Subgroups — #27355782

Order 13568.

626 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283802	C₂	1	2	yes	maximal
2	C₂×C₄×Dic₂₁₂	C₂	1	1	yes	maximal, characteristic
3	#20273613	C₂	1	2	yes	maximal, complemented
4	#20273588	C₂	1	4	yes	maximal, complemented
5	C₈×Dic₂₁₂	C₂	1	2	yes	maximal, complemented
6	#20273478	C₂	1	1	yes	maximal, characteristic
7	#20273163	C₂	1	2	yes	maximal
8	C₈×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₂₁₂	C₄	1	1	yes	characteristic
10	C₂×C₄×Dic₁₀₆	C₂²	1	2	yes	—
11	#16810753	C₂²	1	4	yes	—
12	#16810752	—	2	4	no	—
13	C₈×Dic₁₀₆	C₂²	1	4	yes	—
14	C₄×Dic₂₁₂	C₄	1	2	yes	—
15	C₄×Dic₂₁₂	C₄	1	1	yes	characteristic
16	C₄×Dic₂₁₂	C₄	1	1	yes	characteristic
17	C₄×Dic₂₁₂	C₂²	1	1	yes	characteristic
18	C₄×Dic₂₁₂	C₂²	1	2	yes	—
19	C₄×Dic₂₁₂	C₂²	1	1	yes	characteristic
20	#16810680	C₂²	1	2	yes	—
21	#16810677	—	2	2	no	—
22	#16810673	—	2	2	no	—
23	C₂×#13133151	C₄	1	1	yes	characteristic
24	C₂×#13133142	C₄	1	2	yes	—
25	#16810376	C₂²	1	2	yes	—
26	#16810374	C₂²	1	2	yes	—
27	#16810355	C₂²	1	4	yes	—
28	#16810355	C₂²	1	4	yes	—
29	C₅₃×#431	C₂²	1	1	yes	characteristic
30	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
32	C₅₃×#322	C₂²	1	2	yes	—
33	C₈²×C₅₃	C₂²	1	2	yes	—
34	C₂²×Dic₁₀₆	C₂×C₄	1	2	yes	—
35	C₂×Dic₂₁₂	—	2	4	no	—
36	C₂×Dic₂₁₂	C₂×C₄	1	1	yes	characteristic
37	C₂×Dic₂₁₂	C₂×C₄	1	2	yes	—
38	C₂×Dic₂₁₂	C₂×C₄	1	1	yes	characteristic
39	C₄×Dic₁₀₆	—	2	2	no	—
40	C₄×Dic₁₀₆	—	2	2	no	—
41	C₄×Dic₁₀₆	C₂×C₄	1	2	yes	—
42	C₄×Dic₁₀₆	C₂×C₄	1	2	yes	—
43	C₄×Dic₁₀₆	C₂³	1	2	yes	—
44	C₄×Dic₁₀₆	C₂³	1	2	yes	—
45	C₂×C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
46	C₂×#11719919	—	2	2	no	—
47	C₂×C₄×Dic₅₃	—	2	2	no	—
48	#13133151	C₂×C₄	1	2	yes	—
49	#13133151	C₂×C₄	1	1	yes	characteristic
50	#13133151	C₂×C₄	1	1	yes	characteristic
51	C₄⋊#88322	C₂³	1	4	yes	—
52	#13133146	—	2	4	no	—
53	#13133146	—	2	4	no	—
54	#13133142	C₂×C₄	1	2	yes	—
55	#13133142	C₂×C₄	1	2	yes	—
56	#13133142	C₂×C₄	1	2	yes	—
57	#13133142	C₂×C₄	1	2	yes	—
58	C₂²⋊#88322	—	2	2	no	—
59	#13133128	—	2	2	no	—
60	#13133127	—	2	4	no	—
61	C₈×Dic₅₃	—	2	4	no	—
62	C₂×M₁₆×C₅₃	—	2	1	no	—
63	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
64	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
66	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
67	C₂²⋊C₈×C₅₃	D₄	1	1	yes	characteristic
68	C₂²⋊C₈×C₅₃	D₄	1	1	yes	characteristic
69	#97×C₅₃	—	2	1	no	—
70	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
71	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
72	C₄×C₈×C₅₃	C₂³	1	2	yes	—
73	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₅₃	C₂³	1	2	yes	—
76	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
79	C₂²×Dic₅₃	—	2	2	no	—
80	C₂²×Dic₅₃	—	2	2	no	—
81	C₂×Dic₁₀₆	—	2	4	no	—
82	C₂×Dic₁₀₆	—	2	2	no	—
83	C₂×Dic₁₀₆	—	4	2	no	—
84	C₂×Dic₁₀₆	C₂²×C₄	1	2	yes	—
85	C₂×Dic₁₀₆	C₂²×C₄	1	2	yes	—
86	M₁₆×C₅₃	—	2	4	no	—
87	C₂×C₈×C₅₃	—	2	2	no	—
88	C₂×C₈×C₅₃	—	2	2	no	—
89	C₂×C₈×C₅₃	—	4	1	no	—
90	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
95	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
96	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
97	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₅₃	#41	1	1	yes	characteristic
99	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
100	C₄⋊Dic₅₃	—	2	2	no	—
101	C₄⋊Dic₅₃	C₂²×C₄	1	2	yes	—
102	C₄⋊Dic₅₃	C₂²×C₄	1	2	yes	—
103	#11719919	—	2	2	no	—
104	#11719919	—	2	2	no	—
105	#11719919	—	2	2	no	—
106	#11719919	—	2	2	no	—
107	C₄×Dic₅₃	—	2	2	no	—
108	C₄×Dic₅₃	—	2	2	no	—
109	C₄×Dic₅₃	—	2	2	no	—
110	C₄×Dic₅₃	—	2	2	no	—
111	C₂×#88322	—	2	4	no	—
112	Dic₂₁₂	—	2	4	no	—
113	Dic₂₁₂	—	2	2	no	—
114	Dic₂₁₂	M₁₆	1	4	yes	complemented
115	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
116	C₂×C₄×C₅₃	—	2	2	no	—
117	C₂×C₄×C₅₃	—	2	1	no	—
118	C₂×C₄×C₅₃	—	2	1	no	—
119	C₂×C₄×C₅₃	—	2	2	no	—
120	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
121	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
124	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
126	C₂×Dic₅₃	—	2	2	no	—
127	C₂×Dic₅₃	—	2	2	no	—
128	C₂×Dic₅₃	—	2	2	no	—
129	C₂×Dic₅₃	—	2	2	no	—
130	C₂×Dic₅₃	—	4	4	no	—
131	C₂×Dic₅₃	—	4	4	no	—
132	Dic₁₀₆	—	2	2	no	—
133	Dic₁₀₆	—	2	2	no	—
134	Dic₁₀₆	—	2	2	no	—
135	Dic₁₀₆	—	4	4	no	—
136	Dic₁₀₆	C₂×M₁₆	1	4	yes	—
137	C₈×C₅₃	—	2	4	no	—
138	C₈×C₅₃	—	2	1	no	—
139	C₈×C₅₃	—	2	1	no	—
140	C₈×C₅₃	—	2	1	no	—
141	C₈×C₅₃	—	2	2	no	—
142	C₈×C₅₃	—	2	2	no	—
143	C₈×C₅₃	C₂×M₁₆	1	2	yes	—
144	#88322	—	4	4	no	—
145	#18058	—	53	1	no	maximal, Sylow
146	C₂²×C₅₃	—	2	1	no	—
147	C₂²×C₅₃	—	2	1	no	—
148	C₂²×C₅₃	#443	1	2	yes	—
149	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
150	C₄×C₅₃	—	2	1	no	—
151	C₄×C₅₃	—	2	1	no	—
152	C₄×C₅₃	—	2	1	no	—
153	C₄×C₅₃	—	2	1	no	—
154	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
155	C₄×C₅₃	#568	1	1	yes	characteristic
156	C₄×C₅₃	#578	1	2	yes	—
157	Dic₅₃	—	2	4	no	—
158	Dic₅₃	—	4	2	no	—
159	Dic₅₃	—	4	2	no	—
160	Dic₅₃	—	4	2	no	—
161	#2984	—	53	2	no	—
162	C₂×C₄×Q₁₆	—	53	1	no	—
163	#1591	—	53	2	no	—
164	#1587	—	53	4	no	—
165	C₈×Q₁₆	—	53	2	no	—
166	#1568	—	53	1	no	—
167	#1491	—	53	2	no	—
168	C₈×M₁₆	D₅₃	1	1	yes	characteristic
169	C₂×C₅₃	—	2	2	no	—
170	C₂×C₅₃	D₄×M₁₆	1	1	yes	characteristic
171	C₂×C₅₃	#2950	1	1	yes	characteristic
172	C₂×C₅₃	C₂×#443	1	1	yes	characteristic
173	C₂²×Q₁₆	—	53	1	no	—
174	C₂×C₄×Q₈	—	53	2	no	—
175	#446	—	53	4	no	—
176	#446	—	106	4	no	—
177	C₈×Q₈	—	53	4	no	—
178	C₄×Q₁₆	—	53	1	no	—
179	C₄×Q₁₆	—	53	2	no	—
180	C₄×Q₁₆	—	53	1	no	—
181	C₄×Q₁₆	—	53	1	no	—
182	C₄×Q₁₆	—	53	2	no	—
183	C₄×Q₁₆	—	53	1	no	—
184	#432	—	53	2	no	—
185	#432	—	106	2	no	—
186	#431	—	106	2	no	—
187	#431	C₂×D₅₃	1	1	yes	characteristic
188	C₂×#107	—	53	1	no	—
189	C₂×#103	—	53	2	no	—
190	C₄×M₁₆	C₂×D₅₃	1	1	yes	characteristic
191	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
192	#335	—	53	2	no	—
193	#334	—	53	2	no	—
194	Q₈⋊C₈	—	53	4	no	—
195	Q₈⋊C₈	—	53	4	no	—
196	#322	C₂×D₅₃	1	2	yes	—
197	C₈²	C₂×D₅₃	1	2	yes	—
198	C₅₃	#18058	1	1	yes	characteristic, complemented, Sylow
199	C₂²×Q₈	—	53	2	no	—
200	C₂×Q₁₆	—	53	1	no	—
201	C₂×Q₁₆	—	53	2	no	—
202	C₂×Q₁₆	—	53	1	no	—
203	C₂×Q₁₆	—	106	4	no	—
204	C₂×M₁₆	—	2	1	no	—
205	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
206	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
207	C₄×Q₈	—	53	2	no	—
208	C₄×Q₈	—	53	2	no	—
209	C₄×Q₈	—	53	2	no	—
210	C₄×Q₈	—	53	2	no	—
211	C₄×Q₈	—	106	2	no	—
212	C₄×Q₈	—	106	2	no	—
213	C₂×C₄⋊C₄	—	53	2	no	—
214	C₂×C₄⋊C₄	—	106	2	no	—
215	C₂×C₄²	—	106	2	no	—
216	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
217	#107	—	53	2	no	—
218	#107	—	53	1	no	—
219	#107	—	53	1	no	—
220	C₄⋊C₈	—	53	4	no	—
221	C₄⋊C₈	—	106	4	no	—
222	C₄⋊C₈	—	106	4	no	—
223	C₄⋊C₈	D₂₁₂	1	2	yes	—
224	#103	—	53	2	no	—
225	#103	—	53	2	no	—
226	#103	—	53	2	no	—
227	#103	—	53	2	no	—
228	C₂²⋊C₈	—	106	2	no	—
229	C₂²⋊C₈	D₂₁₂	1	1	yes	characteristic
230	C₂²⋊C₈	D₂₁₂	1	1	yes	characteristic
231	#97	—	2	1	no	—
232	#97	—	106	4	no	—
233	#97	—	106	2	no	—
234	C₄×C₈	—	106	4	no	—
235	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
236	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
237	C₄×C₈	C₂²×D₅₃	1	2	yes	—
238	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
239	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
240	C₄×C₈	C₂²×D₅₃	1	2	yes	—
241	C₂×Q₈	—	53	2	no	—
242	C₂×Q₈	—	53	2	no	—
243	C₂×Q₈	—	106	4	no	—
244	C₂×Q₈	—	106	2	no	—
245	C₂×Q₈	—	212	2	no	—
246	C₂²×C₄	—	106	2	no	—
247	C₂²×C₄	—	106	2	no	—
248	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
249	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
250	C₂²×C₄	#11719945	1	1	yes	characteristic
251	Q₁₆	—	53	4	no	—
252	Q₁₆	—	106	4	no	—
253	Q₁₆	—	106	2	no	—
254	M₁₆	—	2	4	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	4	1	no	—
258	C₂×C₈	—	106	4	no	—
259	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
260	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
261	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
262	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
263	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
264	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
265	C₄⋊C₄	—	53	2	no	—
266	C₄⋊C₄	—	53	2	no	—
267	C₄⋊C₄	—	106	2	no	—
268	C₄⋊C₄	—	106	2	no	—
269	C₄⋊C₄	—	106	2	no	—
270	C₄⋊C₄	—	106	2	no	—
271	C₄⋊C₄	—	106	2	no	—
272	C₄²	—	106	2	no	—
273	C₄²	—	106	2	no	—
274	C₄²	—	106	2	no	—
275	C₄²	—	106	2	no	—
276	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
277	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
278	C₄²	#11719945	1	1	yes	characteristic
279	C₄²	C₂³×D₅₃	1	1	yes	characteristic
280	C₂³	C₄×D₂₁₂	1	1	yes	characteristic
281	Q₈	—	53	4	no	—
282	Q₈	—	106	2	no	—
283	Q₈	—	106	2	no	—
284	Q₈	—	106	2	no	—
285	Q₈	—	212	4	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	106	2	no	—
291	C₂×C₄	—	106	2	no	—
292	C₂×C₄	—	106	2	no	—
293	C₂×C₄	—	106	2	no	—
294	C₂×C₄	—	212	4	no	—
295	C₂×C₄	—	212	4	no	—
296	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
297	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
298	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
299	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
300	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
301	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
302	C₈	—	2	4	no	—
303	C₈	—	2	1	no	—
304	C₈	—	2	1	no	—
305	C₈	—	2	1	no	—
306	C₈	—	2	2	no	—
307	C₈	—	2	2	no	—
308	C₈	—	212	4	no	—
309	C₈	M₁₆×D₅₃	1	2	yes	—
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	#16810741	1	2	yes	—
313	C₂²	C₂×C₄×D₂₁₂	1	1	yes	characteristic
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	106	4	no	—
319	C₄	—	212	2	no	—
320	C₄	—	212	2	no	—
321	C₄	—	212	2	no	—
322	C₄	C₂×M₁₆×D₅₃	1	1	yes	characteristic
323	C₄	#16811389	1	1	yes	characteristic
324	C₄	#16811440	1	2	yes	—
325	C₂	—	2	2	no	—
326	C₂	M₁₆×D₂₁₂	1	1	yes	characteristic
327	C₂	#20283584	1	1	yes	characteristic
328	C₂	C₂×#16810741	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₈×M₁₆×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂³×C₅₃	C₄×D₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	C₈×M₁₆	D₅₃
hypocenter	C₅₃	#18058
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
3. #20273613	C₂	semidirect
4. #20273588	C₂	semidirect
5. C₈×Dic₂₁₂	C₂	semidirect
114. Dic₂₁₂	M₁₆	semidirect
198. C₅₃	#18058	semidirect

Maximal subgroups

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

subgroup	index
#20283802	2
C₂×C₄×Dic₂₁₂	2
#20273613	2
#20273588	2
C₈×Dic₂₁₂	2
#20273478	2
#20273163	2
C₈×M₁₆×C₅₃	2
#18058	53