Subgroups — #27402992

Order 13568.

638 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	M₁₆×Dic₁₀₆	C₂	1	2	yes	maximal
2	C₂×#16810751	C₂	1	1	yes	maximal, characteristic
3	#20275490	C₂	1	2	yes	maximal
4	#20275489	C₂	1	2	yes	maximal
5	#20275489	C₂	1	2	yes	maximal
6	#20274890	C₂	1	1	yes	maximal, characteristic
7	#20274888	C₂	1	1	yes	maximal, characteristic
8	#20274885	C₂	1	1	yes	maximal, characteristic
9	#20274623	C₂	1	1	yes	maximal, characteristic
10	#108×Dic₅₃	C₂	1	1	yes	maximal, characteristic
11	C₅₃×#1778	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
13	#16810753	—	2	2	no	—
14	#16810751	—	2	2	no	—
15	#16810751	—	2	2	no	—
16	#16810751	C₄	1	4	yes	—
17	#16810751	C₄	1	4	yes	—
18	C₈×Dic₁₀₆	—	2	2	no	—
19	C₂×#13133153	C₂²	1	1	yes	characteristic
20	C₂×#13133153	C₂²	1	1	yes	characteristic
21	C₂×#13133153	C₂²	1	2	yes	—
22	C₂×#13133152	C₂²	1	2	yes	—
23	C₂×#13133152	C₂²	1	2	yes	—
24	#16810647	C₂²	1	2	yes	—
25	#16810646	C₂²	1	2	yes	—
26	#16810646	C₂²	1	2	yes	—
27	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
28	C₂×#13133146	C₂²	1	1	yes	characteristic
29	C₂×#13133146	C₂²	1	1	yes	characteristic
30	M₁₆×Dic₅₃	C₂²	1	2	yes	—
31	M₁₆×Dic₅₃	C₂²	1	2	yes	—
32	C₂×#13133127	C₂²	1	1	yes	characteristic
33	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
34	#16810387	C₂²	1	2	yes	—
35	#16810387	C₂²	1	2	yes	—
36	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
37	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
38	C₄×M₁₆×C₅₃	C₂²	1	2	yes	—
39	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
40	C₅₃×#341	C₂²	1	2	yes	—
41	C₂²×Dic₁₀₆	Q₈	1	1	yes	characteristic
42	C₂×#11719917	C₂³	1	2	yes	—
43	C₂×#11719917	C₂³	1	2	yes	—
44	C₂²×#88322	C₂³	1	1	yes	characteristic
45	C₂²×#88322	C₂³	1	1	yes	characteristic
46	C₄×Dic₁₀₆	C₂×C₄	1	2	yes	—
47	C₄×Dic₁₀₆	C₂×C₄	1	2	yes	—
48	C₄×Dic₁₀₆	D₄	1	2	yes	—
49	C₄×Dic₁₀₆	D₄	1	2	yes	—
50	C₂×C₄⋊Dic₅₃	Q₈	1	1	yes	characteristic
51	C₂×#11719919	Q₈	1	1	yes	characteristic
52	C₂×#11719919	Q₈	1	1	yes	characteristic
53	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
54	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
55	#13133153	—	2	2	no	—
56	#13133153	—	2	2	no	—
57	#13133153	—	4	2	no	—
58	#13133152	—	4	2	no	—
59	#13133152	—	4	2	no	—
60	C₄⋊#88322	—	2	2	no	—
61	C₄⋊#88322	—	2	2	no	—
62	C₄⋊#88322	C₂×C₄	1	4	yes	—
63	#13133146	—	2	2	no	—
64	#13133146	—	2	2	no	—
65	#13133146	—	2	2	no	—
66	#13133146	—	2	2	no	—
67	#13133146	C₂×C₄	1	4	yes	—
68	#13133146	C₂×C₄	1	2	yes	—
69	#13133146	C₂×C₄	1	2	yes	—
70	#13133127	—	2	2	no	—
71	#13133127	—	2	2	no	—
72	#13133127	C₂×C₄	1	4	yes	—
73	C₈×Dic₅₃	—	2	2	no	—
74	C₈×Dic₅₃	—	2	2	no	—
75	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
76	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
77	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
78	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
79	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
80	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
81	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
82	#108×C₅₃	—	2	2	no	—
83	#108×C₅₃	Q₈	1	4	yes	complemented
84	#97×C₅₃	—	2	2	no	—
85	#97×C₅₃	C₂×C₄	1	4	yes	—
86	C₄×C₈×C₅₃	—	2	2	no	—
87	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
88	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
89	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
90	C₂²×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
91	C₂²×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
92	C₂²×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
93	C₂²×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
94	C₂×Dic₁₀₆	—	4	2	no	—
95	C₂×Dic₁₀₆	C₄⋊C₄	1	2	yes	—
96	C₂×Dic₁₀₆	C₄⋊C₄	1	2	yes	—
97	M₁₆×C₅₃	—	2	2	no	—
98	M₁₆×C₅₃	—	2	2	no	—
99	M₁₆×C₅₃	—	2	2	no	—
100	M₁₆×C₅₃	C₂×Q₈	1	4	yes	—
101	C₂×C₈×C₅₃	—	2	2	no	—
102	C₂×C₈×C₅₃	—	2	1	no	—
103	C₂×C₈×C₅₃	—	2	1	no	—
104	C₂×C₈×C₅₃	—	2	2	no	—
105	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
106	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
107	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
108	C₂×C₈×C₅₃	#41	1	2	yes	—
109	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
110	C₄²×C₅₃	C₂×D₄	1	2	yes	—
111	C₄⋊Dic₅₃	C₄⋊C₄	1	2	yes	—
112	C₄⋊Dic₅₃	C₄⋊C₄	1	2	yes	—
113	#11719919	C₄⋊C₄	1	2	yes	—
114	#11719919	C₄⋊C₄	1	2	yes	—
115	#11719919	C₄⋊C₄	1	2	yes	—
116	#11719919	C₄⋊C₄	1	2	yes	—
117	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
118	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
119	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
120	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
121	C₄×Dic₅₃	C₂×D₄	1	2	yes	—
122	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
123	#11719917	—	2	2	no	—
124	#11719917	—	2	2	no	—
125	#11719917	—	2	2	no	—
126	#11719917	—	2	2	no	—
127	C₂×#88322	—	2	1	no	—
128	C₂×#88322	—	2	1	no	—
129	C₂×#88322	—	2	1	no	—
130	C₂×#88322	—	2	1	no	—
131	C₂×#88322	—	2	2	no	—
132	C₂×#88322	—	2	2	no	—
133	C₂×#88322	C₂²×C₄	1	2	yes	—
134	C₂×#88322	C₂²×C₄	1	2	yes	—
135	C₂³×C₅₃	C₂²×Q₈	1	1	yes	characteristic
136	C₂×C₄×C₅₃	—	4	1	no	—
137	C₂×C₄×C₅₃	—	4	1	no	—
138	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
139	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
140	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
141	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
142	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
143	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
144	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
145	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
146	C₂×Dic₅₃	—	2	2	no	—
147	C₂×Dic₅₃	—	4	1	no	—
148	C₂×Dic₅₃	—	4	1	no	—
149	C₂×Dic₅₃	—	4	1	no	—
150	C₂×Dic₅₃	C₂×C₄⋊C₄	1	2	yes	—
151	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
152	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
153	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
154	C₂×Dic₅₃	C₂×C₄⋊C₄	1	2	yes	—
155	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
156	Dic₁₀₆	—	2	4	no	—
157	Dic₁₀₆	—	4	2	no	—
158	C₈×C₅₃	—	2	2	no	—
159	C₈×C₅₃	—	2	4	no	—
160	C₈×C₅₃	—	4	2	no	—
161	C₈×C₅₃	C₄×Q₈	1	4	yes	—
162	#88322	—	2	2	no	—
163	#88322	—	2	2	no	—
164	#88322	—	4	2	no	—
165	#88322	—	4	2	no	—
166	#22733	—	53	1	no	maximal, Sylow
167	C₂²×C₅₃	—	2	1	no	—
168	C₂²×C₅₃	—	2	1	no	—
169	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
170	C₂²×C₅₃	#557	1	1	yes	characteristic
171	C₂²×C₅₃	Q₈²	1	1	yes	characteristic
172	C₄×C₅₃	—	2	2	no	—
173	C₄×C₅₃	—	2	2	no	—
174	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
175	C₄×C₅₃	#520	1	1	yes	characteristic
176	C₄×C₅₃	D₄×Q₈	1	1	yes	characteristic
177	C₄×C₅₃	#550	1	1	yes	characteristic
178	Dic₅₃	—	2	2	no	—
179	Dic₅₃	—	2	1	no	—
180	Dic₅₃	—	2	1	no	—
181	Dic₅₃	—	2	1	no	—
182	Dic₅₃	—	2	2	no	—
183	Dic₅₃	#430	1	2	yes	—
184	Q₈×M₁₆	—	53	2	no	—
185	C₂×#446	—	53	1	no	—
186	#1950	—	53	2	no	—
187	#1950	—	53	2	no	—
188	#1950	—	53	2	no	—
189	#1849	—	53	1	no	—
190	#1849	—	53	1	no	—
191	#1848	—	53	1	no	—
192	#1778	—	53	1	no	—
193	#1778	D₅₃	1	1	yes	characteristic
194	C₄×#108	—	53	1	no	—
195	C₂×C₅₃	—	2	1	no	—
196	C₂×C₅₃	—	2	1	no	—
197	C₂×C₅₃	Q₈×C₄⋊C₄	1	1	yes	characteristic
198	C₂×C₅₃	C₂×#430	1	1	yes	characteristic
199	C₂×C₅₃	#2916	1	1	yes	characteristic
200	C₂×C₄×Q₈	—	53	1	no	—
201	#446	—	53	4	no	—
202	#446	—	53	4	no	—
203	#446	—	106	2	no	—
204	#446	—	106	2	no	—
205	#446	—	106	2	no	—
206	C₈×Q₈	—	106	2	no	—
207	C₂×#108	—	53	1	no	—
208	C₂×#108	—	53	1	no	—
209	C₂×#108	—	53	2	no	—
210	C₂×#108	—	53	2	no	—
211	C₂×#108	—	53	2	no	—
212	C₂×#108	C₂×D₅₃	1	1	yes	characteristic
213	C₂×#108	C₂×D₅₃	1	1	yes	characteristic
214	#423	—	53	2	no	—
215	#423	—	53	2	no	—
216	#423	—	53	2	no	—
217	C₂×C₄⋊C₈	—	53	1	no	—
218	C₂×C₄⋊C₈	—	53	1	no	—
219	C₂×C₄⋊C₈	—	53	1	no	—
220	C₄×M₁₆	—	53	2	no	—
221	C₄×M₁₆	—	53	2	no	—
222	C₄×M₁₆	C₂×D₅₃	1	2	yes	—
223	C₂×#97	—	53	1	no	—
224	C₂×#97	C₂×D₅₃	1	1	yes	characteristic
225	C₂×C₄×C₈	—	53	1	no	—
226	#341	—	53	2	no	—
227	#341	—	53	2	no	—
228	#341	C₂×D₅₃	1	2	yes	—
229	C₅₃	#22733	1	1	yes	characteristic, complemented, Sylow
230	C₂²×Q₈	—	53	1	no	—
231	C₂×M₁₆	—	53	2	no	—
232	C₂×M₁₆	—	53	2	no	—
233	C₂×M₁₆	C₂²×D₅₃	1	2	yes	—
234	C₂×M₁₆	C₂²×D₅₃	1	2	yes	—
235	C₂²×C₈	—	53	1	no	—
236	C₂²×C₈	—	53	1	no	—
237	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
238	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
239	C₄×Q₈	—	53	2	no	—
240	C₄×Q₈	—	53	2	no	—
241	C₄×Q₈	—	53	2	no	—
242	C₄×Q₈	—	53	2	no	—
243	C₂×C₄⋊C₄	—	53	1	no	—
244	C₂×C₄⋊C₄	—	53	1	no	—
245	C₂×C₄⋊C₄	—	53	1	no	—
246	C₂×C₄²	—	53	1	no	—
247	C₂×C₄²	—	53	1	no	—
248	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
249	#108	—	2	2	no	—
250	#108	—	106	2	no	—
251	#108	—	106	2	no	—
252	#108	—	212	2	no	—
253	#108	—	212	2	no	—
254	#108	—	212	2	no	—
255	#108	Dic₁₀₆	1	4	yes	complemented
256	C₄⋊C₈	—	53	2	no	—
257	C₄⋊C₈	—	53	2	no	—
258	C₄⋊C₈	—	53	4	no	—
259	C₄⋊C₈	—	53	4	no	—
260	C₄⋊C₈	—	106	2	no	—
261	C₄⋊C₈	—	106	2	no	—
262	C₄⋊C₈	—	106	2	no	—
263	C₄⋊C₈	—	106	2	no	—
264	C₄⋊C₈	—	106	2	no	—
265	C₄⋊C₈	—	106	2	no	—
266	#97	—	2	2	no	—
267	#97	—	53	4	no	—
268	#97	—	106	2	no	—
269	#97	—	106	2	no	—
270	#97	C₄×D₅₃	1	4	yes	—
271	C₄×C₈	—	2	2	no	—
272	C₄×C₈	—	53	2	no	—
273	C₄×C₈	—	53	2	no	—
274	C₄×C₈	—	106	2	no	—
275	C₄×C₈	—	106	2	no	—
276	C₂×Q₈	—	53	2	no	—
277	C₂×Q₈	—	53	2	no	—
278	C₂×Q₈	—	212	2	no	—
279	C₂²×C₄	—	53	1	no	—
280	C₂²×C₄	—	53	1	no	—
281	C₂²×C₄	—	53	1	no	—
282	C₂²×C₄	—	53	1	no	—
283	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
284	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
285	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
286	M₁₆	—	2	2	no	—
287	M₁₆	—	2	2	no	—
288	M₁₆	—	2	2	no	—
289	M₁₆	—	106	2	no	—
290	M₁₆	—	106	2	no	—
291	M₁₆	—	106	2	no	—
292	M₁₆	—	106	2	no	—
293	M₁₆	C₂×Dic₁₀₆	1	4	yes	—
294	C₂×C₈	—	2	2	no	—
295	C₂×C₈	—	2	1	no	—
296	C₂×C₈	—	2	1	no	—
297	C₂×C₈	—	2	2	no	—
298	C₂×C₈	—	53	2	no	—
299	C₂×C₈	—	53	2	no	—
300	C₂×C₈	—	106	1	no	—
301	C₂×C₈	—	106	1	no	—
302	C₂×C₈	—	106	1	no	—
303	C₂×C₈	—	106	1	no	—
304	C₂×C₈	—	106	2	no	—
305	C₂×C₈	—	106	2	no	—
306	C₂×C₈	C₂×Dic₁₀₆	1	2	yes	—
307	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
308	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
309	C₂×C₈	#11719945	1	2	yes	—
310	C₄⋊C₄	—	53	2	no	—
311	C₄⋊C₄	—	53	2	no	—
312	C₄⋊C₄	—	53	2	no	—
313	C₄⋊C₄	—	53	2	no	—
314	C₄⋊C₄	—	53	2	no	—
315	C₄⋊C₄	—	53	2	no	—
316	C₄²	—	53	1	no	—
317	C₄²	—	53	1	no	—
318	C₄²	—	53	2	no	—
319	C₄²	—	53	1	no	—
320	C₄²	—	53	2	no	—
321	C₄²	—	53	1	no	—
322	C₄²	C₂×C₄×D₅₃	1	2	yes	—
323	C₄²	D₄×D₅₃	1	2	yes	—
324	C₂³	C₂×Q₈×D₅₃	1	1	yes	characteristic
325	Q₈	—	106	4	no	—
326	Q₈	—	212	2	no	—
327	C₂×C₄	—	4	1	no	—
328	C₂×C₄	—	4	1	no	—
329	C₂×C₄	—	53	1	no	—
330	C₂×C₄	—	53	1	no	—
331	C₂×C₄	—	53	1	no	—
332	C₂×C₄	—	53	2	no	—
333	C₂×C₄	—	53	1	no	—
334	C₂×C₄	—	53	2	no	—
335	C₂×C₄	—	106	2	no	—
336	C₂×C₄	—	212	1	no	—
337	C₂×C₄	—	212	1	no	—
338	C₂×C₄	—	212	1	no	—
339	C₂×C₄	C₄⋊C₄×D₅₃	1	2	yes	—
340	C₂×C₄	C₄⋊C₄×D₅₃	1	2	yes	—
341	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
342	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
343	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
344	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
345	C₂×C₄	#13133263	1	1	yes	characteristic
346	C₂×C₄	#13133264	1	1	yes	characteristic
347	C₈	—	2	4	no	—
348	C₈	—	2	2	no	—
349	C₈	—	4	2	no	—
350	C₈	—	106	2	no	—
351	C₈	—	106	2	no	—
352	C₈	—	212	2	no	—
353	C₈	—	212	2	no	—
354	C₈	C₄×Dic₁₀₆	1	4	yes	—
355	C₂²	—	2	1	no	—
356	C₂²	—	2	1	no	—
357	C₂²	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
358	C₂²	#16811351	1	1	yes	characteristic
359	C₂²	Q₈×Dic₁₀₆	1	1	yes	characteristic
360	C₄	—	2	2	no	—
361	C₄	—	2	2	no	—
362	C₄	—	53	2	no	—
363	C₄	—	106	1	no	—
364	C₄	—	106	2	no	—
365	C₄	—	106	1	no	—
366	C₄	—	106	1	no	—
367	C₄	—	106	2	no	—
368	C₄	C₂×C₄×Dic₁₀₆	1	1	yes	characteristic
369	C₄	#16811103	1	1	yes	characteristic
370	C₄	D₄×Dic₁₀₆	1	1	yes	characteristic
371	C₄	#16811294	1	1	yes	characteristic
372	C₂	—	2	1	no	—
373	C₂	—	2	1	no	—
374	C₂	C₄⋊C₄×Dic₁₀₆	1	1	yes	characteristic
375	C₂	#430×D₅₃	1	1	yes	characteristic
376	C₂	#20283325	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1778	C₂
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂³×C₅₃	C₂²×Q₈
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1778	D₅₃
hypocenter	C₅₃	#22733
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1778

Complements

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

N	H ≅ G/N	type
83. #108×C₅₃	Q₈	semidirect
229. C₅₃	#22733	semidirect
255. #108	Dic₁₀₆	semidirect

Maximal subgroups

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

subgroup	index
M₁₆×Dic₁₀₆	2
C₂×#16810751	2
#20275490	2
#20275489	2
#20274890	2
#20274888	2
#20274885	2
#20274623	2
#108×Dic₅₃	2
C₅₃×#1778	2
#22733	53