Subgroups — #27537600

Order 13568.

498 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	#20283644	C₂	1	1	yes	maximal, characteristic
2	#20276621	C₂	1	2	yes	maximal, complemented
3	#20276620	C₂	1	1	yes	maximal, characteristic, complemented
4	#20276620	C₂	1	2	yes	maximal, complemented
5	Q₈×#11719908	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₆×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic, complemented
7	#20276472	C₂	1	2	yes	maximal
8	C₄⋊C₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic
9	#20276313	C₂	1	2	yes	maximal
10	C₄×#13133156	C₂	1	1	yes	maximal, characteristic
11	C₅₃×#2183	C₂	1	1	yes	maximal, characteristic, complemented
12	#16811177	C₄	1	1	yes	characteristic
13	C₂×#13133156	C₂²	1	1	yes	characteristic
14	C₂×#13133156	C₂²	1	1	yes	characteristic
15	C₂×C₁₆×D₅₃	C₂²	1	2	yes	—
16	#16810754	C₄	1	1	yes	characteristic
17	#16810752	C₄	1	2	yes	—
18	#16810750	C₄	1	1	yes	characteristic
19	C₈×Dic₁₀₆	C₂²	1	1	yes	characteristic
20	C₈×Dic₁₀₆	C₂²	1	2	yes	—
21	#16810648	C₄	1	2	yes	—
22	#16810645	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	2	yes	—
28	#16810441	C₂²	1	2	yes	—
29	#16810441	C₂²	1	2	yes	—
30	#16810398	C₂²	1	1	yes	characteristic
31	#16810398	C₂²	1	1	yes	characteristic
32	#16810398	C₂²	1	2	yes	—
33	C₄×#11719908	C₂²	1	1	yes	characteristic
34	C₄×#11719908	C₂²	1	2	yes	—
35	C₁₆×Dic₅₃	C₂²	1	2	yes	—
36	C₈×Q₈×C₅₃	C₂²	1	1	yes	characteristic
37	C₅₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
38	C₅₃×C₄⋊C₁₆	C₂²	1	2	yes	—
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	#13133216	C₈	1	1	yes	characteristic
45	#13133215	C₈	1	1	yes	characteristic
46	#13133207	C₈	1	2	yes	—
47	#13133197	C₈	1	1	yes	characteristic
48	#13133195	C₈	1	2	yes	—
49	#13133193	C₈	1	1	yes	characteristic
50	C₄×Dic₁₀₆	C₂×C₄	1	2	yes	—
51	C₄×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
52	#13133174	C₂×C₄	1	2	yes	—
53	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
54	#13133156	—	2	2	no	—
55	#13133156	Q₈	1	4	yes	complemented
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	—	2	4	no	—
60	C₄⋊#88322	C₂×C₄	1	1	yes	characteristic
61	C₄⋊#88322	C₂×C₄	1	2	yes	—
62	#13133146	C₂×C₄	1	2	yes	—
63	#13133146	C₂×C₄	1	2	yes	—
64	#13133146	C₂×C₄	1	2	yes	—
65	#13133128	C₂×C₄	1	2	yes	—
66	#13133127	C₂×C₄	1	2	yes	—
67	C₄×#88322	C₂×C₄	1	1	yes	characteristic
68	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
69	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
70	C₈×Dic₅₃	C₂³	1	2	yes	—
71	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
72	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
73	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
74	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
75	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
76	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
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×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
81	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
82	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
83	C₂×Dic₁₀₆	C₂×C₈	1	1	yes	characteristic
84	C₂×Dic₁₀₆	C₂×C₈	1	2	yes	—
85	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
87	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
88	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
89	C₄⋊C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
90	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
91	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
92	#11719921	C₂×C₈	1	2	yes	—
93	#11719921	C₂×C₈	1	2	yes	—
94	C₄⋊Dic₅₃	C₂×C₈	1	1	yes	characteristic
95	C₄⋊Dic₅₃	C₂×C₈	1	2	yes	—
96	#11719919	C₂×C₈	1	2	yes	—
97	#11719919	C₂×C₈	1	2	yes	—
98	#11719919	C₂×C₈	1	2	yes	—
99	#11719919	C₂×C₈	1	2	yes	—
100	#11719919	C₂×C₈	1	2	yes	—
101	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
102	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
103	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
104	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
105	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
106	C₂×#88322	C₂²×C₄	1	2	yes	—
107	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
108	#11719912	—	2	4	no	—
109	C₈×D₅₃	—	2	2	no	—
110	C₈×D₅₃	C₂×Q₈	1	2	yes	—
111	C₈×D₅₃	#41	1	2	yes	—
112	C₁₆×C₅₃	—	2	2	no	—
113	C₁₆×C₅₃	—	2	1	no	—
114	C₁₆×C₅₃	C₂×Q₈	1	2	yes	—
115	#11719908	—	2	2	no	—
116	#11719908	—	2	1	no	—
117	#11719908	C₂×Q₈	1	2	yes	—
118	C₂²×D₅₃	C₂²×C₈	1	1	yes	characteristic
119	Q₈×C₅₃	—	2	2	no	—
120	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
121	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
123	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
124	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
125	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
126	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
127	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
128	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
129	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
130	C₄×D₅₃	—	2	2	no	—
131	C₄×D₅₃	—	4	1	no	—
132	C₄×D₅₃	C₄×Q₈	1	2	yes	—
133	C₄×D₅₃	#131	1	2	yes	—
134	Dic₁₀₆	—	2	2	no	—
135	Dic₁₀₆	—	4	2	no	—
136	C₈×C₅₃	—	2	1	no	—
137	C₈×C₅₃	—	2	2	no	—
138	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
139	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
140	#88322	—	2	1	no	—
141	#88322	—	2	2	no	—
142	#88322	C₄×Q₈	1	2	yes	—
143	#33476	—	53	1	no	maximal, Sylow
144	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
145	C₂×D₅₃	—	2	1	no	—
146	C₂×D₅₃	—	2	1	no	—
147	C₂×D₅₃	C₈×Q₈	1	2	yes	—
148	C₄×C₅₃	—	2	2	no	—
149	C₄×C₅₃	—	2	2	no	—
150	C₄×C₅₃	—	2	1	no	—
151	C₄×C₅₃	#504	1	2	yes	—
152	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
153	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
154	Dic₅₃	—	2	2	no	—
155	Dic₅₃	—	2	1	no	—
156	Dic₅₃	—	2	1	no	—
157	Dic₅₃	—	2	2	no	—
158	Dic₅₃	C₈×Q₈	1	2	yes	—
159	#2960	—	53	1	no	—
160	#2183	—	53	1	no	—
161	#2183	—	53	2	no	—
162	#2183	—	53	2	no	—
163	#2183	D₅₃	1	1	yes	characteristic, complemented
164	Q₈×C₁₆	—	53	1	no	—
165	Q₈×C₁₆	—	53	1	no	—
166	#2151	—	53	2	no	—
167	C₂×C₄⋊C₁₆	—	53	1	no	—
168	#2108	—	53	2	no	—
169	C₄×M₃₂	—	53	1	no	—
170	C₂×C₅₃	C₂×C₈×Q₈	1	1	yes	characteristic
171	C₂×C₅₃	C₂×#504	1	1	yes	characteristic
172	C₂×C₅₃	#3407	1	1	yes	characteristic
173	D₅₃	—	2	2	no	—
174	#533	—	53	1	no	—
175	C₂×M₃₂	—	53	1	no	—
176	C₂×M₃₂	—	53	1	no	—
177	C₂²×C₁₆	—	53	2	no	—
178	#446	—	53	1	no	—
179	#446	—	53	2	no	—
180	#446	—	53	1	no	—
181	C₈×Q₈	—	53	1	no	—
182	C₈×Q₈	—	53	2	no	—
183	C₈×Q₈	C₂×D₅₃	1	1	yes	characteristic
184	#424	—	53	2	no	—
185	#423	—	53	1	no	—
186	#405	—	53	2	no	—
187	C₂×C₄×C₈	—	53	1	no	—
188	C₄⋊C₁₆	—	53	2	no	—
189	C₄⋊C₁₆	—	53	2	no	—
190	C₄⋊C₁₆	—	53	2	no	—
191	C₄⋊C₁₆	—	53	1	no	—
192	C₄⋊C₁₆	—	53	2	no	—
193	C₄⋊C₁₆	C₂×D₅₃	1	1	yes	characteristic
194	C₄⋊C₁₆	C₂×D₅₃	1	2	yes	—
195	#346	—	53	1	no	—
196	#346	—	53	1	no	—
197	#346	—	53	2	no	—
198	#346	C₂×D₅₃	1	2	yes	—
199	C₄×C₁₆	—	53	2	no	—
200	C₄×C₁₆	—	53	1	no	—
201	C₄×C₁₆	—	53	2	no	—
202	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
203	C₅₃	#33476	1	1	yes	characteristic, complemented, Sylow
204	C₂×M₁₆	—	53	2	no	—
205	C₂²×C₈	—	53	1	no	—
206	C₂²×C₈	—	53	1	no	—
207	C₄⋊Q₈	—	53	1	no	—
208	C₄⋊Q₈	—	53	1	no	—
209	#125	—	53	2	no	—
210	C₂²⋊Q₈	—	53	1	no	—
211	C₂²⋊Q₈	—	53	2	no	—
212	C₂²⋊Q₈	—	53	1	no	—
213	C₄×Q₈	—	53	2	no	—
214	C₄×Q₈	—	53	1	no	—
215	C₄×Q₈	C₄×D₅₃	1	1	yes	characteristic
216	#117	—	53	2	no	—
217	C₂×C₄²	—	53	1	no	—
218	M₃₂	—	53	4	no	—
219	M₃₂	—	106	2	no	—
220	C₂×C₁₆	—	53	1	no	—
221	C₂×C₁₆	—	53	2	no	—
222	C₂×C₁₆	—	53	1	no	—
223	C₂×C₁₆	—	106	4	no	—
224	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
225	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
226	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
227	C₄⋊C₈	—	53	2	no	—
228	C₄⋊C₈	—	53	2	no	—
229	C₄⋊C₈	—	53	2	no	—
230	C₄⋊C₈	—	53	1	no	—
231	C₄⋊C₈	—	53	2	no	—
232	C₄⋊C₈	C₄×D₅₃	1	1	yes	characteristic
233	C₄⋊C₈	C₄×D₅₃	1	2	yes	—
234	#97	—	53	2	no	—
235	#97	—	53	2	no	—
236	C₄×C₈	—	53	1	no	—
237	C₄×C₈	—	53	2	no	—
238	C₄×C₈	—	53	1	no	—
239	C₄×C₈	—	53	1	no	—
240	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
241	C₄×C₈	C₂²×D₅₃	1	2	yes	—
242	C₂×Q₈	—	53	1	no	—
243	C₂×Q₈	—	53	2	no	—
244	C₂×Q₈	C₈×D₅₃	1	1	yes	characteristic
245	C₂²×C₄	—	53	1	no	—
246	C₂²×C₄	—	53	1	no	—
247	C₂²×C₄	—	53	1	no	—
248	M₁₆	—	106	4	no	—
249	C₂×C₈	—	53	1	no	—
250	C₂×C₈	—	53	2	no	—
251	C₂×C₈	—	53	2	no	—
252	C₂×C₈	—	53	1	no	—
253	C₂×C₈	—	53	2	no	—
254	C₂×C₈	—	106	2	no	—
255	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
256	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
257	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
258	C₄⋊C₄	—	53	2	no	—
259	C₄⋊C₄	—	53	2	no	—
260	C₄⋊C₄	—	53	2	no	—
261	C₄⋊C₄	—	53	2	no	—
262	C₄⋊C₄	—	53	2	no	—
263	C₄⋊C₄	—	53	2	no	—
264	C₄⋊C₄	—	53	1	no	—
265	C₄⋊C₄	C₈×D₅₃	1	2	yes	—
266	C₄⋊C₄	C₈×D₅₃	1	1	yes	characteristic
267	C₂²⋊C₄	—	53	2	no	—
268	C₂²⋊C₄	—	53	2	no	—
269	C₄²	—	53	1	no	—
270	C₄²	—	53	1	no	—
271	C₄²	—	53	1	no	—
272	C₄²	—	53	2	no	—
273	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
274	C₄²	C₂×C₄×D₅₃	1	2	yes	—
275	C₁₆	—	2	2	no	—
276	C₁₆	—	2	1	no	—
277	C₁₆	—	53	2	no	—
278	C₁₆	—	106	2	no	—
279	C₁₆	—	106	1	no	—
280	C₁₆	Q₈×D₅₃	1	2	yes	—
281	C₂³	—	53	1	no	—
282	Q₈	—	2	2	no	—
283	Q₈	—	106	2	no	—
284	Q₈	—	212	2	no	—
285	C₂×C₄	—	53	1	no	—
286	C₂×C₄	—	53	2	no	—
287	C₂×C₄	—	53	2	no	—
288	C₂×C₄	—	53	1	no	—
289	C₂×C₄	—	53	1	no	—
290	C₂×C₄	—	53	2	no	—
291	C₂×C₄	—	53	2	no	—
292	C₂×C₄	—	106	2	no	—
293	C₂×C₄	—	212	1	no	—
294	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
295	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
296	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
297	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
298	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
299	C₈	—	2	1	no	—
300	C₈	—	2	2	no	—
301	C₈	—	53	2	no	—
302	C₈	—	106	1	no	—
303	C₈	—	106	2	no	—
304	C₈	C₂×Q₈×D₅₃	1	1	yes	characteristic
305	C₈	#41×D₅₃	1	1	yes	characteristic
306	C₂²	—	53	2	no	—
307	C₂²	—	106	1	no	—
308	C₂²	—	106	1	no	—
309	C₂²	C₂²×C₈×D₅₃	1	1	yes	characteristic
310	C₄	—	2	2	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	2	no	—
313	C₄	—	53	2	no	—
314	C₄	—	106	1	no	—
315	C₄	—	106	2	no	—
316	C₄	—	106	1	no	—
317	C₄	—	106	2	no	—
318	C₄	#16811041	1	2	yes	—
319	C₄	C₄×Q₈×D₅₃	1	1	yes	characteristic
320	C₄	#131×D₅₃	1	1	yes	characteristic
321	C₂	—	106	2	no	—
322	C₂	C₈×Q₈×D₅₃	1	1	yes	characteristic
323	C₂	C₂×#16811041	1	1	yes	characteristic
324	C₂	#20287816	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	C₂²×D₅₃	C₂²×C₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#2183	D₅₃
hypocenter	C₅₃	#33476
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. #20276621	C₂	semidirect
3. #20276620	C₂	semidirect
4. #20276620	C₂	semidirect
5. Q₈×#11719908	C₂	semidirect
6. C₁₆×Dic₁₀₆	C₂	semidirect
11. C₅₃×#2183	C₂	semidirect
55. #13133156	Q₈	semidirect
163. #2183	D₅₃	semidirect
203. C₅₃	#33476	semidirect

Maximal subgroups

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

subgroup	index
#20283644	2
#20276621	2
#20276620	2
Q₈×#11719908	2
C₁₆×Dic₁₀₆	2
#20276472	2
C₄⋊C₁₆×D₅₃	2
#20276313	2
C₄×#13133156	2
C₅₃×#2183	2
#33476	53