Subgroups — C₅₃×#11286 #27218798

Order 13568.

622 conjugacy classes of subgroups.

The subgroup lattice has 294 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	1	yes	maximal, characteristic
2	C₂×C₅₃×#344	C₂	1	2	yes	maximal
3	C₂×C₅₃×#339	C₂	1	2	yes	maximal
4	C₂×C₅₃×H(3,ℤ/4)	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1728	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	1	yes	characteristic
9	C₂×#117×C₅₃	C₂²	1	2	yes	—
10	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
11	C₅₃×#430	—	2	4	no	—
12	C₂×#108×C₅₃	—	2	2	no	—
13	C₅₃×#344	—	2	4	no	—
14	C₅₃×#339	—	2	4	no	—
15	C₅₃×H(3,ℤ/4)	C₄	1	4	yes	complemented
16	C₅₃×#336	—	2	2	no	—
17	C₂³×C₄×C₅₃	—	2	1	no	—
18	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
19	C₂×M₁₆×C₅₃	—	2	2	no	—
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	1	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	C₂×C₄	1	2	yes	—
30	C₂×M₁₆×C₅₃	D₄	1	4	yes	—
31	C₂×M₁₆×C₅₃	D₄	1	2	yes	—
32	C₂×M₁₆×C₅₃	Q₈	1	2	yes	—
33	C₂²×C₈×C₅₃	—	2	1	no	—
34	C₂²×C₈×C₅₃	—	2	2	no	—
35	C₂²×C₈×C₅₃	D₄	1	2	yes	—
36	#117×C₅₃	—	2	2	no	—
37	#117×C₅₃	—	2	2	no	—
38	#117×C₅₃	—	2	2	no	—
39	#117×C₅₃	C₂×C₄	1	4	yes	—
40	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
41	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
42	C₂×C₄²×C₅₃	—	2	1	no	—
43	C₂×C₄²×C₅₃	—	2	2	no	—
44	C₂×C₄²×C₅₃	—	2	2	no	—
45	C₂×C₄²×C₅₃	—	2	2	no	—
46	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
47	#108×C₅₃	—	2	8	no	—
48	C₂⁴×C₅₃	#41	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	—	2	2	no	—
50	C₂²×C₄×C₅₃	—	4	4	no	—
51	C₂²×C₄×C₅₃	—	4	2	no	—
52	C₂²×C₄×C₅₃	—	4	2	no	—
53	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
56	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	#41	1	2	yes	—
60	C₂²×C₄×C₅₃	#41	1	2	yes	—
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	—	4	2	no	—
69	M₁₆×C₅₃	—	4	2	no	—
70	M₁₆×C₅₃	—	4	2	no	—
71	C₂×C₈×C₅₃	—	2	2	no	—
72	C₂×C₈×C₅₃	—	2	2	no	—
73	C₂×C₈×C₅₃	—	2	4	no	—
74	C₂×C₈×C₅₃	—	2	2	no	—
75	C₂×C₈×C₅₃	—	4	2	no	—
76	C₂×C₈×C₅₃	—	4	1	no	—
77	C₂×C₈×C₅₃	—	4	1	no	—
78	C₂×C₈×C₅₃	—	4	2	no	—
79	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
80	C₄⋊C₄×C₅₃	—	2	4	no	—
81	C₄⋊C₄×C₅₃	—	4	2	no	—
82	C₂²⋊C₄×C₅₃	—	2	4	no	—
83	C₂²⋊C₄×C₅₃	—	4	2	no	—
84	C₄²×C₅₃	—	2	4	no	—
85	C₄²×C₅₃	—	2	4	no	—
86	C₄²×C₅₃	—	4	1	no	—
87	C₄²×C₅₃	—	4	2	no	—
88	C₄²×C₅₃	—	4	1	no	—
89	C₂³×C₅₃	—	4	2	no	—
90	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
91	C₂³×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
92	C₂³×C₅₃	#124	1	2	yes	—
93	C₂³×C₅₃	#125	1	1	yes	characteristic
94	C₂³×C₅₃	#126	1	2	yes	—
95	C₂×C₄×C₅₃	—	2	4	no	—
96	C₂×C₄×C₅₃	—	4	2	no	—
97	C₂×C₄×C₅₃	—	4	8	no	—
98	C₂×C₄×C₅₃	—	4	2	no	—
99	C₂×C₄×C₅₃	—	4	4	no	—
100	C₂×C₄×C₅₃	—	8	4	no	—
101	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₅₃	#117	1	2	yes	—
103	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
104	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
105	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
106	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
107	C₂×C₄×C₅₃	#121	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	#123	1	2	yes	—
113	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
114	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
115	C₂×C₄×C₅₃	#125	1	2	yes	—
116	C₂×C₄×C₅₃	#125	1	2	yes	—
117	C₂×C₄×C₅₃	#126	1	2	yes	—
118	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
119	C₈×C₅₃	—	2	4	no	—
120	C₈×C₅₃	—	4	4	no	—
121	C₈×C₅₃	—	4	2	no	—
122	#11286	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
123	C₂²×C₅₃	—	2	2	no	—
124	C₂²×C₅₃	—	2	4	no	—
125	C₂²×C₅₃	—	4	2	no	—
126	C₂²×C₅₃	—	4	1	no	—
127	C₂²×C₅₃	—	4	1	no	—
128	C₂²×C₅₃	#388	1	2	yes	—
129	C₂²×C₅₃	#389	1	1	yes	characteristic
130	C₂²×C₅₃	#393	1	1	yes	characteristic
131	C₂²×C₅₃	#400	1	2	yes	—
132	C₂²×C₅₃	#400	1	1	yes	characteristic
133	C₄×C₅₃	—	4	4	no	—
134	C₄×C₅₃	—	8	4	no	—
135	C₄×C₅₃	#382	1	1	yes	characteristic
136	C₄×C₅₃	#385	1	1	yes	characteristic
137	C₄×C₅₃	#387	1	2	yes	—
138	C₄×C₅₃	#394	1	1	yes	characteristic
139	C₄×C₅₃	#397	1	1	yes	characteristic
140	C₄×C₅₃	#398	1	2	yes	—
141	C₂×#430	C₂×C₅₃	1	1	yes	characteristic
142	C₂×#344	C₂×C₅₃	1	2	yes	—
143	C₂×#339	C₂×C₅₃	1	2	yes	—
144	C₂×H(3,ℤ/4)	C₂×C₅₃	1	1	yes	characteristic
145	#1728	C₂×C₅₃	1	1	yes	characteristic
146	C₂×C₅₃	—	2	1	no	—
147	C₂×C₅₃	—	2	2	no	—
148	C₂×C₅₃	—	2	1	no	—
149	C₂×C₅₃	—	2	2	no	—
150	C₂×C₅₃	#1442	1	1	yes	characteristic
151	C₂×C₅₃	#2102	1	2	yes	—
152	C₂²×M₁₆	C₂²×C₅₃	1	2	yes	—
153	C₂²×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
154	C₂²×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
155	C₂×#117	C₂²×C₅₃	1	2	yes	—
156	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
157	#430	—	2	4	no	—
158	C₂×#108	—	2	2	no	—
159	#344	—	2	4	no	—
160	#339	—	2	4	no	—
161	H(3,ℤ/4)	C₄×C₅₃	1	4	yes	complemented
162	#336	—	2	2	no	—
163	C₅₃	#11286	1	1	yes	characteristic, direct product factor, Sylow
164	C₂³×C₄	—	2	1	no	—
165	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
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	2	yes	—
177	C₂×M₁₆	D₄×C₅₃	1	4	yes	—
178	C₂×M₁₆	D₄×C₅₃	1	2	yes	—
179	C₂×M₁₆	Q₈×C₅₃	1	2	yes	—
180	C₂²×C₈	—	2	1	no	—
181	C₂²×C₈	—	2	2	no	—
182	C₂²×C₈	D₄×C₅₃	1	2	yes	—
183	#117	—	2	2	no	—
184	#117	—	2	2	no	—
185	#117	—	2	2	no	—
186	#117	C₂×C₄×C₅₃	1	4	yes	—
187	C₂×C₄⋊C₄	—	2	2	no	—
188	C₂×C₂²⋊C₄	—	2	2	no	—
189	C₂×C₄²	—	2	2	no	—
190	C₂×C₄²	—	2	2	no	—
191	C₂×C₄²	—	2	2	no	—
192	C₂×C₄²	—	2	1	no	—
193	C₂×C₄²	Q₈×C₅₃	1	2	yes	—
194	#108	—	2	8	no	—
195	C₂⁴	#41×C₅₃	1	1	yes	characteristic
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	—	4	4	no	—
198	C₂²×C₄	—	4	2	no	—
199	C₂²×C₄	—	4	2	no	—
200	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
201	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
202	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
203	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
204	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
205	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
206	C₂²×C₄	#41×C₅₃	1	2	yes	—
207	C₂²×C₄	#41×C₅₃	1	2	yes	—
208	C₂²×C₄	#41×C₅₃	1	2	yes	—
209	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
210	M₁₆	—	2	4	no	—
211	M₁₆	—	2	4	no	—
212	M₁₆	—	2	4	no	—
213	M₁₆	—	2	4	no	—
214	M₁₆	—	2	4	no	—
215	M₁₆	—	4	2	no	—
216	M₁₆	—	4	2	no	—
217	M₁₆	—	4	2	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	2	4	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	4	2	no	—
223	C₂×C₈	—	4	1	no	—
224	C₂×C₈	—	4	2	no	—
225	C₂×C₈	—	4	1	no	—
226	C₂×C₈	C₄⋊C₄×C₅₃	1	4	yes	—
227	C₄⋊C₄	—	2	4	no	—
228	C₄⋊C₄	—	4	2	no	—
229	C₂²⋊C₄	—	2	4	no	—
230	C₂²⋊C₄	—	4	2	no	—
231	C₄²	—	2	4	no	—
232	C₄²	—	2	4	no	—
233	C₄²	—	4	1	no	—
234	C₄²	—	4	2	no	—
235	C₄²	—	4	1	no	—
236	C₂³	—	4	2	no	—
237	C₂³	C₄×D₄×C₅₃	1	1	yes	characteristic
238	C₂³	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
239	C₂³	#124×C₅₃	1	2	yes	—
240	C₂³	#125×C₅₃	1	1	yes	characteristic
241	C₂³	#126×C₅₃	1	2	yes	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	4	8	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	4	no	—
247	C₂×C₄	—	8	4	no	—
248	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
249	C₂×C₄	#117×C₅₃	1	2	yes	—
250	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
251	C₂×C₄	C₄×Q₈×C₅₃	1	1	yes	characteristic
252	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
253	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
254	C₂×C₄	#121×C₅₃	1	2	yes	—
255	C₂×C₄	C₂²⋊Q₈×C₅₃	1	2	yes	—
256	C₂×C₄	C₂²⋊Q₈×C₅₃	1	2	yes	—
257	C₂×C₄	#123×C₅₃	1	1	yes	characteristic
258	C₂×C₄	#123×C₅₃	1	2	yes	—
259	C₂×C₄	#123×C₅₃	1	2	yes	—
260	C₂×C₄	#124×C₅₃	1	1	yes	characteristic
261	C₂×C₄	#124×C₅₃	1	1	yes	characteristic
262	C₂×C₄	#125×C₅₃	1	2	yes	—
263	C₂×C₄	#125×C₅₃	1	2	yes	—
264	C₂×C₄	#126×C₅₃	1	2	yes	—
265	C₂×C₄	C₄⋊Q₈×C₅₃	1	1	yes	characteristic
266	C₈	—	2	4	no	—
267	C₈	—	4	4	no	—
268	C₈	—	4	2	no	—
269	C₂²	—	2	2	no	—
270	C₂²	—	2	4	no	—
271	C₂²	—	4	2	no	—
272	C₂²	—	4	1	no	—
273	C₂²	—	4	1	no	—
274	C₂²	C₅₃×#388	1	2	yes	—
275	C₂²	C₅₃×#389	1	1	yes	characteristic
276	C₂²	C₅₃×#393	1	1	yes	characteristic
277	C₂²	C₅₃×#400	1	2	yes	—
278	C₂²	C₅₃×#400	1	1	yes	characteristic
279	C₄	—	4	4	no	—
280	C₄	—	8	4	no	—
281	C₄	C₅₃×#382	1	1	yes	characteristic
282	C₄	C₅₃×#385	1	1	yes	characteristic
283	C₄	C₅₃×#387	1	2	yes	—
284	C₄	C₅₃×#394	1	1	yes	characteristic
285	C₄	C₅₃×#397	1	1	yes	characteristic
286	C₄	C₅₃×#398	1	2	yes	—
287	C₂	—	2	1	no	—
288	C₂	—	2	2	no	—
289	C₂	—	2	1	no	—
290	C₂	—	2	2	no	—
291	C₂	C₅₃×#1442	1	1	yes	characteristic
292	C₂	C₅₃×#2102	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅₃	#124
commutator	C₂²×C₄	C₂²×C₄×C₅₃
Frattini, cosocle	C₂³×C₄	C₂³×C₅₃
socle	C₂²×C₅₃	#400
prime radical	C₂⁴×C₅₃	#41
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
15. C₅₃×H(3,ℤ/4)	C₄	semidirect
122. #11286	C₅₃	direct (N × H)
161. H(3,ℤ/4)	C₄×C₅₃	semidirect
163. C₅₃	#11286	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
C₂×C₅₃×#344	2
C₂×C₅₃×#339	2
C₂×C₅₃×H(3,ℤ/4)	2
C₅₃×#1728	2
#11286	53

Subgroups — C53×#11286 #27218798

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#11286 #27218798