Subgroups — #27402591

Order 13568.

862 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283591	C₂	1	2	yes	maximal
2	#20283315	C₂	1	2	yes	maximal, complemented
3	#20275501	C₂	1	2	yes	maximal, complemented
4	#20274943	C₂	1	4	yes	maximal, complemented
5	#20274875	C₂	1	1	yes	maximal, characteristic, complemented
6	#20274779	C₂	1	2	yes	maximal, complemented
7	#20274594	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅₃×#443	C₂	1	1	yes	maximal, characteristic
9	#16811106	C₂²	1	4	yes	—
10	#16810724	—	2	4	no	—
11	Q₁₆×Dic₅₃	—	2	2	no	—
12	D₈×Dic₅₃	—	2	2	no	—
13	C₂×#13133153	C₂²	1	1	yes	characteristic
14	#16810665	—	2	4	no	—
15	#16810665	C₂²	1	4	yes	—
16	#16810661	—	2	2	no	—
17	#16810661	C₂²	1	1	yes	characteristic
18	C₂×#13133151	—	2	2	no	—
19	C₂×#13133151	C₂²	1	2	yes	—
20	C₂×#13133151	C₂²	1	2	yes	—
21	#16810624	C₂²	1	4	yes	—
22	#16810578	C₂²	1	2	yes	—
23	#16810493	C₂²	1	2	yes	—
24	#16810391	C₂²	1	4	yes	—
25	#16810383	C₂²	1	2	yes	—
26	C₂×#135×C₅₃	C₂²	1	1	yes	characteristic
27	C₂×#131×C₅₃	C₂²	1	2	yes	—
28	C₅₃×#443	C₄	1	8	yes	complemented
29	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
30	C₂×#104×C₅₃	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
32	C₂×#11719917	C₂³	1	2	yes	—
33	Q₈×Dic₅₃	—	2	4	no	—
34	D₄×Dic₅₃	—	2	4	no	—
35	D₄×Dic₅₃	—	4	4	no	—
36	#13133176	—	2	4	no	—
37	#13133176	C₂³	1	2	yes	—
38	C₂×C₄⋊Dic₅₃	—	2	4	no	—
39	C₂×C₄⋊Dic₅₃	—	4	2	no	—
40	C₂×C₄⋊Dic₅₃	C₂³	1	4	yes	—
41	#13133153	—	2	2	no	—
42	#13133151	—	2	2	no	—
43	#13133151	—	2	2	no	—
44	#13133151	—	2	2	no	—
45	#13133151	—	2	2	no	—
46	#13133151	—	2	4	no	—
47	#13133151	D₄	1	1	yes	characteristic
48	#13133151	D₄	1	1	yes	characteristic
49	#13133149	—	2	4	no	—
50	#13133149	D₄	1	2	yes	—
51	#13133141	—	2	4	no	—
52	#13133138	—	2	4	no	—
53	#13133127	—	2	2	no	—
54	C₈×Dic₅₃	—	2	2	no	—
55	C₂×#41×C₅₃	C₂³	1	2	yes	—
56	#135×C₅₃	C₂×C₄	1	4	yes	—
57	#135×C₅₃	Q₈	1	4	yes	—
58	C₂×Q₁₆×C₅₃	D₄	1	1	yes	characteristic
59	C₂×SD₁₆×C₅₃	D₄	1	2	yes	—
60	C₂×D₈×C₅₃	D₄	1	1	yes	characteristic
61	#131×C₅₃	—	2	4	no	—
62	#131×C₅₃	C₂×C₄	1	8	yes	complemented
63	C₂×M₁₆×C₅₃	—	2	2	no	—
64	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
65	C₂²×C₈×C₅₃	—	2	2	no	—
66	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
67	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
68	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
69	#108×C₅₃	C₂×C₄	1	4	yes	complemented
70	#104×C₅₃	C₂×C₄	1	8	yes	complemented
71	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	complemented
72	#41×C₅₃	—	2	2	no	—
73	#41×C₅₃	—	2	2	no	—
74	#41×C₅₃	C₂²×C₄	1	4	yes	—
75	#41×C₅₃	C₂×Q₈	1	4	yes	—
76	C₂×Q₈×C₅₃	C₂×D₄	1	2	yes	—
77	C₂×D₄×C₅₃	—	2	2	no	—
78	C₂×D₄×C₅₃	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₅₃	—	2	1	no	—
80	C₂²×C₄×C₅₃	—	2	2	no	—
81	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
82	C₂²×Dic₅₃	—	2	4	no	—
83	C₂²×Dic₅₃	—	4	4	no	—
84	Q₁₆×C₅₃	C₄⋊C₄	1	4	yes	complemented
85	SD₁₆×C₅₃	C₄⋊C₄	1	8	yes	complemented
86	D₈×C₅₃	C₄⋊C₄	1	4	yes	complemented
87	M₁₆×C₅₃	—	2	2	no	—
88	M₁₆×C₅₃	—	2	8	no	—
89	M₁₆×C₅₃	C₂²×C₄	1	4	yes	—
90	C₂×C₈×C₅₃	—	2	2	no	—
91	C₂×C₈×C₅₃	—	2	8	no	—
92	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
93	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
94	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
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	C₂×Q₈	1	2	yes	—
99	C₂×C₈×C₅₃	#41	1	2	yes	—
100	C₄²×C₅₃	—	2	1	no	—
101	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
102	C₂²⋊Dic₅₃	—	2	2	no	—
103	C₂²⋊Dic₅₃	—	4	4	no	—
104	C₄⋊Dic₅₃	—	2	4	no	—
105	C₄⋊Dic₅₃	—	2	4	no	—
106	C₄⋊Dic₅₃	—	4	4	no	—
107	C₄⋊Dic₅₃	—	8	4	no	—
108	C₄⋊Dic₅₃	C₂×D₄	1	2	yes	—
109	C₄⋊Dic₅₃	C₂×D₄	1	2	yes	—
110	C₄×Dic₅₃	—	2	2	no	—
111	C₄×Dic₅₃	—	4	4	no	—
112	#11719917	—	2	2	no	—
113	#11719917	—	2	2	no	—
114	C₂×#88322	—	2	2	no	—
115	C₂³×C₅₃	—	2	2	no	—
116	C₂³×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
117	Q₈×C₅₃	—	2	2	no	—
118	Q₈×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
119	D₄×C₅₃	—	2	4	no	—
120	D₄×C₅₃	—	2	4	no	—
121	D₄×C₅₃	—	2	2	no	—
122	D₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
123	C₂×C₄×C₅₃	—	2	2	no	—
124	C₂×C₄×C₅₃	—	2	2	no	—
125	C₂×C₄×C₅₃	—	2	2	no	—
126	C₂×C₄×C₅₃	—	2	2	no	—
127	C₂×C₄×C₅₃	—	2	4	no	—
128	C₂×C₄×C₅₃	—	2	4	no	—
129	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
130	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
133	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
134	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
135	C₂×Dic₅₃	—	2	2	no	—
136	C₂×Dic₅₃	—	2	2	no	—
137	C₂×Dic₅₃	—	4	4	no	—
138	C₂×Dic₅₃	—	8	4	no	—
139	C₂×Dic₅₃	—	16	4	no	—
140	C₈×C₅₃	—	2	4	no	—
141	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
142	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
143	C₈×C₅₃	C₄×D₄	1	2	yes	—
144	C₈×C₅₃	C₄×D₄	1	2	yes	—
145	#88322	—	4	2	no	—
146	#22692	—	53	1	no	maximal, Sylow
147	C₂²×C₅₃	—	2	1	no	—
148	C₂²×C₅₃	—	2	1	no	—
149	C₂²×C₅₃	—	2	4	no	—
150	C₂²×C₅₃	—	4	2	no	—
151	C₂²×C₅₃	—	4	2	no	—
152	C₂²×C₅₃	#520	1	1	yes	characteristic
153	C₂²×C₅₃	D₄²	1	1	yes	characteristic
154	C₂²×C₅₃	#554	1	1	yes	characteristic
155	C₄×C₅₃	—	2	4	no	—
156	C₄×C₅₃	—	2	4	no	—
157	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
158	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
159	C₄×C₅₃	#547	1	1	yes	characteristic
160	C₄×C₅₃	D₄×Q₈	1	1	yes	characteristic
161	Dic₅₃	—	4	2	no	—
162	Dic₅₃	—	4	2	no	—
163	Dic₅₃	—	8	4	no	—
164	C₂×#443	D₅₃	1	1	yes	characteristic
165	#2951	—	53	2	no	—
166	#2913	—	53	2	no	—
167	#1952	—	53	2	no	—
168	#1859	—	53	4	no	—
169	#1845	—	53	1	no	—
170	#1816	—	53	2	no	—
171	#1772	—	53	1	no	—
172	C₂×C₅₃	—	2	1	no	—
173	C₂×C₅₃	—	2	1	no	—
174	C₂×C₅₃	—	4	4	no	—
175	C₂×C₅₃	D₄×C₄⋊C₄	1	1	yes	characteristic
176	C₂×C₅₃	#3292	1	1	yes	characteristic
177	C₂×C₅₃	#3293	1	1	yes	characteristic
178	C₂×#135	C₂×D₅₃	1	1	yes	characteristic
179	C₂×#131	C₂×D₅₃	1	2	yes	—
180	#520	—	53	4	no	—
181	#443	Dic₅₃	1	8	yes	complemented
182	#440	—	106	4	no	—
183	C₄×Q₁₆	—	106	2	no	—
184	C₄×D₈	—	106	2	no	—
185	C₂×#108	—	53	1	no	—
186	C₂×#108	C₂×D₅₃	1	1	yes	characteristic
187	#428	—	53	4	no	—
188	#428	—	106	4	no	—
189	#427	—	53	1	no	—
190	#427	—	106	2	no	—
191	C₂×#107	—	53	2	no	—
192	C₂×#107	—	53	2	no	—
193	C₂×#107	—	106	2	no	—
194	C₂×#104	C₂×D₅₃	1	2	yes	—
195	#417	—	53	4	no	—
196	#405	—	53	2	no	—
197	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
198	#384	—	53	2	no	—
199	#343	—	53	4	no	—
200	#339	—	53	2	no	—
201	C₅₃	#22692	1	1	yes	characteristic, complemented, Sylow
202	C₂×#41	C₂²×D₅₃	1	2	yes	—
203	#135	Dic₁₀₆	1	4	yes	—
204	#135	C₂×Dic₅₃	1	4	yes	—
205	C₂×Q₁₆	D₂₁₂	1	1	yes	characteristic
206	C₂×SD₁₆	D₂₁₂	1	2	yes	—
207	C₂×D₈	D₂₁₂	1	1	yes	characteristic
208	#131	—	2	4	no	—
209	#131	C₂×Dic₅₃	1	8	yes	complemented
210	C₂×M₁₆	—	2	2	no	—
211	C₂×M₁₆	—	53	2	no	—
212	C₂×M₁₆	C₂²×D₅₃	1	2	yes	—
213	C₂²×C₈	—	2	2	no	—
214	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
215	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
216	C₄×Q₈	—	106	4	no	—
217	C₄×D₄	—	106	4	no	—
218	C₄×D₄	—	212	4	no	—
219	#117	—	53	2	no	—
220	#117	—	106	4	no	—
221	C₂×C₄⋊C₄	—	53	4	no	—
222	C₂×C₄⋊C₄	—	106	4	no	—
223	C₂×C₄⋊C₄	—	212	2	no	—
224	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
225	#108	—	106	2	no	—
226	#108	C₂×Dic₅₃	1	4	yes	complemented
227	#107	—	53	1	no	—
228	#107	—	53	1	no	—
229	#107	—	106	4	no	—
230	#107	—	106	2	no	—
231	#107	—	106	2	no	—
232	#107	—	106	2	no	—
233	#107	—	106	2	no	—
234	#106	—	53	2	no	—
235	#106	—	106	4	no	—
236	#104	C₂×Dic₅₃	1	8	yes	complemented
237	#103	—	106	4	no	—
238	#102	—	106	4	no	—
239	#97	—	106	2	no	—
240	C₄×C₈	—	106	2	no	—
241	C₄×C₈	C₂×Dic₅₃	1	4	yes	complemented
242	#41	—	2	2	no	—
243	#41	—	2	2	no	—
244	#41	C₂×Dic₁₀₆	1	4	yes	—
245	#41	C₂²×Dic₅₃	1	4	yes	—
246	C₂×Q₈	C₂×D₂₁₂	1	2	yes	—
247	C₂×D₄	—	2	2	no	—
248	C₂×D₄	C₂×D₂₁₂	1	2	yes	—
249	C₂²×C₄	—	2	1	no	—
250	C₂²×C₄	—	2	2	no	—
251	C₂²×C₄	—	106	4	no	—
252	C₂²×C₄	—	212	4	no	—
253	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
254	Q₁₆	C₄⋊Dic₅₃	1	4	yes	complemented
255	SD₁₆	C₄⋊Dic₅₃	1	8	yes	complemented
256	D₈	C₄⋊Dic₅₃	1	4	yes	complemented
257	M₁₆	—	2	2	no	—
258	M₁₆	—	2	8	no	—
259	M₁₆	—	106	2	no	—
260	M₁₆	—	106	2	no	—
261	M₁₆	C₂²×Dic₅₃	1	4	yes	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	8	no	—
264	C₂×C₈	—	106	2	no	—
265	C₂×C₈	C₂×Dic₁₀₆	1	2	yes	—
266	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
267	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
268	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
269	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
270	C₂×C₈	#11719947	1	2	yes	—
271	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
272	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
273	C₄⋊C₄	—	53	2	no	—
274	C₄⋊C₄	—	53	2	no	—
275	C₄⋊C₄	—	106	4	no	—
276	C₄⋊C₄	—	106	4	no	—
277	C₄⋊C₄	—	424	4	no	—
278	C₄⋊C₄	—	212	4	no	—
279	C₂²⋊C₄	—	106	2	no	—
280	C₂²⋊C₄	—	212	4	no	—
281	C₄²	—	2	1	no	—
282	C₄²	—	106	2	no	—
283	C₄²	—	212	4	no	—
284	C₄²	C₂²×Dic₅₃	1	2	yes	—
285	C₂³	—	2	2	no	—
286	C₂³	#13133263	1	1	yes	characteristic
287	Q₈	—	2	2	no	—
288	Q₈	C₂×C₄⋊Dic₅₃	1	4	yes	—
289	D₄	—	2	4	no	—
290	D₄	—	2	4	no	—
291	D₄	—	2	2	no	—
292	D₄	C₂×C₄⋊Dic₅₃	1	4	yes	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	4	no	—
298	C₂×C₄	—	2	4	no	—
299	C₂×C₄	—	848	4	no	—
300	C₂×C₄	—	106	2	no	—
301	C₂×C₄	—	106	2	no	—
302	C₂×C₄	—	424	4	no	—
303	C₂×C₄	—	212	4	no	—
304	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
305	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
306	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
307	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
308	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
309	C₂×C₄	#13133264	1	1	yes	characteristic
310	C₈	—	2	4	no	—
311	C₈	—	212	2	no	—
312	C₈	C₂×C₄⋊Dic₅₃	1	2	yes	—
313	C₈	C₂×C₄⋊Dic₅₃	1	2	yes	—
314	C₈	D₄×Dic₅₃	1	2	yes	—
315	C₈	D₄×Dic₅₃	1	2	yes	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	4	no	—
319	C₂²	—	4	2	no	—
320	C₂²	—	4	2	no	—
321	C₂²	#16811106	1	1	yes	characteristic
322	C₂²	D₄×D₂₁₂	1	1	yes	characteristic
323	C₂²	#16811330	1	1	yes	characteristic
324	C₄	—	2	4	no	—
325	C₄	—	2	4	no	—
326	C₄	—	424	4	no	—
327	C₄	—	212	2	no	—
328	C₄	—	212	2	no	—
329	C₄	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
330	C₄	C₂×D₄×Dic₅₃	1	1	yes	characteristic
331	C₄	#16811277	1	1	yes	characteristic
332	C₄	D₄×Dic₁₀₆	1	1	yes	characteristic
333	C₂	—	2	1	no	—
334	C₂	—	2	1	no	—
335	C₂	—	4	4	no	—
336	C₂	D₄×C₄⋊Dic₅₃	1	1	yes	characteristic
337	C₂	#20286515	1	1	yes	characteristic
338	C₂	#20286520	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₅₃×#443	C₂
socle	C₂²×C₅₃	D₄²
prime radical	C₂×#135×C₅₃	C₂²
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂×#443	D₅₃
hypocenter	C₅₃	#22692
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#443

Complements

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

N	H ≅ G/N	type
2. #20283315	C₂	semidirect
3. #20275501	C₂	semidirect
4. #20274943	C₂	semidirect
5. #20274875	C₂	semidirect
6. #20274779	C₂	semidirect
7. #20274594	C₂	semidirect
28. C₅₃×#443	C₄	semidirect
62. #131×C₅₃	C₂×C₄	semidirect
69. #108×C₅₃	C₂×C₄	semidirect
70. #104×C₅₃	C₂×C₄	semidirect
71. C₄×C₈×C₅₃	C₂×C₄	semidirect
84. Q₁₆×C₅₃	C₄⋊C₄	semidirect
85. SD₁₆×C₅₃	C₄⋊C₄	semidirect
86. D₈×C₅₃	C₄⋊C₄	semidirect
181. #443	Dic₅₃	semidirect
201. C₅₃	#22692	semidirect
209. #131	C₂×Dic₅₃	semidirect
226. #108	C₂×Dic₅₃	semidirect
236. #104	C₂×Dic₅₃	semidirect
241. C₄×C₈	C₂×Dic₅₃	semidirect
254. Q₁₆	C₄⋊Dic₅₃	semidirect
255. SD₁₆	C₄⋊Dic₅₃	semidirect
256. D₈	C₄⋊Dic₅₃	semidirect

Maximal subgroups

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

subgroup	index
#20283591	2
#20283315	2
#20275501	2
#20274943	2
#20274875	2
#20274779	2
#20274594	2
C₂×C₅₃×#443	2
#22692	53