	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
193	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
194	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
195	C₂×C₄²	—	53	2	no	—
196	C₂×C₄²	—	53	4	no	—
197	C₂×C₄²	—	106	1	no	—
198	C₂×C₄²	—	106	2	no	—
199	M₃₂	—	2	2	no	—
200	M₃₂	—	2	4	no	—
201	M₃₂	—	53	4	no	—
202	M₃₂	—	106	2	no	—
203	M₃₂	—	106	4	no	—
204	M₃₂	C₂×Dic₅₃	1	4	yes	complemented
205	C₂×C₁₆	—	2	2	no	—
206	C₂×C₁₆	—	2	4	no	—
207	C₂×C₁₆	—	106	2	no	—
208	C₂×C₁₆	—	106	4	no	—
209	C₄×C₈	—	106	2	no	—
210	C₄×C₈	—	106	2	no	—
211	C₄×C₈	—	106	4	no	—
212	C₂⁴	#11719912	1	1	yes	characteristic
213	C₂²×C₄	—	106	8	no	—
214	C₂²×C₄	—	106	4	no	—
215	C₂²×C₄	—	106	2	no	—
216	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
217	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
218	C₂²×C₄	C₈×D₅₃	1	2	yes	—
219	C₂²×C₄	#11719912	1	1	yes	characteristic
220	C₂²×C₄	#11719912	1	2	yes	—
221	C₂²×C₄	C₄×Dic₅₃	1	2	yes	—
222	C₂²×C₄	#11719919	1	1	yes	characteristic
223	C₂²×C₄	#11719919	1	1	yes	characteristic
224	C₂²×C₄	#11719921	1	2	yes	—
225	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
226	C₂×C₈	—	53	4	no	—
227	C₂×C₈	—	106	4	no	—
228	C₂×C₈	—	106	4	no	—
229	C₂×C₈	—	106	2	no	—
230	C₂×C₈	—	106	2	no	—
231	C₂×C₈	C₄×Dic₅₃	1	2	yes	—
232	C₂×C₈	#11719919	1	1	yes	characteristic
233	C₂×C₈	#11719919	1	1	yes	characteristic
234	C₂×C₈	#11719919	1	2	yes	—
235	C₂×C₈	#11719919	1	2	yes	—
236	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
237	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
238	C₂×C₈	#11719921	1	2	yes	—
239	C₂×C₈	#11719921	1	2	yes	—
240	C₂×C₈	#11719921	1	2	yes	—
241	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
242	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
243	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
244	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
245	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
246	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
247	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
248	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
249	C₄²	—	53	4	no	—
250	C₄²	—	106	4	no	—
251	C₄²	—	106	2	no	—
252	C₁₆	—	2	4	no	—
253	C₁₆	—	106	4	no	—
254	C₂³	—	2	4	no	—
255	C₂³	#13133127	1	2	yes	—
256	C₂³	#13133129	1	2	yes	—
257	C₂³	#13133146	1	1	yes	characteristic
258	C₂³	#13133146	1	1	yes	characteristic
259	C₂³	C₂×#11719912	1	1	yes	characteristic
260	C₂×C₄	—	106	16	no	—
261	C₂×C₄	—	106	8	no	—
262	C₂×C₄	—	106	4	no	—
263	C₂×C₄	#13133124	1	2	yes	—
264	C₂×C₄	#13133124	1	2	yes	—
265	C₂×C₄	C₈×Dic₅₃	1	2	yes	—
266	C₂×C₄	C₈×Dic₅₃	1	2	yes	—
267	C₂×C₄	#13133127	1	2	yes	—
268	C₂×C₄	#13133129	1	2	yes	—
269	C₂×C₄	#13133129	1	2	yes	—
270	C₂×C₄	#13133129	1	2	yes	—
271	C₂×C₄	#13133146	1	1	yes	characteristic
272	C₂×C₄	#13133146	1	1	yes	characteristic
273	C₂×C₄	#13133146	1	2	yes	—
274	C₂×C₄	#13133146	1	2	yes	—
275	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
276	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
277	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
278	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
279	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
280	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
281	C₈	—	106	4	no	—
282	C₈	#13133124	1	2	yes	—
283	C₈	#13133124	1	2	yes	—
284	C₈	C₂×#11719921	1	1	yes	characteristic
285	C₈	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
286	C₈	C₂×#11719919	1	1	yes	characteristic
287	C₈	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
288	C₂²	—	2	4	no	—
289	C₂²	—	2	4	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	2	2	no	—
292	C₂²	#16810377	1	2	yes	—
293	C₂²	#16810377	1	2	yes	—
294	C₂²	#16810443	1	4	yes	—
295	C₂²	C₂×#13133127	1	1	yes	characteristic
296	C₂²	C₂×#13133129	1	1	yes	characteristic
297	C₂²	C₂×#13133146	1	1	yes	characteristic
298	C₄	—	106	8	no	—
299	C₄	#16810377	1	2	yes	—
300	C₄	#16810377	1	2	yes	—
301	C₄	C₂×#13133124	1	1	yes	characteristic
302	C₄	C₂×C₈×Dic₅₃	1	1	yes	characteristic
303	C₄	C₂×#13133129	1	1	yes	characteristic
304	C₄	C₂×#13133146	1	1	yes	characteristic
305	C₂	—	2	1	no	—
306	C₂	—	2	1	no	—
307	C₂	—	2	2	no	—
308	C₂	#20272796	1	4	yes	direct product factor
309	C₂	C₂×#16810377	1	1	yes	characteristic
310	C₂	C₂×#16810443	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈	C₂²⋊D₅₃
commutator	C₄×C₅₃	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂²×M₃₂×C₅₃	C₂
socle	C₂³×C₅₃	C₄⋊C₈
prime radical	C₂⁴×C₅₃	M₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂²×M₃₂	D₅₃
hypocenter	C₅₃	C₂×#1376
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂³×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
3. #20272796	C₂	direct (N × H)
9. C₂×#13133157	C₄	semidirect
19. C₂×M₃₂×C₅₃	C₄	semidirect
34. #13133157	C₂×C₄	semidirect
53. M₃₂×C₅₃	C₂×C₄	semidirect
157. C₂×C₅₃	#1376	semidirect
169. C₂×M₃₂	Dic₅₃	semidirect
177. C₅₃	C₂×#1376	semidirect
204. M₃₂	C₂×Dic₅₃	semidirect
308. C₂	#20272796	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
#20272796	2
C₂²×M₃₂×C₅₃	2
C₂×#1376	53

Subgroups — C2×#20272796 #27298944

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20272796 #27298944