Subgroups — C₂×#20285228 #28253696

Order 13568.

1942 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811179	C₂	1	2	yes	maximal
2	#20285228	C₂	1	16	yes	maximal, direct product factor
3	C₂×#16811027	C₂	1	2	yes	maximal, complemented
4	C₂×#16811025	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16811022	C₂	1	2	yes	maximal, complemented
6	C₂×#16811020	C₂	1	2	yes	maximal, complemented
7	C₂×#16810660	C₂	1	2	yes	maximal
8	C₂×#16810658	C₂	1	2	yes	maximal
9	C₂×C₄×C₈×D₅₃	C₂	1	1	yes	maximal, characteristic
10	C₂×C₅₃×#499	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
12	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
13	#16811179	—	2	8	no	—
14	#16811179	C₂²	1	16	yes	—
15	C₂×#13133216	C₂²	1	4	yes	—
16	C₂×#13133209	C₂²	1	2	yes	—
17	C₂×#13133195	—	2	4	no	—
18	C₂×#13133182	—	2	4	no	—
19	C₂×#13133175	C₂²	1	4	yes	—
20	C₂×C₄²×D₅₃	C₂²	1	1	yes	characteristic
21	#16811027	C₂²	1	16	yes	complemented
22	#16811025	C₂²	1	8	yes	complemented
23	#16811022	C₂²	1	16	yes	complemented
24	#16811020	C₂²	1	16	yes	complemented
25	#16810660	C₂²	1	16	yes	—
26	#16810658	C₂²	1	16	yes	—
27	C₂×#13133151	C₂²	1	2	yes	—
28	C₂×#13133150	C₂²	1	4	yes	—
29	C₂×#13133149	C₂²	1	2	yes	—
30	C₂×#13133148	C₂²	1	4	yes	—
31	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
32	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
33	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
34	C₄×C₈×D₅₃	C₂²	1	8	yes	—
35	C₂×#125×C₅₃	C₂²	1	2	yes	—
36	C₅₃×#499	C₂²	1	8	yes	complemented
37	C₂×#107×C₅₃	C₂²	1	2	yes	—
38	C₂×#106×C₅₃	C₂²	1	2	yes	—
39	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
40	C₂²×Dic₁₀₆	—	2	4	no	—
41	C₂²×C₄×D₅₃	D₄	1	2	yes	—
42	C₂²×C₄×D₅₃	C₂³	1	1	yes	characteristic
43	C₂²×#88322	C₂³	1	1	yes	characteristic
44	C₂²×#88322	C₂³	1	1	yes	characteristic
45	C₂×C₈×D₅₃	Q₈	1	8	yes	complemented
46	C₂×C₈×D₅₃	Q₈	1	8	yes	complemented
47	C₂×C₈×D₅₃	C₂³	1	4	yes	—
48	C₂×C₈×D₅₃	C₂³	1	4	yes	—
49	#13133216	—	2	8	no	—
50	#13133216	C₂³	1	16	yes	—
51	#13133209	—	2	4	no	—
52	#13133209	C₂³	1	8	yes	—
53	#13133195	—	2	32	no	—
54	#13133182	—	2	32	no	—
55	#13133175	—	2	8	no	—
56	#13133175	C₂³	1	16	yes	—
57	C₂×C₄⋊Dic₅₃	C₂³	1	4	yes	—
58	C₂×#11719919	—	2	4	no	—
59	C₂×#11719919	—	2	4	no	—
60	C₂×#11719921	—	2	4	no	—
61	C₂×C₄×Dic₅₃	—	2	4	no	—
62	C₂×C₄×Dic₅₃	D₄	1	2	yes	—
63	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
64	C₄²×D₅₃	D₄	1	4	yes	—
65	C₄²×D₅₃	C₂³	1	4	yes	—
66	#13133151	C₂³	1	8	yes	—
67	#13133150	C₂³	1	16	yes	—
68	#13133149	C₂³	1	8	yes	—
69	#13133148	C₂³	1	16	yes	—
70	C₄×#88322	C₂³	1	4	yes	—
71	C₈×Dic₅₃	C₂³	1	4	yes	—
72	C₈×Dic₅₃	C₂³	1	4	yes	—
73	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
74	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
75	#125×C₅₃	—	2	4	no	—
76	#125×C₅₃	C₂³	1	8	yes	—
77	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
78	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
79	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
80	#107×C₅₃	C₂³	1	8	yes	—
81	#106×C₅₃	C₂³	1	8	yes	—
82	C₄×C₈×C₅₃	C₂³	1	4	yes	—
83	C₂³×D₅₃	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₅₃	—	2	4	no	—
85	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
86	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
87	C₂²×Dic₅₃	—	2	4	no	—
88	C₂²×Dic₅₃	C₂×D₄	1	1	yes	characteristic
89	C₂²×Dic₅₃	C₂×D₄	1	2	yes	—
90	C₂×C₄×D₅₃	—	2	8	no	—
91	C₂×C₄×D₅₃	C₂×D₄	1	8	yes	—
92	C₂×C₄×D₅₃	C₂×D₄	1	2	yes	—
93	C₂×C₄×D₅₃	C₂×Q₈	1	4	yes	—
94	C₂×C₄×D₅₃	C₂×Q₈	1	4	yes	—
95	C₂×C₄×D₅₃	C₂⁴	1	2	yes	—
96	C₂×Dic₁₀₆	—	2	16	no	—
97	C₂×Dic₁₀₆	—	4	16	no	—
98	C₂×C₈×C₅₃	C₂×Q₈	1	4	yes	—
99	C₂×C₈×C₅₃	C₂×Q₈	1	4	yes	—
100	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
101	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
102	C₄⋊C₄×C₅₃	—	2	16	no	—
103	C₄⋊C₄×C₅₃	—	2	4	no	—
104	C₄⋊C₄×C₅₃	C₂⁴	1	8	yes	—
105	C₄²×C₅₃	C₂×D₄	1	2	yes	—
106	C₄²×C₅₃	C₂⁴	1	2	yes	—
107	#11719921	—	2	16	no	—
108	C₄⋊Dic₅₃	—	2	4	no	—
109	C₄⋊Dic₅₃	C₂⁴	1	8	yes	—
110	#11719919	—	2	16	no	—
111	#11719919	—	2	16	no	—
112	C₄×Dic₅₃	—	2	16	no	—
113	C₄×Dic₅₃	C₂×D₄	1	2	yes	—
114	C₄×Dic₅₃	C₂×D₄	1	8	yes	—
115	C₄×Dic₅₃	C₂⁴	1	2	yes	—
116	C₂×#88322	C₂×Q₈	1	4	yes	—
117	C₂×#88322	C₂×Q₈	1	4	yes	—
118	C₂×#88322	C₂⁴	1	2	yes	—
119	C₂×#88322	C₂⁴	1	2	yes	—
120	C₈×D₅₃	C₂×Q₈	1	16	yes	complemented
121	C₈×D₅₃	C₂×Q₈	1	16	yes	complemented
122	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
123	C₂²×D₅₃	—	2	4	no	—
124	C₂²×D₅₃	C₄⋊Q₈	1	4	yes	—
125	C₂²×D₅₃	C₂²×D₄	1	2	yes	—
126	C₂×C₄×C₅₃	—	2	4	no	—
127	C₂×C₄×C₅₃	—	2	8	no	—
128	C₂×C₄×C₅₃	—	4	8	no	—
129	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
130	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₅₃	C₂²×Q₈	1	2	yes	—
132	C₂×C₄×C₅₃	C₂²×Q₈	1	2	yes	—
133	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
134	C₂×Dic₅₃	—	2	8	no	—
135	C₂×Dic₅₃	—	4	8	no	—
136	C₂×Dic₅₃	C₄⋊Q₈	1	4	yes	—
137	C₂×Dic₅₃	#135	1	8	yes	—
138	C₂×Dic₅₃	C₂²×D₄	1	2	yes	—
139	C₂×Dic₅₃	C₂²×D₄	1	4	yes	—
140	C₄×D₅₃	—	2	16	no	—
141	C₄×D₅₃	C₄⋊Q₈	1	8	yes	—
142	C₄×D₅₃	C₂²×Q₈	1	4	yes	—
143	C₄×D₅₃	C₂²×Q₈	1	4	yes	—
144	Dic₁₀₆	—	4	32	no	—
145	C₈×C₅₃	C₂²×Q₈	1	4	yes	—
146	C₈×C₅₃	C₂²×Q₈	1	4	yes	—
147	#88322	C₂²×Q₈	1	4	yes	—
148	#88322	C₂²×Q₈	1	4	yes	—
149	C₂×#3160	—	53	1	no	maximal, Sylow
150	C₂²×C₅₃	C₂×C₄⋊Q₈	1	2	yes	—
151	C₂²×C₅₃	C₂×#135	1	4	yes	—
152	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
153	C₂×D₅₃	—	2	8	no	—
154	C₂×D₅₃	—	2	4	no	—
155	C₂×D₅₃	C₂×C₄⋊Q₈	1	4	yes	—
156	C₄×C₅₃	—	2	4	no	—
157	C₄×C₅₃	—	4	8	no	—
158	C₄×C₅₃	C₂×C₄⋊Q₈	1	2	yes	—
159	C₄×C₅₃	C₂³×Q₈	1	1	yes	characteristic
160	C₄×C₅₃	C₂³×Q₈	1	1	yes	characteristic
161	Dic₅₃	—	4	8	no	—
162	Dic₅₃	C₂×C₄⋊Q₈	1	4	yes	—
163	Dic₅₃	C₂×#135	1	8	yes	—
164	C₂×#533	—	53	2	no	—
165	#3160	—	53	16	no	—
166	C₂×#500	—	53	2	no	—
167	C₂×#499	—	53	1	no	—
168	C₂×#499	—	53	2	no	—
169	C₂×#499	D₅₃	1	1	yes	characteristic, complemented
170	C₂×#498	—	53	2	no	—
171	C₂×#427	—	53	2	no	—
172	C₂×#427	—	53	2	no	—
173	C₂²×C₄×C₈	—	53	1	no	—
174	C₂×C₅₃	#3160	1	4	yes	complemented
175	C₂×C₅₃	C₂²×C₄⋊Q₈	1	1	yes	characteristic
176	C₂×C₅₃	C₂²×#135	1	2	yes	—
177	D₅₃	—	2	4	no	—
178	C₂³×C₈	—	53	1	no	—
179	C₂³×C₈	—	53	1	no	—
180	#533	—	53	16	no	—
181	#533	—	106	8	no	—
182	C₂×C₄⋊Q₈	—	53	4	no	—
183	C₂×#125	—	53	2	no	—
184	C₂×#125	C₂×D₅₃	1	2	yes	—
185	C₂×C₂²⋊Q₈	—	106	4	no	—
186	C₂×C₄×Q₈	—	106	4	no	—
187	C₂×#117	—	53	4	no	—
188	C₂²×C₄²	—	53	1	no	—
189	#500	—	53	16	no	—
190	#499	—	53	16	no	—
191	#499	—	53	8	no	—
192	#499	C₂×D₅₃	1	8	yes	complemented
193	#498	—	53	16	no	—
194	#427	—	53	16	no	—
195	#427	—	53	16	no	—
196	C₂×#107	—	53	2	no	—
197	C₂×#107	—	53	4	no	—
198	C₂×#107	C₂×D₅₃	1	2	yes	—
199	C₂×#106	—	53	2	no	—
200	C₂×#106	—	53	4	no	—
201	C₂×#106	C₂×D₅₃	1	2	yes	—
202	C₂×C₄×C₈	—	53	1	no	—
203	C₂×C₄×C₈	—	53	1	no	—
204	C₂×C₄×C₈	—	53	1	no	—
205	C₂×C₄×C₈	—	53	8	no	—
206	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
207	C₅₃	C₂×#3160	1	1	yes	characteristic, complemented, Sylow
208	C₂²×Q₈	—	106	4	no	—
209	C₂³×C₄	—	53	1	no	—
210	C₂³×C₄	—	53	2	no	—
211	C₂²×C₈	—	53	1	no	—
212	C₂²×C₈	—	53	1	no	—
213	C₂²×C₈	—	53	8	no	—
214	C₂²×C₈	—	53	8	no	—
215	C₂²×C₈	—	53	4	no	—
216	C₂²×C₈	—	53	4	no	—
217	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
218	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
219	C₄⋊Q₈	—	53	16	no	—
220	C₄⋊Q₈	—	106	8	no	—
221	#125	—	2	4	no	—
222	#125	—	53	8	no	—
223	#125	—	106	4	no	—
224	#125	C₂²×D₅₃	1	8	yes	—
225	C₂²⋊Q₈	—	106	32	no	—
226	C₄×Q₈	—	106	32	no	—
227	#117	—	53	16	no	—
228	#117	—	106	8	no	—
229	C₂×C₄⋊C₄	—	2	4	no	—
230	C₂×C₄⋊C₄	—	53	4	no	—
231	C₂×C₄⋊C₄	—	106	4	no	—
232	C₂×C₄⋊C₄	—	106	4	no	—
233	C₂×C₄⋊C₄	C₂²×D₅₃	1	4	yes	—
234	C₂×C₂²⋊C₄	—	106	4	no	—
235	C₂×C₄²	—	53	2	no	—
236	C₂×C₄²	—	53	1	no	—
237	C₂×C₄²	—	53	4	no	—
238	C₂×C₄²	—	53	4	no	—
239	C₂×C₄²	—	106	4	no	—
240	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
241	#107	—	53	8	no	—
242	#107	—	53	16	no	—
243	#107	C₂²×D₅₃	1	8	yes	—
244	#106	—	53	8	no	—
245	#106	—	53	16	no	—
246	#106	C₂²×D₅₃	1	8	yes	—
247	C₄×C₈	—	53	4	no	—
248	C₄×C₈	—	53	4	no	—
249	C₄×C₈	—	53	4	no	—
250	C₄×C₈	C₂²×D₅₃	1	4	yes	—
251	C₂⁴	—	53	1	no	—
252	C₂×Q₈	—	106	16	no	—
253	C₂×Q₈	—	212	16	no	—
254	C₂²×C₄	—	2	4	no	—
255	C₂²×C₄	—	53	1	no	—
256	C₂²×C₄	—	53	2	no	—
257	C₂²×C₄	—	53	2	no	—
258	C₂²×C₄	—	53	4	no	—
259	C₂²×C₄	—	53	4	no	—
260	C₂²×C₄	—	53	2	no	—
261	C₂²×C₄	—	53	8	no	—
262	C₂²×C₄	—	106	8	no	—
263	C₂²×C₄	—	106	4	no	—
264	C₂²×C₄	D₄×D₅₃	1	2	yes	—
265	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
266	C₂×C₈	—	53	4	no	—
267	C₂×C₈	—	53	4	no	—
268	C₂×C₈	—	53	2	no	—
269	C₂×C₈	—	53	2	no	—
270	C₂×C₈	—	53	16	no	—
271	C₂×C₈	—	53	16	no	—
272	C₂×C₈	Q₈×D₅₃	1	4	yes	—
273	C₂×C₈	Q₈×D₅₃	1	4	yes	—
274	C₂×C₈	C₂³×D₅₃	1	2	yes	—
275	C₂×C₈	C₂³×D₅₃	1	2	yes	—
276	C₄⋊C₄	—	2	16	no	—
277	C₄⋊C₄	—	2	4	no	—
278	C₄⋊C₄	—	53	8	no	—
279	C₄⋊C₄	—	106	16	no	—
280	C₄⋊C₄	—	106	4	no	—
281	C₄⋊C₄	—	106	16	no	—
282	C₄⋊C₄	C₂³×D₅₃	1	8	yes	—
283	C₂²⋊C₄	—	106	16	no	—
284	C₄²	—	53	8	no	—
285	C₄²	—	53	2	no	—
286	C₄²	—	53	2	no	—
287	C₄²	—	106	16	no	—
288	C₄²	D₄×D₅₃	1	2	yes	—
289	C₄²	C₂³×D₅₃	1	2	yes	—
290	C₂³	—	53	4	no	—
291	C₂³	—	53	2	no	—
292	C₂³	—	106	4	no	—
293	C₂³	C₂×D₄×D₅₃	1	1	yes	characteristic
294	Q₈	—	212	32	no	—
295	C₂×C₄	—	2	4	no	—
296	C₂×C₄	—	2	8	no	—
297	C₂×C₄	—	4	8	no	—
298	C₂×C₄	—	53	8	no	—
299	C₂×C₄	—	53	4	no	—
300	C₂×C₄	—	53	2	no	—
301	C₂×C₄	—	53	4	no	—
302	C₂×C₄	—	53	8	no	—
303	C₂×C₄	—	53	4	no	—
304	C₂×C₄	—	53	4	no	—
305	C₂×C₄	—	106	16	no	—
306	C₂×C₄	—	106	8	no	—
307	C₂×C₄	—	212	8	no	—
308	C₂×C₄	C₂×D₄×D₅₃	1	4	yes	—
309	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
310	C₂×C₄	C₂×Q₈×D₅₃	1	2	yes	—
311	C₂×C₄	C₂×Q₈×D₅₃	1	2	yes	—
312	C₂×C₄	C₂⁴×D₅₃	1	1	yes	characteristic
313	C₈	—	53	4	no	—
314	C₈	—	53	4	no	—
315	C₈	C₂×Q₈×D₅₃	1	4	yes	—
316	C₈	C₂×Q₈×D₅₃	1	4	yes	—
317	C₂²	—	53	4	no	—
318	C₂²	—	106	8	no	—
319	C₂²	—	106	4	no	—
320	C₂²	C₄⋊Q₈×D₅₃	1	2	yes	—
321	C₂²	#135×D₅₃	1	4	yes	—
322	C₂²	C₂²×D₄×D₅₃	1	1	yes	characteristic
323	C₄	—	2	4	no	—
324	C₄	—	4	8	no	—
325	C₄	—	53	8	no	—
326	C₄	—	53	4	no	—
327	C₄	—	212	8	no	—
328	C₄	C₄⋊Q₈×D₅₃	1	2	yes	—
329	C₄	C₂²×Q₈×D₅₃	1	1	yes	characteristic
330	C₄	C₂²×Q₈×D₅₃	1	1	yes	characteristic
331	C₂	—	106	4	no	—
332	C₂	#20285228	1	4	yes	direct product factor
333	C₂	C₂×C₄⋊Q₈×D₅₃	1	1	yes	characteristic
334	C₂	C₂×#135×D₅₃	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₅₃
commutator, cosocle	C₂×C₄×C₅₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₅₃
Fitting	C₂×C₅₃×#499	C₂
socle	C₂³×C₅₃	C₂²×D₄
prime radical	C₂³×D₅₃	C₂×D₄
hypercenter	C₂×#499	D₅₃
hypocenter	C₅₃	C₂×#3160
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#499

Complements

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

N	H ≅ G/N	type
2. #20285228	C₂	direct (N × H)
3. C₂×#16811027	C₂	semidirect
4. C₂×#16811025	C₂	semidirect
5. C₂×#16811022	C₂	semidirect
6. C₂×#16811020	C₂	semidirect
10. C₂×C₅₃×#499	C₂	semidirect
21. #16811027	C₂²	semidirect
22. #16811025	C₂²	semidirect
23. #16811022	C₂²	semidirect
24. #16811020	C₂²	semidirect
36. C₅₃×#499	C₂²	semidirect
45. C₂×C₈×D₅₃	Q₈	semidirect
46. C₂×C₈×D₅₃	Q₈	semidirect
120. C₈×D₅₃	C₂×Q₈	semidirect
121. C₈×D₅₃	C₂×Q₈	semidirect
169. C₂×#499	D₅₃	semidirect
174. C₂×C₅₃	#3160	semidirect
192. #499	C₂×D₅₃	semidirect
207. C₅₃	C₂×#3160	semidirect
332. C₂	#20285228	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×#16811179	2
#20285228	2
C₂×#16811027	2
C₂×#16811025	2
C₂×#16811022	2
C₂×#16811020	2
C₂×#16810660	2
C₂×#16810658	2
C₂×C₄×C₈×D₅₃	2
C₂×C₅₃×#499	2
C₂×#3160	53

Subgroups — C2×#20285228 #28253696

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20285228 #28253696