Subgroups — #27538797

Order 13568.

534 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	#20283592	C₂	1	1	yes	maximal, characteristic
2	#20276876	C₂	1	2	yes	maximal, complemented
3	#20276869	C₂	1	2	yes	maximal, complemented
4	#20276864	C₂	1	2	yes	maximal, complemented
5	#20276849	C₂	1	2	yes	maximal, complemented
6	#20276504	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#16810448	C₂	1	1	yes	maximal, characteristic, complemented
8	#20276474	C₂	1	1	yes	maximal, characteristic, complemented
9	#20276425	C₂	1	1	yes	maximal, characteristic
10	#20276420	C₂	1	1	yes	maximal, characteristic
11	C₅₃×#3164	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13133239	C₂²	1	1	yes	characteristic
13	#16811103	—	2	1	no	—
14	C₂×#13133157	C₂²	1	1	yes	characteristic
15	C₂²×#11719908	C₂²	1	1	yes	characteristic
16	#16810725	—	2	2	no	—
17	C₄×Dic₂₁₂	C₂²	1	2	yes	—
18	C₄×D₄₂₄	C₂²	1	2	yes	—
19	C₂×#13133151	C₂²	1	1	yes	characteristic
20	#16810625	—	2	1	no	—
21	#16810450	C₂²	1	2	yes	—
22	#16810448	C₂²	1	2	yes	—
23	#16810448	C₂²	1	2	yes	—
24	#16810448	C₂²	1	2	yes	—
25	C₄⋊#11719908	C₂²	1	2	yes	—
26	C₄⋊#11719908	C₂²	1	2	yes	—
27	#16810426	C₂²	1	1	yes	characteristic
28	#16810426	C₂²	1	2	yes	—
29	#16810426	C₂²	1	1	yes	characteristic
30	#16810423	C₂²	1	1	yes	characteristic
31	#16810423	C₂²	1	2	yes	—
32	#16810423	C₂²	1	1	yes	characteristic
33	C₅₃×#544	—	2	1	no	—
34	C₅₃×#501	—	2	2	no	—
35	C₅₃×#500	C₂²	1	2	yes	—
36	C₅₃×#499	C₂²	1	2	yes	—
37	C₅₃×#428	—	2	1	no	—
38	C₅₃×#427	C₂²	1	1	yes	characteristic
39	C₂×#107×C₅₃	C₂²	1	1	yes	characteristic
40	C₅₃×#405	C₂²	1	1	yes	characteristic
41	C₂×#11719945	—	2	1	no	—
42	#13133239	—	2	2	no	—
43	#13133239	Q₈	1	4	yes	complemented
44	C₂×Dic₂₁₂	C₂³	1	1	yes	characteristic
45	C₂×#11719913	—	2	1	no	—
46	C₂×D₄₂₄	C₂³	1	1	yes	characteristic
47	C₄×Dic₁₀₆	—	2	2	no	—
48	C₄×D₂₁₂	—	2	2	no	—
49	#13133177	—	4	2	no	—
50	#13133175	—	4	1	no	—
51	C₂×C₄⋊Dic₅₃	—	2	1	no	—
52	C₄⋊C₄×D₅₃	—	4	1	no	—
53	#13133157	—	2	2	no	—
54	C₂×#11719908	—	2	2	no	—
55	C₂×#11719908	C₂³	1	1	yes	characteristic
56	C₂×#11719908	C₂³	1	1	yes	characteristic
57	C₂×#11719908	C₂³	1	2	yes	—
58	#13133151	—	2	1	no	—
59	#13133151	C₂³	1	2	yes	—
60	#13133142	—	2	2	no	—
61	#13133139	—	2	2	no	—
62	C₂×M₁₆×C₅₃	D₄	1	1	yes	characteristic
63	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
64	C₄⋊Q₈×C₅₃	—	2	2	no	—
65	C₄⋊Q₈×C₅₃	—	4	1	no	—
66	#125×C₅₃	—	2	2	no	—
67	#125×C₅₃	—	4	1	no	—
68	C₂²⋊Q₈×C₅₃	—	4	2	no	—
69	#117×C₅₃	—	2	1	no	—
70	#117×C₅₃	D₄	1	1	yes	characteristic
71	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
72	#107×C₅₃	—	2	2	no	—
73	#107×C₅₃	—	2	1	no	—
74	#107×C₅₃	C₂³	1	1	yes	characteristic
75	#107×C₅₃	C₂³	1	2	yes	—
76	#107×C₅₃	C₂³	1	1	yes	characteristic
77	#106×C₅₃	—	2	2	no	—
78	#106×C₅₃	—	2	1	no	—
79	#97×C₅₃	D₄	1	2	yes	—
80	C₄×C₈×C₅₃	C₂³	1	2	yes	—
81	C₂×Q₈×C₅₃	—	4	2	no	—
82	C₂²×C₄×C₅₃	—	4	1	no	—
83	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
84	C₂×C₂²⋊D₅₃	—	4	1	no	—
85	C₂²×Dic₅₃	—	4	1	no	—
86	#11719945	—	2	4	no	—
87	#11719945	—	4	2	no	—
88	C₂×D₂₁₂	—	2	1	no	—
89	C₂×C₄×D₅₃	—	4	1	no	—
90	C₂×C₄×D₅₃	—	4	2	no	—
91	C₂×Dic₁₀₆	—	2	1	no	—
92	M₁₆×C₅₃	—	2	2	no	—
93	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
94	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
96	C₂×C₈×C₅₃	#41	1	2	yes	—
97	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
98	C₄⋊C₄×C₅₃	—	2	1	no	—
99	C₄⋊C₄×C₅₃	—	2	2	no	—
100	C₄⋊C₄×C₅₃	—	2	1	no	—
101	C₄⋊C₄×C₅₃	—	4	1	no	—
102	C₄⋊C₄×C₅₃	—	4	2	no	—
103	C₄⋊C₄×C₅₃	—	4	2	no	—
104	C₄⋊C₄×C₅₃	—	4	2	no	—
105	C₄⋊C₄×C₅₃	D₈	1	1	yes	characteristic
106	C₄⋊C₄×C₅₃	D₈	1	1	yes	characteristic
107	C₂²⋊C₄×C₅₃	—	4	1	no	—
108	C₂²⋊C₄×C₅₃	D₈	1	2	yes	—
109	C₄²×C₅₃	—	4	1	no	—
110	C₄²×C₅₃	C₂×D₄	1	2	yes	—
111	#11719921	—	4	2	no	—
112	C₄⋊Dic₅₃	—	2	2	no	—
113	C₄⋊Dic₅₃	—	4	1	no	—
114	#11719919	—	4	2	no	—
115	C₄×Dic₅₃	—	4	2	no	—
116	Dic₂₁₂	—	2	1	no	—
117	Dic₂₁₂	C₂×Q₈	1	2	yes	—
118	D₄₂₄	—	2	1	no	—
119	D₄₂₄	C₂×Q₈	1	2	yes	—
120	#11719913	—	2	4	no	—
121	#11719908	—	2	1	no	—
122	#11719908	—	2	1	no	—
123	#11719908	—	2	2	no	—
124	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
125	C₂²×D₅₃	—	4	1	no	—
126	Q₈×C₅₃	—	16	2	no	—
127	C₂×C₄×C₅₃	—	4	2	no	—
128	C₂×C₄×C₅₃	—	4	1	no	—
129	C₂×C₄×C₅₃	—	4	1	no	—
130	C₂×C₄×C₅₃	—	16	1	no	—
131	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
132	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
133	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
134	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₅₃	#137	1	2	yes	—
136	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
137	C₂²⋊D₅₃	—	4	4	no	—
138	C₂×Dic₅₃	—	4	1	no	—
139	C₂×Dic₅₃	—	4	2	no	—
140	C₂×Dic₅₃	—	16	1	no	—
141	D₂₁₂	—	2	2	no	—
142	D₂₁₂	—	4	1	no	—
143	C₄×D₅₃	—	4	4	no	—
144	C₄×D₅₃	—	16	2	no	—
145	Dic₁₀₆	—	2	2	no	—
146	Dic₁₀₆	—	4	1	no	—
147	C₈×C₅₃	—	2	2	no	—
148	C₈×C₅₃	C₂²⋊Q₈	1	2	yes	—
149	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
150	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
151	#33633	—	53	1	no	maximal, Sylow
152	C₂²×C₅₃	—	2	1	no	—
153	C₂²×C₅₃	—	2	1	no	—
154	C₂²×C₅₃	#474	1	2	yes	—
155	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
156	C₂×D₅₃	—	4	2	no	—
157	C₂×D₅₃	—	8	1	no	—
158	C₂×D₅₃	—	8	1	no	—
159	C₄×C₅₃	—	2	2	no	—
160	C₄×C₅₃	—	2	2	no	—
161	C₄×C₅₃	—	8	2	no	—
162	C₄×C₅₃	—	8	1	no	—
163	C₄×C₅₃	—	8	1	no	—
164	C₄×C₅₃	#474	1	2	yes	—
165	C₄×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
166	C₄×C₅₃	C₂×#137	1	1	yes	characteristic
167	Dic₅₃	—	4	2	no	—
168	Dic₅₃	—	8	2	no	—
169	#3164	D₅₃	1	1	yes	characteristic, complemented
170	#2951	—	53	1	no	—
171	#2229	—	53	2	no	—
172	#2228	—	53	2	no	—
173	#2227	—	53	2	no	—
174	#2224	—	53	2	no	—
175	#2159	—	53	1	no	—
176	C₂×#366	—	53	1	no	—
177	#2151	—	53	1	no	—
178	#2139	—	53	1	no	—
179	#2138	—	53	1	no	—
180	C₂×C₅₃	—	2	2	no	—
181	C₂×C₅₃	C₂×#474	1	1	yes	characteristic
182	C₂×C₅₃	#3415	1	1	yes	characteristic
183	C₂×C₅₃	#3417	1	1	yes	characteristic
184	D₅₃	—	8	2	no	—
185	C₂×#135	—	53	1	no	—
186	#544	—	2	1	no	—
187	#520	—	106	1	no	—
188	C₂×M₃₂	—	53	1	no	—
189	C₂²×C₁₆	—	53	1	no	—
190	#501	—	2	2	no	—
191	#500	C₂×D₅₃	1	2	yes	—
192	#499	C₂×D₅₃	1	2	yes	—
193	#440	—	106	2	no	—
194	C₄×Q₁₆	—	53	2	no	—
195	C₄×D₈	—	53	2	no	—
196	#428	—	2	1	no	—
197	#427	C₂×D₅₃	1	1	yes	characteristic
198	C₂×#107	—	53	1	no	—
199	C₂×#107	C₂×D₅₃	1	1	yes	characteristic
200	#417	—	106	1	no	—
201	#405	C₂×D₅₃	1	1	yes	characteristic
202	#367	—	53	2	no	—
203	#366	—	53	2	no	—
204	#366	—	53	2	no	—
205	#366	—	53	2	no	—
206	C₄⋊C₁₆	—	53	2	no	—
207	C₄⋊C₁₆	—	53	2	no	—
208	#358	—	53	1	no	—
209	#358	—	53	1	no	—
210	#358	—	53	2	no	—
211	#357	—	53	1	no	—
212	#357	—	53	1	no	—
213	#357	—	53	2	no	—
214	C₅₃	#33633	1	1	yes	characteristic, complemented, Sylow
215	C₂×#41	—	106	1	no	—
216	#135	—	53	4	no	—
217	#135	—	106	2	no	—
218	C₂×Q₁₆	—	53	1	no	—
219	C₂×SD₁₆	—	106	1	no	—
220	C₂×D₈	—	53	1	no	—
221	C₂×M₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
222	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
223	C₄⋊Q₈	—	2	2	no	—
224	C₄⋊Q₈	—	4	1	no	—
225	#125	—	2	2	no	—
226	#125	—	4	1	no	—
227	C₂²⋊Q₈	—	4	2	no	—
228	C₄×Q₈	—	106	2	no	—
229	C₄×D₄	—	106	2	no	—
230	C₄×D₄	—	212	2	no	—
231	#117	—	2	1	no	—
232	#117	—	212	1	no	—
233	#117	C₂²⋊D₅₃	1	1	yes	characteristic
234	C₂×C₄⋊C₄	—	2	1	no	—
235	C₂×C₄⋊C₄	—	106	1	no	—
236	C₂×C₄⋊C₄	—	212	1	no	—
237	M₃₂	—	106	2	no	—
238	C₂×C₁₆	—	53	1	no	—
239	C₂×C₁₆	—	53	1	no	—
240	C₂×C₁₆	—	53	2	no	—
241	C₂×C₁₆	—	106	2	no	—
242	#107	—	2	1	no	—
243	#107	—	2	2	no	—
244	#107	—	53	2	no	—
245	#107	—	106	1	no	—
246	#107	C₂²×D₅₃	1	2	yes	—
247	#107	C₂²×D₅₃	1	1	yes	characteristic
248	#107	C₂²×D₅₃	1	1	yes	characteristic
249	#106	—	2	1	no	—
250	#106	—	2	2	no	—
251	#103	—	106	2	no	—
252	#102	—	106	2	no	—
253	#97	C₂²⋊D₅₃	1	2	yes	—
254	C₄×C₈	C₂²×D₅₃	1	2	yes	—
255	#41	—	106	4	no	—
256	#41	—	212	2	no	—
257	C₂×Q₈	—	4	2	no	—
258	C₂×Q₈	—	106	1	no	—
259	C₂×D₄	—	106	1	no	—
260	C₂×D₄	—	212	1	no	—
261	C₂²×C₄	—	4	1	no	—
262	C₂²×C₄	—	212	1	no	—
263	C₂²×C₄	—	212	2	no	—
264	C₂²×C₄	—	212	1	no	—
265	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
266	Q₁₆	—	53	2	no	—
267	Q₁₆	—	106	1	no	—
268	SD₁₆	—	106	4	no	—
269	D₈	—	53	2	no	—
270	D₈	—	106	1	no	—
271	M₁₆	—	2	2	no	—
272	C₂×C₈	Q₈×D₅₃	1	2	yes	—
273	C₂×C₈	#11719949	1	2	yes	—
274	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
275	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
276	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
277	C₄⋊C₄	—	2	1	no	—
278	C₄⋊C₄	—	2	2	no	—
279	C₄⋊C₄	—	2	1	no	—
280	C₄⋊C₄	—	4	1	no	—
281	C₄⋊C₄	—	4	2	no	—
282	C₄⋊C₄	—	4	2	no	—
283	C₄⋊C₄	—	4	2	no	—
284	C₄⋊C₄	—	106	2	no	—
285	C₄⋊C₄	—	212	1	no	—
286	C₄⋊C₄	—	212	2	no	—
287	C₄⋊C₄	#11719922	1	1	yes	characteristic
288	C₄⋊C₄	#11719922	1	1	yes	characteristic
289	C₂²⋊C₄	—	4	1	no	—
290	C₂²⋊C₄	—	212	2	no	—
291	C₂²⋊C₄	#11719922	1	2	yes	—
292	C₄²	—	4	1	no	—
293	C₄²	—	212	2	no	—
294	C₄²	C₂×C₂²⋊D₅₃	1	2	yes	—
295	C₁₆	—	106	1	no	—
296	C₁₆	—	106	1	no	—
297	C₁₆	—	106	2	no	—
298	C₂³	—	212	1	no	—
299	C₂³	C₂×#11719922	1	1	yes	characteristic
300	Q₈	—	16	2	no	—
301	Q₈	—	106	2	no	—
302	Q₈	—	212	1	no	—
303	D₄	—	106	2	no	—
304	D₄	—	212	4	no	—
305	D₄	—	212	1	no	—
306	C₂×C₄	—	4	1	no	—
307	C₂×C₄	—	4	1	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	16	1	no	—
310	C₂×C₄	—	848	2	no	—
311	C₂×C₄	—	848	1	no	—
312	C₂×C₄	—	212	2	no	—
313	C₂×C₄	—	212	1	no	—
314	C₂×C₄	—	212	4	no	—
315	C₂×C₄	#13133193	1	2	yes	—
316	C₂×C₄	C₂×#11719922	1	2	yes	—
317	C₂×C₄	C₂×#11719922	1	2	yes	—
318	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
319	C₂×C₄	#13133248	1	2	yes	—
320	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
321	C₈	—	2	2	no	—
322	C₈	#13133193	1	2	yes	—
323	C₈	C₂×Q₈×D₅₃	1	1	yes	characteristic
324	C₈	C₂×#11719949	1	1	yes	characteristic
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	424	1	no	—
328	C₂²	—	424	1	no	—
329	C₂²	—	212	2	no	—
330	C₂²	#16810900	1	2	yes	—
331	C₂²	C₂²×#11719922	1	1	yes	characteristic
332	C₄	—	2	2	no	—
333	C₄	—	2	2	no	—
334	C₄	—	8	2	no	—
335	C₄	—	8	1	no	—
336	C₄	—	8	1	no	—
337	C₄	—	424	2	no	—
338	C₄	—	212	2	no	—
339	C₄	#16810900	1	2	yes	—
340	C₄	C₂×#13133193	1	1	yes	characteristic
341	C₄	C₂×#13133248	1	1	yes	characteristic
342	C₂	—	2	2	no	—
343	C₂	—	424	2	no	—
344	C₂	C₂×#16810900	1	1	yes	characteristic
345	C₂	#20287854	1	1	yes	characteristic
346	C₂	#20287866	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#11719922
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#3164	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂×#13133239	C₂²
hypercenter	#3164	D₅₃
hypocenter	C₅₃	#33633
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #3164

Complements

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

N	H ≅ G/N	type
2. #20276876	C₂	semidirect
3. #20276869	C₂	semidirect
4. #20276864	C₂	semidirect
5. #20276849	C₂	semidirect
6. #20276504	C₂	semidirect
7. C₂×#16810448	C₂	semidirect
8. #20276474	C₂	semidirect
11. C₅₃×#3164	C₂	semidirect
43. #13133239	Q₈	semidirect
169. #3164	D₅₃	semidirect
214. C₅₃	#33633	semidirect

Maximal subgroups

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

subgroup	index
#20283592	2
#20276876	2
#20276869	2
#20276864	2
#20276849	2
#20276504	2
C₂×#16810448	2
#20276474	2
#20276425	2
#20276420	2
C₅₃×#3164	2
#33633	53