Subgroups — C₅₃×#33707 #27241219

Order 13568.

622 conjugacy classes of subgroups.

The subgroup lattice has 312 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	C₅₃×#3147	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#2243	C₂	1	4	yes	maximal, complemented
4	C₅₃×#2241	C₂	1	2	yes	maximal, complemented
5	C₅₃×#2240	C₂	1	2	yes	maximal, complemented
6	C₅₃×#2139	C₂	1	4	yes	maximal, complemented
7	C₄×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#135×C₅₃	—	2	1	no	—
9	C₂×#135×C₅₃	C₂²	1	2	yes	—
10	C₅₃×#532	—	2	1	no	—
11	C₂×C₄⋊Q₈×C₅₃	—	2	1	no	—
12	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
13	C₅₃×#500	—	2	1	no	—
14	C₅₃×#500	—	2	1	no	—
15	C₅₃×#500	C₂²	1	2	yes	—
16	C₅₃×#500	C₂²	1	2	yes	—
17	C₅₃×#495	—	2	1	no	—
18	C₅₃×#495	C₂²	1	2	yes	—
19	C₅₃×#494	C₂²	1	1	yes	characteristic
20	C₅₃×#493	C₂²	1	1	yes	characteristic
21	C₅₃×#492	—	2	1	no	—
22	C₂×#107×C₅₃	—	2	1	no	—
23	C₂×#107×C₅₃	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
25	C₅₃×#358	C₂²	1	8	yes	—
26	C₅₃×#357	C₂²	1	8	yes	—
27	C₅₃×#346	C₂²	1	2	yes	—
28	C₄×C₁₆×C₅₃	C₂²	1	2	yes	—
29	C₂×#41×C₅₃	—	2	2	no	—
30	C₂²×Q₈×C₅₃	—	4	1	no	—
31	#135×C₅₃	—	2	8	no	—
32	#135×C₅₃	—	4	4	no	—
33	C₂×Q₁₆×C₅₃	—	2	1	no	—
34	C₂×Q₁₆×C₅₃	C₂³	1	2	yes	—
35	C₂×SD₁₆×C₅₃	—	2	2	no	—
36	C₂×SD₁₆×C₅₃	—	2	2	no	—
37	C₂×D₈×C₅₃	—	2	1	no	—
38	C₂×D₈×C₅₃	C₂³	1	2	yes	—
39	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
41	C₄⋊Q₈×C₅₃	—	2	1	no	—
42	C₄⋊Q₈×C₅₃	—	2	2	no	—
43	C₄⋊Q₈×C₅₃	—	2	2	no	—
44	C₄⋊Q₈×C₅₃	—	4	1	no	—
45	C₄⋊Q₈×C₅₃	—	4	1	no	—
46	#127×C₅₃	—	2	1	no	—
47	#124×C₅₃	—	2	2	no	—
48	#121×C₅₃	—	4	1	no	—
49	#121×C₅₃	—	4	1	no	—
50	C₄×D₄×C₅₃	—	4	1	no	—
51	C₄×D₄×C₅₃	—	4	1	no	—
52	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
53	C₂×C₄⋊C₄×C₅₃	—	4	1	no	—
54	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
55	M₃₂×C₅₃	—	2	4	no	—
56	C₂×C₁₆×C₅₃	C₂³	1	4	yes	—
57	#107×C₅₃	—	2	2	no	—
58	#107×C₅₃	—	2	2	no	—
59	#107×C₅₃	—	2	2	no	—
60	#107×C₅₃	C₂³	1	4	yes	—
61	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
62	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
63	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
64	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
65	#41×C₅₃	—	4	4	no	—
66	#41×C₅₃	—	4	4	no	—
67	C₂×Q₈×C₅₃	—	2	2	no	—
68	C₂×Q₈×C₅₃	—	4	2	no	—
69	C₂×Q₈×C₅₃	—	4	2	no	—
70	C₂×Q₈×C₅₃	—	8	4	no	—
71	C₂×D₄×C₅₃	—	2	2	no	—
72	C₂×D₄×C₅₃	—	4	2	no	—
73	C₂×D₄×C₅₃	—	4	1	no	—
74	C₂×D₄×C₅₃	—	4	1	no	—
75	C₂²×C₄×C₅₃	—	4	2	no	—
76	C₂²×C₄×C₅₃	—	4	2	no	—
77	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
78	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
79	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
80	Q₁₆×C₅₃	—	2	4	no	—
81	Q₁₆×C₅₃	—	4	2	no	—
82	SD₁₆×C₅₃	—	4	4	no	—
83	SD₁₆×C₅₃	—	4	4	no	—
84	D₈×C₅₃	—	2	4	no	—
85	D₈×C₅₃	—	4	2	no	—
86	C₂×C₈×C₅₃	SD₁₆	1	4	yes	—
87	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₅₃	#41	1	4	yes	—
91	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
92	C₄⋊C₄×C₅₃	—	2	4	no	—
93	C₄⋊C₄×C₅₃	—	4	2	no	—
94	C₄⋊C₄×C₅₃	—	4	2	no	—
95	C₄⋊C₄×C₅₃	—	4	2	no	—
96	C₄⋊C₄×C₅₃	—	4	1	no	—
97	C₄⋊C₄×C₅₃	—	4	1	no	—
98	C₂²⋊C₄×C₅₃	—	4	2	no	—
99	C₂²⋊C₄×C₅₃	—	4	2	no	—
100	C₄²×C₅₃	D₈	1	1	yes	characteristic
101	C₄²×C₅₃	D₈	1	1	yes	characteristic
102	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
104	C₁₆×C₅₃	—	2	4	no	—
105	C₂³×C₅₃	—	4	2	no	—
106	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
107	Q₈×C₅₃	—	4	2	no	—
108	Q₈×C₅₃	—	4	2	no	—
109	Q₈×C₅₃	—	8	4	no	—
110	Q₈×C₅₃	—	8	4	no	—
111	D₄×C₅₃	—	4	2	no	—
112	D₄×C₅₃	—	4	2	no	—
113	D₄×C₅₃	—	8	4	no	—
114	D₄×C₅₃	—	8	2	no	—
115	D₄×C₅₃	—	8	2	no	—
116	C₂×C₄×C₅₃	—	2	2	no	—
117	C₂×C₄×C₅₃	—	2	2	no	—
118	C₂×C₄×C₅₃	—	4	2	no	—
119	C₂×C₄×C₅₃	—	4	4	no	—
120	C₂×C₄×C₅₃	—	8	4	no	—
121	C₂×C₄×C₅₃	—	8	4	no	—
122	C₂×C₄×C₅₃	#124	1	2	yes	—
123	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
124	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
125	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
127	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
128	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
129	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
130	C₈×C₅₃	#124	1	2	yes	—
131	C₈×C₅₃	C₂×SD₁₆	1	2	yes	—
132	C₈×C₅₃	C₂×SD₁₆	1	2	yes	—
133	C₈×C₅₃	C₂×#41	1	2	yes	—
134	#33707	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
135	C₂²×C₅₃	—	2	1	no	—
136	C₂²×C₅₃	—	2	1	no	—
137	C₂²×C₅₃	—	8	2	no	—
138	C₂²×C₅₃	—	8	2	no	—
139	C₂²×C₅₃	—	8	2	no	—
140	C₂²×C₅₃	#486	1	2	yes	—
141	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
142	C₄×C₅₃	—	2	1	no	—
143	C₄×C₅₃	—	2	1	no	—
144	C₄×C₅₃	—	8	2	no	—
145	C₄×C₅₃	—	8	4	no	—
146	C₄×C₅₃	#486	1	2	yes	—
147	C₄×C₅₃	#509	1	2	yes	—
148	C₄×C₅₃	#510	1	2	yes	—
149	C₄×C₅₃	C₂×#124	1	1	yes	characteristic
150	C₄×C₅₃	C₂²×SD₁₆	1	1	yes	characteristic
151	C₂×#500	C₂×C₅₃	1	1	yes	characteristic, complemented
152	#3147	C₂×C₅₃	1	1	yes	characteristic
153	#2243	C₂×C₅₃	1	4	yes	complemented
154	#2241	C₂×C₅₃	1	2	yes	complemented
155	#2240	C₂×C₅₃	1	2	yes	complemented
156	#2139	C₂×C₅₃	1	4	yes	complemented
157	C₄×M₃₂	C₂×C₅₃	1	1	yes	characteristic, complemented
158	C₂×C₅₃	—	2	2	no	—
159	C₂×C₅₃	—	16	2	no	—
160	C₂×C₅₃	C₂×#486	1	1	yes	characteristic
161	C₂×C₅₃	C₂×#509	1	1	yes	characteristic
162	C₂×C₅₃	C₂×#510	1	1	yes	characteristic
163	C₂×#135	—	2	1	no	—
164	C₂×#135	C₂²×C₅₃	1	2	yes	—
165	#532	—	2	1	no	—
166	C₂×C₄⋊Q₈	—	2	1	no	—
167	C₂×M₃₂	C₂²×C₅₃	1	2	yes	—
168	#500	—	2	1	no	—
169	#500	—	2	1	no	—
170	#500	C₂²×C₅₃	1	2	yes	—
171	#500	C₂²×C₅₃	1	2	yes	—
172	#495	—	2	1	no	—
173	#495	C₂²×C₅₃	1	2	yes	—
174	#494	C₂²×C₅₃	1	1	yes	characteristic
175	#493	C₂²×C₅₃	1	1	yes	characteristic
176	#492	—	2	1	no	—
177	C₂×#107	—	2	1	no	—
178	C₂×#107	C₂²×C₅₃	1	2	yes	—
179	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
180	#358	C₂²×C₅₃	1	8	yes	—
181	#357	C₂²×C₅₃	1	8	yes	—
182	#346	C₂²×C₅₃	1	2	yes	—
183	C₄×C₁₆	C₂²×C₅₃	1	2	yes	—
184	C₅₃	#33707	1	1	yes	characteristic, direct product factor, Sylow
185	C₂×#41	—	2	2	no	—
186	C₂²×Q₈	—	4	1	no	—
187	#135	—	2	8	no	—
188	#135	—	4	4	no	—
189	C₂×Q₁₆	—	2	1	no	—
190	C₂×Q₁₆	C₂³×C₅₃	1	2	yes	—
191	C₂×SD₁₆	—	2	2	no	—
192	C₂×SD₁₆	—	2	2	no	—
193	C₂×D₈	—	2	1	no	—
194	C₂×D₈	C₂³×C₅₃	1	2	yes	—
195	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
196	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
197	C₄⋊Q₈	—	2	2	no	—
198	C₄⋊Q₈	—	2	2	no	—
199	C₄⋊Q₈	—	2	1	no	—
200	C₄⋊Q₈	—	4	1	no	—
201	C₄⋊Q₈	—	4	1	no	—
202	#127	—	2	1	no	—
203	#124	—	2	2	no	—
204	#121	—	4	1	no	—
205	#121	—	4	1	no	—
206	C₄×D₄	—	4	1	no	—
207	C₄×D₄	—	4	1	no	—
208	C₂×C₄⋊C₄	—	2	2	no	—
209	C₂×C₄⋊C₄	—	4	1	no	—
210	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
211	M₃₂	—	2	4	no	—
212	C₂×C₁₆	C₂³×C₅₃	1	4	yes	—
213	#107	—	2	2	no	—
214	#107	—	2	2	no	—
215	#107	—	2	2	no	—
216	#107	C₂³×C₅₃	1	4	yes	—
217	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
218	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
219	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
220	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
221	#41	—	4	4	no	—
222	#41	—	4	4	no	—
223	C₂×Q₈	—	2	2	no	—
224	C₂×Q₈	—	4	2	no	—
225	C₂×Q₈	—	4	2	no	—
226	C₂×Q₈	—	8	4	no	—
227	C₂×D₄	—	2	2	no	—
228	C₂×D₄	—	4	2	no	—
229	C₂×D₄	—	4	1	no	—
230	C₂×D₄	—	4	1	no	—
231	C₂²×C₄	—	4	2	no	—
232	C₂²×C₄	—	4	2	no	—
233	C₂²×C₄	D₈×C₅₃	1	1	yes	characteristic
234	C₂²×C₄	D₈×C₅₃	1	1	yes	characteristic
235	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
236	Q₁₆	—	2	4	no	—
237	Q₁₆	—	4	2	no	—
238	SD₁₆	—	4	4	no	—
239	SD₁₆	—	4	4	no	—
240	D₈	—	2	4	no	—
241	D₈	—	4	2	no	—
242	C₂×C₈	SD₁₆×C₅₃	1	4	yes	—
243	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
244	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
245	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
246	C₂×C₈	#41×C₅₃	1	4	yes	—
247	C₂×C₈	C₂⁴×C₅₃	1	1	yes	characteristic
248	C₄⋊C₄	—	2	4	no	—
249	C₄⋊C₄	—	4	2	no	—
250	C₄⋊C₄	—	4	2	no	—
251	C₄⋊C₄	—	4	2	no	—
252	C₄⋊C₄	—	4	1	no	—
253	C₄⋊C₄	—	4	1	no	—
254	C₂²⋊C₄	—	4	2	no	—
255	C₂²⋊C₄	—	4	2	no	—
256	C₄²	D₈×C₅₃	1	1	yes	characteristic
257	C₄²	D₈×C₅₃	1	1	yes	characteristic
258	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
259	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
260	C₁₆	—	2	4	no	—
261	C₂³	—	4	2	no	—
262	C₂³	C₂×D₈×C₅₃	1	1	yes	characteristic
263	Q₈	—	4	2	no	—
264	Q₈	—	4	2	no	—
265	Q₈	—	8	4	no	—
266	Q₈	—	8	4	no	—
267	D₄	—	4	2	no	—
268	D₄	—	4	2	no	—
269	D₄	—	8	4	no	—
270	D₄	—	8	2	no	—
271	D₄	—	8	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	4	4	no	—
276	C₂×C₄	—	8	4	no	—
277	C₂×C₄	—	8	4	no	—
278	C₂×C₄	#124×C₅₃	1	2	yes	—
279	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
280	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
281	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
282	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
283	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
284	C₂×C₄	C₂×SD₁₆×C₅₃	1	2	yes	—
285	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
286	C₈	#124×C₅₃	1	2	yes	—
287	C₈	C₂×SD₁₆×C₅₃	1	2	yes	—
288	C₈	C₂×SD₁₆×C₅₃	1	2	yes	—
289	C₈	C₂×#41×C₅₃	1	2	yes	—
290	C₂²	—	2	1	no	—
291	C₂²	—	2	1	no	—
292	C₂²	—	8	2	no	—
293	C₂²	—	8	2	no	—
294	C₂²	—	8	2	no	—
295	C₂²	C₅₃×#486	1	2	yes	—
296	C₂²	C₂²×D₈×C₅₃	1	1	yes	characteristic
297	C₄	—	2	1	no	—
298	C₄	—	2	1	no	—
299	C₄	—	8	2	no	—
300	C₄	—	8	4	no	—
301	C₄	C₅₃×#486	1	2	yes	—
302	C₄	C₅₃×#509	1	2	yes	—
303	C₄	C₅₃×#510	1	2	yes	—
304	C₄	C₂×#124×C₅₃	1	1	yes	characteristic
305	C₄	C₂²×SD₁₆×C₅₃	1	1	yes	characteristic
306	C₂	—	2	2	no	—
307	C₂	—	16	2	no	—
308	C₂	C₂×C₅₃×#486	1	1	yes	characteristic
309	C₂	C₂×C₅₃×#509	1	1	yes	characteristic
310	C₂	C₂×C₅₃×#510	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅₃
prime radical	C₅₃×#3147	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅₃×#500	C₂	semidirect
3. C₅₃×#2243	C₂	semidirect
4. C₅₃×#2241	C₂	semidirect
5. C₅₃×#2240	C₂	semidirect
6. C₅₃×#2139	C₂	semidirect
7. C₄×M₃₂×C₅₃	C₂	semidirect
134. #33707	C₅₃	direct (N × H)
151. C₂×#500	C₂×C₅₃	semidirect
153. #2243	C₂×C₅₃	semidirect
154. #2241	C₂×C₅₃	semidirect
155. #2240	C₂×C₅₃	semidirect
156. #2139	C₂×C₅₃	semidirect
157. C₄×M₃₂	C₂×C₅₃	semidirect
184. C₅₃	#33707	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
C₅₃×#3147	2
C₅₃×#2243	2
C₅₃×#2241	2
C₅₃×#2240	2
C₅₃×#2139	2
C₄×M₃₂×C₅₃	2
#33707	53

Subgroups — C53×#33707 #27241219

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#33707 #27241219