Subgroups — C₂×#20273492 #27353156

Order 13568.

670 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810679	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810675	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810674	C₂	1	1	yes	maximal, characteristic
4	#20273492	C₂	1	8	yes	maximal, direct product factor
5	C₂×#16810375	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16810375	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#16810374	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅₃×#334	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₄²×D₅₃	C₂²	1	1	yes	characteristic
10	#16810679	—	2	2	no	—
11	#16810679	C₂²	1	4	yes	—
12	#16810675	—	2	2	no	—
13	#16810675	C₂²	1	4	yes	—
14	#16810674	C₄	1	4	yes	—
15	#16810674	C₂²	1	4	yes	—
16	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
18	C₂×#13133146	C₄	1	1	yes	characteristic
19	C₂×#13133146	C₄	1	1	yes	characteristic
20	C₂×#13133129	—	2	1	no	—
21	C₂×#13133129	—	2	1	no	—
22	C₂×#13133129	C₄	1	1	yes	characteristic
23	C₂×#13133129	C₄	1	1	yes	characteristic
24	C₂×#13133127	—	2	1	no	—
25	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
26	C₂×C₈×Dic₅₃	—	2	1	no	—
27	#16810375	C₂²	1	4	yes	complemented
28	#16810375	C₂²	1	4	yes	complemented
29	#16810374	C₂²	1	4	yes	complemented
30	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
31	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
32	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
33	C₅₃×#334	C₂²	1	4	yes	complemented
34	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×D₅₃	D₄	1	1	yes	characteristic
36	C₂²×C₄×D₅₃	Q₈	1	1	yes	characteristic
37	C₂²×#88322	—	2	1	no	—
38	C₂²×#88322	—	2	1	no	—
39	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
40	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
43	C₂×C₄×Dic₅₃	Q₈	1	1	yes	characteristic
44	C₄²×D₅₃	C₂×C₄	1	2	yes	—
45	C₄²×D₅₃	D₄	1	2	yes	—
46	C₄²×D₅₃	Q₈	1	2	yes	—
47	C₄²×D₅₃	C₂³	1	2	yes	—
48	C₄⋊#88322	—	2	1	no	—
49	C₄⋊#88322	—	2	1	no	—
50	C₄⋊#88322	C₂³	1	2	yes	—
51	C₄⋊#88322	C₂³	1	2	yes	—
52	#13133146	C₂×C₄	1	4	yes	—
53	#13133146	C₂×C₄	1	4	yes	—
54	#13133129	—	2	4	no	—
55	#13133129	—	2	4	no	—
56	#13133129	C₂×C₄	1	4	yes	—
57	#13133129	C₂×C₄	1	4	yes	—
58	#13133127	—	2	4	no	—
59	C₄×#88322	C₂×C₄	1	2	yes	—
60	C₄×#88322	C₂³	1	2	yes	—
61	C₈×Dic₅₃	—	2	4	no	—
62	C₂²×C₈×C₅₃	—	2	1	no	—
63	C₂²×C₈×C₅₃	—	2	1	no	—
64	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
65	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
67	C₄⋊C₈×C₅₃	—	2	1	no	—
68	C₄⋊C₈×C₅₃	—	2	1	no	—
69	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
70	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
71	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
72	C₄×C₈×C₅₃	C₂³	1	2	yes	—
73	C₂³×D₅₃	C₄⋊C₄	1	1	yes	characteristic
74	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
77	C₂²×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
78	C₂²×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
79	C₂²×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
80	C₂×C₄×D₅₃	—	2	2	no	—
81	C₂×C₄×D₅₃	—	2	2	no	—
82	C₂×C₄×D₅₃	—	2	2	no	—
83	C₂×C₄×D₅₃	—	2	2	no	—
84	C₂×C₄×D₅₃	—	4	2	no	—
85	C₂×C₄×D₅₃	C₄⋊C₄	1	2	yes	—
86	C₂×C₄×D₅₃	C₄⋊C₄	1	2	yes	—
87	C₂×C₄×D₅₃	C₄⋊C₄	1	2	yes	—
88	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
89	C₂×C₄×D₅₃	C₂×D₄	1	2	yes	—
90	C₂×C₄×D₅₃	C₂×Q₈	1	2	yes	—
91	C₂×C₈×C₅₃	—	2	2	no	—
92	C₂×C₈×C₅₃	—	2	2	no	—
93	C₂×C₈×C₅₃	—	2	2	no	—
94	C₂×C₈×C₅₃	—	2	2	no	—
95	C₂×C₈×C₅₃	—	4	2	no	—
96	C₂×C₈×C₅₃	—	4	2	no	—
97	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
98	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
99	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
100	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
101	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
102	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
103	C₄×Dic₅₃	C₄⋊C₄	1	2	yes	—
104	C₄×Dic₅₃	C₄⋊C₄	1	2	yes	—
105	C₄×Dic₅₃	C₄⋊C₄	1	2	yes	—
106	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
107	C₄×Dic₅₃	C₂×D₄	1	2	yes	—
108	C₄×Dic₅₃	C₂×Q₈	1	2	yes	—
109	C₂×#88322	—	2	2	no	—
110	C₂×#88322	—	2	2	no	—
111	C₂×#88322	—	2	2	no	—
112	C₂×#88322	—	2	2	no	—
113	C₂×#88322	—	4	2	no	—
114	C₂×#88322	—	4	2	no	—
115	C₂×#88322	C₂²×C₄	1	2	yes	—
116	C₂×#88322	C₂²×C₄	1	2	yes	—
117	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₂²×D₅₃	—	2	2	no	—
119	C₂²×D₅₃	—	2	2	no	—
120	C₂²×D₅₃	—	2	2	no	—
121	C₂²×D₅₃	C₂×C₄⋊C₄	1	2	yes	—
122	C₂×C₄×C₅₃	—	2	2	no	—
123	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₅₃	#131	1	2	yes	—
127	C₂×C₄×C₅₃	#131	1	2	yes	—
128	C₂×C₄×C₅₃	#135	1	2	yes	—
129	C₂×C₄×C₅₃	#135	1	2	yes	—
130	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
131	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
133	C₂×Dic₅₃	—	2	2	no	—
134	C₂×Dic₅₃	—	2	2	no	—
135	C₂×Dic₅₃	—	2	2	no	—
136	C₂×Dic₅₃	C₂×C₄⋊C₄	1	2	yes	—
137	C₂×Dic₅₃	C₂×C₄⋊C₄	1	2	yes	—
138	C₂×Dic₅₃	C₂×C₄⋊C₄	1	2	yes	—
139	C₄×D₅₃	—	2	2	no	—
140	C₄×D₅₃	—	2	2	no	—
141	C₄×D₅₃	—	2	2	no	—
142	C₄×D₅₃	—	2	2	no	—
143	C₄×D₅₃	—	2	4	no	—
144	C₄×D₅₃	—	2	4	no	—
145	C₄×D₅₃	—	4	2	no	—
146	C₄×D₅₃	—	4	2	no	—
147	C₈×C₅₃	—	2	2	no	—
148	C₈×C₅₃	—	2	2	no	—
149	C₈×C₅₃	—	4	2	no	—
150	C₈×C₅₃	—	4	2	no	—
151	#88322	—	2	2	no	—
152	#88322	—	2	2	no	—
153	#88322	—	4	2	no	—
154	#88322	—	4	2	no	—
155	C₂×#1572	—	53	1	no	maximal, Sylow
156	C₂²×C₅₃	#424	1	2	yes	—
157	C₂²×C₅₃	#427	1	2	yes	—
158	C₂²×C₅₃	#430	1	2	yes	—
159	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
160	C₂×D₅₃	—	2	2	no	—
161	C₂×D₅₃	—	2	2	no	—
162	C₂×D₅₃	—	2	2	no	—
163	C₂×D₅₃	—	4	4	no	—
164	C₄×C₅₃	—	2	1	no	—
165	C₄×C₅₃	—	2	1	no	—
166	C₄×C₅₃	#424	1	2	yes	—
167	C₄×C₅₃	#427	1	2	yes	—
168	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
169	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
170	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
171	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
172	Dic₅₃	—	2	2	no	—
173	Dic₅₃	—	2	2	no	—
174	Dic₅₃	—	2	2	no	—
175	C₂×#432	—	53	1	no	—
176	C₂×#431	—	53	1	no	—
177	C₂×#431	—	53	1	no	—
178	#1572	—	53	8	no	—
179	C₂×#335	—	53	1	no	—
180	C₂×#335	—	53	1	no	—
181	C₂×#334	—	53	1	no	—
182	C₂×#334	D₅₃	1	1	yes	characteristic, complemented
183	C₂×C₅₃	#1572	1	4	yes	complemented
184	C₂×C₅₃	C₂×#424	1	1	yes	characteristic
185	C₂×C₅₃	C₂×#427	1	1	yes	characteristic
186	C₂×C₅₃	C₂×#430	1	1	yes	characteristic
187	D₅₃	—	4	2	no	—
188	C₂²×C₄²	—	53	1	no	—
189	#432	—	53	4	no	—
190	#432	—	106	2	no	—
191	#431	—	53	4	no	—
192	#431	—	53	4	no	—
193	#431	—	53	4	no	—
194	#431	—	106	2	no	—
195	C₂×C₄⋊C₈	—	53	1	no	—
196	C₂×C₄⋊C₈	—	53	1	no	—
197	C₂×C₄⋊C₈	—	53	1	no	—
198	C₂×C₄⋊C₈	—	53	1	no	—
199	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
200	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
201	C₂×C₂²⋊C₈	—	53	1	no	—
202	C₂×C₂²⋊C₈	—	53	1	no	—
203	C₂×C₂²⋊C₈	—	106	1	no	—
204	C₂×C₂²⋊C₈	—	106	1	no	—
205	C₂×#97	—	106	1	no	—
206	C₂×C₄×C₈	—	53	1	no	—
207	C₂×C₄×C₈	—	106	1	no	—
208	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
209	#335	—	53	4	no	—
210	#335	—	53	4	no	—
211	#334	—	53	4	no	—
212	#334	C₂×D₅₃	1	4	yes	complemented
213	C₅₃	C₂×#1572	1	1	yes	characteristic, complemented, Sylow
214	C₂³×C₄	—	53	1	no	—
215	C₂³×C₄	—	53	1	no	—
216	C₂³×C₄	—	53	1	no	—
217	C₂²×C₈	—	2	1	no	—
218	C₂²×C₈	—	2	1	no	—
219	C₂²×C₈	—	53	1	no	—
220	C₂²×C₈	—	53	1	no	—
221	C₂²×C₈	—	106	1	no	—
222	C₂²×C₈	—	106	1	no	—
223	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
224	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
225	C₂×C₄²	—	53	1	no	—
226	C₂×C₄²	—	53	1	no	—
227	C₂×C₄²	—	53	1	no	—
228	C₂×C₄²	—	53	2	no	—
229	C₂×C₄²	—	53	2	no	—
230	C₂×C₄²	—	53	2	no	—
231	C₂×C₄²	—	53	2	no	—
232	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
233	C₄⋊C₈	—	2	1	no	—
234	C₄⋊C₈	—	2	1	no	—
235	C₄⋊C₈	—	53	4	no	—
236	C₄⋊C₈	—	53	4	no	—
237	C₄⋊C₈	—	53	2	no	—
238	C₄⋊C₈	—	53	2	no	—
239	C₄⋊C₈	—	106	1	no	—
240	C₄⋊C₈	—	106	1	no	—
241	C₄⋊C₈	C₂²×D₅₃	1	2	yes	—
242	C₄⋊C₈	C₂²×D₅₃	1	2	yes	—
243	C₂²⋊C₈	—	53	4	no	—
244	C₂²⋊C₈	—	53	4	no	—
245	C₂²⋊C₈	—	106	4	no	—
246	C₂²⋊C₈	—	106	4	no	—
247	#97	—	106	4	no	—
248	C₄×C₈	—	53	2	no	—
249	C₄×C₈	—	53	2	no	—
250	C₄×C₈	—	106	4	no	—
251	C₄×C₈	C₄×D₅₃	1	2	yes	—
252	C₄×C₈	C₂²×D₅₃	1	2	yes	—
253	C₂⁴	—	53	1	no	—
254	C₂²×C₄	—	53	1	no	—
255	C₂²×C₄	—	53	1	no	—
256	C₂²×C₄	—	53	1	no	—
257	C₂²×C₄	—	53	2	no	—
258	C₂²×C₄	—	53	2	no	—
259	C₂²×C₄	—	53	2	no	—
260	C₂²×C₄	—	53	2	no	—
261	C₂²×C₄	—	53	2	no	—
262	C₂²×C₄	—	53	2	no	—
263	C₂²×C₄	—	106	2	no	—
264	C₂²×C₄	—	106	2	no	—
265	C₂²×C₄	—	106	2	no	—
266	C₂²×C₄	—	106	2	no	—
267	C₂²×C₄	—	212	2	no	—
268	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
269	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
270	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
271	C₂×C₈	—	2	2	no	—
272	C₂×C₈	—	2	2	no	—
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	—	2	2	no	—
275	C₂×C₈	—	4	2	no	—
276	C₂×C₈	—	4	2	no	—
277	C₂×C₈	—	53	2	no	—
278	C₂×C₈	—	53	2	no	—
279	C₂×C₈	—	106	2	no	—
280	C₂×C₈	—	106	2	no	—
281	C₂×C₈	—	106	2	no	—
282	C₂×C₈	—	106	2	no	—
283	C₂×C₈	—	212	2	no	—
284	C₂×C₈	—	212	2	no	—
285	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
286	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
287	C₄²	—	53	2	no	—
288	C₄²	—	53	2	no	—
289	C₄²	—	53	2	no	—
290	C₄²	—	53	2	no	—
291	C₄²	—	53	2	no	—
292	C₄²	—	53	2	no	—
293	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
294	C₄²	D₄×D₅₃	1	1	yes	characteristic
295	C₄²	Q₈×D₅₃	1	1	yes	characteristic
296	C₄²	C₂³×D₅₃	1	1	yes	characteristic
297	C₂³	—	53	2	no	—
298	C₂³	—	106	2	no	—
299	C₂³	—	106	2	no	—
300	C₂³	—	106	2	no	—
301	C₂³	C₄⋊C₄×D₅₃	1	1	yes	characteristic
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	53	2	no	—
304	C₂×C₄	—	53	2	no	—
305	C₂×C₄	—	53	2	no	—
306	C₂×C₄	—	106	2	no	—
307	C₂×C₄	—	106	2	no	—
308	C₂×C₄	—	106	2	no	—
309	C₂×C₄	—	106	2	no	—
310	C₂×C₄	—	106	2	no	—
311	C₂×C₄	—	106	2	no	—
312	C₂×C₄	—	106	2	no	—
313	C₂×C₄	—	106	4	no	—
314	C₂×C₄	—	106	4	no	—
315	C₂×C₄	—	212	2	no	—
316	C₂×C₄	—	212	2	no	—
317	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
318	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
319	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
320	C₂×C₄	#13133223	1	2	yes	—
321	C₂×C₄	#13133225	1	2	yes	—
322	C₂×C₄	#13133238	1	2	yes	—
323	C₂×C₄	#13133238	1	2	yes	—
324	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
325	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
326	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
327	C₈	—	2	2	no	—
328	C₈	—	2	2	no	—
329	C₈	—	4	2	no	—
330	C₈	—	4	2	no	—
331	C₈	—	106	2	no	—
332	C₈	—	106	2	no	—
333	C₈	—	212	2	no	—
334	C₈	—	212	2	no	—
335	C₂²	—	106	2	no	—
336	C₂²	—	106	2	no	—
337	C₂²	—	106	2	no	—
338	C₂²	—	212	4	no	—
339	C₂²	#16810648	1	2	yes	—
340	C₂²	#16810658	1	2	yes	—
341	C₂²	#16810670	1	2	yes	—
342	C₂²	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
343	C₄	—	2	1	no	—
344	C₄	—	2	1	no	—
345	C₄	—	106	2	no	—
346	C₄	—	106	2	no	—
347	C₄	—	106	2	no	—
348	C₄	#16810648	1	2	yes	—
349	C₄	#16810658	1	2	yes	—
350	C₄	C₂×#13133223	1	1	yes	characteristic
351	C₄	C₂×#13133225	1	1	yes	characteristic
352	C₄	C₂×#13133238	1	1	yes	characteristic
353	C₄	C₂×#13133238	1	1	yes	characteristic
354	C₂	—	212	2	no	—
355	C₂	#20273492	1	4	yes	direct product factor
356	C₂	C₂×#16810648	1	1	yes	characteristic
357	C₂	C₂×#16810658	1	1	yes	characteristic
358	C₂	C₂×#16810670	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₂×C₅₃×#334	C₂
socle	C₂³×C₅₃	C₂×C₄⋊C₄
prime radical	C₂³×D₅₃	C₄⋊C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	C₂×#334	D₅₃
hypocenter	C₅₃	C₂×#1572
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
4. #20273492	C₂	direct (N × H)
5. C₂×#16810375	C₂	semidirect
6. C₂×#16810375	C₂	semidirect
7. C₂×#16810374	C₂	semidirect
8. C₂×C₅₃×#334	C₂	semidirect
27. #16810375	C₂²	semidirect
28. #16810375	C₂²	semidirect
29. #16810374	C₂²	semidirect
33. C₅₃×#334	C₂²	semidirect
182. C₂×#334	D₅₃	semidirect
183. C₂×C₅₃	#1572	semidirect
212. #334	C₂×D₅₃	semidirect
213. C₅₃	C₂×#1572	semidirect
355. C₂	#20273492	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#16810679	2
C₂×#16810675	2
C₂×#16810674	2
#20273492	2
C₂×#16810375	2
C₂×#16810374	2
C₂×C₅₃×#334	2
C₂×#1572	53

Subgroups — C2×#20273492 #27353156

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20273492 #27353156