Subgroups — C₅₃×#11366 #27218878

Order 13568.

634 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#106×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#343	C₂	1	2	yes	maximal
3	C₂×C₅₃×#341	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1736	C₂	1	2	yes	maximal
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	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
9	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
10	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
11	C₂×#106×C₅₃	—	2	2	no	—
12	C₂×#106×C₅₃	—	2	2	no	—
13	C₂×#106×C₅₃	C₄	1	4	yes	complemented
14	C₅₃×#343	—	2	4	no	—
15	C₅₃×#341	—	2	2	no	—
16	C₅₃×#340	—	2	4	no	—
17	C₅₃×#336	—	2	2	no	—
18	C₂³×C₄×C₅₃	—	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	1	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	D₄	1	2	yes	—
29	C₂×M₁₆×C₅₃	Q₈	1	4	yes	—
30	C₂²×C₈×C₅₃	—	2	1	no	—
31	C₂²×C₈×C₅₃	—	2	2	no	—
32	C₂²×C₈×C₅₃	—	2	2	no	—
33	C₂²×C₈×C₅₃	—	2	2	no	—
34	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
35	C₂²×C₈×C₅₃	D₄	1	2	yes	—
36	C₂²×C₈×C₅₃	D₄	1	2	yes	—
37	#117×C₅₃	—	2	2	no	—
38	#117×C₅₃	—	2	2	no	—
39	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
40	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
41	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
42	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
43	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
44	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
45	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
46	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
47	C₂×C₄²×C₅₃	—	2	1	no	—
48	#106×C₅₃	—	2	8	no	—
49	C₂⁴×C₅₃	#41	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	—	2	2	no	—
51	C₂²×C₄×C₅₃	—	2	4	no	—
52	C₂²×C₄×C₅₃	—	4	2	no	—
53	C₂²×C₄×C₅₃	—	4	2	no	—
54	C₂²×C₄×C₅₃	—	8	4	no	—
55	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	C₂×D₄	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	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	4	no	—
72	C₂×C₈×C₅₃	—	2	2	no	—
73	C₂×C₈×C₅₃	—	2	2	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	—	4	1	no	—
78	C₂×C₈×C₅₃	—	4	2	no	—
79	C₂×C₈×C₅₃	—	4	1	no	—
80	C₂×C₈×C₅₃	—	4	2	no	—
81	C₂×C₈×C₅₃	C₂²⋊C₄	1	4	yes	—
82	C₄⋊C₄×C₅₃	—	2	4	no	—
83	C₄⋊C₄×C₅₃	—	2	4	no	—
84	C₄⋊C₄×C₅₃	—	2	4	no	—
85	C₄⋊C₄×C₅₃	—	4	2	no	—
86	C₄⋊C₄×C₅₃	—	4	2	no	—
87	C₂²⋊C₄×C₅₃	—	4	2	no	—
88	C₄²×C₅₃	—	4	1	no	—
89	C₄²×C₅₃	—	4	1	no	—
90	C₂³×C₅₃	—	2	4	no	—
91	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
92	C₂³×C₅₃	#121	1	1	yes	characteristic
93	C₂³×C₅₃	#124	1	1	yes	characteristic
94	C₂³×C₅₃	#125	1	2	yes	—
95	C₂³×C₅₃	#126	1	2	yes	—
96	C₂×C₄×C₅₃	—	4	4	no	—
97	C₂×C₄×C₅₃	—	4	2	no	—
98	C₂×C₄×C₅₃	—	4	4	no	—
99	C₂×C₄×C₅₃	—	8	4	no	—
100	C₂×C₄×C₅₃	—	8	4	no	—
101	C₂×C₄×C₅₃	—	8	4	no	—
102	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₅₃	#117	1	2	yes	—
104	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₄×Q₈	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	1	yes	characteristic
112	C₂×C₄×C₅₃	#123	1	2	yes	—
113	C₂×C₄×C₅₃	#123	1	2	yes	—
114	C₂×C₄×C₅₃	#123	1	2	yes	—
115	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
116	C₂×C₄×C₅₃	#125	1	2	yes	—
117	C₂×C₄×C₅₃	#126	1	2	yes	—
118	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
119	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
120	C₈×C₅₃	—	2	4	no	—
121	C₈×C₅₃	—	4	2	no	—
122	C₈×C₅₃	—	4	4	no	—
123	#11366	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
124	C₂²×C₅₃	—	2	4	no	—
125	C₂²×C₅₃	—	2	4	no	—
126	C₂²×C₅₃	—	2	2	no	—
127	C₂²×C₅₃	—	2	2	no	—
128	C₂²×C₅₃	#387	1	2	yes	—
129	C₂²×C₅₃	#390	1	1	yes	characteristic
130	C₂²×C₅₃	#397	1	1	yes	characteristic
131	C₂²×C₅₃	#398	1	1	yes	characteristic
132	C₂²×C₅₃	#400	1	2	yes	—
133	C₂²×C₅₃	#471	1	4	yes	—
134	C₄×C₅₃	—	8	2	no	—
135	C₄×C₅₃	—	8	4	no	—
136	C₄×C₅₃	#381	1	1	yes	characteristic
137	C₄×C₅₃	#387	1	2	yes	—
138	C₄×C₅₃	#391	1	1	yes	characteristic
139	C₄×C₅₃	#397	1	1	yes	characteristic
140	C₄×C₅₃	#398	1	1	yes	characteristic
141	C₄×C₅₃	#399	1	2	yes	—
142	C₂²×#106	C₂×C₅₃	1	1	yes	characteristic
143	C₂×#343	C₂×C₅₃	1	2	yes	—
144	C₂×#341	C₂×C₅₃	1	1	yes	characteristic
145	#1736	C₂×C₅₃	1	2	yes	—
146	#1729	C₂×C₅₃	1	1	yes	characteristic
147	C₂×C₅₃	—	2	2	no	—
148	C₂×C₅₃	—	2	2	no	—
149	C₂×C₅₃	#1445	1	1	yes	characteristic
150	C₂×C₅₃	#1977	1	2	yes	—
151	C₂×C₅₃	#2079	1	2	yes	—
152	C₂×C₅₃	#2090	1	2	yes	—
153	C₂²×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
154	C₂²×M₁₆	C₂²×C₅₃	1	2	yes	—
155	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
156	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
157	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
158	C₂×#106	—	2	2	no	—
159	C₂×#106	—	2	2	no	—
160	C₂×#106	C₄×C₅₃	1	4	yes	complemented
161	#343	—	2	4	no	—
162	#341	—	2	2	no	—
163	#340	—	2	4	no	—
164	#336	—	2	2	no	—
165	C₅₃	#11366	1	1	yes	characteristic, direct product factor, Sylow
166	C₂³×C₄	—	2	2	no	—
167	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
168	C₂×M₁₆	—	2	1	no	—
169	C₂×M₁₆	—	2	2	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	D₄×C₅₃	1	2	yes	—
177	C₂×M₁₆	Q₈×C₅₃	1	4	yes	—
178	C₂²×C₈	—	2	2	no	—
179	C₂²×C₈	—	2	2	no	—
180	C₂²×C₈	—	2	1	no	—
181	C₂²×C₈	—	2	2	no	—
182	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
183	C₂²×C₈	D₄×C₅₃	1	2	yes	—
184	C₂²×C₈	D₄×C₅₃	1	2	yes	—
185	#117	—	2	2	no	—
186	#117	—	2	2	no	—
187	C₂×C₄⋊C₄	—	2	2	no	—
188	C₂×C₄⋊C₄	—	2	2	no	—
189	C₂×C₄⋊C₄	—	2	2	no	—
190	C₂×C₄⋊C₄	—	2	2	no	—
191	C₂×C₄⋊C₄	—	2	2	no	—
192	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	4	yes	—
193	C₂×C₄⋊C₄	D₄×C₅₃	1	2	yes	—
194	C₂×C₂²⋊C₄	—	2	1	no	—
195	C₂×C₄²	—	2	1	no	—
196	#106	—	2	8	no	—
197	C₂⁴	#41×C₅₃	1	1	yes	characteristic
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	—	2	4	no	—
200	C₂²×C₄	—	4	2	no	—
201	C₂²×C₄	—	4	2	no	—
202	C₂²×C₄	—	8	4	no	—
203	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
204	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
205	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
206	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
207	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
208	C₂²×C₄	C₂×Q₈×C₅₃	1	2	yes	—
209	C₂²×C₄	#41×C₅₃	1	2	yes	—
210	C₂²×C₄	#41×C₅₃	1	2	yes	—
211	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
212	C₂²×C₄	#41×C₅₃	1	2	yes	—
213	M₁₆	—	2	4	no	—
214	M₁₆	—	2	4	no	—
215	M₁₆	—	2	4	no	—
216	M₁₆	—	4	2	no	—
217	M₁₆	—	4	2	no	—
218	M₁₆	—	4	2	no	—
219	C₂×C₈	—	2	4	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	4	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	4	no	—
225	C₂×C₈	—	4	1	no	—
226	C₂×C₈	—	4	2	no	—
227	C₂×C₈	—	4	1	no	—
228	C₂×C₈	—	4	2	no	—
229	C₂×C₈	C₂²⋊C₄×C₅₃	1	4	yes	—
230	C₄⋊C₄	—	2	4	no	—
231	C₄⋊C₄	—	2	4	no	—
232	C₄⋊C₄	—	2	4	no	—
233	C₄⋊C₄	—	4	2	no	—
234	C₄⋊C₄	—	4	2	no	—
235	C₂²⋊C₄	—	4	2	no	—
236	C₄²	—	4	1	no	—
237	C₄²	—	4	1	no	—
238	C₂³	—	2	4	no	—
239	C₂³	C₄×D₄×C₅₃	1	1	yes	characteristic
240	C₂³	#121×C₅₃	1	1	yes	characteristic
241	C₂³	#124×C₅₃	1	1	yes	characteristic
242	C₂³	#125×C₅₃	1	2	yes	—
243	C₂³	#126×C₅₃	1	2	yes	—
244	C₂×C₄	—	4	4	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	4	no	—
247	C₂×C₄	—	8	4	no	—
248	C₂×C₄	—	8	4	no	—
249	C₂×C₄	—	8	4	no	—
250	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
251	C₂×C₄	#117×C₅₃	1	2	yes	—
252	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
253	C₂×C₄	C₄×Q₈×C₅₃	1	2	yes	—
254	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
255	C₂×C₄	C₂²⋊Q₈×C₅₃	1	2	yes	—
256	C₂×C₄	C₂²⋊Q₈×C₅₃	1	2	yes	—
257	C₂×C₄	C₂²⋊Q₈×C₅₃	1	2	yes	—
258	C₂×C₄	#123×C₅₃	1	1	yes	characteristic
259	C₂×C₄	#123×C₅₃	1	1	yes	characteristic
260	C₂×C₄	#123×C₅₃	1	2	yes	—
261	C₂×C₄	#123×C₅₃	1	2	yes	—
262	C₂×C₄	#123×C₅₃	1	2	yes	—
263	C₂×C₄	#124×C₅₃	1	1	yes	characteristic
264	C₂×C₄	#125×C₅₃	1	2	yes	—
265	C₂×C₄	#126×C₅₃	1	2	yes	—
266	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
267	C₂×C₄	C₄⋊Q₈×C₅₃	1	1	yes	characteristic
268	C₈	—	2	4	no	—
269	C₈	—	4	2	no	—
270	C₈	—	4	4	no	—
271	C₂²	—	2	2	no	—
272	C₂²	—	2	2	no	—
273	C₂²	—	2	4	no	—
274	C₂²	—	2	4	no	—
275	C₂²	C₅₃×#387	1	2	yes	—
276	C₂²	C₅₃×#390	1	1	yes	characteristic
277	C₂²	C₅₃×#397	1	1	yes	characteristic
278	C₂²	C₅₃×#398	1	1	yes	characteristic
279	C₂²	C₅₃×#400	1	2	yes	—
280	C₂²	C₅₃×#471	1	4	yes	—
281	C₄	—	8	4	no	—
282	C₄	—	8	2	no	—
283	C₄	C₅₃×#381	1	1	yes	characteristic
284	C₄	C₅₃×#387	1	2	yes	—
285	C₄	C₅₃×#391	1	1	yes	characteristic
286	C₄	C₅₃×#397	1	1	yes	characteristic
287	C₄	C₅₃×#398	1	1	yes	characteristic
288	C₄	C₅₃×#399	1	2	yes	—
289	C₂	—	2	2	no	—
290	C₂	—	2	2	no	—
291	C₂	C₅₃×#1445	1	1	yes	characteristic
292	C₂	C₅₃×#1977	1	2	yes	—
293	C₂	C₅₃×#2079	1	2	yes	—
294	C₂	C₅₃×#2090	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅₃	#121
commutator	C₂²×C₄	C₂²×C₄×C₅₃
Frattini, cosocle	C₂³×C₄	C₂³×C₅₃
prime radical	C₂⁴×C₅₃	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂³×C₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
13. C₂×#106×C₅₃	C₄	semidirect
123. #11366	C₅₃	direct (N × H)
160. C₂×#106	C₄×C₅₃	semidirect
165. C₅₃	#11366	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×#106×C₅₃	2
C₂×C₅₃×#343	2
C₂×C₅₃×#341	2
C₅₃×#1736	2
C₅₃×#1729	2
#11366	53

Subgroups — C53×#11366 #27218878

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#11366 #27218878