Subgroups — C₅₃×#12742 #27220254

Order 13568.

878 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#578	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2145	C₂	1	2	yes	maximal, complemented
3	C₅₃×#2141	C₂	1	2	yes	maximal, complemented
4	C₅₃×#2115	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1812	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×2₋¹⁺⁴×C₅₃	—	2	1	no	—
7	C₅₃×#578	—	2	4	no	—
8	C₅₃×#578	—	2	4	no	—
9	C₂×#137×C₅₃	—	2	2	no	—
10	C₂×#137×C₅₃	—	2	1	no	—
11	C₂×#135×C₅₃	C₄	1	2	yes	complemented
12	C₂×#135×C₅₃	C₂²	1	1	yes	characteristic
13	C₂²×Q₁₆×C₅₃	C₄	1	2	yes	complemented
14	C₂²×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
17	C₅₃×#427	C₂²	1	2	yes	—
18	C₅₃×#410	—	2	1	no	—
19	C₅₃×#360	—	2	4	no	—
20	C₅₃×#358	—	2	4	no	—
21	C₅₃×#349	—	2	2	no	—
22	C₅₃×H(3,ℤ/4)	—	2	1	no	—
23	2₋¹⁺⁴×C₅₃	—	2	8	no	—
24	2₋¹⁺⁴×C₅₃	—	4	4	no	—
25	C₂×#41×C₅₃	—	2	2	no	—
26	C₂×#41×C₅₃	—	2	1	no	—
27	C₂×#41×C₅₃	—	4	2	no	—
28	C₂×#41×C₅₃	—	4	1	no	—
29	C₂×#41×C₅₃	D₄	1	1	yes	characteristic
30	C₂²×Q₈×C₅₃	—	2	2	no	—
31	C₂²×Q₈×C₅₃	—	2	1	no	—
32	C₂²×Q₈×C₅₃	—	4	1	no	—
33	#137×C₅₃	—	2	16	no	—
34	#137×C₅₃	—	2	8	no	—
35	#135×C₅₃	—	2	4	no	—
36	#135×C₅₃	—	2	4	no	—
37	#135×C₅₃	—	2	2	no	—
38	#135×C₅₃	D₄	1	4	yes	—
39	C₂×Q₁₆×C₅₃	—	2	4	no	—
40	C₂×Q₁₆×C₅₃	—	2	4	no	—
41	C₂×Q₁₆×C₅₃	—	2	2	no	—
42	C₂×Q₁₆×C₅₃	—	2	1	no	—
43	C₂×Q₁₆×C₅₃	—	2	2	no	—
44	C₂×Q₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₂×Q₁₆×C₅₃	C₂×C₄	1	2	yes	—
46	C₂×Q₁₆×C₅₃	D₄	1	4	yes	—
47	C₂×SD₁₆×C₅₃	—	2	2	no	—
48	C₂×SD₁₆×C₅₃	—	2	1	no	—
49	C₂×D₈×C₅₃	C₂×C₄	1	1	yes	characteristic
50	#131×C₅₃	—	2	2	no	—
51	#131×C₅₃	—	2	2	no	—
52	C₂×M₁₆×C₅₃	D₄	1	2	yes	—
53	C₂×M₁₆×C₅₃	D₄	1	1	yes	characteristic
54	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
55	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
56	#117×C₅₃	—	2	2	no	—
57	M₃₂×C₅₃	—	2	2	no	—
58	M₃₂×C₅₃	—	2	2	no	—
59	C₂×C₁₆×C₅₃	—	2	2	no	—
60	#107×C₅₃	—	2	2	no	—
61	#106×C₅₃	—	2	2	no	—
62	#99×C₅₃	—	4	2	no	—
63	#41×C₅₃	—	2	2	no	—
64	#41×C₅₃	—	2	2	no	—
65	#41×C₅₃	—	2	8	no	—
66	#41×C₅₃	—	2	4	no	—
67	#41×C₅₃	—	4	4	no	—
68	#41×C₅₃	—	4	16	no	—
69	#41×C₅₃	—	4	8	no	—
70	#41×C₅₃	—	4	2	no	—
71	C₂×Q₈×C₅₃	—	2	8	no	—
72	C₂×Q₈×C₅₃	—	2	2	no	—
73	C₂×Q₈×C₅₃	—	2	1	no	—
74	C₂×Q₈×C₅₃	—	2	4	no	—
75	C₂×Q₈×C₅₃	—	4	4	no	—
76	C₂×Q₈×C₅₃	—	4	8	no	—
77	C₂×Q₈×C₅₃	—	4	2	no	—
78	C₂×Q₈×C₅₃	—	4	1	no	—
79	C₂×Q₈×C₅₃	—	4	2	no	—
80	C₂×Q₈×C₅₃	SD₁₆	1	1	yes	characteristic
81	C₂×D₄×C₅₃	—	2	1	no	—
82	C₂×D₄×C₅₃	—	4	2	no	—
83	C₂×D₄×C₅₃	—	4	1	no	—
84	C₂×D₄×C₅₃	D₈	1	2	yes	—
85	C₂×D₄×C₅₃	SD₁₆	1	1	yes	characteristic
86	C₂²×C₄×C₅₃	—	4	2	no	—
87	C₂²×C₄×C₅₃	—	4	1	no	—
88	C₂²×C₄×C₅₃	—	4	2	no	—
89	C₂²×C₄×C₅₃	—	4	1	no	—
90	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
91	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
92	Q₁₆×C₅₃	—	2	8	no	—
93	Q₁₆×C₅₃	—	2	1	no	—
94	Q₁₆×C₅₃	—	2	2	no	—
95	Q₁₆×C₅₃	—	2	4	no	—
96	Q₁₆×C₅₃	C₂²⋊C₄	1	2	yes	—
97	Q₁₆×C₅₃	C₂²⋊C₄	1	4	yes	—
98	SD₁₆×C₅₃	—	2	8	no	—
99	SD₁₆×C₅₃	—	2	4	no	—
100	D₈×C₅₃	—	2	1	no	—
101	D₈×C₅₃	C₂²⋊C₄	1	2	yes	—
102	M₁₆×C₅₃	—	2	2	no	—
103	M₁₆×C₅₃	—	2	4	no	—
104	C₂×C₈×C₅₃	—	2	2	no	—
105	C₂×C₈×C₅₃	—	2	2	no	—
106	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
107	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
109	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
110	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
111	C₄⋊C₄×C₅₃	—	4	2	no	—
112	C₂²⋊C₄×C₅₃	—	4	2	no	—
113	C₄²×C₅₃	—	4	2	no	—
114	C₁₆×C₅₃	—	4	2	no	—
115	C₂³×C₅₃	—	4	1	no	—
116	C₂³×C₅₃	#102	1	2	yes	—
117	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
118	Q₈×C₅₃	—	2	2	no	—
119	Q₈×C₅₃	—	2	2	no	—
120	Q₈×C₅₃	—	2	4	no	—
121	Q₈×C₅₃	—	4	8	no	—
122	Q₈×C₅₃	—	4	1	no	—
123	Q₈×C₅₃	—	4	2	no	—
124	Q₈×C₅₃	—	4	4	no	—
125	D₄×C₅₃	—	2	4	no	—
126	D₄×C₅₃	—	2	2	no	—
127	D₄×C₅₃	—	2	2	no	—
128	D₄×C₅₃	—	4	8	no	—
129	D₄×C₅₃	—	4	4	no	—
130	D₄×C₅₃	—	4	1	no	—
131	C₂×C₄×C₅₃	—	2	2	no	—
132	C₂×C₄×C₅₃	—	2	2	no	—
133	C₂×C₄×C₅₃	—	4	8	no	—
134	C₂×C₄×C₅₃	—	4	8	no	—
135	C₂×C₄×C₅₃	—	4	1	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	—	8	2	no	—
140	C₂×C₄×C₅₃	#102	1	2	yes	—
141	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
142	C₂×C₄×C₅₃	#120	1	2	yes	—
143	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
144	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
145	C₈×C₅₃	—	2	2	no	—
146	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
147	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
148	C₈×C₅₃	#120	1	2	yes	—
149	#12742	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	2	no	—
154	C₂²×C₅₃	—	8	2	no	—
155	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
156	C₂²×C₅₃	#447	1	2	yes	—
157	C₄×C₅₃	—	2	2	no	—
158	C₄×C₅₃	—	4	4	no	—
159	C₄×C₅₃	—	4	2	no	—
160	C₄×C₅₃	—	16	2	no	—
161	C₄×C₅₃	#379	1	1	yes	characteristic
162	C₄×C₅₃	#418	1	1	yes	characteristic
163	C₄×C₅₃	#450	1	2	yes	—
164	C₂×#578	C₂×C₅₃	1	1	yes	characteristic
165	#2145	C₂×C₅₃	1	2	yes	complemented
166	#2141	C₂×C₅₃	1	2	yes	complemented
167	#2115	C₂×C₅₃	1	1	yes	characteristic, complemented
168	#1812	C₂×C₅₃	1	1	yes	characteristic, complemented
169	C₂×C₅₃	—	2	2	no	—
170	C₂×C₅₃	—	2	1	no	—
171	C₂×C₅₃	—	4	2	no	—
172	C₂×C₅₃	—	8	2	no	—
173	C₂×C₅₃	#1786	1	1	yes	characteristic
174	C₂×2₋¹⁺⁴	—	2	1	no	—
175	#578	—	2	4	no	—
176	#578	—	2	4	no	—
177	C₂×#137	—	2	2	no	—
178	C₂×#137	—	2	1	no	—
179	C₂×#135	C₄×C₅₃	1	2	yes	complemented
180	C₂×#135	C₂²×C₅₃	1	1	yes	characteristic
181	C₂²×Q₁₆	C₄×C₅₃	1	2	yes	complemented
182	C₂²×Q₁₆	C₂²×C₅₃	1	1	yes	characteristic
183	C₂×#131	C₂²×C₅₃	1	1	yes	characteristic
184	C₂×M₃₂	C₂²×C₅₃	1	2	yes	—
185	#427	C₂²×C₅₃	1	2	yes	—
186	#410	—	2	1	no	—
187	#360	—	2	4	no	—
188	#358	—	2	4	no	—
189	#349	—	2	2	no	—
190	H(3,ℤ/4)	—	2	1	no	—
191	C₅₃	#12742	1	1	yes	characteristic, direct product factor, Sylow
192	2₋¹⁺⁴	—	2	8	no	—
193	2₋¹⁺⁴	—	4	4	no	—
194	C₂×#41	—	2	1	no	—
195	C₂×#41	—	2	2	no	—
196	C₂×#41	—	4	2	no	—
197	C₂×#41	—	4	1	no	—
198	C₂×#41	D₄×C₅₃	1	1	yes	characteristic
199	C₂²×Q₈	—	2	1	no	—
200	C₂²×Q₈	—	2	2	no	—
201	C₂²×Q₈	—	4	1	no	—
202	#137	—	2	8	no	—
203	#137	—	2	16	no	—
204	#135	—	2	4	no	—
205	#135	—	2	4	no	—
206	#135	—	2	2	no	—
207	#135	D₄×C₅₃	1	4	yes	—
208	C₂×Q₁₆	—	2	4	no	—
209	C₂×Q₁₆	—	2	4	no	—
210	C₂×Q₁₆	—	2	1	no	—
211	C₂×Q₁₆	—	2	2	no	—
212	C₂×Q₁₆	—	2	2	no	—
213	C₂×Q₁₆	C₂×C₄×C₅₃	1	1	yes	characteristic
214	C₂×Q₁₆	C₂×C₄×C₅₃	1	2	yes	—
215	C₂×Q₁₆	D₄×C₅₃	1	4	yes	—
216	C₂×SD₁₆	—	2	1	no	—
217	C₂×SD₁₆	—	2	2	no	—
218	C₂×D₈	C₂×C₄×C₅₃	1	1	yes	characteristic
219	#131	—	2	2	no	—
220	#131	—	2	2	no	—
221	C₂×M₁₆	D₄×C₅₃	1	2	yes	—
222	C₂×M₁₆	D₄×C₅₃	1	1	yes	characteristic
223	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
224	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
225	#117	—	2	2	no	—
226	M₃₂	—	2	2	no	—
227	M₃₂	—	2	2	no	—
228	C₂×C₁₆	—	2	2	no	—
229	#107	—	2	2	no	—
230	#106	—	2	2	no	—
231	#99	—	4	2	no	—
232	#41	—	2	2	no	—
233	#41	—	2	2	no	—
234	#41	—	2	8	no	—
235	#41	—	2	4	no	—
236	#41	—	4	4	no	—
237	#41	—	4	16	no	—
238	#41	—	4	8	no	—
239	#41	—	4	2	no	—
240	C₂×Q₈	—	2	8	no	—
241	C₂×Q₈	—	2	1	no	—
242	C₂×Q₈	—	2	2	no	—
243	C₂×Q₈	—	2	4	no	—
244	C₂×Q₈	—	4	4	no	—
245	C₂×Q₈	—	4	8	no	—
246	C₂×Q₈	—	4	2	no	—
247	C₂×Q₈	—	4	1	no	—
248	C₂×Q₈	—	4	2	no	—
249	C₂×Q₈	SD₁₆×C₅₃	1	1	yes	characteristic
250	C₂×D₄	—	2	1	no	—
251	C₂×D₄	—	4	2	no	—
252	C₂×D₄	—	4	1	no	—
253	C₂×D₄	D₈×C₅₃	1	2	yes	—
254	C₂×D₄	SD₁₆×C₅₃	1	1	yes	characteristic
255	C₂²×C₄	—	4	2	no	—
256	C₂²×C₄	—	4	1	no	—
257	C₂²×C₄	—	4	2	no	—
258	C₂²×C₄	—	4	1	no	—
259	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
260	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
261	Q₁₆	—	2	8	no	—
262	Q₁₆	—	2	1	no	—
263	Q₁₆	—	2	2	no	—
264	Q₁₆	—	2	4	no	—
265	Q₁₆	C₂²⋊C₄×C₅₃	1	2	yes	—
266	Q₁₆	C₂²⋊C₄×C₅₃	1	4	yes	—
267	SD₁₆	—	2	8	no	—
268	SD₁₆	—	2	4	no	—
269	D₈	—	2	1	no	—
270	D₈	C₂²⋊C₄×C₅₃	1	2	yes	—
271	M₁₆	—	2	2	no	—
272	M₁₆	—	2	4	no	—
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	—	2	2	no	—
275	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
276	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
277	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
278	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
279	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
280	C₄⋊C₄	—	4	2	no	—
281	C₂²⋊C₄	—	4	2	no	—
282	C₄²	—	4	2	no	—
283	C₁₆	—	4	2	no	—
284	C₂³	—	4	1	no	—
285	C₂³	#102×C₅₃	1	2	yes	—
286	C₂³	C₂×D₈×C₅₃	1	1	yes	characteristic
287	Q₈	—	2	2	no	—
288	Q₈	—	2	4	no	—
289	Q₈	—	2	2	no	—
290	Q₈	—	4	8	no	—
291	Q₈	—	4	1	no	—
292	Q₈	—	4	2	no	—
293	Q₈	—	4	4	no	—
294	D₄	—	2	4	no	—
295	D₄	—	2	2	no	—
296	D₄	—	2	2	no	—
297	D₄	—	4	8	no	—
298	D₄	—	4	4	no	—
299	D₄	—	4	1	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	4	8	no	—
303	C₂×C₄	—	4	8	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	4	4	no	—
307	C₂×C₄	—	4	4	no	—
308	C₂×C₄	—	8	2	no	—
309	C₂×C₄	#102×C₅₃	1	2	yes	—
310	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
311	C₂×C₄	#120×C₅₃	1	2	yes	—
312	C₂×C₄	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
313	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
314	C₈	—	2	2	no	—
315	C₈	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
316	C₈	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
317	C₈	#120×C₅₃	1	2	yes	—
318	C₂²	—	2	2	no	—
319	C₂²	—	4	1	no	—
320	C₂²	—	4	2	no	—
321	C₂²	—	4	2	no	—
322	C₂²	—	8	2	no	—
323	C₂²	C₂×#102×C₅₃	1	1	yes	characteristic
324	C₂²	C₅₃×#447	1	2	yes	—
325	C₄	—	2	2	no	—
326	C₄	—	4	4	no	—
327	C₄	—	4	2	no	—
328	C₄	—	16	2	no	—
329	C₄	C₅₃×#379	1	1	yes	characteristic
330	C₄	C₅₃×#418	1	1	yes	characteristic
331	C₄	C₅₃×#450	1	2	yes	—
332	C₂	—	2	2	no	—
333	C₂	—	2	1	no	—
334	C₂	—	4	2	no	—
335	C₂	—	8	2	no	—
336	C₂	C₅₃×#1786	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₅₃	#1786
commutator	C₂×C₈	C₂²×C₄×C₅₃
Frattini, cosocle	C₂²×C₈	C₂³×C₅₃
prime radical	C₂×C₅₃×#578	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×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×#2145	C₂	semidirect
3. C₅₃×#2141	C₂	semidirect
4. C₅₃×#2115	C₂	semidirect
5. C₅₃×#1812	C₂	semidirect
11. C₂×#135×C₅₃	C₄	semidirect
13. C₂²×Q₁₆×C₅₃	C₄	semidirect
149. #12742	C₅₃	direct (N × H)
165. #2145	C₂×C₅₃	semidirect
166. #2141	C₂×C₅₃	semidirect
167. #2115	C₂×C₅₃	semidirect
168. #1812	C₂×C₅₃	semidirect
179. C₂×#135	C₄×C₅₃	semidirect
181. C₂²×Q₁₆	C₄×C₅₃	semidirect
191. C₅₃	#12742	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×#578	2
C₅₃×#2145	2
C₅₃×#2141	2
C₅₃×#2115	2
C₅₃×#1812	2
#12742	53

Subgroups — C53×#12742 #27220254

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#12742 #27220254