Subgroups — #27289787

Order 13568.

674 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810394	C₂	1	2	yes	maximal
2	C₂×#16810383	C₂	1	2	yes	maximal
3	C₂×#16810381	C₂	1	1	yes	maximal, characteristic
4	C₂×#16810380	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719917	C₂²	1	2	yes	—
7	C₂×#13133176	C₂²	1	2	yes	—
8	#16810394	—	2	4	no	—
9	#16810383	—	2	4	no	—
10	#16810381	C₄	1	4	yes	complemented
11	#16810380	C₄	1	4	yes	complemented
12	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
13	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
14	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
15	C₄×M₁₆×C₅₃	—	2	4	no	—
16	C₂×#97×C₅₃	C₄	1	2	yes	complemented
17	C₂×C₄×C₈×C₅₃	C₄	1	2	yes	complemented
18	C₂×#11719917	—	2	2	no	—
19	C₂×#11719917	—	2	2	no	—
20	C₂×#11719917	—	2	2	no	—
21	C₂×#11719917	—	2	2	no	—
22	C₂×#11719917	—	2	2	no	—
23	C₂×#11719917	C₂×C₄	1	4	yes	—
24	C₂²×#88322	—	2	2	no	—
25	#13133176	—	2	2	no	—
26	#13133176	—	2	2	no	—
27	#13133176	—	2	2	no	—
28	#13133176	C₂×C₄	1	4	yes	—
29	C₂×C₄⋊Dic₅₃	—	2	2	no	—
30	C₂×C₂²⋊Dic₅₃	—	2	2	no	—
31	C₂×C₄×Dic₅₃	—	2	2	no	—
32	C₂³×C₄×C₅₃	—	2	1	no	—
33	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
34	C₂×M₁₆×C₅₃	—	2	2	no	—
35	C₂×M₁₆×C₅₃	—	2	2	no	—
36	C₂×M₁₆×C₅₃	—	2	2	no	—
37	C₂×M₁₆×C₅₃	—	2	2	no	—
38	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
39	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
40	C₂×M₁₆×C₅₃	D₄	1	2	yes	—
41	C₂×M₁₆×C₅₃	Q₈	1	2	yes	—
42	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
43	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₅₃	—	2	1	no	—
45	C₂×C₄²×C₅₃	—	2	2	no	—
46	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₅₃	D₄	1	4	yes	—
49	#97×C₅₃	—	2	4	no	—
50	C₄×C₈×C₅₃	—	2	4	no	—
51	C₂⁴×C₅₃	#41	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	—	2	1	no	—
53	C₂²×C₄×C₅₃	—	2	1	no	—
54	C₂²×C₄×C₅₃	—	2	4	no	—
55	C₂²×C₄×C₅₃	—	4	4	no	—
56	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
60	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	#41	1	2	yes	—
63	C₂²×C₄×C₅₃	#41	1	2	yes	—
64	C₂²×C₄×C₅₃	#41	1	2	yes	—
65	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
66	C₂²×Dic₅₃	—	4	2	no	—
67	M₁₆×C₅₃	—	2	4	no	—
68	M₁₆×C₅₃	—	2	4	no	—
69	M₁₆×C₅₃	—	2	4	no	—
70	M₁₆×C₅₃	—	2	4	no	—
71	C₂×C₈×C₅₃	—	2	2	no	—
72	C₂×C₈×C₅₃	—	2	2	no	—
73	C₂×C₈×C₅₃	—	2	2	no	—
74	C₂×C₈×C₅₃	—	2	2	no	—
75	C₂×C₈×C₅₃	C₄²	1	4	yes	—
76	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
77	C₄²×C₅₃	—	2	1	no	—
78	C₄²×C₅₃	—	2	1	no	—
79	C₄²×C₅₃	—	2	4	no	—
80	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
81	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
82	C₂²⋊Dic₅₃	—	2	4	no	—
83	C₂²⋊Dic₅₃	—	4	2	no	—
84	C₄⋊Dic₅₃	—	2	4	no	—
85	C₄⋊Dic₅₃	—	4	2	no	—
86	C₄×Dic₅₃	—	2	4	no	—
87	C₄×Dic₅₃	—	4	2	no	—
88	#11719917	—	2	4	no	—
89	#11719917	—	2	4	no	—
90	#11719917	—	4	2	no	—
91	#11719917	—	4	2	no	—
92	C₂×#88322	—	2	4	no	—
93	C₂×#88322	—	4	2	no	—
94	C₂×#88322	—	4	2	no	—
95	C₂³×C₅₃	—	4	2	no	—
96	C₂³×C₅₃	#117	1	1	yes	characteristic
97	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
98	C₂³×C₅₃	C₄×Q₈	1	1	yes	characteristic
99	C₂³×C₅₃	#123	1	2	yes	—
100	C₂³×C₅₃	#126	1	2	yes	—
101	C₂×C₄×C₅₃	—	2	4	no	—
102	C₂×C₄×C₅₃	—	2	2	no	—
103	C₂×C₄×C₅₃	—	2	2	no	—
104	C₂×C₄×C₅₃	—	4	8	no	—
105	C₂×C₄×C₅₃	—	4	2	no	—
106	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
107	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₅₃	#117	1	2	yes	—
110	C₂×C₄×C₅₃	#117	1	2	yes	—
111	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
112	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
113	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
114	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
115	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
117	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
118	C₂×C₄×C₅₃	#121	1	2	yes	—
119	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
120	C₂×C₄×C₅₃	#123	1	2	yes	—
121	C₂×C₄×C₅₃	#124	1	2	yes	—
122	C₂×C₄×C₅₃	#125	1	2	yes	—
123	C₂×C₄×C₅₃	#126	1	2	yes	—
124	C₂×Dic₅₃	—	4	4	no	—
125	C₂×Dic₅₃	—	8	4	no	—
126	C₈×C₅₃	—	2	4	no	—
127	C₈×C₅₃	—	2	4	no	—
128	#88322	—	4	4	no	—
129	#10813	—	53	1	no	maximal, Sylow
130	C₂²×C₅₃	—	2	4	no	—
131	C₂²×C₅₃	—	2	2	no	—
132	C₂²×C₅₃	—	4	2	no	—
133	C₂²×C₅₃	—	4	1	no	—
134	C₂²×C₅₃	—	4	1	no	—
135	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
136	C₂²×C₅₃	#381	1	1	yes	characteristic
137	C₂²×C₅₃	#383	1	1	yes	characteristic
138	C₂²×C₅₃	#387	1	2	yes	—
139	C₂²×C₅₃	#388	1	2	yes	—
140	C₄×C₅₃	—	2	4	no	—
141	C₄×C₅₃	—	4	2	no	—
142	C₄×C₅₃	#376	1	1	yes	characteristic
143	C₄×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
144	C₄×C₅₃	#380	1	1	yes	characteristic
145	C₄×C₅₃	#382	1	1	yes	characteristic
146	C₄×C₅₃	#387	1	2	yes	—
147	C₄×C₅₃	#388	1	2	yes	—
148	Dic₅₃	—	8	4	no	—
149	C₂×C₄×M₁₆	D₅₃	1	1	yes	characteristic
150	C₂×#344	—	53	2	no	—
151	C₂×#339	—	53	2	no	—
152	C₂×#338	—	53	1	no	—
153	C₂×H(3,ℤ/4)	—	53	1	no	—
154	C₂×C₅₃	—	2	2	no	—
155	C₂×C₅₃	—	2	1	no	—
156	C₂×C₅₃	—	2	1	no	—
157	C₂×C₅₃	—	2	2	no	—
158	C₂×C₅₃	#1434	1	1	yes	characteristic
159	C₂×C₅₃	#1841	1	2	yes	—
160	C₂²×M₁₆	—	53	2	no	—
161	C₂²×M₁₆	C₂×D₅₃	1	1	yes	characteristic
162	C₂²×M₁₆	C₂×D₅₃	1	1	yes	characteristic
163	C₂×#117	—	53	2	no	—
164	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
165	C₄×M₁₆	—	2	4	no	—
166	C₂×#97	Dic₅₃	1	2	yes	complemented
167	C₂×C₄×C₈	Dic₅₃	1	2	yes	complemented
168	#344	—	106	4	no	—
169	#339	—	106	4	no	—
170	#338	—	53	4	no	—
171	H(3,ℤ/4)	—	53	4	no	—
172	C₅₃	#10813	1	1	yes	characteristic, complemented, Sylow
173	C₂³×C₄	—	2	1	no	—
174	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
175	C₂×M₁₆	—	2	2	no	—
176	C₂×M₁₆	—	2	2	no	—
177	C₂×M₁₆	—	2	2	no	—
178	C₂×M₁₆	—	2	2	no	—
179	C₂×M₁₆	—	53	4	no	—
180	C₂×M₁₆	—	106	2	no	—
181	C₂×M₁₆	—	106	2	no	—
182	C₂×M₁₆	—	106	2	no	—
183	C₂×M₁₆	—	106	2	no	—
184	C₂×M₁₆	—	106	2	no	—
185	C₂×M₁₆	Dic₁₀₆	1	2	yes	—
186	C₂×M₁₆	C₄×D₅₃	1	2	yes	—
187	C₂×M₁₆	C₄×D₅₃	1	2	yes	—
188	C₂×M₁₆	D₂₁₂	1	2	yes	—
189	C₂²×C₈	—	106	2	no	—
190	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
191	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
192	#117	—	53	4	no	—
193	#117	—	106	2	no	—
194	#117	—	106	2	no	—
195	#117	—	106	2	no	—
196	C₂×C₄⋊C₄	—	106	2	no	—
197	C₂×C₂²⋊C₄	—	106	2	no	—
198	C₂×C₄²	—	2	2	no	—
199	C₂×C₄²	—	2	1	no	—
200	C₂×C₄²	—	106	2	no	—
201	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
202	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
203	C₂×C₄²	C₂²⋊D₅₃	1	4	yes	—
204	#97	—	2	4	no	—
205	C₄×C₈	—	2	4	no	—
206	C₂⁴	#11719945	1	1	yes	characteristic
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	—	2	1	no	—
209	C₂²×C₄	—	2	4	no	—
210	C₂²×C₄	—	4	4	no	—
211	C₂²×C₄	—	212	2	no	—
212	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
213	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
214	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
215	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
216	C₂²×C₄	#11719945	1	2	yes	—
217	C₂²×C₄	#11719945	1	2	yes	—
218	C₂²×C₄	#11719945	1	2	yes	—
219	C₂²×C₄	#11719945	1	1	yes	characteristic
220	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
221	C₂²×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
222	M₁₆	—	2	4	no	—
223	M₁₆	—	2	4	no	—
224	M₁₆	—	2	4	no	—
225	M₁₆	—	2	4	no	—
226	M₁₆	—	106	4	no	—
227	M₁₆	—	106	4	no	—
228	M₁₆	—	212	2	no	—
229	M₁₆	—	212	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	106	4	no	—
235	C₂×C₈	—	212	2	no	—
236	C₂×C₈	—	212	2	no	—
237	C₂×C₈	C₄×Dic₅₃	1	4	yes	—
238	C₂×C₈	C₄⋊Dic₅₃	1	4	yes	—
239	C₄⋊C₄	—	106	4	no	—
240	C₄⋊C₄	—	212	2	no	—
241	C₂²⋊C₄	—	106	4	no	—
242	C₂²⋊C₄	—	212	2	no	—
243	C₄²	—	2	1	no	—
244	C₄²	—	2	1	no	—
245	C₄²	—	2	4	no	—
246	C₄²	—	106	4	no	—
247	C₄²	—	212	2	no	—
248	C₄²	C₂²⋊Dic₅₃	1	2	yes	—
249	C₄²	C₂²⋊Dic₅₃	1	2	yes	—
250	C₂³	—	4	2	no	—
251	C₂³	#13133176	1	1	yes	characteristic
252	C₂³	C₄×D₂₁₂	1	1	yes	characteristic
253	C₂³	C₄×Dic₁₀₆	1	1	yes	characteristic
254	C₂³	#13133200	1	2	yes	—
255	C₂³	#13133211	1	2	yes	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	4	8	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	424	4	no	—
262	C₂×C₄	—	212	4	no	—
263	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
264	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
265	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
266	C₂×C₄	#13133174	1	2	yes	—
267	C₂×C₄	#13133174	1	2	yes	—
268	C₂×C₄	#13133176	1	1	yes	characteristic
269	C₂×C₄	#13133176	1	1	yes	characteristic
270	C₂×C₄	#13133176	1	1	yes	characteristic
271	C₂×C₄	C₄×C₂²⋊D₅₃	1	2	yes	—
272	C₂×C₄	C₄×C₂²⋊D₅₃	1	2	yes	—
273	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
274	C₂×C₄	C₄×Dic₁₀₆	1	1	yes	characteristic
275	C₂×C₄	#13133191	1	2	yes	—
276	C₂×C₄	#13133196	1	2	yes	—
277	C₂×C₄	#13133200	1	2	yes	—
278	C₂×C₄	#13133205	1	2	yes	—
279	C₂×C₄	#13133209	1	2	yes	—
280	C₂×C₄	#13133211	1	2	yes	—
281	C₈	—	2	4	no	—
282	C₈	—	2	4	no	—
283	C₈	—	212	4	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	2	4	no	—
286	C₂²	—	4	2	no	—
287	C₂²	—	4	1	no	—
288	C₂²	—	4	1	no	—
289	C₂²	C₄×C₄⋊Dic₅₃	1	1	yes	characteristic
290	C₂²	#16810486	1	1	yes	characteristic
291	C₂²	#16810491	1	1	yes	characteristic
292	C₂²	#16810507	1	2	yes	—
293	C₂²	#16810514	1	2	yes	—
294	C₄	—	2	4	no	—
295	C₄	—	4	2	no	—
296	C₄	—	424	4	no	—
297	C₄	#16810469	1	1	yes	characteristic
298	C₄	C₄×C₂²⋊Dic₅₃	1	1	yes	characteristic
299	C₄	#16810483	1	1	yes	characteristic
300	C₄	#16810489	1	1	yes	characteristic
301	C₄	#16810507	1	2	yes	—
302	C₄	#16810514	1	2	yes	—
303	C₂	—	2	1	no	—
304	C₂	—	2	2	no	—
305	C₂	—	2	2	no	—
306	C₂	—	2	1	no	—
307	C₂	#20272942	1	1	yes	characteristic
308	C₂	#20274864	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#13133176
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₂×C₄×M₁₆×C₅₃	C₂
socle	C₂²×C₅₃	#383
prime radical	C₂⁴×C₅₃	#41
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	C₂×C₄×M₁₆	D₅₃
hypocenter	C₅₃	#10813
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆ ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
10. #16810381	C₄	semidirect
11. #16810380	C₄	semidirect
16. C₂×#97×C₅₃	C₄	semidirect
17. C₂×C₄×C₈×C₅₃	C₄	semidirect
166. C₂×#97	Dic₅₃	semidirect
167. C₂×C₄×C₈	Dic₅₃	semidirect
172. C₅₃	#10813	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810394	2
C₂×#16810383	2
C₂×#16810381	2
C₂×#16810380	2
C₂×C₄×M₁₆×C₅₃	2
#10813	53