Subgroups — C₂×C₅₃×#1673 #27226615

Order 13568.

790 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#500	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅₃×#461	C₂	1	2	yes	maximal
3	C₂×C₈×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1673	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₅₃×#335	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅₃×#332	C₂	1	2	yes	maximal, complemented
7	C₂²×SD₁₆×C₅₃	—	2	2	no	—
8	C₂³×C₈×C₅₃	—	2	1	no	—
9	C₂×C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
10	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
11	C₅₃×#500	—	2	2	no	—
12	C₅₃×#500	C₂²	1	4	yes	complemented
13	C₅₃×#461	—	2	4	no	—
14	C₅₃×#461	C₂²	1	8	yes	—
15	C₈×D₄×C₅₃	—	2	2	no	—
16	C₈×D₄×C₅₃	C₂²	1	4	yes	—
17	C₂×#107×C₅₃	—	2	1	no	—
18	C₂×#107×C₅₃	—	2	1	no	—
19	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
20	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×#103×C₅₃	—	2	2	no	—
22	C₂×C₂²⋊C₈×C₅₃	—	2	1	no	—
23	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₅₃×#335	C₂²	1	4	yes	complemented
25	C₅₃×#332	C₂²	1	8	yes	complemented
26	C₂²×Q₈×C₅₃	—	2	2	no	—
27	C₂²×D₄×C₅₃	D₄	1	1	yes	characteristic
28	C₂³×C₄×C₅₃	—	2	1	no	—
29	C₂×SD₁₆×C₅₃	—	2	8	no	—
30	C₂×SD₁₆×C₅₃	—	4	8	no	—
31	C₂²×C₈×C₅₃	—	2	4	no	—
32	C₂²×C₈×C₅₃	—	2	1	no	—
33	C₂²×C₈×C₅₃	—	2	2	no	—
34	C₂²×C₈×C₅₃	—	4	4	no	—
35	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
37	C₄⋊Q₈×C₅₃	—	2	2	no	—
38	C₄⋊Q₈×C₅₃	—	2	2	no	—
39	C₄⋊Q₈×C₅₃	—	2	2	no	—
40	C₄⋊Q₈×C₅₃	C₂³	1	4	yes	—
41	C₄×D₄×C₅₃	—	2	1	no	—
42	C₄×D₄×C₅₃	—	2	1	no	—
43	C₄×D₄×C₅₃	D₄	1	2	yes	—
44	C₄×D₄×C₅₃	C₂³	1	2	yes	—
45	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
46	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
47	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
48	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
49	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
50	#107×C₅₃	—	2	4	no	—
51	#107×C₅₃	—	2	4	no	—
52	C₄⋊C₈×C₅₃	—	2	1	no	—
53	C₄⋊C₈×C₅₃	—	2	2	no	—
54	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
55	C₄⋊C₈×C₅₃	C₂³	1	4	yes	—
56	#103×C₅₃	—	2	8	no	—
57	C₂²⋊C₈×C₅₃	—	2	4	no	—
58	C₄×C₈×C₅₃	D₄	1	2	yes	—
59	C₄×C₈×C₅₃	C₂³	1	2	yes	—
60	C₂⁴×C₅₃	—	2	1	no	—
61	C₂×Q₈×C₅₃	—	2	4	no	—
62	C₂×Q₈×C₅₃	—	4	2	no	—
63	C₂×Q₈×C₅₃	—	4	4	no	—
64	C₂×Q₈×C₅₃	—	4	4	no	—
65	C₂×D₄×C₅₃	—	2	1	no	—
66	C₂×D₄×C₅₃	—	2	2	no	—
67	C₂×D₄×C₅₃	D₈	1	4	yes	—
68	C₂×D₄×C₅₃	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₅₃	—	2	1	no	—
70	C₂²×C₄×C₅₃	—	2	2	no	—
71	C₂²×C₄×C₅₃	—	2	2	no	—
72	C₂²×C₄×C₅₃	—	4	2	no	—
73	C₂²×C₄×C₅₃	—	4	2	no	—
74	C₂²×C₄×C₅₃	—	4	2	no	—
75	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
78	SD₁₆×C₅₃	—	4	16	no	—
79	C₂×C₈×C₅₃	—	2	2	no	—
80	C₂×C₈×C₅₃	—	2	2	no	—
81	C₂×C₈×C₅₃	—	2	4	no	—
82	C₂×C₈×C₅₃	—	4	4	no	—
83	C₂×C₈×C₅₃	—	4	2	no	—
84	C₂×C₈×C₅₃	—	4	8	no	—
85	C₂×C₈×C₅₃	SD₁₆	1	4	yes	complemented
86	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
87	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
88	C₄⋊C₄×C₅₃	—	2	1	no	—
89	C₄⋊C₄×C₅₃	—	2	4	no	—
90	C₄⋊C₄×C₅₃	—	2	4	no	—
91	C₄⋊C₄×C₅₃	—	4	2	no	—
92	C₄⋊C₄×C₅₃	—	4	2	no	—
93	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
94	C₂²⋊C₄×C₅₃	—	2	2	no	—
95	C₂²⋊C₄×C₅₃	—	2	2	no	—
96	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₅₃	#41	1	1	yes	characteristic
99	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
100	C₂³×C₅₃	—	2	2	no	—
101	C₂³×C₅₃	—	2	4	no	—
102	C₂³×C₅₃	—	4	2	no	—
103	C₂³×C₅₃	—	4	2	no	—
104	C₂³×C₅₃	#121	1	1	yes	characteristic
105	Q₈×C₅₃	—	4	4	no	—
106	Q₈×C₅₃	—	4	4	no	—
107	Q₈×C₅₃	—	8	4	no	—
108	D₄×C₅₃	—	2	2	no	—
109	D₄×C₅₃	—	2	4	no	—
110	D₄×C₅₃	C₂×D₈	1	4	yes	—
111	C₂×C₄×C₅₃	—	2	2	no	—
112	C₂×C₄×C₅₃	—	2	2	no	—
113	C₂×C₄×C₅₃	—	4	2	no	—
114	C₂×C₄×C₅₃	—	4	2	no	—
115	C₂×C₄×C₅₃	—	4	2	no	—
116	C₂×C₄×C₅₃	—	4	4	no	—
117	C₂×C₄×C₅₃	—	4	4	no	—
118	C₂×C₄×C₅₃	—	8	4	no	—
119	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
120	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
121	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
123	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
124	C₂×C₄×C₅₃	#135	1	2	yes	—
125	C₂×C₄×C₅₃	#137	1	2	yes	—
126	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
128	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
129	C₈×C₅₃	—	2	2	no	—
130	C₈×C₅₃	—	4	4	no	—
131	C₈×C₅₃	—	4	2	no	—
132	C₈×C₅₃	C₂×SD₁₆	1	4	yes	complemented
133	C₂×#1673	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
134	C₂²×C₅₃	—	2	4	no	—
135	C₂²×C₅₃	—	4	8	no	—
136	C₂²×C₅₃	—	4	2	no	—
137	C₂²×C₅₃	—	4	2	no	—
138	C₂²×C₅₃	#461	1	2	yes	—
139	C₂²×C₅₃	#466	1	2	yes	—
140	C₂²×C₅₃	#473	1	2	yes	—
141	C₂²×C₅₃	C₂×#121	1	1	yes	characteristic
142	C₄×C₅₃	—	2	1	no	—
143	C₄×C₅₃	—	2	1	no	—
144	C₄×C₅₃	—	4	2	no	—
145	C₄×C₅₃	—	8	4	no	—
146	C₄×C₅₃	#461	1	2	yes	—
147	C₄×C₅₃	#466	1	2	yes	—
148	C₄×C₅₃	C₂²×D₈	1	1	yes	characteristic
149	C₄×C₅₃	C₂²×SD₁₆	1	1	yes	characteristic
150	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
151	C₄×C₅₃	C₂×#137	1	1	yes	characteristic
152	C₂×#500	C₂×C₅₃	1	1	yes	characteristic, complemented
153	C₂×#461	C₂×C₅₃	1	2	yes	—
154	C₂×C₈×D₄	C₂×C₅₃	1	1	yes	characteristic
155	#1673	C₂×C₅₃	1	8	yes	direct product factor
156	C₂×#335	C₂×C₅₃	1	1	yes	characteristic, complemented
157	C₂×#332	C₂×C₅₃	1	2	yes	complemented
158	C₂×C₅₃	—	4	4	no	—
159	C₂×C₅₃	#1673	1	4	yes	direct product factor
160	C₂×C₅₃	C₂×#461	1	1	yes	characteristic
161	C₂×C₅₃	C₂×#466	1	1	yes	characteristic
162	C₂×C₅₃	C₂×#473	1	1	yes	characteristic
163	C₂²×SD₁₆	—	2	2	no	—
164	C₂³×C₈	—	2	1	no	—
165	C₂×C₄⋊Q₈	C₂²×C₅₃	1	2	yes	—
166	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
167	#500	—	2	2	no	—
168	#500	C₂²×C₅₃	1	4	yes	complemented
169	#461	—	2	4	no	—
170	#461	C₂²×C₅₃	1	8	yes	—
171	C₈×D₄	—	2	2	no	—
172	C₈×D₄	C₂²×C₅₃	1	4	yes	—
173	C₂×#107	—	2	1	no	—
174	C₂×#107	—	2	1	no	—
175	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
176	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
177	C₂×#103	—	2	2	no	—
178	C₂×C₂²⋊C₈	—	2	1	no	—
179	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
180	#335	C₂²×C₅₃	1	4	yes	complemented
181	#332	C₂²×C₅₃	1	8	yes	complemented
182	C₅₃	C₂×#1673	1	1	yes	characteristic, direct product factor, Sylow
183	C₂²×Q₈	—	2	2	no	—
184	C₂²×D₄	D₄×C₅₃	1	1	yes	characteristic
185	C₂³×C₄	—	2	1	no	—
186	C₂×SD₁₆	—	2	8	no	—
187	C₂×SD₁₆	—	4	8	no	—
188	C₂²×C₈	—	2	1	no	—
189	C₂²×C₈	—	2	2	no	—
190	C₂²×C₈	—	2	4	no	—
191	C₂²×C₈	—	4	4	no	—
192	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
193	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
194	C₄⋊Q₈	—	2	2	no	—
195	C₄⋊Q₈	—	2	2	no	—
196	C₄⋊Q₈	—	2	2	no	—
197	C₄⋊Q₈	C₂³×C₅₃	1	4	yes	—
198	C₄×D₄	—	2	1	no	—
199	C₄×D₄	—	2	1	no	—
200	C₄×D₄	D₄×C₅₃	1	2	yes	—
201	C₄×D₄	C₂³×C₅₃	1	2	yes	—
202	C₂×C₄⋊C₄	—	2	2	no	—
203	C₂×C₄⋊C₄	—	2	2	no	—
204	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
205	C₂×C₂²⋊C₄	—	2	1	no	—
206	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
207	#107	—	2	4	no	—
208	#107	—	2	4	no	—
209	C₄⋊C₈	—	2	1	no	—
210	C₄⋊C₈	—	2	2	no	—
211	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
212	C₄⋊C₈	C₂³×C₅₃	1	4	yes	—
213	#103	—	2	8	no	—
214	C₂²⋊C₈	—	2	4	no	—
215	C₄×C₈	D₄×C₅₃	1	2	yes	—
216	C₄×C₈	C₂³×C₅₃	1	2	yes	—
217	C₂⁴	—	2	1	no	—
218	C₂×Q₈	—	2	4	no	—
219	C₂×Q₈	—	4	2	no	—
220	C₂×Q₈	—	4	4	no	—
221	C₂×Q₈	—	4	4	no	—
222	C₂×D₄	—	2	1	no	—
223	C₂×D₄	—	2	2	no	—
224	C₂×D₄	D₈×C₅₃	1	4	yes	—
225	C₂×D₄	C₂×D₄×C₅₃	1	2	yes	—
226	C₂²×C₄	—	2	1	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	4	2	no	—
230	C₂²×C₄	—	4	2	no	—
231	C₂²×C₄	—	4	2	no	—
232	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
233	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
234	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
235	SD₁₆	—	4	16	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	4	no	—
239	C₂×C₈	—	4	4	no	—
240	C₂×C₈	—	4	2	no	—
241	C₂×C₈	—	4	8	no	—
242	C₂×C₈	SD₁₆×C₅₃	1	4	yes	complemented
243	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
244	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
245	C₄⋊C₄	—	2	1	no	—
246	C₄⋊C₄	—	2	4	no	—
247	C₄⋊C₄	—	2	4	no	—
248	C₄⋊C₄	—	4	2	no	—
249	C₄⋊C₄	—	4	2	no	—
250	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
251	C₂²⋊C₄	—	2	2	no	—
252	C₂²⋊C₄	—	2	2	no	—
253	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
254	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
255	C₄²	#41×C₅₃	1	1	yes	characteristic
256	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
257	C₂³	—	2	2	no	—
258	C₂³	—	2	4	no	—
259	C₂³	—	4	2	no	—
260	C₂³	—	4	2	no	—
261	C₂³	#121×C₅₃	1	1	yes	characteristic
262	Q₈	—	4	4	no	—
263	Q₈	—	4	4	no	—
264	Q₈	—	8	4	no	—
265	D₄	—	2	2	no	—
266	D₄	—	2	4	no	—
267	D₄	C₂×D₈×C₅₃	1	4	yes	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	4	no	—
274	C₂×C₄	—	4	4	no	—
275	C₂×C₄	—	8	4	no	—
276	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
277	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
278	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
279	C₂×C₄	C₂×D₈×C₅₃	1	2	yes	—
280	C₂×C₄	C₂×SD₁₆×C₅₃	1	2	yes	—
281	C₂×C₄	#135×C₅₃	1	2	yes	—
282	C₂×C₄	#137×C₅₃	1	2	yes	—
283	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
284	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
285	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
286	C₈	—	2	2	no	—
287	C₈	—	4	4	no	—
288	C₈	—	4	2	no	—
289	C₈	C₂×SD₁₆×C₅₃	1	4	yes	complemented
290	C₂²	—	2	4	no	—
291	C₂²	—	4	8	no	—
292	C₂²	—	4	2	no	—
293	C₂²	—	4	2	no	—
294	C₂²	C₅₃×#461	1	2	yes	—
295	C₂²	C₅₃×#466	1	2	yes	—
296	C₂²	C₅₃×#473	1	2	yes	—
297	C₂²	C₂×#121×C₅₃	1	1	yes	characteristic
298	C₄	—	2	1	no	—
299	C₄	—	2	1	no	—
300	C₄	—	4	2	no	—
301	C₄	—	8	4	no	—
302	C₄	C₅₃×#461	1	2	yes	—
303	C₄	C₅₃×#466	1	2	yes	—
304	C₄	C₂²×D₈×C₅₃	1	1	yes	characteristic
305	C₄	C₂²×SD₁₆×C₅₃	1	1	yes	characteristic
306	C₄	C₂×#135×C₅₃	1	1	yes	characteristic
307	C₄	C₂×#137×C₅₃	1	1	yes	characteristic
308	C₂	—	4	4	no	—
309	C₂	C₅₃×#1673	1	4	yes	direct product factor
310	C₂	C₂×C₅₃×#461	1	1	yes	characteristic
311	C₂	C₂×C₅₃×#466	1	1	yes	characteristic
312	C₂	C₂×C₅₃×#473	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅₃	#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅₃
prime radical	C₂²×D₄×C₅₃	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅₃×#500	C₂	semidirect
4. C₅₃×#1673	C₂	direct (N × H)
5. C₂×C₅₃×#335	C₂	semidirect
6. C₂×C₅₃×#332	C₂	semidirect
12. C₅₃×#500	C₂²	semidirect
24. C₅₃×#335	C₂²	semidirect
25. C₅₃×#332	C₂²	semidirect
85. C₂×C₈×C₅₃	SD₁₆	semidirect
132. C₈×C₅₃	C₂×SD₁₆	semidirect
133. C₂×#1673	C₅₃	direct (N × H)
152. C₂×#500	C₂×C₅₃	semidirect
155. #1673	C₂×C₅₃	direct (N × H)
156. C₂×#335	C₂×C₅₃	semidirect
157. C₂×#332	C₂×C₅₃	semidirect
159. C₂×C₅₃	#1673	direct (N × H)
168. #500	C₂²×C₅₃	semidirect
180. #335	C₂²×C₅₃	semidirect
181. #332	C₂²×C₅₃	semidirect
182. C₅₃	C₂×#1673	direct (N × H)
242. C₂×C₈	SD₁₆×C₅₃	semidirect
289. C₈	C₂×SD₁₆×C₅₃	semidirect
309. C₂	C₅₃×#1673	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×#500	2
C₂×C₅₃×#461	2
C₂×C₈×D₄×C₅₃	2
C₅₃×#1673	2
C₂×C₅₃×#335	2
C₂×C₅₃×#332	2
C₂×#1673	53

Subgroups — C2×C53×#1673 #27226615

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#1673 #27226615