Subgroups — #27382742

Order 13568.

938 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810754	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	#20274592	C₂	1	2	yes	maximal
4	#20274587	C₂	1	2	yes	maximal
5	#20274580	C₂	1	1	yes	maximal, characteristic
6	C₈×C₄⋊Dic₅₃	C₂	1	1	yes	maximal, characteristic
7	#20274570	C₂	1	1	yes	maximal, characteristic
8	#20274570	C₂	1	2	yes	maximal
9	C₄×#13133127	C₂	1	1	yes	maximal, characteristic
10	#97×Dic₅₃	C₂	1	2	yes	maximal
11	C₂×C₅₃×#446	C₂	1	1	yes	maximal, characteristic
12	C₂×Q₈×Dic₅₃	C₄	1	2	yes	—
13	#16810754	C₄	1	8	yes	complemented
14	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊#88322	C₂²	1	2	yes	—
16	C₂×#13133128	C₂²	1	2	yes	—
17	C₂×#13133127	C₂²	1	2	yes	—
18	C₂×#13133127	C₂²	1	1	yes	characteristic
19	C₂×#13133127	C₂²	1	1	yes	characteristic
20	C₂×#13133127	C₂²	1	1	yes	characteristic
21	C₂×#13133127	C₂²	1	1	yes	characteristic
22	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
23	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
24	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
25	C₄⋊C₄×Dic₅₃	C₄	1	4	yes	—
26	C₄⋊C₄×Dic₅₃	C₄	1	2	yes	—
27	C₄×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
28	C₄×C₄⋊Dic₅₃	C₂²	1	2	yes	—
29	C₄²×Dic₅₃	C₂²	1	2	yes	—
30	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
31	#16810377	C₂²	1	2	yes	—
32	#16810377	C₂²	1	2	yes	—
33	#16810377	C₂²	1	2	yes	—
34	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
35	C₅₃×#446	C₄	1	8	yes	complemented
36	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
37	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
38	C₂×#97×C₅₃	C₂²	1	2	yes	—
39	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
40	C₂²×#88322	C₂³	1	2	yes	—
41	C₂²×#88322	C₂³	1	1	yes	characteristic
42	C₂²×#88322	C₂³	1	1	yes	characteristic
43	Q₈×Dic₅₃	—	2	8	no	—
44	C₂×C₄⋊Dic₅₃	C₂×C₄	1	4	yes	—
45	C₂×C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
46	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
47	C₂×C₄×Dic₅₃	C₂×C₄	1	4	yes	—
48	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
49	C₂×C₄×Dic₅₃	C₂×C₄	1	4	yes	—
50	C₂×C₄×Dic₅₃	C₂³	1	2	yes	—
51	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
52	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
53	C₄⋊#88322	C₂×C₄	1	8	yes	—
54	C₄⋊#88322	C₂×C₄	1	4	yes	—
55	#13133128	C₂×C₄	1	8	yes	—
56	#13133127	—	2	2	no	—
57	#13133127	—	2	4	no	—
58	#13133127	—	2	2	no	—
59	#13133127	—	2	2	no	—
60	#13133127	Q₈	1	4	yes	complemented
61	C₄×#88322	C₂×C₄	1	4	yes	—
62	C₈×Dic₅₃	—	2	4	no	—
63	C₈×Dic₅₃	—	2	4	no	—
64	#13133124	C₂×C₄	1	4	yes	—
65	#13133124	C₂×C₄	1	2	yes	—
66	C₂²×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
67	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
68	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
69	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
70	C₄×Q₈×C₅₃	C₂×C₄	1	4	yes	—
71	C₄×Q₈×C₅₃	C₂×C₄	1	4	yes	—
72	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
73	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
74	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
75	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
76	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
77	C₄⋊C₈×C₅₃	C₂×C₄	1	8	yes	—
78	#97×C₅₃	C₂×C₄	1	8	yes	—
79	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
80	C₂×Q₈×C₅₃	—	2	4	no	—
81	C₂×Q₈×C₅₃	C₄²	1	4	yes	—
82	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
84	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
85	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
87	C₂²×Dic₅₃	C₂²×C₄	1	2	yes	—
88	C₂²×Dic₅₃	C₂²×C₄	1	2	yes	—
89	C₂²×Dic₅₃	C₂²×C₄	1	4	yes	—
90	C₂×C₈×C₅₃	—	2	2	no	—
91	C₂×C₈×C₅₃	—	2	1	no	—
92	C₂×C₈×C₅₃	—	2	2	no	—
93	C₂×C₈×C₅₃	—	2	1	no	—
94	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
95	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
96	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
97	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
98	C₂×C₈×C₅₃	#41	1	2	yes	—
99	C₄⋊C₄×C₅₃	C₄²	1	8	yes	—
100	C₄⋊C₄×C₅₃	C₄²	1	4	yes	—
101	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
102	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
103	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
104	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
105	C₄⋊Dic₅₃	—	2	8	no	—
106	C₄⋊Dic₅₃	M₁₆	1	8	yes	—
107	C₄×Dic₅₃	—	2	4	no	—
108	C₄×Dic₅₃	—	2	2	no	—
109	C₄×Dic₅₃	—	2	2	no	—
110	C₄×Dic₅₃	—	2	1	no	—
111	C₄×Dic₅₃	—	2	1	no	—
112	C₄×Dic₅₃	—	2	8	no	—
113	C₄×Dic₅₃	—	4	4	no	—
114	C₄×Dic₅₃	—	4	2	no	—
115	C₄×Dic₅₃	C₂×Q₈	1	2	yes	—
116	C₄×Dic₅₃	#41	1	2	yes	—
117	C₂×#88322	—	2	2	no	—
118	C₂×#88322	—	2	2	no	—
119	C₂×#88322	—	2	1	no	—
120	C₂×#88322	—	2	1	no	—
121	C₂×#88322	C₂²×C₄	1	2	yes	—
122	C₂×#88322	C₂²×C₄	1	4	yes	—
123	C₂×#88322	C₂²×C₄	1	2	yes	—
124	C₂×#88322	C₂×Q₈	1	2	yes	—
125	C₂×#88322	#41	1	2	yes	—
126	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
127	Q₈×C₅₃	—	2	8	no	—
128	C₂×C₄×C₅₃	—	2	4	no	—
129	C₂×C₄×C₅₃	—	2	2	no	—
130	C₂×C₄×C₅₃	—	2	4	no	—
131	C₂×C₄×C₅₃	C₂×C₄²	1	4	yes	—
132	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
133	C₂×C₄×C₅₃	C₂×C₄²	1	4	yes	—
134	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
135	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
136	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
137	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
138	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
139	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
140	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
141	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
142	C₂×Dic₅₃	—	2	4	no	—
143	C₂×Dic₅₃	—	2	2	no	—
144	C₂×Dic₅₃	—	2	4	no	—
145	C₂×Dic₅₃	—	2	2	no	—
146	C₂×Dic₅₃	—	2	2	no	—
147	C₂×Dic₅₃	—	2	2	no	—
148	C₂×Dic₅₃	—	2	2	no	—
149	C₂×Dic₅₃	C₄×Q₈	1	4	yes	—
150	C₂×Dic₅₃	C₂×M₁₆	1	8	yes	—
151	C₈×C₅₃	—	2	4	no	—
152	C₈×C₅₃	—	2	2	no	—
153	C₈×C₅₃	C₄×Q₈	1	4	yes	complemented
154	#88322	—	2	4	no	—
155	#88322	—	2	2	no	—
156	#88322	C₄×Q₈	1	4	yes	complemented
157	#20520	—	53	1	no	maximal, Sylow
158	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
159	C₂²×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
160	C₂²×C₅₃	C₄×#41	1	1	yes	characteristic
161	C₂²×C₅₃	C₂²×M₁₆	1	2	yes	—
162	C₂²×C₅₃	#568	1	2	yes	—
163	C₄×C₅₃	—	2	4	no	—
164	C₄×C₅₃	—	2	2	no	—
165	C₄×C₅₃	—	2	4	no	—
166	C₄×C₅₃	C₄×M₁₆	1	4	yes	—
167	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
168	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
169	C₄×C₅₃	C₄×#41	1	1	yes	characteristic
170	C₄×C₅₃	C₄×#41	1	1	yes	characteristic
171	Dic₅₃	—	2	4	no	—
172	Dic₅₃	—	2	8	no	—
173	Dic₅₃	—	4	2	no	—
174	C₂×#446	—	53	1	no	—
175	C₂×#446	D₅₃	1	1	yes	characteristic
176	C₄²×Q₈	—	53	1	no	—
177	#1771	—	53	2	no	—
178	#1771	—	53	2	no	—
179	C₈×C₄⋊C₄	—	53	1	no	—
180	C₈×C₄⋊C₄	—	53	1	no	—
181	#1767	—	53	2	no	—
182	#1767	—	53	1	no	—
183	C₄×#97	—	53	2	no	—
184	C₄×#97	—	53	1	no	—
185	C₂×C₅₃	C₄²×Q₈	1	1	yes	characteristic
186	C₂×C₅₃	C₂×C₄×M₁₆	1	1	yes	characteristic
187	C₂×C₅₃	#2875	1	1	yes	characteristic
188	C₂×C₅₃	Q₈×M₁₆	1	2	yes	—
189	C₂×C₅₃	#2970	1	2	yes	—
190	C₂×C₄×Q₈	—	53	2	no	—
191	C₂×C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
192	#446	—	53	8	no	—
193	#446	Dic₅₃	1	8	yes	complemented
194	C₂×C₄⋊C₈	—	53	1	no	—
195	C₂×C₄⋊C₈	—	53	2	no	—
196	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
197	C₂×C₄⋊C₈	C₂×D₅₃	1	2	yes	—
198	C₂×#97	—	53	1	no	—
199	C₂×#97	—	53	2	no	—
200	C₂×#97	—	53	2	no	—
201	C₂×#97	—	53	1	no	—
202	C₂×#97	—	53	1	no	—
203	C₂×#97	—	53	1	no	—
204	C₂×#97	C₂×D₅₃	1	2	yes	—
205	C₂×C₄×C₈	—	53	1	no	—
206	C₂×C₄×C₈	—	53	2	no	—
207	C₂×C₄×C₈	—	53	2	no	—
208	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
209	C₄×C₄⋊C₄	—	53	2	no	—
210	C₄×C₄⋊C₄	—	53	4	no	—
211	C₄×C₄⋊C₄	—	53	1	no	—
212	C₄×C₄⋊C₄	—	53	2	no	—
213	C₄³	—	53	2	no	—
214	C₄³	—	53	1	no	—
215	#336	—	53	2	no	—
216	#336	—	53	2	no	—
217	#336	—	53	2	no	—
218	C₅₃	#20520	1	1	yes	characteristic, complemented, Sylow
219	C₂²×Q₈	C₄×D₅₃	1	1	yes	characteristic
220	C₂²×C₈	—	53	1	no	—
221	C₂²×C₈	—	53	1	no	—
222	C₂²×C₈	—	53	2	no	—
223	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
224	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
225	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
226	C₄×Q₈	—	106	8	no	—
227	C₄×Q₈	C₄×D₅₃	1	4	yes	—
228	C₄×Q₈	C₂×Dic₅₃	1	4	yes	—
229	C₂×C₄⋊C₄	—	53	2	no	—
230	C₂×C₄⋊C₄	—	53	4	no	—
231	C₂×C₄⋊C₄	C₄×D₅₃	1	2	yes	—
232	C₂×C₄⋊C₄	C₄×D₅₃	1	1	yes	characteristic
233	C₂×C₄²	—	53	1	no	—
234	C₂×C₄²	—	53	4	no	—
235	C₂×C₄²	—	53	2	no	—
236	C₂×C₄²	—	53	2	no	—
237	C₂×C₄²	—	53	1	no	—
238	C₂×C₄²	—	53	4	no	—
239	C₂×C₄²	—	53	2	no	—
240	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
241	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
242	C₄⋊C₈	—	53	4	no	—
243	C₄⋊C₈	—	53	8	no	—
244	C₄⋊C₈	C₂×Dic₅₃	1	4	yes	—
245	C₄⋊C₈	C₂×Dic₅₃	1	8	yes	—
246	#97	—	53	4	no	—
247	#97	—	53	8	no	—
248	#97	—	106	4	no	—
249	#97	—	106	2	no	—
250	#97	—	106	2	no	—
251	#97	—	106	2	no	—
252	#97	C₂×Dic₅₃	1	8	yes	—
253	C₄×C₈	—	53	4	no	—
254	C₄×C₈	—	106	4	no	—
255	C₄×C₈	—	106	4	no	—
256	C₄×C₈	C₂×Dic₅₃	1	4	yes	—
257	#95	—	53	2	no	—
258	#95	—	53	4	no	—
259	C₂×Q₈	—	2	4	no	—
260	C₂×Q₈	C₄×Dic₅₃	1	4	yes	—
261	C₂²×C₄	—	53	4	no	—
262	C₂²×C₄	—	53	2	no	—
263	C₂²×C₄	—	53	2	no	—
264	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
265	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
266	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
267	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
268	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
269	C₂×C₈	—	2	2	no	—
270	C₂×C₈	—	2	1	no	—
271	C₂×C₈	—	2	2	no	—
272	C₂×C₈	—	2	1	no	—
273	C₂×C₈	—	53	2	no	—
274	C₂×C₈	—	53	2	no	—
275	C₂×C₈	—	53	2	no	—
276	C₂×C₈	—	53	2	no	—
277	C₂×C₈	—	53	4	no	—
278	C₂×C₈	—	106	1	no	—
279	C₂×C₈	—	106	2	no	—
280	C₂×C₈	—	106	2	no	—
281	C₂×C₈	—	106	1	no	—
282	C₂×C₈	Q₈×D₅₃	1	2	yes	—
283	C₂×C₈	#11719949	1	2	yes	—
284	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
285	C₂×C₈	C₂²×Dic₅₃	1	4	yes	—
286	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
287	C₄⋊C₄	—	53	8	no	—
288	C₄⋊C₄	—	106	8	no	—
289	C₄⋊C₄	C₄×Dic₅₃	1	8	yes	—
290	C₄⋊C₄	C₄×Dic₅₃	1	4	yes	—
291	C₄²	—	53	2	no	—
292	C₄²	—	53	2	no	—
293	C₄²	—	106	8	no	—
294	C₄²	—	106	4	no	—
295	C₄²	—	106	2	no	—
296	C₄²	—	106	1	no	—
297	C₄²	—	106	2	no	—
298	C₄²	—	106	1	no	—
299	C₄²	—	212	4	no	—
300	C₄²	—	212	2	no	—
301	C₄²	C₂×C₄×D₅₃	1	2	yes	—
302	C₄²	C₂×C₄×D₅₃	1	4	yes	—
303	C₄²	C₂²×Dic₅₃	1	2	yes	—
304	C₄²	C₂²×Dic₅₃	1	4	yes	—
305	C₂³	C₂²×C₄×D₅₃	1	1	yes	characteristic
306	Q₈	—	2	8	no	—
307	C₂×C₄	—	2	4	no	—
308	C₂×C₄	—	2	4	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	53	4	no	—
311	C₂×C₄	—	53	8	no	—
312	C₂×C₄	—	106	4	no	—
313	C₂×C₄	—	106	2	no	—
314	C₂×C₄	—	106	2	no	—
315	C₂×C₄	—	106	4	no	—
316	C₂×C₄	—	106	2	no	—
317	C₂×C₄	—	106	2	no	—
318	C₂×C₄	—	106	2	no	—
319	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
320	C₂×C₄	C₂×C₄×Dic₅₃	1	4	yes	—
321	C₂×C₄	C₂×C₄×Dic₅₃	1	4	yes	—
322	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
323	C₂×C₄	M₁₆×D₅₃	1	4	yes	—
324	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
325	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
326	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
327	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
328	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
329	C₂×C₄	C₂×#11719949	1	1	yes	characteristic
330	C₈	—	2	4	no	—
331	C₈	—	2	2	no	—
332	C₈	—	53	4	no	—
333	C₈	—	106	2	no	—
334	C₈	—	106	4	no	—
335	C₈	Q₈×Dic₅₃	1	4	yes	complemented
336	C₂²	C₂²×C₄×Dic₅₃	1	1	yes	characteristic
337	C₂²	C₄×Q₈×D₅₃	1	1	yes	characteristic
338	C₂²	C₄×#11719949	1	1	yes	characteristic
339	C₂²	C₂×M₁₆×D₅₃	1	2	yes	—
340	C₂²	#16811388	1	2	yes	—
341	C₄	—	2	2	no	—
342	C₄	—	2	4	no	—
343	C₄	—	2	4	no	—
344	C₄	—	106	4	no	—
345	C₄	—	106	8	no	—
346	C₄	—	212	2	no	—
347	C₄	M₁₆×Dic₅₃	1	4	yes	—
348	C₄	C₄×Q₈×D₅₃	1	1	yes	characteristic
349	C₄	C₂×Q₈×Dic₅₃	1	1	yes	characteristic
350	C₄	C₄×#11719949	1	1	yes	characteristic
351	C₄	#41×Dic₅₃	1	1	yes	characteristic
352	C₂	C₄×Q₈×Dic₅₃	1	1	yes	characteristic
353	C₂	C₂×M₁₆×Dic₅₃	1	1	yes	characteristic
354	C₂	#20283141	1	1	yes	characteristic
355	C₂	#20283667	1	2	yes	—
356	C₂	#20283728	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₅₃
commutator	C₂²×C₅₃	C₂²×C₄²
Fitting	C₂×C₅₃×#446	C₂
socle, prime radical	C₂³×C₅₃	C₂³×C₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂×#446	D₅₃
hypocenter	C₅₃	#20520
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#446

Complements

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

N	H ≅ G/N	type
13. #16810754	C₄	semidirect
35. C₅₃×#446	C₄	semidirect
60. #13133127	Q₈	semidirect
153. C₈×C₅₃	C₄×Q₈	semidirect
156. #88322	C₄×Q₈	semidirect
193. #446	Dic₅₃	semidirect
218. C₅₃	#20520	semidirect
335. C₈	Q₈×Dic₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810754	2
C₄×Q₈×Dic₅₃	2
#20274592	2
#20274587	2
#20274580	2
C₈×C₄⋊Dic₅₃	2
#20274570	2
C₄×#13133127	2
#97×Dic₅₃	2
C₂×C₅₃×#446	2
#20520	53