Subgroups — C₅₃×#11332 #27218844

Order 13568.

662 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#430	C₂	1	2	yes	maximal
2	C₂²×#108×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₃×#344	C₂	1	2	yes	maximal
4	C₂×C₅₃×#341	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1729	C₂	1	1	yes	maximal, characteristic
6	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
7	C₂²×M₁₆×C₅₃	C₂²	1	2	yes	—
8	C₂²×M₁₆×C₅₃	C₂²	1	2	yes	—
9	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
11	C₅₃×#430	—	2	4	no	—
12	C₅₃×#430	—	2	4	no	—
13	C₂×#108×C₅₃	—	2	4	no	—
14	C₂×#108×C₅₃	—	2	4	no	—
15	C₂×#108×C₅₃	—	2	2	no	—
16	C₅₃×#344	—	2	4	no	—
17	C₅₃×#341	C₄	1	4	yes	complemented
18	C₅₃×#336	—	2	2	no	—
19	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
20	C₂×M₁₆×C₅₃	—	2	2	no	—
21	C₂×M₁₆×C₅₃	—	2	2	no	—
22	C₂×M₁₆×C₅₃	—	2	2	no	—
23	C₂×M₁₆×C₅₃	—	2	2	no	—
24	C₂×M₁₆×C₅₃	—	2	2	no	—
25	C₂×M₁₆×C₅₃	—	2	2	no	—
26	C₂×M₁₆×C₅₃	—	2	2	no	—
27	C₂×M₁₆×C₅₃	—	2	2	no	—
28	C₂×M₁₆×C₅₃	—	2	2	no	—
29	C₂×M₁₆×C₅₃	—	2	2	no	—
30	C₂×M₁₆×C₅₃	—	2	1	no	—
31	C₂×M₁₆×C₅₃	C₂×C₄	1	4	yes	—
32	C₂×M₁₆×C₅₃	D₄	1	2	yes	—
33	C₂×M₁₆×C₅₃	D₄	1	4	yes	—
34	C₂×M₁₆×C₅₃	D₄	1	4	yes	—
35	C₂×M₁₆×C₅₃	Q₈	1	4	yes	—
36	C₂²×C₈×C₅₃	—	2	2	no	—
37	C₂²×C₈×C₅₃	—	2	2	no	—
38	C₂²×C₈×C₅₃	—	2	1	no	—
39	C₂²×C₈×C₅₃	—	2	2	no	—
40	C₂²×C₈×C₅₃	—	2	2	no	—
41	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
42	C₂²×C₈×C₅₃	D₄	1	2	yes	—
43	C₂²×C₈×C₅₃	Q₈	1	2	yes	—
44	#117×C₅₃	—	2	2	no	—
45	#117×C₅₃	—	2	2	no	—
46	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
47	C₂×C₂²⋊C₄×C₅₃	Q₈	1	2	yes	—
48	C₂×C₄²×C₅₃	—	2	1	no	—
49	#108×C₅₃	—	2	8	no	—
50	#108×C₅₃	—	2	16	no	—
51	C₂⁴×C₅₃	C₂×Q₈	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	—	2	2	no	—
53	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
55	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
57	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
60	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	#41	1	2	yes	—
62	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
63	M₁₆×C₅₃	—	2	4	no	—
64	M₁₆×C₅₃	—	2	4	no	—
65	M₁₆×C₅₃	—	2	4	no	—
66	M₁₆×C₅₃	—	2	4	no	—
67	M₁₆×C₅₃	—	2	4	no	—
68	M₁₆×C₅₃	—	2	4	no	—
69	M₁₆×C₅₃	—	4	2	no	—
70	M₁₆×C₅₃	—	4	2	no	—
71	M₁₆×C₅₃	—	4	2	no	—
72	M₁₆×C₅₃	C₄⋊C₄	1	8	yes	—
73	C₂×C₈×C₅₃	—	2	4	no	—
74	C₂×C₈×C₅₃	—	2	4	no	—
75	C₂×C₈×C₅₃	—	2	2	no	—
76	C₂×C₈×C₅₃	—	2	4	no	—
77	C₂×C₈×C₅₃	—	2	4	no	—
78	C₂×C₈×C₅₃	—	2	4	no	—
79	C₂×C₈×C₅₃	—	2	2	no	—
80	C₂×C₈×C₅₃	—	2	4	no	—
81	C₂×C₈×C₅₃	—	2	2	no	—
82	C₂×C₈×C₅₃	—	4	1	no	—
83	C₂×C₈×C₅₃	—	4	2	no	—
84	C₂×C₈×C₅₃	—	4	2	no	—
85	C₂×C₈×C₅₃	—	4	1	no	—
86	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
87	C₄⋊C₄×C₅₃	—	4	2	no	—
88	C₂²⋊C₄×C₅₃	—	2	4	no	—
89	C₄²×C₅₃	—	4	1	no	—
90	C₄²×C₅₃	—	4	1	no	—
91	C₂³×C₅₃	—	4	2	no	—
92	C₂³×C₅₃	C₄×Q₈	1	1	yes	characteristic
93	C₂³×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
94	C₂³×C₅₃	C₂²⋊Q₈	1	2	yes	—
95	C₂³×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
96	C₂³×C₅₃	C₄⋊Q₈	1	2	yes	—
97	C₂×C₄×C₅₃	—	4	4	no	—
98	C₂×C₄×C₅₃	—	4	2	no	—
99	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
101	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
102	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
103	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
104	C₂×C₄×C₅₃	#120	1	2	yes	—
105	C₂×C₄×C₅₃	#121	1	2	yes	—
106	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
107	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
108	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
109	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
110	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
111	C₂×C₄×C₅₃	#123	1	2	yes	—
112	C₂×C₄×C₅₃	#125	1	1	yes	characteristic
113	C₂×C₄×C₅₃	#125	1	2	yes	—
114	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
115	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
116	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
117	C₈×C₅₃	—	2	4	no	—
118	C₈×C₅₃	—	2	8	no	—
119	C₈×C₅₃	—	4	4	no	—
120	C₈×C₅₃	—	4	2	no	—
121	#11332	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
122	C₂²×C₅₃	—	2	2	no	—
123	C₂²×C₅₃	—	2	2	no	—
124	C₂²×C₅₃	—	2	2	no	—
125	C₂²×C₅₃	—	4	2	no	—
126	C₂²×C₅₃	—	4	2	no	—
127	C₂²×C₅₃	#389	1	2	yes	—
128	C₂²×C₅₃	#389	1	1	yes	characteristic
129	C₂²×C₅₃	#395	1	1	yes	characteristic
130	C₂²×C₅₃	#395	1	2	yes	—
131	C₂²×C₅₃	#399	1	1	yes	characteristic
132	C₄×C₅₃	—	8	2	no	—
133	C₄×C₅₃	#384	1	1	yes	characteristic
134	C₄×C₅₃	#385	1	1	yes	characteristic
135	C₄×C₅₃	#385	1	2	yes	—
136	C₄×C₅₃	#392	1	1	yes	characteristic
137	C₄×C₅₃	#398	1	2	yes	—
138	C₄×C₅₃	#398	1	1	yes	characteristic
139	C₂×#430	C₂×C₅₃	1	2	yes	—
140	C₂²×#108	C₂×C₅₃	1	1	yes	characteristic
141	C₂×#344	C₂×C₅₃	1	2	yes	—
142	C₂×#341	C₂×C₅₃	1	1	yes	characteristic
143	#1729	C₂×C₅₃	1	1	yes	characteristic
144	C₂×C₅₃	—	2	1	no	—
145	C₂×C₅₃	—	2	2	no	—
146	C₂×C₅₃	—	2	1	no	—
147	C₂×C₅₃	—	2	2	no	—
148	C₂×C₅₃	#1444	1	1	yes	characteristic
149	C₂×C₅₃	#1953	1	2	yes	—
150	C₂²×M₁₆	C₂²×C₅₃	1	2	yes	—
151	C₂²×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
152	C₂²×M₁₆	C₂²×C₅₃	1	2	yes	—
153	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
154	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
155	#430	—	2	4	no	—
156	#430	—	2	4	no	—
157	C₂×#108	—	2	4	no	—
158	C₂×#108	—	2	4	no	—
159	C₂×#108	—	2	2	no	—
160	#344	—	2	4	no	—
161	#341	C₄×C₅₃	1	4	yes	complemented
162	#336	—	2	2	no	—
163	C₅₃	#11332	1	1	yes	characteristic, direct product factor, Sylow
164	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
165	C₂×M₁₆	—	2	2	no	—
166	C₂×M₁₆	—	2	2	no	—
167	C₂×M₁₆	—	2	2	no	—
168	C₂×M₁₆	—	2	2	no	—
169	C₂×M₁₆	—	2	1	no	—
170	C₂×M₁₆	—	2	2	no	—
171	C₂×M₁₆	—	2	2	no	—
172	C₂×M₁₆	—	2	2	no	—
173	C₂×M₁₆	—	2	2	no	—
174	C₂×M₁₆	—	2	2	no	—
175	C₂×M₁₆	—	2	2	no	—
176	C₂×M₁₆	C₂×C₄×C₅₃	1	4	yes	—
177	C₂×M₁₆	D₄×C₅₃	1	2	yes	—
178	C₂×M₁₆	D₄×C₅₃	1	4	yes	—
179	C₂×M₁₆	D₄×C₅₃	1	4	yes	—
180	C₂×M₁₆	Q₈×C₅₃	1	4	yes	—
181	C₂²×C₈	—	2	2	no	—
182	C₂²×C₈	—	2	2	no	—
183	C₂²×C₈	—	2	2	no	—
184	C₂²×C₈	—	2	1	no	—
185	C₂²×C₈	—	2	2	no	—
186	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
187	C₂²×C₈	D₄×C₅₃	1	2	yes	—
188	C₂²×C₈	Q₈×C₅₃	1	2	yes	—
189	#117	—	2	2	no	—
190	#117	—	2	2	no	—
191	C₂×C₄⋊C₄	—	2	1	no	—
192	C₂×C₂²⋊C₄	Q₈×C₅₃	1	2	yes	—
193	C₂×C₄²	—	2	1	no	—
194	#108	—	2	8	no	—
195	#108	—	2	16	no	—
196	C₂⁴	C₂×Q₈×C₅₃	1	1	yes	characteristic
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
199	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
200	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
201	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
202	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
203	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
204	C₂²×C₄	C₂×Q₈×C₅₃	1	2	yes	—
205	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
206	C₂²×C₄	#41×C₅₃	1	2	yes	—
207	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
208	M₁₆	—	2	4	no	—
209	M₁₆	—	2	4	no	—
210	M₁₆	—	2	4	no	—
211	M₁₆	—	2	4	no	—
212	M₁₆	—	2	4	no	—
213	M₁₆	—	2	4	no	—
214	M₁₆	—	4	2	no	—
215	M₁₆	—	4	2	no	—
216	M₁₆	—	4	2	no	—
217	M₁₆	C₄⋊C₄×C₅₃	1	8	yes	—
218	C₂×C₈	—	2	4	no	—
219	C₂×C₈	—	2	4	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	2	4	no	—
222	C₂×C₈	—	2	4	no	—
223	C₂×C₈	—	2	4	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	4	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	4	1	no	—
228	C₂×C₈	—	4	2	no	—
229	C₂×C₈	—	4	2	no	—
230	C₂×C₈	—	4	1	no	—
231	C₂×C₈	C₄⋊C₄×C₅₃	1	4	yes	—
232	C₄⋊C₄	—	4	2	no	—
233	C₂²⋊C₄	—	2	4	no	—
234	C₄²	—	4	1	no	—
235	C₄²	—	4	1	no	—
236	C₂³	—	4	2	no	—
237	C₂³	C₄×Q₈×C₅₃	1	1	yes	characteristic
238	C₂³	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
239	C₂³	C₂²⋊Q₈×C₅₃	1	2	yes	—
240	C₂³	C₄⋊Q₈×C₅₃	1	1	yes	characteristic
241	C₂³	C₄⋊Q₈×C₅₃	1	2	yes	—
242	C₂×C₄	—	4	4	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
245	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
246	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
247	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
248	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
249	C₂×C₄	#120×C₅₃	1	2	yes	—
250	C₂×C₄	#121×C₅₃	1	2	yes	—
251	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
252	C₂×C₄	C₂²⋊Q₈×C₅₃	1	2	yes	—
253	C₂×C₄	C₂²⋊Q₈×C₅₃	1	2	yes	—
254	C₂×C₄	C₂²⋊Q₈×C₅₃	1	2	yes	—
255	C₂×C₄	#123×C₅₃	1	1	yes	characteristic
256	C₂×C₄	#123×C₅₃	1	2	yes	—
257	C₂×C₄	#125×C₅₃	1	2	yes	—
258	C₂×C₄	#125×C₅₃	1	1	yes	characteristic
259	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
260	C₂×C₄	C₄⋊Q₈×C₅₃	1	1	yes	characteristic
261	C₂×C₄	C₄⋊Q₈×C₅₃	1	2	yes	—
262	C₈	—	2	4	no	—
263	C₈	—	2	8	no	—
264	C₈	—	4	2	no	—
265	C₈	—	4	4	no	—
266	C₂²	—	2	2	no	—
267	C₂²	—	2	2	no	—
268	C₂²	—	2	2	no	—
269	C₂²	—	4	2	no	—
270	C₂²	—	4	2	no	—
271	C₂²	C₅₃×#389	1	2	yes	—
272	C₂²	C₅₃×#389	1	1	yes	characteristic
273	C₂²	C₅₃×#395	1	1	yes	characteristic
274	C₂²	C₅₃×#395	1	2	yes	—
275	C₂²	C₅₃×#399	1	1	yes	characteristic
276	C₄	—	8	2	no	—
277	C₄	C₅₃×#384	1	1	yes	characteristic
278	C₄	C₅₃×#385	1	1	yes	characteristic
279	C₄	C₅₃×#385	1	2	yes	—
280	C₄	C₅₃×#392	1	1	yes	characteristic
281	C₄	C₅₃×#398	1	2	yes	—
282	C₄	C₅₃×#398	1	1	yes	characteristic
283	C₂	—	2	1	no	—
284	C₂	—	2	2	no	—
285	C₂	—	2	1	no	—
286	C₂	—	2	2	no	—
287	C₂	C₅₃×#1444	1	1	yes	characteristic
288	C₂	C₅₃×#1953	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅₃	#121
commutator	C₂²×C₄	C₂²×C₄×C₅₃
Frattini, cosocle	C₂³×C₄	C₂³×C₅₃
socle	C₂²×C₅₃	#389
prime radical	C₂⁴×C₅₃	C₂×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂³×C₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
17. C₅₃×#341	C₄	semidirect
121. #11332	C₅₃	direct (N × H)
161. #341	C₄×C₅₃	semidirect
163. C₅₃	#11332	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×#430	2
C₂²×#108×C₅₃	2
C₂×C₅₃×#344	2
C₂×C₅₃×#341	2
C₅₃×#1729	2
#11332	53

Subgroups — C53×#11332 #27218844

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#11332 #27218844