Subgroups — #27537815

Order 13568.

502 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283692	C₂	1	1	yes	maximal, characteristic
2	Q₈×#11719908	C₂	1	1	yes	maximal, characteristic, complemented
3	#20276546	C₂	1	1	yes	maximal, characteristic, complemented
4	#20276542	C₂	1	2	yes	maximal, complemented
5	C₁₆×D₂₁₂	C₂	1	1	yes	maximal, characteristic, complemented
6	#20276534	C₂	1	2	yes	maximal, complemented
7	#20276523	C₂	1	2	yes	maximal, complemented
8	#20276512	C₂	1	1	yes	maximal, characteristic, complemented
9	#20276472	C₂	1	2	yes	maximal, complemented
10	C₄⋊C₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#2183	C₂	1	1	yes	maximal, characteristic, complemented
12	#16811160	C₄	1	1	yes	characteristic
13	C₂×#13133156	C₂²	1	2	yes	—
14	C₂×C₁₆×D₅₃	C₂²	1	2	yes	—
15	C₂×C₁₆×D₅₃	C₂²	1	2	yes	—
16	#16810754	C₄	1	1	yes	characteristic
17	#16810700	C₄	1	1	yes	characteristic
18	#16810693	C₄	1	2	yes	—
19	C₈×D₂₁₂	C₂²	1	2	yes	—
20	C₈×D₂₁₂	C₂²	1	1	yes	characteristic
21	#16810683	C₄	1	2	yes	—
22	#16810679	C₄	1	1	yes	characteristic
23	#16810577	C₂²	1	2	yes	—
24	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
25	C₄⋊#11719908	C₂²	1	1	yes	characteristic
26	C₄⋊#11719908	C₂²	1	2	yes	—
27	#16810441	C₂²	1	1	yes	characteristic
28	#16810441	C₂²	1	1	yes	characteristic
29	#16810441	C₂²	1	2	yes	—
30	#16810402	C₂²	1	1	yes	characteristic
31	#16810402	C₂²	1	1	yes	characteristic
32	#16810402	C₂²	1	2	yes	—
33	#16810402	C₂²	1	2	yes	—
34	C₄×#11719908	C₂²	1	1	yes	characteristic
35	C₄×#11719908	C₂²	1	2	yes	—
36	C₈×Q₈×C₅₃	C₂²	1	1	yes	characteristic
37	C₅₃×C₄⋊C₁₆	C₂²	1	2	yes	—
38	C₅₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
39	C₅₃×#346	C₂²	1	2	yes	—
40	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
41	C₂×#11719912	C₂×C₄	1	2	yes	—
42	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
44	C₂×C₈×D₅₃	C₂³	1	2	yes	—
45	#13133212	C₈	1	2	yes	—
46	#13133208	C₈	1	1	yes	characteristic
47	#13133206	C₈	1	1	yes	characteristic
48	#13133202	C₈	1	2	yes	—
49	#13133193	C₈	1	1	yes	characteristic
50	#13133192	C₈	1	1	yes	characteristic
51	C₄×D₂₁₂	C₂×C₄	1	2	yes	—
52	C₄×D₂₁₂	C₂×C₄	1	1	yes	characteristic
53	#13133174	C₂×C₄	1	2	yes	—
54	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
55	#13133156	—	4	2	no	—
56	C₂×#11719908	C₂³	1	2	yes	—
57	C₂×#11719908	C₂³	1	1	yes	characteristic
58	C₂×#11719908	C₂³	1	1	yes	characteristic
59	C₁₆×D₅₃	—	4	2	no	—
60	C₁₆×D₅₃	—	4	2	no	—
61	C₄⋊#88322	C₂×C₄	1	1	yes	characteristic
62	C₄⋊#88322	C₂×C₄	1	2	yes	—
63	#13133129	C₂×C₄	1	2	yes	—
64	#13133129	C₂×C₄	1	2	yes	—
65	#13133129	C₂×C₄	1	2	yes	—
66	#13133128	C₂×C₄	1	2	yes	—
67	#13133127	C₂×C₄	1	2	yes	—
68	C₄×#88322	C₂×C₄	1	1	yes	characteristic
69	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
70	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
71	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
72	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
73	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
74	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
75	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
76	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
77	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₅₃	C₂³	1	2	yes	—
79	C₂×Q₈×C₅₃	C₂×C₈	1	1	yes	characteristic
80	C₂×D₂₁₂	C₂×C₈	1	2	yes	—
81	C₂×D₂₁₂	C₂×C₈	1	1	yes	characteristic
82	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
83	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
84	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
86	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
88	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
89	C₄⋊C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
90	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
91	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
92	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
93	#11719921	C₂×C₈	1	2	yes	—
94	#11719921	C₂×C₈	1	2	yes	—
95	#11719921	C₂×C₈	1	2	yes	—
96	#11719921	C₂×C₈	1	2	yes	—
97	#11719921	C₂×C₈	1	2	yes	—
98	C₄⋊Dic₅₃	C₂×C₈	1	1	yes	characteristic
99	C₄⋊Dic₅₃	C₂×C₈	1	2	yes	—
100	#11719919	C₂×C₈	1	2	yes	—
101	#11719919	C₂×C₈	1	2	yes	—
102	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
103	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
104	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
105	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
106	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
107	C₂×#88322	C₂²×C₄	1	2	yes	—
108	#11719912	—	4	2	no	—
109	C₈×D₅₃	—	2	1	no	—
110	C₈×D₅₃	—	2	2	no	—
111	C₈×D₅₃	—	2	1	no	—
112	C₈×D₅₃	—	2	2	no	—
113	C₈×D₅₃	—	4	1	no	—
114	C₁₆×C₅₃	—	2	1	no	—
115	C₁₆×C₅₃	—	2	2	no	—
116	C₁₆×C₅₃	—	2	1	no	—
117	#11719908	—	2	2	no	—
118	#11719908	—	2	1	no	—
119	#11719908	—	2	1	no	—
120	C₂²×D₅₃	C₂²×C₈	1	2	yes	—
121	C₂²×D₅₃	C₂²×C₈	1	1	yes	characteristic
122	Q₈×C₅₃	—	2	2	no	—
123	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
124	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
125	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
126	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
127	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
128	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
129	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
130	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
131	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
132	D₂₁₂	—	2	4	no	—
133	D₂₁₂	—	2	2	no	—
134	C₄×D₅₃	—	2	2	no	—
135	C₄×D₅₃	—	2	1	no	—
136	C₄×D₅₃	—	2	2	no	—
137	C₄×D₅₃	—	2	1	no	—
138	C₄×D₅₃	—	2	2	no	—
139	C₄×D₅₃	—	2	2	no	—
140	C₈×C₅₃	—	2	1	no	—
141	C₈×C₅₃	—	2	2	no	—
142	C₈×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
143	C₈×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
144	#88322	—	2	1	no	—
145	#88322	—	2	2	no	—
146	#88322	—	2	1	no	—
147	#33508	—	53	1	no	maximal, Sylow
148	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
149	C₂×D₅₃	—	2	1	no	—
150	C₂×D₅₃	—	2	1	no	—
151	C₂×D₅₃	—	2	2	no	—
152	C₂×D₅₃	—	2	2	no	—
153	C₂×D₅₃	—	2	2	no	—
154	C₂×D₅₃	#504	1	2	yes	—
155	C₄×C₅₃	—	2	1	no	—
156	C₄×C₅₃	—	2	2	no	—
157	C₄×C₅₃	—	2	2	no	—
158	C₄×C₅₃	#504	1	2	yes	—
159	C₄×C₅₃	#520	1	1	yes	characteristic
160	C₄×C₅₃	#568	1	1	yes	characteristic
161	Dic₅₃	—	2	1	no	—
162	Dic₅₃	—	2	1	no	—
163	Dic₅₃	—	2	1	no	—
164	Dic₅₃	—	2	2	no	—
165	#2966	—	53	1	no	—
166	#2183	D₅₃	1	1	yes	characteristic, complemented
167	Q₈×C₁₆	—	53	1	no	—
168	#2169	—	53	1	no	—
169	#2168	—	53	2	no	—
170	D₄×C₁₆	—	53	2	no	—
171	D₄×C₁₆	—	53	1	no	—
172	#2164	—	53	2	no	—
173	#2162	—	53	1	no	—
174	#2151	—	53	2	no	—
175	C₂×C₄⋊C₁₆	—	53	1	no	—
176	C₂×C₅₃	#2988	1	1	yes	characteristic
177	C₂×C₅₃	C₂×#504	1	1	yes	characteristic
178	C₂×C₅₃	C₂×#504	1	1	yes	characteristic
179	D₅₃	—	2	2	no	—
180	D₅₃	—	4	2	no	—
181	#529	—	53	1	no	—
182	C₂×M₃₂	—	53	2	no	—
183	C₂²×C₁₆	—	53	2	no	—
184	C₂²×C₁₆	—	53	2	no	—
185	#446	—	53	1	no	—
186	C₈×Q₈	C₂×D₅₃	1	1	yes	characteristic
187	#436	—	53	1	no	—
188	#435	—	53	2	no	—
189	C₈×D₄	—	53	2	no	—
190	C₈×D₄	—	53	1	no	—
191	#433	—	53	2	no	—
192	#432	—	53	1	no	—
193	#405	—	53	2	no	—
194	C₂×C₄×C₈	—	53	1	no	—
195	C₄⋊C₁₆	—	53	2	no	—
196	C₄⋊C₁₆	—	53	1	no	—
197	C₄⋊C₁₆	—	53	1	no	—
198	C₄⋊C₁₆	—	53	1	no	—
199	C₄⋊C₁₆	—	53	2	no	—
200	C₄⋊C₁₆	C₂×D₅₃	1	1	yes	characteristic
201	C₄⋊C₁₆	C₂×D₅₃	1	2	yes	—
202	C₂²⋊C₁₆	—	53	1	no	—
203	C₂²⋊C₁₆	—	53	1	no	—
204	C₂²⋊C₁₆	—	53	2	no	—
205	C₂²⋊C₁₆	—	53	2	no	—
206	#346	C₂×D₅₃	1	2	yes	—
207	C₄×C₁₆	—	53	1	no	—
208	C₄×C₁₆	—	53	2	no	—
209	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
210	C₅₃	#33508	1	1	yes	characteristic, complemented, Sylow
211	C₂×M₁₆	—	53	2	no	—
212	C₂²×C₈	—	53	1	no	—
213	C₂²×C₈	—	53	2	no	—
214	C₂²×C₈	—	53	1	no	—
215	#126	—	53	2	no	—
216	#125	—	53	1	no	—
217	#124	—	53	1	no	—
218	#123	—	53	2	no	—
219	C₂²⋊Q₈	—	53	1	no	—
220	#121	—	53	1	no	—
221	C₄×Q₈	C₄×D₅₃	1	1	yes	characteristic
222	C₄×D₄	—	53	2	no	—
223	C₄×D₄	—	53	1	no	—
224	#117	—	53	2	no	—
225	C₂×C₄²	—	53	1	no	—
226	M₃₂	—	212	2	no	—
227	C₂×C₁₆	—	53	1	no	—
228	C₂×C₁₆	—	53	2	no	—
229	C₂×C₁₆	—	53	1	no	—
230	C₂×C₁₆	—	212	2	no	—
231	C₂×C₁₆	—	212	2	no	—
232	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
233	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
234	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
235	C₄⋊C₈	—	53	1	no	—
236	C₄⋊C₈	—	53	2	no	—
237	C₄⋊C₈	C₄×D₅₃	1	1	yes	characteristic
238	C₄⋊C₈	C₄×D₅₃	1	2	yes	—
239	C₂²⋊C₈	—	53	2	no	—
240	C₂²⋊C₈	—	53	2	no	—
241	C₂²⋊C₈	—	53	2	no	—
242	#97	—	53	2	no	—
243	#97	—	53	2	no	—
244	C₄×C₈	—	53	1	no	—
245	C₄×C₈	—	53	1	no	—
246	C₄×C₈	—	53	1	no	—
247	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
248	C₄×C₈	C₂²×D₅₃	1	2	yes	—
249	C₂×Q₈	C₈×D₅₃	1	1	yes	characteristic
250	C₂×D₄	—	53	2	no	—
251	C₂×D₄	—	53	1	no	—
252	C₂²×C₄	—	53	1	no	—
253	C₂²×C₄	—	53	1	no	—
254	C₂²×C₄	—	53	1	no	—
255	C₂²×C₄	—	53	2	no	—
256	M₁₆	—	212	2	no	—
257	C₂×C₈	—	53	1	no	—
258	C₂×C₈	—	53	1	no	—
259	C₂×C₈	—	53	2	no	—
260	C₂×C₈	—	106	1	no	—
261	C₂×C₈	—	106	2	no	—
262	C₂×C₈	—	106	1	no	—
263	C₂×C₈	—	106	2	no	—
264	C₂×C₈	—	212	1	no	—
265	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
266	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
267	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
268	C₄⋊C₄	—	53	2	no	—
269	C₄⋊C₄	—	53	2	no	—
270	C₄⋊C₄	—	53	1	no	—
271	C₄⋊C₄	—	53	2	no	—
272	C₄⋊C₄	C₈×D₅₃	1	1	yes	characteristic
273	C₄⋊C₄	C₈×D₅₃	1	2	yes	—
274	C₂²⋊C₄	—	53	2	no	—
275	C₂²⋊C₄	—	53	2	no	—
276	C₂²⋊C₄	—	53	2	no	—
277	C₂²⋊C₄	—	53	2	no	—
278	C₂²⋊C₄	—	53	2	no	—
279	C₄²	—	53	1	no	—
280	C₄²	—	53	1	no	—
281	C₄²	—	53	1	no	—
282	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
283	C₄²	C₂×C₄×D₅₃	1	2	yes	—
284	C₁₆	—	2	1	no	—
285	C₁₆	—	2	2	no	—
286	C₁₆	—	2	1	no	—
287	C₁₆	—	106	2	no	—
288	C₁₆	—	106	1	no	—
289	C₁₆	—	106	1	no	—
290	C₂³	—	53	1	no	—
291	C₂³	—	53	2	no	—
292	Q₈	—	2	2	no	—
293	D₄	—	106	4	no	—
294	D₄	—	106	2	no	—
295	C₂×C₄	—	53	1	no	—
296	C₂×C₄	—	53	1	no	—
297	C₂×C₄	—	53	1	no	—
298	C₂×C₄	—	53	2	no	—
299	C₂×C₄	—	106	2	no	—
300	C₂×C₄	—	106	2	no	—
301	C₂×C₄	—	106	1	no	—
302	C₂×C₄	—	106	2	no	—
303	C₂×C₄	—	106	1	no	—
304	C₂×C₄	—	106	2	no	—
305	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
306	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
307	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
308	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
309	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
310	C₈	—	2	1	no	—
311	C₈	—	2	2	no	—
312	C₈	—	106	1	no	—
313	C₈	—	106	2	no	—
314	C₈	—	106	1	no	—
315	C₈	#13133263	1	1	yes	characteristic
316	C₈	#13133265	1	1	yes	characteristic
317	C₂²	—	53	2	no	—
318	C₂²	—	106	1	no	—
319	C₂²	—	106	1	no	—
320	C₂²	—	106	2	no	—
321	C₂²	—	106	2	no	—
322	C₂²	—	106	2	no	—
323	C₂²	C₂²×C₈×D₅₃	1	1	yes	characteristic
324	C₄	—	2	1	no	—
325	C₄	—	2	2	no	—
326	C₄	—	2	2	no	—
327	C₄	—	106	1	no	—
328	C₄	—	106	1	no	—
329	C₄	—	106	1	no	—
330	C₄	—	106	2	no	—
331	C₄	#16811041	1	2	yes	—
332	C₄	#16811105	1	1	yes	characteristic
333	C₄	#16811388	1	1	yes	characteristic
334	C₂	—	106	2	no	—
335	C₂	—	212	2	no	—
336	C₂	#20283825	1	1	yes	characteristic
337	C₂	#504×D₅₃	1	1	yes	characteristic
338	C₂	C₂×#16811041	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	C₅₃×#2183	C₂
prime radical	#13133192	C₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#2183	D₅₃
hypocenter	C₅₃	#33508
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ #2183

Complements

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

N	H ≅ G/N	type
2. Q₈×#11719908	C₂	semidirect
3. #20276546	C₂	semidirect
4. #20276542	C₂	semidirect
5. C₁₆×D₂₁₂	C₂	semidirect
6. #20276534	C₂	semidirect
7. #20276523	C₂	semidirect
8. #20276512	C₂	semidirect
9. #20276472	C₂	semidirect
10. C₄⋊C₁₆×D₅₃	C₂	semidirect
11. C₅₃×#2183	C₂	semidirect
166. #2183	D₅₃	semidirect
210. C₅₃	#33508	semidirect

Maximal subgroups

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

subgroup	index
#20283692	2
Q₈×#11719908	2
#20276546	2
#20276542	2
C₁₆×D₂₁₂	2
#20276534	2
#20276523	2
#20276512	2
#20276472	2
C₄⋊C₁₆×D₅₃	2
C₅₃×#2183	2
#33508	53