Subgroups — #27353310

Order 13568.

572 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283832	C₂	1	2	yes	maximal
2	#20273504	C₂	1	2	yes	maximal, complemented
3	#20273500	C₂	1	2	yes	maximal, complemented
4	#20273495	C₂	1	2	yes	maximal, complemented
5	#20273490	C₂	1	2	yes	maximal, complemented
6	#20273482	C₂	1	2	yes	maximal, complemented
7	#20273478	C₂	1	1	yes	maximal, characteristic, complemented
8	#20273474	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#2922	C₂	1	1	yes	maximal, characteristic
10	#16810754	—	2	2	no	—
11	#16810701	—	2	2	no	—
12	#16810695	—	2	2	no	—
13	#16810695	—	2	2	no	—
14	#16810680	C₂²	1	2	yes	—
15	#16810680	C₂²	1	2	yes	—
16	#16810676	C₂²	1	2	yes	—
17	#16810647	C₂²	1	2	yes	—
18	#16810647	C₂²	1	2	yes	—
19	C₂×C₄⋊#88322	C₂²	1	2	yes	—
20	#16810376	C₂²	1	2	yes	—
21	#16810376	C₂²	1	1	yes	characteristic
22	#16810376	C₂²	1	1	yes	characteristic
23	#16810376	C₂²	1	1	yes	characteristic
24	#16810376	C₂²	1	1	yes	characteristic
25	#16810376	C₂²	1	2	yes	—
26	#16810376	C₂²	1	2	yes	—
27	#16810374	C₂²	1	2	yes	—
28	#16810374	C₂²	1	2	yes	—
29	#16810374	C₂²	1	2	yes	—
30	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
31	C₅₃×#433	C₄	1	4	yes	—
32	C₅₃×#431	C₂²	1	2	yes	—
33	C₅₃×#431	C₂²	1	2	yes	—
34	C₅₃×#424	C₄	1	2	yes	—
35	C₅₃×#408	C₄	1	2	yes	—
36	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
37	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
38	C₂×#11719917	—	2	2	no	—
39	C₂×#11719917	—	2	2	no	—
40	C₂²×#88322	—	2	2	no	—
41	C₄⋊#88322	—	2	2	no	—
42	C₄⋊#88322	—	2	2	no	—
43	C₄⋊#88322	—	2	2	no	—
44	C₄⋊#88322	C₂³	1	2	yes	—
45	C₄⋊#88322	C₂³	1	2	yes	—
46	C₄⋊#88322	C₂³	1	2	yes	—
47	C₄⋊#88322	C₂³	1	2	yes	—
48	C₂²⋊#88322	—	2	2	no	—
49	C₂²⋊#88322	—	2	2	no	—
50	C₂²⋊#88322	—	2	2	no	—
51	#13133128	—	2	2	no	—
52	#13133128	—	2	2	no	—
53	C₄×#88322	—	2	2	no	—
54	C₂×#41×C₅₃	C₂×C₄	1	1	yes	characteristic
55	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
56	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
57	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
58	C₄×Q₈×C₅₃	Q₈	1	1	yes	characteristic
59	C₄×D₄×C₅₃	D₄	1	1	yes	characteristic
60	C₄×D₄×C₅₃	D₄	1	2	yes	—
61	C₄×D₄×C₅₃	Q₈	1	1	yes	characteristic
62	C₄×D₄×C₅₃	Q₈	1	2	yes	—
63	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
64	#117×C₅₃	C₂×C₄	1	2	yes	—
65	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
66	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
67	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
68	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
69	C₂²⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
70	C₂²⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
71	#97×C₅₃	C₂×C₄	1	2	yes	—
72	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
73	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₅₃	C₂³	1	2	yes	—
75	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
76	#41×C₅₃	—	4	2	no	—
77	C₂×Q₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
78	C₂×D₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
79	C₂×D₄×C₅₃	C₄⋊C₄	1	2	yes	—
80	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
85	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
86	M₁₆×C₅₃	—	4	2	no	—
87	C₂×C₈×C₅₃	—	4	2	no	—
88	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
89	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
90	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
91	C₄⋊C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
92	C₄⋊C₄×C₅₃	C₄⋊C₄	1	2	yes	—
93	C₄⋊C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
94	C₄⋊C₄×C₅₃	C₄⋊C₄	1	2	yes	—
95	C₂²⋊C₄×C₅₃	C₄⋊C₄	1	2	yes	—
96	C₂²⋊C₄×C₅₃	C₄⋊C₄	1	2	yes	—
97	C₂²⋊C₄×C₅₃	C₄⋊C₄	1	2	yes	—
98	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
99	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
100	C₄²×C₅₃	C₂×D₄	1	2	yes	—
101	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
103	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
104	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
105	#11719917	—	8	2	no	—
106	#11719917	—	8	2	no	—
107	C₂×#88322	—	2	2	no	—
108	C₂×#88322	—	2	2	no	—
109	C₂×#88322	—	2	2	no	—
110	C₂×#88322	—	2	2	no	—
111	C₂×#88322	—	8	2	no	—
112	C₂³×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
113	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
114	Q₈×C₅₃	—	4	1	no	—
115	D₄×C₅₃	—	4	2	no	—
116	D₄×C₅₃	—	4	1	no	—
117	C₂×C₄×C₅₃	—	2	4	no	—
118	C₂×C₄×C₅₃	—	2	2	no	—
119	C₂×C₄×C₅₃	—	2	2	no	—
120	C₂×C₄×C₅₃	—	2	1	no	—
121	C₂×C₄×C₅₃	—	2	2	no	—
122	C₂×C₄×C₅₃	—	2	1	no	—
123	C₂×C₄×C₅₃	—	4	2	no	—
124	C₂×C₄×C₅₃	—	4	1	no	—
125	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
126	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
128	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
131	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
132	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
133	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
134	C₈×C₅₃	—	2	2	no	—
135	C₈×C₅₃	—	2	2	no	—
136	C₈×C₅₃	—	2	4	no	—
137	#88322	—	4	2	no	—
138	#88322	—	4	2	no	—
139	#88322	—	4	2	no	—
140	#88322	—	4	2	no	—
141	#17825	—	53	1	no	maximal, Sylow
142	C₂²×C₅₃	—	2	2	no	—
143	C₂²×C₅₃	—	2	1	no	—
144	C₂²×C₅₃	—	2	2	no	—
145	C₂²×C₅₃	—	2	1	no	—
146	C₂²×C₅₃	—	2	2	no	—
147	C₂²×C₅₃	—	2	1	no	—
148	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
149	C₄×C₅₃	—	2	2	no	—
150	C₄×C₅₃	—	2	1	no	—
151	C₄×C₅₃	—	2	2	no	—
152	C₄×C₅₃	—	2	1	no	—
153	C₄×C₅₃	—	2	1	no	—
154	C₄×C₅₃	—	2	1	no	—
155	C₄×C₅₃	—	2	2	no	—
156	C₄×C₅₃	#568	1	2	yes	—
157	C₄×C₅₃	#576	1	1	yes	characteristic
158	C₄×C₅₃	#578	1	1	yes	characteristic
159	#2989	—	53	2	no	—
160	#2922	D₅₃	1	1	yes	characteristic
161	#1574	—	53	2	no	—
162	#1573	—	53	2	no	—
163	#1572	—	53	2	no	—
164	#1571	—	53	2	no	—
165	#1569	—	53	2	no	—
166	#1568	—	53	1	no	—
167	#1567	—	53	1	no	—
168	C₂×C₅₃	—	4	2	no	—
169	C₂×C₅₃	—	4	1	no	—
170	C₂×C₅₃	#2905	1	1	yes	characteristic
171	C₂×C₅₃	#2913	1	1	yes	characteristic
172	C₂×C₅₃	#2916	1	1	yes	characteristic
173	C₄×#41	C₂×D₅₃	1	1	yes	characteristic
174	#446	—	106	2	no	—
175	#436	—	106	2	no	—
176	#435	—	106	2	no	—
177	#435	—	106	2	no	—
178	#433	Dic₅₃	1	4	yes	—
179	#432	—	53	2	no	—
180	#432	—	53	2	no	—
181	#431	—	53	2	no	—
182	#431	C₂×D₅₃	1	2	yes	—
183	#431	C₂×D₅₃	1	2	yes	—
184	#424	Dic₅₃	1	2	yes	—
185	#423	—	53	2	no	—
186	#423	—	53	2	no	—
187	C₂×C₄⋊C₈	—	53	2	no	—
188	#408	Dic₅₃	1	2	yes	—
189	C₄×M₁₆	C₂×D₅₃	1	1	yes	characteristic
190	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
191	#335	—	53	2	no	—
192	#335	—	53	1	no	—
193	#335	—	53	1	no	—
194	#335	—	53	1	no	—
195	#335	—	53	1	no	—
196	#335	—	53	2	no	—
197	#335	—	53	2	no	—
198	#334	—	53	2	no	—
199	#334	—	53	2	no	—
200	#334	—	53	2	no	—
201	C₅₃	#17825	1	1	yes	characteristic, complemented, Sylow
202	C₂×#41	C₂×Dic₅₃	1	1	yes	characteristic
203	C₂×M₁₆	—	106	2	no	—
204	C₂×M₁₆	—	106	2	no	—
205	C₂×M₁₆	C₂×Dic₅₃	1	2	yes	—
206	C₂²×C₈	—	106	2	no	—
207	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
208	C₄×Q₈	Dic₁₀₆	1	1	yes	characteristic
209	C₄×Q₈	D₂₁₂	1	1	yes	characteristic
210	C₄×D₄	Dic₁₀₆	1	1	yes	characteristic
211	C₄×D₄	Dic₁₀₆	1	2	yes	—
212	C₄×D₄	D₂₁₂	1	1	yes	characteristic
213	C₄×D₄	D₂₁₂	1	2	yes	—
214	#117	C₂×Dic₅₃	1	1	yes	characteristic
215	#117	C₂×Dic₅₃	1	2	yes	—
216	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
217	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
218	C₄⋊C₈	—	53	2	no	—
219	C₄⋊C₈	—	53	2	no	—
220	C₄⋊C₈	—	53	2	no	—
221	C₄⋊C₈	—	53	2	no	—
222	C₄⋊C₈	—	106	2	no	—
223	C₄⋊C₈	—	106	2	no	—
224	C₄⋊C₈	—	106	2	no	—
225	C₄⋊C₈	C₂×Dic₅₃	1	4	yes	—
226	C₄⋊C₈	C₂×Dic₅₃	1	4	yes	—
227	C₂²⋊C₈	—	106	2	no	—
228	C₂²⋊C₈	—	106	2	no	—
229	C₂²⋊C₈	—	106	2	no	—
230	C₂²⋊C₈	C₂×Dic₅₃	1	4	yes	—
231	C₂²⋊C₈	C₂×Dic₅₃	1	4	yes	—
232	#97	—	106	2	no	—
233	#97	—	106	2	no	—
234	#97	C₂×Dic₅₃	1	2	yes	—
235	C₄×C₈	—	106	2	no	—
236	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
237	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
238	C₄×C₈	C₂²×D₅₃	1	2	yes	—
239	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
240	#41	—	4	2	no	—
241	C₂×Q₈	C₄⋊Dic₅₃	1	1	yes	characteristic
242	C₂×D₄	C₄⋊Dic₅₃	1	1	yes	characteristic
243	C₂×D₄	C₄⋊Dic₅₃	1	2	yes	—
244	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
245	C₂²×C₄	C₂×Dic₁₀₆	1	2	yes	—
246	C₂²×C₄	C₂×D₂₁₂	1	2	yes	—
247	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
248	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
249	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
250	M₁₆	—	4	2	no	—
251	M₁₆	—	424	2	no	—
252	M₁₆	—	424	2	no	—
253	C₂×C₈	—	4	2	no	—
254	C₂×C₈	—	106	2	no	—
255	C₂×C₈	—	106	2	no	—
256	C₂×C₈	—	106	2	no	—
257	C₂×C₈	—	106	2	no	—
258	C₂×C₈	—	424	2	no	—
259	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
260	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
261	C₂×C₈	C₂²×Dic₅₃	1	4	yes	—
262	C₄⋊C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
263	C₄⋊C₄	C₄⋊Dic₅₃	1	2	yes	—
264	C₄⋊C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
265	C₄⋊C₄	C₄⋊Dic₅₃	1	2	yes	—
266	C₂²⋊C₄	C₄⋊Dic₅₃	1	2	yes	—
267	C₂²⋊C₄	C₄⋊Dic₅₃	1	2	yes	—
268	C₂²⋊C₄	C₄⋊Dic₅₃	1	2	yes	—
269	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
270	C₄²	C₂×Dic₁₀₆	1	2	yes	—
271	C₄²	C₂×D₂₁₂	1	2	yes	—
272	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
273	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
274	C₄²	C₂²×Dic₅₃	1	2	yes	—
275	C₄²	C₂³×D₅₃	1	1	yes	characteristic
276	C₂³	C₂×C₄⋊Dic₅₃	1	2	yes	—
277	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
278	Q₈	—	4	1	no	—
279	D₄	—	4	1	no	—
280	D₄	—	4	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	4	1	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
290	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
291	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
292	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
293	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
294	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
295	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
296	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
297	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
298	C₈	—	2	2	no	—
299	C₈	—	2	2	no	—
300	C₈	—	2	4	no	—
301	C₈	—	212	2	no	—
302	C₈	—	212	2	no	—
303	C₈	—	212	2	no	—
304	C₈	—	212	2	no	—
305	C₂²	—	2	1	no	—
306	C₂²	—	2	2	no	—
307	C₂²	—	2	2	no	—
308	C₂²	—	2	1	no	—
309	C₂²	—	2	2	no	—
310	C₂²	—	2	1	no	—
311	C₂²	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
312	C₄	—	2	1	no	—
313	C₄	—	2	2	no	—
314	C₄	—	2	2	no	—
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	2	2	no	—
318	C₄	—	2	1	no	—
319	C₄	#16811390	1	2	yes	—
320	C₄	#16811428	1	1	yes	characteristic
321	C₄	#16811440	1	1	yes	characteristic
322	C₂	—	4	1	no	—
323	C₂	—	4	2	no	—
324	C₂	#20283279	1	1	yes	characteristic
325	C₂	#20283315	1	1	yes	characteristic
326	C₂	#20283326	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#2922	C₂
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂×#41×C₅₃	C₂×C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#2922	D₅₃
hypocenter	C₅₃	#17825
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41 ◁ #2922

Complements

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

N	H ≅ G/N	type
2. #20273504	C₂	semidirect
3. #20273500	C₂	semidirect
4. #20273495	C₂	semidirect
5. #20273490	C₂	semidirect
6. #20273482	C₂	semidirect
7. #20273478	C₂	semidirect
8. #20273474	C₂	semidirect
201. C₅₃	#17825	semidirect

Maximal subgroups

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

subgroup	index
#20283832	2
#20273504	2
#20273500	2
#20273495	2
#20273490	2
#20273482	2
#20273478	2
#20273474	2
C₅₃×#2922	2
#17825	53