Subgroups — C₂×#498×D₅₃ #28253678

Order 13568.

2398 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊Q₈×D₅₃	C₂	1	2	yes	maximal
2	C₂×#16811021	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16811020	C₂	1	4	yes	maximal, complemented
4	C₂×#16811019	C₂	1	2	yes	maximal, complemented
5	#498×D₅₃	C₂	1	16	yes	maximal, direct product factor
6	C₂×#106×D₅₃	C₂	1	4	yes	maximal
7	C₂×C₄×C₈×D₅₃	C₂	1	1	yes	maximal, characteristic
8	C₂×C₅₃×#498	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×Q₈×D₅₃	—	2	2	no	—
10	C₂²×C₈×D₅₃	C₂²	1	2	yes	—
11	C₂×#13133217	C₂²	1	2	yes	—
12	C₂×#13133216	—	2	2	no	—
13	C₂×#13133216	C₂²	1	4	yes	—
14	C₂×#13133215	—	2	2	no	—
15	C₄⋊Q₈×D₅₃	—	2	8	no	—
16	C₄⋊Q₈×D₅₃	C₂²	1	16	yes	—
17	C₂×C₄⋊C₄×D₅₃	—	2	2	no	—
18	C₂×C₄⋊C₄×D₅₃	C₂²	1	4	yes	—
19	C₂×C₄²×D₅₃	C₂²	1	1	yes	characteristic
20	#16811021	C₂²	1	8	yes	complemented
21	#16811020	C₂²	1	32	yes	complemented
22	#16811019	C₂²	1	16	yes	complemented
23	C₂×#13133149	C₂²	1	4	yes	—
24	C₂×#13133148	C₂²	1	8	yes	—
25	#106×D₅₃	C₂²	1	32	yes	—
26	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
27	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
28	C₄×C₈×D₅₃	C₂²	1	8	yes	—
29	C₂×C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
30	C₅₃×#498	C₂²	1	8	yes	complemented
31	C₂×#106×C₅₃	C₂²	1	4	yes	—
32	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
33	C₂²×Dic₁₀₆	—	2	2	no	—
34	C₂²×Dic₁₀₆	—	2	4	no	—
35	C₂×Q₈×D₅₃	—	2	16	no	—
36	C₂×Q₈×D₅₃	—	4	16	no	—
37	C₂²×C₄×D₅₃	—	2	4	no	—
38	C₂²×C₄×D₅₃	D₄	1	2	yes	—
39	C₂²×C₄×D₅₃	C₂³	1	1	yes	characteristic
40	C₂²×#88322	C₂³	1	2	yes	—
41	C₂×C₈×D₅₃	Q₈	1	16	yes	complemented
42	C₂×C₈×D₅₃	C₂³	1	8	yes	—
43	#13133217	—	2	4	no	—
44	#13133217	C₂³	1	8	yes	—
45	#13133216	—	2	8	no	—
46	#13133216	—	2	16	no	—
47	#13133216	C₂³	1	16	yes	—
48	#13133215	—	2	16	no	—
49	C₂×C₄⋊Dic₅₃	—	2	2	no	—
50	C₂×C₄⋊Dic₅₃	C₂³	1	4	yes	—
51	C₂×#11719919	—	2	4	no	—
52	C₂×#11719919	—	2	4	no	—
53	C₄⋊C₄×D₅₃	—	2	16	no	—
54	C₄⋊C₄×D₅₃	—	2	8	no	—
55	C₄⋊C₄×D₅₃	C₂³	1	16	yes	—
56	C₂×C₄×Dic₅₃	D₄	1	2	yes	—
57	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
58	C₄²×D₅₃	D₄	1	4	yes	—
59	C₄²×D₅₃	C₂³	1	4	yes	—
60	#13133149	C₂³	1	16	yes	—
61	#13133148	C₂³	1	32	yes	—
62	C₄×#88322	C₂³	1	4	yes	—
63	C₈×Dic₅₃	C₂³	1	8	yes	—
64	C₂²×Q₈×C₅₃	—	2	2	no	—
65	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
66	C₄⋊Q₈×C₅₃	—	2	4	no	—
67	C₄⋊Q₈×C₅₃	C₂³	1	8	yes	—
68	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
69	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
70	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
71	#106×C₅₃	C₂³	1	16	yes	—
72	C₄×C₈×C₅₃	C₂³	1	4	yes	—
73	C₂³×D₅₃	C₂×D₄	1	1	yes	characteristic
74	C₂×Q₈×C₅₃	—	2	8	no	—
75	C₂×Q₈×C₅₃	—	4	8	no	—
76	C₂²×C₄×C₅₃	—	2	4	no	—
77	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
79	C₂²×Dic₅₃	—	2	4	no	—
80	C₂²×Dic₅₃	C₂×D₄	1	1	yes	characteristic
81	C₂²×Dic₅₃	C₂×D₄	1	2	yes	—
82	Q₈×D₅₃	—	4	64	no	—
83	C₂×C₄×D₅₃	—	2	16	no	—
84	C₂×C₄×D₅₃	—	4	16	no	—
85	C₂×C₄×D₅₃	SD₁₆	1	16	yes	—
86	C₂×C₄×D₅₃	C₂×D₄	1	2	yes	—
87	C₂×C₄×D₅₃	C₂×D₄	1	8	yes	—
88	C₂×C₄×D₅₃	C₂×Q₈	1	8	yes	—
89	C₂×C₄×D₅₃	C₂⁴	1	2	yes	—
90	C₂×Dic₁₀₆	—	2	8	no	—
91	C₂×Dic₁₀₆	—	2	16	no	—
92	C₂×Dic₁₀₆	—	4	8	no	—
93	C₂×Dic₁₀₆	—	4	16	no	—
94	C₂×C₈×C₅₃	C₂×Q₈	1	8	yes	complemented
95	C₂×C₈×C₅₃	C₂⁴	1	4	yes	—
96	C₄⋊C₄×C₅₃	—	2	8	no	—
97	C₄⋊C₄×C₅₃	—	2	4	no	—
98	C₄⋊C₄×C₅₃	C₂⁴	1	8	yes	—
99	C₄²×C₅₃	C₂×D₄	1	2	yes	—
100	C₄²×C₅₃	C₂⁴	1	2	yes	—
101	C₄⋊Dic₅₃	—	2	8	no	—
102	C₄⋊Dic₅₃	—	2	4	no	—
103	C₄⋊Dic₅₃	C₂⁴	1	8	yes	—
104	#11719919	—	2	16	no	—
105	#11719919	—	2	16	no	—
106	C₄×Dic₅₃	C₂×D₄	1	2	yes	—
107	C₄×Dic₅₃	C₂×D₄	1	8	yes	—
108	C₄×Dic₅₃	C₂⁴	1	2	yes	—
109	C₂×#88322	C₂×Q₈	1	8	yes	complemented
110	C₂×#88322	C₂⁴	1	4	yes	—
111	C₈×D₅₃	C₂×Q₈	1	32	yes	complemented
112	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
113	C₂²×D₅₃	C₄⋊Q₈	1	4	yes	—
114	C₂²×D₅₃	C₂×SD₁₆	1	8	yes	—
115	C₂²×D₅₃	C₂²×D₄	1	2	yes	—
116	Q₈×C₅₃	—	4	16	no	—
117	C₂×C₄×C₅₃	—	2	8	no	—
118	C₂×C₄×C₅₃	—	4	8	no	—
119	C₂×C₄×C₅₃	C₂×SD₁₆	1	8	yes	—
120	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
121	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
122	C₂×C₄×C₅₃	C₂²×Q₈	1	4	yes	—
123	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
124	C₂×Dic₅₃	—	2	8	no	—
125	C₂×Dic₅₃	—	4	8	no	—
126	C₂×Dic₅₃	C₄⋊Q₈	1	4	yes	—
127	C₂×Dic₅₃	C₂×SD₁₆	1	8	yes	—
128	C₂×Dic₅₃	C₂²×D₄	1	2	yes	—
129	C₂×Dic₅₃	C₂²×D₄	1	4	yes	—
130	C₄×D₅₃	—	4	32	no	—
131	C₄×D₅₃	C₄⋊Q₈	1	8	yes	—
132	C₄×D₅₃	C₂×SD₁₆	1	32	yes	—
133	C₄×D₅₃	C₂²×Q₈	1	8	yes	—
134	Dic₁₀₆	—	4	16	no	—
135	Dic₁₀₆	—	4	32	no	—
136	C₈×C₅₃	C₂²×Q₈	1	8	yes	complemented
137	#88322	C₂²×Q₈	1	8	yes	complemented
138	C₂²×#498	—	53	1	no	maximal, Sylow
139	C₂²×C₅₃	C₂×C₄⋊Q₈	1	2	yes	—
140	C₂²×C₅₃	C₂²×SD₁₆	1	4	yes	—
141	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
142	C₂×D₅₃	#498	1	16	yes	direct product factor
143	C₂×D₅₃	C₂×C₄⋊Q₈	1	4	yes	—
144	C₂×D₅₃	C₂²×SD₁₆	1	8	yes	—
145	C₄×C₅₃	—	4	8	no	—
146	C₄×C₅₃	C₂×C₄⋊Q₈	1	2	yes	—
147	C₄×C₅₃	C₂²×SD₁₆	1	8	yes	—
148	C₄×C₅₃	C₂³×Q₈	1	2	yes	—
149	Dic₅₃	—	4	8	no	—
150	Dic₅₃	C₂×C₄⋊Q₈	1	4	yes	—
151	Dic₅₃	C₂²×SD₁₆	1	8	yes	—
152	C₂²×C₄⋊Q₈	—	53	2	no	—
153	C₂×#498	—	53	16	no	—
154	C₂×#498	—	53	4	no	—
155	C₂×#498	—	53	1	no	—
156	C₂×#498	—	53	2	no	—
157	C₂×#498	D₅₃	1	1	yes	characteristic, direct product factor
158	C₂²×#106	—	53	4	no	—
159	C₂²×C₄×C₈	—	53	1	no	—
160	C₂×C₅₃	C₂×#498	1	4	yes	complemented
161	C₂×C₅₃	C₂²×C₄⋊Q₈	1	1	yes	characteristic
162	C₂×C₅₃	C₂³×SD₁₆	1	2	yes	—
163	D₅₃	C₂×#498	1	8	yes	direct product factor
164	C₂³×Q₈	—	106	2	no	—
165	C₂³×C₈	—	53	2	no	—
166	C₂×C₄⋊Q₈	—	53	16	no	—
167	C₂×C₄⋊Q₈	—	53	4	no	—
168	C₂×C₄⋊Q₈	—	53	2	no	—
169	C₂×C₄⋊Q₈	—	106	2	no	—
170	C₂×C₄⋊Q₈	—	106	8	no	—
171	C₂×C₄⋊Q₈	—	106	2	no	—
172	C₂×C₄⋊Q₈	C₂×D₅₃	1	2	yes	—
173	C₂²×C₄⋊C₄	—	53	4	no	—
174	C₂²×C₄⋊C₄	—	106	2	no	—
175	C₂²×C₄²	—	53	1	no	—
176	#498	—	53	16	no	—
177	#498	—	53	32	no	—
178	#498	—	53	8	no	—
179	#498	C₂×D₅₃	1	8	yes	direct product factor
180	C₂×#106	—	53	4	no	—
181	C₂×#106	—	53	8	no	—
182	C₂×#106	—	53	32	no	—
183	C₂×#106	C₂×D₅₃	1	4	yes	—
184	C₂×C₄×C₈	—	53	2	no	—
185	C₂×C₄×C₈	—	53	1	no	—
186	C₂×C₄×C₈	—	53	8	no	—
187	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
188	C₅₃	C₂²×#498	1	1	yes	characteristic, complemented, Sylow
189	C₂²×Q₈	—	2	2	no	—
190	C₂²×Q₈	—	106	4	no	—
191	C₂²×Q₈	—	106	16	no	—
192	C₂²×Q₈	—	106	2	no	—
193	C₂²×Q₈	—	212	16	no	—
194	C₂³×C₄	—	53	2	no	—
195	C₂³×C₄	—	53	1	no	—
196	C₂³×C₄	—	106	4	no	—
197	C₂²×C₈	—	53	2	no	—
198	C₂²×C₈	—	53	16	no	—
199	C₂²×C₈	—	53	8	no	—
200	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
201	C₄⋊Q₈	—	2	4	no	—
202	C₄⋊Q₈	—	53	16	no	—
203	C₄⋊Q₈	—	53	8	no	—
204	C₄⋊Q₈	—	106	16	no	—
205	C₄⋊Q₈	—	106	8	no	—
206	C₄⋊Q₈	—	106	16	no	—
207	C₄⋊Q₈	—	106	4	no	—
208	C₄⋊Q₈	C₂²×D₅₃	1	8	yes	—
209	C₂×C₄⋊C₄	—	2	2	no	—
210	C₂×C₄⋊C₄	—	53	4	no	—
211	C₂×C₄⋊C₄	—	53	16	no	—
212	C₂×C₄⋊C₄	—	106	8	no	—
213	C₂×C₄⋊C₄	—	106	16	no	—
214	C₂×C₄⋊C₄	—	106	4	no	—
215	C₂×C₄⋊C₄	—	106	4	no	—
216	C₂×C₄⋊C₄	—	106	2	no	—
217	C₂×C₄⋊C₄	C₂²×D₅₃	1	4	yes	—
218	C₂×C₄²	—	53	2	no	—
219	C₂×C₄²	—	53	1	no	—
220	C₂×C₄²	—	53	4	no	—
221	C₂×C₄²	—	53	4	no	—
222	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
223	#106	—	53	16	no	—
224	#106	—	53	32	no	—
225	#106	C₂²×D₅₃	1	16	yes	—
226	C₄×C₈	—	53	8	no	—
227	C₄×C₈	—	53	4	no	—
228	C₄×C₈	C₂²×D₅₃	1	4	yes	—
229	C₂⁴	—	53	1	no	—
230	C₂×Q₈	—	2	8	no	—
231	C₂×Q₈	—	4	8	no	—
232	C₂×Q₈	—	106	16	no	—
233	C₂×Q₈	—	106	8	no	—
234	C₂×Q₈	—	212	16	no	—
235	C₂×Q₈	—	212	64	no	—
236	C₂×Q₈	—	212	8	no	—
237	C₂²×C₄	—	2	4	no	—
238	C₂²×C₄	—	53	1	no	—
239	C₂²×C₄	—	53	2	no	—
240	C₂²×C₄	—	53	16	no	—
241	C₂²×C₄	—	53	2	no	—
242	C₂²×C₄	—	53	8	no	—
243	C₂²×C₄	—	53	8	no	—
244	C₂²×C₄	—	53	2	no	—
245	C₂²×C₄	—	106	16	no	—
246	C₂²×C₄	—	106	4	no	—
247	C₂²×C₄	—	212	16	no	—
248	C₂²×C₄	D₄×D₅₃	1	2	yes	—
249	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
250	C₂×C₈	—	53	8	no	—
251	C₂×C₈	—	53	4	no	—
252	C₂×C₈	—	53	32	no	—
253	C₂×C₈	Q₈×D₅₃	1	8	yes	complemented
254	C₂×C₈	C₂³×D₅₃	1	4	yes	—
255	C₄⋊C₄	—	2	8	no	—
256	C₄⋊C₄	—	2	4	no	—
257	C₄⋊C₄	—	53	8	no	—
258	C₄⋊C₄	—	106	16	no	—
259	C₄⋊C₄	—	106	16	no	—
260	C₄⋊C₄	—	106	4	no	—
261	C₄⋊C₄	—	106	8	no	—
262	C₄⋊C₄	C₂³×D₅₃	1	8	yes	—
263	C₄²	—	53	8	no	—
264	C₄²	—	53	2	no	—
265	C₄²	—	53	2	no	—
266	C₄²	D₄×D₅₃	1	2	yes	—
267	C₄²	C₂³×D₅₃	1	2	yes	—
268	C₂³	—	53	8	no	—
269	C₂³	—	53	4	no	—
270	C₂³	—	53	2	no	—
271	C₂³	C₂×D₄×D₅₃	1	1	yes	characteristic
272	Q₈	—	4	16	no	—
273	Q₈	—	212	32	no	—
274	Q₈	—	212	16	no	—
275	C₂×C₄	—	2	8	no	—
276	C₂×C₄	—	4	8	no	—
277	C₂×C₄	—	53	8	no	—
278	C₂×C₄	—	53	4	no	—
279	C₂×C₄	—	53	2	no	—
280	C₂×C₄	—	53	4	no	—
281	C₂×C₄	—	53	32	no	—
282	C₂×C₄	—	53	8	no	—
283	C₂×C₄	—	53	8	no	—
284	C₂×C₄	—	106	8	no	—
285	C₂×C₄	—	212	32	no	—
286	C₂×C₄	—	212	8	no	—
287	C₂×C₄	SD₁₆×D₅₃	1	8	yes	—
288	C₂×C₄	C₂×D₄×D₅₃	1	4	yes	—
289	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
290	C₂×C₄	C₂×Q₈×D₅₃	1	4	yes	—
291	C₂×C₄	C₂⁴×D₅₃	1	1	yes	characteristic
292	C₈	—	53	8	no	—
293	C₈	C₂×Q₈×D₅₃	1	8	yes	complemented
294	C₂²	—	53	16	no	—
295	C₂²	—	53	8	no	—
296	C₂²	—	53	4	no	—
297	C₂²	C₄⋊Q₈×D₅₃	1	2	yes	—
298	C₂²	C₂×SD₁₆×D₅₃	1	4	yes	—
299	C₂²	C₂²×D₄×D₅₃	1	1	yes	characteristic
300	C₄	—	4	8	no	—
301	C₄	—	53	8	no	—
302	C₄	—	53	4	no	—
303	C₄	—	212	8	no	—
304	C₄	C₄⋊Q₈×D₅₃	1	2	yes	—
305	C₄	C₂×SD₁₆×D₅₃	1	8	yes	—
306	C₄	C₂²×Q₈×D₅₃	1	2	yes	—
307	C₂	—	53	8	no	—
308	C₂	#498×D₅₃	1	4	yes	direct product factor
309	C₂	C₂×C₄⋊Q₈×D₅₃	1	1	yes	characteristic
310	C₂	C₂²×SD₁₆×D₅₃	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₅₃
commutator, cosocle	C₂×C₄×C₅₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₅₃
Fitting	C₂×C₅₃×#498	C₂
socle	C₂³×C₅₃	C₂²×D₄
prime radical	C₂³×D₅₃	C₂×D₄
hypercenter	C₂×#498	D₅₃
hypocenter	C₅₃	C₂²×#498
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#498

Complements

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

N	H ≅ G/N	type
2. C₂×#16811021	C₂	semidirect
3. C₂×#16811020	C₂	semidirect
4. C₂×#16811019	C₂	semidirect
5. #498×D₅₃	C₂	direct (N × H)
8. C₂×C₅₃×#498	C₂	semidirect
20. #16811021	C₂²	semidirect
21. #16811020	C₂²	semidirect
22. #16811019	C₂²	semidirect
30. C₅₃×#498	C₂²	semidirect
41. C₂×C₈×D₅₃	Q₈	semidirect
94. C₂×C₈×C₅₃	C₂×Q₈	semidirect
109. C₂×#88322	C₂×Q₈	semidirect
111. C₈×D₅₃	C₂×Q₈	semidirect
136. C₈×C₅₃	C₂²×Q₈	semidirect
137. #88322	C₂²×Q₈	semidirect
142. C₂×D₅₃	#498	direct (N × H)
157. C₂×#498	D₅₃	direct (N × H)
160. C₂×C₅₃	C₂×#498	semidirect
163. D₅₃	C₂×#498	direct (N × H)
179. #498	C₂×D₅₃	direct (N × H)
188. C₅₃	C₂²×#498	semidirect
253. C₂×C₈	Q₈×D₅₃	semidirect
293. C₈	C₂×Q₈×D₅₃	semidirect
308. C₂	#498×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×C₄⋊Q₈×D₅₃	2
C₂×#16811021	2
C₂×#16811020	2
C₂×#16811019	2
#498×D₅₃	2
C₂×#106×D₅₃	2
C₂×C₄×C₈×D₅₃	2
C₂×C₅₃×#498	2
C₂²×#498	53

Subgroups — C2×#498×D53 #28253678

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#498×D₅₃ #28253678