Subgroups — C₂×#20276480 #27537091

Order 13568.

698 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	C₂×#16810578	C₂	1	1	yes	maximal, characteristic
3	#20276480	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16810443	C₂	1	4	yes	maximal, complemented
5	C₂²×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719917	C₄	1	1	yes	characteristic
7	C₂×#13133176	C₄	1	1	yes	characteristic
8	C₂×#13133157	—	2	1	no	—
9	C₂×#13133157	—	2	2	no	—
10	C₂×#13133157	C₂²	1	4	yes	—
11	C₂×#13133157	C₂²	1	2	yes	—
12	C₂²×#11719908	—	2	1	no	—
13	#16810578	C₄	1	4	yes	—
14	#16810578	C₂²	1	4	yes	—
15	C₂×#13133127	C₄	1	2	yes	—
16	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
17	#16810443	C₂²	1	16	yes	complemented
18	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×M₃₂×C₅₃	—	2	1	no	—
20	C₂×M₃₂×C₅₃	—	2	2	no	—
21	C₂×M₃₂×C₅₃	C₂²	1	4	yes	—
22	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
23	C₂²×C₁₆×C₅₃	—	2	1	no	—
24	C₂×#11719917	C₈	1	2	yes	—
25	C₂×#11719917	C₈	1	2	yes	—
26	C₂×#11719917	C₈	1	4	yes	—
27	C₂×#11719917	C₂×C₄	1	4	yes	—
28	C₂²×#88322	C₂×C₄	1	2	yes	—
29	#13133176	—	2	2	no	—
30	#13133176	C₂×C₄	1	4	yes	—
31	C₂×C₄⋊Dic₅₃	—	2	1	no	—
32	C₂×C₂²⋊Dic₅₃	—	2	1	no	—
33	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
34	#13133157	—	2	4	no	—
35	#13133157	—	2	2	no	—
36	#13133157	C₂³	1	4	yes	—
37	C₂×#11719908	—	2	4	no	—
38	C₂×#11719908	—	2	2	no	—
39	#13133127	C₂×C₄	1	8	yes	—
40	C₈×Dic₅₃	C₂×C₄	1	4	yes	—
41	C₈×Dic₅₃	C₂³	1	4	yes	—
42	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
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	1	yes	characteristic
49	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
50	C₂²×C₈×C₅₃	Q₈	1	2	yes	—
51	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
53	M₃₂×C₅₃	—	2	4	no	—
54	M₃₂×C₅₃	—	2	2	no	—
55	M₃₂×C₅₃	C₂³	1	4	yes	—
56	C₂×C₁₆×C₅₃	—	2	4	no	—
57	C₂×C₁₆×C₅₃	—	2	2	no	—
58	C₂⁴×C₅₃	M₁₆	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
62	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
65	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	M₁₆	1	2	yes	—
67	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
69	C₂²×Dic₅₃	—	2	2	no	—
70	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
73	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
74	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
75	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
76	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
77	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
80	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
81	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
84	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
85	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
86	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
87	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
88	C₂²⋊Dic₅₃	—	2	4	no	—
89	C₄⋊Dic₅₃	—	2	4	no	—
90	C₄×Dic₅₃	—	2	2	no	—
91	C₄×Dic₅₃	C₂²×C₄	1	4	yes	—
92	#11719917	C₂×C₈	1	8	yes	—
93	#11719917	C₂×C₈	1	8	yes	—
94	C₂×#88322	C₂×C₈	1	4	yes	—
95	C₂×#88322	C₂×C₈	1	4	yes	—
96	C₂×#88322	C₂²×C₄	1	4	yes	—
97	C₁₆×C₅₃	—	2	4	no	—
98	#11719908	—	2	4	no	—
99	C₂³×C₅₃	—	4	2	no	—
100	C₂³×C₅₃	C₄⋊C₈	1	1	yes	characteristic
101	C₂³×C₅₃	C₄⋊C₈	1	1	yes	characteristic
102	C₂³×C₅₃	C₄⋊C₈	1	2	yes	—
103	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
104	C₂³×C₅₃	C₂×M₁₆	1	2	yes	—
105	C₂×C₄×C₅₃	C₄⋊C₈	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	C₄⋊C₈	1	2	yes	—
109	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
110	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
111	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
112	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
114	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
115	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
117	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
118	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
119	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
120	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
121	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
122	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
123	C₂×Dic₅₃	—	2	4	no	—
124	C₂×Dic₅₃	—	4	4	no	—
125	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
127	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
128	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₈×C₅₃	C₂²×D₄	1	1	yes	characteristic
130	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
131	#88322	C₂²×C₈	1	8	yes	—
132	C₂×#2153	—	53	1	no	maximal, Sylow
133	C₂²×C₅₃	—	2	2	no	—
134	C₂²×C₅₃	—	2	2	no	—
135	C₂²×C₅₃	—	2	2	no	—
136	C₂²×C₅₃	—	4	2	no	—
137	C₂²×C₅₃	—	4	2	no	—
138	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
139	C₂²×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
140	C₂²×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
141	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
142	C₂²×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
143	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
144	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
145	C₄×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
146	C₄×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
147	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
148	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
149	Dic₅₃	—	4	4	no	—
150	C₂²×M₃₂	—	53	1	no	—
151	C₂²×M₃₂	D₅₃	1	1	yes	characteristic
152	C₂×#405	—	53	1	no	—
153	#2153	—	53	8	no	—
154	C₂×#364	—	53	4	no	—
155	C₂×C₅₃	—	2	1	no	—
156	C₂×C₅₃	—	2	1	no	—
157	C₂×C₅₃	—	2	2	no	—
158	C₂×C₅₃	—	2	2	no	—
159	C₂×C₅₃	#2153	1	2	yes	complemented
160	C₂×C₅₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
161	C₂²×M₁₆	—	53	1	no	—
162	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
163	C₂×#117	—	53	1	no	—
164	C₂×M₃₂	—	2	2	no	—
165	C₂×M₃₂	—	2	1	no	—
166	C₂×M₃₂	—	53	2	no	—
167	C₂×M₃₂	—	53	4	no	—
168	C₂×M₃₂	—	106	2	no	—
169	C₂×M₃₂	—	106	1	no	—
170	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
171	C₂×M₃₂	C₂×D₅₃	1	4	yes	—
172	C₂²×C₁₆	—	2	1	no	—
173	C₂²×C₁₆	—	106	1	no	—
174	#405	—	53	4	no	—
175	#405	—	53	4	no	—
176	C₂×#97	—	53	2	no	—
177	C₂×C₄×C₈	—	53	2	no	—
178	#364	—	53	16	no	—
179	C₅₃	C₂×#2153	1	1	yes	characteristic, complemented, Sylow
180	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
181	C₂×M₁₆	—	53	4	no	—
182	C₂×M₁₆	—	53	2	no	—
183	C₂×M₁₆	—	53	2	no	—
184	C₂×M₁₆	—	53	4	no	—
185	C₂²×C₈	—	53	2	no	—
186	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
187	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
188	C₂²×C₈	Dic₁₀₆	1	2	yes	—
189	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
190	C₂²×C₈	C₄×D₅₃	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	2	yes	—
195	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
196	#117	—	53	4	no	—
197	#117	—	106	2	no	—
198	C₂×C₄⋊C₄	—	106	1	no	—
199	C₂×C₂²⋊C₄	—	106	1	no	—
200	C₂×C₄²	—	53	2	no	—
201	M₃₂	—	2	4	no	—
202	M₃₂	—	2	2	no	—
203	M₃₂	—	53	4	no	—
204	M₃₂	—	106	4	no	—
205	M₃₂	—	106	2	no	—
206	M₃₂	C₂²×D₅₃	1	4	yes	—
207	C₂×C₁₆	—	2	4	no	—
208	C₂×C₁₆	—	2	2	no	—
209	C₂×C₁₆	—	106	4	no	—
210	C₂×C₁₆	—	106	2	no	—
211	#97	—	53	8	no	—
212	C₄×C₈	—	53	4	no	—
213	C₄×C₈	—	53	4	no	—
214	C₂⁴	#11719912	1	1	yes	characteristic
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	#11719919	1	2	yes	—
222	C₂²×C₄	#11719919	1	1	yes	characteristic
223	C₂²×C₄	#11719919	1	1	yes	characteristic
224	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
225	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
226	M₁₆	—	53	8	no	—
227	M₁₆	—	53	8	no	—
228	C₂×C₈	—	53	4	no	—
229	C₂×C₈	—	53	4	no	—
230	C₂×C₈	—	53	4	no	—
231	C₂×C₈	#11719919	1	2	yes	—
232	C₂×C₈	#11719919	1	2	yes	—
233	C₂×C₈	#11719919	1	2	yes	—
234	C₂×C₈	#11719919	1	1	yes	characteristic
235	C₂×C₈	#11719919	1	1	yes	characteristic
236	C₂×C₈	#11719919	1	2	yes	—
237	C₂×C₈	#11719919	1	2	yes	—
238	C₂×C₈	C₂×Dic₁₀₆	1	2	yes	—
239	C₂×C₈	C₂×Dic₁₀₆	1	2	yes	—
240	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
241	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
242	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
243	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
244	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
245	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
246	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
247	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
248	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
249	C₄⋊C₄	—	106	4	no	—
250	C₂²⋊C₄	—	106	4	no	—
251	C₄²	—	53	4	no	—
252	C₄²	—	106	2	no	—
253	C₁₆	—	2	4	no	—
254	C₁₆	—	106	4	no	—
255	C₂³	—	4	2	no	—
256	C₂³	#13133146	1	1	yes	characteristic
257	C₂³	#13133146	1	1	yes	characteristic
258	C₂³	#13133146	1	2	yes	—
259	C₂³	C₂×#11719912	1	1	yes	characteristic
260	C₂³	C₂×#11719912	1	2	yes	—
261	C₂×C₄	—	106	4	no	—
262	C₂×C₄	—	212	4	no	—
263	C₂×C₄	#13133146	1	2	yes	—
264	C₂×C₄	#13133146	1	2	yes	—
265	C₂×C₄	#13133146	1	2	yes	—
266	C₂×C₄	#13133146	1	2	yes	—
267	C₂×C₄	#13133146	1	2	yes	—
268	C₂×C₄	#13133146	1	1	yes	characteristic
269	C₂×C₄	#13133146	1	1	yes	characteristic
270	C₂×C₄	C₂×#11719919	1	2	yes	—
271	C₂×C₄	C₂×#11719919	1	2	yes	—
272	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
273	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
274	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
275	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
276	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
277	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
278	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
279	C₂×C₄	C₂×#11719912	1	2	yes	—
280	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
281	C₈	—	53	8	no	—
282	C₈	C₂×#11719919	1	1	yes	characteristic
283	C₈	C₂×#11719919	1	2	yes	—
284	C₈	C₂×#11719919	1	2	yes	—
285	C₈	C₂×#11719919	1	1	yes	characteristic
286	C₈	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
287	C₈	C₂²×Dic₁₀₆	1	1	yes	characteristic
288	C₂²	—	2	2	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	4	2	no	—
292	C₂²	—	4	2	no	—
293	C₂²	C₂×#13133146	1	1	yes	characteristic
294	C₂²	C₂×#13133146	1	2	yes	—
295	C₂²	C₂×#13133146	1	2	yes	—
296	C₂²	C₂×#13133146	1	1	yes	characteristic
297	C₂²	C₂²×#11719912	1	1	yes	characteristic
298	C₄	—	212	4	no	—
299	C₄	C₂×#13133146	1	1	yes	characteristic
300	C₄	C₂×#13133146	1	2	yes	—
301	C₄	C₂×#13133146	1	2	yes	—
302	C₄	C₂×#13133146	1	1	yes	characteristic
303	C₄	C₂²×#11719919	1	1	yes	characteristic
304	C₄	C₂²×C₈×D₅₃	1	1	yes	characteristic
305	C₂	—	2	1	no	—
306	C₂	—	2	1	no	—
307	C₂	—	2	2	no	—
308	C₂	—	2	2	no	—
309	C₂	#20276480	1	2	yes	direct product factor
310	C₂	C₂²×#13133146	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₅₃
commutator	C₄×C₅₃	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂²×M₃₂×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₄⋊C₈
prime radical	C₂⁴×C₅₃	M₁₆
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂²×M₃₂	D₅₃
hypocenter	C₅₃	C₂×#2153
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
3. #20276480	C₂	direct (N × H)
4. C₂×#16810443	C₂	semidirect
17. #16810443	C₂²	semidirect
159. C₂×C₅₃	#2153	semidirect
179. C₅₃	C₂×#2153	semidirect
309. C₂	#20276480	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
C₂×#16810578	2
#20276480	2
C₂×#16810443	2
C₂²×M₃₂×C₅₃	2
C₂×#2153	53

Subgroups — C2×#20276480 #27537091

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20276480 #27537091