Subgroups — C₂²×#108×D₅₃ #28247779

Order 13568.

2846 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×M₁₆×D₅₃	C₂	1	2	yes	maximal
2	C₂³×C₈×D₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂²×#13133153	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×#13133152	C₂	1	2	yes	maximal, complemented
5	C₂×#108×D₅₃	C₂	1	24	yes	maximal, direct product factor
6	C₂²×#108×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×C₄×D₅₃	C₂²	1	1	yes	characteristic
8	C₂²×#11719917	C₂²	1	2	yes	—
9	C₂²×#11719912	—	2	2	no	—
10	C₂×M₁₆×D₅₃	—	2	12	no	—
11	C₂×M₁₆×D₅₃	C₂²	1	24	yes	—
12	C₂³×#88322	C₂²	1	1	yes	characteristic
13	C₂²×C₈×D₅₃	—	2	2	no	—
14	C₂²×C₈×D₅₃	C₄	1	12	yes	—
15	C₂²×C₈×D₅₃	C₄	1	2	yes	—
16	C₂²×C₈×D₅₃	C₄	1	2	yes	—
17	C₂²×C₈×D₅₃	C₂²	1	12	yes	—
18	C₂×#13133153	C₂²	1	12	yes	complemented
19	C₂×#13133152	C₂²	1	24	yes	complemented
20	#108×D₅₃	C₂²	1	64	yes	direct product factor
21	C₂²×M₁₆×C₅₃	C₂²	1	2	yes	—
22	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×#108×C₅₃	C₂²	1	12	yes	complemented
24	C₂⁴×D₅₃	Q₈	1	1	yes	characteristic
25	C₂³×Dic₅₃	Q₈	1	1	yes	characteristic
26	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×D₅₃	C₂×C₄	1	6	yes	—
28	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×D₅₃	D₄	1	1	yes	characteristic
30	C₂²×C₄×D₅₃	D₄	1	6	yes	—
31	C₂²×C₄×D₅₃	D₄	1	1	yes	characteristic
32	C₂²×C₄×D₅₃	Q₈	1	6	yes	—
33	C₂²×C₄×D₅₃	C₂³	1	6	yes	—
34	C₂×#11719917	—	2	6	no	—
35	C₂×#11719917	C₂³	1	12	yes	—
36	C₂×#11719912	—	2	24	no	—
37	M₁₆×D₅₃	—	2	48	no	—
38	M₁₆×D₅₃	C₂³	1	32	yes	—
39	C₂²×#88322	—	2	2	no	—
40	C₂²×#88322	C₂×C₄	1	6	yes	—
41	C₂²×#88322	C₂×C₄	1	2	yes	—
42	C₂²×#88322	C₂³	1	6	yes	—
43	C₂×C₈×D₅₃	—	2	24	no	—
44	C₂×C₈×D₅₃	C₂×C₄	1	24	yes	—
45	C₂×C₈×D₅₃	C₂×C₄	1	48	yes	—
46	C₂×C₈×D₅₃	C₂×C₄	1	24	yes	—
47	C₂×C₈×D₅₃	C₂³	1	16	yes	—
48	#13133153	C₂³	1	16	yes	complemented
49	#13133152	C₂³	1	32	yes	complemented
50	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
51	C₂×M₁₆×C₅₃	—	2	6	no	—
52	C₂×M₁₆×C₅₃	C₂³	1	12	yes	—
53	C₂²×C₈×C₅₃	—	2	2	no	—
54	C₂²×C₈×C₅₃	C₂×C₄	1	6	yes	—
55	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
56	C₂²×C₈×C₅₃	C₂³	1	6	yes	—
57	#108×C₅₃	C₂³	1	16	yes	complemented
58	C₂⁴×C₅₃	C₂×Q₈	1	1	yes	characteristic
59	C₂³×D₅₃	—	2	8	no	—
60	C₂³×D₅₃	C₄⋊C₄	1	1	yes	characteristic
61	C₂³×D₅₃	C₄⋊C₄	1	6	yes	—
62	C₂³×D₅₃	C₄⋊C₄	1	1	yes	characteristic
63	C₂³×D₅₃	C₂×Q₈	1	6	yes	—
64	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
65	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
67	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	C₂×Q₈	1	3	yes	—
69	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
70	C₂²×Dic₅₃	C₄⋊C₄	1	6	yes	—
71	C₂²×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
72	C₂²×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
73	C₂²×Dic₅₃	C₂×Q₈	1	6	yes	—
74	C₂×C₄×D₅₃	C₄⋊C₄	1	6	yes	—
75	C₂×C₄×D₅₃	C₄⋊C₄	1	24	yes	—
76	C₂×C₄×D₅₃	C₄⋊C₄	1	6	yes	—
77	C₂×C₄×D₅₃	C₄⋊C₄	1	6	yes	—
78	C₂×C₄×D₅₃	C₄⋊C₄	1	6	yes	—
79	C₂×C₄×D₅₃	C₂²×C₄	1	6	yes	—
80	C₂×C₄×D₅₃	C₂²×C₄	1	12	yes	—
81	C₂×C₄×D₅₃	C₂²×C₄	1	6	yes	—
82	C₂×C₄×D₅₃	C₂×D₄	1	6	yes	—
83	C₂×C₄×D₅₃	C₂×D₄	1	12	yes	—
84	C₂×C₄×D₅₃	C₂×D₄	1	6	yes	—
85	C₂×C₄×D₅₃	C₂×Q₈	1	12	yes	—
86	C₂×C₄×D₅₃	C₂⁴	1	4	yes	—
87	M₁₆×C₅₃	—	2	12	no	—
88	M₁₆×C₅₃	C₂⁴	1	8	yes	—
89	C₂×C₈×C₅₃	—	2	12	no	—
90	C₂×C₈×C₅₃	C₂²×C₄	1	12	yes	—
91	C₂×C₈×C₅₃	C₂²×C₄	1	12	yes	—
92	C₂×C₈×C₅₃	C₂⁴	1	4	yes	—
93	#11719917	—	2	12	no	—
94	#11719917	C₂⁴	1	8	yes	—
95	C₂×#88322	—	2	12	no	—
96	C₂×#88322	C₂²×C₄	1	12	yes	—
97	C₂×#88322	C₂²×C₄	1	12	yes	—
98	C₂×#88322	C₂⁴	1	4	yes	—
99	#11719912	—	2	32	no	—
100	C₈×D₅₃	—	2	32	no	—
101	C₈×D₅₃	C₂²×C₄	1	32	yes	—
102	C₈×D₅₃	C₂²×C₄	1	32	yes	—
103	C₂³×C₅₃	—	2	4	no	—
104	C₂³×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
105	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₂³×C₅₃	C₂²×Q₈	1	3	yes	—
107	C₂²×D₅₃	—	2	4	no	—
108	C₂²×D₅₃	—	2	24	no	—
109	C₂²×D₅₃	—	2	24	no	—
110	C₂²×D₅₃	#108	1	8	yes	direct product factor
111	C₂²×D₅₃	C₂×C₄⋊C₄	1	6	yes	—
112	C₂²×D₅₃	C₂×C₄⋊C₄	1	12	yes	—
113	C₂²×D₅₃	C₂×C₄⋊C₄	1	6	yes	—
114	C₂²×D₅₃	C₂²×Q₈	1	4	yes	—
115	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	6	yes	—
116	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
117	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
118	C₂×C₄×C₅₃	C₂³×C₄	1	3	yes	—
119	C₂×C₄×C₅₃	C₂³×C₄	1	3	yes	—
120	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
121	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
122	C₂×C₄×C₅₃	C₂²×Q₈	1	3	yes	—
123	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
124	C₂×Dic₅₃	C₂×C₄⋊C₄	1	12	yes	—
125	C₂×Dic₅₃	C₂×C₄⋊C₄	1	6	yes	—
126	C₂×Dic₅₃	C₂×C₄⋊C₄	1	6	yes	—
127	C₂×Dic₅₃	C₂²×Q₈	1	4	yes	—
128	C₄×D₅₃	C₂×C₄⋊C₄	1	12	yes	—
129	C₄×D₅₃	C₂×C₄⋊C₄	1	12	yes	—
130	C₄×D₅₃	C₂×C₄⋊C₄	1	12	yes	—
131	C₄×D₅₃	C₂×C₄⋊C₄	1	12	yes	—
132	C₄×D₅₃	C₂³×C₄	1	4	yes	—
133	C₄×D₅₃	C₂³×C₄	1	4	yes	—
134	C₄×D₅₃	C₂²×D₄	1	4	yes	—
135	C₄×D₅₃	C₂²×D₄	1	4	yes	—
136	C₈×C₅₃	—	2	8	no	—
137	C₈×C₅₃	C₂³×C₄	1	8	yes	—
138	#88322	—	2	8	no	—
139	#88322	C₂³×C₄	1	8	yes	—
140	C₂³×#108	—	53	1	no	maximal, Sylow
141	C₂²×C₅₃	—	2	6	no	—
142	C₂²×C₅₃	—	2	6	no	—
143	C₂²×C₅₃	C₂×#108	1	4	yes	complemented
144	C₂²×C₅₃	C₂²×C₄⋊C₄	1	3	yes	—
145	C₂²×C₅₃	C₂²×C₄⋊C₄	1	3	yes	—
146	C₂²×C₅₃	C₂³×Q₈	1	1	yes	characteristic
147	C₂×D₅₃	—	2	12	no	—
148	C₂×D₅₃	—	2	24	no	—
149	C₂×D₅₃	—	2	8	no	—
150	C₂×D₅₃	C₂×#108	1	24	yes	direct product factor
151	C₂×D₅₃	C₂²×C₄⋊C₄	1	4	yes	—
152	C₂×D₅₃	C₂²×C₄⋊C₄	1	4	yes	—
153	C₄×C₅₃	C₂²×C₄⋊C₄	1	3	yes	—
154	C₄×C₅₃	C₂²×C₄⋊C₄	1	3	yes	—
155	C₄×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
156	C₄×C₅₃	C₂³×D₄	1	1	yes	characteristic
157	Dic₅₃	C₂²×C₄⋊C₄	1	4	yes	—
158	Dic₅₃	C₂²×C₄⋊C₄	1	4	yes	—
159	C₂³×M₁₆	—	53	2	no	—
160	C₂⁴×C₈	—	53	1	no	—
161	C₂²×#108	—	53	24	no	—
162	C₂²×#108	—	53	2	no	—
163	C₂²×#108	—	53	1	no	—
164	C₂²×#108	D₅₃	1	1	yes	characteristic, direct product factor
165	C₂×C₅₃	—	2	3	no	—
166	C₂×C₅₃	—	2	1	no	—
167	C₂×C₅₃	C₂²×#108	1	6	yes	complemented
168	C₂×C₅₃	C₂³×C₄⋊C₄	1	1	yes	characteristic
169	D₅₃	—	2	4	no	—
170	D₅₃	C₂²×#108	1	8	yes	direct product factor
171	C₂⁴×C₄	—	53	1	no	—
172	C₂²×M₁₆	—	53	2	no	—
173	C₂²×M₁₆	—	53	24	no	—
174	C₂²×M₁₆	—	106	2	no	—
175	C₂²×M₁₆	—	106	12	no	—
176	C₂²×M₁₆	C₂×D₅₃	1	2	yes	—
177	C₂³×C₈	—	53	1	no	—
178	C₂³×C₈	—	53	2	no	—
179	C₂³×C₈	—	53	2	no	—
180	C₂³×C₈	—	53	12	no	—
181	C₂³×C₈	—	53	12	no	—
182	C₂³×C₈	—	106	2	no	—
183	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
184	C₂×#108	—	53	24	no	—
185	C₂×#108	—	53	12	no	—
186	C₂×#108	—	53	64	no	—
187	C₂×#108	C₂×D₅₃	1	12	yes	direct product factor
188	C₅₃	C₂³×#108	1	1	yes	characteristic, complemented, Sylow
189	C₂⁵	—	53	1	no	—
190	C₂³×C₄	—	53	1	no	—
191	C₂³×C₄	—	53	1	no	—
192	C₂³×C₄	—	53	6	no	—
193	C₂³×C₄	—	53	1	no	—
194	C₂³×C₄	—	53	6	no	—
195	C₂³×C₄	—	53	1	no	—
196	C₂³×C₄	—	53	1	no	—
197	C₂³×C₄	—	53	6	no	—
198	C₂³×C₄	—	53	6	no	—
199	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
200	C₂×M₁₆	—	2	6	no	—
201	C₂×M₁₆	—	53	12	no	—
202	C₂×M₁₆	—	53	32	no	—
203	C₂×M₁₆	—	106	24	no	—
204	C₂×M₁₆	—	106	48	no	—
205	C₂×M₁₆	—	106	6	no	—
206	C₂×M₁₆	C₂²×D₅₃	1	12	yes	—
207	C₂²×C₈	—	2	2	no	—
208	C₂²×C₈	—	53	2	no	—
209	C₂²×C₈	—	53	6	no	—
210	C₂²×C₈	—	53	6	no	—
211	C₂²×C₈	—	53	24	no	—
212	C₂²×C₈	—	53	48	no	—
213	C₂²×C₈	—	53	24	no	—
214	C₂²×C₈	—	53	16	no	—
215	C₂²×C₈	—	106	24	no	—
216	C₂²×C₈	—	106	2	no	—
217	C₂²×C₈	C₄×D₅₃	1	2	yes	—
218	C₂²×C₈	C₄×D₅₃	1	6	yes	—
219	C₂²×C₈	C₂²×D₅₃	1	6	yes	—
220	#108	—	53	32	no	—
221	#108	—	53	16	no	—
222	#108	C₂²×D₅₃	1	16	yes	direct product factor
223	C₂⁴	—	53	1	no	—
224	C₂⁴	—	53	6	no	—
225	C₂⁴	—	53	1	no	—
226	C₂⁴	—	53	6	no	—
227	C₂⁴	—	106	8	no	—
228	C₂⁴	Q₈×D₅₃	1	1	yes	characteristic
229	C₂²×C₄	—	53	6	no	—
230	C₂²×C₄	—	53	1	no	—
231	C₂²×C₄	—	53	1	no	—
232	C₂²×C₄	—	53	6	no	—
233	C₂²×C₄	—	53	6	no	—
234	C₂²×C₄	—	53	24	no	—
235	C₂²×C₄	—	53	6	no	—
236	C₂²×C₄	—	53	6	no	—
237	C₂²×C₄	—	53	6	no	—
238	C₂²×C₄	—	53	6	no	—
239	C₂²×C₄	—	53	12	no	—
240	C₂²×C₄	—	53	6	no	—
241	C₂²×C₄	—	53	6	no	—
242	C₂²×C₄	—	53	12	no	—
243	C₂²×C₄	—	53	6	no	—
244	C₂²×C₄	—	53	12	no	—
245	C₂²×C₄	—	53	4	no	—
246	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
247	C₂²×C₄	C₂×C₄×D₅₃	1	3	yes	—
248	C₂²×C₄	D₄×D₅₃	1	3	yes	—
249	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
250	C₂²×C₄	Q₈×D₅₃	1	3	yes	—
251	C₂²×C₄	C₂³×D₅₃	1	3	yes	—
252	M₁₆	—	2	12	no	—
253	M₁₆	—	53	8	no	—
254	M₁₆	—	106	32	no	—
255	M₁₆	—	106	12	no	—
256	M₁₆	C₂³×D₅₃	1	8	yes	—
257	C₂×C₈	—	2	12	no	—
258	C₂×C₈	—	53	12	no	—
259	C₂×C₈	—	53	12	no	—
260	C₂×C₈	—	53	4	no	—
261	C₂×C₈	—	53	32	no	—
262	C₂×C₈	—	53	32	no	—
263	C₂×C₈	—	106	32	no	—
264	C₂×C₈	—	106	12	no	—
265	C₂×C₈	C₂×C₄×D₅₃	1	12	yes	—
266	C₂×C₈	C₂×C₄×D₅₃	1	12	yes	—
267	C₂×C₈	C₂³×D₅₃	1	4	yes	—
268	C₂³	—	2	4	no	—
269	C₂³	—	53	6	no	—
270	C₂³	—	53	8	no	—
271	C₂³	—	53	12	no	—
272	C₂³	—	53	6	no	—
273	C₂³	—	53	4	no	—
274	C₂³	—	106	24	no	—
275	C₂³	—	106	24	no	—
276	C₂³	—	106	4	no	—
277	C₂³	C₄⋊C₄×D₅₃	1	3	yes	—
278	C₂³	C₄⋊C₄×D₅₃	1	1	yes	characteristic
279	C₂³	C₂×Q₈×D₅₃	1	3	yes	—
280	C₂×C₄	—	53	12	no	—
281	C₂×C₄	—	53	6	no	—
282	C₂×C₄	—	53	6	no	—
283	C₂×C₄	—	53	4	no	—
284	C₂×C₄	—	53	12	no	—
285	C₂×C₄	—	53	12	no	—
286	C₂×C₄	—	53	12	no	—
287	C₂×C₄	—	53	12	no	—
288	C₂×C₄	—	53	4	no	—
289	C₂×C₄	—	53	4	no	—
290	C₂×C₄	—	53	4	no	—
291	C₂×C₄	—	53	4	no	—
292	C₂×C₄	C₄⋊C₄×D₅₃	1	6	yes	—
293	C₂×C₄	C₄⋊C₄×D₅₃	1	3	yes	—
294	C₂×C₄	C₄⋊C₄×D₅₃	1	3	yes	—
295	C₂×C₄	C₂²×C₄×D₅₃	1	3	yes	—
296	C₂×C₄	C₂²×C₄×D₅₃	1	3	yes	—
297	C₂×C₄	C₂×D₄×D₅₃	1	3	yes	—
298	C₂×C₄	C₂×D₄×D₅₃	1	3	yes	—
299	C₂×C₄	C₂×Q₈×D₅₃	1	3	yes	—
300	C₂×C₄	C₂⁴×D₅₃	1	1	yes	characteristic
301	C₈	—	2	8	no	—
302	C₈	—	53	8	no	—
303	C₈	—	106	8	no	—
304	C₈	C₂²×C₄×D₅₃	1	8	yes	—
305	C₂²	—	2	6	no	—
306	C₂²	—	2	6	no	—
307	C₂²	—	53	4	no	—
308	C₂²	—	53	24	no	—
309	C₂²	—	53	4	no	—
310	C₂²	—	106	24	no	—
311	C₂²	—	106	8	no	—
312	C₂²	—	106	12	no	—
313	C₂²	#108×D₅₃	1	4	yes	direct product factor
314	C₂²	C₂×C₄⋊C₄×D₅₃	1	3	yes	—
315	C₂²	C₂×C₄⋊C₄×D₅₃	1	3	yes	—
316	C₂²	C₂²×Q₈×D₅₃	1	1	yes	characteristic
317	C₄	—	53	4	no	—
318	C₄	—	53	4	no	—
319	C₄	C₂×C₄⋊C₄×D₅₃	1	3	yes	—
320	C₄	C₂×C₄⋊C₄×D₅₃	1	3	yes	—
321	C₄	C₂³×C₄×D₅₃	1	1	yes	characteristic
322	C₄	C₂²×D₄×D₅₃	1	1	yes	characteristic
323	C₂	—	2	3	no	—
324	C₂	—	2	1	no	—
325	C₂	—	53	8	no	—
326	C₂	—	106	4	no	—
327	C₂	C₂×#108×D₅₃	1	6	yes	direct product factor
328	C₂	C₂²×C₄⋊C₄×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₅₃
commutator	C₄×C₅₃	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₅₃
Fitting	C₂²×#108×C₅₃	C₂
socle	C₂³×C₅₃	C₂×C₄⋊C₄
prime radical	C₂⁴×D₅₃	Q₈
cosocle	C₂×C₄×C₅₃	C₂⁵
hypercenter	C₂²×#108	D₅₃
hypocenter	C₅₃	C₂³×#108
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂²×#108

Complements

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

N	H ≅ G/N	type
3. C₂²×#13133153	C₂	semidirect
4. C₂²×#13133152	C₂	semidirect
5. C₂×#108×D₅₃	C₂	direct (N × H)
6. C₂²×#108×C₅₃	C₂	semidirect
18. C₂×#13133153	C₂²	semidirect
19. C₂×#13133152	C₂²	semidirect
20. #108×D₅₃	C₂²	direct (N × H)
23. C₂×#108×C₅₃	C₂²	semidirect
48. #13133153	C₂³	semidirect
49. #13133152	C₂³	semidirect
57. #108×C₅₃	C₂³	semidirect
110. C₂²×D₅₃	#108	direct (N × H)
143. C₂²×C₅₃	C₂×#108	semidirect
150. C₂×D₅₃	C₂×#108	direct (N × H)
164. C₂²×#108	D₅₃	direct (N × H)
167. C₂×C₅₃	C₂²×#108	semidirect
170. D₅₃	C₂²×#108	direct (N × H)
187. C₂×#108	C₂×D₅₃	direct (N × H)
188. C₅₃	C₂³×#108	semidirect
222. #108	C₂²×D₅₃	direct (N × H)
313. C₂²	#108×D₅₃	direct (N × H)
327. C₂	C₂×#108×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
C₂²×M₁₆×D₅₃	2
C₂³×C₈×D₅₃	2
C₂²×#13133153	2
C₂²×#13133152	2
C₂×#108×D₅₃	2
C₂²×#108×C₅₃	2
C₂³×#108	53

Subgroups — C22×#108×D53 #28247779

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#108×D₅₃ #28247779