Subgroups — #27283326

Order 13568.

586 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810996	C₂	1	1	yes	maximal, characteristic
2	#20275667	C₂	1	1	yes	maximal, characteristic, complemented
3	#20275642	C₂	1	1	yes	maximal, characteristic
4	C₂×#16810374	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16810370	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16810366	C₂	1	1	yes	maximal, characteristic
7	C₈×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#11719913	—	2	1	no	—
9	C₂²×#11719913	—	2	1	no	—
10	C₂×#13133217	C₂²	1	1	yes	characteristic
11	C₂×#13133214	C₂²	1	1	yes	characteristic
12	#16810996	—	2	2	no	—
13	#16810996	C₄	1	4	yes	complemented
14	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
16	C₂×#13133137	—	2	2	no	—
17	#16810385	—	2	2	no	—
18	#16810374	—	2	2	no	—
19	#16810370	—	2	2	no	—
20	#16810366	—	2	2	no	—
21	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₅₃	—	2	1	no	—
23	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
25	C₅₃×#336	—	2	1	no	—
26	C₂²×Dic₁₀₆	—	2	1	no	—
27	C₂²×D₂₁₂	—	2	1	no	—
28	C₂²×D₂₁₂	—	2	1	no	—
29	C₂²×D₂₁₂	—	2	1	no	—
30	C₂×#11719913	—	2	4	no	—
31	C₂×#11719913	—	2	4	no	—
32	C₂×#11719913	—	4	2	no	—
33	C₂×#11719913	—	4	2	no	—
34	C₂×#11719913	—	4	2	no	—
35	C₂×#11719913	—	4	2	no	—
36	C₂²×#88322	—	2	1	no	—
37	C₂²×#88322	—	2	1	no	—
38	#13133217	—	2	1	no	—
39	#13133217	—	2	1	no	—
40	#13133217	—	2	1	no	—
41	#13133217	C₂×C₄	1	2	yes	—
42	#13133214	—	2	1	no	—
43	#13133214	—	2	1	no	—
44	#13133214	—	2	1	no	—
45	#13133214	C₂×C₄	1	2	yes	—
46	C₂×C₄⋊Dic₅₃	—	2	1	no	—
47	C₂×C₄⋊Dic₅₃	—	2	1	no	—
48	C₄⋊#88322	—	2	1	no	—
49	C₄⋊#88322	—	2	1	no	—
50	C₄⋊#88322	—	2	1	no	—
51	C₄⋊#88322	—	2	1	no	—
52	#13133137	—	4	4	no	—
53	C₂²×C₈×C₅₃	—	2	1	no	—
54	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
55	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
56	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
57	C₂×C₄²×C₅₃	—	2	1	no	—
58	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
59	C₄⋊C₈×C₅₃	—	2	1	no	—
60	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
61	C₄×C₈×C₅₃	—	4	2	no	—
62	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
63	C₄×C₈×C₅₃	D₄	1	2	yes	—
64	#95×C₅₃	—	2	1	no	—
65	C₂³×D₅₃	—	4	1	no	—
66	C₂²×C₄×C₅₃	—	2	2	no	—
67	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
70	C₂²×Dic₅₃	—	4	1	no	—
71	C₂×D₂₁₂	—	2	2	no	—
72	C₂×D₂₁₂	—	2	2	no	—
73	C₂×D₂₁₂	—	2	2	no	—
74	C₂×D₂₁₂	—	4	1	no	—
75	C₂×D₂₁₂	—	4	2	no	—
76	C₂×D₂₁₂	—	4	1	no	—
77	C₂×D₂₁₂	—	4	2	no	—
78	C₂×D₂₁₂	—	4	1	no	—
79	C₂×D₂₁₂	—	4	1	no	—
80	C₂×D₂₁₂	—	4	1	no	—
81	C₂×D₂₁₂	—	4	1	no	—
82	C₂×D₂₁₂	—	4	1	no	—
83	C₂×D₂₁₂	—	8	2	no	—
84	C₂×Dic₁₀₆	—	2	2	no	—
85	C₂×Dic₁₀₆	—	4	1	no	—
86	C₂×Dic₁₀₆	—	4	1	no	—
87	C₂×Dic₁₀₆	—	4	1	no	—
88	C₂×Dic₁₀₆	—	4	1	no	—
89	C₂×Dic₁₀₆	—	4	1	no	—
90	C₂×C₈×C₅₃	—	2	1	no	—
91	C₂×C₈×C₅₃	—	2	1	no	—
92	C₂×C₈×C₅₃	—	2	2	no	—
93	C₂×C₈×C₅₃	—	4	1	no	—
94	C₂×C₈×C₅₃	—	4	1	no	—
95	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
96	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
97	C₂×C₈×C₅₃	D₈	1	2	yes	—
98	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
99	C₄⋊C₄×C₅₃	—	2	2	no	—
100	C₄⋊C₄×C₅₃	SD₁₆	1	4	yes	complemented
101	C₄²×C₅₃	—	4	1	no	—
102	C₄²×C₅₃	—	4	1	no	—
103	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
104	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
106	C₄²×C₅₃	#41	1	1	yes	characteristic
107	C₄⋊Dic₅₃	—	2	2	no	—
108	C₄⋊Dic₅₃	—	2	2	no	—
109	C₄⋊Dic₅₃	—	4	1	no	—
110	C₄⋊Dic₅₃	—	4	1	no	—
111	C₂×#88322	—	4	1	no	—
112	C₂×#88322	—	4	1	no	—
113	C₂×#88322	—	4	2	no	—
114	C₂×#88322	—	4	2	no	—
115	#11719913	—	4	8	no	—
116	#11719913	—	4	8	no	—
117	C₂³×C₅₃	#121	1	1	yes	characteristic
118	C₂²×D₅₃	—	4	2	no	—
119	C₂²×D₅₃	—	8	1	no	—
120	C₂²×D₅₃	—	8	1	no	—
121	C₂²×D₅₃	—	8	2	no	—
122	C₂²×D₅₃	—	8	2	no	—
123	C₂×C₄×C₅₃	—	2	2	no	—
124	C₂×C₄×C₅₃	—	2	4	no	—
125	C₂×C₄×C₅₃	—	4	2	no	—
126	C₂×C₄×C₅₃	—	4	2	no	—
127	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
129	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
130	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
131	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
132	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
133	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
135	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
136	C₂×C₄×C₅₃	#135	1	2	yes	—
137	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
138	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
139	C₂×Dic₅₃	—	4	2	no	—
140	C₂×Dic₅₃	—	8	1	no	—
141	C₂×Dic₅₃	—	8	1	no	—
142	D₂₁₂	—	4	2	no	—
143	D₂₁₂	—	4	2	no	—
144	D₂₁₂	—	4	4	no	—
145	D₂₁₂	—	8	2	no	—
146	D₂₁₂	—	8	2	no	—
147	D₂₁₂	—	8	1	no	—
148	D₂₁₂	—	8	1	no	—
149	D₂₁₂	—	8	1	no	—
150	D₂₁₂	—	8	1	no	—
151	Dic₁₀₆	—	4	2	no	—
152	Dic₁₀₆	—	4	2	no	—
153	Dic₁₀₆	—	8	1	no	—
154	Dic₁₀₆	—	8	1	no	—
155	C₈×C₅₃	—	2	2	no	—
156	C₈×C₅₃	—	4	2	no	—
157	C₈×C₅₃	#102	1	4	yes	complemented
158	#88322	—	8	1	no	—
159	#88322	—	8	1	no	—
160	#9732	—	53	1	no	maximal, Sylow
161	C₂²×C₅₃	#390	1	1	yes	characteristic
162	C₂²×C₅₃	#459	1	1	yes	characteristic
163	C₂²×C₅₃	#460	1	1	yes	characteristic
164	C₂²×C₅₃	#465	1	2	yes	—
165	C₂²×C₅₃	#471	1	2	yes	—
166	C₂×D₅₃	—	8	2	no	—
167	C₂×D₅₃	—	8	2	no	—
168	C₂×D₅₃	—	8	2	no	—
169	C₂×D₅₃	—	16	2	no	—
170	C₂×D₅₃	—	16	2	no	—
171	C₄×C₅₃	—	2	1	no	—
172	C₄×C₅₃	—	2	1	no	—
173	C₄×C₅₃	—	4	4	no	—
174	C₄×C₅₃	C₂×#102	1	1	yes	characteristic
175	C₄×C₅₃	#418	1	1	yes	characteristic
176	C₄×C₅₃	C₄×SD₁₆	1	2	yes	—
177	C₄×C₅₃	#459	1	1	yes	characteristic
178	C₄×C₅₃	#460	1	1	yes	characteristic
179	C₄×C₅₃	#492	1	1	yes	characteristic
180	C₄×C₅₃	#495	1	1	yes	characteristic
181	Dic₅₃	—	8	2	no	—
182	C₂×#492	—	53	1	no	—
183	#1988	—	53	1	no	—
184	#1982	—	53	1	no	—
185	C₈×C₄⋊C₄	D₅₃	1	1	yes	characteristic, complemented
186	C₂×#334	—	53	1	no	—
187	C₂×#332	—	53	1	no	—
188	C₂×#331	—	53	1	no	—
189	C₂×C₅₃	#1665	1	2	yes	—
190	C₂×C₅₃	#1666	1	2	yes	—
191	C₂×C₅₃	#1968	1	1	yes	characteristic
192	C₂×C₅₃	#1977	1	1	yes	characteristic
193	C₂×C₅₃	#1982	1	1	yes	characteristic
194	D₅₃	—	16	2	no	—
195	C₂²×SD₁₆	—	106	1	no	—
196	C₂²×SD₁₆	—	106	1	no	—
197	C₂×C₄⋊Q₈	—	53	1	no	—
198	C₂×#127	—	53	1	no	—
199	#492	—	53	4	no	—
200	#492	—	106	2	no	—
201	C₂×C₄⋊C₈	—	53	1	no	—
202	C₂×C₄⋊C₈	—	53	1	no	—
203	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
204	C₂×#102	—	106	2	no	—
205	C₂×C₄×C₈	—	2	1	no	—
206	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
207	C₄×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
208	#340	—	106	2	no	—
209	#336	—	2	1	no	—
210	#334	—	106	2	no	—
211	#332	—	106	2	no	—
212	#331	—	106	2	no	—
213	C₅₃	#9732	1	1	yes	characteristic, complemented, Sylow
214	C₂²×Q₈	—	106	1	no	—
215	C₂²×D₄	—	106	1	no	—
216	C₂²×D₄	—	106	1	no	—
217	C₂²×D₄	—	106	1	no	—
218	C₂×SD₁₆	—	106	4	no	—
219	C₂×SD₁₆	—	106	4	no	—
220	C₂×SD₁₆	—	212	2	no	—
221	C₂×SD₁₆	—	212	2	no	—
222	C₂×SD₁₆	—	212	2	no	—
223	C₂×SD₁₆	—	212	2	no	—
224	C₂²×C₈	—	2	1	no	—
225	C₂²×C₈	—	106	1	no	—
226	C₂²×C₈	—	106	1	no	—
227	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
228	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
229	C₄⋊Q₈	—	53	2	no	—
230	C₄⋊Q₈	—	106	1	no	—
231	C₄⋊Q₈	—	106	1	no	—
232	C₄⋊Q₈	—	106	1	no	—
233	#127	—	53	2	no	—
234	#127	—	106	1	no	—
235	#127	—	106	1	no	—
236	#127	—	106	1	no	—
237	C₂×C₄⋊C₄	—	106	1	no	—
238	C₂×C₄⋊C₄	—	106	1	no	—
239	C₂×C₄⋊C₄	D₂₁₂	1	2	yes	—
240	C₂×C₄²	—	2	1	no	—
241	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
242	C₄⋊C₈	—	2	1	no	—
243	C₄⋊C₈	—	106	1	no	—
244	C₄⋊C₈	—	106	1	no	—
245	C₄⋊C₈	—	106	1	no	—
246	C₄⋊C₈	—	106	1	no	—
247	C₄⋊C₈	D₂₁₂	1	2	yes	—
248	#102	—	212	4	no	—
249	C₄×C₈	—	4	2	no	—
250	C₄×C₈	C₄×D₅₃	1	2	yes	—
251	C₄×C₈	D₂₁₂	1	2	yes	—
252	#95	—	2	1	no	—
253	C₂⁴	—	212	1	no	—
254	C₂×Q₈	—	106	2	no	—
255	C₂×Q₈	—	212	1	no	—
256	C₂×Q₈	—	212	1	no	—
257	C₂×Q₈	—	212	1	no	—
258	C₂×Q₈	—	212	1	no	—
259	C₂×Q₈	—	212	1	no	—
260	C₂×D₄	—	106	2	no	—
261	C₂×D₄	—	106	2	no	—
262	C₂×D₄	—	106	2	no	—
263	C₂×D₄	—	424	2	no	—
264	C₂×D₄	—	212	1	no	—
265	C₂×D₄	—	212	1	no	—
266	C₂×D₄	—	212	1	no	—
267	C₂×D₄	—	212	1	no	—
268	C₂×D₄	—	212	1	no	—
269	C₂×D₄	—	212	2	no	—
270	C₂×D₄	—	212	2	no	—
271	C₂×D₄	—	212	1	no	—
272	C₂×D₄	—	212	1	no	—
273	C₂²×C₄	—	2	2	no	—
274	C₂²×C₄	—	212	1	no	—
275	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
276	C₂²×C₄	#11719945	1	1	yes	characteristic
277	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
278	SD₁₆	—	212	8	no	—
279	SD₁₆	—	212	8	no	—
280	C₂×C₈	—	2	1	no	—
281	C₂×C₈	—	2	1	no	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	—	4	1	no	—
284	C₂×C₈	—	4	1	no	—
285	C₂×C₈	—	212	1	no	—
286	C₂×C₈	—	212	1	no	—
287	C₂×C₈	—	212	2	no	—
288	C₂×C₈	—	212	2	no	—
289	C₂×C₈	#11719921	1	2	yes	—
290	C₂×C₈	#11719921	1	2	yes	—
291	C₂×C₈	#11719922	1	2	yes	—
292	C₂×C₈	#11719924	1	2	yes	—
293	C₄⋊C₄	—	2	2	no	—
294	C₄⋊C₄	—	106	2	no	—
295	C₄⋊C₄	—	106	2	no	—
296	C₄⋊C₄	—	212	1	no	—
297	C₄⋊C₄	—	212	1	no	—
298	C₄⋊C₄	#11719913	1	4	yes	complemented
299	C₄²	—	4	1	no	—
300	C₄²	—	4	1	no	—
301	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
302	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
303	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
304	C₄²	#11719945	1	1	yes	characteristic
305	C₂³	—	424	1	no	—
306	C₂³	—	424	1	no	—
307	C₂³	—	424	2	no	—
308	C₂³	—	424	2	no	—
309	C₂³	—	212	2	no	—
310	C₂³	#13133191	1	1	yes	characteristic
311	Q₈	—	424	1	no	—
312	Q₈	—	424	1	no	—
313	Q₈	—	212	2	no	—
314	Q₈	—	212	2	no	—
315	D₄	—	424	2	no	—
316	D₄	—	424	2	no	—
317	D₄	—	424	1	no	—
318	D₄	—	424	1	no	—
319	D₄	—	424	1	no	—
320	D₄	—	424	1	no	—
321	D₄	—	212	2	no	—
322	D₄	—	212	2	no	—
323	D₄	—	212	4	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	2	4	no	—
326	C₂×C₄	—	4	2	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	—	424	1	no	—
329	C₂×C₄	—	424	1	no	—
330	C₂×C₄	—	212	2	no	—
331	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
332	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
333	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
334	C₂×C₄	#13133191	1	1	yes	characteristic
335	C₂×C₄	#13133205	1	1	yes	characteristic
336	C₂×C₄	#13133214	1	1	yes	characteristic
337	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
338	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
339	C₂×C₄	C₂×#11719913	1	2	yes	—
340	C₂×C₄	#13133239	1	2	yes	—
341	C₂×C₄	#13133242	1	1	yes	characteristic
342	C₂×C₄	#13133242	1	1	yes	characteristic
343	C₈	—	2	2	no	—
344	C₈	—	4	2	no	—
345	C₈	—	424	1	no	—
346	C₈	—	424	1	no	—
347	C₈	#13133137	1	4	yes	complemented
348	C₂²	—	848	2	no	—
349	C₂²	—	848	2	no	—
350	C₂²	—	424	2	no	—
351	C₂²	—	424	2	no	—
352	C₂²	—	424	2	no	—
353	C₂²	#16810525	1	1	yes	characteristic
354	C₂²	#16810818	1	1	yes	characteristic
355	C₂²	#16810823	1	1	yes	characteristic
356	C₂²	#16810852	1	2	yes	—
357	C₂²	#16810886	1	2	yes	—
358	C₄	—	2	1	no	—
359	C₄	—	2	1	no	—
360	C₄	—	4	4	no	—
361	C₄	—	424	2	no	—
362	C₄	C₂×#13133137	1	1	yes	characteristic
363	C₄	#16810627	1	1	yes	characteristic
364	C₄	C₄×#11719913	1	2	yes	—
365	C₄	#16810818	1	1	yes	characteristic
366	C₄	#16810823	1	1	yes	characteristic
367	C₄	#16810996	1	1	yes	characteristic
368	C₄	#16811007	1	1	yes	characteristic
369	C₂	—	848	2	no	—
370	C₂	#20274056	1	2	yes	—
371	C₂	#20274063	1	2	yes	—
372	C₂	#20275569	1	1	yes	characteristic
373	C₂	#20275618	1	1	yes	characteristic
374	C₂	#20275642	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13133191
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₈×C₄⋊C₄×C₅₃	C₂
socle	C₂³×C₅₃	#121
prime radical	C₂×#13133214	C₂²
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₈×C₄⋊C₄	D₅₃
hypocenter	C₅₃	#9732
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₈×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. #20275667	C₂	semidirect
4. C₂×#16810374	C₂	semidirect
5. C₂×#16810370	C₂	semidirect
7. C₈×C₄⋊C₄×C₅₃	C₂	semidirect
13. #16810996	C₄	semidirect
100. C₄⋊C₄×C₅₃	SD₁₆	semidirect
157. C₈×C₅₃	#102	semidirect
185. C₈×C₄⋊C₄	D₅₃	semidirect
213. C₅₃	#9732	semidirect
298. C₄⋊C₄	#11719913	semidirect
347. C₈	#13133137	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810996	2
#20275667	2
#20275642	2
C₂×#16810374	2
C₂×#16810370	2
C₂×#16810366	2
C₈×C₄⋊C₄×C₅₃	2
#9732	53