Subgroups — C₂²×#16810483 #27542352

Order 13568.

8670 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄⋊Dic₅₃	C₂	1	2	yes	maximal, complemented
2	C₂³×C₂²⋊Dic₅₃	C₂	1	2	yes	maximal
3	C₂×#16810483	C₂	1	24	yes	maximal, direct product factor
4	C₂²×#13133124	C₂	1	2	yes	maximal, complemented
5	C₂⁵×C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂⁴×Dic₅₃	C₂²	1	4	yes	—
7	C₂²×C₄⋊Dic₅₃	—	2	16	no	—
8	C₂²×C₄⋊Dic₅₃	C₂²	1	24	yes	complemented
9	C₂²×C₂²⋊Dic₅₃	—	2	16	no	—
10	C₂²×C₂²⋊Dic₅₃	C₂²	1	24	yes	—
11	#16810483	C₂²	1	64	yes	direct product factor
12	C₂×#13133124	C₂²	1	24	yes	complemented
13	C₂⁶×C₅₃	C₂²	1	1	yes	characteristic
14	C₂⁴×C₄×C₅₃	—	2	16	no	—
15	C₂⁴×C₄×C₅₃	C₄	1	16	yes	complemented
16	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
17	C₂⁴×C₄×C₅₃	C₂²	1	12	yes	—
18	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
19	C₂³×Dic₅₃	—	2	16	no	—
20	C₂³×Dic₅₃	—	2	16	no	—
21	C₂³×Dic₅₃	—	2	16	no	—
22	C₂³×Dic₅₃	C₂³	1	24	yes	—
23	C₂×C₄⋊Dic₅₃	—	2	96	no	—
24	C₂×C₄⋊Dic₅₃	C₂³	1	32	yes	complemented
25	C₂×C₂²⋊Dic₅₃	—	2	96	no	—
26	C₂×C₂²⋊Dic₅₃	C₂³	1	32	yes	—
27	#13133124	C₂³	1	32	yes	complemented
28	C₂⁵×C₅₃	—	2	8	no	—
29	C₂⁵×C₅₃	—	2	8	no	—
30	C₂⁵×C₅₃	C₂×C₄	1	8	yes	—
31	C₂⁵×C₅₃	D₄	1	8	yes	—
32	C₂⁵×C₅₃	Q₈	1	8	yes	—
33	C₂⁵×C₅₃	C₂³	1	6	yes	—
34	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
35	C₂³×C₄×C₅₃	—	2	8	no	—
36	C₂³×C₄×C₅₃	—	2	128	no	—
37	C₂³×C₄×C₅₃	—	2	96	no	—
38	C₂³×C₄×C₅₃	C₂×C₄	1	8	yes	—
39	C₂³×C₄×C₅₃	C₂×C₄	1	8	yes	—
40	C₂³×C₄×C₅₃	C₂×C₄	1	96	yes	complemented
41	C₂³×C₄×C₅₃	D₄	1	16	yes	—
42	C₂³×C₄×C₅₃	C₂³	1	6	yes	—
43	C₂³×C₄×C₅₃	C₂³	1	16	yes	—
44	C₂³×C₄×C₅₃	C₂³	1	6	yes	—
45	C₂⁴×C₅₃	—	2	32	no	—
46	C₂⁴×C₅₃	—	2	32	no	—
47	C₂⁴×C₅₃	—	2	64	no	—
48	C₂⁴×C₅₃	—	2	64	no	—
49	C₂⁴×C₅₃	—	2	24	no	—
50	C₂⁴×C₅₃	—	2	24	no	—
51	C₂⁴×C₅₃	C₄⋊C₄	1	64	yes	complemented
52	C₂⁴×C₅₃	C₂²×C₄	1	24	yes	—
53	C₂⁴×C₅₃	C₂²×C₄	1	4	yes	—
54	C₂⁴×C₅₃	C₂×D₄	1	24	yes	—
55	C₂⁴×C₅₃	C₂×D₄	1	8	yes	—
56	C₂⁴×C₅₃	C₂×D₄	1	4	yes	—
57	C₂⁴×C₅₃	C₂×Q₈	1	24	yes	—
58	C₂⁴×C₅₃	C₂×Q₈	1	4	yes	—
59	C₂⁴×C₅₃	#41	1	8	yes	—
60	C₂⁴×C₅₃	C₂⁴	1	4	yes	—
61	C₂⁴×C₅₃	C₂⁴	1	3	yes	—
62	C₂²×C₄×C₅₃	—	2	32	no	—
63	C₂²×C₄×C₅₃	—	2	24	no	—
64	C₂²×C₄×C₅₃	—	2	384	no	—
65	C₂²×C₄×C₅₃	—	2	64	no	—
66	C₂²×C₄×C₅₃	C₂²⋊C₄	1	64	yes	complemented
67	C₂²×C₄×C₅₃	C₂²×C₄	1	24	yes	—
68	C₂²×C₄×C₅₃	C₂²×C₄	1	24	yes	—
69	C₂²×C₄×C₅₃	C₂²×C₄	1	64	yes	complemented
70	C₂²×C₄×C₅₃	C₂×D₄	1	48	yes	—
71	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
72	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
73	C₂²×Dic₅₃	—	2	48	no	—
74	C₂²×Dic₅₃	—	2	48	no	—
75	C₂²×Dic₅₃	—	2	48	no	—
76	C₂²×Dic₅₃	—	4	64	no	—
77	C₂²×Dic₅₃	C₂⁴	1	16	yes	—
78	C₂²⋊Dic₅₃	—	2	64	no	—
79	C₄⋊Dic₅₃	—	2	64	no	—
80	C₂³×C₅₃	—	2	48	no	—
81	C₂³×C₅₃	—	2	48	no	—
82	C₂³×C₅₃	—	2	96	no	—
83	C₂³×C₅₃	—	2	96	no	—
84	C₂³×C₅₃	—	2	8	no	—
85	C₂³×C₅₃	—	2	8	no	—
86	C₂³×C₅₃	—	2	256	no	—
87	C₂³×C₅₃	C₂×C₂²⋊C₄	1	16	yes	—
88	C₂³×C₅₃	C₂×C₄⋊C₄	1	16	yes	—
89	C₂³×C₅₃	C₂×C₄⋊C₄	1	96	yes	complemented
90	C₂³×C₅₃	#117	1	16	yes	—
91	C₂³×C₅₃	#121	1	32	yes	—
92	C₂³×C₅₃	C₂²⋊Q₈	1	32	yes	—
93	C₂³×C₅₃	C₂³×C₄	1	8	yes	—
94	C₂³×C₅₃	C₂³×C₄	1	6	yes	—
95	C₂³×C₅₃	C₂²×D₄	1	12	yes	—
96	C₂³×C₅₃	C₂²×D₄	1	8	yes	—
97	C₂³×C₅₃	C₂²×D₄	1	6	yes	—
98	C₂³×C₅₃	C₂²×Q₈	1	8	yes	—
99	C₂³×C₅₃	C₂²×Q₈	1	6	yes	—
100	C₂³×C₅₃	C₂×#41	1	12	yes	—
101	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
102	C₂×C₄×C₅₃	—	2	48	no	—
103	C₂×C₄×C₅₃	—	2	8	no	—
104	C₂×C₄×C₅₃	—	2	128	no	—
105	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	96	yes	complemented
106	C₂×C₄×C₅₃	C₂³×C₄	1	8	yes	—
107	C₂×C₄×C₅₃	C₂³×C₄	1	8	yes	—
108	C₂×C₄×C₅₃	C₂²×D₄	1	16	yes	—
109	C₂×Dic₅₃	—	2	16	no	—
110	C₂×Dic₅₃	—	2	16	no	—
111	C₂×Dic₅₃	—	2	16	no	—
112	C₂×Dic₅₃	—	4	96	no	—
113	C₂²×#380	—	53	1	no	maximal, Sylow
114	C₂²×C₅₃	—	2	8	no	—
115	C₂²×C₅₃	—	2	8	no	—
116	C₂²×C₅₃	—	2	16	no	—
117	C₂²×C₅₃	—	2	16	no	—
118	C₂²×C₅₃	—	2	192	no	—
119	C₂²×C₅₃	#380	1	64	yes	complemented
120	C₂²×C₅₃	C₂²×C₂²⋊C₄	1	12	yes	—
121	C₂²×C₅₃	C₂²×C₄⋊C₄	1	12	yes	—
122	C₂²×C₅₃	C₂²×C₄⋊C₄	1	16	yes	complemented
123	C₂²×C₅₃	C₂×#117	1	12	yes	—
124	C₂²×C₅₃	C₂×#121	1	24	yes	—
125	C₂²×C₅₃	C₂×C₂²⋊Q₈	1	24	yes	—
126	C₂²×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
127	C₂²×C₅₃	C₂³×D₄	1	2	yes	—
128	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
129	C₂²×C₅₃	C₂³×Q₈	1	1	yes	characteristic
130	C₂²×C₅₃	C₂²×#41	1	2	yes	—
131	C₄×C₅₃	—	2	8	no	—
132	C₄×C₅₃	C₂²×C₂²⋊C₄	1	16	yes	complemented
133	Dic₅₃	—	4	16	no	—
134	C₂⁵×C₄	D₅₃	1	1	yes	characteristic
135	C₂³×C₄⋊C₄	—	53	2	no	—
136	C₂³×C₂²⋊C₄	—	53	2	no	—
137	C₂×#380	—	53	24	no	—
138	C₂²×#95	—	53	2	no	—
139	C₂×C₅₃	—	2	16	no	—
140	C₂×C₅₃	C₂×#380	1	24	yes	complemented
141	C₂×C₅₃	C₂³×C₂²⋊C₄	1	1	yes	characteristic
142	C₂×C₅₃	C₂³×C₄⋊C₄	1	1	yes	characteristic
143	C₂×C₅₃	C₂²×#117	1	1	yes	characteristic
144	C₂×C₅₃	C₂²×#121	1	2	yes	—
145	C₂×C₅₃	C₂²×C₂²⋊Q₈	1	2	yes	—
146	C₂⁶	C₂×D₅₃	1	1	yes	characteristic
147	C₂⁴×C₄	—	2	16	no	—
148	C₂⁴×C₄	—	53	4	no	—
149	C₂⁴×C₄	Dic₅₃	1	16	yes	complemented
150	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic
151	C₂⁴×C₄	C₂×D₅₃	1	12	yes	—
152	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic
153	C₂²×C₄⋊C₄	—	53	24	no	—
154	C₂²×C₄⋊C₄	—	106	16	no	—
155	C₂²×C₂²⋊C₄	—	53	24	no	—
156	C₂²×C₂²⋊C₄	—	106	16	no	—
157	#380	—	53	64	no	—
158	C₂×#95	—	53	24	no	—
159	C₅₃	C₂²×#380	1	1	yes	characteristic, complemented, Sylow
160	C₂⁵	—	2	8	no	—
161	C₂⁵	—	2	8	no	—
162	C₂⁵	Dic₁₀₆	1	8	yes	—
163	C₂⁵	D₂₁₂	1	8	yes	—
164	C₂⁵	C₂×Dic₅₃	1	8	yes	—
165	C₂⁵	C₂²×D₅₃	1	6	yes	—
166	C₂⁵	C₂²×D₅₃	1	1	yes	characteristic
167	C₂³×C₄	—	2	8	no	—
168	C₂³×C₄	—	2	96	no	—
169	C₂³×C₄	—	2	128	no	—
170	C₂³×C₄	—	53	24	no	—
171	C₂³×C₄	—	106	16	no	—
172	C₂³×C₄	—	106	16	no	—
173	C₂³×C₄	—	106	16	no	—
174	C₂³×C₄	C₂×Dic₅₃	1	8	yes	—
175	C₂³×C₄	C₂×Dic₅₃	1	8	yes	—
176	C₂³×C₄	C₂×Dic₅₃	1	96	yes	complemented
177	C₂³×C₄	C₂²⋊D₅₃	1	16	yes	—
178	C₂³×C₄	C₂²×D₅₃	1	6	yes	—
179	C₂³×C₄	C₂²×D₅₃	1	16	yes	—
180	C₂³×C₄	C₂²×D₅₃	1	6	yes	—
181	C₂×C₄⋊C₄	—	53	32	no	—
182	C₂×C₄⋊C₄	—	106	96	no	—
183	C₂×C₂²⋊C₄	—	53	32	no	—
184	C₂×C₂²⋊C₄	—	106	96	no	—
185	#95	—	53	32	no	—
186	C₂⁴	—	2	32	no	—
187	C₂⁴	—	2	32	no	—
188	C₂⁴	—	2	64	no	—
189	C₂⁴	—	2	64	no	—
190	C₂⁴	—	2	24	no	—
191	C₂⁴	—	2	24	no	—
192	C₂⁴	C₄⋊Dic₅₃	1	64	yes	complemented
193	C₂⁴	C₂×Dic₁₀₆	1	24	yes	—
194	C₂⁴	C₂×Dic₁₀₆	1	4	yes	—
195	C₂⁴	C₂×D₂₁₂	1	24	yes	—
196	C₂⁴	C₂×D₂₁₂	1	4	yes	—
197	C₂⁴	#11719945	1	8	yes	—
198	C₂⁴	C₂²×Dic₅₃	1	24	yes	—
199	C₂⁴	C₂²×Dic₅₃	1	4	yes	—
200	C₂⁴	C₂×C₂²⋊D₅₃	1	8	yes	—
201	C₂⁴	C₂³×D₅₃	1	4	yes	—
202	C₂⁴	C₂³×D₅₃	1	3	yes	—
203	C₂²×C₄	—	2	32	no	—
204	C₂²×C₄	—	2	24	no	—
205	C₂²×C₄	—	2	384	no	—
206	C₂²×C₄	—	2	64	no	—
207	C₂²×C₄	—	53	16	no	—
208	C₂²×C₄	—	106	48	no	—
209	C₂²×C₄	—	106	48	no	—
210	C₂²×C₄	—	106	48	no	—
211	C₂²×C₄	—	212	64	no	—
212	C₂²×C₄	C₂²⋊Dic₅₃	1	64	yes	complemented
213	C₂²×C₄	C₂²×Dic₅₃	1	24	yes	—
214	C₂²×C₄	C₂²×Dic₅₃	1	24	yes	—
215	C₂²×C₄	C₂²×Dic₅₃	1	64	yes	complemented
216	C₂²×C₄	C₂×C₂²⋊D₅₃	1	48	yes	—
217	C₂²×C₄	C₂³×D₅₃	1	4	yes	—
218	C₂²×C₄	C₂³×D₅₃	1	4	yes	—
219	C₄⋊C₄	—	106	64	no	—
220	C₂²⋊C₄	—	106	64	no	—
221	C₂³	—	2	48	no	—
222	C₂³	—	2	48	no	—
223	C₂³	—	2	96	no	—
224	C₂³	—	2	96	no	—
225	C₂³	—	2	8	no	—
226	C₂³	—	2	8	no	—
227	C₂³	—	2	256	no	—
228	C₂³	C₂×C₂²⋊Dic₅₃	1	16	yes	—
229	C₂³	C₂×C₄⋊Dic₅₃	1	16	yes	—
230	C₂³	C₂×C₄⋊Dic₅₃	1	96	yes	complemented
231	C₂³	#13133176	1	16	yes	—
232	C₂³	#13133191	1	32	yes	—
233	C₂³	#13133196	1	32	yes	—
234	C₂³	C₂³×Dic₅₃	1	8	yes	—
235	C₂³	C₂³×Dic₅₃	1	6	yes	—
236	C₂³	C₂²×C₂²⋊D₅₃	1	12	yes	—
237	C₂³	C₂²×D₂₁₂	1	8	yes	—
238	C₂³	C₂²×D₂₁₂	1	6	yes	—
239	C₂³	C₂²×Dic₁₀₆	1	8	yes	—
240	C₂³	C₂²×Dic₁₀₆	1	6	yes	—
241	C₂³	C₂×#11719945	1	12	yes	—
242	C₂³	C₂⁴×D₅₃	1	1	yes	characteristic
243	C₂×C₄	—	2	48	no	—
244	C₂×C₄	—	2	8	no	—
245	C₂×C₄	—	2	128	no	—
246	C₂×C₄	—	106	16	no	—
247	C₂×C₄	—	106	16	no	—
248	C₂×C₄	—	106	16	no	—
249	C₂×C₄	—	212	96	no	—
250	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	96	yes	complemented
251	C₂×C₄	C₂³×Dic₅₃	1	8	yes	—
252	C₂×C₄	C₂³×Dic₅₃	1	8	yes	—
253	C₂×C₄	C₂²×C₂²⋊D₅₃	1	16	yes	—
254	C₂²	—	2	192	no	—
255	C₂²	—	2	8	no	—
256	C₂²	—	2	8	no	—
257	C₂²	—	2	16	no	—
258	C₂²	—	2	16	no	—
259	C₂²	#16810483	1	64	yes	direct product factor
260	C₂²	C₂²×C₂²⋊Dic₅₃	1	12	yes	—
261	C₂²	C₂²×C₄⋊Dic₅₃	1	12	yes	—
262	C₂²	C₂²×C₄⋊Dic₅₃	1	16	yes	complemented
263	C₂²	C₂×#13133176	1	12	yes	—
264	C₂²	C₂×#13133191	1	24	yes	—
265	C₂²	C₂×#13133196	1	24	yes	—
266	C₂²	C₂⁴×Dic₅₃	1	1	yes	characteristic
267	C₂²	C₂³×C₂²⋊D₅₃	1	2	yes	—
268	C₂²	C₂³×D₂₁₂	1	1	yes	characteristic
269	C₂²	C₂³×Dic₁₀₆	1	1	yes	characteristic
270	C₂²	C₂²×#11719945	1	2	yes	—
271	C₄	—	2	8	no	—
272	C₄	—	212	16	no	—
273	C₄	C₂²×C₂²⋊Dic₅₃	1	16	yes	complemented
274	C₂	—	2	16	no	—
275	C₂	C₂×#16810483	1	24	yes	direct product factor
276	C₂	C₂³×C₂²⋊Dic₅₃	1	1	yes	characteristic
277	C₂	C₂³×C₄⋊Dic₅₃	1	1	yes	characteristic
278	C₂	C₂²×#13133176	1	1	yes	characteristic
279	C₂	C₂²×#13133191	1	2	yes	—
280	C₂	C₂²×#13133196	1	2	yes	—

	subgroup	G/H
center	C₂⁵	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂⁴×C₄
Frattini	C₂³	C₂⁴×D₅₃
Fitting	C₂⁵×C₄×C₅₃	C₂
socle	C₂⁵×C₅₃	C₂³
prime radical	C₂⁶×C₅₃	C₂²
cosocle	C₂³×C₅₃	C₂⁵
hypercenter	C₂⁵×C₄	D₅₃
hypocenter	C₅₃	C₂²×#380
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁵ ◁ C₂⁵×C₄

Complements

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

N	H ≅ G/N	type
1. C₂³×C₄⋊Dic₅₃	C₂	semidirect
3. C₂×#16810483	C₂	direct (N × H)
4. C₂²×#13133124	C₂	semidirect
8. C₂²×C₄⋊Dic₅₃	C₂²	semidirect
11. #16810483	C₂²	direct (N × H)
12. C₂×#13133124	C₂²	semidirect
15. C₂⁴×C₄×C₅₃	C₄	semidirect
24. C₂×C₄⋊Dic₅₃	C₂³	semidirect
27. #13133124	C₂³	semidirect
40. C₂³×C₄×C₅₃	C₂×C₄	semidirect
51. C₂⁴×C₅₃	C₄⋊C₄	semidirect
66. C₂²×C₄×C₅₃	C₂²⋊C₄	semidirect
69. C₂²×C₄×C₅₃	C₂²×C₄	semidirect
89. C₂³×C₅₃	C₂×C₄⋊C₄	semidirect
105. C₂×C₄×C₅₃	C₂×C₂²⋊C₄	semidirect
119. C₂²×C₅₃	#380	semidirect
122. C₂²×C₅₃	C₂²×C₄⋊C₄	semidirect
132. C₄×C₅₃	C₂²×C₂²⋊C₄	semidirect
140. C₂×C₅₃	C₂×#380	semidirect
149. C₂⁴×C₄	Dic₅₃	semidirect
159. C₅₃	C₂²×#380	semidirect
176. C₂³×C₄	C₂×Dic₅₃	semidirect
192. C₂⁴	C₄⋊Dic₅₃	semidirect
212. C₂²×C₄	C₂²⋊Dic₅₃	semidirect
215. C₂²×C₄	C₂²×Dic₅₃	semidirect
230. C₂³	C₂×C₄⋊Dic₅₃	semidirect
250. C₂×C₄	C₂×C₂²⋊Dic₅₃	semidirect
259. C₂²	#16810483	direct (N × H)
262. C₂²	C₂²×C₄⋊Dic₅₃	semidirect
273. C₄	C₂²×C₂²⋊Dic₅₃	semidirect
275. C₂	C₂×#16810483	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×C₄⋊Dic₅₃	2
C₂³×C₂²⋊Dic₅₃	2
C₂×#16810483	2
C₂²×#13133124	2
C₂⁵×C₄×C₅₃	2
C₂²×#380	53

Subgroups — C22×#16810483 #27542352

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16810483 #27542352