Subgroups — C₂²×#16810525 #27542444

Order 13568.

9534 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×D₂₁₂	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₄⋊Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂³×#11719921	C₂	1	4	yes	maximal, complemented
4	C₂×#16810525	C₂	1	24	yes	maximal, direct product factor
5	C₂³×C₄²×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂⁵×D₅₃	C₂²	1	2	yes	—
7	C₂³×D₂₁₂	—	2	8	no	—
8	C₂³×D₂₁₂	—	2	8	no	—
9	C₂³×D₂₁₂	C₄	1	16	yes	complemented
10	C₂³×D₂₁₂	C₂²	1	12	yes	—
11	C₂⁴×Dic₅₃	C₂²	1	2	yes	—
12	C₂²×C₄⋊Dic₅₃	—	2	8	no	—
13	C₂²×C₄⋊Dic₅₃	C₂²	1	12	yes	complemented
14	C₂²×#11719921	—	2	32	no	—
15	C₂²×#11719921	C₂²	1	48	yes	complemented
16	#16810525	C₂²	1	64	yes	direct product factor
17	C₂⁴×C₄×C₅₃	C₂²	1	2	yes	—
18	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
19	C₂²×C₄²×C₅₃	—	2	8	no	—
20	C₂²×C₄²×C₅₃	C₂²	1	12	yes	complemented
21	C₂⁴×D₅₃	—	2	16	no	—
22	C₂⁴×D₅₃	—	2	8	no	—
23	C₂⁴×D₅₃	—	2	8	no	—
24	C₂⁴×D₅₃	—	2	16	no	—
25	C₂⁴×D₅₃	C₂×C₄	1	16	yes	—
26	C₂⁴×D₅₃	C₂³	1	12	yes	—
27	C₂²×D₂₁₂	—	2	128	no	—
28	C₂²×D₂₁₂	—	2	48	no	—
29	C₂²×D₂₁₂	—	2	48	no	—
30	C₂²×D₂₁₂	C₂×C₄	1	96	yes	complemented
31	C₂²×D₂₁₂	C₂³	1	16	yes	—
32	C₂³×Dic₅₃	—	2	16	no	—
33	C₂³×Dic₅₃	—	2	8	no	—
34	C₂³×Dic₅₃	C₂³	1	12	yes	—
35	C₂×C₄⋊Dic₅₃	—	2	48	no	—
36	C₂×C₄⋊Dic₅₃	C₂³	1	16	yes	complemented
37	C₂×#11719921	—	2	192	no	—
38	C₂×#11719921	C₂³	1	64	yes	complemented
39	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
40	C₂³×C₄×C₅₃	—	2	8	no	—
41	C₂³×C₄×C₅₃	—	2	8	no	—
42	C₂³×C₄×C₅₃	C₂×C₄	1	8	yes	—
43	C₂³×C₄×C₅₃	D₄	1	8	yes	—
44	C₂³×C₄×C₅₃	D₄	1	8	yes	—
45	C₂³×C₄×C₅₃	D₄	1	16	yes	complemented
46	C₂³×C₄×C₅₃	C₂³	1	12	yes	—
47	C₂³×C₄×C₅₃	C₂³	1	6	yes	—
48	C₂×C₄²×C₅₃	—	2	48	no	—
49	C₂×C₄²×C₅₃	C₂³	1	16	yes	complemented
50	C₂⁴×C₅₃	C₂²×C₄	1	4	yes	—
51	C₂⁴×C₅₃	C₂×D₄	1	4	yes	—
52	C₂⁴×C₅₃	C₂×D₄	1	4	yes	—
53	C₂⁴×C₅₃	C₂×D₄	1	8	yes	—
54	C₂⁴×C₅₃	#41	1	8	yes	—
55	C₂⁴×C₅₃	C₂⁴	1	3	yes	—
56	C₂³×D₅₃	—	2	64	no	—
57	C₂³×D₅₃	—	2	128	no	—
58	C₂³×D₅₃	—	2	48	no	—
59	C₂³×D₅₃	—	2	24	no	—
60	C₂³×D₅₃	—	2	24	no	—
61	C₂³×D₅₃	—	2	48	no	—
62	C₂³×D₅₃	—	4	64	no	—
63	C₂³×D₅₃	—	4	32	no	—
64	C₂³×D₅₃	—	4	32	no	—
65	C₂³×D₅₃	C₂²×C₄	1	48	yes	—
66	C₂³×D₅₃	C₂⁴	1	8	yes	—
67	C₂²×C₄×C₅₃	—	2	64	no	—
68	C₂²×C₄×C₅₃	—	2	24	no	—
69	C₂²×C₄×C₅₃	—	2	24	no	—
70	C₂²×C₄×C₅₃	C₂²⋊C₄	1	64	yes	complemented
71	C₂²×C₄×C₅₃	C₂²×C₄	1	24	yes	—
72	C₂²×C₄×C₅₃	C₂×D₄	1	24	yes	—
73	C₂²×C₄×C₅₃	C₂×D₄	1	24	yes	—
74	C₂²×C₄×C₅₃	C₂×D₄	1	48	yes	complemented
75	C₂²×C₄×C₅₃	C₂⁴	1	8	yes	—
76	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
77	C₂²×Dic₅₃	—	2	48	no	—
78	C₂²×Dic₅₃	—	2	24	no	—
79	C₂²×Dic₅₃	—	4	32	no	—
80	C₂²×Dic₅₃	C₂⁴	1	8	yes	—
81	C₂×D₂₁₂	—	2	384	no	—
82	C₂×D₂₁₂	—	2	32	no	—
83	C₂×D₂₁₂	—	2	32	no	—
84	C₂×D₂₁₂	C₂²×C₄	1	64	yes	complemented
85	C₄²×C₅₃	—	2	32	no	—
86	#11719921	—	2	128	no	—
87	C₄⋊Dic₅₃	—	2	32	no	—
88	C₂³×C₅₃	C₂×C₂²⋊C₄	1	16	yes	—
89	C₂³×C₅₃	C₄×D₄	1	32	yes	—
90	C₂³×C₅₃	#121	1	32	yes	—
91	C₂³×C₅₃	#124	1	16	yes	—
92	C₂³×C₅₃	#127	1	16	yes	—
93	C₂³×C₅₃	C₂³×C₄	1	6	yes	—
94	C₂³×C₅₃	C₂²×D₄	1	6	yes	—
95	C₂³×C₅₃	C₂²×D₄	1	6	yes	—
96	C₂³×C₅₃	C₂²×D₄	1	12	yes	—
97	C₂³×C₅₃	C₂×#41	1	12	yes	—
98	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
99	C₂²×D₅₃	—	2	96	no	—
100	C₂²×D₅₃	—	2	192	no	—
101	C₂²×D₅₃	—	2	8	no	—
102	C₂²×D₅₃	—	2	8	no	—
103	C₂²×D₅₃	—	2	16	no	—
104	C₂²×D₅₃	—	2	16	no	—
105	C₂²×D₅₃	—	4	96	no	—
106	C₂²×D₅₃	—	4	48	no	—
107	C₂²×D₅₃	—	4	48	no	—
108	C₂²×D₅₃	—	4	256	no	—
109	C₂²×D₅₃	C₂³×C₄	1	16	yes	—
110	C₂×C₄×C₅₃	—	2	96	no	—
111	C₂×C₄×C₅₃	—	2	8	no	—
112	C₂×C₄×C₅₃	—	2	8	no	—
113	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	96	yes	complemented
114	C₂×C₄×C₅₃	C₂³×C₄	1	8	yes	—
115	C₂×C₄×C₅₃	C₂²×D₄	1	8	yes	—
116	C₂×C₄×C₅₃	C₂²×D₄	1	8	yes	—
117	C₂×C₄×C₅₃	C₂²×D₄	1	16	yes	complemented
118	C₂×Dic₅₃	—	2	8	no	—
119	C₂×Dic₅₃	—	2	16	no	—
120	C₂×Dic₅₃	—	4	48	no	—
121	D₂₁₂	—	2	128	no	—
122	C₂²×#390	—	53	1	no	maximal, Sylow
123	C₂²×C₅₃	#390	1	64	yes	complemented
124	C₂²×C₅₃	C₂²×C₂²⋊C₄	1	12	yes	—
125	C₂²×C₅₃	C₂×C₄×D₄	1	24	yes	—
126	C₂²×C₅₃	C₂×#121	1	24	yes	—
127	C₂²×C₅₃	C₂×#124	1	12	yes	—
128	C₂²×C₅₃	C₂×#127	1	12	yes	—
129	C₂²×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
130	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
131	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
132	C₂²×C₅₃	C₂³×D₄	1	2	yes	—
133	C₂²×C₅₃	C₂²×#41	1	2	yes	—
134	C₂×D₅₃	—	2	16	no	—
135	C₂×D₅₃	—	2	32	no	—
136	C₂×D₅₃	—	4	16	no	—
137	C₂×D₅₃	—	4	8	no	—
138	C₂×D₅₃	—	4	8	no	—
139	C₂×D₅₃	—	4	192	no	—
140	C₄×C₅₃	—	2	16	no	—
141	C₄×C₅₃	C₂²×C₂²⋊C₄	1	16	yes	complemented
142	Dic₅₃	—	4	8	no	—
143	C₂⁴×D₄	—	53	1	no	—
144	C₂³×C₄⋊C₄	—	53	1	no	—
145	C₂³×C₂²⋊C₄	—	53	4	no	—
146	C₂³×C₄²	D₅₃	1	1	yes	characteristic, complemented
147	C₂×#390	—	53	24	no	—
148	C₂×C₅₃	C₂×#390	1	24	yes	complemented
149	C₂×C₅₃	C₂³×C₂²⋊C₄	1	1	yes	characteristic
150	C₂×C₅₃	C₂²×C₄×D₄	1	2	yes	—
151	C₂×C₅₃	C₂²×#121	1	2	yes	—
152	C₂×C₅₃	C₂²×#124	1	1	yes	characteristic
153	C₂×C₅₃	C₂²×#127	1	1	yes	characteristic
154	D₅₃	—	4	16	no	—
155	C₂⁶	—	53	2	no	—
156	C₂³×D₄	—	53	16	no	—
157	C₂³×D₄	—	53	12	no	—
158	C₂³×D₄	—	106	8	no	—
159	C₂³×D₄	—	106	8	no	—
160	C₂⁴×C₄	—	53	2	no	—
161	C₂⁴×C₄	C₂×D₅₃	1	2	yes	—
162	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic
163	C₂²×C₄⋊C₄	—	53	12	no	—
164	C₂²×C₄⋊C₄	—	106	8	no	—
165	C₂²×C₂²⋊C₄	—	53	48	no	—
166	C₂²×C₂²⋊C₄	—	106	32	no	—
167	C₂²×C₄²	—	2	8	no	—
168	C₂²×C₄²	C₂×D₅₃	1	12	yes	complemented
169	#390	—	53	64	no	—
170	C₅₃	C₂²×#390	1	1	yes	characteristic, complemented, Sylow
171	C₂⁵	—	53	16	no	—
172	C₂⁵	—	53	12	no	—
173	C₂⁵	—	106	16	no	—
174	C₂⁵	—	106	8	no	—
175	C₂⁵	—	106	16	no	—
176	C₂⁵	—	106	8	no	—
177	C₂⁵	C₂²×D₅₃	1	1	yes	characteristic
178	C₂²×D₄	—	53	96	no	—
179	C₂²×D₄	—	53	16	no	—
180	C₂²×D₄	—	106	48	no	—
181	C₂²×D₄	—	106	48	no	—
182	C₂²×D₄	—	106	128	no	—
183	C₂³×C₄	—	2	8	no	—
184	C₂³×C₄	—	2	8	no	—
185	C₂³×C₄	—	53	12	no	—
186	C₂³×C₄	—	106	16	no	—
187	C₂³×C₄	—	106	8	no	—
188	C₂³×C₄	C₄×D₅₃	1	8	yes	—
189	C₂³×C₄	D₂₁₂	1	8	yes	—
190	C₂³×C₄	D₂₁₂	1	16	yes	complemented
191	C₂³×C₄	C₂²⋊D₅₃	1	8	yes	—
192	C₂³×C₄	C₂²×D₅₃	1	12	yes	—
193	C₂³×C₄	C₂²×D₅₃	1	6	yes	—
194	C₂×C₄⋊C₄	—	53	16	no	—
195	C₂×C₄⋊C₄	—	106	48	no	—
196	C₂×C₂²⋊C₄	—	53	64	no	—
197	C₂×C₂²⋊C₄	—	106	192	no	—
198	C₂×C₄²	—	2	48	no	—
199	C₂×C₄²	C₂²×D₅₃	1	16	yes	complemented
200	C₂⁴	—	53	48	no	—
201	C₂⁴	—	53	8	no	—
202	C₂⁴	—	106	64	no	—
203	C₂⁴	—	106	128	no	—
204	C₂⁴	—	106	24	no	—
205	C₂⁴	—	106	48	no	—
206	C₂⁴	—	106	24	no	—
207	C₂⁴	—	106	48	no	—
208	C₂⁴	—	212	64	no	—
209	C₂⁴	—	212	32	no	—
210	C₂⁴	—	212	32	no	—
211	C₂⁴	C₂×C₄×D₅₃	1	4	yes	—
212	C₂⁴	C₂×D₂₁₂	1	4	yes	—
213	C₂⁴	C₂×D₂₁₂	1	8	yes	—
214	C₂⁴	#11719945	1	8	yes	—
215	C₂⁴	C₂×C₂²⋊D₅₃	1	4	yes	—
216	C₂⁴	C₂³×D₅₃	1	3	yes	—
217	C₂×D₄	—	53	64	no	—
218	C₂×D₄	—	106	384	no	—
219	C₂×D₄	—	106	32	no	—
220	C₂×D₄	—	106	32	no	—
221	C₂²×C₄	—	2	64	no	—
222	C₂²×C₄	—	2	24	no	—
223	C₂²×C₄	—	2	24	no	—
224	C₂²×C₄	—	53	8	no	—
225	C₂²×C₄	—	106	48	no	—
226	C₂²×C₄	—	106	24	no	—
227	C₂²×C₄	—	212	32	no	—
228	C₂²×C₄	#11719921	1	64	yes	complemented
229	C₂²×C₄	C₂×C₄×D₅₃	1	24	yes	—
230	C₂²×C₄	C₂×D₂₁₂	1	24	yes	—
231	C₂²×C₄	C₂×D₂₁₂	1	48	yes	complemented
232	C₂²×C₄	C₂×C₂²⋊D₅₃	1	24	yes	—
233	C₂²×C₄	C₂³×D₅₃	1	8	yes	—
234	C₂²×C₄	C₂³×D₅₃	1	4	yes	—
235	C₄⋊C₄	—	106	32	no	—
236	C₂²⋊C₄	—	106	128	no	—
237	C₄²	—	2	32	no	—
238	C₂³	—	53	16	no	—
239	C₂³	—	106	96	no	—
240	C₂³	—	106	192	no	—
241	C₂³	—	106	8	no	—
242	C₂³	—	106	16	no	—
243	C₂³	—	106	8	no	—
244	C₂³	—	106	16	no	—
245	C₂³	—	212	256	no	—
246	C₂³	—	212	96	no	—
247	C₂³	—	212	48	no	—
248	C₂³	—	212	48	no	—
249	C₂³	C₂×#11719921	1	16	yes	—
250	C₂³	C₄×D₂₁₂	1	32	yes	—
251	C₂³	#13133191	1	32	yes	—
252	C₂³	#13133205	1	16	yes	—
253	C₂³	#13133214	1	16	yes	—
254	C₂³	C₂²×C₄×D₅₃	1	6	yes	—
255	C₂³	C₂²×C₂²⋊D₅₃	1	6	yes	—
256	C₂³	C₂²×D₂₁₂	1	6	yes	—
257	C₂³	C₂²×D₂₁₂	1	12	yes	—
258	C₂³	C₂×#11719945	1	12	yes	—
259	C₂³	C₂⁴×D₅₃	1	1	yes	characteristic
260	D₄	—	106	128	no	—
261	C₂×C₄	—	2	96	no	—
262	C₂×C₄	—	2	8	no	—
263	C₂×C₄	—	2	8	no	—
264	C₂×C₄	—	106	16	no	—
265	C₂×C₄	—	106	8	no	—
266	C₂×C₄	—	212	48	no	—
267	C₂×C₄	C₂×#11719921	1	96	yes	complemented
268	C₂×C₄	C₂²×C₄×D₅₃	1	8	yes	—
269	C₂×C₄	C₂²×C₂²⋊D₅₃	1	8	yes	—
270	C₂×C₄	C₂²×D₂₁₂	1	8	yes	—
271	C₂×C₄	C₂²×D₂₁₂	1	16	yes	complemented
272	C₂²	—	106	16	no	—
273	C₂²	—	106	32	no	—
274	C₂²	—	212	192	no	—
275	C₂²	—	212	16	no	—
276	C₂²	—	212	8	no	—
277	C₂²	—	212	8	no	—
278	C₂²	#16810525	1	64	yes	direct product factor
279	C₂²	C₂²×#11719921	1	12	yes	—
280	C₂²	C₂×C₄×D₂₁₂	1	24	yes	—
281	C₂²	C₂×#13133191	1	24	yes	—
282	C₂²	C₂×#13133205	1	12	yes	—
283	C₂²	C₂×#13133214	1	12	yes	—
284	C₂²	C₂³×C₄×D₅₃	1	1	yes	characteristic
285	C₂²	C₂³×C₂²⋊D₅₃	1	1	yes	characteristic
286	C₂²	C₂³×D₂₁₂	1	1	yes	characteristic
287	C₂²	C₂³×D₂₁₂	1	2	yes	—
288	C₂²	C₂²×#11719945	1	2	yes	—
289	C₄	—	2	16	no	—
290	C₄	—	212	8	no	—
291	C₄	C₂²×#11719921	1	16	yes	complemented
292	C₂	—	212	16	no	—
293	C₂	C₂×#16810525	1	24	yes	direct product factor
294	C₂	C₂³×#11719921	1	1	yes	characteristic
295	C₂	C₂²×C₄×D₂₁₂	1	2	yes	—
296	C₂	C₂²×#13133191	1	2	yes	—
297	C₂	C₂²×#13133205	1	1	yes	characteristic
298	C₂	C₂²×#13133214	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂⁴×C₄
Frattini	C₂³	C₂⁴×D₅₃
Fitting	C₂³×C₄²×C₅₃	C₂
socle	C₂⁵×C₅₃	C₂³
prime radical	C₂⁴×D₂₁₂	C₂
cosocle	C₂³×C₅₃	C₂⁵
hypercenter	C₂³×C₄²	D₅₃
hypocenter	C₅₃	C₂²×#390
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁵ ◁ C₂³×C₄²

Complements

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

N	H ≅ G/N	type
2. C₂³×C₄⋊Dic₅₃	C₂	semidirect
3. C₂³×#11719921	C₂	semidirect
4. C₂×#16810525	C₂	direct (N × H)
5. C₂³×C₄²×C₅₃	C₂	semidirect
9. C₂³×D₂₁₂	C₄	semidirect
13. C₂²×C₄⋊Dic₅₃	C₂²	semidirect
15. C₂²×#11719921	C₂²	semidirect
16. #16810525	C₂²	direct (N × H)
20. C₂²×C₄²×C₅₃	C₂²	semidirect
30. C₂²×D₂₁₂	C₂×C₄	semidirect
36. C₂×C₄⋊Dic₅₃	C₂³	semidirect
38. C₂×#11719921	C₂³	semidirect
45. C₂³×C₄×C₅₃	D₄	semidirect
49. C₂×C₄²×C₅₃	C₂³	semidirect
70. C₂²×C₄×C₅₃	C₂²⋊C₄	semidirect
74. C₂²×C₄×C₅₃	C₂×D₄	semidirect
84. C₂×D₂₁₂	C₂²×C₄	semidirect
113. C₂×C₄×C₅₃	C₂×C₂²⋊C₄	semidirect
117. C₂×C₄×C₅₃	C₂²×D₄	semidirect
123. C₂²×C₅₃	#390	semidirect
141. C₄×C₅₃	C₂²×C₂²⋊C₄	semidirect
146. C₂³×C₄²	D₅₃	semidirect
148. C₂×C₅₃	C₂×#390	semidirect
168. C₂²×C₄²	C₂×D₅₃	semidirect
170. C₅₃	C₂²×#390	semidirect
190. C₂³×C₄	D₂₁₂	semidirect
199. C₂×C₄²	C₂²×D₅₃	semidirect
228. C₂²×C₄	#11719921	semidirect
231. C₂²×C₄	C₂×D₂₁₂	semidirect
267. C₂×C₄	C₂×#11719921	semidirect
271. C₂×C₄	C₂²×D₂₁₂	semidirect
278. C₂²	#16810525	direct (N × H)
291. C₄	C₂²×#11719921	semidirect
293. C₂	C₂×#16810525	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂⁴×D₂₁₂	2
C₂³×C₄⋊Dic₅₃	2
C₂³×#11719921	2
C₂×#16810525	2
C₂³×C₄²×C₅₃	2
C₂²×#390	53

Subgroups — C22×#16810525 #27542444

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16810525 #27542444