Subgroups — C₂×C₅₃×#1349 #27219219

Order 13568.

498 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#466	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅₃×#341	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#1349	C₂	1	4	yes	maximal, direct product factor
5	C₂²×D₈×C₅₃	C₄	1	1	yes	characteristic
6	C₂²×M₁₆×C₅₃	—	2	1	no	—
7	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
8	C₂×#121×C₅₃	—	2	1	no	—
9	C₂²×C₁₆×C₅₃	—	2	1	no	—
10	C₅₃×#466	—	2	1	no	—
11	C₅₃×#466	—	2	1	no	—
12	C₅₃×#466	C₄	1	2	yes	—
13	C₅₃×#466	C₂²	1	2	yes	—
14	C₂×#107×C₅₃	C₄	1	1	yes	characteristic
15	C₂×#102×C₅₃	—	2	1	no	—
16	C₅₃×C₂²⋊C₁₆	—	2	1	no	—
17	C₅₃×C₂²⋊C₁₆	C₂²	1	2	yes	complemented
18	C₅₃×#341	—	2	1	no	—
19	C₅₃×#341	C₂²	1	2	yes	complemented
20	C₂²×D₄×C₅₃	—	2	1	no	—
21	C₂²×D₄×C₅₃	—	4	1	no	—
22	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
23	C₂×D₈×C₅₃	—	2	1	no	—
24	C₂×D₈×C₅₃	—	4	2	no	—
25	C₂×D₈×C₅₃	C₂×C₄	1	2	yes	—
26	C₂×M₁₆×C₅₃	—	2	4	no	—
27	C₂×M₁₆×C₅₃	—	2	2	no	—
28	C₂×M₁₆×C₅₃	—	4	2	no	—
29	C₂×M₁₆×C₅₃	—	4	1	no	—
30	C₂²×C₈×C₅₃	—	4	2	no	—
31	C₂²×C₈×C₅₃	—	4	1	no	—
32	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
38	#121×C₅₃	—	2	2	no	—
39	#121×C₅₃	—	2	2	no	—
40	#121×C₅₃	—	4	1	no	—
41	#121×C₅₃	—	4	1	no	—
42	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
43	C₂×C₂²⋊C₄×C₅₃	—	4	1	no	—
44	C₂×C₁₆×C₅₃	—	2	2	no	—
45	C₂×C₁₆×C₅₃	—	4	2	no	—
46	#107×C₅₃	—	2	1	no	—
47	#107×C₅₃	C₂×C₄	1	2	yes	—
48	#102×C₅₃	—	2	2	no	—
49	#102×C₅₃	—	2	2	no	—
50	C₂⁴×C₅₃	—	4	1	no	—
51	C₂⁴×C₅₃	SD₁₆	1	1	yes	characteristic
52	C₂×D₄×C₅₃	—	2	2	no	—
53	C₂×D₄×C₅₃	—	4	2	no	—
54	C₂×D₄×C₅₃	—	4	2	no	—
55	C₂×D₄×C₅₃	—	4	2	no	—
56	C₂×D₄×C₅₃	—	4	1	no	—
57	C₂×D₄×C₅₃	—	4	2	no	—
58	C₂×D₄×C₅₃	—	8	4	no	—
59	C₂²×C₄×C₅₃	—	2	2	no	—
60	C₂²×C₄×C₅₃	—	2	2	no	—
61	C₂²×C₄×C₅₃	—	4	1	no	—
62	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
68	D₈×C₅₃	—	4	2	no	—
69	D₈×C₅₃	—	4	2	no	—
70	M₁₆×C₅₃	—	2	4	no	—
71	M₁₆×C₅₃	—	4	4	no	—
72	M₁₆×C₅₃	—	4	2	no	—
73	C₂×C₈×C₅₃	—	2	2	no	—
74	C₂×C₈×C₅₃	—	2	2	no	—
75	C₂×C₈×C₅₃	—	4	1	no	—
76	C₂×C₈×C₅₃	—	4	1	no	—
77	C₂×C₈×C₅₃	—	4	1	no	—
78	C₂×C₈×C₅₃	—	4	1	no	—
79	C₂×C₈×C₅₃	—	4	4	no	—
80	C₂×C₈×C₅₃	—	4	2	no	—
81	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
85	C₄⋊C₄×C₅₃	—	2	2	no	—
86	C₄⋊C₄×C₅₃	—	4	1	no	—
87	C₂²⋊C₄×C₅₃	—	4	2	no	—
88	C₂²⋊C₄×C₅₃	—	4	2	no	—
89	C₁₆×C₅₃	—	4	2	no	—
90	C₂³×C₅₃	—	2	2	no	—
91	C₂³×C₅₃	—	2	2	no	—
92	C₂³×C₅₃	—	2	2	no	—
93	C₂³×C₅₃	—	4	2	no	—
94	C₂³×C₅₃	—	8	2	no	—
95	C₂³×C₅₃	—	8	2	no	—
96	C₂³×C₅₃	—	8	2	no	—
97	C₂³×C₅₃	#102	1	1	yes	characteristic
98	C₂³×C₅₃	#102	1	1	yes	characteristic
99	C₂³×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
100	D₄×C₅₃	—	4	2	no	—
101	D₄×C₅₃	—	4	2	no	—
102	D₄×C₅₃	—	8	4	no	—
103	D₄×C₅₃	—	8	4	no	—
104	D₄×C₅₃	—	8	2	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	1	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	2	no	—
113	C₂×C₄×C₅₃	—	8	2	no	—
114	C₂×C₄×C₅₃	#99	1	2	yes	—
115	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
116	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
117	C₂×C₄×C₅₃	#104	1	2	yes	—
118	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
120	C₈×C₅₃	—	2	1	no	—
121	C₈×C₅₃	—	2	1	no	—
122	C₈×C₅₃	—	2	1	no	—
123	C₈×C₅₃	—	2	1	no	—
124	C₈×C₅₃	—	4	4	no	—
125	C₂×#1349	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
126	C₂²×C₅₃	—	2	1	no	—
127	C₂²×C₅₃	—	2	1	no	—
128	C₂²×C₅₃	—	2	1	no	—
129	C₂²×C₅₃	—	2	1	no	—
130	C₂²×C₅₃	—	2	1	no	—
131	C₂²×C₅₃	—	2	1	no	—
132	C₂²×C₅₃	—	4	4	no	—
133	C₂²×C₅₃	—	8	2	no	—
134	C₂²×C₅₃	—	8	2	no	—
135	C₂²×C₅₃	—	8	2	no	—
136	C₂²×C₅₃	—	16	4	no	—
137	C₂²×C₅₃	#327	1	2	yes	—
138	C₂²×C₅₃	#359	1	2	yes	—
139	C₂²×C₅₃	#361	1	2	yes	—
140	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
141	C₄×C₅₃	—	2	1	no	—
142	C₄×C₅₃	—	2	1	no	—
143	C₄×C₅₃	—	8	2	no	—
144	C₄×C₅₃	#327	1	2	yes	—
145	C₄×C₅₃	C₂×#99	1	1	yes	characteristic
146	C₄×C₅₃	C₂×#104	1	1	yes	characteristic
147	C₂×#466	C₂×C₅₃	1	1	yes	characteristic
148	C₂×C₂²⋊C₁₆	C₂×C₅₃	1	1	yes	characteristic, complemented
149	C₂×#341	C₂×C₅₃	1	1	yes	characteristic, complemented
150	#1349	C₂×C₅₃	1	4	yes	direct product factor
151	C₂×C₅₃	—	4	1	no	—
152	C₂×C₅₃	—	4	1	no	—
153	C₂×C₅₃	—	16	2	no	—
154	C₂×C₅₃	#1349	1	4	yes	direct product factor
155	C₂×C₅₃	C₂×#327	1	1	yes	characteristic
156	C₂×C₅₃	C₂×#359	1	1	yes	characteristic
157	C₂×C₅₃	C₂×#361	1	1	yes	characteristic
158	C₂²×D₈	C₄×C₅₃	1	1	yes	characteristic
159	C₂²×M₁₆	—	2	1	no	—
160	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
161	C₂×#121	—	2	1	no	—
162	C₂²×C₁₆	—	2	1	no	—
163	#466	—	2	1	no	—
164	#466	—	2	1	no	—
165	#466	C₄×C₅₃	1	2	yes	—
166	#466	C₂²×C₅₃	1	2	yes	—
167	C₂×#107	C₄×C₅₃	1	1	yes	characteristic
168	C₂×#102	—	2	1	no	—
169	C₂²⋊C₁₆	—	2	1	no	—
170	C₂²⋊C₁₆	C₂²×C₅₃	1	2	yes	complemented
171	#341	—	2	1	no	—
172	#341	C₂²×C₅₃	1	2	yes	complemented
173	C₅₃	C₂×#1349	1	1	yes	characteristic, direct product factor, Sylow
174	C₂²×D₄	—	2	1	no	—
175	C₂²×D₄	—	4	1	no	—
176	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
177	C₂×D₈	—	2	1	no	—
178	C₂×D₈	—	4	2	no	—
179	C₂×D₈	C₂×C₄×C₅₃	1	2	yes	—
180	C₂×M₁₆	—	2	4	no	—
181	C₂×M₁₆	—	2	2	no	—
182	C₂×M₁₆	—	4	2	no	—
183	C₂×M₁₆	—	4	1	no	—
184	C₂²×C₈	—	4	2	no	—
185	C₂²×C₈	—	4	1	no	—
186	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
187	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
188	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
189	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
190	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
191	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
192	#121	—	2	2	no	—
193	#121	—	2	2	no	—
194	#121	—	4	1	no	—
195	#121	—	4	1	no	—
196	C₂×C₄⋊C₄	—	2	1	no	—
197	C₂×C₂²⋊C₄	—	4	1	no	—
198	C₂×C₁₆	—	2	2	no	—
199	C₂×C₁₆	—	4	2	no	—
200	#107	—	2	1	no	—
201	#107	C₂×C₄×C₅₃	1	2	yes	—
202	#102	—	2	2	no	—
203	#102	—	2	2	no	—
204	C₂⁴	—	4	1	no	—
205	C₂⁴	SD₁₆×C₅₃	1	1	yes	characteristic
206	C₂×D₄	—	2	2	no	—
207	C₂×D₄	—	4	2	no	—
208	C₂×D₄	—	4	2	no	—
209	C₂×D₄	—	4	2	no	—
210	C₂×D₄	—	4	1	no	—
211	C₂×D₄	—	4	2	no	—
212	C₂×D₄	—	8	4	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	4	1	no	—
216	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
217	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
218	C₂²×C₄	D₈×C₅₃	1	1	yes	characteristic
219	C₂²×C₄	D₈×C₅₃	1	1	yes	characteristic
220	C₂²×C₄	SD₁₆×C₅₃	1	1	yes	characteristic
221	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
222	D₈	—	4	2	no	—
223	D₈	—	4	2	no	—
224	M₁₆	—	2	4	no	—
225	M₁₆	—	4	4	no	—
226	M₁₆	—	4	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	4	1	no	—
230	C₂×C₈	—	4	1	no	—
231	C₂×C₈	—	4	1	no	—
232	C₂×C₈	—	4	1	no	—
233	C₂×C₈	—	4	4	no	—
234	C₂×C₈	—	4	2	no	—
235	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
236	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
237	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
238	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
239	C₄⋊C₄	—	2	2	no	—
240	C₄⋊C₄	—	4	1	no	—
241	C₂²⋊C₄	—	4	2	no	—
242	C₂²⋊C₄	—	4	2	no	—
243	C₁₆	—	4	2	no	—
244	C₂³	—	2	2	no	—
245	C₂³	—	2	2	no	—
246	C₂³	—	2	2	no	—
247	C₂³	—	4	2	no	—
248	C₂³	—	8	2	no	—
249	C₂³	—	8	2	no	—
250	C₂³	—	8	2	no	—
251	C₂³	#102×C₅₃	1	1	yes	characteristic
252	C₂³	#102×C₅₃	1	1	yes	characteristic
253	C₂³	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
254	D₄	—	4	2	no	—
255	D₄	—	4	2	no	—
256	D₄	—	8	4	no	—
257	D₄	—	8	4	no	—
258	D₄	—	8	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	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	—	4	2	no	—
267	C₂×C₄	—	8	2	no	—
268	C₂×C₄	#99×C₅₃	1	2	yes	—
269	C₂×C₄	#102×C₅₃	1	1	yes	characteristic
270	C₂×C₄	#102×C₅₃	1	1	yes	characteristic
271	C₂×C₄	#104×C₅₃	1	2	yes	—
272	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
273	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
274	C₈	—	2	1	no	—
275	C₈	—	2	1	no	—
276	C₈	—	2	1	no	—
277	C₈	—	2	1	no	—
278	C₈	—	4	4	no	—
279	C₂²	—	2	1	no	—
280	C₂²	—	2	1	no	—
281	C₂²	—	2	1	no	—
282	C₂²	—	2	1	no	—
283	C₂²	—	2	1	no	—
284	C₂²	—	2	1	no	—
285	C₂²	—	4	4	no	—
286	C₂²	—	8	2	no	—
287	C₂²	—	8	2	no	—
288	C₂²	—	8	2	no	—
289	C₂²	—	16	4	no	—
290	C₂²	C₅₃×#327	1	2	yes	—
291	C₂²	C₅₃×#359	1	2	yes	—
292	C₂²	C₅₃×#361	1	2	yes	—
293	C₂²	C₂×#102×C₅₃	1	1	yes	characteristic
294	C₄	—	2	1	no	—
295	C₄	—	2	1	no	—
296	C₄	—	8	2	no	—
297	C₄	C₅₃×#327	1	2	yes	—
298	C₄	C₂×#99×C₅₃	1	1	yes	characteristic
299	C₄	C₂×#104×C₅₃	1	1	yes	characteristic
300	C₂	—	4	1	no	—
301	C₂	—	4	1	no	—
302	C₂	—	16	2	no	—
303	C₂	C₅₃×#1349	1	4	yes	direct product factor
304	C₂	C₂×C₅₃×#327	1	1	yes	characteristic
305	C₂	C₂×C₅₃×#359	1	1	yes	characteristic
306	C₂	C₂×C₅₃×#361	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅₃	#102
commutator	C₂×C₈	C₂²×C₄×C₅₃
Frattini, cosocle	C₂²×C₈	C₂³×C₅₃
prime radical	C₂×C₅₃×#466	C₂
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂³×C₄×C₅₃ ◁ C₂³×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅₃×C₂²⋊C₁₆	C₂	semidirect
3. C₂×C₅₃×#341	C₂	semidirect
4. C₅₃×#1349	C₂	direct (N × H)
17. C₅₃×C₂²⋊C₁₆	C₂²	semidirect
19. C₅₃×#341	C₂²	semidirect
125. C₂×#1349	C₅₃	direct (N × H)
148. C₂×C₂²⋊C₁₆	C₂×C₅₃	semidirect
149. C₂×#341	C₂×C₅₃	semidirect
150. #1349	C₂×C₅₃	direct (N × H)
154. C₂×C₅₃	#1349	direct (N × H)
170. C₂²⋊C₁₆	C₂²×C₅₃	semidirect
172. #341	C₂²×C₅₃	semidirect
173. C₅₃	C₂×#1349	direct (N × H)
303. C₂	C₅₃×#1349	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#466	2
C₂×C₅₃×C₂²⋊C₁₆	2
C₂×C₅₃×#341	2
C₅₃×#1349	2
C₂×#1349	53

Subgroups — C2×C53×#1349 #27219219

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#1349 #27219219