Subgroups — C₅₃×#12746 #27220258

Order 13568.

886 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3584	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2148	C₂	1	2	yes	maximal, complemented
3	C₂×C₅₃×#359	C₂	1	2	yes	maximal, complemented
4	C₅₃×#2113	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1813	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#585	—	2	1	no	—
7	C₅₃×#578	—	2	2	no	—
8	C₅₃×#577	—	2	4	no	—
9	C₅₃×#576	—	2	2	no	—
10	C₅₃×#575	—	2	1	no	—
11	C₅₃×#575	C₄	1	4	yes	—
12	C₂×#135×C₅₃	—	2	2	no	—
13	C₂×#135×C₅₃	C₂²	1	2	yes	—
14	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic, complemented
16	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic, complemented
17	C₂×#108×C₅₃	C₂²	1	2	yes	complemented
18	C₅₃×#413	—	2	1	no	—
19	C₅₃×#362	—	2	2	no	—
20	C₅₃×#361	—	2	2	no	—
21	C₅₃×#359	—	2	4	no	—
22	C₅₃×C₂²⋊C₁₆	—	2	2	no	—
23	C₅₃×#338	—	2	1	no	—
24	2₋¹⁺⁴×C₅₃	—	2	4	no	—
25	2₋¹⁺⁴×C₅₃	—	4	1	no	—
26	H(5,𝔽₂)×C₅₃	—	2	4	no	—
27	H(5,𝔽₂)×C₅₃	—	4	1	no	—
28	H(5,𝔽₂)×C₅₃	—	4	2	no	—
29	C₂×#41×C₅₃	—	2	4	no	—
30	C₂×#41×C₅₃	—	2	2	no	—
31	C₂×#41×C₅₃	—	4	4	no	—
32	C₂×#41×C₅₃	D₄	1	1	yes	characteristic
33	#137×C₅₃	—	2	4	no	—
34	#137×C₅₃	—	2	8	no	—
35	AGL(1,ℤ/8)×C₅₃	—	2	4	no	—
36	AGL(1,ℤ/8)×C₅₃	—	2	8	no	—
37	#135×C₅₃	—	2	4	no	—
38	#135×C₅₃	—	2	4	no	—
39	#135×C₅₃	—	2	4	no	—
40	#135×C₅₃	—	2	2	no	—
41	#135×C₅₃	—	2	2	no	—
42	#135×C₅₃	C₂×C₄	1	4	yes	—
43	#135×C₅₃	D₄	1	4	yes	—
44	C₂×Q₁₆×C₅₃	—	2	2	no	—
45	C₂×Q₁₆×C₅₃	D₄	1	2	yes	—
46	C₂×SD₁₆×C₅₃	—	2	4	no	—
47	C₂×SD₁₆×C₅₃	—	2	2	no	—
48	C₂×D₈×C₅₃	—	2	2	no	—
49	C₂×D₈×C₅₃	D₄	1	2	yes	—
50	#131×C₅₃	—	2	2	no	—
51	#131×C₅₃	—	2	2	no	—
52	C₂×M₁₆×C₅₃	—	2	2	no	—
53	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
54	C₂×M₁₆×C₅₃	D₄	1	1	yes	characteristic
55	C₂²×C₈×C₅₃	D₄	1	2	yes	—
56	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
57	M₃₂×C₅₃	—	2	2	no	—
58	C₂×C₁₆×C₅₃	—	2	1	no	—
59	C₂×C₁₆×C₅₃	—	2	2	no	—
60	C₂×C₁₆×C₅₃	—	2	1	no	—
61	#108×C₅₃	—	2	2	no	—
62	#108×C₅₃	—	2	2	no	—
63	#101×C₅₃	—	4	1	no	—
64	#100×C₅₃	—	4	1	no	—
65	#41×C₅₃	—	2	2	no	—
66	#41×C₅₃	—	2	2	no	—
67	#41×C₅₃	—	2	8	no	—
68	#41×C₅₃	—	2	4	no	—
69	#41×C₅₃	—	2	4	no	—
70	#41×C₅₃	—	4	4	no	—
71	#41×C₅₃	—	4	8	no	—
72	#41×C₅₃	—	4	4	no	—
73	#41×C₅₃	—	4	4	no	—
74	#41×C₅₃	—	4	4	no	—
75	#41×C₅₃	—	4	2	no	—
76	#41×C₅₃	—	4	2	no	—
77	C₂×Q₈×C₅₃	—	2	4	no	—
78	C₂×Q₈×C₅₃	—	2	2	no	—
79	C₂×Q₈×C₅₃	—	4	4	no	—
80	C₂×Q₈×C₅₃	D₈	1	1	yes	characteristic
81	C₂×D₄×C₅₃	—	2	4	no	—
82	C₂×D₄×C₅₃	—	2	2	no	—
83	C₂×D₄×C₅₃	—	4	8	no	—
84	C₂×D₄×C₅₃	—	4	4	no	—
85	C₂×D₄×C₅₃	—	4	4	no	—
86	C₂×D₄×C₅₃	—	4	2	no	—
87	C₂×D₄×C₅₃	D₈	1	1	yes	characteristic
88	C₂×D₄×C₅₃	SD₁₆	1	2	yes	—
89	C₂²×C₄×C₅₃	—	4	4	no	—
90	C₂²×C₄×C₅₃	—	4	2	no	—
91	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
92	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
93	Q₁₆×C₅₃	—	2	4	no	—
94	Q₁₆×C₅₃	—	2	2	no	—
95	Q₁₆×C₅₃	C₂²⋊C₄	1	4	yes	—
96	SD₁₆×C₅₃	—	2	8	no	—
97	SD₁₆×C₅₃	—	2	4	no	—
98	SD₁₆×C₅₃	—	2	4	no	—
99	D₈×C₅₃	—	2	4	no	—
100	D₈×C₅₃	—	2	2	no	—
101	D₈×C₅₃	C₂²⋊C₄	1	4	yes	—
102	M₁₆×C₅₃	—	2	2	no	—
103	M₁₆×C₅₃	—	2	4	no	—
104	M₁₆×C₅₃	—	4	2	no	—
105	M₁₆×C₅₃	—	4	2	no	—
106	C₂×C₈×C₅₃	—	2	2	no	—
107	C₂×C₈×C₅₃	—	2	2	no	—
108	C₂×C₈×C₅₃	—	4	2	no	—
109	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
110	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
112	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
113	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
114	C₁₆×C₅₃	—	4	1	no	—
115	C₁₆×C₅₃	—	4	1	no	—
116	C₂³×C₅₃	—	4	4	no	—
117	C₂³×C₅₃	—	4	2	no	—
118	C₂³×C₅₃	#102	1	2	yes	—
119	C₂³×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
120	Q₈×C₅₃	—	2	2	no	—
121	Q₈×C₅₃	—	2	4	no	—
122	Q₈×C₅₃	—	4	4	no	—
123	Q₈×C₅₃	—	4	2	no	—
124	D₄×C₅₃	—	2	4	no	—
125	D₄×C₅₃	—	2	4	no	—
126	D₄×C₅₃	—	2	2	no	—
127	D₄×C₅₃	—	4	4	no	—
128	D₄×C₅₃	—	4	8	no	—
129	D₄×C₅₃	—	4	4	no	—
130	D₄×C₅₃	—	4	4	no	—
131	D₄×C₅₃	—	4	2	no	—
132	C₂×C₄×C₅₃	—	2	2	no	—
133	C₂×C₄×C₅₃	—	2	2	no	—
134	C₂×C₄×C₅₃	—	4	8	no	—
135	C₂×C₄×C₅₃	—	4	4	no	—
136	C₂×C₄×C₅₃	—	4	2	no	—
137	C₂×C₄×C₅₃	—	4	4	no	—
138	C₂×C₄×C₅₃	—	4	4	no	—
139	C₂×C₄×C₅₃	#102	1	2	yes	—
140	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
141	C₂×C₄×C₅₃	#120	1	2	yes	—
142	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
143	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
144	C₈×C₅₃	—	2	2	no	—
145	C₈×C₅₃	—	8	2	no	—
146	C₈×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
147	C₈×C₅₃	#120	1	1	yes	characteristic
148	C₈×C₅₃	#120	1	1	yes	characteristic
149	#12746	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
150	C₂²×C₅₃	—	2	2	no	—
151	C₂²×C₅₃	—	4	1	no	—
152	C₂²×C₅₃	—	4	2	no	—
153	C₂²×C₅₃	—	4	4	no	—
154	C₂²×C₅₃	—	16	2	no	—
155	C₂²×C₅₃	—	16	4	no	—
156	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
157	C₂²×C₅₃	#450	1	2	yes	—
158	C₄×C₅₃	—	2	2	no	—
159	C₄×C₅₃	—	4	4	no	—
160	C₄×C₅₃	#379	1	1	yes	characteristic
161	C₄×C₅₃	#418	1	1	yes	characteristic
162	C₄×C₅₃	#447	1	2	yes	—
163	#3584	C₂×C₅₃	1	1	yes	characteristic
164	#2148	C₂×C₅₃	1	2	yes	complemented
165	C₂×#359	C₂×C₅₃	1	2	yes	complemented
166	#2113	C₂×C₅₃	1	1	yes	characteristic, complemented
167	#1813	C₂×C₅₃	1	1	yes	characteristic, complemented
168	C₂×C₅₃	—	2	1	no	—
169	C₂×C₅₃	—	2	2	no	—
170	C₂×C₅₃	—	4	2	no	—
171	C₂×C₅₃	—	8	4	no	—
172	C₂×C₅₃	#1786	1	1	yes	characteristic
173	#585	—	2	1	no	—
174	#578	—	2	2	no	—
175	#577	—	2	4	no	—
176	#576	—	2	2	no	—
177	#575	—	2	1	no	—
178	#575	C₄×C₅₃	1	4	yes	—
179	C₂×#135	—	2	2	no	—
180	C₂×#135	C₂²×C₅₃	1	2	yes	—
181	C₂×#131	C₂²×C₅₃	1	1	yes	characteristic
182	C₂×M₃₂	C₂²×C₅₃	1	1	yes	characteristic, complemented
183	C₂²×C₁₆	C₂²×C₅₃	1	1	yes	characteristic, complemented
184	C₂×#108	C₂²×C₅₃	1	2	yes	complemented
185	#413	—	2	1	no	—
186	#362	—	2	2	no	—
187	#361	—	2	2	no	—
188	#359	—	2	4	no	—
189	C₂²⋊C₁₆	—	2	2	no	—
190	#338	—	2	1	no	—
191	C₅₃	#12746	1	1	yes	characteristic, direct product factor, Sylow
192	2₋¹⁺⁴	—	2	4	no	—
193	2₋¹⁺⁴	—	4	1	no	—
194	H(5,𝔽₂)	—	2	4	no	—
195	H(5,𝔽₂)	—	4	1	no	—
196	H(5,𝔽₂)	—	4	2	no	—
197	C₂×#41	—	2	2	no	—
198	C₂×#41	—	2	4	no	—
199	C₂×#41	—	4	4	no	—
200	C₂×#41	D₄×C₅₃	1	1	yes	characteristic
201	#137	—	2	4	no	—
202	#137	—	2	8	no	—
203	AGL(1,ℤ/8)	—	2	4	no	—
204	AGL(1,ℤ/8)	—	2	8	no	—
205	#135	—	2	4	no	—
206	#135	—	2	4	no	—
207	#135	—	2	2	no	—
208	#135	—	2	2	no	—
209	#135	—	2	4	no	—
210	#135	C₂×C₄×C₅₃	1	4	yes	—
211	#135	D₄×C₅₃	1	4	yes	—
212	C₂×Q₁₆	—	2	2	no	—
213	C₂×Q₁₆	D₄×C₅₃	1	2	yes	—
214	C₂×SD₁₆	—	2	4	no	—
215	C₂×SD₁₆	—	2	2	no	—
216	C₂×D₈	—	2	2	no	—
217	C₂×D₈	D₄×C₅₃	1	2	yes	—
218	#131	—	2	2	no	—
219	#131	—	2	2	no	—
220	C₂×M₁₆	—	2	2	no	—
221	C₂×M₁₆	C₂×C₄×C₅₃	1	2	yes	—
222	C₂×M₁₆	D₄×C₅₃	1	1	yes	characteristic
223	C₂²×C₈	D₄×C₅₃	1	2	yes	—
224	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
225	M₃₂	—	2	2	no	—
226	C₂×C₁₆	—	2	1	no	—
227	C₂×C₁₆	—	2	1	no	—
228	C₂×C₁₆	—	2	2	no	—
229	#108	—	2	2	no	—
230	#108	—	2	2	no	—
231	#101	—	4	1	no	—
232	#100	—	4	1	no	—
233	#41	—	2	2	no	—
234	#41	—	2	2	no	—
235	#41	—	2	8	no	—
236	#41	—	2	4	no	—
237	#41	—	2	4	no	—
238	#41	—	4	4	no	—
239	#41	—	4	8	no	—
240	#41	—	4	4	no	—
241	#41	—	4	4	no	—
242	#41	—	4	4	no	—
243	#41	—	4	2	no	—
244	#41	—	4	2	no	—
245	C₂×Q₈	—	2	4	no	—
246	C₂×Q₈	—	2	2	no	—
247	C₂×Q₈	—	4	4	no	—
248	C₂×Q₈	D₈×C₅₃	1	1	yes	characteristic
249	C₂×D₄	—	2	4	no	—
250	C₂×D₄	—	2	2	no	—
251	C₂×D₄	—	4	8	no	—
252	C₂×D₄	—	4	4	no	—
253	C₂×D₄	—	4	4	no	—
254	C₂×D₄	—	4	2	no	—
255	C₂×D₄	D₈×C₅₃	1	1	yes	characteristic
256	C₂×D₄	SD₁₆×C₅₃	1	2	yes	—
257	C₂²×C₄	—	4	4	no	—
258	C₂²×C₄	—	4	2	no	—
259	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
260	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
261	Q₁₆	—	2	4	no	—
262	Q₁₆	—	2	2	no	—
263	Q₁₆	C₂²⋊C₄×C₅₃	1	4	yes	—
264	SD₁₆	—	2	8	no	—
265	SD₁₆	—	2	4	no	—
266	SD₁₆	—	2	4	no	—
267	D₈	—	2	4	no	—
268	D₈	—	2	2	no	—
269	D₈	C₂²⋊C₄×C₅₃	1	4	yes	—
270	M₁₆	—	2	2	no	—
271	M₁₆	—	2	4	no	—
272	M₁₆	—	4	2	no	—
273	M₁₆	—	4	2	no	—
274	C₂×C₈	—	2	2	no	—
275	C₂×C₈	—	2	2	no	—
276	C₂×C₈	—	4	2	no	—
277	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
278	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
279	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
280	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
281	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
282	C₁₆	—	4	1	no	—
283	C₁₆	—	4	1	no	—
284	C₂³	—	4	4	no	—
285	C₂³	—	4	2	no	—
286	C₂³	#102×C₅₃	1	2	yes	—
287	C₂³	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
288	Q₈	—	2	2	no	—
289	Q₈	—	2	4	no	—
290	Q₈	—	4	4	no	—
291	Q₈	—	4	2	no	—
292	D₄	—	2	4	no	—
293	D₄	—	2	4	no	—
294	D₄	—	2	2	no	—
295	D₄	—	4	4	no	—
296	D₄	—	4	8	no	—
297	D₄	—	4	4	no	—
298	D₄	—	4	4	no	—
299	D₄	—	4	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	4	8	no	—
303	C₂×C₄	—	4	4	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	4	4	no	—
306	C₂×C₄	—	4	4	no	—
307	C₂×C₄	#102×C₅₃	1	2	yes	—
308	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
309	C₂×C₄	#120×C₅₃	1	2	yes	—
310	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
311	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
312	C₈	—	2	2	no	—
313	C₈	—	8	2	no	—
314	C₈	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
315	C₈	#120×C₅₃	1	1	yes	characteristic
316	C₈	#120×C₅₃	1	1	yes	characteristic
317	C₂²	—	2	2	no	—
318	C₂²	—	4	1	no	—
319	C₂²	—	4	2	no	—
320	C₂²	—	4	4	no	—
321	C₂²	—	16	2	no	—
322	C₂²	—	16	4	no	—
323	C₂²	C₂×#102×C₅₃	1	1	yes	characteristic
324	C₂²	C₅₃×#450	1	2	yes	—
325	C₄	—	2	2	no	—
326	C₄	—	4	4	no	—
327	C₄	C₅₃×#379	1	1	yes	characteristic
328	C₄	C₅₃×#418	1	1	yes	characteristic
329	C₄	C₅₃×#447	1	2	yes	—
330	C₂	—	2	1	no	—
331	C₂	—	2	2	no	—
332	C₂	—	4	2	no	—
333	C₂	—	8	4	no	—
334	C₂	C₅₃×#1786	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₅₃	#418
commutator	C₂×C₈	C₂²×C₄×C₅₃
Frattini, cosocle	C₂²×C₈	C₂³×C₅₃
socle	C₂×C₅₃	#1786
prime radical	C₅₃×#3584	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ C₂×#131×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#2148	C₂	semidirect
3. C₂×C₅₃×#359	C₂	semidirect
4. C₅₃×#2113	C₂	semidirect
5. C₅₃×#1813	C₂	semidirect
15. C₂×M₃₂×C₅₃	C₂²	semidirect
16. C₂²×C₁₆×C₅₃	C₂²	semidirect
17. C₂×#108×C₅₃	C₂²	semidirect
149. #12746	C₅₃	direct (N × H)
164. #2148	C₂×C₅₃	semidirect
165. C₂×#359	C₂×C₅₃	semidirect
166. #2113	C₂×C₅₃	semidirect
167. #1813	C₂×C₅₃	semidirect
182. C₂×M₃₂	C₂²×C₅₃	semidirect
183. C₂²×C₁₆	C₂²×C₅₃	semidirect
184. C₂×#108	C₂²×C₅₃	semidirect
191. C₅₃	#12746	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅₃×#3584	2
C₅₃×#2148	2
C₂×C₅₃×#359	2
C₅₃×#2113	2
C₅₃×#1813	2
#12746	53

Subgroups — C53×#12746 #27220258

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#12746 #27220258