Subgroups — C₂×C₅₃×#1693 #27226647

Order 13568.

758 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×C₅₃×#500	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅₃×#461	C₂	1	2	yes	maximal
3	C₂×C₅₃×#436	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1693	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₅₃×#334	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	C₂²×M₁₆×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	C₅₃×#436	—	2	2	no	—
16	C₅₃×#436	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×#334	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	C₂×M₁₆×C₅₃	—	2	4	no	—
32	C₂×M₁₆×C₅₃	—	4	4	no	—
33	C₂²×C₈×C₅₃	—	2	1	no	—
34	C₂²×C₈×C₅₃	—	2	2	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	4	no	—
67	C₂×D₄×C₅₃	C₂×D₄	1	2	yes	—
68	C₂²×C₄×C₅₃	—	2	1	no	—
69	C₂²×C₄×C₅₃	—	2	2	no	—
70	C₂²×C₄×C₅₃	—	2	2	no	—
71	C₂²×C₄×C₅₃	—	4	2	no	—
72	C₂²×C₄×C₅₃	—	4	2	no	—
73	C₂²×C₄×C₅₃	—	4	2	no	—
74	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
77	SD₁₆×C₅₃	—	4	16	no	—
78	M₁₆×C₅₃	—	4	8	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	2	no	—
83	C₂×C₈×C₅₃	—	4	4	no	—
84	C₂×C₈×C₅₃	SD₁₆	1	4	yes	complemented
85	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
86	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
87	C₄⋊C₄×C₅₃	—	2	1	no	—
88	C₄⋊C₄×C₅₃	—	2	4	no	—
89	C₄⋊C₄×C₅₃	—	2	4	no	—
90	C₄⋊C₄×C₅₃	—	4	2	no	—
91	C₄⋊C₄×C₅₃	—	4	2	no	—
92	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
93	C₂²⋊C₄×C₅₃	—	2	2	no	—
94	C₂²⋊C₄×C₅₃	—	2	2	no	—
95	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₅₃	#41	1	1	yes	characteristic
98	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
99	C₂³×C₅₃	—	2	2	no	—
100	C₂³×C₅₃	—	4	2	no	—
101	C₂³×C₅₃	—	4	2	no	—
102	C₂³×C₅₃	—	4	2	no	—
103	C₂³×C₅₃	#121	1	1	yes	characteristic
104	Q₈×C₅₃	—	4	4	no	—
105	Q₈×C₅₃	—	4	4	no	—
106	Q₈×C₅₃	—	8	4	no	—
107	D₄×C₅₃	—	2	4	no	—
108	D₄×C₅₃	—	4	2	no	—
109	C₂×C₄×C₅₃	—	2	2	no	—
110	C₂×C₄×C₅₃	—	2	2	no	—
111	C₂×C₄×C₅₃	—	4	2	no	—
112	C₂×C₄×C₅₃	—	4	2	no	—
113	C₂×C₄×C₅₃	—	4	2	no	—
114	C₂×C₄×C₅₃	—	4	4	no	—
115	C₂×C₄×C₅₃	—	4	4	no	—
116	C₂×C₄×C₅₃	—	8	4	no	—
117	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
118	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
119	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
120	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
121	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	4	yes	—
122	C₂×C₄×C₅₃	#137	1	2	yes	—
123	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
126	C₈×C₅₃	—	2	2	no	—
127	C₈×C₅₃	—	4	2	no	—
128	C₈×C₅₃	—	4	4	no	—
129	C₈×C₅₃	C₂×SD₁₆	1	4	yes	complemented
130	C₂×#1693	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
131	C₂²×C₅₃	—	4	2	no	—
132	C₂²×C₅₃	—	4	2	no	—
133	C₂²×C₅₃	—	4	2	no	—
134	C₂²×C₅₃	—	8	4	no	—
135	C₂²×C₅₃	#461	1	2	yes	—
136	C₂²×C₅₃	#469	1	2	yes	—
137	C₂²×C₅₃	#473	1	2	yes	—
138	C₂²×C₅₃	C₂×#121	1	1	yes	characteristic
139	C₄×C₅₃	—	2	1	no	—
140	C₄×C₅₃	—	2	1	no	—
141	C₄×C₅₃	—	4	2	no	—
142	C₄×C₅₃	—	8	4	no	—
143	C₄×C₅₃	#461	1	2	yes	—
144	C₄×C₅₃	#469	1	2	yes	—
145	C₄×C₅₃	C₂²×SD₁₆	1	1	yes	characteristic
146	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
147	C₄×C₅₃	C₂×#137	1	1	yes	characteristic
148	C₂×#500	C₂×C₅₃	1	1	yes	characteristic, complemented
149	C₂×#461	C₂×C₅₃	1	2	yes	—
150	C₂×#436	C₂×C₅₃	1	1	yes	characteristic
151	#1693	C₂×C₅₃	1	8	yes	direct product factor
152	C₂×#334	C₂×C₅₃	1	1	yes	characteristic, complemented
153	C₂×#332	C₂×C₅₃	1	2	yes	complemented
154	C₂×C₅₃	—	8	2	no	—
155	C₂×C₅₃	#1693	1	4	yes	direct product factor
156	C₂×C₅₃	C₂×#461	1	1	yes	characteristic
157	C₂×C₅₃	C₂×#469	1	1	yes	characteristic
158	C₂×C₅₃	C₂×#473	1	1	yes	characteristic
159	C₂²×SD₁₆	—	2	2	no	—
160	C₂²×M₁₆	—	2	1	no	—
161	C₂×C₄⋊Q₈	C₂²×C₅₃	1	2	yes	—
162	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
163	#500	—	2	2	no	—
164	#500	C₂²×C₅₃	1	4	yes	complemented
165	#461	—	2	4	no	—
166	#461	C₂²×C₅₃	1	8	yes	—
167	#436	—	2	2	no	—
168	#436	C₂²×C₅₃	1	4	yes	—
169	C₂×#107	—	2	1	no	—
170	C₂×#107	—	2	1	no	—
171	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
172	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
173	C₂×#103	—	2	2	no	—
174	C₂×C₂²⋊C₈	—	2	1	no	—
175	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
176	#334	C₂²×C₅₃	1	4	yes	complemented
177	#332	C₂²×C₅₃	1	8	yes	complemented
178	C₅₃	C₂×#1693	1	1	yes	characteristic, direct product factor, Sylow
179	C₂²×Q₈	—	2	2	no	—
180	C₂²×D₄	D₄×C₅₃	1	1	yes	characteristic
181	C₂³×C₄	—	2	1	no	—
182	C₂×SD₁₆	—	2	8	no	—
183	C₂×SD₁₆	—	4	8	no	—
184	C₂×M₁₆	—	2	4	no	—
185	C₂×M₁₆	—	4	4	no	—
186	C₂²×C₈	—	2	1	no	—
187	C₂²×C₈	—	2	2	no	—
188	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
189	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
190	C₄⋊Q₈	—	2	2	no	—
191	C₄⋊Q₈	—	2	2	no	—
192	C₄⋊Q₈	—	2	2	no	—
193	C₄⋊Q₈	C₂³×C₅₃	1	4	yes	—
194	C₄×D₄	—	2	1	no	—
195	C₄×D₄	—	2	1	no	—
196	C₄×D₄	D₄×C₅₃	1	2	yes	—
197	C₄×D₄	C₂³×C₅₃	1	2	yes	—
198	C₂×C₄⋊C₄	—	2	2	no	—
199	C₂×C₄⋊C₄	—	2	2	no	—
200	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
201	C₂×C₂²⋊C₄	—	2	1	no	—
202	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
203	#107	—	2	4	no	—
204	#107	—	2	4	no	—
205	C₄⋊C₈	—	2	1	no	—
206	C₄⋊C₈	—	2	2	no	—
207	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
208	C₄⋊C₈	C₂³×C₅₃	1	4	yes	—
209	#103	—	2	8	no	—
210	C₂²⋊C₈	—	2	4	no	—
211	C₄×C₈	D₄×C₅₃	1	2	yes	—
212	C₄×C₈	C₂³×C₅₃	1	2	yes	—
213	C₂⁴	—	2	1	no	—
214	C₂×Q₈	—	2	4	no	—
215	C₂×Q₈	—	4	2	no	—
216	C₂×Q₈	—	4	4	no	—
217	C₂×Q₈	—	4	4	no	—
218	C₂×D₄	—	2	1	no	—
219	C₂×D₄	—	2	4	no	—
220	C₂×D₄	C₂×D₄×C₅₃	1	2	yes	—
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	—	4	2	no	—
225	C₂²×C₄	—	4	2	no	—
226	C₂²×C₄	—	4	2	no	—
227	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
228	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
229	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
230	SD₁₆	—	4	16	no	—
231	M₁₆	—	4	8	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	—	4	2	no	—
236	C₂×C₈	—	4	4	no	—
237	C₂×C₈	SD₁₆×C₅₃	1	4	yes	complemented
238	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
239	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
240	C₄⋊C₄	—	2	1	no	—
241	C₄⋊C₄	—	2	4	no	—
242	C₄⋊C₄	—	2	4	no	—
243	C₄⋊C₄	—	4	2	no	—
244	C₄⋊C₄	—	4	2	no	—
245	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
246	C₂²⋊C₄	—	2	2	no	—
247	C₂²⋊C₄	—	2	2	no	—
248	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
249	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
250	C₄²	#41×C₅₃	1	1	yes	characteristic
251	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
252	C₂³	—	2	2	no	—
253	C₂³	—	4	2	no	—
254	C₂³	—	4	2	no	—
255	C₂³	—	4	2	no	—
256	C₂³	#121×C₅₃	1	1	yes	characteristic
257	Q₈	—	4	4	no	—
258	Q₈	—	4	4	no	—
259	Q₈	—	8	4	no	—
260	D₄	—	2	4	no	—
261	D₄	—	4	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	4	4	no	—
268	C₂×C₄	—	4	4	no	—
269	C₂×C₄	—	8	4	no	—
270	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
271	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
272	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
273	C₂×C₄	C₂×SD₁₆×C₅₃	1	2	yes	—
274	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	4	yes	—
275	C₂×C₄	#137×C₅₃	1	2	yes	—
276	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
277	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
278	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
279	C₈	—	2	2	no	—
280	C₈	—	4	2	no	—
281	C₈	—	4	4	no	—
282	C₈	C₂×SD₁₆×C₅₃	1	4	yes	complemented
283	C₂²	—	4	2	no	—
284	C₂²	—	4	2	no	—
285	C₂²	—	4	2	no	—
286	C₂²	—	8	4	no	—
287	C₂²	C₅₃×#461	1	2	yes	—
288	C₂²	C₅₃×#469	1	2	yes	—
289	C₂²	C₅₃×#473	1	2	yes	—
290	C₂²	C₂×#121×C₅₃	1	1	yes	characteristic
291	C₄	—	2	1	no	—
292	C₄	—	2	1	no	—
293	C₄	—	4	2	no	—
294	C₄	—	8	4	no	—
295	C₄	C₅₃×#461	1	2	yes	—
296	C₄	C₅₃×#469	1	2	yes	—
297	C₄	C₂²×SD₁₆×C₅₃	1	1	yes	characteristic
298	C₄	C₂×AGL(1,ℤ/8)×C₅₃	1	2	yes	—
299	C₄	C₂×#137×C₅₃	1	1	yes	characteristic
300	C₂	—	8	2	no	—
301	C₂	C₅₃×#1693	1	4	yes	direct product factor
302	C₂	C₂×C₅₃×#461	1	1	yes	characteristic
303	C₂	C₂×C₅₃×#469	1	1	yes	characteristic
304	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×#1693	C₂	direct (N × H)
5. C₂×C₅₃×#334	C₂	semidirect
6. C₂×C₅₃×#332	C₂	semidirect
12. C₅₃×#500	C₂²	semidirect
24. C₅₃×#334	C₂²	semidirect
25. C₅₃×#332	C₂²	semidirect
84. C₂×C₈×C₅₃	SD₁₆	semidirect
129. C₈×C₅₃	C₂×SD₁₆	semidirect
130. C₂×#1693	C₅₃	direct (N × H)
148. C₂×#500	C₂×C₅₃	semidirect
151. #1693	C₂×C₅₃	direct (N × H)
152. C₂×#334	C₂×C₅₃	semidirect
153. C₂×#332	C₂×C₅₃	semidirect
155. C₂×C₅₃	#1693	direct (N × H)
164. #500	C₂²×C₅₃	semidirect
176. #334	C₂²×C₅₃	semidirect
177. #332	C₂²×C₅₃	semidirect
178. C₅₃	C₂×#1693	direct (N × H)
237. C₂×C₈	SD₁₆×C₅₃	semidirect
282. C₈	C₂×SD₁₆×C₅₃	semidirect
301. C₂	C₅₃×#1693	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×C₅₃×#436	2
C₅₃×#1693	2
C₂×C₅₃×#334	2
C₂×C₅₃×#332	2
C₂×#1693	53

Subgroups — C2×C53×#1693 #27226647

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#1693 #27226647