Subgroups — C₄×#108×D₅₃ #27382186

Order 13568.

1086 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#108×D₅₃	C₂	1	2	yes	maximal
2	C₄×M₁₆×D₅₃	C₂	1	2	yes	maximal
3	C₂×C₄×C₈×D₅₃	C₂	1	1	yes	maximal, characteristic
4	#108×Dic₅₃	C₂	1	2	yes	maximal, complemented
5	C₄×#13133152	C₂	1	2	yes	maximal, complemented
6	C₄×#13133153	C₂	1	1	yes	maximal, characteristic, complemented
7	#20274616	C₂	1	2	yes	maximal, complemented
8	#341×D₅₃	C₂	1	2	yes	maximal
9	C₄×#108×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×M₁₆×D₅₃	C₂²	1	4	yes	—
11	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
12	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
13	C₂×C₄²×D₅₃	C₂²	1	1	yes	characteristic
14	C₂×#13133153	C₂²	1	2	yes	—
15	C₂×#13133152	C₂²	1	4	yes	—
16	#108×D₅₃	C₄	1	16	yes	direct product factor
17	C₄×#11719917	C₂²	1	2	yes	—
18	#16810575	—	2	2	no	—
19	M₁₆×Dic₅₃	C₂²	1	4	yes	—
20	C₄×#11719912	—	2	2	no	—
21	#97×D₅₃	—	2	2	no	—
22	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
23	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
24	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
25	C₄×C₈×D₅₃	—	2	2	no	—
26	C₄×C₈×D₅₃	C₄	1	4	yes	—
27	C₄×C₈×D₅₃	C₄	1	4	yes	—
28	#16810388	C₂²	1	2	yes	—
29	#16810387	C₂²	1	4	yes	—
30	C₂×#108×C₅₃	C₂²	1	2	yes	—
31	C₄×M₁₆×C₅₃	C₂²	1	2	yes	—
32	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
33	C₅₃×#341	C₂²	1	2	yes	—
34	C₂²×C₄×D₅₃	Q₈	1	2	yes	—
35	C₂²×C₄×D₅₃	C₂³	1	1	yes	characteristic
36	C₂×#11719917	C₂³	1	4	yes	—
37	C₂×#11719912	—	2	4	no	—
38	M₁₆×D₅₃	—	2	8	no	—
39	M₁₆×D₅₃	C₂×C₄	1	16	yes	—
40	C₂²×#88322	C₂³	1	1	yes	characteristic
41	C₂²×#88322	C₂³	1	1	yes	characteristic
42	C₂×C₈×D₅₃	—	2	4	no	—
43	C₂×C₈×D₅₃	—	2	4	no	—
44	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
45	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
46	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
47	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
48	C₂×C₈×D₅₃	C₂×C₄	1	8	yes	—
49	C₂×C₄×Dic₅₃	Q₈	1	2	yes	—
50	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
51	C₄²×D₅₃	C₂×C₄	1	2	yes	—
52	C₄²×D₅₃	C₂×C₄	1	2	yes	—
53	C₄²×D₅₃	D₄	1	1	yes	characteristic
54	C₄²×D₅₃	D₄	1	1	yes	characteristic
55	C₄²×D₅₃	D₄	1	1	yes	characteristic
56	C₄²×D₅₃	D₄	1	1	yes	characteristic
57	#13133153	C₂×C₄	1	8	yes	complemented
58	#13133152	C₂×C₄	1	16	yes	complemented
59	#13133128	—	2	2	no	—
60	#13133127	—	2	4	no	—
61	C₄×#88322	—	2	2	no	—
62	C₄×#88322	C₂×C₄	1	4	yes	—
63	C₈×Dic₅₃	—	2	4	no	—
64	C₈×Dic₅₃	C₂×C₄	1	4	yes	—
65	C₈×Dic₅₃	C₂×C₄	1	4	yes	—
66	C₂×M₁₆×C₅₃	C₂³	1	4	yes	—
67	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
68	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
69	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
70	#108×C₅₃	C₂×C₄	1	8	yes	complemented
71	#97×C₅₃	—	2	2	no	—
72	C₄×C₈×C₅₃	—	2	2	no	—
73	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
74	C₂³×D₅₃	C₂×Q₈	1	1	yes	characteristic
75	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
76	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
77	C₂²×Dic₅₃	C₂×Q₈	1	2	yes	—
78	C₂²×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
79	C₂×C₄×D₅₃	—	2	8	no	—
80	C₂×C₄×D₅₃	C₄⋊C₄	1	2	yes	—
81	C₂×C₄×D₅₃	C₄⋊C₄	1	2	yes	—
82	C₂×C₄×D₅₃	C₄⋊C₄	1	2	yes	—
83	C₂×C₄×D₅₃	C₄⋊C₄	1	2	yes	—
84	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₄×D₅₃	C₂²×C₄	1	4	yes	—
87	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
88	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
89	C₂×C₄×D₅₃	#41	1	4	yes	—
90	M₁₆×C₅₃	—	2	4	no	—
91	M₁₆×C₅₃	C₂²×C₄	1	8	yes	—
92	C₂×C₈×C₅₃	—	2	4	no	—
93	C₂×C₈×C₅₃	—	2	2	no	—
94	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
95	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
96	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
97	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
98	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
100	C₄×Dic₅₃	C₄⋊C₄	1	2	yes	—
101	C₄×Dic₅₃	C₄⋊C₄	1	2	yes	—
102	C₄×Dic₅₃	C₄⋊C₄	1	2	yes	—
103	C₄×Dic₅₃	C₄⋊C₄	1	2	yes	—
104	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
105	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
106	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
107	#11719917	—	2	4	no	—
108	#11719917	C₂²×C₄	1	8	yes	—
109	C₂×#88322	—	2	4	no	—
110	C₂×#88322	—	2	2	no	—
111	C₂×#88322	C₂²×C₄	1	2	yes	—
112	C₂×#88322	C₂²×C₄	1	2	yes	—
113	C₂×#88322	C₂²×C₄	1	4	yes	—
114	#11719912	—	2	16	no	—
115	C₈×D₅₃	—	2	4	no	—
116	C₈×D₅₃	—	2	4	no	—
117	C₈×D₅₃	—	2	16	no	—
118	C₈×D₅₃	C₄²	1	8	yes	—
119	C₈×D₅₃	C₄²	1	8	yes	—
120	C₂³×C₅₃	C₂²×Q₈	1	1	yes	characteristic
121	C₂²×D₅₃	—	2	4	no	—
122	C₂²×D₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂²×D₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₂²×D₅₃	C₄×Q₈	1	4	yes	—
125	C₂×C₄×C₅₃	—	2	4	no	—
126	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
127	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
128	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
129	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
130	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
132	C₂×Dic₅₃	—	2	4	no	—
133	C₂×Dic₅₃	C₂×C₄⋊C₄	1	2	yes	—
134	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
135	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
136	C₂×Dic₅₃	C₂×C₄⋊C₄	1	2	yes	—
137	C₂×Dic₅₃	C₄×Q₈	1	4	yes	—
138	C₄×D₅₃	—	2	4	no	—
139	C₄×D₅₃	—	2	4	no	—
140	C₄×D₅₃	—	2	4	no	—
141	C₄×D₅₃	#108	1	8	yes	direct product factor
142	C₄×D₅₃	C₂×C₄²	1	2	yes	—
143	C₄×D₅₃	C₂×C₄²	1	2	yes	—
144	C₄×D₅₃	#117	1	2	yes	—
145	C₄×D₅₃	#117	1	2	yes	—
146	C₄×D₅₃	C₄×D₄	1	4	yes	—
147	C₄×D₅₃	C₄×D₄	1	4	yes	—
148	C₈×C₅₃	—	2	2	no	—
149	C₈×C₅₃	—	2	8	no	—
150	C₈×C₅₃	C₂×C₄²	1	4	yes	—
151	#88322	—	2	2	no	—
152	#88322	—	2	8	no	—
153	#88322	C₂×C₄²	1	4	yes	—
154	C₂×C₄×#108	—	53	1	no	maximal, Sylow
155	C₂²×C₅₃	—	2	2	no	—
156	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
157	C₂²×C₅₃	C₂×C₄×Q₈	1	2	yes	—
158	C₂×D₅₃	—	2	2	no	—
159	C₂×D₅₃	—	2	8	no	—
160	C₂×D₅₃	C₄×C₄⋊C₄	1	2	yes	—
161	C₂×D₅₃	C₄×C₄⋊C₄	1	2	yes	—
162	C₂×D₅₃	C₂×#108	1	4	yes	—
163	C₄×C₅₃	—	2	2	no	—
164	C₄×C₅₃	C₂×#108	1	4	yes	complemented
165	C₄×C₅₃	C₂²×C₄²	1	1	yes	characteristic
166	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
167	C₄×C₅₃	C₂×C₄×D₄	1	2	yes	—
168	Dic₅₃	—	2	2	no	—
169	Dic₅₃	C₄×C₄⋊C₄	1	2	yes	—
170	Dic₅₃	C₄×C₄⋊C₄	1	2	yes	—
171	Dic₅₃	C₂×#108	1	4	yes	complemented
172	C₂²×#108	—	53	2	no	—
173	C₂×C₄×M₁₆	—	53	2	no	—
174	C₂²×C₄×C₈	—	53	1	no	—
175	C₄×#108	—	53	2	no	—
176	C₄×#108	—	53	2	no	—
177	C₄×#108	—	53	1	no	—
178	C₄×#108	—	53	2	no	—
179	C₄×#108	D₅₃	1	1	yes	characteristic, direct product factor
180	C₂×#341	—	53	2	no	—
181	C₂×C₅₃	—	2	2	no	—
182	C₂×C₅₃	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
183	C₂×C₅₃	C₂²×#108	1	2	yes	—
184	D₅₃	—	2	2	no	—
185	D₅₃	C₄×#108	1	4	yes	direct product factor
186	C₂²×M₁₆	—	53	4	no	—
187	C₂³×C₈	—	53	1	no	—
188	C₂³×C₈	—	53	1	no	—
189	C₂²×C₄²	—	53	1	no	—
190	C₂×#108	—	53	4	no	—
191	C₂×#108	—	53	2	no	—
192	C₂×#108	—	53	16	no	—
193	C₂×#108	C₂×D₅₃	1	2	yes	—
194	C₄×M₁₆	—	53	4	no	—
195	C₄×M₁₆	—	53	2	no	—
196	C₄×M₁₆	—	106	2	no	—
197	C₄×M₁₆	—	106	2	no	—
198	C₄×M₁₆	C₂×D₅₃	1	2	yes	—
199	C₂×#97	—	106	2	no	—
200	C₂×C₄×C₈	—	53	1	no	—
201	C₂×C₄×C₈	—	53	1	no	—
202	C₂×C₄×C₈	—	53	1	no	—
203	C₂×C₄×C₈	—	53	4	no	—
204	C₂×C₄×C₈	—	53	4	no	—
205	C₂×C₄×C₈	—	106	2	no	—
206	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
207	#341	—	53	4	no	—
208	#341	—	53	2	no	—
209	#341	C₂×D₅₃	1	2	yes	—
210	C₅₃	C₂×C₄×#108	1	1	yes	characteristic, complemented, Sylow
211	C₂³×C₄	—	53	1	no	—
212	C₂³×C₄	—	53	2	no	—
213	C₂×M₁₆	—	53	4	no	—
214	C₂×M₁₆	—	53	16	no	—
215	C₂×M₁₆	—	106	4	no	—
216	C₂×M₁₆	—	106	8	no	—
217	C₂×M₁₆	C₂²×D₅₃	1	4	yes	—
218	C₂²×C₈	—	53	1	no	—
219	C₂²×C₈	—	53	2	no	—
220	C₂²×C₈	—	53	2	no	—
221	C₂²×C₈	—	53	8	no	—
222	C₂²×C₈	—	53	1	no	—
223	C₂²×C₈	—	53	2	no	—
224	C₂²×C₈	—	53	2	no	—
225	C₂²×C₈	—	106	4	no	—
226	C₂²×C₈	—	106	4	no	—
227	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
228	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
229	C₂×C₄²	—	53	2	no	—
230	C₂×C₄²	—	53	1	no	—
231	C₂×C₄²	—	53	2	no	—
232	C₂×C₄²	—	53	2	no	—
233	C₂×C₄²	—	53	1	no	—
234	C₂×C₄²	—	53	1	no	—
235	C₂×C₄²	—	53	1	no	—
236	C₂×C₄²	—	53	1	no	—
237	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
238	#108	—	53	16	no	—
239	#108	—	53	8	no	—
240	#108	C₄×D₅₃	1	8	yes	direct product factor
241	#97	—	2	2	no	—
242	#97	—	106	4	no	—
243	#97	—	106	2	no	—
244	C₄×C₈	—	2	2	no	—
245	C₄×C₈	—	53	4	no	—
246	C₄×C₈	—	53	4	no	—
247	C₄×C₈	—	53	4	no	—
248	C₄×C₈	—	106	4	no	—
249	C₄×C₈	—	106	2	no	—
250	C₄×C₈	C₄×D₅₃	1	4	yes	—
251	C₂⁴	—	53	1	no	—
252	C₂²×C₄	—	53	1	no	—
253	C₂²×C₄	—	53	2	no	—
254	C₂²×C₄	—	53	1	no	—
255	C₂²×C₄	—	53	1	no	—
256	C₂²×C₄	—	53	4	no	—
257	C₂²×C₄	—	53	1	no	—
258	C₂²×C₄	—	53	1	no	—
259	C₂²×C₄	—	53	4	no	—
260	C₂²×C₄	—	53	2	no	—
261	C₂²×C₄	—	53	2	no	—
262	C₂²×C₄	—	53	2	no	—
263	C₂²×C₄	—	53	2	no	—
264	C₂²×C₄	—	106	8	no	—
265	C₂²×C₄	Q₈×D₅₃	1	2	yes	—
266	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
267	M₁₆	—	2	4	no	—
268	M₁₆	—	53	8	no	—
269	M₁₆	—	106	16	no	—
270	M₁₆	—	106	4	no	—
271	M₁₆	C₂×C₄×D₅₃	1	8	yes	—
272	C₂×C₈	—	2	4	no	—
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	—	53	2	no	—
275	C₂×C₈	—	53	4	no	—
276	C₂×C₈	—	53	8	no	—
277	C₂×C₈	—	53	8	no	—
278	C₂×C₈	—	53	2	no	—
279	C₂×C₈	—	106	4	no	—
280	C₂×C₈	—	106	4	no	—
281	C₂×C₈	—	106	16	no	—
282	C₂×C₈	—	106	4	no	—
283	C₂×C₈	—	106	2	no	—
284	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
285	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
286	C₂×C₈	C₂×C₄×D₅₃	1	4	yes	—
287	C₄²	—	53	2	no	—
288	C₄²	—	53	2	no	—
289	C₄²	—	53	2	no	—
290	C₄²	—	53	2	no	—
291	C₄²	—	53	2	no	—
292	C₄²	—	53	1	no	—
293	C₄²	—	53	1	no	—
294	C₄²	C₂×C₄×D₅₃	1	2	yes	—
295	C₄²	D₄×D₅₃	1	1	yes	characteristic
296	C₄²	D₄×D₅₃	1	1	yes	characteristic
297	C₂³	—	53	1	no	—
298	C₂³	—	53	1	no	—
299	C₂³	—	53	4	no	—
300	C₂³	—	106	4	no	—
301	C₂³	C₂×Q₈×D₅₃	1	1	yes	characteristic
302	C₂×C₄	—	2	4	no	—
303	C₂×C₄	—	53	1	no	—
304	C₂×C₄	—	53	1	no	—
305	C₂×C₄	—	53	2	no	—
306	C₂×C₄	—	53	4	no	—
307	C₂×C₄	—	53	2	no	—
308	C₂×C₄	—	53	2	no	—
309	C₂×C₄	—	53	2	no	—
310	C₂×C₄	—	53	2	no	—
311	C₂×C₄	—	53	4	no	—
312	C₂×C₄	—	53	4	no	—
313	C₂×C₄	—	53	8	no	—
314	C₂×C₄	—	53	2	no	—
315	C₂×C₄	—	106	4	no	—
316	C₂×C₄	—	106	4	no	—
317	C₂×C₄	—	106	4	no	—
318	C₂×C₄	—	106	4	no	—
319	C₂×C₄	C₄⋊C₄×D₅₃	1	2	yes	—
320	C₂×C₄	C₄⋊C₄×D₅₃	1	2	yes	—
321	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
322	C₂×C₄	C₂²×C₄×D₅₃	1	2	yes	—
323	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
324	C₂×C₄	#41×D₅₃	1	2	yes	—
325	C₈	—	2	2	no	—
326	C₈	—	2	8	no	—
327	C₈	—	53	4	no	—
328	C₈	—	106	2	no	—
329	C₈	—	106	8	no	—
330	C₈	C₄²×D₅₃	1	4	yes	—
331	C₂²	—	2	2	no	—
332	C₂²	—	53	2	no	—
333	C₂²	—	53	4	no	—
334	C₂²	—	53	2	no	—
335	C₂²	—	106	8	no	—
336	C₂²	—	106	2	no	—
337	C₂²	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
338	C₂²	C₄×Q₈×D₅₃	1	2	yes	—
339	C₄	—	2	2	no	—
340	C₄	—	53	2	no	—
341	C₄	—	53	2	no	—
342	C₄	—	53	4	no	—
343	C₄	—	106	2	no	—
344	C₄	#108×D₅₃	1	4	yes	direct product factor
345	C₄	C₂×C₄²×D₅₃	1	1	yes	characteristic
346	C₄	#117×D₅₃	1	1	yes	characteristic
347	C₄	C₄×D₄×D₅₃	1	2	yes	—
348	C₂	—	2	2	no	—
349	C₂	—	53	4	no	—
350	C₂	—	106	2	no	—
351	C₂	C₄×C₄⋊C₄×D₅₃	1	1	yes	characteristic
352	C₂	C₂×#108×D₅₃	1	2	yes	—

	subgroup	G/H
center	C₄²	D₄×D₅₃
commutator	C₄×C₅₃	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₄×#108×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂³×D₅₃	C₂×Q₈
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₄×#108	D₅₃
hypocenter	C₅₃	C₂×C₄×#108
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₄×#108

Complements

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

N	H ≅ G/N	type
4. #108×Dic₅₃	C₂	semidirect
5. C₄×#13133152	C₂	semidirect
6. C₄×#13133153	C₂	semidirect
7. #20274616	C₂	semidirect
9. C₄×#108×C₅₃	C₂	semidirect
16. #108×D₅₃	C₄	direct (N × H)
57. #13133153	C₂×C₄	semidirect
58. #13133152	C₂×C₄	semidirect
70. #108×C₅₃	C₂×C₄	semidirect
141. C₄×D₅₃	#108	direct (N × H)
164. C₄×C₅₃	C₂×#108	semidirect
171. Dic₅₃	C₂×#108	semidirect
179. C₄×#108	D₅₃	direct (N × H)
185. D₅₃	C₄×#108	direct (N × H)
210. C₅₃	C₂×C₄×#108	semidirect
240. #108	C₄×D₅₃	direct (N × H)
344. C₄	#108×D₅₃	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×#108×D₅₃	2
C₄×M₁₆×D₅₃	2
C₂×C₄×C₈×D₅₃	2
#108×Dic₅₃	2
C₄×#13133152	2
C₄×#13133153	2
#20274616	2
#341×D₅₃	2
C₄×#108×C₅₃	2
C₂×C₄×#108	53

Subgroups — C4×#108×D53 #27382186

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#108×D₅₃ #27382186