Subgroups — C₂×#20274257 #27366742

Order 13568.

702 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810971	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810823	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810754	C₂	1	1	yes	maximal, characteristic, complemented
4	#20274257	C₂	1	8	yes	maximal, direct product factor
5	C₂×#16810375	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16810372	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#16810366	C₂	1	1	yes	maximal, characteristic
8	C₂×C₅₃×#477	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#11719924	—	2	1	no	—
10	C₂×#13133214	C₂²	1	1	yes	characteristic
11	#16810971	—	2	2	no	—
12	#16810971	C₂²	1	4	yes	—
13	#16810823	—	2	2	no	—
14	#16810823	C₂²	1	4	yes	—
15	#16810754	—	2	2	no	—
16	#16810754	C₂²	1	4	yes	complemented
17	C₂×C₄⋊#88322	—	2	1	no	—
18	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
20	C₂×#13133137	—	2	1	no	—
21	C₂×#13133137	—	2	1	no	—
22	C₂×#13133137	—	2	1	no	—
23	C₂×#13133128	—	2	1	no	—
24	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
25	#16810375	C₂²	1	4	yes	complemented
26	#16810372	C₂²	1	4	yes	complemented
27	#16810366	C₂²	1	4	yes	—
28	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
29	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
30	C₅₃×#477	—	2	2	no	—
31	C₅₃×#477	C₂²	1	4	yes	complemented
32	C₂×#107×C₅₃	—	2	1	no	—
33	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
34	C₂×#103×C₅₃	—	2	1	no	—
35	C₂²×D₂₁₂	—	2	1	no	—
36	C₂²×D₂₁₂	—	2	1	no	—
37	C₂²×D₂₁₂	—	2	1	no	—
38	C₂×#11719924	—	2	4	no	—
39	C₂×#11719924	—	4	4	no	—
40	C₂²×#88322	—	2	1	no	—
41	C₂²×#88322	—	2	1	no	—
42	C₂²×#88322	D₄	1	1	yes	characteristic
43	C₂²×#88322	D₄	1	1	yes	characteristic
44	#13133214	—	2	1	no	—
45	#13133214	—	2	1	no	—
46	#13133214	—	2	1	no	—
47	#13133214	C₂³	1	2	yes	—
48	C₄⋊#88322	—	2	1	no	—
49	C₄⋊#88322	—	2	1	no	—
50	C₄⋊#88322	—	2	4	no	—
51	C₄⋊#88322	C₂³	1	2	yes	—
52	C₄⋊#88322	C₂³	1	2	yes	—
53	#13133137	—	2	4	no	—
54	#13133137	—	2	4	no	—
55	#13133137	—	2	4	no	—
56	#13133128	—	2	4	no	—
57	C₄×#88322	D₄	1	2	yes	—
58	C₄×#88322	C₂³	1	2	yes	—
59	C₂²×Q₈×C₅₃	—	2	1	no	—
60	C₂²×Q₈×C₅₃	D₄	1	1	yes	characteristic
61	C₂²×C₈×C₅₃	—	2	1	no	—
62	C₄⋊Q₈×C₅₃	—	2	1	no	—
63	C₄⋊Q₈×C₅₃	—	2	1	no	—
64	C₄⋊Q₈×C₅₃	—	2	1	no	—
65	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
66	C₄×Q₈×C₅₃	—	2	1	no	—
67	C₄×Q₈×C₅₃	—	2	1	no	—
68	C₄×Q₈×C₅₃	D₄	1	2	yes	—
69	C₄×Q₈×C₅₃	C₂³	1	2	yes	—
70	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
71	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
72	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
73	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
74	C₂×C₄²×C₅₃	—	2	1	no	—
75	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
76	#107×C₅₃	—	2	4	no	—
77	C₄⋊C₈×C₅₃	—	2	1	no	—
78	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
79	#103×C₅₃	—	2	4	no	—
80	C₂³×D₅₃	—	4	1	no	—
81	C₂×Q₈×C₅₃	—	2	1	no	—
82	C₂×Q₈×C₅₃	—	2	2	no	—
83	C₂×Q₈×C₅₃	—	2	2	no	—
84	C₂×Q₈×C₅₃	—	2	2	no	—
85	C₂×Q₈×C₅₃	—	4	1	no	—
86	C₂×Q₈×C₅₃	—	8	2	no	—
87	C₂×Q₈×C₅₃	C₂×D₄	1	2	yes	—
88	C₂²×C₄×C₅₃	—	2	1	no	—
89	C₂²×C₄×C₅₃	—	2	1	no	—
90	C₂²×C₄×C₅₃	—	4	1	no	—
91	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
94	C₂×D₂₁₂	—	2	2	no	—
95	C₂×D₂₁₂	—	2	2	no	—
96	C₂×D₂₁₂	—	2	2	no	—
97	C₂×D₂₁₂	—	4	2	no	—
98	C₂×D₂₁₂	—	4	2	no	—
99	C₂×D₂₁₂	—	4	2	no	—
100	C₂×D₂₁₂	—	4	2	no	—
101	C₂×D₂₁₂	—	4	1	no	—
102	C₂×D₂₁₂	—	4	1	no	—
103	C₂×D₂₁₂	—	4	1	no	—
104	C₂×D₂₁₂	—	8	2	no	—
105	C₂×C₈×C₅₃	—	2	2	no	—
106	C₂×C₈×C₅₃	—	4	2	no	—
107	C₄⋊C₄×C₅₃	—	2	2	no	—
108	C₄⋊C₄×C₅₃	—	2	1	no	—
109	C₄⋊C₄×C₅₃	—	2	2	no	—
110	C₄⋊C₄×C₅₃	—	2	2	no	—
111	C₄⋊C₄×C₅₃	—	2	2	no	—
112	C₄⋊C₄×C₅₃	—	4	1	no	—
113	C₄⋊C₄×C₅₃	—	4	1	no	—
114	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
115	C₄²×C₅₃	—	2	2	no	—
116	C₄²×C₅₃	—	2	2	no	—
117	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
118	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
119	C₄²×C₅₃	#41	1	1	yes	characteristic
120	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
121	#11719924	—	4	8	no	—
122	C₂×#88322	—	2	2	no	—
123	C₂×#88322	—	2	2	no	—
124	C₂×#88322	—	2	2	no	—
125	C₂×#88322	—	2	2	no	—
126	C₂×#88322	—	4	2	no	—
127	C₂×#88322	—	4	2	no	—
128	C₂×#88322	C₂×D₄	1	2	yes	—
129	C₂×#88322	C₂×D₄	1	2	yes	—
130	C₂³×C₅₃	#121	1	1	yes	characteristic
131	C₂²×D₅₃	—	4	2	no	—
132	C₂²×D₅₃	—	8	2	no	—
133	C₂²×D₅₃	—	8	2	no	—
134	C₂²×D₅₃	—	8	2	no	—
135	Q₈×C₅₃	—	2	2	no	—
136	Q₈×C₅₃	—	2	2	no	—
137	Q₈×C₅₃	—	4	2	no	—
138	Q₈×C₅₃	—	8	2	no	—
139	Q₈×C₅₃	—	8	2	no	—
140	C₂×C₄×C₅₃	—	2	2	no	—
141	C₂×C₄×C₅₃	—	2	2	no	—
142	C₂×C₄×C₅₃	—	2	2	no	—
143	C₂×C₄×C₅₃	—	4	2	no	—
144	C₂×C₄×C₅₃	—	4	2	no	—
145	C₂×C₄×C₅₃	—	4	2	no	—
146	C₂×C₄×C₅₃	—	8	2	no	—
147	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
148	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
149	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
150	C₂×C₄×C₅₃	#135	1	2	yes	—
151	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
152	C₂×C₄×C₅₃	#137	1	2	yes	—
153	C₂×C₄×C₅₃	#137	1	2	yes	—
154	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
155	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
156	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
157	D₂₁₂	—	4	2	no	—
158	D₂₁₂	—	4	2	no	—
159	D₂₁₂	—	4	2	no	—
160	D₂₁₂	—	4	2	no	—
161	D₂₁₂	—	8	2	no	—
162	D₂₁₂	—	8	2	no	—
163	D₂₁₂	—	8	2	no	—
164	D₂₁₂	—	8	2	no	—
165	C₈×C₅₃	—	4	2	no	—
166	#88322	—	2	2	no	—
167	#88322	—	2	2	no	—
168	#88322	—	4	2	no	—
169	#88322	—	4	2	no	—
170	C₂×#1698	—	53	1	no	maximal, Sylow
171	C₂²×C₅₃	#464	1	2	yes	—
172	C₂²×C₅₃	#468	1	2	yes	—
173	C₂²×C₅₃	#472	1	2	yes	—
174	C₂²×C₅₃	C₂×#121	1	1	yes	characteristic
175	C₂×D₅₃	—	8	2	no	—
176	C₂×D₅₃	—	8	2	no	—
177	C₂×D₅₃	—	8	2	no	—
178	C₂×D₅₃	—	16	4	no	—
179	C₄×C₅₃	—	2	1	no	—
180	C₄×C₅₃	—	2	1	no	—
181	C₄×C₅₃	—	4	2	no	—
182	C₄×C₅₃	—	4	2	no	—
183	C₄×C₅₃	—	8	2	no	—
184	C₄×C₅₃	#464	1	2	yes	—
185	C₄×C₅₃	#468	1	2	yes	—
186	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
187	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
188	C₄×C₅₃	C₂×#137	1	1	yes	characteristic
189	C₄×C₅₃	C₂×#137	1	1	yes	characteristic
190	C₂×#486	—	53	1	no	—
191	C₂×#477	D₅₃	1	1	yes	characteristic, complemented
192	C₂×#460	—	53	1	no	—
193	C₂×#446	—	53	1	no	—
194	#1698	—	53	8	no	—
195	C₂×#335	—	53	1	no	—
196	C₂×#333	—	53	1	no	—
197	C₂×#331	—	53	1	no	—
198	C₂×C₅₃	#1698	1	4	yes	complemented
199	C₂×C₅₃	C₂×#464	1	1	yes	characteristic
200	C₂×C₅₃	C₂×#468	1	1	yes	characteristic
201	C₂×C₅₃	C₂×#472	1	1	yes	characteristic
202	D₅₃	—	16	2	no	—
203	C₂²×SD₁₆	—	106	1	no	—
204	C₂×C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
205	C₂×#127	—	53	1	no	—
206	C₂×C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
207	#486	—	53	4	no	—
208	#486	—	106	2	no	—
209	#477	—	2	2	no	—
210	#477	C₂×D₅₃	1	4	yes	complemented
211	#460	—	53	4	no	—
212	#460	—	106	2	no	—
213	#446	—	53	4	no	—
214	#446	—	106	2	no	—
215	C₂×#107	—	2	1	no	—
216	C₂×C₄⋊C₈	—	53	1	no	—
217	C₂×C₄⋊C₈	—	53	1	no	—
218	C₂×C₄⋊C₈	—	106	1	no	—
219	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
220	C₂×#103	—	2	1	no	—
221	C₂×#102	—	106	1	no	—
222	C₂×#102	—	106	1	no	—
223	C₂×#102	—	106	1	no	—
224	C₂×#97	—	106	1	no	—
225	C₂×C₄×C₈	—	53	1	no	—
226	#335	—	53	4	no	—
227	#333	—	53	4	no	—
228	#331	—	53	4	no	—
229	C₅₃	C₂×#1698	1	1	yes	characteristic, complemented, Sylow
230	C₂²×Q₈	—	2	1	no	—
231	C₂²×Q₈	C₂²⋊D₅₃	1	1	yes	characteristic
232	C₂²×D₄	—	106	1	no	—
233	C₂²×D₄	—	106	1	no	—
234	C₂²×D₄	—	106	1	no	—
235	C₂×SD₁₆	—	106	4	no	—
236	C₂×SD₁₆	—	212	4	no	—
237	C₂²×C₈	—	2	1	no	—
238	C₂²×C₈	—	53	1	no	—
239	C₂²×C₈	—	53	1	no	—
240	C₂²×C₈	—	106	1	no	—
241	C₂²×C₈	—	106	1	no	—
242	C₄⋊Q₈	—	2	1	no	—
243	C₄⋊Q₈	—	2	1	no	—
244	C₄⋊Q₈	—	2	1	no	—
245	C₄⋊Q₈	C₂²×D₅₃	1	2	yes	—
246	#127	—	53	2	no	—
247	#127	—	106	1	no	—
248	#127	—	106	1	no	—
249	#127	—	106	1	no	—
250	C₄×Q₈	—	2	1	no	—
251	C₄×Q₈	—	2	1	no	—
252	C₄×Q₈	C₂²⋊D₅₃	1	2	yes	—
253	C₄×Q₈	C₂²×D₅₃	1	2	yes	—
254	C₂×C₄⋊C₄	—	2	1	no	—
255	C₂×C₄⋊C₄	—	2	1	no	—
256	C₂×C₄⋊C₄	—	2	1	no	—
257	C₂×C₄⋊C₄	C₂²⋊D₅₃	1	1	yes	characteristic
258	C₂×C₄²	—	2	1	no	—
259	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
260	#107	—	2	4	no	—
261	C₄⋊C₈	—	2	1	no	—
262	C₄⋊C₈	—	53	2	no	—
263	C₄⋊C₈	—	53	2	no	—
264	C₄⋊C₈	—	106	4	no	—
265	C₄⋊C₈	—	106	1	no	—
266	C₄⋊C₈	—	106	1	no	—
267	C₄⋊C₈	C₂²×D₅₃	1	2	yes	—
268	#103	—	2	4	no	—
269	#102	—	106	4	no	—
270	#102	—	106	4	no	—
271	#102	—	106	4	no	—
272	#97	—	106	4	no	—
273	C₄×C₈	—	53	2	no	—
274	C₄×C₈	—	53	2	no	—
275	C₂⁴	—	212	1	no	—
276	C₂×Q₈	—	2	1	no	—
277	C₂×Q₈	—	2	2	no	—
278	C₂×Q₈	—	2	2	no	—
279	C₂×Q₈	—	2	2	no	—
280	C₂×Q₈	—	4	1	no	—
281	C₂×Q₈	—	8	2	no	—
282	C₂×Q₈	C₂×C₂²⋊D₅₃	1	2	yes	—
283	C₂×D₄	—	106	2	no	—
284	C₂×D₄	—	106	2	no	—
285	C₂×D₄	—	106	2	no	—
286	C₂×D₄	—	424	2	no	—
287	C₂×D₄	—	212	1	no	—
288	C₂×D₄	—	212	1	no	—
289	C₂×D₄	—	212	1	no	—
290	C₂×D₄	—	212	2	no	—
291	C₂×D₄	—	212	2	no	—
292	C₂×D₄	—	212	2	no	—
293	C₂×D₄	—	212	2	no	—
294	C₂²×C₄	—	2	1	no	—
295	C₂²×C₄	—	2	1	no	—
296	C₂²×C₄	—	4	1	no	—
297	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
298	C₂²×C₄	#11719947	1	1	yes	characteristic
299	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
300	SD₁₆	—	212	8	no	—
301	C₂×C₈	—	2	2	no	—
302	C₂×C₈	—	4	2	no	—
303	C₂×C₈	—	53	2	no	—
304	C₂×C₈	—	53	2	no	—
305	C₂×C₈	—	106	2	no	—
306	C₂×C₈	—	106	2	no	—
307	C₂×C₈	—	106	2	no	—
308	C₂×C₈	—	106	2	no	—
309	C₂×C₈	—	212	2	no	—
310	C₂×C₈	—	212	2	no	—
311	C₄⋊C₄	—	2	2	no	—
312	C₄⋊C₄	—	2	1	no	—
313	C₄⋊C₄	—	2	2	no	—
314	C₄⋊C₄	—	2	2	no	—
315	C₄⋊C₄	—	2	2	no	—
316	C₄⋊C₄	—	4	1	no	—
317	C₄⋊C₄	—	4	1	no	—
318	C₄⋊C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
319	C₄²	—	2	2	no	—
320	C₄²	—	2	2	no	—
321	C₄²	D₄×D₅₃	1	1	yes	characteristic
322	C₄²	#11719947	1	1	yes	characteristic
323	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
324	C₄²	C₂³×D₅₃	1	1	yes	characteristic
325	C₂³	—	424	2	no	—
326	C₂³	—	424	2	no	—
327	C₂³	—	424	2	no	—
328	C₂³	—	212	2	no	—
329	C₂³	#13133189	1	1	yes	characteristic
330	Q₈	—	2	2	no	—
331	Q₈	—	2	2	no	—
332	Q₈	—	4	2	no	—
333	Q₈	—	8	2	no	—
334	Q₈	—	8	2	no	—
335	D₄	—	424	2	no	—
336	D₄	—	424	2	no	—
337	D₄	—	424	2	no	—
338	D₄	—	424	2	no	—
339	D₄	—	212	2	no	—
340	D₄	—	212	2	no	—
341	D₄	—	212	2	no	—
342	D₄	—	212	2	no	—
343	C₂×C₄	—	2	2	no	—
344	C₂×C₄	—	2	2	no	—
345	C₂×C₄	—	2	2	no	—
346	C₂×C₄	—	4	2	no	—
347	C₂×C₄	—	4	2	no	—
348	C₂×C₄	—	4	2	no	—
349	C₂×C₄	—	8	2	no	—
350	C₂×C₄	#13133189	1	1	yes	characteristic
351	C₂×C₄	#13133189	1	1	yes	characteristic
352	C₂×C₄	#13133189	1	1	yes	characteristic
353	C₂×C₄	#13133240	1	2	yes	—
354	C₂×C₄	#13133242	1	2	yes	—
355	C₂×C₄	#13133248	1	2	yes	—
356	C₂×C₄	#13133249	1	2	yes	—
357	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
358	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
359	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
360	C₈	—	4	2	no	—
361	C₈	—	106	2	no	—
362	C₈	—	106	2	no	—
363	C₈	—	212	2	no	—
364	C₈	—	212	2	no	—
365	C₂²	—	848	4	no	—
366	C₂²	—	424	2	no	—
367	C₂²	—	424	2	no	—
368	C₂²	—	424	2	no	—
369	C₂²	#16810847	1	2	yes	—
370	C₂²	#16810871	1	2	yes	—
371	C₂²	#16810891	1	2	yes	—
372	C₂²	C₂×#13133189	1	1	yes	characteristic
373	C₄	—	2	1	no	—
374	C₄	—	2	1	no	—
375	C₄	—	4	2	no	—
376	C₄	—	4	2	no	—
377	C₄	—	8	2	no	—
378	C₄	#16810847	1	2	yes	—
379	C₄	#16810871	1	2	yes	—
380	C₄	C₂×#13133240	1	1	yes	characteristic
381	C₄	C₂×#13133242	1	1	yes	characteristic
382	C₄	C₂×#13133248	1	1	yes	characteristic
383	C₄	C₂×#13133249	1	1	yes	characteristic
384	C₂	—	848	2	no	—
385	C₂	#20274257	1	4	yes	direct product factor
386	C₂	C₂×#16810847	1	1	yes	characteristic
387	C₂	C₂×#16810871	1	1	yes	characteristic
388	C₂	C₂×#16810891	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13133189
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₂×C₅₃×#477	C₂
socle	C₂³×C₅₃	#121
prime radical	C₂×#13133214	C₂²
hypercenter	C₂×#477	D₅₃
hypocenter	C₅₃	C₂×#1698
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#477

Complements

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

N	H ≅ G/N	type
3. C₂×#16810754	C₂	semidirect
4. #20274257	C₂	direct (N × H)
5. C₂×#16810375	C₂	semidirect
6. C₂×#16810372	C₂	semidirect
8. C₂×C₅₃×#477	C₂	semidirect
16. #16810754	C₂²	semidirect
25. #16810375	C₂²	semidirect
26. #16810372	C₂²	semidirect
31. C₅₃×#477	C₂²	semidirect
191. C₂×#477	D₅₃	semidirect
198. C₂×C₅₃	#1698	semidirect
210. #477	C₂×D₅₃	semidirect
229. C₅₃	C₂×#1698	semidirect
385. C₂	#20274257	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×#16810971	2
C₂×#16810823	2
C₂×#16810754	2
#20274257	2
C₂×#16810375	2
C₂×#16810372	2
C₂×#16810366	2
C₂×C₅₃×#477	2
C₂×#1698	53

Subgroups — C2×#20274257 #27366742

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20274257 #27366742