Subgroups — C₅₃×#12825 #27220337

Order 13568.

400 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#443	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#364	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#2148	C₂	1	2	yes	maximal
4	C₅₃×#1848	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1385	C₂	1	2	yes	maximal, complemented
6	C₂×#135×C₅₃	C₂²	1	1	yes	characteristic
7	C₂×#131×C₅₃	—	2	1	no	—
8	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
9	C₅₃×#443	—	2	1	no	—
10	C₅₃×#443	—	2	1	no	—
11	C₅₃×#443	C₄	1	2	yes	—
12	C₅₃×#443	C₄	1	2	yes	—
13	C₂×#108×C₅₃	—	2	1	no	—
14	C₂×#108×C₅₃	C₂²	1	2	yes	—
15	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#423	—	2	1	no	—
17	C₂×#104×C₅₃	—	2	1	no	—
18	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₅₃×#364	—	2	1	no	—
20	C₅₃×#364	—	2	1	no	—
21	C₅₃×#362	—	2	2	no	—
22	C₅₃×#361	—	2	2	no	—
23	C₂×#41×C₅₃	—	2	1	no	—
24	#135×C₅₃	—	2	1	no	—
25	#135×C₅₃	—	2	1	no	—
26	#135×C₅₃	C₂×C₄	1	2	yes	—
27	#135×C₅₃	Q₈	1	2	yes	—
28	C₂×Q₁₆×C₅₃	D₄	1	1	yes	characteristic
29	C₂×SD₁₆×C₅₃	—	2	1	no	—
30	C₂×D₈×C₅₃	D₄	1	1	yes	characteristic
31	#131×C₅₃	—	2	2	no	—
32	#131×C₅₃	—	2	2	no	—
33	#131×C₅₃	—	4	1	no	—
34	#131×C₅₃	—	4	1	no	—
35	C₂×M₁₆×C₅₃	—	2	1	no	—
36	C₂×M₁₆×C₅₃	—	2	2	no	—
37	C₂×M₁₆×C₅₃	—	4	1	no	—
38	C₂²×C₈×C₅₃	—	4	1	no	—
39	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
41	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
42	M₃₂×C₅₃	—	2	2	no	—
43	M₃₂×C₅₃	—	2	2	no	—
44	C₂×C₁₆×C₅₃	—	2	2	no	—
45	#108×C₅₃	—	2	1	no	—
46	#108×C₅₃	—	2	2	no	—
47	#108×C₅₃	—	2	2	no	—
48	#108×C₅₃	—	4	1	no	—
49	#108×C₅₃	—	4	1	no	—
50	#108×C₅₃	C₂×C₄	1	2	yes	—
51	C₄⋊C₈×C₅₃	—	4	2	no	—
52	#104×C₅₃	—	2	2	no	—
53	#104×C₅₃	—	2	2	no	—
54	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
55	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
56	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
57	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
58	#41×C₅₃	—	2	2	no	—
59	#41×C₅₃	—	2	2	no	—
60	#41×C₅₃	—	4	1	no	—
61	#41×C₅₃	—	4	1	no	—
62	C₂×Q₈×C₅₃	—	2	1	no	—
63	C₂×D₄×C₅₃	—	2	1	no	—
64	C₂×D₄×C₅₃	—	4	1	no	—
65	C₂²×C₄×C₅₃	—	2	1	no	—
66	C₂²×C₄×C₅₃	—	4	1	no	—
67	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
68	Q₁₆×C₅₃	—	2	1	no	—
69	Q₁₆×C₅₃	C₄⋊C₄	1	2	yes	—
70	SD₁₆×C₅₃	—	2	2	no	—
71	SD₁₆×C₅₃	—	2	2	no	—
72	D₈×C₅₃	—	2	1	no	—
73	D₈×C₅₃	C₄⋊C₄	1	2	yes	—
74	M₁₆×C₅₃	—	2	2	no	—
75	M₁₆×C₅₃	—	4	1	no	—
76	M₁₆×C₅₃	—	4	2	no	—
77	M₁₆×C₅₃	—	4	2	no	—
78	M₁₆×C₅₃	—	4	2	no	—
79	M₁₆×C₅₃	—	4	2	no	—
80	C₂×C₈×C₅₃	—	4	1	no	—
81	C₂×C₈×C₅₃	—	4	2	no	—
82	C₂×C₈×C₅₃	—	4	2	no	—
83	C₂×C₈×C₅₃	—	4	2	no	—
84	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₈×C₅₃	D₈	1	1	yes	characteristic
87	C₂×C₈×C₅₃	D₈	1	1	yes	characteristic
88	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
89	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
90	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
94	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
95	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
96	C₄²×C₅₃	—	2	1	no	—
97	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
98	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
99	C₁₆×C₅₃	—	4	2	no	—
100	C₂³×C₅₃	—	4	1	no	—
101	C₂³×C₅₃	#137	1	1	yes	characteristic
102	Q₈×C₅₃	—	2	2	no	—
103	Q₈×C₅₃	—	4	1	no	—
104	D₄×C₅₃	—	2	2	no	—
105	D₄×C₅₃	—	4	2	no	—
106	D₄×C₅₃	—	4	2	no	—
107	D₄×C₅₃	—	4	1	no	—
108	C₂×C₄×C₅₃	—	2	1	no	—
109	C₂×C₄×C₅₃	—	2	1	no	—
110	C₂×C₄×C₅₃	—	4	2	no	—
111	C₂×C₄×C₅₃	—	4	1	no	—
112	C₂×C₄×C₅₃	—	4	2	no	—
113	C₂×C₄×C₅₃	—	4	2	no	—
114	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
116	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
117	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
118	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
119	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
120	C₈×C₅₃	—	4	2	no	—
121	C₈×C₅₃	—	8	2	no	—
122	C₈×C₅₃	#102	1	1	yes	characteristic
123	C₈×C₅₃	#102	1	1	yes	characteristic
124	C₈×C₅₃	#102	1	1	yes	characteristic
125	C₈×C₅₃	#102	1	1	yes	characteristic
126	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₈×C₅₃	#117	1	1	yes	characteristic
128	C₈×C₅₃	#121	1	1	yes	characteristic
129	C₈×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
130	#12825	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
131	C₂²×C₅₃	—	4	1	no	—
132	C₂²×C₅₃	—	4	2	no	—
133	C₂²×C₅₃	—	8	2	no	—
134	C₂²×C₅₃	#417	1	1	yes	characteristic
135	C₂²×C₅₃	#461	1	1	yes	characteristic
136	C₂²×C₅₃	#474	1	1	yes	characteristic
137	C₄×C₅₃	—	2	2	no	—
138	C₄×C₅₃	—	4	1	no	—
139	C₄×C₅₃	—	4	2	no	—
140	C₄×C₅₃	#380	1	1	yes	characteristic
141	C₄×C₅₃	C₂×#102	1	1	yes	characteristic
142	C₄×C₅₃	#459	1	1	yes	characteristic
143	C₄×C₅₃	#476	1	1	yes	characteristic
144	C₂×#443	C₂×C₅₃	1	1	yes	characteristic
145	C₂×#364	C₂×C₅₃	1	1	yes	characteristic, complemented
146	#2148	C₂×C₅₃	1	2	yes	—
147	#1848	C₂×C₅₃	1	1	yes	characteristic, complemented
148	#1385	C₂×C₅₃	1	2	yes	complemented
149	C₂×C₅₃	—	2	1	no	—
150	C₂×C₅₃	—	2	1	no	—
151	C₂×C₅₃	—	2	1	no	—
152	C₂×C₅₃	—	8	2	no	—
153	C₂×C₅₃	#1802	1	1	yes	characteristic
154	C₂×#135	C₂²×C₅₃	1	1	yes	characteristic
155	C₂×#131	—	2	1	no	—
156	C₂×M₃₂	C₂²×C₅₃	1	2	yes	—
157	#443	—	2	1	no	—
158	#443	—	2	1	no	—
159	#443	C₄×C₅₃	1	2	yes	—
160	#443	C₄×C₅₃	1	2	yes	—
161	C₂×#108	—	2	1	no	—
162	C₂×#108	C₂²×C₅₃	1	2	yes	—
163	C₂×#108	C₂²×C₅₃	1	1	yes	characteristic
164	#423	—	2	1	no	—
165	C₂×#104	—	2	1	no	—
166	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
167	#364	—	2	1	no	—
168	#364	—	2	1	no	—
169	#362	—	2	2	no	—
170	#361	—	2	2	no	—
171	C₅₃	#12825	1	1	yes	characteristic, direct product factor, Sylow
172	C₂×#41	—	2	1	no	—
173	#135	—	2	1	no	—
174	#135	—	2	1	no	—
175	#135	C₂×C₄×C₅₃	1	2	yes	—
176	#135	Q₈×C₅₃	1	2	yes	—
177	C₂×Q₁₆	D₄×C₅₃	1	1	yes	characteristic
178	C₂×SD₁₆	—	2	1	no	—
179	C₂×D₈	D₄×C₅₃	1	1	yes	characteristic
180	#131	—	2	2	no	—
181	#131	—	2	2	no	—
182	#131	—	4	1	no	—
183	#131	—	4	1	no	—
184	C₂×M₁₆	—	2	1	no	—
185	C₂×M₁₆	—	2	2	no	—
186	C₂×M₁₆	—	4	1	no	—
187	C₂²×C₈	—	4	1	no	—
188	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
189	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
190	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
191	M₃₂	—	2	2	no	—
192	M₃₂	—	2	2	no	—
193	C₂×C₁₆	—	2	2	no	—
194	#108	—	2	1	no	—
195	#108	—	2	2	no	—
196	#108	—	2	2	no	—
197	#108	—	4	1	no	—
198	#108	—	4	1	no	—
199	#108	C₂×C₄×C₅₃	1	2	yes	—
200	C₄⋊C₈	—	4	2	no	—
201	#104	—	2	2	no	—
202	#104	—	2	2	no	—
203	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
204	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
205	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
206	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
207	#41	—	2	2	no	—
208	#41	—	2	2	no	—
209	#41	—	4	1	no	—
210	#41	—	4	1	no	—
211	C₂×Q₈	—	2	1	no	—
212	C₂×D₄	—	2	1	no	—
213	C₂×D₄	—	4	1	no	—
214	C₂²×C₄	—	2	1	no	—
215	C₂²×C₄	—	4	1	no	—
216	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
217	Q₁₆	—	2	1	no	—
218	Q₁₆	C₄⋊C₄×C₅₃	1	2	yes	—
219	SD₁₆	—	2	2	no	—
220	SD₁₆	—	2	2	no	—
221	D₈	—	2	1	no	—
222	D₈	C₄⋊C₄×C₅₃	1	2	yes	—
223	M₁₆	—	2	2	no	—
224	M₁₆	—	4	1	no	—
225	M₁₆	—	4	2	no	—
226	M₁₆	—	4	2	no	—
227	M₁₆	—	4	2	no	—
228	M₁₆	—	4	2	no	—
229	C₂×C₈	—	4	1	no	—
230	C₂×C₈	—	4	2	no	—
231	C₂×C₈	—	4	2	no	—
232	C₂×C₈	—	4	2	no	—
233	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
234	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
235	C₂×C₈	D₈×C₅₃	1	1	yes	characteristic
236	C₂×C₈	D₈×C₅₃	1	1	yes	characteristic
237	C₂×C₈	SD₁₆×C₅₃	1	1	yes	characteristic
238	C₂×C₈	SD₁₆×C₅₃	1	1	yes	characteristic
239	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
240	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
241	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
242	C₂×C₈	C₂×Q₈×C₅₃	1	1	yes	characteristic
243	C₂×C₈	#41×C₅₃	1	1	yes	characteristic
244	C₂×C₈	#41×C₅₃	1	1	yes	characteristic
245	C₄²	—	2	1	no	—
246	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
247	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
248	C₁₆	—	4	2	no	—
249	C₂³	—	4	1	no	—
250	C₂³	#137×C₅₃	1	1	yes	characteristic
251	Q₈	—	2	2	no	—
252	Q₈	—	4	1	no	—
253	D₄	—	2	2	no	—
254	D₄	—	4	2	no	—
255	D₄	—	4	2	no	—
256	D₄	—	4	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	4	1	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
264	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
265	C₂×C₄	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
266	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
267	C₂×C₄	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
268	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	1	yes	characteristic
269	C₈	—	4	2	no	—
270	C₈	—	8	2	no	—
271	C₈	#102×C₅₃	1	1	yes	characteristic
272	C₈	#102×C₅₃	1	1	yes	characteristic
273	C₈	#102×C₅₃	1	1	yes	characteristic
274	C₈	#102×C₅₃	1	1	yes	characteristic
275	C₈	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
276	C₈	#117×C₅₃	1	1	yes	characteristic
277	C₈	#121×C₅₃	1	1	yes	characteristic
278	C₈	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
279	C₂²	—	4	1	no	—
280	C₂²	—	4	2	no	—
281	C₂²	—	8	2	no	—
282	C₂²	C₅₃×#417	1	1	yes	characteristic
283	C₂²	C₅₃×#461	1	1	yes	characteristic
284	C₂²	C₅₃×#474	1	1	yes	characteristic
285	C₄	—	2	2	no	—
286	C₄	—	4	1	no	—
287	C₄	—	4	2	no	—
288	C₄	C₅₃×#380	1	1	yes	characteristic
289	C₄	C₂×#102×C₅₃	1	1	yes	characteristic
290	C₄	C₅₃×#459	1	1	yes	characteristic
291	C₄	C₅₃×#476	1	1	yes	characteristic
292	C₂	—	2	1	no	—
293	C₂	—	2	1	no	—
294	C₂	—	2	1	no	—
295	C₂	—	8	2	no	—
296	C₂	C₅₃×#1802	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₅₃	#476
commutator	C₂×C₈	C₂²×C₄×C₅₃
Frattini, cosocle	C₂²×C₈	C₂³×C₅₃
socle	C₂×C₅₃	#1802
prime radical	C₂×#135×C₅₃	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅₃×#364	C₂	semidirect
4. C₅₃×#1848	C₂	semidirect
5. C₅₃×#1385	C₂	semidirect
130. #12825	C₅₃	direct (N × H)
145. C₂×#364	C₂×C₅₃	semidirect
147. #1848	C₂×C₅₃	semidirect
148. #1385	C₂×C₅₃	semidirect
171. C₅₃	#12825	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×#443	2
C₂×C₅₃×#364	2
C₅₃×#2148	2
C₅₃×#1848	2
C₅₃×#1385	2
#12825	53

Subgroups — C53×#12825 #27220337

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#12825 #27220337