Subgroups — C₅₃×#12609 #27220121

Order 13568.

444 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	C₅₃×#2875	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2117	C₂	1	2	yes	maximal
3	C₂×C₅₃×#347	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅₃×#347	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1378	C₂	1	2	yes	maximal, complemented
6	C₂×#131×C₅₃	C₄	1	1	yes	characteristic
7	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
8	C₄×#41×C₅₃	C₄	1	1	yes	characteristic
9	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
10	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
11	C₅₃×#405	—	2	1	no	—
12	C₅₃×#405	—	2	1	no	—
13	C₅₃×#405	C₄	1	1	yes	characteristic
14	C₅₃×#405	C₂²	1	1	yes	characteristic
15	C₄×M₁₆×C₅₃	—	2	1	no	—
16	C₄×M₁₆×C₅₃	C₄	1	1	yes	characteristic
17	C₂×#97×C₅₃	—	2	1	no	—
18	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
19	C₅₃×#350	C₄	1	8	yes	complemented
20	C₅₃×#347	—	2	1	no	—
21	C₅₃×#347	—	2	1	no	—
22	C₅₃×#347	—	2	1	no	—
23	C₅₃×#347	—	2	1	no	—
24	C₂×#41×C₅₃	C₂×C₄	1	1	yes	characteristic
25	#131×C₅₃	—	2	2	no	—
26	#131×C₅₃	—	2	2	no	—
27	#131×C₅₃	—	2	2	no	—
28	#131×C₅₃	C₂×C₄	1	4	yes	—
29	C₂×M₁₆×C₅₃	—	2	1	no	—
30	C₂×M₁₆×C₅₃	—	2	1	no	—
31	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₅₃	—	2	1	no	—
34	C₂²×C₈×C₅₃	—	2	1	no	—
35	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
37	C₄×Q₈×C₅₃	—	2	1	no	—
38	C₄×D₄×C₅₃	—	2	2	no	—
39	C₄×D₄×C₅₃	—	2	1	no	—
40	#117×C₅₃	—	2	1	no	—
41	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₅₃	—	2	1	no	—
43	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
44	M₃₂×C₅₃	—	2	2	no	—
45	M₃₂×C₅₃	—	2	2	no	—
46	M₃₂×C₅₃	—	2	2	no	—
47	M₃₂×C₅₃	C₂×C₄	1	4	yes	complemented
48	C₂×C₁₆×C₅₃	—	2	2	no	—
49	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	complemented
50	#97×C₅₃	—	2	1	no	—
51	#97×C₅₃	—	2	1	no	—
52	#97×C₅₃	—	2	1	no	—
53	#97×C₅₃	D₄	1	1	yes	characteristic
54	#97×C₅₃	D₄	1	1	yes	characteristic
55	#97×C₅₃	D₄	1	1	yes	characteristic
56	#97×C₅₃	D₄	1	1	yes	characteristic
57	C₄×C₈×C₅₃	—	2	1	no	—
58	C₄×C₈×C₅₃	—	2	1	no	—
59	C₄×C₈×C₅₃	—	2	1	no	—
60	#41×C₅₃	—	2	1	no	—
61	#41×C₅₃	—	2	1	no	—
62	#41×C₅₃	C₄²	1	2	yes	—
63	#41×C₅₃	M₁₆	1	2	yes	—
64	C₂×Q₈×C₅₃	M₁₆	1	1	yes	characteristic
65	C₂×D₄×C₅₃	—	2	1	no	—
66	C₂×D₄×C₅₃	M₁₆	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	—	2	1	no	—
68	C₂²×C₄×C₅₃	—	2	2	no	—
69	C₂²×C₄×C₅₃	—	2	1	no	—
70	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
71	M₁₆×C₅₃	—	2	4	no	—
72	M₁₆×C₅₃	—	2	1	no	—
73	M₁₆×C₅₃	—	2	1	no	—
74	M₁₆×C₅₃	—	2	4	no	—
75	M₁₆×C₅₃	C₄²	1	2	yes	—
76	M₁₆×C₅₃	M₁₆	1	2	yes	—
77	C₂×C₈×C₅₃	—	2	4	no	—
78	C₂×C₈×C₅₃	—	2	1	no	—
79	C₂×C₈×C₅₃	—	2	1	no	—
80	C₂×C₈×C₅₃	—	2	1	no	—
81	C₂×C₈×C₅₃	—	2	1	no	—
82	C₂×C₈×C₅₃	—	2	4	no	—
83	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
86	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₅₃	#41	1	2	yes	—
89	C₄⋊C₄×C₅₃	—	2	2	no	—
90	C₄⋊C₄×C₅₃	—	2	1	no	—
91	C₂²⋊C₄×C₅₃	—	2	2	no	—
92	C₂²⋊C₄×C₅₃	M₁₆	1	2	yes	—
93	C₄²×C₅₃	—	2	2	no	—
94	C₄²×C₅₃	—	2	1	no	—
95	C₄²×C₅₃	—	2	1	no	—
96	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
97	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
98	C₁₆×C₅₃	—	2	4	no	—
99	C₁₆×C₅₃	—	2	4	no	—
100	C₂³×C₅₃	—	2	1	no	—
101	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
102	Q₈×C₅₃	—	2	1	no	—
103	Q₈×C₅₃	#97	1	2	yes	—
104	D₄×C₅₃	—	2	2	no	—
105	D₄×C₅₃	—	2	2	no	—
106	D₄×C₅₃	—	2	1	no	—
107	D₄×C₅₃	#97	1	2	yes	—
108	C₂×C₄×C₅₃	—	2	1	no	—
109	C₂×C₄×C₅₃	—	2	2	no	—
110	C₂×C₄×C₅₃	—	2	2	no	—
111	C₂×C₄×C₅₃	—	2	1	no	—
112	C₂×C₄×C₅₃	—	2	2	no	—
113	C₂×C₄×C₅₃	—	2	1	no	—
114	C₂×C₄×C₅₃	—	8	1	no	—
115	C₂×C₄×C₅₃	—	8	2	no	—
116	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
117	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
119	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
120	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
121	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
122	C₈×C₅₃	—	2	2	no	—
123	C₈×C₅₃	—	2	1	no	—
124	C₈×C₅₃	—	2	2	no	—
125	C₈×C₅₃	—	2	1	no	—
126	C₈×C₅₃	C₄×D₄	1	2	yes	—
127	C₈×C₅₃	C₄×D₄	1	2	yes	—
128	#12609	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
129	C₂²×C₅₃	—	2	2	no	—
130	C₂²×C₅₃	—	4	1	no	—
131	C₂²×C₅₃	—	4	2	no	—
132	C₂²×C₅₃	C₄×M₁₆	1	1	yes	characteristic
133	C₂²×C₅₃	#407	1	1	yes	characteristic
134	C₂²×C₅₃	#432	1	1	yes	characteristic
135	C₄×C₅₃	—	2	2	no	—
136	C₄×C₅₃	—	4	2	no	—
137	C₄×C₅₃	—	4	1	no	—
138	C₄×C₅₃	—	4	1	no	—
139	C₄×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
140	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
141	C₄×C₅₃	#407	1	1	yes	characteristic
142	C₄×C₅₃	#432	1	1	yes	characteristic
143	#2875	C₂×C₅₃	1	1	yes	characteristic
144	#2117	C₂×C₅₃	1	2	yes	—
145	C₂×#347	C₂×C₅₃	1	1	yes	characteristic, complemented
146	C₂×#347	C₂×C₅₃	1	1	yes	characteristic, complemented
147	#1378	C₂×C₅₃	1	2	yes	complemented
148	C₂×C₅₃	—	2	1	no	—
149	C₂×C₅₃	—	2	1	no	—
150	C₂×C₅₃	—	2	1	no	—
151	C₂×C₅₃	—	4	2	no	—
152	C₂×C₅₃	#1749	1	1	yes	characteristic
153	C₂×#131	C₄×C₅₃	1	1	yes	characteristic
154	C₂×#131	C₂²×C₅₃	1	1	yes	characteristic
155	C₄×#41	C₄×C₅₃	1	1	yes	characteristic
156	C₂×M₃₂	C₂²×C₅₃	1	2	yes	—
157	C₂×M₃₂	C₂²×C₅₃	1	2	yes	—
158	#405	—	2	1	no	—
159	#405	—	2	1	no	—
160	#405	C₄×C₅₃	1	1	yes	characteristic
161	#405	C₂²×C₅₃	1	1	yes	characteristic
162	C₄×M₁₆	—	2	1	no	—
163	C₄×M₁₆	C₄×C₅₃	1	1	yes	characteristic
164	C₂×#97	—	2	1	no	—
165	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
166	#350	C₄×C₅₃	1	8	yes	complemented
167	#347	—	2	1	no	—
168	#347	—	2	1	no	—
169	#347	—	2	1	no	—
170	#347	—	2	1	no	—
171	C₅₃	#12609	1	1	yes	characteristic, direct product factor, Sylow
172	C₂×#41	C₂×C₄×C₅₃	1	1	yes	characteristic
173	#131	—	2	2	no	—
174	#131	—	2	2	no	—
175	#131	—	2	2	no	—
176	#131	C₂×C₄×C₅₃	1	4	yes	—
177	C₂×M₁₆	—	2	1	no	—
178	C₂×M₁₆	—	2	1	no	—
179	C₂×M₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
180	C₂×M₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
181	C₂²×C₈	—	2	1	no	—
182	C₂²×C₈	—	2	1	no	—
183	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
184	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
185	C₄×Q₈	—	2	1	no	—
186	C₄×D₄	—	2	1	no	—
187	C₄×D₄	—	2	2	no	—
188	#117	—	2	1	no	—
189	#117	C₂×C₄×C₅₃	1	1	yes	characteristic
190	C₂×C₄²	—	2	1	no	—
191	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
192	M₃₂	—	2	2	no	—
193	M₃₂	—	2	2	no	—
194	M₃₂	—	2	2	no	—
195	M₃₂	C₂×C₄×C₅₃	1	4	yes	complemented
196	C₂×C₁₆	—	2	2	no	—
197	C₂×C₁₆	C₂×C₄×C₅₃	1	4	yes	complemented
198	#97	—	2	1	no	—
199	#97	—	2	1	no	—
200	#97	—	2	1	no	—
201	#97	D₄×C₅₃	1	1	yes	characteristic
202	#97	D₄×C₅₃	1	1	yes	characteristic
203	#97	D₄×C₅₃	1	1	yes	characteristic
204	#97	D₄×C₅₃	1	1	yes	characteristic
205	C₄×C₈	—	2	1	no	—
206	C₄×C₈	—	2	1	no	—
207	C₄×C₈	—	2	1	no	—
208	#41	—	2	1	no	—
209	#41	—	2	1	no	—
210	#41	C₄²×C₅₃	1	2	yes	—
211	#41	M₁₆×C₅₃	1	2	yes	—
212	C₂×Q₈	M₁₆×C₅₃	1	1	yes	characteristic
213	C₂×D₄	—	2	1	no	—
214	C₂×D₄	M₁₆×C₅₃	1	1	yes	characteristic
215	C₂²×C₄	—	2	1	no	—
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	—	2	1	no	—
218	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
219	M₁₆	—	2	4	no	—
220	M₁₆	—	2	1	no	—
221	M₁₆	—	2	1	no	—
222	M₁₆	—	2	4	no	—
223	M₁₆	C₄²×C₅₃	1	2	yes	—
224	M₁₆	M₁₆×C₅₃	1	2	yes	—
225	C₂×C₈	—	2	4	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	2	1	no	—
228	C₂×C₈	—	2	1	no	—
229	C₂×C₈	—	2	1	no	—
230	C₂×C₈	—	2	4	no	—
231	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
232	C₂×C₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
233	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
234	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
235	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
236	C₂×C₈	#41×C₅₃	1	2	yes	—
237	C₄⋊C₄	—	2	2	no	—
238	C₄⋊C₄	—	2	1	no	—
239	C₂²⋊C₄	—	2	2	no	—
240	C₂²⋊C₄	M₁₆×C₅₃	1	2	yes	—
241	C₄²	—	2	2	no	—
242	C₄²	—	2	1	no	—
243	C₄²	—	2	1	no	—
244	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
245	C₄²	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
246	C₁₆	—	2	4	no	—
247	C₁₆	—	2	4	no	—
248	C₂³	—	2	1	no	—
249	C₂³	C₂×M₁₆×C₅₃	1	1	yes	characteristic
250	Q₈	—	2	1	no	—
251	Q₈	#97×C₅₃	1	2	yes	—
252	D₄	—	2	2	no	—
253	D₄	—	2	2	no	—
254	D₄	—	2	1	no	—
255	D₄	#97×C₅₃	1	2	yes	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	8	1	no	—
263	C₂×C₄	—	8	2	no	—
264	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
265	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
266	C₂×C₄	#117×C₅₃	1	1	yes	characteristic
267	C₂×C₄	C₂×M₁₆×C₅₃	1	1	yes	characteristic
268	C₂×C₄	C₂×M₁₆×C₅₃	1	1	yes	characteristic
269	C₂×C₄	C₂×M₁₆×C₅₃	1	1	yes	characteristic
270	C₈	—	2	2	no	—
271	C₈	—	2	1	no	—
272	C₈	—	2	2	no	—
273	C₈	—	2	1	no	—
274	C₈	C₄×D₄×C₅₃	1	2	yes	—
275	C₈	C₄×D₄×C₅₃	1	2	yes	—
276	C₂²	—	2	2	no	—
277	C₂²	—	4	1	no	—
278	C₂²	—	4	2	no	—
279	C₂²	C₄×M₁₆×C₅₃	1	1	yes	characteristic
280	C₂²	C₅₃×#407	1	1	yes	characteristic
281	C₂²	C₅₃×#432	1	1	yes	characteristic
282	C₄	—	2	2	no	—
283	C₄	—	4	1	no	—
284	C₄	—	4	2	no	—
285	C₄	—	4	1	no	—
286	C₄	C₄×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
287	C₄	C₂×#97×C₅₃	1	1	yes	characteristic
288	C₄	C₅₃×#407	1	1	yes	characteristic
289	C₄	C₅₃×#432	1	1	yes	characteristic
290	C₂	—	2	1	no	—
291	C₂	—	2	1	no	—
292	C₂	—	2	1	no	—
293	C₂	—	4	2	no	—
294	C₂	C₅₃×#1749	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₅₃	C₄×C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄²×C₅₃
Frattini, cosocle	C₂²×C₈	C₂³×C₅₃
socle	C₂×C₅₃	#1749
prime radical	C₂×#41×C₅₃	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅₃ ◁ C₂×#97×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₅₃×#347	C₂	semidirect
4. C₂×C₅₃×#347	C₂	semidirect
5. C₅₃×#1378	C₂	semidirect
19. C₅₃×#350	C₄	semidirect
47. M₃₂×C₅₃	C₂×C₄	semidirect
49. C₂×C₁₆×C₅₃	C₂×C₄	semidirect
128. #12609	C₅₃	direct (N × H)
145. C₂×#347	C₂×C₅₃	semidirect
146. C₂×#347	C₂×C₅₃	semidirect
147. #1378	C₂×C₅₃	semidirect
166. #350	C₄×C₅₃	semidirect
171. C₅₃	#12609	direct (N × H)
195. M₃₂	C₂×C₄×C₅₃	semidirect
197. C₂×C₁₆	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×#2875	2
C₅₃×#2117	2
C₂×C₅₃×#347	2
C₅₃×#1378	2
#12609	53

Subgroups — C53×#12609 #27220121

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#12609 #27220121