Subgroups — C₅₃×#12949 #27220461

Order 13568.

394 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#2952	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#365	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#2148	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#2143	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1953	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅₃×#1383	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1378	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#135×C₅₃	C₂²	1	1	yes	characteristic
9	C₅₃×#520	—	2	1	no	—
10	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
11	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
12	C₅₃×#442	—	2	1	no	—
13	C₅₃×#441	—	2	1	no	—
14	C₄×SD₁₆×C₅₃	—	2	2	no	—
15	C₅₃×#430	—	2	1	no	—
16	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
18	C₂×#106×C₅₃	C₂²	1	1	yes	characteristic
19	C₅₃×#417	—	2	1	no	—
20	C₅₃×#405	C₂²	1	1	yes	characteristic
21	C₅₃×#365	—	2	2	no	—
22	C₅₃×#362	—	2	1	no	—
23	C₅₃×#361	—	2	1	no	—
24	C₅₃×#359	C₄	1	4	yes	complemented
25	C₅₃×#344	—	2	1	no	—
26	C₂×#41×C₅₃	—	2	1	no	—
27	#135×C₅₃	—	2	1	no	—
28	#135×C₅₃	—	2	1	no	—
29	#135×C₅₃	C₂×C₄	1	2	yes	—
30	#135×C₅₃	Q₈	1	2	yes	—
31	C₂×Q₁₆×C₅₃	D₄	1	1	yes	characteristic
32	C₂×SD₁₆×C₅₃	—	2	1	no	—
33	C₂×D₈×C₅₃	D₄	1	1	yes	characteristic
34	C₂×M₁₆×C₅₃	—	2	1	no	—
35	C₂×M₁₆×C₅₃	—	2	1	no	—
36	C₂×M₁₆×C₅₃	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
38	C₄×Q₈×C₅₃	—	2	2	no	—
39	C₄×D₄×C₅₃	—	2	2	no	—
40	C₄×D₄×C₅₃	—	4	2	no	—
41	#117×C₅₃	—	4	1	no	—
42	#117×C₅₃	D₄	1	1	yes	characteristic
43	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
44	C₂×C₄⋊C₄×C₅₃	—	4	1	no	—
45	M₃₂×C₅₃	—	2	1	no	—
46	M₃₂×C₅₃	—	2	1	no	—
47	M₃₂×C₅₃	—	2	1	no	—
48	M₃₂×C₅₃	—	2	1	no	—
49	C₂×C₁₆×C₅₃	—	2	1	no	—
50	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	complemented
51	#108×C₅₃	—	2	1	no	—
52	#108×C₅₃	—	2	1	no	—
53	#108×C₅₃	—	2	1	no	—
54	#108×C₅₃	—	4	2	no	—
55	#108×C₅₃	C₂×C₄	1	2	yes	complemented
56	#106×C₅₃	—	2	1	no	—
57	#106×C₅₃	D₄	1	2	yes	—
58	#103×C₅₃	—	2	2	no	—
59	#102×C₅₃	—	2	2	no	—
60	#97×C₅₃	—	2	1	no	—
61	C₄×C₈×C₅₃	—	2	1	no	—
62	#41×C₅₃	—	2	2	no	—
63	#41×C₅₃	—	2	2	no	—
64	#41×C₅₃	—	4	1	no	—
65	#41×C₅₃	—	4	1	no	—
66	C₂×Q₈×C₅₃	—	2	1	no	—
67	C₂×D₄×C₅₃	—	2	1	no	—
68	C₂×D₄×C₅₃	—	4	1	no	—
69	C₂²×C₄×C₅₃	—	4	2	no	—
70	C₂²×C₄×C₅₃	—	4	1	no	—
71	C₂²×C₄×C₅₃	—	4	1	no	—
72	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
73	Q₁₆×C₅₃	—	2	1	no	—
74	Q₁₆×C₅₃	C₄⋊C₄	1	2	yes	—
75	SD₁₆×C₅₃	—	2	2	no	—
76	SD₁₆×C₅₃	—	2	2	no	—
77	D₈×C₅₃	—	2	1	no	—
78	D₈×C₅₃	C₄⋊C₄	1	2	yes	—
79	M₁₆×C₅₃	—	2	2	no	—
80	M₁₆×C₅₃	—	2	1	no	—
81	M₁₆×C₅₃	—	2	1	no	—
82	M₁₆×C₅₃	—	4	1	no	—
83	M₁₆×C₅₃	—	4	1	no	—
84	M₁₆×C₅₃	—	4	1	no	—
85	C₂×C₈×C₅₃	—	2	1	no	—
86	C₂×C₈×C₅₃	—	4	1	no	—
87	C₂×C₈×C₅₃	—	4	1	no	—
88	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
92	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
93	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
94	C₄⋊C₄×C₅₃	—	2	1	no	—
95	C₄⋊C₄×C₅₃	—	2	2	no	—
96	C₄⋊C₄×C₅₃	—	4	2	no	—
97	C₄⋊C₄×C₅₃	—	4	1	no	—
98	C₂²⋊C₄×C₅₃	—	4	2	no	—
99	C₂²⋊C₄×C₅₃	SD₁₆	1	2	yes	—
100	C₄²×C₅₃	—	2	1	no	—
101	C₄²×C₅₃	—	4	2	no	—
102	C₁₆×C₅₃	—	2	2	no	—
103	C₁₆×C₅₃	—	4	1	no	—
104	C₂³×C₅₃	—	4	1	no	—
105	C₂³×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
106	Q₈×C₅₃	—	2	2	no	—
107	Q₈×C₅₃	—	4	1	no	—
108	D₄×C₅₃	—	2	2	no	—
109	D₄×C₅₃	—	4	2	no	—
110	D₄×C₅₃	—	4	2	no	—
111	D₄×C₅₃	—	4	1	no	—
112	C₂×C₄×C₅₃	—	2	2	no	—
113	C₂×C₄×C₅₃	—	4	1	no	—
114	C₂×C₄×C₅₃	—	4	2	no	—
115	C₂×C₄×C₅₃	—	4	2	no	—
116	C₂×C₄×C₅₃	—	4	2	no	—
117	C₂×C₄×C₅₃	—	16	2	no	—
118	C₂×C₄×C₅₃	—	16	1	no	—
119	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
121	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
122	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
123	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
124	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
125	C₈×C₅₃	—	4	2	no	—
126	C₈×C₅₃	—	4	1	no	—
127	C₈×C₅₃	—	8	1	no	—
128	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
130	C₈×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
131	C₈×C₅₃	#123	1	1	yes	characteristic
132	#12949	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
133	C₂²×C₅₃	—	4	1	no	—
134	C₂²×C₅₃	—	4	2	no	—
135	C₂²×C₅₃	—	8	1	no	—
136	C₂²×C₅₃	—	8	1	no	—
137	C₂²×C₅₃	C₄×SD₁₆	1	1	yes	characteristic
138	C₂²×C₅₃	#450	1	1	yes	characteristic
139	C₂²×C₅₃	#476	1	1	yes	characteristic
140	C₄×C₅₃	—	4	2	no	—
141	C₄×C₅₃	—	4	2	no	—
142	C₄×C₅₃	—	8	2	no	—
143	C₄×C₅₃	#385	1	1	yes	characteristic
144	C₄×C₅₃	#442	1	1	yes	characteristic
145	C₄×C₅₃	#449	1	1	yes	characteristic
146	C₄×C₅₃	#478	1	1	yes	characteristic
147	#2952	C₂×C₅₃	1	1	yes	characteristic
148	C₂×#365	C₂×C₅₃	1	1	yes	characteristic, complemented
149	#2148	C₂×C₅₃	1	1	yes	characteristic
150	#2143	C₂×C₅₃	1	1	yes	characteristic
151	#1953	C₂×C₅₃	1	1	yes	characteristic, complemented
152	#1383	C₂×C₅₃	1	1	yes	characteristic, complemented
153	#1378	C₂×C₅₃	1	1	yes	characteristic, complemented
154	C₂×C₅₃	—	2	1	no	—
155	C₂×C₅₃	—	2	1	no	—
156	C₂×C₅₃	—	2	1	no	—
157	C₂×C₅₃	—	8	2	no	—
158	C₂×C₅₃	#1864	1	1	yes	characteristic
159	C₂×#135	C₂²×C₅₃	1	1	yes	characteristic
160	#520	—	2	1	no	—
161	C₂×M₃₂	C₂²×C₅₃	1	1	yes	characteristic
162	C₂×M₃₂	C₂²×C₅₃	1	1	yes	characteristic
163	#442	—	2	1	no	—
164	#441	—	2	1	no	—
165	C₄×SD₁₆	—	2	2	no	—
166	#430	—	2	1	no	—
167	C₂×#108	C₂²×C₅₃	1	1	yes	characteristic
168	C₂×#108	C₂²×C₅₃	1	1	yes	characteristic
169	C₂×#106	C₂²×C₅₃	1	1	yes	characteristic
170	#417	—	2	1	no	—
171	#405	C₂²×C₅₃	1	1	yes	characteristic
172	#365	—	2	2	no	—
173	#362	—	2	1	no	—
174	#361	—	2	1	no	—
175	#359	C₄×C₅₃	1	4	yes	complemented
176	#344	—	2	1	no	—
177	C₅₃	#12949	1	1	yes	characteristic, direct product factor, Sylow
178	C₂×#41	—	2	1	no	—
179	#135	—	2	1	no	—
180	#135	—	2	1	no	—
181	#135	C₂×C₄×C₅₃	1	2	yes	—
182	#135	Q₈×C₅₃	1	2	yes	—
183	C₂×Q₁₆	D₄×C₅₃	1	1	yes	characteristic
184	C₂×SD₁₆	—	2	1	no	—
185	C₂×D₈	D₄×C₅₃	1	1	yes	characteristic
186	C₂×M₁₆	—	2	1	no	—
187	C₂×M₁₆	—	2	1	no	—
188	C₂×M₁₆	D₄×C₅₃	1	1	yes	characteristic
189	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
190	C₄×Q₈	—	2	2	no	—
191	C₄×D₄	—	2	2	no	—
192	C₄×D₄	—	4	2	no	—
193	#117	—	4	1	no	—
194	#117	D₄×C₅₃	1	1	yes	characteristic
195	C₂×C₄⋊C₄	—	2	1	no	—
196	C₂×C₄⋊C₄	—	4	1	no	—
197	M₃₂	—	2	1	no	—
198	M₃₂	—	2	1	no	—
199	M₃₂	—	2	1	no	—
200	M₃₂	—	2	1	no	—
201	C₂×C₁₆	—	2	1	no	—
202	C₂×C₁₆	C₂×C₄×C₅₃	1	2	yes	complemented
203	#108	—	2	1	no	—
204	#108	—	2	1	no	—
205	#108	—	2	1	no	—
206	#108	—	4	2	no	—
207	#108	C₂×C₄×C₅₃	1	2	yes	complemented
208	#106	—	2	1	no	—
209	#106	D₄×C₅₃	1	2	yes	—
210	#103	—	2	2	no	—
211	#102	—	2	2	no	—
212	#97	—	2	1	no	—
213	C₄×C₈	—	2	1	no	—
214	#41	—	2	2	no	—
215	#41	—	2	2	no	—
216	#41	—	4	1	no	—
217	#41	—	4	1	no	—
218	C₂×Q₈	—	2	1	no	—
219	C₂×D₄	—	2	1	no	—
220	C₂×D₄	—	4	1	no	—
221	C₂²×C₄	—	4	2	no	—
222	C₂²×C₄	—	4	1	no	—
223	C₂²×C₄	—	4	1	no	—
224	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
225	Q₁₆	—	2	1	no	—
226	Q₁₆	C₄⋊C₄×C₅₃	1	2	yes	—
227	SD₁₆	—	2	2	no	—
228	SD₁₆	—	2	2	no	—
229	D₈	—	2	1	no	—
230	D₈	C₄⋊C₄×C₅₃	1	2	yes	—
231	M₁₆	—	2	2	no	—
232	M₁₆	—	2	1	no	—
233	M₁₆	—	2	1	no	—
234	M₁₆	—	4	1	no	—
235	M₁₆	—	4	1	no	—
236	M₁₆	—	4	1	no	—
237	C₂×C₈	—	2	1	no	—
238	C₂×C₈	—	4	1	no	—
239	C₂×C₈	—	4	1	no	—
240	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
241	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
242	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
243	C₂×C₈	C₂×Q₈×C₅₃	1	1	yes	characteristic
244	C₂×C₈	#41×C₅₃	1	1	yes	characteristic
245	C₂×C₈	#41×C₅₃	1	1	yes	characteristic
246	C₄⋊C₄	—	2	1	no	—
247	C₄⋊C₄	—	2	2	no	—
248	C₄⋊C₄	—	4	2	no	—
249	C₄⋊C₄	—	4	1	no	—
250	C₂²⋊C₄	—	4	2	no	—
251	C₂²⋊C₄	SD₁₆×C₅₃	1	2	yes	—
252	C₄²	—	2	1	no	—
253	C₄²	—	4	2	no	—
254	C₁₆	—	2	2	no	—
255	C₁₆	—	4	1	no	—
256	C₂³	—	4	1	no	—
257	C₂³	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
258	Q₈	—	2	2	no	—
259	Q₈	—	4	1	no	—
260	D₄	—	2	2	no	—
261	D₄	—	4	2	no	—
262	D₄	—	4	2	no	—
263	D₄	—	4	1	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	4	1	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	16	2	no	—
270	C₂×C₄	—	16	1	no	—
271	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
272	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
273	C₂×C₄	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
274	C₂×C₄	#135×C₅₃	1	1	yes	characteristic
275	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
276	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
277	C₈	—	4	2	no	—
278	C₈	—	4	1	no	—
279	C₈	—	8	1	no	—
280	C₈	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
281	C₈	C₄×D₄×C₅₃	1	1	yes	characteristic
282	C₈	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
283	C₈	#123×C₅₃	1	1	yes	characteristic
284	C₂²	—	4	1	no	—
285	C₂²	—	4	2	no	—
286	C₂²	—	8	1	no	—
287	C₂²	—	8	1	no	—
288	C₂²	C₄×SD₁₆×C₅₃	1	1	yes	characteristic
289	C₂²	C₅₃×#450	1	1	yes	characteristic
290	C₂²	C₅₃×#476	1	1	yes	characteristic
291	C₄	—	4	2	no	—
292	C₄	—	4	2	no	—
293	C₄	—	8	2	no	—
294	C₄	C₅₃×#385	1	1	yes	characteristic
295	C₄	C₅₃×#442	1	1	yes	characteristic
296	C₄	C₅₃×#449	1	1	yes	characteristic
297	C₄	C₅₃×#478	1	1	yes	characteristic
298	C₂	—	2	1	no	—
299	C₂	—	2	1	no	—
300	C₂	—	2	1	no	—
301	C₂	—	8	2	no	—
302	C₂	C₅₃×#1864	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₅₃	#1864
commutator	C₂×C₈	C₂²×C₄×C₅₃
Frattini, cosocle	C₂²×C₈	C₂³×C₅₃
prime radical	C₂×#135×C₅₃	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ C₅₃×#405 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅₃×#365	C₂	semidirect
5. C₅₃×#1953	C₂	semidirect
6. C₅₃×#1383	C₂	semidirect
7. C₅₃×#1378	C₂	semidirect
24. C₅₃×#359	C₄	semidirect
50. C₂×C₁₆×C₅₃	C₂×C₄	semidirect
55. #108×C₅₃	C₂×C₄	semidirect
132. #12949	C₅₃	direct (N × H)
148. C₂×#365	C₂×C₅₃	semidirect
151. #1953	C₂×C₅₃	semidirect
152. #1383	C₂×C₅₃	semidirect
153. #1378	C₂×C₅₃	semidirect
175. #359	C₄×C₅₃	semidirect
177. C₅₃	#12949	direct (N × H)
202. C₂×C₁₆	C₂×C₄×C₅₃	semidirect
207. #108	C₂×C₄×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₅₃×#2952	2
C₂×C₅₃×#365	2
C₅₃×#2148	2
C₅₃×#2143	2
C₅₃×#1953	2
C₅₃×#1383	2
C₅₃×#1378	2
#12949	53

Subgroups — C53×#12949 #27220461

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#12949 #27220461