Subgroups — C₅₃×#11364 #27218876

Order 13568.

606 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×#427	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1740	C₂	1	2	yes	maximal
3	C₅₃×#1735	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1733	C₂	1	2	yes	maximal
5	C₅₃×#1729	C₂	1	1	yes	maximal, characteristic
6	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
7	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
8	C₂×#117×C₅₃	C₂²	1	2	yes	—
9	C₂×#117×C₅₃	C₂²	1	2	yes	—
10	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
11	C₅₃×#427	—	2	2	no	—
12	C₅₃×#427	—	2	2	no	—
13	C₂×#107×C₅₃	C₄	1	2	yes	complemented
14	C₂×#106×C₅₃	C₄	1	2	yes	complemented
15	C₅₃×#342	—	2	4	no	—
16	C₅₃×#340	—	2	2	no	—
17	C₅₃×#339	—	2	4	no	—
18	C₅₃×#336	—	2	2	no	—
19	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
20	C₂×M₁₆×C₅₃	—	2	1	no	—
21	C₂×M₁₆×C₅₃	—	2	2	no	—
22	C₂×M₁₆×C₅₃	—	2	2	no	—
23	C₂×M₁₆×C₅₃	D₄	1	2	yes	—
24	C₂²×C₈×C₅₃	—	2	1	no	—
25	C₂²×C₈×C₅₃	—	2	2	no	—
26	C₂²×C₈×C₅₃	—	2	2	no	—
27	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₅₃	D₄	1	2	yes	—
30	C₂²×C₈×C₅₃	D₄	1	2	yes	—
31	#117×C₅₃	—	2	2	no	—
32	#117×C₅₃	—	2	2	no	—
33	#117×C₅₃	—	2	2	no	—
34	#117×C₅₃	—	2	2	no	—
35	#117×C₅₃	—	2	2	no	—
36	#117×C₅₃	—	2	2	no	—
37	#117×C₅₃	—	2	2	no	—
38	#117×C₅₃	—	2	2	no	—
39	#117×C₅₃	—	2	2	no	—
40	#117×C₅₃	Q₈	1	4	yes	—
41	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
42	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
43	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
44	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
45	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
46	C₂×C₂²⋊C₄×C₅₃	D₄	1	2	yes	—
47	C₂×C₄²×C₅₃	—	2	1	no	—
48	C₂×C₄²×C₅₃	—	2	2	no	—
49	C₂×C₄²×C₅₃	—	2	2	no	—
50	#107×C₅₃	—	2	4	no	—
51	#106×C₅₃	—	2	4	no	—
52	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	—	2	2	no	—
54	C₂²×C₄×C₅₃	—	2	4	no	—
55	C₂²×C₄×C₅₃	—	4	2	no	—
56	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
61	C₂²×C₄×C₅₃	#41	1	2	yes	—
62	C₂²×C₄×C₅₃	#41	1	2	yes	—
63	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	#41	1	2	yes	—
65	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
66	M₁₆×C₅₃	—	2	4	no	—
67	M₁₆×C₅₃	—	4	2	no	—
68	C₂×C₈×C₅₃	—	2	2	no	—
69	C₂×C₈×C₅₃	—	2	2	no	—
70	C₂×C₈×C₅₃	—	2	2	no	—
71	C₂×C₈×C₅₃	—	2	2	no	—
72	C₂×C₈×C₅₃	—	2	4	no	—
73	C₂×C₈×C₅₃	—	2	2	no	—
74	C₂×C₈×C₅₃	—	4	1	no	—
75	C₂×C₈×C₅₃	—	4	1	no	—
76	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
77	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
78	C₄⋊C₄×C₅₃	—	2	4	no	—
79	C₄⋊C₄×C₅₃	—	2	4	no	—
80	C₄⋊C₄×C₅₃	—	2	4	no	—
81	C₄⋊C₄×C₅₃	—	4	2	no	—
82	C₄⋊C₄×C₅₃	—	4	2	no	—
83	C₂²⋊C₄×C₅₃	—	2	4	no	—
84	C₂²⋊C₄×C₅₃	—	2	4	no	—
85	C₂²⋊C₄×C₅₃	—	4	2	no	—
86	C₂²⋊C₄×C₅₃	—	4	2	no	—
87	C₂²⋊C₄×C₅₃	—	4	2	no	—
88	C₄²×C₅₃	—	2	4	no	—
89	C₄²×C₅₃	—	4	2	no	—
90	C₄²×C₅₃	—	4	1	no	—
91	C₄²×C₅₃	—	4	2	no	—
92	C₄²×C₅₃	—	4	2	no	—
93	C₄²×C₅₃	—	4	1	no	—
94	C₂³×C₅₃	—	4	2	no	—
95	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
96	C₂³×C₅₃	#121	1	1	yes	characteristic
97	C₂³×C₅₃	C₂²⋊Q₈	1	2	yes	—
98	C₂³×C₅₃	#123	1	2	yes	—
99	C₂³×C₅₃	#127	1	1	yes	characteristic
100	C₂×C₄×C₅₃	—	4	4	no	—
101	C₂×C₄×C₅₃	—	4	4	no	—
102	C₂×C₄×C₅₃	—	4	2	no	—
103	C₂×C₄×C₅₃	—	4	4	no	—
104	C₂×C₄×C₅₃	—	8	4	no	—
105	C₂×C₄×C₅₃	—	8	4	no	—
106	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₅₃	#117	1	2	yes	—
108	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
110	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
111	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
112	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
113	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
114	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
115	C₂×C₄×C₅₃	#123	1	2	yes	—
116	C₂×C₄×C₅₃	#123	1	2	yes	—
117	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
118	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
119	C₂×C₄×C₅₃	#125	1	2	yes	—
120	C₂×C₄×C₅₃	#125	1	2	yes	—
121	C₂×C₄×C₅₃	#126	1	2	yes	—
122	C₂×C₄×C₅₃	#126	1	2	yes	—
123	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
124	C₈×C₅₃	—	2	2	no	—
125	C₈×C₅₃	—	2	2	no	—
126	C₈×C₅₃	—	4	2	no	—
127	#11364	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
128	C₂²×C₅₃	—	2	2	no	—
129	C₂²×C₅₃	—	2	2	no	—
130	C₂²×C₅₃	—	2	2	no	—
131	C₂²×C₅₃	—	4	2	no	—
132	C₂²×C₅₃	—	4	2	no	—
133	C₂²×C₅₃	#387	1	2	yes	—
134	C₂²×C₅₃	#390	1	1	yes	characteristic
135	C₂²×C₅₃	#397	1	1	yes	characteristic
136	C₂²×C₅₃	#398	1	1	yes	characteristic
137	C₂²×C₅₃	#399	1	2	yes	—
138	C₄×C₅₃	—	8	2	no	—
139	C₄×C₅₃	—	8	4	no	—
140	C₄×C₅₃	—	8	4	no	—
141	C₄×C₅₃	#381	1	1	yes	characteristic
142	C₄×C₅₃	#387	1	2	yes	—
143	C₄×C₅₃	#391	1	1	yes	characteristic
144	C₄×C₅₃	#397	1	1	yes	characteristic
145	C₄×C₅₃	#398	1	1	yes	characteristic
146	C₄×C₅₃	#400	1	2	yes	—
147	C₂×#427	C₂×C₅₃	1	1	yes	characteristic
148	#1740	C₂×C₅₃	1	2	yes	—
149	#1735	C₂×C₅₃	1	1	yes	characteristic
150	#1733	C₂×C₅₃	1	2	yes	—
151	#1729	C₂×C₅₃	1	1	yes	characteristic
152	C₂×C₅₃	—	2	1	no	—
153	C₂×C₅₃	—	2	2	no	—
154	C₂×C₅₃	—	2	2	no	—
155	C₂×C₅₃	—	2	1	no	—
156	C₂×C₅₃	#1445	1	1	yes	characteristic
157	C₂×C₅₃	#1976	1	2	yes	—
158	C₂²×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
159	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
160	C₂×#117	C₂²×C₅₃	1	2	yes	—
161	C₂×#117	C₂²×C₅₃	1	2	yes	—
162	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
163	#427	—	2	2	no	—
164	#427	—	2	2	no	—
165	C₂×#107	C₄×C₅₃	1	2	yes	complemented
166	C₂×#106	C₄×C₅₃	1	2	yes	complemented
167	#342	—	2	4	no	—
168	#340	—	2	2	no	—
169	#339	—	2	4	no	—
170	#336	—	2	2	no	—
171	C₅₃	#11364	1	1	yes	characteristic, direct product factor, Sylow
172	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
173	C₂×M₁₆	—	2	1	no	—
174	C₂×M₁₆	—	2	2	no	—
175	C₂×M₁₆	—	2	2	no	—
176	C₂×M₁₆	D₄×C₅₃	1	2	yes	—
177	C₂²×C₈	—	2	2	no	—
178	C₂²×C₈	—	2	2	no	—
179	C₂²×C₈	—	2	1	no	—
180	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
181	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
182	C₂²×C₈	D₄×C₅₃	1	2	yes	—
183	C₂²×C₈	D₄×C₅₃	1	2	yes	—
184	#117	—	2	2	no	—
185	#117	—	2	2	no	—
186	#117	—	2	2	no	—
187	#117	—	2	2	no	—
188	#117	—	2	2	no	—
189	#117	—	2	2	no	—
190	#117	—	2	2	no	—
191	#117	—	2	2	no	—
192	#117	—	2	2	no	—
193	#117	Q₈×C₅₃	1	4	yes	—
194	C₂×C₄⋊C₄	—	2	2	no	—
195	C₂×C₄⋊C₄	—	2	1	no	—
196	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	4	yes	—
197	C₂×C₂²⋊C₄	—	2	2	no	—
198	C₂×C₂²⋊C₄	—	2	2	no	—
199	C₂×C₂²⋊C₄	D₄×C₅₃	1	2	yes	—
200	C₂×C₄²	—	2	2	no	—
201	C₂×C₄²	—	2	1	no	—
202	C₂×C₄²	—	2	2	no	—
203	#107	—	2	4	no	—
204	#106	—	2	4	no	—
205	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
206	C₂²×C₄	—	2	2	no	—
207	C₂²×C₄	—	2	4	no	—
208	C₂²×C₄	—	4	2	no	—
209	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
210	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
211	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
212	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
213	C₂²×C₄	C₂×Q₈×C₅₃	1	2	yes	—
214	C₂²×C₄	#41×C₅₃	1	2	yes	—
215	C₂²×C₄	#41×C₅₃	1	2	yes	—
216	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
217	C₂²×C₄	#41×C₅₃	1	2	yes	—
218	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
219	M₁₆	—	2	4	no	—
220	M₁₆	—	4	2	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	2	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	1	no	—
229	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
230	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
231	C₄⋊C₄	—	2	4	no	—
232	C₄⋊C₄	—	2	4	no	—
233	C₄⋊C₄	—	2	4	no	—
234	C₄⋊C₄	—	4	2	no	—
235	C₄⋊C₄	—	4	2	no	—
236	C₂²⋊C₄	—	2	4	no	—
237	C₂²⋊C₄	—	2	4	no	—
238	C₂²⋊C₄	—	4	2	no	—
239	C₂²⋊C₄	—	4	2	no	—
240	C₂²⋊C₄	—	4	2	no	—
241	C₄²	—	2	4	no	—
242	C₄²	—	4	2	no	—
243	C₄²	—	4	1	no	—
244	C₄²	—	4	2	no	—
245	C₄²	—	4	2	no	—
246	C₄²	—	4	1	no	—
247	C₂³	—	4	2	no	—
248	C₂³	C₄×D₄×C₅₃	1	1	yes	characteristic
249	C₂³	#121×C₅₃	1	1	yes	characteristic
250	C₂³	C₂²⋊Q₈×C₅₃	1	2	yes	—
251	C₂³	#123×C₅₃	1	2	yes	—
252	C₂³	#127×C₅₃	1	1	yes	characteristic
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	4	4	no	—
257	C₂×C₄	—	8	4	no	—
258	C₂×C₄	—	8	4	no	—
259	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
260	C₂×C₄	#117×C₅₃	1	2	yes	—
261	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
262	C₂×C₄	C₄×Q₈×C₅₃	1	2	yes	—
263	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
264	C₂×C₄	C₂²⋊Q₈×C₅₃	1	2	yes	—
265	C₂×C₄	C₂²⋊Q₈×C₅₃	1	2	yes	—
266	C₂×C₄	#123×C₅₃	1	1	yes	characteristic
267	C₂×C₄	#123×C₅₃	1	1	yes	characteristic
268	C₂×C₄	#123×C₅₃	1	2	yes	—
269	C₂×C₄	#123×C₅₃	1	2	yes	—
270	C₂×C₄	#124×C₅₃	1	1	yes	characteristic
271	C₂×C₄	#124×C₅₃	1	1	yes	characteristic
272	C₂×C₄	#125×C₅₃	1	2	yes	—
273	C₂×C₄	#125×C₅₃	1	2	yes	—
274	C₂×C₄	#126×C₅₃	1	2	yes	—
275	C₂×C₄	#126×C₅₃	1	2	yes	—
276	C₂×C₄	C₄⋊Q₈×C₅₃	1	1	yes	characteristic
277	C₈	—	2	2	no	—
278	C₈	—	2	2	no	—
279	C₈	—	4	2	no	—
280	C₂²	—	2	2	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	2	2	no	—
283	C₂²	—	4	2	no	—
284	C₂²	—	4	2	no	—
285	C₂²	C₅₃×#387	1	2	yes	—
286	C₂²	C₅₃×#390	1	1	yes	characteristic
287	C₂²	C₅₃×#397	1	1	yes	characteristic
288	C₂²	C₅₃×#398	1	1	yes	characteristic
289	C₂²	C₅₃×#399	1	2	yes	—
290	C₄	—	8	2	no	—
291	C₄	—	8	4	no	—
292	C₄	—	8	4	no	—
293	C₄	C₅₃×#381	1	1	yes	characteristic
294	C₄	C₅₃×#387	1	2	yes	—
295	C₄	C₅₃×#391	1	1	yes	characteristic
296	C₄	C₅₃×#397	1	1	yes	characteristic
297	C₄	C₅₃×#398	1	1	yes	characteristic
298	C₄	C₅₃×#400	1	2	yes	—
299	C₂	—	2	1	no	—
300	C₂	—	2	2	no	—
301	C₂	—	2	2	no	—
302	C₂	—	2	1	no	—
303	C₂	C₅₃×#1445	1	1	yes	characteristic
304	C₂	C₅₃×#1976	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	#390
prime radical	C₂⁴×C₅₃	C₂×D₄
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
13. C₂×#107×C₅₃	C₄	semidirect
14. C₂×#106×C₅₃	C₄	semidirect
127. #11364	C₅₃	direct (N × H)
165. C₂×#107	C₄×C₅₃	semidirect
166. C₂×#106	C₄×C₅₃	semidirect
171. C₅₃	#11364	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×#427	2
C₅₃×#1740	2
C₅₃×#1735	2
C₅₃×#1733	2
C₅₃×#1729	2
#11364	53

Subgroups — C53×#11364 #27218876

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#11364 #27218876