Subgroups — C₂²×C₄⋊C₈×D₅₃ #28247698

Order 13568.

3870 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	C₂³×C₈×D₅₃	C₂	1	2	yes	maximal
2	C₂²×C₄²×D₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×#13133146	C₂	1	2	yes	maximal, complemented
5	C₂×C₄⋊C₈×D₅₃	C₂	1	24	yes	maximal, direct product factor
6	C₂²×C₄⋊C₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×C₄×D₅₃	C₄	1	1	yes	characteristic
8	C₂³×C₄×D₅₃	C₄	1	1	yes	characteristic
9	C₂³×C₄×D₅₃	C₂²	1	1	yes	characteristic
10	C₂³×#88322	C₂²	1	2	yes	—
11	C₂²×C₈×D₅₃	—	2	16	no	—
12	C₂²×C₈×D₅₃	C₂²	1	24	yes	—
13	C₂²×C₄×Dic₅₃	C₄	1	1	yes	characteristic
14	C₂²×C₄×Dic₅₃	C₄	1	1	yes	characteristic
15	C₂²×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₄²×D₅₃	C₄	1	12	yes	—
17	C₂×C₄²×D₅₃	C₂²	1	12	yes	—
18	C₂×C₄⋊#88322	C₂²	1	12	yes	complemented
19	C₂×#13133146	C₂²	1	24	yes	complemented
20	C₄⋊C₈×D₅₃	C₂²	1	64	yes	direct product factor
21	C₂³×C₈×C₅₃	C₂²	1	2	yes	—
22	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₈×C₅₃	C₂²	1	12	yes	complemented
24	C₂⁴×D₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂³×Dic₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂³×Dic₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂³×Dic₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×D₅₃	—	2	8	no	—
29	C₂²×C₄×D₅₃	C₈	1	16	yes	complemented
30	C₂²×C₄×D₅₃	C₂×C₄	1	12	yes	—
31	C₂²×C₄×D₅₃	C₂×C₄	1	6	yes	—
32	C₂²×C₄×D₅₃	C₂×C₄	1	12	yes	—
33	C₂²×C₄×D₅₃	D₄	1	8	yes	—
34	C₂²×C₄×D₅₃	Q₈	1	8	yes	—
35	C₂²×C₄×D₅₃	C₂³	1	6	yes	—
36	C₂²×#88322	—	2	8	no	—
37	C₂²×#88322	C₂³	1	12	yes	—
38	C₂×C₈×D₅₃	—	2	96	no	—
39	C₂×C₈×D₅₃	C₂³	1	32	yes	—
40	C₂×C₄×Dic₅₃	C₂×C₄	1	6	yes	—
41	C₂×C₄×Dic₅₃	C₂×C₄	1	12	yes	—
42	C₂×C₄×Dic₅₃	C₂×C₄	1	12	yes	—
43	C₂×C₄×Dic₅₃	C₂³	1	6	yes	—
44	C₄²×D₅₃	C₂×C₄	1	48	yes	—
45	C₄²×D₅₃	C₂³	1	16	yes	—
46	C₄⋊#88322	C₂³	1	16	yes	complemented
47	#13133146	C₂³	1	32	yes	complemented
48	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
49	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
50	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₅₃	—	2	8	no	—
52	C₂²×C₈×C₅₃	C₂³	1	12	yes	—
53	C₂×C₄²×C₅₃	C₂×C₄	1	6	yes	—
54	C₂×C₄²×C₅₃	C₂³	1	6	yes	—
55	C₄⋊C₈×C₅₃	C₂³	1	16	yes	complemented
56	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
57	C₂³×D₅₃	C₄⋊C₄	1	8	yes	—
58	C₂³×D₅₃	C₂×C₈	1	8	yes	—
59	C₂³×D₅₃	M₁₆	1	8	yes	—
60	C₂³×D₅₃	C₂²×C₄	1	6	yes	—
61	C₂²×C₄×C₅₃	—	2	4	no	—
62	C₂²×C₄×C₅₃	C₂×C₈	1	8	yes	complemented
63	C₂²×C₄×C₅₃	C₂²×C₄	1	6	yes	—
64	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
65	C₂²×C₄×C₅₃	C₂²×C₄	1	6	yes	—
66	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
67	C₂²×C₄×C₅₃	C₂×Q₈	1	4	yes	—
68	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
69	C₂²×Dic₅₃	—	2	4	no	—
70	C₂²×Dic₅₃	C₄⋊C₄	1	8	yes	—
71	C₂²×Dic₅₃	C₂×C₈	1	8	yes	complemented
72	C₂²×Dic₅₃	C₂²×C₄	1	6	yes	—
73	C₂²×Dic₅₃	C₂²×C₄	1	6	yes	—
74	C₂²×Dic₅₃	C₂²×C₄	1	6	yes	—
75	C₂×C₄×D₅₃	—	2	48	no	—
76	C₂×C₄×D₅₃	C₄⋊C₄	1	48	yes	—
77	C₂×C₄×D₅₃	C₂×C₈	1	96	yes	complemented
78	C₂×C₄×D₅₃	C₂²×C₄	1	16	yes	—
79	C₂×C₄×D₅₃	C₂²×C₄	1	12	yes	—
80	C₂×C₄×D₅₃	C₂²×C₄	1	16	yes	—
81	C₂×C₄×D₅₃	C₂×D₄	1	24	yes	—
82	C₂×C₄×D₅₃	C₂×Q₈	1	24	yes	—
83	C₂×C₄×D₅₃	C₂⁴	1	4	yes	—
84	C₂×C₈×C₅₃	—	2	24	no	—
85	C₂×C₈×C₅₃	C₂⁴	1	8	yes	—
86	C₄²×C₅₃	C₂²×C₄	1	12	yes	—
87	C₄²×C₅₃	C₂⁴	1	4	yes	—
88	C₄×Dic₅₃	C₂²×C₄	1	12	yes	—
89	C₄×Dic₅₃	C₂²×C₄	1	16	yes	—
90	C₄×Dic₅₃	C₂²×C₄	1	16	yes	—
91	C₄×Dic₅₃	C₂⁴	1	4	yes	—
92	C₂×#88322	—	2	24	no	—
93	C₂×#88322	C₂⁴	1	8	yes	—
94	C₈×D₅₃	—	2	64	no	—
95	C₂³×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
96	C₂³×C₅₃	C₂²×C₈	1	4	yes	—
97	C₂³×C₅₃	C₂×M₁₆	1	4	yes	—
98	C₂³×C₅₃	C₂³×C₄	1	3	yes	—
99	C₂²×D₅₃	C₄⋊C₈	1	64	yes	direct product factor
100	C₂²×D₅₃	C₂×C₄⋊C₄	1	24	yes	—
101	C₂²×D₅₃	C₂²×C₈	1	24	yes	—
102	C₂²×D₅₃	C₂×M₁₆	1	24	yes	—
103	C₂²×D₅₃	C₂³×C₄	1	4	yes	—
104	C₂×C₄×C₅₃	—	2	12	no	—
105	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	12	yes	—
106	C₂×C₄×C₅₃	C₂²×C₈	1	24	yes	complemented
107	C₂×C₄×C₅₃	C₂³×C₄	1	4	yes	—
108	C₂×C₄×C₅₃	C₂³×C₄	1	3	yes	—
109	C₂×C₄×C₅₃	C₂³×C₄	1	4	yes	—
110	C₂×C₄×C₅₃	C₂²×D₄	1	6	yes	—
111	C₂×C₄×C₅₃	C₂²×Q₈	1	6	yes	—
112	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
113	C₂×Dic₅₃	—	2	12	no	—
114	C₂×Dic₅₃	C₂×C₄⋊C₄	1	24	yes	—
115	C₂×Dic₅₃	C₂²×C₈	1	24	yes	complemented
116	C₂×Dic₅₃	C₂³×C₄	1	4	yes	—
117	C₂×Dic₅₃	C₂³×C₄	1	4	yes	—
118	C₂×Dic₅₃	C₂³×C₄	1	4	yes	—
119	C₄×D₅₃	—	2	32	no	—
120	C₄×D₅₃	C₂×C₄⋊C₄	1	48	yes	—
121	C₄×D₅₃	C₂²×C₈	1	64	yes	complemented
122	C₄×D₅₃	C₂²×D₄	1	8	yes	—
123	C₄×D₅₃	C₂²×Q₈	1	8	yes	—
124	C₈×C₅₃	—	2	8	no	—
125	#88322	—	2	8	no	—
126	C₂³×C₄⋊C₈	—	53	1	no	maximal, Sylow
127	C₂²×C₅₃	C₂×C₄⋊C₈	1	16	yes	complemented
128	C₂²×C₅₃	C₂²×C₄⋊C₄	1	6	yes	—
129	C₂²×C₅₃	C₂³×C₈	1	6	yes	—
130	C₂²×C₅₃	C₂²×M₁₆	1	6	yes	—
131	C₂²×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
132	C₂×D₅₃	C₂×C₄⋊C₈	1	96	yes	direct product factor
133	C₂×D₅₃	C₂²×C₄⋊C₄	1	8	yes	—
134	C₂×D₅₃	C₂³×C₈	1	8	yes	—
135	C₂×D₅₃	C₂²×M₁₆	1	8	yes	—
136	C₄×C₅₃	—	2	4	no	—
137	C₄×C₅₃	C₂²×C₄⋊C₄	1	6	yes	—
138	C₄×C₅₃	C₂³×C₈	1	8	yes	complemented
139	C₄×C₅₃	C₂³×D₄	1	1	yes	characteristic
140	C₄×C₅₃	C₂³×Q₈	1	1	yes	characteristic
141	Dic₅₃	—	2	4	no	—
142	Dic₅₃	C₂²×C₄⋊C₄	1	8	yes	—
143	Dic₅₃	C₂³×C₈	1	8	yes	complemented
144	C₂⁴×C₈	—	53	2	no	—
145	C₂³×C₄²	—	53	1	no	—
146	C₂²×C₄⋊C₈	—	53	1	no	—
147	C₂²×C₄⋊C₈	—	53	24	no	—
148	C₂²×C₄⋊C₈	—	53	2	no	—
149	C₂²×C₄⋊C₈	D₅₃	1	1	yes	characteristic, direct product factor
150	C₂×C₅₃	C₂²×C₄⋊C₈	1	12	yes	complemented
151	C₂×C₅₃	C₂³×C₄⋊C₄	1	1	yes	characteristic
152	C₂×C₅₃	C₂⁴×C₈	1	1	yes	characteristic
153	C₂×C₅₃	C₂³×M₁₆	1	1	yes	characteristic
154	D₅₃	C₂²×C₄⋊C₈	1	16	yes	direct product factor
155	C₂⁴×C₄	—	53	1	no	—
156	C₂⁴×C₄	—	53	1	no	—
157	C₂⁴×C₄	—	53	1	no	—
158	C₂³×C₈	—	53	2	no	—
159	C₂³×C₈	—	53	24	no	—
160	C₂³×C₈	—	106	16	no	—
161	C₂³×C₈	C₂×D₅₃	1	2	yes	—
162	C₂²×C₄²	—	53	1	no	—
163	C₂²×C₄²	—	53	1	no	—
164	C₂²×C₄²	—	53	1	no	—
165	C₂²×C₄²	—	53	12	no	—
166	C₂²×C₄²	—	53	12	no	—
167	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
168	C₂×C₄⋊C₈	—	53	24	no	—
169	C₂×C₄⋊C₈	—	53	12	no	—
170	C₂×C₄⋊C₈	—	53	64	no	—
171	C₂×C₄⋊C₈	C₂×D₅₃	1	12	yes	direct product factor
172	C₅₃	C₂³×C₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
173	C₂⁵	—	53	1	no	—
174	C₂³×C₄	—	53	1	no	—
175	C₂³×C₄	—	53	1	no	—
176	C₂³×C₄	—	53	6	no	—
177	C₂³×C₄	—	53	12	no	—
178	C₂³×C₄	—	53	8	no	—
179	C₂³×C₄	—	53	8	no	—
180	C₂³×C₄	—	53	6	no	—
181	C₂³×C₄	—	53	16	no	—
182	C₂³×C₄	—	53	1	no	—
183	C₂³×C₄	—	53	12	no	—
184	C₂³×C₄	—	106	8	no	—
185	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
186	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
187	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
188	C₂²×C₈	—	2	8	no	—
189	C₂²×C₈	—	53	12	no	—
190	C₂²×C₈	—	53	32	no	—
191	C₂²×C₈	—	106	96	no	—
192	C₂²×C₈	—	106	8	no	—
193	C₂²×C₈	C₂²×D₅₃	1	12	yes	—
194	C₂×C₄²	—	53	12	no	—
195	C₂×C₄²	—	53	6	no	—
196	C₂×C₄²	—	53	12	no	—
197	C₂×C₄²	—	53	6	no	—
198	C₂×C₄²	—	53	48	no	—
199	C₂×C₄²	—	53	16	no	—
200	C₂×C₄²	C₄×D₅₃	1	6	yes	—
201	C₂×C₄²	C₂²×D₅₃	1	6	yes	—
202	C₄⋊C₈	—	53	32	no	—
203	C₄⋊C₈	—	53	16	no	—
204	C₄⋊C₈	C₂²×D₅₃	1	16	yes	direct product factor
205	C₂⁴	—	53	8	no	—
206	C₂⁴	—	53	8	no	—
207	C₂⁴	—	53	8	no	—
208	C₂⁴	—	53	6	no	—
209	C₂⁴	C₂×C₄×D₅₃	1	1	yes	characteristic
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	53	8	no	—
212	C₂²×C₄	—	53	8	no	—
213	C₂²×C₄	—	53	6	no	—
214	C₂²×C₄	—	53	6	no	—
215	C₂²×C₄	—	53	48	no	—
216	C₂²×C₄	—	53	96	no	—
217	C₂²×C₄	—	53	12	no	—
218	C₂²×C₄	—	53	16	no	—
219	C₂²×C₄	—	53	24	no	—
220	C₂²×C₄	—	53	24	no	—
221	C₂²×C₄	—	53	4	no	—
222	C₂²×C₄	—	53	6	no	—
223	C₂²×C₄	—	53	16	no	—
224	C₂²×C₄	—	106	48	no	—
225	C₂²×C₄	—	106	4	no	—
226	C₂²×C₄	C₈×D₅₃	1	8	yes	complemented
227	C₂²×C₄	C₂×C₄×D₅₃	1	6	yes	—
228	C₂²×C₄	C₂×C₄×D₅₃	1	3	yes	—
229	C₂²×C₄	C₂×C₄×D₅₃	1	6	yes	—
230	C₂²×C₄	D₄×D₅₃	1	4	yes	—
231	C₂²×C₄	Q₈×D₅₃	1	4	yes	—
232	C₂²×C₄	C₂³×D₅₃	1	3	yes	—
233	C₂×C₈	—	2	24	no	—
234	C₂×C₈	—	53	8	no	—
235	C₂×C₈	—	106	64	no	—
236	C₂×C₈	—	106	24	no	—
237	C₂×C₈	C₂³×D₅₃	1	8	yes	—
238	C₄²	—	53	16	no	—
239	C₄²	—	53	12	no	—
240	C₄²	—	53	16	no	—
241	C₄²	—	53	4	no	—
242	C₄²	C₂×C₄×D₅₃	1	12	yes	—
243	C₄²	C₂³×D₅₃	1	4	yes	—
244	C₂³	—	53	24	no	—
245	C₂³	—	53	24	no	—
246	C₂³	—	53	24	no	—
247	C₂³	—	53	64	no	—
248	C₂³	—	53	4	no	—
249	C₂³	C₄⋊C₄×D₅₃	1	4	yes	—
250	C₂³	C₂×C₈×D₅₃	1	4	yes	—
251	C₂³	M₁₆×D₅₃	1	4	yes	—
252	C₂³	C₂²×C₄×D₅₃	1	3	yes	—
253	C₂×C₄	—	2	12	no	—
254	C₂×C₄	—	53	24	no	—
255	C₂×C₄	—	53	24	no	—
256	C₂×C₄	—	53	4	no	—
257	C₂×C₄	—	53	4	no	—
258	C₂×C₄	—	53	48	no	—
259	C₂×C₄	—	53	64	no	—
260	C₂×C₄	—	53	8	no	—
261	C₂×C₄	—	53	8	no	—
262	C₂×C₄	—	53	4	no	—
263	C₂×C₄	—	106	32	no	—
264	C₂×C₄	—	106	12	no	—
265	C₂×C₄	C₄⋊C₄×D₅₃	1	12	yes	—
266	C₂×C₄	C₂×C₈×D₅₃	1	24	yes	complemented
267	C₂×C₄	C₂²×C₄×D₅₃	1	4	yes	—
268	C₂×C₄	C₂²×C₄×D₅₃	1	3	yes	—
269	C₂×C₄	C₂²×C₄×D₅₃	1	4	yes	—
270	C₂×C₄	C₂×D₄×D₅₃	1	6	yes	—
271	C₂×C₄	C₂×Q₈×D₅₃	1	6	yes	—
272	C₂×C₄	C₂⁴×D₅₃	1	1	yes	characteristic
273	C₈	—	2	8	no	—
274	C₈	—	106	8	no	—
275	C₂²	—	53	96	no	—
276	C₂²	—	53	8	no	—
277	C₂²	—	53	8	no	—
278	C₂²	—	53	8	no	—
279	C₂²	C₄⋊C₈×D₅₃	1	16	yes	direct product factor
280	C₂²	C₂×C₄⋊C₄×D₅₃	1	6	yes	—
281	C₂²	C₂²×C₈×D₅₃	1	6	yes	—
282	C₂²	C₂×M₁₆×D₅₃	1	6	yes	—
283	C₂²	C₂³×C₄×D₅₃	1	1	yes	characteristic
284	C₄	—	2	4	no	—
285	C₄	—	53	8	no	—
286	C₄	—	53	8	no	—
287	C₄	—	106	4	no	—
288	C₄	C₂×C₄⋊C₄×D₅₃	1	6	yes	—
289	C₄	C₂²×C₈×D₅₃	1	8	yes	complemented
290	C₄	C₂²×D₄×D₅₃	1	1	yes	characteristic
291	C₄	C₂²×Q₈×D₅₃	1	1	yes	characteristic
292	C₂	—	53	16	no	—
293	C₂	C₂×C₄⋊C₈×D₅₃	1	12	yes	direct product factor
294	C₂	C₂²×C₄⋊C₄×D₅₃	1	1	yes	characteristic
295	C₂	C₂³×C₈×D₅₃	1	1	yes	characteristic
296	C₂	C₂²×M₁₆×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	C₂²×D₅₃
commutator	C₂×C₅₃	C₂⁴×C₈
Frattini	C₂×C₄	C₂⁴×D₅₃
Fitting	C₂²×C₄⋊C₈×C₅₃	C₂
socle	C₂⁴×C₅₃	C₂²×C₄
prime radical	C₂⁴×D₅₃	C₂×C₄
cosocle	C₂×C₄×C₅₃	C₂⁵
hypercenter	C₂²×C₄⋊C₈	D₅₃
hypocenter	C₅₃	C₂³×C₄⋊C₈
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
3. C₂²×C₄⋊#88322	C₂	semidirect
4. C₂²×#13133146	C₂	semidirect
5. C₂×C₄⋊C₈×D₅₃	C₂	direct (N × H)
6. C₂²×C₄⋊C₈×C₅₃	C₂	semidirect
18. C₂×C₄⋊#88322	C₂²	semidirect
19. C₂×#13133146	C₂²	semidirect
20. C₄⋊C₈×D₅₃	C₂²	direct (N × H)
23. C₂×C₄⋊C₈×C₅₃	C₂²	semidirect
29. C₂²×C₄×D₅₃	C₈	semidirect
46. C₄⋊#88322	C₂³	semidirect
47. #13133146	C₂³	semidirect
55. C₄⋊C₈×C₅₃	C₂³	semidirect
62. C₂²×C₄×C₅₃	C₂×C₈	semidirect
71. C₂²×Dic₅₃	C₂×C₈	semidirect
77. C₂×C₄×D₅₃	C₂×C₈	semidirect
99. C₂²×D₅₃	C₄⋊C₈	direct (N × H)
106. C₂×C₄×C₅₃	C₂²×C₈	semidirect
115. C₂×Dic₅₃	C₂²×C₈	semidirect
121. C₄×D₅₃	C₂²×C₈	semidirect
127. C₂²×C₅₃	C₂×C₄⋊C₈	semidirect
132. C₂×D₅₃	C₂×C₄⋊C₈	direct (N × H)
138. C₄×C₅₃	C₂³×C₈	semidirect
143. Dic₅₃	C₂³×C₈	semidirect
149. C₂²×C₄⋊C₈	D₅₃	direct (N × H)
150. C₂×C₅₃	C₂²×C₄⋊C₈	semidirect
154. D₅₃	C₂²×C₄⋊C₈	direct (N × H)
171. C₂×C₄⋊C₈	C₂×D₅₃	direct (N × H)
172. C₅₃	C₂³×C₄⋊C₈	semidirect
204. C₄⋊C₈	C₂²×D₅₃	direct (N × H)
226. C₂²×C₄	C₈×D₅₃	semidirect
266. C₂×C₄	C₂×C₈×D₅₃	semidirect
279. C₂²	C₄⋊C₈×D₅₃	direct (N × H)
289. C₄	C₂²×C₈×D₅₃	semidirect
293. C₂	C₂×C₄⋊C₈×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
C₂³×C₈×D₅₃	2
C₂²×C₄²×D₅₃	2
C₂²×C₄⋊#88322	2
C₂²×#13133146	2
C₂×C₄⋊C₈×D₅₃	2
C₂²×C₄⋊C₈×C₅₃	2
C₂³×C₄⋊C₈	53

Subgroups — C22×C4⋊C8×D53 #28247698

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄⋊C₈×D₅₃ #28247698