Subgroups — #27541751

Order 13568.

486 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	#20287704	C₂	1	1	yes	maximal, characteristic, complemented
2	#20276949	C₂	1	2	yes	maximal, complemented
3	#20276944	C₂	1	1	yes	maximal, characteristic, complemented
4	#20276876	C₂	1	2	yes	maximal, complemented
5	#20276859	C₂	1	1	yes	maximal, characteristic, complemented
6	#20276770	C₂	1	2	yes	maximal
7	#20276760	C₂	1	1	yes	maximal, characteristic, complemented
8	Q₈×#11719908	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#13133161	C₂	1	2	yes	maximal, complemented
10	C₄×#13133158	C₂	1	1	yes	maximal, characteristic
11	C₅₃×#3380	C₂	1	1	yes	maximal, characteristic, complemented
12	#16811346	—	2	1	no	—
13	C₂×#13133161	C₂²	1	2	yes	—
14	C₂×#13133158	C₂²	1	1	yes	characteristic
15	#16811007	C₂²	1	2	yes	—
16	#16811003	C₂²	1	1	yes	characteristic
17	#16810846	—	2	2	no	—
18	#16810829	—	2	1	no	—
19	C₄×Dic₂₁₂	C₂²	1	2	yes	—
20	C₄×D₄₂₄	C₂²	1	1	yes	characteristic
21	#16810450	C₂²	1	2	yes	—
22	#16810448	C₂²	1	1	yes	characteristic
23	C₄⋊#11719908	C₂²	1	1	yes	characteristic
24	C₄⋊#11719908	C₂²	1	2	yes	—
25	#16810426	C₂²	1	2	yes	—
26	#16810426	C₂²	1	2	yes	—
27	#16810426	C₂²	1	2	yes	—
28	#16810424	C₂²	1	1	yes	characteristic
29	#16810423	C₂²	1	2	yes	—
30	#16810423	C₂²	1	1	yes	characteristic
31	C₄×#11719908	C₂²	1	1	yes	characteristic
32	C₄×#11719908	C₂²	1	2	yes	—
33	C₅₃×#554	—	2	1	no	—
34	C₅₃×#500	C₂²	1	1	yes	characteristic
35	C₅₃×#499	C₂²	1	2	yes	—
36	C₅₃×#478	—	2	2	no	—
37	C₅₃×#476	—	2	1	no	—
38	C₈×Q₈×C₅₃	C₂²	1	1	yes	characteristic
39	C₄×D₈×C₅₃	C₂²	1	1	yes	characteristic
40	C₄×D₈×C₅₃	C₂²	1	2	yes	—
41	C₂×Dic₂₁₂	—	2	1	no	—
42	C₂×Dic₂₁₂	C₂³	1	2	yes	—
43	C₂×#11719913	—	2	2	no	—
44	C₂×D₄₂₄	C₂³	1	1	yes	characteristic
45	#13133217	—	2	1	no	—
46	#13133212	—	4	2	no	—
47	#13133205	—	2	2	no	—
48	#13133193	—	4	1	no	—
49	C₄×Dic₁₀₆	—	2	2	no	—
50	C₄×D₂₁₂	—	2	1	no	—
51	#13133175	—	4	1	no	—
52	#13133161	—	2	4	no	—
53	#13133158	—	2	2	no	—
54	C₂×#11719908	C₂³	1	1	yes	characteristic
55	C₂×#11719908	C₂³	1	2	yes	—
56	C₂×#11719908	C₂³	1	1	yes	characteristic
57	#13133151	C₂³	1	1	yes	characteristic
58	#13133142	—	2	2	no	—
59	#13133139	—	2	1	no	—
60	C₂×D₈×C₅₃	C₂³	1	1	yes	characteristic
61	C₄⋊Q₈×C₅₃	—	2	1	no	—
62	#125×C₅₃	—	2	2	no	—
63	C₂²⋊Q₈×C₅₃	—	4	2	no	—
64	C₂²⋊Q₈×C₅₃	—	4	1	no	—
65	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
66	C₄×D₄×C₅₃	—	2	1	no	—
67	C₄×D₄×C₅₃	—	2	2	no	—
68	C₂×C₄⋊C₄×C₅₃	—	4	1	no	—
69	#107×C₅₃	—	2	1	no	—
70	#107×C₅₃	C₂³	1	1	yes	characteristic
71	#107×C₅₃	C₂³	1	2	yes	—
72	#106×C₅₃	—	2	2	no	—
73	C₄⋊C₈×C₅₃	D₄	1	1	yes	characteristic
74	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
75	#102×C₅₃	—	2	2	no	—
76	#102×C₅₃	—	2	1	no	—
77	C₄×C₈×C₅₃	C₂³	1	2	yes	—
78	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
79	C₂×Q₈×C₅₃	—	4	1	no	—
80	C₂×Q₈×C₅₃	D₈	1	1	yes	characteristic
81	C₂×D₄×C₅₃	—	2	1	no	—
82	C₂²×C₄×C₅₃	—	4	1	no	—
83	C₂²×C₄×C₅₃	—	4	2	no	—
84	C₂×D₂₁₂	—	2	1	no	—
85	C₂×C₄×D₅₃	—	4	1	no	—
86	C₂×Dic₁₀₆	—	2	2	no	—
87	D₈×C₅₃	—	2	1	no	—
88	D₈×C₅₃	#41	1	2	yes	—
89	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
90	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
92	C₄⋊C₄×C₅₃	—	2	1	no	—
93	C₄⋊C₄×C₅₃	—	2	2	no	—
94	C₄⋊C₄×C₅₃	—	4	2	no	—
95	C₄⋊C₄×C₅₃	—	4	1	no	—
96	C₄⋊C₄×C₅₃	—	4	2	no	—
97	C₄⋊C₄×C₅₃	D₈	1	1	yes	characteristic
98	C₄⋊C₄×C₅₃	D₈	1	2	yes	—
99	C₂²⋊C₄×C₅₃	—	4	2	no	—
100	C₂²⋊C₄×C₅₃	—	4	1	no	—
101	C₄²×C₅₃	C₂×D₄	1	2	yes	—
102	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
103	#11719921	—	4	2	no	—
104	#11719921	—	4	1	no	—
105	#11719921	—	4	2	no	—
106	C₄⋊Dic₅₃	—	2	1	no	—
107	C₄⋊Dic₅₃	—	4	1	no	—
108	C₄⋊Dic₅₃	—	4	1	no	—
109	#11719919	—	4	2	no	—
110	C₄×Dic₅₃	—	4	2	no	—
111	Dic₂₁₂	—	2	2	no	—
112	Dic₂₁₂	—	2	2	no	—
113	Dic₂₁₂	—	8	1	no	—
114	D₄₂₄	—	2	1	no	—
115	D₄₂₄	—	2	1	no	—
116	#11719913	—	8	2	no	—
117	#11719908	—	2	1	no	—
118	#11719908	—	2	2	no	—
119	#11719908	—	2	1	no	—
120	C₂³×C₅₃	—	4	1	no	—
121	C₂²×D₅₃	—	4	1	no	—
122	Q₈×C₅₃	—	2	2	no	—
123	Q₈×C₅₃	—	16	1	no	—
124	D₄×C₅₃	—	2	2	no	—
125	D₄×C₅₃	—	4	1	no	—
126	C₂×C₄×C₅₃	—	4	2	no	—
127	C₂×C₄×C₅₃	—	4	1	no	—
128	C₂×C₄×C₅₃	—	8	2	no	—
129	C₂×C₄×C₅₃	—	16	2	no	—
130	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
131	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
132	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
133	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
135	C₂×Dic₅₃	—	4	2	no	—
136	C₂×Dic₅₃	—	4	1	no	—
137	D₂₁₂	—	4	1	no	—
138	D₂₁₂	—	4	1	no	—
139	C₄×D₅₃	—	8	2	no	—
140	Dic₁₀₆	—	4	2	no	—
141	Dic₁₀₆	—	4	2	no	—
142	C₈×C₅₃	—	2	2	no	—
143	C₈×C₅₃	—	2	1	no	—
144	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
145	C₈×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
146	#33927	—	53	1	no	maximal, Sylow
147	C₂²×C₅₃	—	4	2	no	—
148	C₂²×C₅₃	—	8	1	no	—
149	C₂²×C₅₃	—	8	1	no	—
150	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
151	C₂×D₅₃	—	8	1	no	—
152	C₂×D₅₃	—	8	1	no	—
153	C₂×D₅₃	—	8	1	no	—
154	C₄×C₅₃	—	2	2	no	—
155	C₄×C₅₃	—	2	2	no	—
156	C₄×C₅₃	—	2	1	no	—
157	C₄×C₅₃	—	8	1	no	—
158	C₄×C₅₃	—	8	2	no	—
159	C₄×C₅₃	#508	1	2	yes	—
160	C₄×C₅₃	#548	1	1	yes	characteristic
161	C₄×C₅₃	#577	1	1	yes	characteristic
162	Dic₅₃	—	8	2	no	—
163	Dic₅₃	—	8	1	no	—
164	#3393	—	53	1	no	—
165	#3380	D₅₃	1	1	yes	characteristic, complemented
166	#2242	—	53	2	no	—
167	#2241	—	53	1	no	—
168	#2229	—	53	2	no	—
169	#2226	—	53	1	no	—
170	#2210	—	53	2	no	—
171	#2208	—	53	1	no	—
172	Q₈×C₁₆	—	53	1	no	—
173	C₄×SD₃₂	—	53	2	no	—
174	C₄×D₁₆	—	53	1	no	—
175	C₂×C₅₃	—	8	2	no	—
176	C₂×C₅₃	#3358	1	1	yes	characteristic
177	C₂×C₅₃	C₂×#508	1	1	yes	characteristic
178	C₂×C₅₃	#3416	1	1	yes	characteristic
179	D₅₃	—	16	1	no	—
180	#556	—	106	1	no	—
181	#554	—	2	1	no	—
182	C₂×SD₃₂	—	53	2	no	—
183	C₂×D₁₆	—	53	1	no	—
184	#500	C₂×D₅₃	1	1	yes	characteristic
185	#499	C₂×D₅₃	1	2	yes	—
186	#495	—	53	2	no	—
187	#494	—	53	1	no	—
188	#478	—	2	2	no	—
189	#476	—	2	1	no	—
190	#464	—	106	2	no	—
191	#461	—	106	1	no	—
192	C₈×Q₈	C₂×D₅₃	1	1	yes	characteristic
193	C₄×Q₁₆	—	53	2	no	—
194	C₄×D₈	—	53	1	no	—
195	C₄×D₈	C₂×D₅₃	1	1	yes	characteristic
196	C₄×D₈	C₂×D₅₃	1	2	yes	—
197	#367	—	53	2	no	—
198	#366	—	53	1	no	—
199	C₄⋊C₁₆	—	53	1	no	—
200	C₄⋊C₁₆	—	53	2	no	—
201	#358	—	53	2	no	—
202	#358	—	53	2	no	—
203	#358	—	53	2	no	—
204	#357	—	53	1	no	—
205	#357	—	53	1	no	—
206	#357	—	53	2	no	—
207	C₄×C₁₆	—	53	1	no	—
208	C₄×C₁₆	—	53	2	no	—
209	C₅₃	#33927	1	1	yes	characteristic, complemented, Sylow
210	C₂×Q₁₆	—	53	2	no	—
211	C₂×Q₁₆	—	106	1	no	—
212	C₂×SD₁₆	—	106	2	no	—
213	C₂×D₈	—	53	1	no	—
214	C₂×D₈	C₂²×D₅₃	1	1	yes	characteristic
215	C₄⋊Q₈	—	2	1	no	—
216	C₄⋊Q₈	—	106	1	no	—
217	#126	—	212	2	no	—
218	#125	—	2	2	no	—
219	#124	—	106	2	no	—
220	C₂²⋊Q₈	—	4	1	no	—
221	C₂²⋊Q₈	—	4	2	no	—
222	C₂²⋊Q₈	—	212	1	no	—
223	C₄×Q₈	—	106	2	no	—
224	C₄×Q₈	C₂²⋊D₅₃	1	1	yes	characteristic
225	C₄×D₄	—	2	1	no	—
226	C₄×D₄	—	2	2	no	—
227	C₄×D₄	—	106	1	no	—
228	#117	—	212	1	no	—
229	C₂×C₄⋊C₄	—	4	1	no	—
230	SD₃₂	—	106	4	no	—
231	D₁₆	—	106	2	no	—
232	C₂×C₁₆	—	53	1	no	—
233	C₂×C₁₆	—	53	2	no	—
234	C₂×C₁₆	—	53	1	no	—
235	#107	—	2	1	no	—
236	#107	—	53	1	no	—
237	#107	C₂²×D₅₃	1	1	yes	characteristic
238	#107	C₂²×D₅₃	1	2	yes	—
239	#106	—	2	2	no	—
240	C₄⋊C₈	C₂²⋊D₅₃	1	1	yes	characteristic
241	C₄⋊C₈	C₂²⋊D₅₃	1	2	yes	—
242	#103	—	106	2	no	—
243	#102	—	2	2	no	—
244	#102	—	2	1	no	—
245	#102	—	106	1	no	—
246	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
247	C₄×C₈	C₂²×D₅₃	1	2	yes	—
248	C₂×Q₈	—	4	1	no	—
249	C₂×Q₈	—	106	2	no	—
250	C₂×Q₈	#11719922	1	1	yes	characteristic
251	C₂×D₄	—	2	1	no	—
252	C₂×D₄	—	106	1	no	—
253	C₂²×C₄	—	4	1	no	—
254	C₂²×C₄	—	4	2	no	—
255	C₂²×C₄	—	212	1	no	—
256	Q₁₆	—	106	2	no	—
257	Q₁₆	—	106	2	no	—
258	Q₁₆	—	424	1	no	—
259	SD₁₆	—	424	2	no	—
260	D₈	—	2	1	no	—
261	D₈	—	106	1	no	—
262	D₈	—	106	1	no	—
263	D₈	#11719949	1	2	yes	—
264	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
265	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
266	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
267	C₄⋊C₄	—	2	1	no	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	4	2	no	—
270	C₄⋊C₄	—	4	1	no	—
271	C₄⋊C₄	—	4	2	no	—
272	C₄⋊C₄	—	106	1	no	—
273	C₄⋊C₄	—	212	1	no	—
274	C₄⋊C₄	—	212	2	no	—
275	C₄⋊C₄	—	212	1	no	—
276	C₄⋊C₄	#11719922	1	1	yes	characteristic
277	C₄⋊C₄	#11719922	1	2	yes	—
278	C₂²⋊C₄	—	4	1	no	—
279	C₂²⋊C₄	—	4	2	no	—
280	C₂²⋊C₄	—	212	2	no	—
281	C₂²⋊C₄	—	212	1	no	—
282	C₂²⋊C₄	—	212	2	no	—
283	C₄²	—	212	2	no	—
284	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
285	C₄²	C₂×C₂²⋊D₅₃	1	2	yes	—
286	C₁₆	—	106	1	no	—
287	C₁₆	—	106	2	no	—
288	C₁₆	—	106	1	no	—
289	C₂³	—	4	1	no	—
290	C₂³	—	212	1	no	—
291	Q₈	—	2	2	no	—
292	Q₈	—	16	1	no	—
293	Q₈	—	212	2	no	—
294	Q₈	—	212	2	no	—
295	D₄	—	2	2	no	—
296	D₄	—	4	1	no	—
297	D₄	—	212	1	no	—
298	D₄	—	212	1	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	8	2	no	—
302	C₂×C₄	—	16	2	no	—
303	C₂×C₄	—	424	2	no	—
304	C₂×C₄	—	212	1	no	—
305	C₂×C₄	—	212	2	no	—
306	C₂×C₄	C₂×#11719922	1	2	yes	—
307	C₂×C₄	C₂×#11719922	1	2	yes	—
308	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
309	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
310	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
311	C₈	—	2	2	no	—
312	C₈	—	2	1	no	—
313	C₈	C₂×#11719949	1	1	yes	characteristic
314	C₈	#13133265	1	1	yes	characteristic
315	C₂²	—	4	2	no	—
316	C₂²	—	8	1	no	—
317	C₂²	—	8	1	no	—
318	C₂²	—	424	1	no	—
319	C₂²	—	424	1	no	—
320	C₂²	—	424	1	no	—
321	C₂²	C₂²×#11719922	1	1	yes	characteristic
322	C₄	—	2	2	no	—
323	C₄	—	2	2	no	—
324	C₄	—	2	1	no	—
325	C₄	—	8	1	no	—
326	C₄	—	8	2	no	—
327	C₄	—	424	1	no	—
328	C₄	—	424	2	no	—
329	C₄	#16811053	1	2	yes	—
330	C₄	#16811284	1	1	yes	characteristic
331	C₄	#16811431	1	1	yes	characteristic
332	C₂	—	8	2	no	—
333	C₂	—	848	1	no	—
334	C₂	#20287350	1	1	yes	characteristic
335	C₂	C₂×#16811053	1	1	yes	characteristic
336	C₂	#20287862	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#11719922
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#3380	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂×#13133158	C₂²
hypercenter	#3380	D₅₃
hypocenter	C₅₃	#33927
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ #3380

Complements

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

N	H ≅ G/N	type
1. #20287704	C₂	semidirect
2. #20276949	C₂	semidirect
3. #20276944	C₂	semidirect
4. #20276876	C₂	semidirect
5. #20276859	C₂	semidirect
7. #20276760	C₂	semidirect
8. Q₈×#11719908	C₂	semidirect
9. C₄×#13133161	C₂	semidirect
11. C₅₃×#3380	C₂	semidirect
165. #3380	D₅₃	semidirect
209. C₅₃	#33927	semidirect

Maximal subgroups

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

subgroup	index
#20287704	2
#20276949	2
#20276944	2
#20276876	2
#20276859	2
#20276770	2
#20276760	2
Q₈×#11719908	2
C₄×#13133161	2
C₄×#13133158	2
C₅₃×#3380	2
#33927	53