Subgroups — C₂×#20273461 #27353064

Order 13568.

830 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810676	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×#88322	C₂	1	1	yes	maximal, characteristic
3	#20273461	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16810375	C₂	1	2	yes	maximal, complemented
5	C₂×#16810373	C₂	1	2	yes	maximal, complemented
6	C₂×C₅₃×#431	C₂	1	1	yes	maximal, characteristic
7	C₂³×#88322	C₄	1	1	yes	characteristic
8	C₂³×#88322	C₄	1	1	yes	characteristic
9	#16810676	—	2	2	no	—
10	#16810676	C₂²	1	4	yes	—
11	C₂×C₄⋊#88322	C₂²	1	2	yes	—
12	C₂×C₂²⋊#88322	—	2	1	no	—
13	C₂×C₄×#88322	—	2	1	no	—
14	C₂×C₄×#88322	C₄	1	1	yes	characteristic
15	C₂×C₄×#88322	C₄	1	1	yes	characteristic
16	C₂×C₄×#88322	C₄	1	4	yes	—
17	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
18	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
19	C₂×C₄×#88322	C₂²	1	4	yes	—
20	#16810375	C₂²	1	8	yes	complemented
21	#16810373	C₂²	1	8	yes	complemented
22	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
23	C₅₃×#431	—	2	2	no	—
24	C₅₃×#431	C₂²	1	4	yes	—
25	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
26	C₂×C₂²⋊C₈×C₅₃	—	2	1	no	—
27	C₂×C₄×C₈×C₅₃	—	2	1	no	—
28	C₂²×#88322	—	2	4	no	—
29	C₂²×#88322	—	2	2	no	—
30	C₂²×#88322	C₈	1	8	yes	complemented
31	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
32	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
33	C₂²×#88322	C₂×C₄	1	4	yes	—
34	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
35	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
36	C₂²×#88322	C₂×C₄	1	4	yes	—
37	C₄⋊#88322	—	2	2	no	—
38	C₄⋊#88322	C₂³	1	4	yes	—
39	C₂²⋊#88322	—	2	4	no	—
40	C₄×#88322	—	2	4	no	—
41	C₄×#88322	C₂×C₄	1	2	yes	—
42	C₄×#88322	C₂×C₄	1	2	yes	—
43	C₄×#88322	C₂×C₄	1	4	yes	—
44	C₄×#88322	C₂×C₄	1	4	yes	—
45	C₄×#88322	C₂³	1	2	yes	—
46	C₄×#88322	C₂³	1	2	yes	—
47	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
48	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
49	C₂³×C₄×C₅₃	Q₈	1	1	yes	characteristic
50	C₂²×C₈×C₅₃	—	2	2	no	—
51	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
52	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄²×C₅₃	D₄	1	2	yes	—
54	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
55	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
56	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
57	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
58	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
59	C₄⋊C₈×C₅₃	—	2	2	no	—
60	C₄⋊C₈×C₅₃	C₂³	1	4	yes	—
61	C₂²⋊C₈×C₅₃	—	2	4	no	—
62	C₄×C₈×C₅₃	—	2	4	no	—
63	C₂⁴×C₅₃	C₄⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	—	2	4	no	—
65	C₂²×C₄×C₅₃	—	2	4	no	—
66	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
67	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
69	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
70	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₅₃	C₂×C₈	1	4	yes	—
73	C₂²×C₄×C₅₃	M₁₆	1	4	yes	—
74	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
76	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
79	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
80	C₂×C₈×C₅₃	—	2	4	no	—
81	C₂×C₈×C₅₃	—	4	4	no	—
82	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
83	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
84	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
85	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
86	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
87	C₄²×C₅₃	C₂×D₄	1	2	yes	—
88	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
90	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
91	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
92	C₂×#88322	—	2	8	no	—
93	C₂×#88322	—	2	2	no	—
94	C₂×#88322	—	2	2	no	—
95	C₂×#88322	—	2	4	no	—
96	C₂×#88322	—	4	4	no	—
97	C₂×#88322	C₂×C₈	1	4	yes	—
98	C₂×#88322	C₂×C₈	1	4	yes	—
99	C₂×#88322	C₂×C₈	1	16	yes	complemented
100	C₂×#88322	C₂²×C₄	1	2	yes	—
101	C₂×#88322	C₂²×C₄	1	2	yes	—
102	C₂×#88322	C₂²×C₄	1	2	yes	—
103	C₂×#88322	C₂²×C₄	1	2	yes	—
104	C₂³×C₅₃	—	2	2	no	—
105	C₂³×C₅₃	—	2	2	no	—
106	C₂³×C₅₃	C₄⋊C₈	1	4	yes	—
107	C₂³×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
108	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₅₃	—	2	2	no	—
110	C₂×C₄×C₅₃	—	2	2	no	—
111	C₂×C₄×C₅₃	—	2	4	no	—
112	C₂×C₄×C₅₃	—	2	4	no	—
113	C₂×C₄×C₅₃	—	2	4	no	—
114	C₂×C₄×C₅₃	—	2	4	no	—
115	C₂×C₄×C₅₃	C₄⋊C₈	1	8	yes	—
116	C₂×C₄×C₅₃	C₄⋊C₈	1	4	yes	—
117	C₂×C₄×C₅₃	#108	1	4	yes	—
118	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
119	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
121	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
124	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
125	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
126	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
127	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
128	C₂×C₄×C₅₃	#135	1	4	yes	—
129	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
130	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
132	C₈×C₅₃	—	4	4	no	—
133	#88322	—	2	2	no	—
134	#88322	—	2	2	no	—
135	#88322	—	4	4	no	—
136	#88322	C₂²×C₈	1	4	yes	—
137	#88322	C₂²×C₈	1	4	yes	—
138	C₂×#1564	—	53	1	no	maximal, Sylow
139	C₂²×C₅₃	—	2	8	no	—
140	C₂²×C₅₃	—	2	2	no	—
141	C₂²×C₅₃	—	2	2	no	—
142	C₂²×C₅₃	C₂×C₄⋊C₈	1	4	yes	—
143	C₂²×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
144	C₂²×C₅₃	#427	1	2	yes	—
145	C₂²×C₅₃	C₂×#108	1	2	yes	—
146	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
147	C₄×C₅₃	—	2	2	no	—
148	C₄×C₅₃	—	2	1	no	—
149	C₄×C₅₃	—	2	1	no	—
150	C₄×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
151	C₄×C₅₃	C₂×C₄⋊C₈	1	4	yes	—
152	C₄×C₅₃	#427	1	2	yes	—
153	C₄×C₅₃	C₂×#108	1	4	yes	—
154	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
155	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
156	C₄×C₅₃	C₂×#135	1	2	yes	—
157	C₂×#431	—	53	1	no	—
158	C₂×#431	D₅₃	1	1	yes	characteristic
159	C₂²×C₄×C₈	—	53	1	no	—
160	#1564	—	53	8	no	—
161	C₂×#335	—	53	2	no	—
162	C₂×#334	—	53	2	no	—
163	C₂×C₅₃	—	2	4	no	—
164	C₂×C₅₃	#1564	1	4	yes	complemented
165	C₂×C₅₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
166	C₂×C₅₃	C₂×#427	1	1	yes	characteristic
167	C₂×C₅₃	C₂²×#108	1	1	yes	characteristic
168	C₂³×C₈	—	53	1	no	—
169	C₂³×C₈	—	53	1	no	—
170	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
171	#431	—	2	2	no	—
172	#431	—	53	4	no	—
173	#431	—	106	2	no	—
174	#431	C₂×D₅₃	1	4	yes	—
175	C₂×C₄⋊C₈	—	53	2	no	—
176	C₂×C₄⋊C₈	C₂×D₅₃	1	2	yes	—
177	C₂×C₂²⋊C₈	—	2	1	no	—
178	C₂×C₂²⋊C₈	—	106	1	no	—
179	C₂×C₄×C₈	—	2	1	no	—
180	C₂×C₄×C₈	—	53	4	no	—
181	C₂×C₄×C₈	—	53	1	no	—
182	C₂×C₄×C₈	—	53	1	no	—
183	C₂×C₄×C₈	—	53	1	no	—
184	C₂×C₄×C₈	—	53	1	no	—
185	C₂×C₄×C₈	—	53	4	no	—
186	C₂×C₄×C₈	—	106	1	no	—
187	#335	—	53	8	no	—
188	#334	—	53	8	no	—
189	C₅₃	C₂×#1564	1	1	yes	characteristic, complemented, Sylow
190	C₂³×C₄	Dic₁₀₆	1	1	yes	characteristic
191	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
192	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
193	C₂²×C₈	—	2	2	no	—
194	C₂²×C₈	—	53	1	no	—
195	C₂²×C₈	—	53	1	no	—
196	C₂²×C₈	—	53	4	no	—
197	C₂²×C₈	—	53	8	no	—
198	C₂²×C₈	—	53	1	no	—
199	C₂²×C₈	—	53	1	no	—
200	C₂²×C₈	—	53	4	no	—
201	C₂²×C₈	—	106	4	no	—
202	C₂²×C₈	—	106	2	no	—
203	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
204	C₂×C₄²	Dic₁₀₆	1	2	yes	—
205	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
206	C₂×C₄²	C₄×D₅₃	1	2	yes	—
207	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
208	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
209	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
210	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
211	C₄⋊C₈	—	2	2	no	—
212	C₄⋊C₈	—	53	4	no	—
213	C₄⋊C₈	—	106	2	no	—
214	C₄⋊C₈	C₂²×D₅₃	1	4	yes	—
215	C₂²⋊C₈	—	2	4	no	—
216	C₂²⋊C₈	—	106	4	no	—
217	C₄×C₈	—	2	4	no	—
218	C₄×C₈	—	53	2	no	—
219	C₄×C₈	—	53	2	no	—
220	C₄×C₈	—	53	4	no	—
221	C₄×C₈	—	53	4	no	—
222	C₄×C₈	—	53	2	no	—
223	C₄×C₈	—	53	2	no	—
224	C₄×C₈	—	106	4	no	—
225	C₂⁴	#11719919	1	1	yes	characteristic
226	C₂²×C₄	—	2	4	no	—
227	C₂²×C₄	—	2	4	no	—
228	C₂²×C₄	C₈×D₅₃	1	4	yes	—
229	C₂²×C₄	#11719912	1	4	yes	—
230	C₂²×C₄	#11719919	1	1	yes	characteristic
231	C₂²×C₄	#11719919	1	2	yes	—
232	C₂²×C₄	#11719919	1	2	yes	—
233	C₂²×C₄	#11719919	1	2	yes	—
234	C₂²×C₄	#11719919	1	1	yes	characteristic
235	C₂²×C₄	#11719919	1	1	yes	characteristic
236	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
237	C₂²×C₄	C₂×Dic₁₀₆	1	2	yes	—
238	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
239	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
240	C₂²×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
241	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
242	C₂×C₈	—	2	4	no	—
243	C₂×C₈	—	4	4	no	—
244	C₂×C₈	—	53	4	no	—
245	C₂×C₈	—	53	4	no	—
246	C₂×C₈	—	53	16	no	—
247	C₂×C₈	—	53	2	no	—
248	C₂×C₈	—	53	2	no	—
249	C₂×C₈	—	53	2	no	—
250	C₂×C₈	—	53	2	no	—
251	C₂×C₈	—	106	8	no	—
252	C₂×C₈	—	106	2	no	—
253	C₂×C₈	—	106	2	no	—
254	C₂×C₈	—	106	4	no	—
255	C₂×C₈	—	212	4	no	—
256	C₄²	#11719919	1	2	yes	—
257	C₄²	#11719919	1	2	yes	—
258	C₄²	#11719919	1	2	yes	—
259	C₄²	C₂×Dic₁₀₆	1	2	yes	—
260	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
261	C₄²	C₂×C₄×D₅₃	1	2	yes	—
262	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
263	C₄²	C₂×C₂²⋊D₅₃	1	2	yes	—
264	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
265	C₄²	C₂³×D₅₃	1	1	yes	characteristic
266	C₂³	—	2	2	no	—
267	C₂³	—	2	2	no	—
268	C₂³	#13133146	1	4	yes	—
269	C₂³	C₂×#11719919	1	2	yes	—
270	C₂³	C₂×#11719919	1	1	yes	characteristic
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	#13133146	1	8	yes	—
278	C₂×C₄	#13133146	1	4	yes	—
279	C₂×C₄	#13133152	1	4	yes	—
280	C₂×C₄	C₂×#11719919	1	2	yes	—
281	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
282	C₂×C₄	C₂×#11719919	1	2	yes	—
283	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
284	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
285	C₂×C₄	C₂×#11719919	1	2	yes	—
286	C₂×C₄	C₂×C₈×D₅₃	1	4	yes	—
287	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
288	C₂×C₄	C₂×#11719912	1	4	yes	—
289	C₂×C₄	C₂×#11719912	1	2	yes	—
290	C₂×C₄	#13133237	1	4	yes	—
291	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
292	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
293	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
294	C₈	—	4	4	no	—
295	C₈	—	53	4	no	—
296	C₈	—	53	4	no	—
297	C₈	—	106	2	no	—
298	C₈	—	106	2	no	—
299	C₈	—	212	4	no	—
300	C₂²	—	2	8	no	—
301	C₂²	—	2	2	no	—
302	C₂²	—	2	2	no	—
303	C₂²	C₂×#13133146	1	4	yes	—
304	C₂²	C₂×#13133146	1	2	yes	—
305	C₂²	#16810659	1	2	yes	—
306	C₂²	C₂×#13133152	1	2	yes	—
307	C₂²	C₂²×#11719919	1	1	yes	characteristic
308	C₄	—	2	2	no	—
309	C₄	—	2	1	no	—
310	C₄	—	2	1	no	—
311	C₄	C₂×#13133146	1	2	yes	—
312	C₄	C₂×#13133146	1	4	yes	—
313	C₄	#16810659	1	2	yes	—
314	C₄	C₂×#13133152	1	4	yes	—
315	C₄	C₂²×C₈×D₅₃	1	1	yes	characteristic
316	C₄	C₂²×#11719912	1	1	yes	characteristic
317	C₄	C₂×#13133237	1	2	yes	—
318	C₂	—	2	4	no	—
319	C₂	#20273461	1	4	yes	direct product factor
320	C₂	C₂²×#13133146	1	1	yes	characteristic
321	C₂	C₂×#16810659	1	1	yes	characteristic
322	C₂	C₂²×#13133152	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²	C₂²⋊D₅₃
commutator	C₄×C₅₃	C₂³×C₈
Frattini	C₄²	C₂³×D₅₃
Fitting	C₂×C₅₃×#431	C₂
socle	C₂³×C₅₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₅₃	C₄⋊C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	C₂×#431	D₅₃
hypocenter	C₅₃	C₂×#1564
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄² ◁ C₂²×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
3. #20273461	C₂	direct (N × H)
4. C₂×#16810375	C₂	semidirect
5. C₂×#16810373	C₂	semidirect
20. #16810375	C₂²	semidirect
21. #16810373	C₂²	semidirect
30. C₂²×#88322	C₈	semidirect
99. C₂×#88322	C₂×C₈	semidirect
164. C₂×C₅₃	#1564	semidirect
189. C₅₃	C₂×#1564	semidirect
319. C₂	#20273461	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×#16810676	2
C₂²×C₄×#88322	2
#20273461	2
C₂×#16810375	2
C₂×#16810373	2
C₂×C₅₃×#431	2
C₂×#1564	53

Subgroups — C2×#20273461 #27353064

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20273461 #27353064