Subgroups — C₂×#20274470 #27381595

Order 13568.

930 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810578	C₂	1	2	yes	maximal
2	#20274470	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16810381	C₂	1	1	yes	maximal, characteristic
4	C₂×#16810380	C₂	1	1	yes	maximal, characteristic
5	C₂×#16810379	C₂	1	2	yes	maximal
6	C₂×C₅₃×#405	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11719917	C₂²	1	2	yes	—
8	C₂×#13133176	C₂²	1	2	yes	—
9	#16810578	C₄	1	8	yes	complemented
10	#16810578	C₂²	1	8	yes	—
11	C₂×#13133127	C₄	1	4	yes	complemented
12	C₂×C₈×Dic₅₃	C₄	1	4	yes	complemented
13	#16810381	C₂²	1	4	yes	—
14	#16810380	C₂²	1	4	yes	—
15	#16810379	C₂²	1	8	yes	—
16	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
17	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
19	C₅₃×#405	C₄	1	4	yes	complemented
20	C₅₃×#405	C₂²	1	4	yes	—
21	C₂×#97×C₅₃	C₄	1	2	yes	complemented
22	C₂×C₄×C₈×C₅₃	C₄	1	2	yes	complemented
23	C₂×#11719917	—	2	2	no	—
24	C₂×#11719917	—	2	2	no	—
25	C₂×#11719917	—	2	2	no	—
26	C₂×#11719917	—	2	2	no	—
27	C₂×#11719917	C₂×C₄	1	4	yes	—
28	C₂×#11719917	C₂³	1	4	yes	—
29	C₂²×#88322	C₂×C₄	1	4	yes	—
30	#13133176	—	2	2	no	—
31	#13133176	—	2	2	no	—
32	#13133176	C₂×C₄	1	4	yes	—
33	#13133176	C₂³	1	4	yes	—
34	C₂×C₄⋊Dic₅₃	—	2	2	no	—
35	C₂×C₂²⋊Dic₅₃	—	2	2	no	—
36	C₂×C₄×Dic₅₃	C₂×C₄	1	4	yes	—
37	#13133127	C₂×C₄	1	16	yes	complemented
38	C₈×Dic₅₃	C₂×C₄	1	16	yes	complemented
39	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
40	C₂×M₁₆×C₅₃	—	2	2	no	—
41	C₂×M₁₆×C₅₃	—	2	2	no	—
42	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
43	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
44	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
45	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
46	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
47	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
48	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
50	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
51	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
52	#117×C₅₃	—	2	1	no	—
53	#117×C₅₃	—	2	1	no	—
54	#117×C₅₃	C₂×C₄	1	2	yes	—
55	#117×C₅₃	C₂³	1	2	yes	—
56	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
57	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
58	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
59	#97×C₅₃	C₂×C₄	1	8	yes	complemented
60	C₄×C₈×C₅₃	C₂×C₄	1	8	yes	complemented
61	C₂⁴×C₅₃	#41	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	—	2	2	no	—
63	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
69	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
70	C₂²×C₄×C₅₃	#41	1	2	yes	—
71	C₂²×C₄×C₅₃	#41	1	2	yes	—
72	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
73	C₂²×Dic₅₃	—	2	4	no	—
74	M₁₆×C₅₃	—	2	4	no	—
75	M₁₆×C₅₃	—	2	4	no	—
76	C₂×C₈×C₅₃	—	2	2	no	—
77	C₂×C₈×C₅₃	—	2	2	no	—
78	C₂×C₈×C₅₃	C₄²	1	2	yes	—
79	C₂×C₈×C₅₃	C₄²	1	4	yes	—
80	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
81	C₂×C₈×C₅₃	C₄²	1	4	yes	—
82	C₂×C₈×C₅₃	C₄²	1	2	yes	—
83	C₂×C₈×C₅₃	C₄²	1	2	yes	—
84	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
85	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
86	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
87	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
88	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
89	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
90	C₄⋊C₄×C₅₃	—	2	2	no	—
91	C₄⋊C₄×C₅₃	—	2	2	no	—
92	C₂²⋊C₄×C₅₃	—	2	2	no	—
93	C₂²⋊C₄×C₅₃	—	2	2	no	—
94	C₄²×C₅₃	—	2	2	no	—
95	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
96	C₂²⋊Dic₅₃	—	2	4	no	—
97	C₂²⋊Dic₅₃	—	2	4	no	—
98	C₄⋊Dic₅₃	—	2	4	no	—
99	C₄⋊Dic₅₃	—	2	4	no	—
100	C₄×Dic₅₃	—	2	4	no	—
101	C₄×Dic₅₃	C₂²×C₄	1	8	yes	—
102	#11719917	—	2	4	no	—
103	#11719917	—	2	4	no	—
104	#11719917	—	2	4	no	—
105	#11719917	—	2	4	no	—
106	C₂×#88322	—	2	4	no	—
107	C₂×#88322	—	2	4	no	—
108	C₂×#88322	C₂²×C₄	1	8	yes	—
109	C₂³×C₅₃	—	4	2	no	—
110	C₂³×C₅₃	#117	1	2	yes	—
111	C₂³×C₅₃	#117	1	2	yes	—
112	C₂³×C₅₃	#117	1	1	yes	characteristic
113	C₂³×C₅₃	#117	1	1	yes	characteristic
114	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
115	C₂×C₄×C₅₃	—	2	4	no	—
116	C₂×C₄×C₅₃	—	4	4	no	—
117	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
118	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
119	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
120	C₂×C₄×C₅₃	#117	1	2	yes	—
121	C₂×C₄×C₅₃	#117	1	2	yes	—
122	C₂×C₄×C₅₃	#117	1	2	yes	—
123	C₂×C₄×C₅₃	#117	1	2	yes	—
124	C₂×C₄×C₅₃	#117	1	2	yes	—
125	C₂×C₄×C₅₃	#117	1	2	yes	—
126	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
127	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
128	C₂×C₄×C₅₃	#117	1	2	yes	—
129	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
130	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
131	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
132	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
133	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
134	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
135	C₂×Dic₅₃	—	2	8	no	—
136	C₂×Dic₅₃	—	4	8	no	—
137	C₈×C₅₃	—	2	4	no	—
138	C₈×C₅₃	C₂×C₄²	1	4	yes	—
139	C₈×C₅₃	C₂×C₄²	1	2	yes	—
140	C₈×C₅₃	C₂×C₄²	1	2	yes	—
141	#88322	—	2	8	no	—
142	C₂×#1747	—	53	1	no	maximal, Sylow
143	C₂²×C₅₃	—	2	4	no	—
144	C₂²×C₅₃	—	2	2	no	—
145	C₂²×C₅₃	—	4	2	no	—
146	C₂²×C₅₃	—	4	1	no	—
147	C₂²×C₅₃	—	4	1	no	—
148	C₂²×C₅₃	#376	1	2	yes	—
149	C₂²×C₅₃	#376	1	1	yes	characteristic
150	C₂²×C₅₃	#376	1	1	yes	characteristic
151	C₂²×C₅₃	C₂×#117	1	2	yes	—
152	C₂²×C₅₃	C₂×#117	1	1	yes	characteristic
153	C₄×C₅₃	—	4	4	no	—
154	C₄×C₅₃	#376	1	1	yes	characteristic
155	C₄×C₅₃	#376	1	2	yes	—
156	C₄×C₅₃	#376	1	1	yes	characteristic
157	C₄×C₅₃	C₂²×C₄²	1	1	yes	characteristic
158	C₄×C₅₃	C₂×#117	1	2	yes	—
159	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
160	Dic₅₃	—	4	8	no	—
161	C₂×#405	—	53	2	no	—
162	C₂×#405	D₅₃	1	1	yes	characteristic
163	#1747	—	53	8	no	—
164	C₂×#338	—	53	1	no	—
165	C₂×H(3,ℤ/4)	—	53	1	no	—
166	C₂×H(3,ℤ/4)	—	53	2	no	—
167	C₂×C₅₃	—	2	2	no	—
168	C₂×C₅₃	—	2	1	no	—
169	C₂×C₅₃	—	2	2	no	—
170	C₂×C₅₃	—	2	1	no	—
171	C₂×C₅₃	#1747	1	2	yes	complemented
172	C₂×C₅₃	C₂×#376	1	1	yes	characteristic
173	C₂²×M₁₆	—	53	2	no	—
174	C₂²×M₁₆	C₂×D₅₃	1	1	yes	characteristic
175	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
176	C₂×#117	—	53	2	no	—
177	C₂×#117	C₂×D₅₃	1	1	yes	characteristic
178	#405	—	53	8	no	—
179	#405	—	53	8	no	—
180	#405	Dic₅₃	1	4	yes	complemented
181	#405	C₂×D₅₃	1	4	yes	—
182	C₂×#97	—	53	4	no	—
183	C₂×#97	Dic₅₃	1	2	yes	complemented
184	C₂×C₄×C₈	—	53	4	no	—
185	C₂×C₄×C₈	Dic₅₃	1	2	yes	complemented
186	#338	—	53	4	no	—
187	H(3,ℤ/4)	—	53	4	no	—
188	H(3,ℤ/4)	—	53	8	no	—
189	C₅₃	C₂×#1747	1	1	yes	characteristic, complemented, Sylow
190	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
191	C₂×M₁₆	—	2	2	no	—
192	C₂×M₁₆	—	2	2	no	—
193	C₂×M₁₆	—	53	4	no	—
194	C₂×M₁₆	—	53	4	no	—
195	C₂×M₁₆	—	106	2	no	—
196	C₂×M₁₆	—	106	2	no	—
197	C₂×M₁₆	—	106	2	no	—
198	C₂×M₁₆	—	106	2	no	—
199	C₂×M₁₆	C₂×Dic₅₃	1	2	yes	—
200	C₂×M₁₆	C₂²×D₅₃	1	2	yes	—
201	C₂²×C₈	—	53	4	no	—
202	C₂²×C₈	C₄×D₅₃	1	2	yes	—
203	C₂²×C₈	C₄×D₅₃	1	4	yes	—
204	C₂²×C₈	C₄×D₅₃	1	2	yes	—
205	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
206	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
207	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
208	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
209	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
210	#117	—	2	1	no	—
211	#117	—	2	1	no	—
212	#117	—	53	4	no	—
213	#117	—	53	4	no	—
214	#117	—	106	2	no	—
215	#117	—	106	2	no	—
216	#117	C₂×Dic₅₃	1	2	yes	—
217	#117	C₂²×D₅₃	1	2	yes	—
218	C₂×C₄⋊C₄	—	2	1	no	—
219	C₂×C₄⋊C₄	—	106	2	no	—
220	C₂×C₂²⋊C₄	—	2	1	no	—
221	C₂×C₂²⋊C₄	—	106	2	no	—
222	C₂×C₄²	—	53	4	no	—
223	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
224	#97	—	53	16	no	—
225	#97	C₂×Dic₅₃	1	8	yes	complemented
226	C₄×C₈	—	53	16	no	—
227	C₄×C₈	C₂×Dic₅₃	1	8	yes	complemented
228	C₂⁴	#11719945	1	1	yes	characteristic
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	106	4	no	—
231	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
232	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
233	C₂²×C₄	#11719945	1	1	yes	characteristic
234	C₂²×C₄	#11719945	1	1	yes	characteristic
235	C₂²×C₄	#11719945	1	1	yes	characteristic
236	C₂²×C₄	#11719945	1	2	yes	—
237	C₂²×C₄	#11719945	1	2	yes	—
238	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
239	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
240	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
241	M₁₆	—	2	4	no	—
242	M₁₆	—	2	4	no	—
243	M₁₆	—	106	4	no	—
244	M₁₆	—	106	4	no	—
245	M₁₆	—	106	4	no	—
246	M₁₆	—	106	4	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	53	8	no	—
250	C₂×C₈	—	106	4	no	—
251	C₂×C₈	—	106	4	no	—
252	C₂×C₈	C₄×Dic₅₃	1	4	yes	—
253	C₂×C₈	C₄×Dic₅₃	1	2	yes	—
254	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
255	C₂×C₈	C₄×Dic₅₃	1	2	yes	—
256	C₂×C₈	C₄×Dic₅₃	1	2	yes	—
257	C₂×C₈	C₄×Dic₅₃	1	4	yes	—
258	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
259	C₂×C₈	C₂×C₄×D₅₃	1	4	yes	—
260	C₂×C₈	C₂×C₄×D₅₃	1	4	yes	—
261	C₂×C₈	C₂²×Dic₅₃	1	4	yes	—
262	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
263	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	106	4	no	—
267	C₄⋊C₄	—	106	4	no	—
268	C₂²⋊C₄	—	2	2	no	—
269	C₂²⋊C₄	—	2	2	no	—
270	C₂²⋊C₄	—	106	4	no	—
271	C₂²⋊C₄	—	106	4	no	—
272	C₄²	—	2	2	no	—
273	C₄²	—	53	8	no	—
274	C₄²	—	106	4	no	—
275	C₄²	C₂²×Dic₅₃	1	4	yes	—
276	C₂³	—	4	2	no	—
277	C₂³	#13133174	1	2	yes	—
278	C₂³	#13133174	1	2	yes	—
279	C₂³	#13133176	1	1	yes	characteristic
280	C₂³	#13133176	1	1	yes	characteristic
281	C₂³	C₂×#11719945	1	1	yes	characteristic
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	—	4	4	no	—
284	C₂×C₄	—	106	8	no	—
285	C₂×C₄	—	212	8	no	—
286	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
287	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
288	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
289	C₂×C₄	#13133174	1	2	yes	—
290	C₂×C₄	#13133174	1	2	yes	—
291	C₂×C₄	#13133174	1	2	yes	—
292	C₂×C₄	#13133174	1	2	yes	—
293	C₂×C₄	#13133174	1	2	yes	—
294	C₂×C₄	#13133174	1	2	yes	—
295	C₂×C₄	#13133176	1	1	yes	characteristic
296	C₂×C₄	#13133176	1	1	yes	characteristic
297	C₂×C₄	#13133176	1	2	yes	—
298	C₂×C₄	#13133176	1	1	yes	characteristic
299	C₂×C₄	#13133176	1	1	yes	characteristic
300	C₂×C₄	C₂²×C₄×D₅₃	1	2	yes	—
301	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
302	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
303	C₂×C₄	C₂×#11719945	1	2	yes	—
304	C₈	—	2	4	no	—
305	C₈	—	106	8	no	—
306	C₈	C₂×C₄×Dic₅₃	1	4	yes	—
307	C₈	C₂×C₄×Dic₅₃	1	2	yes	—
308	C₈	C₂×C₄×Dic₅₃	1	2	yes	—
309	C₂²	—	2	2	no	—
310	C₂²	—	2	4	no	—
311	C₂²	—	4	2	no	—
312	C₂²	—	4	1	no	—
313	C₂²	—	4	1	no	—
314	C₂²	#16810469	1	2	yes	—
315	C₂²	#16810469	1	1	yes	characteristic
316	C₂²	#16810469	1	1	yes	characteristic
317	C₂²	C₂×#13133174	1	2	yes	—
318	C₂²	C₂×#13133176	1	1	yes	characteristic
319	C₄	—	4	4	no	—
320	C₄	—	212	8	no	—
321	C₄	#16810469	1	2	yes	—
322	C₄	#16810469	1	1	yes	characteristic
323	C₄	#16810469	1	1	yes	characteristic
324	C₄	C₂²×C₄×Dic₅₃	1	1	yes	characteristic
325	C₄	C₂×#13133174	1	2	yes	—
326	C₄	C₂×#13133176	1	1	yes	characteristic
327	C₂	—	2	1	no	—
328	C₂	—	2	2	no	—
329	C₂	—	2	1	no	—
330	C₂	—	2	2	no	—
331	C₂	#20274470	1	2	yes	direct product factor
332	C₂	C₂×#16810469	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₄×Dic₅₃
commutator	C₄×C₅₃	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₅₃×#405	C₂
socle	C₂²×C₅₃	#376
prime radical	C₂⁴×C₅₃	#41
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂×#405	D₅₃
hypocenter	C₅₃	C₂×#1747
LCS	G ▷ C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
2. #20274470	C₂	direct (N × H)
9. #16810578	C₄	semidirect
11. C₂×#13133127	C₄	semidirect
12. C₂×C₈×Dic₅₃	C₄	semidirect
19. C₅₃×#405	C₄	semidirect
21. C₂×#97×C₅₃	C₄	semidirect
22. C₂×C₄×C₈×C₅₃	C₄	semidirect
37. #13133127	C₂×C₄	semidirect
38. C₈×Dic₅₃	C₂×C₄	semidirect
59. #97×C₅₃	C₂×C₄	semidirect
60. C₄×C₈×C₅₃	C₂×C₄	semidirect
171. C₂×C₅₃	#1747	semidirect
180. #405	Dic₅₃	semidirect
183. C₂×#97	Dic₅₃	semidirect
185. C₂×C₄×C₈	Dic₅₃	semidirect
189. C₅₃	C₂×#1747	semidirect
225. #97	C₂×Dic₅₃	semidirect
227. C₄×C₈	C₂×Dic₅₃	semidirect
331. C₂	#20274470	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×#16810578	2
#20274470	2
C₂×#16810381	2
C₂×#16810380	2
C₂×#16810379	2
C₂×C₅₃×#405	2
C₂×#1747	53

Subgroups — C2×#20274470 #27381595

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20274470 #27381595