Subgroups — #27539625

Order 13568.

602 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20285294	C₂	1	1	yes	maximal, characteristic, complemented
2	#20276988	C₂	1	1	yes	maximal, characteristic, complemented
3	#20276964	C₂	1	1	yes	maximal, characteristic, complemented
4	#20276940	C₂	1	1	yes	maximal, characteristic, complemented
5	#20276938	C₂	1	1	yes	maximal, characteristic, complemented
6	#20276799	C₂	1	4	yes	maximal, complemented
7	C₄×#13133158	C₂	1	4	yes	maximal, complemented
8	C₂×C₄×#11719908	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#3169	C₂	1	1	yes	maximal, characteristic, complemented
10	#16811173	—	2	1	no	—
11	C₂×#13133158	C₂²	1	4	yes	—
12	C₂²×#11719908	C₂²	1	2	yes	—
13	#16811028	C₂²	1	1	yes	characteristic
14	#16811000	C₂²	1	1	yes	characteristic
15	#16810970	—	2	1	no	—
16	#16810859	—	2	1	no	—
17	#16810859	C₂²	1	2	yes	—
18	C₄×D₄₂₄	—	2	1	no	—
19	C₄×D₄₂₄	C₂²	1	2	yes	—
20	#16810448	C₂²	1	4	yes	—
21	#16810424	C₂²	1	4	yes	—
22	#16810423	C₂²	1	4	yes	—
23	C₄×#11719908	C₂²	1	2	yes	—
24	C₄×#11719908	C₂²	1	1	yes	characteristic
25	C₄×#11719908	C₂²	1	1	yes	characteristic
26	C₅₃×#532	—	2	1	no	—
27	C₅₃×#500	C₂²	1	1	yes	characteristic
28	C₅₃×#493	C₂²	1	1	yes	characteristic
29	C₅₃×#486	—	2	1	no	—
30	C₅₃×#466	—	2	1	no	—
31	C₅₃×#466	C₂²	1	2	yes	—
32	C₄×D₈×C₅₃	—	2	1	no	—
33	C₄×D₈×C₅₃	C₂²	1	2	yes	—
34	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
35	C₂×#11719945	—	4	1	no	—
36	C₂×D₄₂₄	—	2	1	no	—
37	C₂×D₄₂₄	C₂³	1	2	yes	—
38	#13133217	—	2	1	no	—
39	#13133214	—	2	1	no	—
40	#13133205	—	4	1	no	—
41	#13133191	—	2	2	no	—
42	#13133191	—	4	1	no	—
43	C₄×D₂₁₂	—	2	2	no	—
44	C₄×D₂₁₂	—	4	1	no	—
45	#13133158	—	2	8	no	—
46	C₂×#11719908	—	2	4	no	—
47	C₂×#11719908	C₂³	1	2	yes	—
48	C₂×#11719908	C₂³	1	2	yes	—
49	#13133151	—	2	1	no	—
50	#13133151	C₂³	1	2	yes	—
51	#13133139	—	2	2	no	—
52	#13133139	—	2	2	no	—
53	C₂×#41×C₅₃	—	4	1	no	—
54	C₂×D₈×C₅₃	—	2	1	no	—
55	C₂×D₈×C₅₃	C₂³	1	2	yes	—
56	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
57	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
58	C₄⋊Q₈×C₅₃	—	2	1	no	—
59	#127×C₅₃	—	2	1	no	—
60	#124×C₅₃	—	4	1	no	—
61	#121×C₅₃	—	2	2	no	—
62	#121×C₅₃	—	4	1	no	—
63	C₄×D₄×C₅₃	—	2	2	no	—
64	C₄×D₄×C₅₃	—	4	1	no	—
65	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
66	#107×C₅₃	—	2	1	no	—
67	#107×C₅₃	C₂³	1	2	yes	—
68	#102×C₅₃	—	2	2	no	—
69	#102×C₅₃	—	2	2	no	—
70	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
71	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
72	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
73	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
74	#41×C₅₃	—	8	4	no	—
75	C₂×Q₈×C₅₃	—	4	1	no	—
76	C₂×D₄×C₅₃	—	2	2	no	—
77	C₂×D₄×C₅₃	—	4	2	no	—
78	C₂×D₄×C₅₃	—	4	1	no	—
79	C₂×D₄×C₅₃	—	4	1	no	—
80	C₂²×C₄×C₅₃	—	4	2	no	—
81	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
82	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
83	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
84	C₂×C₂²⋊D₅₃	—	4	2	no	—
85	#11719945	—	8	4	no	—
86	C₂×D₂₁₂	—	2	2	no	—
87	C₂×D₂₁₂	—	4	1	no	—
88	C₂×D₂₁₂	—	4	1	no	—
89	C₂×C₄×D₅₃	—	4	2	no	—
90	C₂×Dic₁₀₆	—	4	1	no	—
91	D₈×C₅₃	—	2	4	no	—
92	D₈×C₅₃	—	4	2	no	—
93	C₂×C₈×C₅₃	—	2	2	no	—
94	C₂×C₈×C₅₃	—	2	2	no	—
95	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
99	C₄⋊C₄×C₅₃	—	2	2	no	—
100	C₄⋊C₄×C₅₃	—	4	1	no	—
101	C₂²⋊C₄×C₅₃	—	4	2	no	—
102	C₂²⋊C₄×C₅₃	—	4	2	no	—
103	C₄²×C₅₃	D₈	1	1	yes	characteristic
104	C₄²×C₅₃	D₈	1	1	yes	characteristic
105	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
106	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
107	#11719921	—	4	2	no	—
108	#11719921	—	4	2	no	—
109	C₄⋊Dic₅₃	—	2	2	no	—
110	C₄⋊Dic₅₃	—	4	1	no	—
111	D₄₂₄	—	2	4	no	—
112	D₄₂₄	—	4	2	no	—
113	#11719908	—	2	2	no	—
114	#11719908	#41	1	4	yes	complemented
115	C₂³×C₅₃	—	4	2	no	—
116	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
117	C₂²×D₅₃	—	4	2	no	—
118	Q₈×C₅₃	—	8	2	no	—
119	D₄×C₅₃	—	4	2	no	—
120	D₄×C₅₃	—	4	2	no	—
121	D₄×C₅₃	—	8	4	no	—
122	D₄×C₅₃	—	8	2	no	—
123	D₄×C₅₃	—	8	2	no	—
124	C₂×C₄×C₅₃	—	2	1	no	—
125	C₂×C₄×C₅₃	—	2	1	no	—
126	C₂×C₄×C₅₃	—	2	2	no	—
127	C₂×C₄×C₅₃	—	4	2	no	—
128	C₂×C₄×C₅₃	—	8	4	no	—
129	C₂×C₄×C₅₃	D₁₆	1	4	yes	complemented
130	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
131	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
132	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
136	C₂²⋊D₅₃	—	8	4	no	—
137	C₂×Dic₅₃	—	4	2	no	—
138	D₂₁₂	—	4	2	no	—
139	D₂₁₂	—	4	2	no	—
140	D₂₁₂	—	8	2	no	—
141	D₂₁₂	—	8	2	no	—
142	C₄×D₅₃	—	8	4	no	—
143	Dic₁₀₆	—	8	2	no	—
144	C₈×C₅₃	—	2	1	no	—
145	C₈×C₅₃	—	2	1	no	—
146	C₈×C₅₃	#135	1	2	yes	—
147	C₈×C₅₃	C₂×#41	1	2	yes	—
148	#33733	—	53	1	no	maximal, Sylow
149	C₂²×C₅₃	—	2	1	no	—
150	C₂²×C₅₃	—	2	1	no	—
151	C₂²×C₅₃	—	8	2	no	—
152	C₂²×C₅₃	—	8	2	no	—
153	C₂²×C₅₃	—	8	2	no	—
154	C₂²×C₅₃	C₂×D₁₆	1	2	yes	—
155	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
156	C₂×D₅₃	—	8	2	no	—
157	C₂×D₅₃	—	8	2	no	—
158	C₂×D₅₃	—	8	2	no	—
159	C₄×C₅₃	—	2	1	no	—
160	C₄×C₅₃	—	2	1	no	—
161	C₄×C₅₃	—	8	2	no	—
162	C₄×C₅₃	C₂×D₁₆	1	2	yes	—
163	C₄×C₅₃	C₂×D₁₆	1	2	yes	—
164	C₄×C₅₃	#508	1	2	yes	—
165	C₄×C₅₃	#532	1	1	yes	characteristic
166	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
167	Dic₅₃	—	8	2	no	—
168	#3169	—	53	1	no	—
169	#3169	D₅₃	1	1	yes	characteristic, complemented
170	#2252	—	53	1	no	—
171	#2246	—	53	1	no	—
172	#2240	—	53	1	no	—
173	#2240	—	53	1	no	—
174	#2215	—	53	4	no	—
175	C₄×D₁₆	—	53	4	no	—
176	C₂×C₄×C₁₆	—	53	1	no	—
177	C₂×C₅₃	—	2	2	no	—
178	C₂×C₅₃	—	16	2	no	—
179	C₂×C₅₃	#3169	1	1	yes	characteristic
180	C₂×C₅₃	C₂²×D₁₆	1	1	yes	characteristic
181	C₂×C₅₃	C₂×#508	1	1	yes	characteristic
182	D₅₃	—	16	2	no	—
183	#532	—	2	1	no	—
184	#532	—	106	1	no	—
185	C₂×D₁₆	—	53	4	no	—
186	C₂²×C₁₆	—	53	2	no	—
187	#500	—	53	1	no	—
188	#500	C₂×D₅₃	1	1	yes	characteristic
189	#493	—	53	1	no	—
190	#493	C₂×D₅₃	1	1	yes	characteristic
191	#486	—	2	1	no	—
192	#486	—	106	1	no	—
193	#466	—	2	1	no	—
194	#466	—	53	2	no	—
195	#466	—	106	1	no	—
196	#466	C₂×D₅₃	1	2	yes	—
197	C₄×D₈	—	2	1	no	—
198	C₄×D₈	—	53	2	no	—
199	C₄×D₈	—	106	1	no	—
200	C₄×D₈	C₂×D₅₃	1	2	yes	—
201	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
202	#366	—	53	4	no	—
203	#357	—	53	4	no	—
204	#357	—	53	4	no	—
205	C₄×C₁₆	—	53	2	no	—
206	C₄×C₁₆	—	53	1	no	—
207	C₄×C₁₆	—	53	1	no	—
208	C₅₃	#33733	1	1	yes	characteristic, complemented, Sylow
209	C₂×#41	—	4	1	no	—
210	C₂×#41	—	212	1	no	—
211	C₂×D₈	—	2	1	no	—
212	C₂×D₈	—	53	2	no	—
213	C₂×D₈	—	106	1	no	—
214	C₂×D₈	C₂²×D₅₃	1	2	yes	—
215	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
216	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
217	C₄⋊Q₈	—	2	1	no	—
218	C₄⋊Q₈	—	106	1	no	—
219	#127	—	2	1	no	—
220	#127	—	106	1	no	—
221	#124	—	4	1	no	—
222	#124	—	212	1	no	—
223	#121	—	2	2	no	—
224	#121	—	4	1	no	—
225	#121	—	106	2	no	—
226	#121	—	212	1	no	—
227	C₄×D₄	—	2	2	no	—
228	C₄×D₄	—	4	1	no	—
229	C₄×D₄	—	106	2	no	—
230	C₄×D₄	—	212	1	no	—
231	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
232	D₁₆	—	106	8	no	—
233	C₂×C₁₆	—	53	2	no	—
234	C₂×C₁₆	—	53	2	no	—
235	C₂×C₁₆	—	106	4	no	—
236	#107	—	2	1	no	—
237	#107	—	53	2	no	—
238	#107	—	106	1	no	—
239	#107	C₂²×D₅₃	1	2	yes	—
240	#102	—	2	2	no	—
241	#102	—	2	2	no	—
242	#102	—	106	2	no	—
243	#102	—	106	2	no	—
244	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
245	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
246	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
247	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
248	#41	—	8	4	no	—
249	#41	—	424	4	no	—
250	C₂×Q₈	—	4	1	no	—
251	C₂×Q₈	—	212	1	no	—
252	C₂×D₄	—	2	2	no	—
253	C₂×D₄	—	4	2	no	—
254	C₂×D₄	—	4	1	no	—
255	C₂×D₄	—	4	1	no	—
256	C₂×D₄	—	106	2	no	—
257	C₂×D₄	—	212	2	no	—
258	C₂×D₄	—	212	1	no	—
259	C₂×D₄	—	212	1	no	—
260	C₂²×C₄	—	4	2	no	—
261	C₂²×C₄	—	212	2	no	—
262	C₂²×C₄	#11719922	1	1	yes	characteristic
263	C₂²×C₄	#11719922	1	1	yes	characteristic
264	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
265	D₈	—	2	4	no	—
266	D₈	—	4	2	no	—
267	D₈	—	106	4	no	—
268	D₈	—	212	2	no	—
269	C₂×C₈	—	2	2	no	—
270	C₂×C₈	—	2	2	no	—
271	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
272	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
273	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
274	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
275	C₄⋊C₄	—	2	2	no	—
276	C₄⋊C₄	—	4	1	no	—
277	C₄⋊C₄	—	106	2	no	—
278	C₄⋊C₄	—	212	1	no	—
279	C₂²⋊C₄	—	4	2	no	—
280	C₂²⋊C₄	—	4	2	no	—
281	C₂²⋊C₄	—	212	2	no	—
282	C₂²⋊C₄	—	212	2	no	—
283	C₄²	#11719922	1	1	yes	characteristic
284	C₄²	#11719922	1	1	yes	characteristic
285	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
286	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
287	C₁₆	—	53	4	no	—
288	C₁₆	—	106	2	no	—
289	C₂³	—	4	2	no	—
290	C₂³	—	212	2	no	—
291	C₂³	C₂×#11719922	1	1	yes	characteristic
292	Q₈	—	8	2	no	—
293	Q₈	—	424	2	no	—
294	D₄	—	4	2	no	—
295	D₄	—	4	2	no	—
296	D₄	—	8	4	no	—
297	D₄	—	8	2	no	—
298	D₄	—	8	2	no	—
299	D₄	—	424	4	no	—
300	D₄	—	424	2	no	—
301	D₄	—	424	2	no	—
302	D₄	—	212	2	no	—
303	D₄	—	212	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	8	4	no	—
309	C₂×C₄	—	424	4	no	—
310	C₂×C₄	—	212	2	no	—
311	C₂×C₄	#13133158	1	4	yes	complemented
312	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
313	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
314	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
315	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
316	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
317	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
318	C₈	—	2	1	no	—
319	C₈	—	2	1	no	—
320	C₈	#13133237	1	2	yes	—
321	C₈	#41×D₅₃	1	2	yes	—
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	8	2	no	—
325	C₂²	—	8	2	no	—
326	C₂²	—	8	2	no	—
327	C₂²	—	424	2	no	—
328	C₂²	—	424	2	no	—
329	C₂²	—	424	2	no	—
330	C₂²	C₂×#13133158	1	2	yes	—
331	C₂²	C₂²×#11719922	1	1	yes	characteristic
332	C₄	—	2	1	no	—
333	C₄	—	2	1	no	—
334	C₄	—	8	2	no	—
335	C₄	—	424	2	no	—
336	C₄	C₂×#13133158	1	2	yes	—
337	C₄	C₂×#13133158	1	2	yes	—
338	C₄	#16811053	1	2	yes	—
339	C₄	#16811169	1	1	yes	characteristic
340	C₄	C₂×#13133237	1	1	yes	characteristic
341	C₂	—	2	2	no	—
342	C₂	—	16	2	no	—
343	C₂	—	848	2	no	—
344	C₂	#20285290	1	1	yes	characteristic
345	C₂	C₂²×#13133158	1	1	yes	characteristic
346	C₂	C₂×#16811053	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#11719922
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#3169	C₂
socle	C₂²×C₅₃	C₂²×D₈
hypercenter	#3169	D₅₃
hypocenter	C₅₃	#33733
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3169

Complements

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

N	H ≅ G/N	type
1. #20285294	C₂	semidirect
2. #20276988	C₂	semidirect
3. #20276964	C₂	semidirect
4. #20276940	C₂	semidirect
5. #20276938	C₂	semidirect
6. #20276799	C₂	semidirect
7. C₄×#13133158	C₂	semidirect
8. C₂×C₄×#11719908	C₂	semidirect
9. C₅₃×#3169	C₂	semidirect
114. #11719908	#41	semidirect
129. C₂×C₄×C₅₃	D₁₆	semidirect
169. #3169	D₅₃	semidirect
208. C₅₃	#33733	semidirect
311. C₂×C₄	#13133158	semidirect

Maximal subgroups

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

subgroup	index
#20285294	2
#20276988	2
#20276964	2
#20276940	2
#20276938	2
#20276799	2
C₄×#13133158	2
C₂×C₄×#11719908	2
C₅₃×#3169	2
#33733	53