Subgroups — #27384556

Order 13568.

994 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810671	C₂	1	4	yes	maximal
2	C₂×#16810647	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×#11719917	C₂	1	1	yes	maximal, characteristic
4	#20274889	C₂	1	4	yes	maximal, complemented
5	#20274886	C₂	1	4	yes	maximal, complemented
6	C₂×C₅₃×#423	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11719917	C₂²	1	2	yes	—
8	C₂²×#11719917	C₂²	1	2	yes	—
9	#16810671	—	2	16	no	—
10	C₂×#13133152	C₂²	1	16	yes	—
11	#16810647	—	2	2	no	—
12	#16810647	C₂²	1	2	yes	—
13	#16810647	C₂²	1	2	yes	—
14	C₂×C₄⋊#88322	—	2	2	no	—
15	C₄×#11719917	C₄	1	2	yes	—
16	C₄×#11719917	C₄	1	2	yes	—
17	C₄×#11719917	C₄	1	2	yes	—
18	C₄×#11719917	C₄	1	2	yes	—
19	C₂×#13133128	C₂²	1	2	yes	—
20	C₂×C₄×#88322	C₂²	1	2	yes	—
21	C₂²×M₁₆×C₅₃	C₂²	1	2	yes	—
22	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
23	C₅₃×#423	—	2	2	no	—
24	C₅₃×#423	C₂²	1	2	yes	—
25	C₅₃×#423	C₂²	1	2	yes	—
26	C₂×C₄⋊C₈×C₅₃	—	2	2	no	—
27	C₂×#11719917	—	2	2	no	—
28	C₂×#11719917	—	2	8	no	—
29	C₂×#11719917	C₂×C₄	1	4	yes	—
30	C₂×#11719917	C₂×C₄	1	4	yes	—
31	C₂×#11719917	D₄	1	8	yes	—
32	C₂×#11719917	Q₈	1	8	yes	—
33	C₂×#11719917	C₂³	1	4	yes	—
34	C₂²×#88322	—	2	2	no	—
35	C₂²×#88322	C₂³	1	4	yes	—
36	#13133152	—	2	32	no	—
37	C₄⋊#88322	—	2	4	no	—
38	C₄⋊#88322	—	2	4	no	—
39	#13133128	C₂×C₄	1	4	yes	—
40	#13133128	C₂×C₄	1	4	yes	—
41	C₄×#88322	C₂×C₄	1	4	yes	—
42	C₄×#88322	C₂×C₄	1	4	yes	—
43	C₂³×C₄×C₅₃	Q₈	1	2	yes	—
44	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
45	C₂×M₁₆×C₅₃	—	2	2	no	—
46	C₂×M₁₆×C₅₃	—	2	8	no	—
47	C₂×M₁₆×C₅₃	C₂³	1	4	yes	—
48	C₂²×C₈×C₅₃	—	2	2	no	—
49	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄²×C₅₃	D₄	1	2	yes	—
54	C₂×C₄²×C₅₃	D₄	1	2	yes	—
55	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
56	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
57	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
58	C₄⋊C₈×C₅₃	—	2	4	no	—
59	C₄⋊C₈×C₅₃	—	2	4	no	—
60	C₂⁴×C₅₃	C₂×Q₈	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	—	4	4	no	—
62	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
63	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
64	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
65	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
66	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
69	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
72	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
73	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
74	C₂²×C₄×C₅₃	C₂×Q₈	1	4	yes	—
75	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
76	M₁₆×C₅₃	—	2	8	no	—
77	M₁₆×C₅₃	—	4	4	no	—
78	C₂×C₈×C₅₃	—	2	4	no	—
79	C₂×C₈×C₅₃	—	4	4	no	—
80	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
81	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
82	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
83	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
84	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
85	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
86	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
87	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₅₃	C₂×D₄	1	2	yes	—
89	C₄²×C₅₃	C₂×D₄	1	2	yes	—
90	#11719917	—	2	8	no	—
91	#11719917	—	4	4	no	—
92	#11719917	C₄⋊C₄	1	16	yes	—
93	#11719917	C₄⋊C₄	1	16	yes	—
94	C₂×#88322	—	2	4	no	—
95	C₂×#88322	—	4	4	no	—
96	C₂×#88322	C₂²×C₄	1	8	yes	—
97	C₂×#88322	C₂×D₄	1	8	yes	—
98	C₂×#88322	C₂×Q₈	1	8	yes	—
99	C₂³×C₅₃	—	4	2	no	—
100	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂³×C₅₃	C₄⋊Q₈	1	4	yes	—
103	C₂³×C₅₃	C₂²×Q₈	1	1	yes	characteristic
104	C₂×C₄×C₅₃	—	2	4	no	—
105	C₂×C₄×C₅₃	—	2	4	no	—
106	C₂×C₄×C₅₃	—	2	4	no	—
107	C₂×C₄×C₅₃	—	4	2	no	—
108	C₂×C₄×C₅₃	—	4	2	no	—
109	C₂×C₄×C₅₃	—	4	2	no	—
110	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
111	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
112	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
113	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
114	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
115	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
118	C₂×C₄×C₅₃	#127	1	2	yes	—
119	C₂×C₄×C₅₃	#127	1	2	yes	—
120	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
121	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
122	C₂×C₄×C₅₃	C₄⋊Q₈	1	4	yes	—
123	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
124	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
125	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₄×C₅₃	C₂²×Q₈	1	2	yes	—
127	C₈×C₅₃	—	4	4	no	—
128	#88322	—	4	4	no	—
129	#88322	C₂×C₄⋊C₄	1	16	yes	—
130	#20806	—	53	1	no	maximal, Sylow
131	C₂²×C₅₃	—	2	2	no	—
132	C₂²×C₅₃	—	2	2	no	—
133	C₂²×C₅₃	—	2	2	no	—
134	C₂²×C₅₃	—	4	4	no	—
135	C₂²×C₅₃	#384	1	2	yes	—
136	C₂²×C₅₃	#384	1	2	yes	—
137	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
138	C₂²×C₅₃	C₂×C₄⋊Q₈	1	2	yes	—
139	C₄×C₅₃	—	2	2	no	—
140	C₄×C₅₃	—	2	2	no	—
141	C₄×C₅₃	—	2	2	no	—
142	C₄×C₅₃	#384	1	2	yes	—
143	C₄×C₅₃	#384	1	2	yes	—
144	C₄×C₅₃	#430	1	4	yes	complemented
145	C₄×C₅₃	C₂²×C₄⋊C₄	1	2	yes	—
146	C₄×C₅₃	C₂×#127	1	1	yes	characteristic
147	C₄×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
148	C₂×#430	—	53	4	no	—
149	C₂×#423	—	53	1	no	—
150	C₂×#423	D₅₃	1	1	yes	characteristic
151	C₂×C₄×M₁₆	—	53	1	no	—
152	#1849	—	53	4	no	—
153	#1848	—	53	4	no	—
154	C₂×C₅₃	—	2	2	no	—
155	C₂×C₅₃	—	2	2	no	—
156	C₂×C₅₃	—	2	2	no	—
157	C₂×C₅₃	C₂×#384	1	1	yes	characteristic
158	C₂×C₅₃	C₂×#430	1	2	yes	—
159	C₂²×M₁₆	—	53	2	no	—
160	C₂²×M₁₆	—	53	2	no	—
161	C₂²×M₁₆	C₂×D₅₃	1	2	yes	—
162	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
163	#430	—	106	16	no	—
164	C₂×#108	—	53	16	no	—
165	#423	—	2	2	no	—
166	#423	—	53	2	no	—
167	#423	—	53	2	no	—
168	#423	—	106	2	no	—
169	#423	C₂×D₅₃	1	2	yes	—
170	#423	C₂×D₅₃	1	2	yes	—
171	C₂×C₄⋊C₈	—	2	2	no	—
172	C₂×C₄⋊C₈	—	106	2	no	—
173	C₄×M₁₆	—	53	2	no	—
174	C₄×M₁₆	—	53	2	no	—
175	C₄×M₁₆	—	53	2	no	—
176	C₄×M₁₆	—	53	2	no	—
177	C₂×#97	—	53	2	no	—
178	C₂×C₄×C₈	—	53	2	no	—
179	C₅₃	#20806	1	1	yes	characteristic, complemented, Sylow
180	C₂³×C₄	Dic₁₀₆	1	2	yes	—
181	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
182	C₂×M₁₆	—	2	2	no	—
183	C₂×M₁₆	—	2	8	no	—
184	C₂×M₁₆	—	53	4	no	—
185	C₂×M₁₆	—	53	4	no	—
186	C₂×M₁₆	—	53	8	no	—
187	C₂×M₁₆	—	53	8	no	—
188	C₂×M₁₆	—	53	4	no	—
189	C₂×M₁₆	—	106	2	no	—
190	C₂×M₁₆	—	106	8	no	—
191	C₂×M₁₆	C₂²×D₅₃	1	4	yes	—
192	C₂²×C₈	—	2	2	no	—
193	C₂²×C₈	—	53	4	no	—
194	C₂²×C₈	—	106	2	no	—
195	C₂×C₄²	Dic₁₀₆	1	2	yes	—
196	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
197	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
198	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
199	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
200	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
201	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
202	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
203	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
204	#108	—	106	32	no	—
205	C₄⋊C₈	—	2	4	no	—
206	C₄⋊C₈	—	2	4	no	—
207	C₄⋊C₈	—	106	4	no	—
208	C₄⋊C₈	—	106	4	no	—
209	#97	—	53	4	no	—
210	#97	—	53	4	no	—
211	C₄×C₈	—	53	4	no	—
212	C₄×C₈	—	53	4	no	—
213	C₂⁴	C₂×Dic₁₀₆	1	1	yes	characteristic
214	C₂²×C₄	—	4	4	no	—
215	C₂²×C₄	#11719919	1	2	yes	—
216	C₂²×C₄	#11719919	1	2	yes	—
217	C₂²×C₄	#11719919	1	2	yes	—
218	C₂²×C₄	#11719919	1	2	yes	—
219	C₂²×C₄	C₂×Dic₁₀₆	1	2	yes	—
220	C₂²×C₄	C₂×Dic₁₀₆	1	2	yes	—
221	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
222	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
223	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
224	C₂²×C₄	D₄×D₅₃	1	4	yes	—
225	C₂²×C₄	Q₈×D₅₃	1	4	yes	—
226	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
227	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
228	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
229	M₁₆	—	2	8	no	—
230	M₁₆	—	4	4	no	—
231	M₁₆	—	53	16	no	—
232	M₁₆	—	53	16	no	—
233	M₁₆	—	106	8	no	—
234	M₁₆	—	212	4	no	—
235	C₂×C₈	—	2	4	no	—
236	C₂×C₈	—	4	4	no	—
237	C₂×C₈	—	53	8	no	—
238	C₂×C₈	—	53	8	no	—
239	C₂×C₈	—	53	8	no	—
240	C₂×C₈	—	106	4	no	—
241	C₂×C₈	—	212	4	no	—
242	C₄²	#11719919	1	2	yes	—
243	C₄²	#11719919	1	2	yes	—
244	C₄²	#11719919	1	2	yes	—
245	C₄²	#11719919	1	2	yes	—
246	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
247	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
248	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
249	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
250	C₄²	C₂×C₂²⋊D₅₃	1	2	yes	—
251	C₄²	C₂×C₂²⋊D₅₃	1	2	yes	—
252	C₂³	—	4	2	no	—
253	C₂³	C₂×#11719919	1	1	yes	characteristic
254	C₂³	C₂×#11719919	1	1	yes	characteristic
255	C₂³	#13133216	1	4	yes	—
256	C₂³	C₂²×Dic₁₀₆	1	1	yes	characteristic
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	C₄⋊C₄×D₅₃	1	4	yes	—
264	C₂×C₄	C₄⋊C₄×D₅₃	1	4	yes	—
265	C₂×C₄	C₂×#11719919	1	2	yes	—
266	C₂×C₄	C₂×#11719919	1	2	yes	—
267	C₂×C₄	C₂×#11719919	1	2	yes	—
268	C₂×C₄	C₂×#11719919	1	2	yes	—
269	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
270	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
271	C₂×C₄	#13133213	1	2	yes	—
272	C₂×C₄	#13133213	1	2	yes	—
273	C₂×C₄	#13133215	1	2	yes	—
274	C₂×C₄	#13133215	1	2	yes	—
275	C₂×C₄	#13133216	1	4	yes	—
276	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
277	C₂×C₄	C₂×D₄×D₅₃	1	2	yes	—
278	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
279	C₂×C₄	C₂×Q₈×D₅₃	1	2	yes	—
280	C₈	—	4	4	no	—
281	C₈	—	53	16	no	—
282	C₈	—	212	4	no	—
283	C₂²	—	2	2	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	4	4	no	—
287	C₂²	#16810492	1	2	yes	—
288	C₂²	#16810492	1	2	yes	—
289	C₂²	C₂²×#11719919	1	1	yes	characteristic
290	C₂²	C₂×#13133216	1	2	yes	—
291	C₄	—	2	2	no	—
292	C₄	—	2	2	no	—
293	C₄	—	2	2	no	—
294	C₄	#16810492	1	2	yes	—
295	C₄	#16810492	1	2	yes	—
296	C₄	#16810671	1	4	yes	complemented
297	C₄	C₂×C₄⋊C₄×D₅₃	1	2	yes	—
298	C₄	C₂×#13133213	1	1	yes	characteristic
299	C₄	C₂×#13133215	1	1	yes	characteristic
300	C₂	—	2	2	no	—
301	C₂	—	2	2	no	—
302	C₂	—	2	2	no	—
303	C₂	C₂×#16810492	1	1	yes	characteristic
304	C₂	C₂×#16810671	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₅₃×#423	C₂
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₅₃	C₂×Q₈
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂×#423	D₅₃
hypocenter	C₅₃	#20806
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂×#423

Complements

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

N	H ≅ G/N	type
4. #20274889	C₂	semidirect
5. #20274886	C₂	semidirect
144. C₄×C₅₃	#430	semidirect
179. C₅₃	#20806	semidirect
296. C₄	#16810671	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810671	2
C₂×#16810647	2
C₂×C₄×#11719917	2
#20274889	2
#20274886	2
C₂×C₅₃×#423	2
#20806	53