Subgroups — C₂×#20274596 #27382100

Order 13568.

1102 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133153	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810579	C₂	1	2	yes	maximal
3	C₂²×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
4	#20274596	C₂	1	8	yes	maximal, direct product factor
5	C₂×#16810382	C₂	1	2	yes	maximal
6	C₂×C₅₃×#427	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11719917	C₂²	1	2	yes	—
8	C₂³×#88322	C₂²	1	1	yes	characteristic
9	C₂²×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
10	C₂×#13133153	C₄	1	8	yes	complemented
11	C₂×#13133153	C₂²	1	4	yes	—
12	#16810579	—	2	4	no	—
13	#16810579	C₂²	1	8	yes	—
14	C₂×#13133128	—	2	2	no	—
15	C₂×C₄×#88322	—	2	2	no	—
16	C₂×C₈×Dic₅₃	C₄	1	8	yes	complemented
17	C₂×C₈×Dic₅₃	C₂²	1	4	yes	—
18	#16810382	C₂²	1	8	yes	—
19	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×#117×C₅₃	C₂²	1	2	yes	—
21	C₅₃×#427	C₄	1	4	yes	complemented
22	C₅₃×#427	C₂²	1	4	yes	—
23	C₂×#107×C₅₃	C₄	1	2	yes	complemented
24	C₂×#106×C₅₃	C₄	1	2	yes	complemented
25	C₂³×Dic₅₃	—	2	1	no	—
26	C₂×#11719917	—	2	2	no	—
27	C₂×#11719917	—	2	4	no	—
28	C₂×#11719917	C₂×C₄	1	8	yes	—
29	C₂×#11719917	C₂³	1	4	yes	—
30	C₂²×#88322	—	2	2	no	—
31	C₂²×#88322	C₂×C₄	1	4	yes	—
32	C₂²×#88322	D₄	1	1	yes	characteristic
33	C₂²×#88322	D₄	1	1	yes	characteristic
34	C₂²×#88322	D₄	1	2	yes	—
35	C₂²×#88322	Q₈	1	4	yes	—
36	C₂²×#88322	C₂³	1	2	yes	—
37	C₂×C₄×Dic₅₃	—	2	2	no	—
38	C₂×C₄×Dic₅₃	—	2	1	no	—
39	C₂×C₄×Dic₅₃	C₂×C₄	1	4	yes	—
40	C₂×C₄×Dic₅₃	C₂³	1	2	yes	—
41	#13133153	C₂×C₄	1	16	yes	complemented
42	#13133128	—	2	8	no	—
43	C₄×#88322	—	2	8	no	—
44	C₈×Dic₅₃	C₂×C₄	1	16	yes	complemented
45	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
46	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
47	C₂²×C₈×C₅₃	C₂×C₄	1	4	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	2	no	—
53	#117×C₅₃	—	2	2	no	—
54	#117×C₅₃	C₂×C₄	1	4	yes	—
55	#117×C₅₃	C₂³	1	4	yes	—
56	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
57	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
58	C₂×C₄²×C₅₃	—	2	2	no	—
59	#107×C₅₃	C₂×C₄	1	8	yes	complemented
60	#106×C₅₃	C₂×C₄	1	8	yes	complemented
61	C₂⁴×C₅₃	#41	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	—	2	4	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	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
70	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	#41	1	2	yes	—
72	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
73	C₂²×Dic₅₃	—	2	8	no	—
74	C₂²×Dic₅₃	—	2	4	no	—
75	C₂²×Dic₅₃	—	2	2	no	—
76	C₂×C₈×C₅₃	C₄²	1	4	yes	—
77	C₂×C₈×C₅₃	C₄²	1	4	yes	—
78	C₂×C₈×C₅₃	C₄²	1	8	yes	—
79	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
80	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
81	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
83	C₄⋊C₄×C₅₃	—	2	4	no	—
84	C₄⋊C₄×C₅₃	C₂²×C₄	1	8	yes	—
85	C₂²⋊C₄×C₅₃	—	2	4	no	—
86	C₂²⋊C₄×C₅₃	—	2	4	no	—
87	C₄²×C₅₃	—	2	4	no	—
88	C₄²×C₅₃	—	2	4	no	—
89	C₄×Dic₅₃	—	2	4	no	—
90	C₄×Dic₅₃	—	2	2	no	—
91	C₄×Dic₅₃	C₂²×C₄	1	4	yes	—
92	#11719917	—	2	8	no	—
93	#11719917	—	2	4	no	—
94	#11719917	C₂²×C₄	1	8	yes	—
95	C₂×#88322	—	2	8	no	—
96	C₂×#88322	—	2	4	no	—
97	C₂×#88322	C₄⋊C₄	1	4	yes	—
98	C₂×#88322	C₄⋊C₄	1	4	yes	—
99	C₂×#88322	C₄⋊C₄	1	8	yes	—
100	C₂×#88322	C₂²×C₄	1	4	yes	—
101	C₂×#88322	C₂×D₄	1	2	yes	—
102	C₂×#88322	C₂×D₄	1	2	yes	—
103	C₂×#88322	C₂×Q₈	1	4	yes	—
104	C₂³×C₅₃	—	2	4	no	—
105	C₂³×C₅₃	#117	1	2	yes	—
106	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
107	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
108	C₂³×C₅₃	C₄×Q₈	1	2	yes	—
109	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
110	C₂×C₄×C₅₃	—	2	8	no	—
111	C₂×C₄×C₅₃	—	4	8	no	—
112	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
113	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
114	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
115	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
116	C₂×C₄×C₅₃	#117	1	2	yes	—
117	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
118	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
119	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
121	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
122	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
123	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
124	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
125	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
126	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
127	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
128	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
129	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
130	C₂×Dic₅₃	—	2	8	no	—
131	C₂×Dic₅₃	—	2	16	no	—
132	C₂×Dic₅₃	—	2	4	no	—
133	C₈×C₅₃	C₂×C₄²	1	4	yes	—
134	C₈×C₅₃	C₂×C₄²	1	4	yes	—
135	#88322	—	2	8	no	—
136	#88322	C₂×C₄⋊C₄	1	4	yes	—
137	#88322	C₂×C₄⋊C₄	1	4	yes	—
138	C₂×#1772	—	53	1	no	maximal, Sylow
139	C₂²×C₅₃	—	2	2	no	—
140	C₂²×C₅₃	—	2	2	no	—
141	C₂²×C₅₃	—	2	4	no	—
142	C₂²×C₅₃	—	2	4	no	—
143	C₂²×C₅₃	C₄×C₄⋊C₄	1	2	yes	—
144	C₂²×C₅₃	C₄×C₄⋊C₄	1	2	yes	—
145	C₂²×C₅₃	#443	1	4	yes	—
146	C₂²×C₅₃	C₂×#117	1	1	yes	characteristic
147	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
148	C₂²×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
149	C₄×C₅₃	—	4	8	no	—
150	C₄×C₅₃	C₄×C₄⋊C₄	1	2	yes	—
151	C₄×C₅₃	C₄×C₄⋊C₄	1	2	yes	—
152	C₄×C₅₃	C₂²×C₄²	1	1	yes	characteristic
153	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
154	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
155	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
156	Dic₅₃	—	2	8	no	—
157	C₂²×#108	—	53	1	no	—
158	C₂×#427	D₅₃	1	1	yes	characteristic
159	C₂×#405	—	53	2	no	—
160	C₂²×C₄×C₈	—	53	1	no	—
161	#1772	—	53	8	no	—
162	C₂×#339	—	53	2	no	—
163	C₂×C₅₃	—	2	1	no	—
164	C₂×C₅₃	—	2	1	no	—
165	C₂×C₅₃	—	2	2	no	—
166	C₂×C₅₃	#1772	1	4	yes	complemented
167	C₂×C₅₃	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
168	C₂×C₅₃	C₂×#443	1	2	yes	—
169	C₂²×M₁₆	—	53	2	no	—
170	C₂³×C₈	—	53	1	no	—
171	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
172	C₂×#117	C₂×D₅₃	1	2	yes	—
173	C₂²×C₄²	—	53	1	no	—
174	C₂×#108	—	53	8	no	—
175	C₂×#108	—	53	4	no	—
176	#427	Dic₅₃	1	4	yes	complemented
177	#427	C₂×D₅₃	1	4	yes	—
178	C₂×#107	Dic₅₃	1	2	yes	complemented
179	C₂×#106	Dic₅₃	1	2	yes	complemented
180	#405	—	53	8	no	—
181	#405	—	106	4	no	—
182	C₂×#97	—	106	2	no	—
183	C₂×C₄×C₈	—	53	8	no	—
184	C₂×C₄×C₈	—	53	4	no	—
185	C₂×C₄×C₈	—	106	2	no	—
186	#339	—	53	8	no	—
187	C₅₃	C₂×#1772	1	1	yes	characteristic, complemented, Sylow
188	C₂³×C₄	—	106	1	no	—
189	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
190	C₂×M₁₆	—	53	8	no	—
191	C₂×M₁₆	—	53	4	no	—
192	C₂×M₁₆	—	106	2	no	—
193	C₂×M₁₆	—	106	4	no	—
194	C₂²×C₈	—	53	4	no	—
195	C₂²×C₈	—	53	2	no	—
196	C₂²×C₈	—	53	1	no	—
197	C₂²×C₈	—	53	1	no	—
198	C₂²×C₈	—	53	4	no	—
199	C₂²×C₈	—	53	2	no	—
200	C₂²×C₈	—	106	2	no	—
201	C₂²×C₈	C₄×D₅₃	1	4	yes	—
202	C₂²×C₈	C₄×D₅₃	1	4	yes	—
203	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
204	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
205	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
206	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
207	#117	—	2	2	no	—
208	#117	—	2	2	no	—
209	#117	C₂×Dic₅₃	1	4	yes	—
210	#117	C₂²×D₅₃	1	4	yes	—
211	C₂×C₄⋊C₄	C₂×Dic₅₃	1	4	yes	—
212	C₂×C₂²⋊C₄	—	2	2	no	—
213	C₂×C₄²	—	2	2	no	—
214	C₂×C₄²	—	53	4	no	—
215	C₂×C₄²	—	53	2	no	—
216	C₂×C₄²	—	106	1	no	—
217	C₂×C₄²	—	106	2	no	—
218	#108	—	53	16	no	—
219	#107	C₂×Dic₅₃	1	8	yes	complemented
220	#106	C₂×Dic₅₃	1	8	yes	complemented
221	#97	—	106	8	no	—
222	C₄×C₈	—	53	16	no	—
223	C₄×C₈	—	106	8	no	—
224	C₂⁴	#11719947	1	1	yes	characteristic
225	C₂²×C₄	—	2	4	no	—
226	C₂²×C₄	—	106	8	no	—
227	C₂²×C₄	—	106	4	no	—
228	C₂²×C₄	—	106	2	no	—
229	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
230	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
231	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
232	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
233	C₂²×C₄	#11719947	1	1	yes	characteristic
234	C₂²×C₄	Q₈×D₅₃	1	2	yes	—
235	C₂²×C₄	#11719949	1	2	yes	—
236	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
237	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
238	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
239	M₁₆	—	53	8	no	—
240	M₁₆	—	106	8	no	—
241	M₁₆	—	106	4	no	—
242	C₂×C₈	—	53	8	no	—
243	C₂×C₈	—	53	4	no	—
244	C₂×C₈	—	53	4	no	—
245	C₂×C₈	—	53	4	no	—
246	C₂×C₈	—	53	2	no	—
247	C₂×C₈	—	53	2	no	—
248	C₂×C₈	—	53	4	no	—
249	C₂×C₈	—	106	8	no	—
250	C₂×C₈	—	106	4	no	—
251	C₂×C₈	C₄×Dic₅₃	1	4	yes	—
252	C₂×C₈	C₄×Dic₅₃	1	4	yes	—
253	C₂×C₈	C₄×Dic₅₃	1	8	yes	—
254	C₂×C₈	C₂×C₄×D₅₃	1	4	yes	—
255	C₂×C₈	C₂×C₄×D₅₃	1	4	yes	—
256	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
257	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
258	C₄⋊C₄	—	2	4	no	—
259	C₄⋊C₄	C₂²×Dic₅₃	1	8	yes	—
260	C₂²⋊C₄	—	2	4	no	—
261	C₂²⋊C₄	—	2	4	no	—
262	C₄²	—	2	4	no	—
263	C₄²	—	2	4	no	—
264	C₄²	—	53	4	no	—
265	C₄²	—	106	4	no	—
266	C₄²	—	106	2	no	—
267	C₂³	—	2	4	no	—
268	C₂³	#13133175	1	2	yes	—
269	C₂³	D₄×Dic₅₃	1	1	yes	characteristic
270	C₂³	D₄×Dic₅₃	1	1	yes	characteristic
271	C₂³	#13133182	1	2	yes	—
272	C₂³	C₂×#11719947	1	1	yes	characteristic
273	C₂×C₄	—	2	8	no	—
274	C₂×C₄	—	4	8	no	—
275	C₂×C₄	—	106	16	no	—
276	C₂×C₄	—	106	8	no	—
277	C₂×C₄	—	106	4	no	—
278	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
279	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
280	C₂×C₄	C₄⋊C₄×D₅₃	1	2	yes	—
281	C₂×C₄	C₄⋊C₄×D₅₃	1	2	yes	—
282	C₂×C₄	#13133175	1	2	yes	—
283	C₂×C₄	#13133179	1	2	yes	—
284	C₂×C₄	#13133179	1	2	yes	—
285	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
286	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
287	C₂×C₄	#13133182	1	2	yes	—
288	C₂×C₄	Q₈×Dic₅₃	1	2	yes	—
289	C₂×C₄	Q₈×Dic₅₃	1	2	yes	—
290	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
291	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
292	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
293	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
294	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
295	C₂×C₄	C₂×#11719949	1	1	yes	characteristic
296	C₈	—	53	4	no	—
297	C₈	—	53	4	no	—
298	C₈	—	106	8	no	—
299	C₈	C₂×C₄×Dic₅₃	1	4	yes	—
300	C₈	C₂×C₄×Dic₅₃	1	4	yes	—
301	C₂²	—	2	4	no	—
302	C₂²	—	2	4	no	—
303	C₂²	—	2	2	no	—
304	C₂²	—	2	2	no	—
305	C₂²	C₄⋊C₄×Dic₅₃	1	2	yes	—
306	C₂²	C₄⋊C₄×Dic₅₃	1	2	yes	—
307	C₂²	#16810740	1	4	yes	—
308	C₂²	C₂×#13133175	1	1	yes	characteristic
309	C₂²	C₂×D₄×Dic₅₃	1	1	yes	characteristic
310	C₂²	C₂×#13133182	1	1	yes	characteristic
311	C₄	—	4	8	no	—
312	C₄	—	106	8	no	—
313	C₄	C₄⋊C₄×Dic₅₃	1	2	yes	—
314	C₄	C₄⋊C₄×Dic₅₃	1	2	yes	—
315	C₄	C₂²×C₄×Dic₅₃	1	1	yes	characteristic
316	C₄	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
317	C₄	C₂×#13133179	1	1	yes	characteristic
318	C₄	C₂×Q₈×Dic₅₃	1	1	yes	characteristic
319	C₂	—	2	1	no	—
320	C₂	—	2	1	no	—
321	C₂	—	2	2	no	—
322	C₂	#20274596	1	4	yes	direct product factor
323	C₂	C₂×C₄⋊C₄×Dic₅₃	1	1	yes	characteristic
324	C₂	C₂×#16810740	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	D₄×D₅₃
commutator	C₄×C₅₃	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₅₃×#427	C₂
socle	C₂³×C₅₃	C₄×D₄
prime radical	C₂⁴×C₅₃	#41
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂×#427	D₅₃
hypocenter	C₅₃	C₂×#1772
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#427

Complements

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

N	H ≅ G/N	type
4. #20274596	C₂	direct (N × H)
10. C₂×#13133153	C₄	semidirect
16. C₂×C₈×Dic₅₃	C₄	semidirect
21. C₅₃×#427	C₄	semidirect
23. C₂×#107×C₅₃	C₄	semidirect
24. C₂×#106×C₅₃	C₄	semidirect
41. #13133153	C₂×C₄	semidirect
44. C₈×Dic₅₃	C₂×C₄	semidirect
59. #107×C₅₃	C₂×C₄	semidirect
60. #106×C₅₃	C₂×C₄	semidirect
166. C₂×C₅₃	#1772	semidirect
176. #427	Dic₅₃	semidirect
178. C₂×#107	Dic₅₃	semidirect
179. C₂×#106	Dic₅₃	semidirect
187. C₅₃	C₂×#1772	semidirect
219. #107	C₂×Dic₅₃	semidirect
220. #106	C₂×Dic₅₃	semidirect
322. C₂	#20274596	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#13133153	2
C₂×#16810579	2
C₂²×C₈×Dic₅₃	2
#20274596	2
C₂×#16810382	2
C₂×C₅₃×#427	2
C₂×#1772	53

Subgroups — C2×#20274596 #27382100

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20274596 #27382100