Subgroups — #27355551

Order 13568.

638 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283666	C₂	1	2	yes	maximal
2	C₂×C₄×Dic₂₁₂	C₂	1	1	yes	maximal, characteristic
3	#20273611	C₂	1	2	yes	maximal, complemented
4	#20273564	C₂	1	4	yes	maximal, complemented
5	#20273521	C₂	1	2	yes	maximal, complemented
6	#20273143	C₂	1	2	yes	maximal
7	C₈×#11719917	C₂	1	1	yes	maximal, characteristic
8	C₅₃×#1567	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₂₁₂	C₄	1	1	yes	characteristic
10	C₂×C₄×Dic₁₀₆	C₂²	1	2	yes	—
11	#16810752	—	2	4	no	—
12	#16810750	C₂²	1	4	yes	—
13	#16810747	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	#16810676	C₂²	1	1	yes	characteristic
21	C₂×#13133151	C₄	1	1	yes	characteristic
22	#16810646	—	2	2	no	—
23	C₂×#13133142	C₄	1	2	yes	—
24	C₄×#11719917	C₂²	1	1	yes	characteristic
25	M₁₆×Dic₅₃	—	2	2	no	—
26	#16810353	C₂²	1	4	yes	—
27	#16810353	C₂²	1	4	yes	—
28	#16810345	C₂²	1	2	yes	—
29	C₈×#88322	C₂²	1	2	yes	—
30	C₅₃×#423	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
32	C₅₃×#335	C₂²	1	2	yes	—
33	C₅₃×#335	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	C₂×#11719917	D₄	1	1	yes	characteristic
40	C₂×#11719917	D₄	1	1	yes	characteristic
41	C₄×Dic₁₀₆	—	2	2	no	—
42	C₄×Dic₁₀₆	—	2	2	no	—
43	C₄×Dic₁₀₆	C₂×C₄	1	2	yes	—
44	C₄×Dic₁₀₆	C₂×C₄	1	2	yes	—
45	C₄×Dic₁₀₆	C₂³	1	2	yes	—
46	C₄×Dic₁₀₆	C₂³	1	2	yes	—
47	C₂×C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
48	C₂×#11719919	—	2	2	no	—
49	C₂×C₄×Dic₅₃	—	2	2	no	—
50	#13133151	C₂×C₄	1	2	yes	—
51	#13133151	C₂×C₄	1	1	yes	characteristic
52	#13133151	C₂×C₄	1	1	yes	characteristic
53	C₄⋊#88322	—	2	1	no	—
54	#13133146	—	2	4	no	—
55	#13133146	—	2	4	no	—
56	#13133142	C₂×C₄	1	2	yes	—
57	#13133142	C₂×C₄	1	2	yes	—
58	#13133142	C₂×C₄	1	2	yes	—
59	#13133142	C₂×C₄	1	2	yes	—
60	C₂²⋊#88322	—	2	1	no	—
61	#13133128	D₄	1	2	yes	—
62	#13133127	—	2	4	no	—
63	C₄×#88322	C₂³	1	2	yes	—
64	C₄×#88322	C₂³	1	2	yes	—
65	C₈×Dic₅₃	—	2	4	no	—
66	C₂×M₁₆×C₅₃	—	2	2	no	—
67	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
68	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
69	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
70	C₄⋊C₈×C₅₃	—	2	2	no	—
71	C₄⋊C₈×C₅₃	C₂³	1	4	yes	—
72	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
73	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
74	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
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	—	4	4	no	—
87	C₂×C₈×C₅₃	—	2	2	no	—
88	C₂×C₈×C₅₃	—	2	4	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	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
95	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₅₃	#41	1	1	yes	characteristic
97	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
98	C₄⋊Dic₅₃	—	2	2	no	—
99	C₄⋊Dic₅₃	C₂²×C₄	1	2	yes	—
100	C₄⋊Dic₅₃	C₂²×C₄	1	2	yes	—
101	#11719919	—	2	2	no	—
102	#11719919	—	2	2	no	—
103	#11719919	—	2	2	no	—
104	#11719919	—	2	2	no	—
105	C₄×Dic₅₃	—	2	2	no	—
106	C₄×Dic₅₃	—	2	2	no	—
107	C₄×Dic₅₃	—	2	2	no	—
108	C₄×Dic₅₃	—	2	2	no	—
109	#11719917	—	2	2	no	—
110	#11719917	Q₁₆	1	4	yes	complemented
111	C₂×#88322	—	2	2	no	—
112	C₂×#88322	C₂×D₄	1	2	yes	—
113	C₂×#88322	C₂×D₄	1	2	yes	—
114	Dic₂₁₂	—	2	4	no	—
115	Dic₂₁₂	—	2	2	no	—
116	Dic₂₁₂	M₁₆	1	4	yes	complemented
117	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
118	C₂×C₄×C₅₃	—	2	1	no	—
119	C₂×C₄×C₅₃	—	2	1	no	—
120	C₂×C₄×C₅₃	—	4	1	no	—
121	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
125	C₂×C₄×C₅₃	#135	1	2	yes	—
126	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
127	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
128	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
129	C₂×Dic₅₃	—	2	2	no	—
130	C₂×Dic₅₃	—	2	2	no	—
131	C₂×Dic₅₃	—	2	2	no	—
132	C₂×Dic₅₃	—	2	2	no	—
133	C₂×Dic₅₃	—	4	4	no	—
134	C₂×Dic₅₃	—	8	2	no	—
135	Dic₁₀₆	—	2	2	no	—
136	Dic₁₀₆	—	2	2	no	—
137	Dic₁₀₆	—	2	2	no	—
138	Dic₁₀₆	—	4	4	no	—
139	Dic₁₀₆	C₂×M₁₆	1	4	yes	—
140	C₈×C₅₃	—	2	1	no	—
141	C₈×C₅₃	—	2	1	no	—
142	C₈×C₅₃	—	2	1	no	—
143	C₈×C₅₃	—	4	4	no	—
144	C₈×C₅₃	C₂×M₁₆	1	2	yes	—
145	#88322	—	2	2	no	—
146	#88322	—	4	2	no	—
147	#88322	C₂×Q₁₆	1	4	yes	complemented
148	M₁₆×Q₁₆	—	53	1	no	maximal, Sylow
149	C₂²×C₅₃	—	2	1	no	—
150	C₂²×C₅₃	—	2	1	no	—
151	C₂²×C₅₃	C₄×Q₁₆	1	2	yes	—
152	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
153	C₄×C₅₃	—	2	1	no	—
154	C₄×C₅₃	—	2	1	no	—
155	C₄×C₅₃	—	2	1	no	—
156	C₄×C₅₃	C₄×Q₁₆	1	2	yes	—
157	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
158	C₄×C₅₃	#568	1	1	yes	characteristic
159	C₄×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
160	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
161	Dic₅₃	—	2	4	no	—
162	Dic₅₃	—	4	2	no	—
163	Dic₅₃	—	4	2	no	—
164	Dic₅₃	—	4	2	no	—
165	Q₈×M₁₆	—	53	2	no	—
166	C₂×C₄×Q₁₆	—	53	1	no	—
167	#1591	—	53	2	no	—
168	#1584	—	53	4	no	—
169	C₈×Q₁₆	—	53	2	no	—
170	#1567	D₅₃	1	1	yes	characteristic
171	#1487	—	53	2	no	—
172	C₈×M₁₆	—	53	1	no	—
173	C₂×C₅₃	—	2	2	no	—
174	C₂×C₅₃	D₄×M₁₆	1	1	yes	characteristic
175	C₂×C₅₃	C₂×C₄×Q₁₆	1	1	yes	characteristic
176	C₂×C₅₃	#2957	1	1	yes	characteristic
177	C₂²×Q₁₆	—	53	1	no	—
178	C₂×C₄×Q₈	—	53	2	no	—
179	#446	—	53	4	no	—
180	#446	—	106	4	no	—
181	C₈×Q₈	—	53	4	no	—
182	C₄×Q₁₆	—	53	1	no	—
183	C₄×Q₁₆	—	53	2	no	—
184	C₄×Q₁₆	—	53	1	no	—
185	C₄×Q₁₆	—	53	1	no	—
186	C₄×Q₁₆	—	53	2	no	—
187	C₄×Q₁₆	—	53	1	no	—
188	#431	—	53	1	no	—
189	C₂×#107	—	53	1	no	—
190	#423	—	106	2	no	—
191	#423	C₂×D₅₃	1	2	yes	—
192	C₂×#103	—	53	2	no	—
193	C₄×M₁₆	—	53	1	no	—
194	C₄×M₁₆	—	106	2	no	—
195	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
196	#335	C₂×D₅₃	1	2	yes	—
197	#335	C₂×D₅₃	1	2	yes	—
198	Q₈⋊C₈	—	53	4	no	—
199	Q₈⋊C₈	—	53	4	no	—
200	#322	—	53	2	no	—
201	C₈²	—	53	2	no	—
202	C₅₃	M₁₆×Q₁₆	1	1	yes	characteristic, complemented, Sylow
203	C₂²×Q₈	—	53	2	no	—
204	C₂×Q₁₆	—	53	1	no	—
205	C₂×Q₁₆	—	53	2	no	—
206	C₂×Q₁₆	—	53	1	no	—
207	C₂×Q₁₆	—	106	4	no	—
208	C₂×M₁₆	—	2	2	no	—
209	C₂×M₁₆	—	53	1	no	—
210	C₂×M₁₆	—	53	1	no	—
211	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
212	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
213	C₄×Q₈	—	53	2	no	—
214	C₄×Q₈	—	53	2	no	—
215	C₄×Q₈	—	53	2	no	—
216	C₄×Q₈	—	53	2	no	—
217	C₄×Q₈	—	106	2	no	—
218	C₄×Q₈	—	106	2	no	—
219	C₂×C₄⋊C₄	—	53	2	no	—
220	C₂×C₄⋊C₄	—	106	2	no	—
221	C₂×C₄²	—	106	2	no	—
222	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
223	#107	—	53	2	no	—
224	#107	—	53	1	no	—
225	#107	—	53	1	no	—
226	C₄⋊C₈	—	2	2	no	—
227	C₄⋊C₈	—	106	4	no	—
228	C₄⋊C₈	—	106	4	no	—
229	C₄⋊C₈	—	106	1	no	—
230	C₄⋊C₈	C₂²×D₅₃	1	4	yes	—
231	#103	—	53	2	no	—
232	#103	—	53	2	no	—
233	#103	—	53	2	no	—
234	#103	—	53	2	no	—
235	C₂²⋊C₈	—	106	1	no	—
236	#97	—	53	2	no	—
237	#97	—	106	4	no	—
238	C₄×C₈	—	53	2	no	—
239	C₄×C₈	—	53	2	no	—
240	C₄×C₈	—	106	4	no	—
241	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
242	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
243	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
244	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
245	C₂×Q₈	—	53	2	no	—
246	C₂×Q₈	—	53	2	no	—
247	C₂×Q₈	—	106	4	no	—
248	C₂×Q₈	—	106	2	no	—
249	C₂×Q₈	—	212	2	no	—
250	C₂²×C₄	—	106	2	no	—
251	C₂²×C₄	—	106	2	no	—
252	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
253	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
254	C₂²×C₄	#11719949	1	1	yes	characteristic
255	Q₁₆	—	53	4	no	—
256	Q₁₆	—	106	4	no	—
257	Q₁₆	—	106	2	no	—
258	M₁₆	—	4	4	no	—
259	M₁₆	—	53	4	no	—
260	M₁₆	—	106	2	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	2	4	no	—
263	C₂×C₈	—	4	1	no	—
264	C₂×C₈	—	53	2	no	—
265	C₂×C₈	—	53	2	no	—
266	C₂×C₈	—	106	2	no	—
267	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
268	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
269	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
270	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
271	C₄⋊C₄	—	53	2	no	—
272	C₄⋊C₄	—	53	2	no	—
273	C₄⋊C₄	—	106	2	no	—
274	C₄⋊C₄	—	106	2	no	—
275	C₄⋊C₄	—	106	2	no	—
276	C₄⋊C₄	—	106	2	no	—
277	C₄⋊C₄	—	106	2	no	—
278	C₄²	—	106	2	no	—
279	C₄²	—	106	2	no	—
280	C₄²	—	106	2	no	—
281	C₄²	—	106	2	no	—
282	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
283	C₄²	D₄×D₅₃	1	1	yes	characteristic
284	C₄²	#11719949	1	1	yes	characteristic
285	C₄²	C₂³×D₅₃	1	1	yes	characteristic
286	C₂³	#13133179	1	1	yes	characteristic
287	Q₈	—	53	4	no	—
288	Q₈	—	106	2	no	—
289	Q₈	—	106	2	no	—
290	Q₈	—	106	2	no	—
291	Q₈	—	212	4	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	106	2	no	—
296	C₂×C₄	—	106	2	no	—
297	C₂×C₄	—	106	2	no	—
298	C₂×C₄	—	106	2	no	—
299	C₂×C₄	—	424	2	no	—
300	C₂×C₄	—	212	4	no	—
301	C₂×C₄	#13133179	1	1	yes	characteristic
302	C₂×C₄	#13133179	1	1	yes	characteristic
303	C₂×C₄	#13133179	1	1	yes	characteristic
304	C₂×C₄	Q₁₆×D₅₃	1	2	yes	—
305	C₂×C₄	#13133236	1	2	yes	—
306	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
307	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
308	C₂×C₄	C₂×#11719949	1	1	yes	characteristic
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₈	—	2	1	no	—
312	C₈	—	4	4	no	—
313	C₈	—	53	4	no	—
314	C₈	—	106	2	no	—
315	C₈	—	212	2	no	—
316	C₈	M₁₆×D₅₃	1	2	yes	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	#16810717	1	2	yes	—
320	C₂²	C₂×#13133179	1	1	yes	characteristic
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	106	4	no	—
325	C₄	—	212	2	no	—
326	C₄	—	212	2	no	—
327	C₄	—	212	2	no	—
328	C₄	#16810717	1	2	yes	—
329	C₄	C₂×M₁₆×D₅₃	1	1	yes	characteristic
330	C₄	#16811389	1	1	yes	characteristic
331	C₄	C₂×Q₁₆×D₅₃	1	1	yes	characteristic
332	C₄	C₂×#13133236	1	1	yes	characteristic
333	C₂	—	2	2	no	—
334	C₂	#20283450	1	1	yes	characteristic
335	C₂	C₂×#16810717	1	1	yes	characteristic
336	C₂	#20283631	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1567	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂³×C₅₃	C₄×D₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#1567	D₅₃
hypocenter	C₅₃	M₁₆×Q₁₆
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1567

Complements

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

N	H ≅ G/N	type
3. #20273611	C₂	semidirect
4. #20273564	C₂	semidirect
5. #20273521	C₂	semidirect
110. #11719917	Q₁₆	semidirect
116. Dic₂₁₂	M₁₆	semidirect
147. #88322	C₂×Q₁₆	semidirect
202. C₅₃	M₁₆×Q₁₆	semidirect

Maximal subgroups

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

subgroup	index
#20283666	2
C₂×C₄×Dic₂₁₂	2
#20273611	2
#20273564	2
#20273521	2
#20273143	2
C₈×#11719917	2
C₅₃×#1567	2
M₁₆×Q₁₆	53