Subgroups — C₂×#543×D₅₃ #28290812

Order 13568.

9310 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	C₂×#16811249	C₂	1	3	yes	maximal, complemented
2	C₂×#16811248	C₂	1	12	yes	maximal, complemented
3	#543×D₅₃	C₂	1	32	yes	maximal, direct product factor
4	C₂×C₂²⋊Q₈×D₅₃	C₂	1	12	yes	maximal, complemented
5	C₂²×C₂²⋊C₄×D₅₃	C₂	1	3	yes	maximal
6	C₂×C₅₃×#543	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁵×D₅₃	C₂²	1	1	yes	characteristic
8	C₂²×Q₈×D₅₃	C₂²	1	4	yes	complemented
9	C₂³×C₄×D₅₃	C₂²	1	6	yes	—
10	#16811249	C₂²	1	48	yes	complemented
11	#16811248	C₂²	1	192	yes	complemented
12	C₂×#13133197	C₂²	1	12	yes	complemented
13	C₂×#13133196	C₂²	1	12	yes	complemented
14	C₂×#13133195	C₂²	1	24	yes	complemented
15	C₂×#13133194	C₂²	1	24	yes	complemented
16	C₂×#13133193	C₂²	1	12	yes	complemented
17	C₂²⋊Q₈×D₅₃	C₂²	1	192	yes	complemented
18	C₂×C₄⋊C₄×D₅₃	C₂²	1	12	yes	complemented
19	C₂²×C₂²⋊Dic₅₃	C₂²	1	3	yes	—
20	C₂²×#11719921	C₂²	1	6	yes	—
21	C₂×C₂²⋊C₄×D₅₃	C₂²	1	48	yes	—
22	C₂×C₂²⋊C₄×D₅₃	C₂²	1	12	yes	—
23	C₂×C₂²⋊C₄×D₅₃	C₂²	1	12	yes	—
24	C₅₃×#543	C₂²	1	16	yes	complemented
25	C₂×C₂²⋊Q₈×C₅₃	C₂²	1	12	yes	complemented
26	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	3	yes	—
27	C₂⁴×D₅₃	—	4	8	no	—
28	C₂⁴×D₅₃	Q₈	1	16	yes	complemented
29	C₂⁴×D₅₃	C₂³	1	3	yes	—
30	C₂⁴×D₅₃	C₂³	1	8	yes	—
31	C₂⁴×D₅₃	C₂³	1	3	yes	—
32	C₂²×Dic₁₀₆	C₂³	1	12	yes	complemented
33	C₂×Q₈×D₅₃	—	4	16	no	—
34	C₂×Q₈×D₅₃	C₂³	1	32	yes	complemented
35	C₂³×Dic₅₃	C₂³	1	6	yes	—
36	C₂²×C₄×D₅₃	—	4	24	no	—
37	C₂²×C₄×D₅₃	C₂³	1	12	yes	—
38	C₂²×C₄×D₅₃	C₂³	1	48	yes	—
39	C₂²×C₄×D₅₃	C₂³	1	12	yes	—
40	#13133197	C₂³	1	96	yes	complemented
41	#13133196	C₂³	1	96	yes	complemented
42	#13133195	C₂³	1	192	yes	complemented
43	#13133194	C₂³	1	192	yes	complemented
44	#13133193	C₂³	1	96	yes	complemented
45	C₂×C₄⋊Dic₅₃	C₂³	1	12	yes	complemented
46	C₂×#11719919	C₂³	1	24	yes	complemented
47	C₄⋊C₄×D₅₃	C₂³	1	96	yes	complemented
48	C₂×C₂²⋊Dic₅₃	C₂³	1	12	yes	—
49	C₂×C₂²⋊Dic₅₃	C₂³	1	24	yes	—
50	C₂×#11719921	C₂³	1	24	yes	—
51	C₂×#11719921	C₂³	1	48	yes	—
52	C₂²⋊C₄×D₅₃	C₂³	1	96	yes	—
53	C₂²⋊C₄×D₅₃	C₂³	1	96	yes	—
54	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
55	C₂²×Q₈×C₅₃	C₂³	1	4	yes	complemented
56	C₂³×C₄×C₅₃	C₂³	1	6	yes	—
57	C₂²⋊Q₈×C₅₃	C₂³	1	96	yes	complemented
58	C₂×C₄⋊C₄×C₅₃	C₂³	1	12	yes	complemented
59	C₂×C₂²⋊C₄×C₅₃	C₂³	1	12	yes	—
60	C₂×C₂²⋊C₄×C₅₃	C₂³	1	24	yes	—
61	C₂⁴×C₅₃	—	4	4	no	—
62	C₂⁴×C₅₃	C₂×Q₈	1	8	yes	complemented
63	C₂⁴×C₅₃	C₂⁴	1	3	yes	—
64	C₂⁴×C₅₃	C₂⁴	1	4	yes	—
65	C₂³×D₅₃	—	2	24	no	—
66	C₂³×D₅₃	—	4	64	no	—
67	C₂³×D₅₃	—	4	32	no	—
68	C₂³×D₅₃	—	4	12	no	—
69	C₂³×D₅₃	C₂×Q₈	1	24	yes	—
70	C₂³×D₅₃	C₂×Q₈	1	64	yes	complemented
71	C₂³×D₅₃	C₂×Q₈	1	24	yes	complemented
72	C₂³×D₅₃	C₂⁴	1	12	yes	—
73	C₂³×D₅₃	C₂⁴	1	3	yes	—
74	C₂³×D₅₃	C₂⁴	1	12	yes	—
75	C₂³×D₅₃	C₂⁴	1	1	yes	characteristic
76	C₂×Q₈×C₅₃	—	4	8	no	—
77	C₂×Q₈×C₅₃	C₂⁴	1	16	yes	complemented
78	C₂²×C₄×C₅₃	—	4	12	no	—
79	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
80	C₂²×C₄×C₅₃	C₂⁴	1	24	yes	—
81	C₂²×Dic₅₃	—	4	12	no	—
82	C₂²×Dic₅₃	C₂⁴	1	12	yes	—
83	C₂²×Dic₅₃	C₂⁴	1	24	yes	—
84	Q₈×D₅₃	—	4	64	no	—
85	C₂×C₄×D₅₃	—	2	48	no	—
86	C₂×C₄×D₅₃	—	4	96	no	—
87	C₂×C₄×D₅₃	—	4	24	no	—
88	C₂×C₄×D₅₃	C₂⁴	1	48	yes	—
89	C₂×C₄×D₅₃	C₂⁴	1	48	yes	—
90	C₂×Dic₁₀₆	—	4	24	no	—
91	C₂×Dic₁₀₆	C₂⁴	1	48	yes	complemented
92	C₄⋊C₄×C₅₃	C₂⁴	1	48	yes	complemented
93	C₂²⋊C₄×C₅₃	C₂⁴	1	48	yes	—
94	C₂²⋊Dic₅₃	C₂⁴	1	48	yes	—
95	#11719921	C₂⁴	1	96	yes	—
96	C₄⋊Dic₅₃	C₂⁴	1	48	yes	complemented
97	#11719919	C₂⁴	1	96	yes	complemented
98	C₂³×C₅₃	—	2	12	no	—
99	C₂³×C₅₃	—	4	16	no	—
100	C₂³×C₅₃	—	4	8	no	—
101	C₂³×C₅₃	C₂²×Q₈	1	12	yes	—
102	C₂³×C₅₃	C₂²×Q₈	1	16	yes	complemented
103	C₂³×C₅₃	C₂⁵	1	6	yes	—
104	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
105	C₂²×D₅₃	—	2	12	no	—
106	C₂²×D₅₃	—	2	96	no	—
107	C₂²×D₅₃	—	2	48	no	—
108	C₂²×D₅₃	—	4	128	no	—
109	C₂²×D₅₃	—	4	48	no	—
110	C₂²×D₅₃	—	4	24	no	—
111	C₂²×D₅₃	C₂²×Q₈	1	12	yes	—
112	C₂²×D₅₃	C₂²×Q₈	1	48	yes	—
113	C₂²×D₅₃	C₂²×Q₈	1	4	yes	—
114	C₂²×D₅₃	C₂²×Q₈	1	48	yes	complemented
115	C₂²×D₅₃	H(5,𝔽₂)	1	8	yes	—
116	C₂²×D₅₃	C₂⁵	1	6	yes	—
117	C₂²×D₅₃	C₂⁵	1	2	yes	—
118	Q₈×C₅₃	—	4	16	no	—
119	C₂×C₄×C₅₃	—	2	24	no	—
120	C₂×C₄×C₅₃	—	4	24	no	—
121	C₂×C₄×C₅₃	C₂⁵	1	24	yes	—
122	C₂×Dic₅₃	—	2	24	no	—
123	C₂×Dic₅₃	—	4	24	no	—
124	C₂×Dic₅₃	C₂⁵	1	24	yes	—
125	C₄×D₅₃	—	2	96	no	—
126	C₄×D₅₃	—	4	48	no	—
127	Dic₁₀₆	—	4	48	no	—
128	C₂²×#543	—	53	1	no	maximal, Sylow
129	C₂²×C₅₃	—	2	12	no	—
130	C₂²×C₅₃	—	2	24	no	—
131	C₂²×C₅₃	—	4	16	no	—
132	C₂²×C₅₃	C₂³×Q₈	1	12	yes	—
133	C₂²×C₅₃	C₂³×Q₈	1	2	yes	—
134	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	4	yes	—
135	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
136	C₂×D₅₃	—	2	12	no	—
137	C₂×D₅₃	—	2	24	no	—
138	C₂×D₅₃	—	2	96	no	—
139	C₂×D₅₃	—	4	48	no	—
140	C₂×D₅₃	#543	1	16	yes	direct product factor
141	C₂×D₅₃	C₂³×Q₈	1	12	yes	—
142	C₂×D₅₃	C₂³×Q₈	1	4	yes	—
143	C₂×D₅₃	C₂×H(5,𝔽₂)	1	8	yes	—
144	C₄×C₅₃	—	2	24	no	—
145	Dic₅₃	—	2	24	no	—
146	C₂×#543	—	53	32	no	—
147	C₂×#543	—	53	12	no	—
148	C₂×#543	—	53	3	no	—
149	C₂×#543	D₅₃	1	1	yes	characteristic, direct product factor
150	C₂²×C₂²⋊Q₈	—	53	12	no	—
151	C₂³×C₂²⋊C₄	—	53	3	no	—
152	C₂×C₅₃	—	2	12	no	—
153	C₂×C₅₃	C₂×#543	1	4	yes	complemented
154	C₂×C₅₃	C₂⁴×Q₈	1	1	yes	characteristic
155	C₂×C₅₃	C₂²×H(5,𝔽₂)	1	2	yes	—
156	D₅₃	—	2	12	no	—
157	D₅₃	C₂×#543	1	8	yes	direct product factor
158	C₂⁶	—	53	1	no	—
159	C₂³×Q₈	—	53	4	no	—
160	C₂⁴×C₄	—	53	6	no	—
161	#543	—	53	192	no	—
162	#543	—	53	48	no	—
163	#543	C₂×D₅₃	1	16	yes	direct product factor
164	C₂×C₂²⋊Q₈	—	53	12	no	—
165	C₂×C₂²⋊Q₈	—	53	12	no	—
166	C₂×C₂²⋊Q₈	—	53	192	no	—
167	C₂×C₂²⋊Q₈	—	53	24	no	—
168	C₂×C₂²⋊Q₈	—	53	12	no	—
169	C₂×C₂²⋊Q₈	—	53	24	no	—
170	C₂×C₂²⋊Q₈	C₂×D₅₃	1	12	yes	complemented
171	C₂²×C₄⋊C₄	—	53	12	no	—
172	C₂²×C₂²⋊C₄	—	53	3	no	—
173	C₂²×C₂²⋊C₄	—	53	6	no	—
174	C₂²×C₂²⋊C₄	—	53	12	no	—
175	C₂²×C₂²⋊C₄	—	53	12	no	—
176	C₂²×C₂²⋊C₄	—	53	48	no	—
177	C₂²×C₂²⋊C₄	C₂×D₅₃	1	3	yes	—
178	C₅₃	C₂²×#543	1	1	yes	characteristic, complemented, Sylow
179	C₂⁵	—	53	3	no	—
180	C₂⁵	—	53	16	no	—
181	C₂⁵	—	53	8	no	—
182	C₂⁵	—	53	3	no	—
183	C₂⁵	—	212	8	no	—
184	C₂⁵	C₂²×D₅₃	1	1	yes	characteristic
185	C₂²×Q₈	—	53	32	no	—
186	C₂²×Q₈	—	53	12	no	—
187	C₂²×Q₈	—	212	16	no	—
188	C₂²×Q₈	C₂²×D₅₃	1	4	yes	complemented
189	C₂³×C₄	—	53	6	no	—
190	C₂³×C₄	—	53	12	no	—
191	C₂³×C₄	—	53	12	no	—
192	C₂³×C₄	—	53	48	no	—
193	C₂³×C₄	—	212	24	no	—
194	C₂³×C₄	C₂²×D₅₃	1	6	yes	—
195	C₂²⋊Q₈	—	53	96	no	—
196	C₂²⋊Q₈	—	53	96	no	—
197	C₂²⋊Q₈	—	53	192	no	—
198	C₂²⋊Q₈	—	53	96	no	—
199	C₂²⋊Q₈	—	53	192	no	—
200	C₂²⋊Q₈	C₂²×D₅₃	1	96	yes	complemented
201	C₂×C₄⋊C₄	—	53	24	no	—
202	C₂×C₄⋊C₄	—	53	12	no	—
203	C₂×C₄⋊C₄	—	53	96	no	—
204	C₂×C₄⋊C₄	C₂²×D₅₃	1	12	yes	complemented
205	C₂×C₂²⋊C₄	—	53	12	no	—
206	C₂×C₂²⋊C₄	—	53	24	no	—
207	C₂×C₂²⋊C₄	—	53	24	no	—
208	C₂×C₂²⋊C₄	—	53	48	no	—
209	C₂×C₂²⋊C₄	—	53	96	no	—
210	C₂×C₂²⋊C₄	—	53	96	no	—
211	C₂×C₂²⋊C₄	C₂²×D₅₃	1	12	yes	—
212	C₂×C₂²⋊C₄	C₂²×D₅₃	1	24	yes	—
213	C₂⁴	—	4	4	no	—
214	C₂⁴	—	53	1	no	—
215	C₂⁴	—	53	24	no	—
216	C₂⁴	—	53	12	no	—
217	C₂⁴	—	53	3	no	—
218	C₂⁴	—	53	24	no	—
219	C₂⁴	—	53	64	no	—
220	C₂⁴	—	53	12	no	—
221	C₂⁴	—	106	24	no	—
222	C₂⁴	—	212	12	no	—
223	C₂⁴	—	212	64	no	—
224	C₂⁴	—	212	32	no	—
225	C₂⁴	Q₈×D₅₃	1	8	yes	complemented
226	C₂⁴	C₂³×D₅₃	1	3	yes	—
227	C₂⁴	C₂³×D₅₃	1	4	yes	—
228	C₂×Q₈	—	4	8	no	—
229	C₂×Q₈	—	53	48	no	—
230	C₂×Q₈	—	212	24	no	—
231	C₂×Q₈	—	212	64	no	—
232	C₂×Q₈	C₂³×D₅₃	1	16	yes	complemented
233	C₂²×C₄	—	4	12	no	—
234	C₂²×C₄	—	53	12	no	—
235	C₂²×C₄	—	53	24	no	—
236	C₂²×C₄	—	53	48	no	—
237	C₂²×C₄	—	53	48	no	—
238	C₂²×C₄	—	106	48	no	—
239	C₂²×C₄	—	212	24	no	—
240	C₂²×C₄	—	212	96	no	—
241	C₂²×C₄	—	212	12	no	—
242	C₂²×C₄	C₂³×D₅₃	1	12	yes	—
243	C₂²×C₄	C₂³×D₅₃	1	24	yes	—
244	C₄⋊C₄	—	53	96	no	—
245	C₄⋊C₄	—	53	48	no	—
246	C₄⋊C₄	C₂³×D₅₃	1	48	yes	complemented
247	C₂²⋊C₄	—	53	48	no	—
248	C₂²⋊C₄	—	53	96	no	—
249	C₂²⋊C₄	C₂³×D₅₃	1	48	yes	—
250	C₂³	—	2	12	no	—
251	C₂³	—	4	16	no	—
252	C₂³	—	4	8	no	—
253	C₂³	—	53	4	no	—
254	C₂³	—	53	8	no	—
255	C₂³	—	53	2	no	—
256	C₂³	—	53	48	no	—
257	C₂³	—	53	12	no	—
258	C₂³	—	53	6	no	—
259	C₂³	—	53	48	no	—
260	C₂³	—	106	48	no	—
261	C₂³	—	106	96	no	—
262	C₂³	—	106	12	no	—
263	C₂³	—	212	48	no	—
264	C₂³	—	212	24	no	—
265	C₂³	—	212	128	no	—
266	C₂³	C₂×Q₈×D₅₃	1	12	yes	—
267	C₂³	C₂×Q₈×D₅₃	1	16	yes	complemented
268	C₂³	C₂⁴×D₅₃	1	6	yes	—
269	C₂³	C₂⁴×D₅₃	1	1	yes	characteristic
270	Q₈	—	4	16	no	—
271	Q₈	—	212	48	no	—
272	C₂×C₄	—	2	24	no	—
273	C₂×C₄	—	4	24	no	—
274	C₂×C₄	—	53	24	no	—
275	C₂×C₄	—	106	96	no	—
276	C₂×C₄	—	106	24	no	—
277	C₂×C₄	—	212	48	no	—
278	C₂×C₄	—	212	24	no	—
279	C₂×C₄	C₂⁴×D₅₃	1	24	yes	—
280	C₂²	—	2	24	no	—
281	C₂²	—	2	12	no	—
282	C₂²	—	4	16	no	—
283	C₂²	—	53	16	no	—
284	C₂²	—	53	4	no	—
285	C₂²	—	53	8	no	—
286	C₂²	—	53	12	no	—
287	C₂²	—	106	96	no	—
288	C₂²	—	106	24	no	—
289	C₂²	—	106	12	no	—
290	C₂²	—	212	48	no	—
291	C₂²	C₂²×Q₈×D₅₃	1	12	yes	—
292	C₂²	C₂²×Q₈×D₅₃	1	2	yes	—
293	C₂²	H(5,𝔽₂)×D₅₃	1	4	yes	—
294	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
295	C₄	—	2	24	no	—
296	C₄	—	106	24	no	—
297	C₂	—	2	12	no	—
298	C₂	—	53	8	no	—
299	C₂	—	106	12	no	—
300	C₂	#543×D₅₃	1	4	yes	direct product factor
301	C₂	C₂³×Q₈×D₅₃	1	1	yes	characteristic
302	C₂	C₂×H(5,𝔽₂)×D₅₃	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂⁴×D₅₃
commutator, cosocle	C₂²×C₅₃	C₂⁶
Frattini	C₂²	C₂⁵×D₅₃
Fitting	C₂×C₅₃×#543	C₂
socle	C₂³×C₅₃	C₂⁵
prime radical	C₂⁵×D₅₃	C₂²
hypercenter	C₂×#543	D₅₃
hypocenter	C₅₃	C₂²×#543
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×#543

Complements

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

N	H ≅ G/N	type
1. C₂×#16811249	C₂	semidirect
2. C₂×#16811248	C₂	semidirect
3. #543×D₅₃	C₂	direct (N × H)
4. C₂×C₂²⋊Q₈×D₅₃	C₂	semidirect
6. C₂×C₅₃×#543	C₂	semidirect
8. C₂²×Q₈×D₅₃	C₂²	semidirect
10. #16811249	C₂²	semidirect
11. #16811248	C₂²	semidirect
12. C₂×#13133197	C₂²	semidirect
13. C₂×#13133196	C₂²	semidirect
14. C₂×#13133195	C₂²	semidirect
15. C₂×#13133194	C₂²	semidirect
16. C₂×#13133193	C₂²	semidirect
17. C₂²⋊Q₈×D₅₃	C₂²	semidirect
18. C₂×C₄⋊C₄×D₅₃	C₂²	semidirect
24. C₅₃×#543	C₂²	semidirect
25. C₂×C₂²⋊Q₈×C₅₃	C₂²	semidirect
28. C₂⁴×D₅₃	Q₈	semidirect
32. C₂²×Dic₁₀₆	C₂³	semidirect
34. C₂×Q₈×D₅₃	C₂³	semidirect
40. #13133197	C₂³	semidirect
41. #13133196	C₂³	semidirect
42. #13133195	C₂³	semidirect
43. #13133194	C₂³	semidirect
44. #13133193	C₂³	semidirect
45. C₂×C₄⋊Dic₅₃	C₂³	semidirect
46. C₂×#11719919	C₂³	semidirect
47. C₄⋊C₄×D₅₃	C₂³	semidirect
55. C₂²×Q₈×C₅₃	C₂³	semidirect
57. C₂²⋊Q₈×C₅₃	C₂³	semidirect
58. C₂×C₄⋊C₄×C₅₃	C₂³	semidirect
62. C₂⁴×C₅₃	C₂×Q₈	semidirect
70. C₂³×D₅₃	C₂×Q₈	semidirect
71. C₂³×D₅₃	C₂×Q₈	semidirect
77. C₂×Q₈×C₅₃	C₂⁴	semidirect
91. C₂×Dic₁₀₆	C₂⁴	semidirect
92. C₄⋊C₄×C₅₃	C₂⁴	semidirect
96. C₄⋊Dic₅₃	C₂⁴	semidirect
97. #11719919	C₂⁴	semidirect
102. C₂³×C₅₃	C₂²×Q₈	semidirect
114. C₂²×D₅₃	C₂²×Q₈	semidirect
140. C₂×D₅₃	#543	direct (N × H)
149. C₂×#543	D₅₃	direct (N × H)
153. C₂×C₅₃	C₂×#543	semidirect
157. D₅₃	C₂×#543	direct (N × H)
163. #543	C₂×D₅₃	direct (N × H)
170. C₂×C₂²⋊Q₈	C₂×D₅₃	semidirect
178. C₅₃	C₂²×#543	semidirect
188. C₂²×Q₈	C₂²×D₅₃	semidirect
200. C₂²⋊Q₈	C₂²×D₅₃	semidirect
204. C₂×C₄⋊C₄	C₂²×D₅₃	semidirect
225. C₂⁴	Q₈×D₅₃	semidirect
232. C₂×Q₈	C₂³×D₅₃	semidirect
246. C₄⋊C₄	C₂³×D₅₃	semidirect
267. C₂³	C₂×Q₈×D₅₃	semidirect
300. C₂	#543×D₅₃	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×#16811249	2
C₂×#16811248	2
#543×D₅₃	2
C₂×C₂²⋊Q₈×D₅₃	2
C₂²×C₂²⋊C₄×D₅₃	2
C₂×C₅₃×#543	2
C₂²×#543	53

Subgroups — C2×#543×D53 #28290812

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#543×D₅₃ #28290812