Subgroups — C₂²×#16811364 #28290962

Order 13568.

9606 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811364	C₂	1	48	yes	maximal, direct product factor
2	C₂²×#13133214	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#13133211	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×#13133202	C₂	1	3	yes	maximal, complemented
5	C₂²×#13133192	C₂	1	3	yes	maximal, complemented
6	C₂²×#13133190	C₂	1	3	yes	maximal, complemented
7	C₂²×#13133187	C₂	1	3	yes	maximal, complemented
8	C₂²×#126×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂⁵×D₅₃	C₂²	1	1	yes	characteristic
10	C₂³×D₂₁₂	C₂²	1	3	yes	—
11	C₂³×D₂₁₂	C₂²	1	3	yes	—
12	C₂³×D₂₁₂	C₂²	1	3	yes	complemented
13	C₂³×C₂²⋊D₅₃	C₂²	1	3	yes	—
14	C₂³×C₄×D₅₃	C₂²	1	3	yes	complemented
15	#16811364	C₂²	1	256	yes	direct product factor
16	C₂×#13133214	C₂²	1	24	yes	complemented
17	C₂×#13133211	C₂²	1	24	yes	complemented
18	C₂×#13133202	C₂²	1	72	yes	complemented
19	C₂×#13133192	C₂²	1	72	yes	complemented
20	C₂×#13133190	C₂²	1	72	yes	complemented
21	C₂×#13133187	C₂²	1	72	yes	complemented
22	C₂²×#11719919	C₂²	1	3	yes	complemented
23	C₂²×#11719921	C₂²	1	3	yes	—
24	C₂²×#11719921	C₂²	1	3	yes	—
25	C₂²×#11719921	C₂²	1	3	yes	complemented
26	C₂×#126×C₅₃	C₂²	1	24	yes	complemented
27	C₂²×C₄⋊C₄×C₅₃	C₂²	1	3	yes	complemented
28	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	3	yes	—
29	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic, complemented
30	C₂⁴×D₅₃	—	4	12	no	—
31	C₂⁴×D₅₃	C₂³	1	1	yes	characteristic
32	C₂⁴×D₅₃	C₂³	1	1	yes	characteristic
33	C₂⁴×D₅₃	C₂³	1	3	yes	—
34	C₂⁴×D₅₃	C₂³	1	12	yes	—
35	C₂²×D₂₁₂	—	4	12	no	—
36	C₂²×D₂₁₂	—	4	12	no	—
37	C₂²×D₂₁₂	—	4	12	no	—
38	C₂²×D₂₁₂	C₂³	1	36	yes	—
39	C₂²×D₂₁₂	C₂³	1	36	yes	—
40	C₂²×D₂₁₂	C₂³	1	36	yes	complemented
41	C₂²×C₂²⋊D₅₃	—	4	12	no	—
42	C₂²×C₂²⋊D₅₃	C₂³	1	36	yes	—
43	C₂³×Dic₅₃	C₂³	1	3	yes	—
44	C₂²×C₄×D₅₃	—	4	12	no	—
45	C₂²×C₄×D₅₃	C₂³	1	36	yes	complemented
46	#13133214	C₂³	1	64	yes	complemented
47	#13133211	C₂³	1	64	yes	complemented
48	#13133202	C₂³	1	192	yes	complemented
49	#13133192	C₂³	1	192	yes	complemented
50	#13133190	C₂³	1	192	yes	complemented
51	#13133187	C₂³	1	192	yes	complemented
52	C₂×#11719919	C₂³	1	36	yes	complemented
53	C₂×#11719921	C₂³	1	36	yes	—
54	C₂×#11719921	C₂³	1	36	yes	—
55	C₂×#11719921	C₂³	1	36	yes	complemented
56	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
57	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
58	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
59	#126×C₅₃	C₂³	1	64	yes	complemented
60	C₂×C₄⋊C₄×C₅₃	C₂³	1	36	yes	complemented
61	C₂×C₂²⋊C₄×C₅₃	C₂³	1	36	yes	—
62	C₂×C₄²×C₅₃	C₂³	1	12	yes	complemented
63	C₂⁴×C₅₃	—	2	12	no	—
64	C₂⁴×C₅₃	C₂⁴	1	6	yes	—
65	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
66	C₂³×D₅₃	—	2	12	no	—
67	C₂³×D₅₃	—	2	12	no	—
68	C₂³×D₅₃	—	2	12	no	—
69	C₂³×D₅₃	—	2	12	no	—
70	C₂³×D₅₃	—	2	12	no	—
71	C₂³×D₅₃	—	4	72	no	—
72	C₂³×D₅₃	—	4	64	no	—
73	C₂³×D₅₃	C₂⁴	1	6	yes	—
74	C₂³×D₅₃	C₂⁴	1	6	yes	—
75	C₂³×D₅₃	C₂⁴	1	18	yes	—
76	C₂³×D₅₃	C₂⁴	1	16	yes	—
77	C₂²×C₄×C₅₃	—	2	12	no	—
78	C₂²×C₄×C₅₃	—	2	12	no	—
79	C₂²×C₄×C₅₃	C₂⁴	1	18	yes	—
80	C₂²×C₄×C₅₃	C₂⁴	1	18	yes	—
81	C₂×C₂²⋊D₅₃	—	4	72	no	—
82	C₂×C₂²⋊D₅₃	C₂⁴	1	48	yes	—
83	C₂²×Dic₅₃	—	2	12	no	—
84	C₂²×Dic₅₃	C₂⁴	1	18	yes	—
85	C₂×D₂₁₂	—	4	72	no	—
86	C₂×D₂₁₂	—	4	72	no	—
87	C₂×D₂₁₂	—	4	72	no	—
88	C₂×D₂₁₂	C₂⁴	1	48	yes	—
89	C₂×D₂₁₂	C₂⁴	1	48	yes	—
90	C₂×D₂₁₂	C₂⁴	1	48	yes	complemented
91	C₂×C₄×D₅₃	—	4	72	no	—
92	C₂×C₄×D₅₃	C₂⁴	1	48	yes	complemented
93	C₄⋊C₄×C₅₃	C₂⁴	1	48	yes	complemented
94	C₂²⋊C₄×C₅₃	C₂⁴	1	48	yes	—
95	C₄²×C₅₃	C₂⁴	1	16	yes	complemented
96	#11719921	C₂⁴	1	48	yes	—
97	#11719921	C₂⁴	1	48	yes	—
98	#11719921	C₂⁴	1	48	yes	complemented
99	#11719919	C₂⁴	1	48	yes	complemented
100	C₂³×C₅₃	—	2	36	no	—
101	C₂³×C₅₃	—	4	16	no	—
102	C₂³×C₅₃	H(5,𝔽₂)	1	12	yes	—
103	C₂³×C₅₃	C₂⁵	1	4	yes	—
104	C₂³×C₅₃	C₂⁵	1	3	yes	—
105	C₂²×D₅₃	—	2	36	no	—
106	C₂²×D₅₃	—	2	36	no	—
107	C₂²×D₅₃	—	2	36	no	—
108	C₂²×D₅₃	—	2	36	no	—
109	C₂²×D₅₃	—	2	36	no	—
110	C₂²×D₅₃	—	4	48	no	—
111	C₂²×D₅₃	—	4	192	no	—
112	C₂²×D₅₃	—	4	48	no	—
113	C₂²×D₅₃	—	4	16	no	—
114	C₂²×D₅₃	—	4	16	no	—
115	C₂²×D₅₃	C₂⁵	1	12	yes	—
116	C₂²×D₅₃	C₂⁵	1	4	yes	—
117	C₂²×D₅₃	C₂⁵	1	4	yes	—
118	C₂×C₄×C₅₃	—	2	36	no	—
119	C₂×C₄×C₅₃	—	2	36	no	—
120	C₂×C₄×C₅₃	C₂⁵	1	12	yes	—
121	C₂×C₄×C₅₃	C₂⁵	1	12	yes	—
122	C₂²⋊D₅₃	—	4	48	no	—
123	C₂×Dic₅₃	—	2	36	no	—
124	C₂×Dic₅₃	C₂⁵	1	12	yes	—
125	D₂₁₂	—	4	48	no	—
126	D₂₁₂	—	4	48	no	—
127	D₂₁₂	—	4	48	no	—
128	C₄×D₅₃	—	4	48	no	—
129	C₂²×#560	—	53	1	no	maximal, Sylow
130	C₂²×C₅₃	—	2	12	no	—
131	C₂²×C₅₃	—	4	24	no	—
132	C₂²×C₅₃	#560	1	16	yes	complemented
133	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	18	yes	—
134	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
135	C₂×D₅₃	—	2	12	no	—
136	C₂×D₅₃	—	2	12	no	—
137	C₂×D₅₃	—	2	12	no	—
138	C₂×D₅₃	—	2	12	no	—
139	C₂×D₅₃	—	2	12	no	—
140	C₂×D₅₃	—	4	64	no	—
141	C₂×D₅₃	—	4	72	no	—
142	C₂×D₅₃	—	4	24	no	—
143	C₂×D₅₃	—	4	24	no	—
144	C₄×C₅₃	—	2	12	no	—
145	C₄×C₅₃	—	2	12	no	—
146	Dic₅₃	—	2	12	no	—
147	C₂×#560	—	53	48	no	—
148	C₂²×#127	—	53	1	no	—
149	C₂²×#126	—	53	1	no	—
150	C₂²×#126	D₅₃	1	1	yes	characteristic, complemented
151	C₂²×#123	—	53	3	no	—
152	C₂²×#121	—	53	3	no	—
153	C₂²×#121	—	53	3	no	—
154	C₂²×#120	—	53	3	no	—
155	C₂×C₅₃	—	4	4	no	—
156	C₂×C₅₃	C₂×#560	1	12	yes	complemented
157	C₂×C₅₃	C₂²×H(5,𝔽₂)	1	3	yes	—
158	D₅₃	—	4	12	no	—
159	D₅₃	—	4	4	no	—
160	D₅₃	—	4	4	no	—
161	C₂⁶	—	53	1	no	—
162	C₂³×D₄	—	53	3	no	—
163	C₂³×D₄	—	53	3	no	—
164	C₂³×D₄	—	53	3	no	—
165	C₂³×D₄	—	53	3	no	—
166	C₂⁴×C₄	—	53	3	no	—
167	#560	—	53	256	no	—
168	C₂×#127	—	53	24	no	—
169	C₂×#126	—	53	24	no	—
170	C₂×#126	C₂×D₅₃	1	24	yes	complemented
171	C₂×#123	—	53	72	no	—
172	C₂×#121	—	53	72	no	—
173	C₂×#121	—	53	72	no	—
174	C₂×#120	—	53	72	no	—
175	C₂²×C₄⋊C₄	—	53	3	no	—
176	C₂²×C₄⋊C₄	C₂×D₅₃	1	3	yes	complemented
177	C₂²×C₂²⋊C₄	—	53	3	no	—
178	C₂²×C₂²⋊C₄	—	53	3	no	—
179	C₂²×C₂²⋊C₄	—	53	3	no	—
180	C₂²×C₂²⋊C₄	C₂×D₅₃	1	3	yes	—
181	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic, complemented
182	C₅₃	C₂²×#560	1	1	yes	characteristic, complemented, Sylow
183	C₂⁵	—	53	1	no	—
184	C₂⁵	—	53	1	no	—
185	C₂⁵	—	53	3	no	—
186	C₂⁵	—	53	12	no	—
187	C₂⁵	—	212	12	no	—
188	C₂⁵	C₂²×D₅₃	1	1	yes	characteristic
189	C₂²×D₄	—	53	36	no	—
190	C₂²×D₄	—	53	36	no	—
191	C₂²×D₄	—	53	36	no	—
192	C₂²×D₄	—	53	36	no	—
193	C₂²×D₄	—	212	12	no	—
194	C₂²×D₄	—	212	12	no	—
195	C₂²×D₄	—	212	12	no	—
196	C₂²×D₄	—	212	12	no	—
197	C₂³×C₄	—	53	3	no	—
198	C₂³×C₄	—	53	36	no	—
199	C₂³×C₄	—	212	12	no	—
200	C₂³×C₄	C₂²×D₅₃	1	3	yes	—
201	C₂³×C₄	C₂²×D₅₃	1	3	yes	—
202	#127	—	53	64	no	—
203	#126	—	53	64	no	—
204	#126	C₂²×D₅₃	1	64	yes	complemented
205	#123	—	53	192	no	—
206	#121	—	53	192	no	—
207	#121	—	53	192	no	—
208	#120	—	53	192	no	—
209	C₂×C₄⋊C₄	—	53	36	no	—
210	C₂×C₄⋊C₄	C₂²×D₅₃	1	36	yes	complemented
211	C₂×C₂²⋊C₄	—	53	36	no	—
212	C₂×C₂²⋊C₄	—	53	36	no	—
213	C₂×C₂²⋊C₄	—	53	36	no	—
214	C₂×C₂²⋊C₄	C₂²×D₅₃	1	36	yes	—
215	C₂×C₄²	C₂²×D₅₃	1	12	yes	complemented
216	C₂⁴	—	2	12	no	—
217	C₂⁴	—	53	6	no	—
218	C₂⁴	—	53	6	no	—
219	C₂⁴	—	53	18	no	—
220	C₂⁴	—	53	16	no	—
221	C₂⁴	—	106	12	no	—
222	C₂⁴	—	106	12	no	—
223	C₂⁴	—	106	12	no	—
224	C₂⁴	—	106	12	no	—
225	C₂⁴	—	106	12	no	—
226	C₂⁴	—	212	72	no	—
227	C₂⁴	—	212	64	no	—
228	C₂⁴	C₂³×D₅₃	1	6	yes	—
229	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
230	C₂×D₄	—	53	48	no	—
231	C₂×D₄	—	53	48	no	—
232	C₂×D₄	—	53	48	no	—
233	C₂×D₄	—	53	48	no	—
234	C₂×D₄	—	212	72	no	—
235	C₂×D₄	—	212	72	no	—
236	C₂×D₄	—	212	72	no	—
237	C₂×D₄	—	212	72	no	—
238	C₂²×C₄	—	2	12	no	—
239	C₂²×C₄	—	2	12	no	—
240	C₂²×C₄	—	53	18	no	—
241	C₂²×C₄	—	53	48	no	—
242	C₂²×C₄	—	106	12	no	—
243	C₂²×C₄	—	212	72	no	—
244	C₂²×C₄	C₂³×D₅₃	1	18	yes	—
245	C₂²×C₄	C₂³×D₅₃	1	18	yes	—
246	C₄⋊C₄	—	53	48	no	—
247	C₄⋊C₄	C₂³×D₅₃	1	48	yes	complemented
248	C₂²⋊C₄	—	53	48	no	—
249	C₂²⋊C₄	—	53	48	no	—
250	C₂²⋊C₄	—	53	48	no	—
251	C₂²⋊C₄	C₂³×D₅₃	1	48	yes	—
252	C₄²	C₂³×D₅₃	1	16	yes	complemented
253	C₂³	—	2	36	no	—
254	C₂³	—	4	16	no	—
255	C₂³	—	53	4	no	—
256	C₂³	—	53	4	no	—
257	C₂³	—	53	12	no	—
258	C₂³	—	106	36	no	—
259	C₂³	—	106	36	no	—
260	C₂³	—	106	36	no	—
261	C₂³	—	106	36	no	—
262	C₂³	—	106	36	no	—
263	C₂³	—	212	48	no	—
264	C₂³	—	212	192	no	—
265	C₂³	—	212	48	no	—
266	C₂³	—	212	16	no	—
267	C₂³	—	212	16	no	—
268	C₂³	#13133263	1	12	yes	—
269	C₂³	C₂⁴×D₅₃	1	4	yes	—
270	C₂³	C₂⁴×D₅₃	1	3	yes	—
271	D₄	—	212	48	no	—
272	D₄	—	212	48	no	—
273	D₄	—	212	48	no	—
274	D₄	—	212	48	no	—
275	C₂×C₄	—	2	36	no	—
276	C₂×C₄	—	2	36	no	—
277	C₂×C₄	—	53	12	no	—
278	C₂×C₄	—	106	36	no	—
279	C₂×C₄	—	212	48	no	—
280	C₂×C₄	C₂⁴×D₅₃	1	12	yes	—
281	C₂×C₄	C₂⁴×D₅₃	1	12	yes	—
282	C₂²	—	2	12	no	—
283	C₂²	—	4	24	no	—
284	C₂²	—	106	12	no	—
285	C₂²	—	106	12	no	—
286	C₂²	—	106	12	no	—
287	C₂²	—	106	12	no	—
288	C₂²	—	106	12	no	—
289	C₂²	—	212	64	no	—
290	C₂²	—	212	24	no	—
291	C₂²	—	212	24	no	—
292	C₂²	—	212	72	no	—
293	C₂²	#16811364	1	16	yes	direct product factor
294	C₂²	C₂×#13133263	1	18	yes	—
295	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
296	C₄	—	2	12	no	—
297	C₄	—	2	12	no	—
298	C₄	—	106	12	no	—
299	C₂	—	4	4	no	—
300	C₂	—	212	4	no	—
301	C₂	—	212	4	no	—
302	C₂	—	212	12	no	—
303	C₂	C₂×#16811364	1	12	yes	direct product factor
304	C₂	C₂²×#13133263	1	3	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₅₃
commutator, cosocle	C₂²×C₅₃	C₂⁶
Frattini	C₂²	C₂⁵×D₅₃
Fitting	C₂²×#126×C₅₃	C₂
socle	C₂⁴×C₅₃	C₂⁴
hypercenter	C₂²×#126	D₅₃
hypocenter	C₅₃	C₂²×#560
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#126

Complements

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

N	H ≅ G/N	type
1. C₂×#16811364	C₂	direct (N × H)
2. C₂²×#13133214	C₂	semidirect
3. C₂²×#13133211	C₂	semidirect
4. C₂²×#13133202	C₂	semidirect
5. C₂²×#13133192	C₂	semidirect
6. C₂²×#13133190	C₂	semidirect
7. C₂²×#13133187	C₂	semidirect
8. C₂²×#126×C₅₃	C₂	semidirect
12. C₂³×D₂₁₂	C₂²	semidirect
14. C₂³×C₄×D₅₃	C₂²	semidirect
15. #16811364	C₂²	direct (N × H)
16. C₂×#13133214	C₂²	semidirect
17. C₂×#13133211	C₂²	semidirect
18. C₂×#13133202	C₂²	semidirect
19. C₂×#13133192	C₂²	semidirect
20. C₂×#13133190	C₂²	semidirect
21. C₂×#13133187	C₂²	semidirect
22. C₂²×#11719919	C₂²	semidirect
25. C₂²×#11719921	C₂²	semidirect
26. C₂×#126×C₅₃	C₂²	semidirect
27. C₂²×C₄⋊C₄×C₅₃	C₂²	semidirect
29. C₂²×C₄²×C₅₃	C₂²	semidirect
40. C₂²×D₂₁₂	C₂³	semidirect
45. C₂²×C₄×D₅₃	C₂³	semidirect
46. #13133214	C₂³	semidirect
47. #13133211	C₂³	semidirect
48. #13133202	C₂³	semidirect
49. #13133192	C₂³	semidirect
50. #13133190	C₂³	semidirect
51. #13133187	C₂³	semidirect
52. C₂×#11719919	C₂³	semidirect
55. C₂×#11719921	C₂³	semidirect
59. #126×C₅₃	C₂³	semidirect
60. C₂×C₄⋊C₄×C₅₃	C₂³	semidirect
62. C₂×C₄²×C₅₃	C₂³	semidirect
90. C₂×D₂₁₂	C₂⁴	semidirect
92. C₂×C₄×D₅₃	C₂⁴	semidirect
93. C₄⋊C₄×C₅₃	C₂⁴	semidirect
95. C₄²×C₅₃	C₂⁴	semidirect
98. #11719921	C₂⁴	semidirect
99. #11719919	C₂⁴	semidirect
132. C₂²×C₅₃	#560	semidirect
150. C₂²×#126	D₅₃	semidirect
156. C₂×C₅₃	C₂×#560	semidirect
170. C₂×#126	C₂×D₅₃	semidirect
176. C₂²×C₄⋊C₄	C₂×D₅₃	semidirect
181. C₂²×C₄²	C₂×D₅₃	semidirect
182. C₅₃	C₂²×#560	semidirect
204. #126	C₂²×D₅₃	semidirect
210. C₂×C₄⋊C₄	C₂²×D₅₃	semidirect
215. C₂×C₄²	C₂²×D₅₃	semidirect
247. C₄⋊C₄	C₂³×D₅₃	semidirect
252. C₄²	C₂³×D₅₃	semidirect
293. C₂²	#16811364	direct (N × H)
303. C₂	C₂×#16811364	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×#16811364	2
C₂²×#13133214	2
C₂²×#13133211	2
C₂²×#13133202	2
C₂²×#13133192	2
C₂²×#13133190	2
C₂²×#13133187	2
C₂²×#126×C₅₃	2
C₂²×#560	53

Subgroups — C22×#16811364 #28290962

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16811364 #28290962