Subgroups — #27277924

Order 13568.

556 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂²⋊#88322	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810373	C₂	1	2	yes	maximal, complemented
4	#20272549	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1797	C₂	1	1	yes	maximal, characteristic
6	C₂³×#88322	—	2	1	no	—
7	C₂×C₄⋊#88322	—	2	4	no	—
8	C₂×C₄⋊#88322	—	2	1	no	—
9	C₂×C₄⋊#88322	C₂²	1	2	yes	—
10	C₂×C₂²⋊#88322	C₄	1	1	yes	characteristic
11	C₂×C₂²⋊#88322	C₄	1	1	yes	characteristic
12	C₂×C₄×#88322	C₂²	1	2	yes	—
13	#16810378	C₄	1	2	yes	—
14	#16810373	—	2	4	no	—
15	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₂×C₂²⋊C₈×C₅₃	—	2	1	no	—
19	C₅₃×#336	—	2	1	no	—
20	C₂²×#88322	—	2	4	no	—
21	C₂²×#88322	—	2	2	no	—
22	C₂²×#88322	—	4	2	no	—
23	C₂²×#88322	—	4	2	no	—
24	C₂²×#88322	C₂×C₄	1	2	yes	—
25	C₂²×#88322	C₂×C₄	1	2	yes	—
26	C₄⋊#88322	—	2	4	no	—
27	C₄⋊#88322	—	2	2	no	—
28	C₄⋊#88322	D₄	1	4	yes	—
29	C₂²⋊#88322	—	2	2	no	—
30	C₂²⋊#88322	C₈	1	4	yes	complemented
31	C₄×#88322	—	2	2	no	—
32	C₄×#88322	—	2	2	no	—
33	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
35	C₂³×C₄×C₅₃	Q₈	1	1	yes	characteristic
36	C₂²×C₈×C₅₃	—	2	1	no	—
37	C₂²×C₈×C₅₃	—	2	1	no	—
38	C₂×C₄²×C₅₃	—	2	2	no	—
39	C₂×C₄²×C₅₃	—	2	2	no	—
40	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
43	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
44	C₄⋊C₈×C₅₃	—	2	2	no	—
45	C₄⋊C₈×C₅₃	—	2	2	no	—
46	C₂²⋊C₈×C₅₃	—	4	2	no	—
47	C₂⁴×C₅₃	C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	—	2	2	no	—
49	C₂²×C₄×C₅₃	—	2	1	no	—
50	C₂²×C₄×C₅₃	—	2	1	no	—
51	C₂²×C₄×C₅₃	—	2	2	no	—
52	C₂²×C₄×C₅₃	—	2	1	no	—
53	C₂²×C₄×C₅₃	—	2	1	no	—
54	C₂²×C₄×C₅₃	—	2	2	no	—
55	C₂²×C₄×C₅₃	—	2	2	no	—
56	C₂²×C₄×C₅₃	—	4	1	no	—
57	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
61	C₂²×C₄×C₅₃	M₁₆	1	2	yes	—
62	C₂²×C₄×C₅₃	SD₁₆	1	2	yes	—
63	C₂²×C₄×C₅₃	SD₁₆	1	2	yes	—
64	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
67	C₂×C₈×C₅₃	—	4	1	no	—
68	C₂×C₈×C₅₃	—	4	1	no	—
69	C₂×C₈×C₅₃	—	4	1	no	—
70	C₂×C₈×C₅₃	—	4	1	no	—
71	C₂×C₈×C₅₃	—	4	1	no	—
72	C₂×C₈×C₅₃	—	4	1	no	—
73	C₄²×C₅₃	—	2	2	no	—
74	C₄²×C₅₃	—	2	1	no	—
75	C₄²×C₅₃	—	2	2	no	—
76	C₄²×C₅₃	—	2	1	no	—
77	C₄²×C₅₃	C₂×D₄	1	2	yes	—
78	C₄²×C₅₃	#41	1	2	yes	—
79	C₂×#88322	—	2	2	no	—
80	C₂×#88322	—	2	2	no	—
81	C₂×#88322	—	2	2	no	—
82	C₂×#88322	—	2	4	no	—
83	C₂×#88322	—	2	2	no	—
84	C₂×#88322	—	2	2	no	—
85	C₂×#88322	—	4	8	no	—
86	C₂×#88322	—	4	2	no	—
87	C₂×#88322	—	4	2	no	—
88	C₂×#88322	C₂×C₈	1	4	yes	complemented
89	C₂³×C₅₃	—	2	1	no	—
90	C₂³×C₅₃	—	2	1	no	—
91	C₂³×C₅₃	—	2	1	no	—
92	C₂³×C₅₃	—	2	1	no	—
93	C₂³×C₅₃	C₄⋊C₈	1	2	yes	—
94	C₂³×C₅₃	#106	1	2	yes	—
95	C₂³×C₅₃	#108	1	2	yes	—
96	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₅₃	—	2	1	no	—
98	C₂×C₄×C₅₃	—	2	1	no	—
99	C₂×C₄×C₅₃	—	2	1	no	—
100	C₂×C₄×C₅₃	—	2	1	no	—
101	C₂×C₄×C₅₃	—	2	1	no	—
102	C₂×C₄×C₅₃	—	2	1	no	—
103	C₂×C₄×C₅₃	—	2	4	no	—
104	C₂×C₄×C₅₃	—	2	4	no	—
105	C₂×C₄×C₅₃	—	2	2	no	—
106	C₂×C₄×C₅₃	—	2	2	no	—
107	C₂×C₄×C₅₃	—	2	2	no	—
108	C₂×C₄×C₅₃	—	2	2	no	—
109	C₂×C₄×C₅₃	—	2	1	no	—
110	C₂×C₄×C₅₃	—	2	1	no	—
111	C₂×C₄×C₅₃	—	2	1	no	—
112	C₂×C₄×C₅₃	—	4	4	no	—
113	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
114	C₂×C₄×C₅₃	#106	1	2	yes	—
115	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
117	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
118	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
119	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
120	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
121	C₂×C₄×C₅₃	#131	1	2	yes	—
122	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
123	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
124	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
125	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
126	C₈×C₅₃	—	8	1	no	—
127	C₈×C₅₃	—	8	1	no	—
128	#88322	—	2	4	no	—
129	#88322	—	4	2	no	—
130	#88322	—	4	4	no	—
131	#8830	—	53	1	no	maximal, Sylow
132	C₂²×C₅₃	—	2	2	no	—
133	C₂²×C₅₃	—	2	2	no	—
134	C₂²×C₅₃	—	2	2	no	—
135	C₂²×C₅₃	—	2	2	no	—
136	C₂²×C₅₃	—	2	2	no	—
137	C₂²×C₅₃	—	2	2	no	—
138	C₂²×C₅₃	#334	1	4	yes	complemented
139	C₂²×C₅₃	#385	1	1	yes	characteristic
140	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
141	C₂²×C₅₃	#424	1	1	yes	characteristic
142	C₂²×C₅₃	C₂×#106	1	1	yes	characteristic
143	C₂²×C₅₃	#428	1	1	yes	characteristic
144	C₂²×C₅₃	C₂×#108	1	1	yes	characteristic
145	C₂²×C₅₃	#430	1	1	yes	characteristic
146	C₄×C₅₃	—	2	2	no	—
147	C₄×C₅₃	—	2	2	no	—
148	C₄×C₅₃	—	2	2	no	—
149	C₄×C₅₃	—	4	1	no	—
150	C₄×C₅₃	C₈×D₄	1	2	yes	—
151	C₄×C₅₃	#435	1	2	yes	—
152	C₄×C₅₃	#448	1	1	yes	characteristic
153	C₄×C₅₃	#450	1	1	yes	characteristic
154	C₄×C₅₃	#481	1	1	yes	characteristic
155	C₄×C₅₃	#483	1	1	yes	characteristic
156	C₂²×C₄⋊C₈	—	53	1	no	—
157	#1797	D₅₃	1	1	yes	characteristic
158	C₄×C₂²⋊C₈	—	53	1	no	—
159	C₂×#334	—	53	2	no	—
160	#1277	—	53	2	no	—
161	C₂×C₅₃	—	2	4	no	—
162	C₂×C₅₃	C₂×#334	1	1	yes	characteristic
163	C₂×C₅₃	#1565	1	1	yes	characteristic
164	C₂×C₅₃	#1570	1	1	yes	characteristic
165	C₂×C₅₃	#1573	1	1	yes	characteristic
166	C₂×C₅₃	#1850	1	1	yes	characteristic
167	C₂×C₅₃	#1853	1	1	yes	characteristic
168	C₂×C₅₃	#1855	1	1	yes	characteristic
169	C₂³×C₈	—	106	1	no	—
170	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
171	C₂×C₄⋊C₈	—	53	2	no	—
172	C₂×C₄⋊C₈	—	106	4	no	—
173	C₂×C₄⋊C₈	—	106	1	no	—
174	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
175	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
176	C₂×C₂²⋊C₈	—	2	1	no	—
177	C₂×C₂²⋊C₈	—	53	1	no	—
178	C₂×C₂²⋊C₈	—	53	1	no	—
179	C₂×C₄×C₈	—	53	2	no	—
180	#336	—	2	1	no	—
181	#336	—	53	2	no	—
182	#334	—	106	4	no	—
183	C₅₃	#8830	1	1	yes	characteristic, complemented, Sylow
184	C₂³×C₄	Dic₁₀₆	1	1	yes	characteristic
185	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
186	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
187	C₂²×C₈	—	2	1	no	—
188	C₂²×C₈	—	2	1	no	—
189	C₂²×C₈	—	53	2	no	—
190	C₂²×C₈	—	53	2	no	—
191	C₂²×C₈	—	106	4	no	—
192	C₂²×C₈	—	106	2	no	—
193	C₂²×C₈	—	212	2	no	—
194	C₂²×C₈	—	212	2	no	—
195	C₂×C₄²	—	2	2	no	—
196	C₂×C₄²	—	2	2	no	—
197	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
198	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
199	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
200	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
201	C₄⋊C₈	—	2	2	no	—
202	C₄⋊C₈	—	2	2	no	—
203	C₄⋊C₈	—	53	4	no	—
204	C₄⋊C₈	—	106	4	no	—
205	C₄⋊C₈	—	106	2	no	—
206	C₂²⋊C₈	—	4	2	no	—
207	C₂²⋊C₈	—	53	4	no	—
208	C₂²⋊C₈	—	106	2	no	—
209	C₄×C₈	—	106	2	no	—
210	C₄×C₈	—	106	2	no	—
211	C₂⁴	#11719919	1	1	yes	characteristic
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	2	1	no	—
214	C₂²×C₄	—	2	1	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	—	2	1	no	—
217	C₂²×C₄	—	2	1	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	4	1	no	—
221	C₂²×C₄	C₈×D₅₃	1	2	yes	—
222	C₂²×C₄	#11719912	1	2	yes	—
223	C₂²×C₄	#11719919	1	1	yes	characteristic
224	C₂²×C₄	#11719919	1	1	yes	characteristic
225	C₂²×C₄	#11719919	1	1	yes	characteristic
226	C₂²×C₄	#11719923	1	2	yes	—
227	C₂²×C₄	#11719924	1	2	yes	—
228	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
229	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
230	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
231	C₂×C₈	—	4	1	no	—
232	C₂×C₈	—	4	1	no	—
233	C₂×C₈	—	4	1	no	—
234	C₂×C₈	—	4	1	no	—
235	C₂×C₈	—	4	1	no	—
236	C₂×C₈	—	4	1	no	—
237	C₂×C₈	—	53	4	no	—
238	C₂×C₈	—	106	2	no	—
239	C₂×C₈	—	106	2	no	—
240	C₂×C₈	—	106	2	no	—
241	C₂×C₈	—	106	4	no	—
242	C₂×C₈	—	106	2	no	—
243	C₂×C₈	—	106	2	no	—
244	C₂×C₈	—	212	8	no	—
245	C₂×C₈	—	212	2	no	—
246	C₂×C₈	—	212	2	no	—
247	C₄²	—	2	2	no	—
248	C₄²	—	2	1	no	—
249	C₄²	—	2	2	no	—
250	C₄²	—	2	1	no	—
251	C₄²	D₄×D₅₃	1	2	yes	—
252	C₄²	#11719947	1	2	yes	—
253	C₂³	—	2	1	no	—
254	C₂³	—	2	1	no	—
255	C₂³	—	2	1	no	—
256	C₂³	—	2	1	no	—
257	C₂³	#13133146	1	2	yes	—
258	C₂³	#13133148	1	2	yes	—
259	C₂³	#13133152	1	2	yes	—
260	C₂³	C₂×#11719919	1	1	yes	characteristic
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	4	4	no	—
277	C₂×C₄	#13133146	1	2	yes	—
278	C₂×C₄	#13133148	1	2	yes	—
279	C₂×C₄	#13133177	1	2	yes	—
280	C₂×C₄	#13133185	1	1	yes	characteristic
281	C₂×C₄	#13133194	1	2	yes	—
282	C₂×C₄	#13133198	1	1	yes	characteristic
283	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
284	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
285	C₂×C₄	#13133225	1	2	yes	—
286	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
287	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
288	C₂×C₄	#13133242	1	1	yes	characteristic
289	C₂×C₄	#13133247	1	1	yes	characteristic
290	C₈	—	8	1	no	—
291	C₈	—	8	1	no	—
292	C₈	—	106	4	no	—
293	C₈	—	212	4	no	—
294	C₈	—	212	2	no	—
295	C₂²	—	2	2	no	—
296	C₂²	—	2	2	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	2	no	—
301	C₂²	#16810373	1	4	yes	complemented
302	C₂²	#16810495	1	1	yes	characteristic
303	C₂²	C₂×#13133146	1	1	yes	characteristic
304	C₂²	#16810651	1	1	yes	characteristic
305	C₂²	C₂×#13133148	1	1	yes	characteristic
306	C₂²	#16810666	1	1	yes	characteristic
307	C₂²	C₂×#13133152	1	1	yes	characteristic
308	C₂²	#16810671	1	1	yes	characteristic
309	C₄	—	2	2	no	—
310	C₄	—	2	2	no	—
311	C₄	—	2	2	no	—
312	C₄	—	4	1	no	—
313	C₄	#16810686	1	2	yes	—
314	C₄	#16810691	1	2	yes	—
315	C₄	#16810761	1	1	yes	characteristic
316	C₄	#16810773	1	1	yes	characteristic
317	C₄	#16810939	1	1	yes	characteristic
318	C₄	#16810951	1	1	yes	characteristic
319	C₂	—	2	4	no	—
320	C₂	C₂×#16810373	1	1	yes	characteristic
321	C₂	#20273468	1	1	yes	characteristic
322	C₂	#20273487	1	1	yes	characteristic
323	C₂	#20273501	1	1	yes	characteristic
324	C₂	#20274892	1	1	yes	characteristic
325	C₂	#20274907	1	1	yes	characteristic
326	C₂	#20274919	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1797	C₂
socle	C₂³×C₅₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₅₃	C₄⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1797	D₅₃
hypocenter	C₅₃	#8830
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ #1797

Complements

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

N	H ≅ G/N	type
3. C₂×#16810373	C₂	semidirect
4. #20272549	C₂	semidirect
30. C₂²⋊#88322	C₈	semidirect
88. C₂×#88322	C₂×C₈	semidirect
138. C₂²×C₅₃	#334	semidirect
183. C₅₃	#8830	semidirect
301. C₂²	#16810373	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊#88322	2
C₄×C₂²⋊#88322	2
C₂×#16810373	2
#20272549	2
C₅₃×#1797	2
#8830	53