Subgroups — C₂³×#13133202 #28290742

Order 13568.

15150 conjugacy classes of subgroups.

The subgroup lattice has 280 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	C₂⁴×C₄×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#13133202	C₂	1	56	yes	maximal, direct product factor
4	C₂³×#11719919	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×#11719921	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×#11719921	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×#11719921	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂⁵×D₅₃	C₂²	1	1	yes	characteristic
10	C₂⁵×D₅₃	C₂²	1	1	yes	characteristic
11	C₂³×D₂₁₂	—	2	16	no	—
12	C₂³×D₂₁₂	C₂²	1	28	yes	—
13	C₂⁴×Dic₅₃	C₂²	1	1	yes	characteristic
14	C₂⁴×Dic₅₃	C₂²	1	1	yes	characteristic
15	C₂³×C₄×D₅₃	—	2	16	no	—
16	C₂³×C₄×D₅₃	C₂²	1	28	yes	complemented
17	C₂×#13133202	C₂²	1	448	yes	direct product factor
18	C₂²×#11719919	C₂²	1	28	yes	complemented
19	C₂²×#11719921	C₂²	1	28	yes	complemented
20	C₂²×#11719921	C₂²	1	28	yes	complemented
21	C₂²×#11719921	C₂²	1	28	yes	complemented
22	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
23	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
24	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
25	C₂²×C₄⋊C₄×C₅₃	C₂²	1	28	yes	complemented
26	C₂⁴×D₅₃	—	2	8	no	—
27	C₂⁴×D₅₃	—	2	8	no	—
28	C₂⁴×D₅₃	—	2	8	no	—
29	C₂⁴×D₅₃	—	2	8	no	—
30	C₂⁴×D₅₃	—	2	8	no	—
31	C₂⁴×D₅₃	D₄	1	16	yes	—
32	C₂⁴×D₅₃	C₂³	1	14	yes	—
33	C₂⁴×D₅₃	C₂³	1	14	yes	—
34	C₂²×D₂₁₂	—	2	224	no	—
35	C₂²×D₂₁₂	C₂³	1	112	yes	—
36	C₂³×Dic₅₃	—	2	8	no	—
37	C₂³×Dic₅₃	—	2	8	no	—
38	C₂³×Dic₅₃	C₂³	1	14	yes	—
39	C₂³×Dic₅₃	C₂³	1	14	yes	—
40	C₂²×C₄×D₅₃	—	2	224	no	—
41	C₂²×C₄×D₅₃	C₂³	1	112	yes	complemented
42	#13133202	C₂³	1	512	yes	direct product factor
43	C₂×#11719919	C₂³	1	112	yes	complemented
44	C₂×#11719921	C₂³	1	112	yes	complemented
45	C₂×#11719921	C₂³	1	112	yes	complemented
46	C₂×#11719921	C₂³	1	112	yes	complemented
47	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
48	C₂³×C₄×C₅₃	—	2	8	no	—
49	C₂³×C₄×C₅₃	—	2	8	no	—
50	C₂³×C₄×C₅₃	—	2	8	no	—
51	C₂³×C₄×C₅₃	C₂³	1	14	yes	—
52	C₂³×C₄×C₅₃	C₂³	1	14	yes	—
53	C₂³×C₄×C₅₃	C₂³	1	14	yes	—
54	C₂×C₄⋊C₄×C₅₃	C₂³	1	112	yes	complemented
55	C₂⁴×C₅₃	C₂×D₄	1	8	yes	—
56	C₂⁴×C₅₃	#41	1	8	yes	—
57	C₂⁴×C₅₃	#41	1	8	yes	—
58	C₂⁴×C₅₃	C₂⁴	1	7	yes	—
59	C₂³×D₅₃	—	2	128	no	—
60	C₂³×D₅₃	—	2	56	no	—
61	C₂³×D₅₃	—	2	56	no	—
62	C₂³×D₅₃	—	2	56	no	—
63	C₂³×D₅₃	—	2	56	no	—
64	C₂³×D₅₃	—	2	56	no	—
65	C₂³×D₅₃	—	4	64	no	—
66	C₂³×D₅₃	C₂×D₄	1	112	yes	—
67	C₂³×D₅₃	C₂⁴	1	28	yes	—
68	C₂³×D₅₃	C₂⁴	1	28	yes	—
69	C₂²×C₄×C₅₃	—	2	56	no	—
70	C₂²×C₄×C₅₃	—	2	56	no	—
71	C₂²×C₄×C₅₃	—	2	56	no	—
72	C₂²×C₄×C₅₃	C₂⁴	1	28	yes	—
73	C₂²×C₄×C₅₃	C₂⁴	1	28	yes	—
74	C₂²×C₄×C₅₃	C₂⁴	1	28	yes	—
75	C₂²×Dic₅₃	—	2	56	no	—
76	C₂²×Dic₅₃	—	2	56	no	—
77	C₂²×Dic₅₃	C₂⁴	1	28	yes	—
78	C₂²×Dic₅₃	C₂⁴	1	28	yes	—
79	C₂×D₂₁₂	—	2	448	no	—
80	C₂×D₂₁₂	C₂⁴	1	64	yes	—
81	C₂×C₄×D₅₃	—	2	448	no	—
82	C₂×C₄×D₅₃	C₂⁴	1	64	yes	complemented
83	C₄⋊C₄×C₅₃	C₂⁴	1	64	yes	complemented
84	#11719921	C₂⁴	1	64	yes	complemented
85	#11719921	C₂⁴	1	64	yes	complemented
86	#11719921	C₂⁴	1	64	yes	complemented
87	#11719919	C₂⁴	1	64	yes	complemented
88	C₂³×C₅₃	#123	1	64	yes	complemented
89	C₂³×C₅₃	C₂²×D₄	1	28	yes	—
90	C₂³×C₅₃	C₂×#41	1	28	yes	—
91	C₂³×C₅₃	C₂×#41	1	28	yes	—
92	C₂³×C₅₃	C₂⁵	1	7	yes	—
93	C₂²×D₅₃	—	2	448	no	—
94	C₂²×D₅₃	—	2	56	no	—
95	C₂²×D₅₃	—	2	56	no	—
96	C₂²×D₅₃	—	2	56	no	—
97	C₂²×D₅₃	—	2	56	no	—
98	C₂²×D₅₃	—	2	56	no	—
99	C₂²×D₅₃	—	4	224	no	—
100	C₂²×D₅₃	C₂²×D₄	1	112	yes	—
101	C₂²×D₅₃	C₂⁵	1	8	yes	—
102	C₂²×D₅₃	C₂⁵	1	8	yes	—
103	C₂×C₄×C₅₃	—	2	56	no	—
104	C₂×C₄×C₅₃	—	2	56	no	—
105	C₂×C₄×C₅₃	—	2	56	no	—
106	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
107	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
108	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
109	C₂×Dic₅₃	—	2	56	no	—
110	C₂×Dic₅₃	—	2	56	no	—
111	C₂×Dic₅₃	C₂⁵	1	8	yes	—
112	C₂×Dic₅₃	C₂⁵	1	8	yes	—
113	D₂₁₂	—	2	128	no	—
114	C₄×D₅₃	—	2	128	no	—
115	C₂³×#123	—	53	1	no	maximal, Sylow
116	C₂²×C₅₃	C₂×#123	1	112	yes	complemented
117	C₂²×C₅₃	C₂³×D₄	1	14	yes	—
118	C₂²×C₅₃	C₂²×#41	1	14	yes	—
119	C₂²×C₅₃	C₂²×#41	1	14	yes	—
120	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
121	C₂×D₅₃	—	2	8	no	—
122	C₂×D₅₃	—	2	8	no	—
123	C₂×D₅₃	—	2	224	no	—
124	C₂×D₅₃	—	2	8	no	—
125	C₂×D₅₃	—	2	8	no	—
126	C₂×D₅₃	—	2	8	no	—
127	C₂×D₅₃	—	4	112	no	—
128	C₂×D₅₃	C₂³×D₄	1	16	yes	—
129	C₄×C₅₃	—	2	8	no	—
130	C₄×C₅₃	—	2	8	no	—
131	C₄×C₅₃	—	2	8	no	—
132	Dic₅₃	—	2	8	no	—
133	Dic₅₃	—	2	8	no	—
134	C₂⁴×D₄	—	53	1	no	—
135	C₂⁵×C₄	—	53	1	no	—
136	C₂²×#123	—	53	56	no	—
137	C₂³×C₄⋊C₄	—	53	1	no	—
138	C₂³×C₄⋊C₄	D₅₃	1	1	yes	characteristic, complemented
139	C₂³×C₂²⋊C₄	—	53	1	no	—
140	C₂³×C₂²⋊C₄	—	53	1	no	—
141	C₂³×C₂²⋊C₄	—	53	1	no	—
142	C₂×C₅₃	C₂²×#123	1	28	yes	complemented
143	C₂×C₅₃	C₂⁴×D₄	1	1	yes	characteristic
144	C₂×C₅₃	C₂³×#41	1	1	yes	characteristic
145	C₂×C₅₃	C₂³×#41	1	1	yes	characteristic
146	D₅₃	—	2	16	no	—
147	D₅₃	—	4	8	no	—
148	C₂⁶	—	53	1	no	—
149	C₂⁶	—	53	1	no	—
150	C₂³×D₄	—	53	28	no	—
151	C₂³×D₄	—	106	16	no	—
152	C₂⁴×C₄	—	53	1	no	—
153	C₂⁴×C₄	—	53	1	no	—
154	C₂⁴×C₄	—	53	28	no	—
155	C₂⁴×C₄	—	106	16	no	—
156	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic
157	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic
158	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic
159	C₂×#123	—	53	448	no	—
160	C₂²×C₄⋊C₄	—	53	28	no	—
161	C₂²×C₄⋊C₄	C₂×D₅₃	1	28	yes	complemented
162	C₂²×C₂²⋊C₄	—	53	28	no	—
163	C₂²×C₂²⋊C₄	—	53	28	no	—
164	C₂²×C₂²⋊C₄	—	53	28	no	—
165	C₅₃	C₂³×#123	1	1	yes	characteristic, complemented, Sylow
166	C₂⁵	—	53	16	no	—
167	C₂⁵	—	53	14	no	—
168	C₂⁵	—	53	14	no	—
169	C₂⁵	—	106	8	no	—
170	C₂⁵	—	106	8	no	—
171	C₂⁵	—	106	8	no	—
172	C₂⁵	—	106	8	no	—
173	C₂⁵	—	106	8	no	—
174	C₂⁵	C₂²×D₅₃	1	1	yes	characteristic
175	C₂²×D₄	—	53	112	no	—
176	C₂²×D₄	—	106	224	no	—
177	C₂³×C₄	—	2	8	no	—
178	C₂³×C₄	—	2	8	no	—
179	C₂³×C₄	—	2	8	no	—
180	C₂³×C₄	—	53	14	no	—
181	C₂³×C₄	—	53	14	no	—
182	C₂³×C₄	—	53	112	no	—
183	C₂³×C₄	—	106	224	no	—
184	C₂³×C₄	—	106	8	no	—
185	C₂³×C₄	—	106	8	no	—
186	C₂³×C₄	C₂²×D₅₃	1	14	yes	—
187	C₂³×C₄	C₂²×D₅₃	1	14	yes	—
188	C₂³×C₄	C₂²×D₅₃	1	14	yes	—
189	#123	—	53	512	no	—
190	C₂×C₄⋊C₄	—	53	112	no	—
191	C₂×C₄⋊C₄	C₂²×D₅₃	1	112	yes	complemented
192	C₂×C₂²⋊C₄	—	53	112	no	—
193	C₂×C₂²⋊C₄	—	53	112	no	—
194	C₂×C₂²⋊C₄	—	53	112	no	—
195	C₂⁴	—	53	112	no	—
196	C₂⁴	—	53	28	no	—
197	C₂⁴	—	53	28	no	—
198	C₂⁴	—	106	128	no	—
199	C₂⁴	—	106	56	no	—
200	C₂⁴	—	106	56	no	—
201	C₂⁴	—	106	56	no	—
202	C₂⁴	—	106	56	no	—
203	C₂⁴	—	106	56	no	—
204	C₂⁴	—	212	64	no	—
205	C₂⁴	#11719945	1	8	yes	—
206	C₂⁴	D₄×D₅₃	1	8	yes	—
207	C₂⁴	#11719949	1	8	yes	—
208	C₂⁴	C₂³×D₅₃	1	7	yes	—
209	C₂×D₄	—	53	64	no	—
210	C₂×D₄	—	106	448	no	—
211	C₂²×C₄	—	2	56	no	—
212	C₂²×C₄	—	2	56	no	—
213	C₂²×C₄	—	2	56	no	—
214	C₂²×C₄	—	53	28	no	—
215	C₂²×C₄	—	53	28	no	—
216	C₂²×C₄	—	53	64	no	—
217	C₂²×C₄	—	106	448	no	—
218	C₂²×C₄	—	106	56	no	—
219	C₂²×C₄	—	106	56	no	—
220	C₂²×C₄	C₂³×D₅₃	1	28	yes	—
221	C₂²×C₄	C₂³×D₅₃	1	28	yes	—
222	C₂²×C₄	C₂³×D₅₃	1	28	yes	—
223	C₄⋊C₄	—	53	64	no	—
224	C₄⋊C₄	C₂³×D₅₃	1	64	yes	complemented
225	C₂²⋊C₄	—	53	64	no	—
226	C₂²⋊C₄	—	53	64	no	—
227	C₂²⋊C₄	—	53	64	no	—
228	C₂³	—	53	112	no	—
229	C₂³	—	53	8	no	—
230	C₂³	—	53	8	no	—
231	C₂³	—	106	448	no	—
232	C₂³	—	106	56	no	—
233	C₂³	—	106	56	no	—
234	C₂³	—	106	56	no	—
235	C₂³	—	106	56	no	—
236	C₂³	—	106	56	no	—
237	C₂³	—	212	224	no	—
238	C₂³	#13133202	1	64	yes	direct product factor
239	C₂³	C₂×D₄×D₅₃	1	28	yes	—
240	C₂³	C₂×#11719945	1	28	yes	—
241	C₂³	C₂×#11719949	1	28	yes	—
242	C₂³	C₂⁴×D₅₃	1	7	yes	—
243	D₄	—	106	128	no	—
244	C₂×C₄	—	2	56	no	—
245	C₂×C₄	—	2	56	no	—
246	C₂×C₄	—	2	56	no	—
247	C₂×C₄	—	53	8	no	—
248	C₂×C₄	—	53	8	no	—
249	C₂×C₄	—	106	128	no	—
250	C₂×C₄	—	106	56	no	—
251	C₂×C₄	—	106	56	no	—
252	C₂×C₄	C₂⁴×D₅₃	1	8	yes	—
253	C₂×C₄	C₂⁴×D₅₃	1	8	yes	—
254	C₂×C₄	C₂⁴×D₅₃	1	8	yes	—
255	C₂²	—	53	16	no	—
256	C₂²	—	106	224	no	—
257	C₂²	—	106	8	no	—
258	C₂²	—	106	8	no	—
259	C₂²	—	106	8	no	—
260	C₂²	—	106	8	no	—
261	C₂²	—	106	8	no	—
262	C₂²	—	212	112	no	—
263	C₂²	C₂×#13133202	1	112	yes	direct product factor
264	C₂²	C₂²×D₄×D₅₃	1	14	yes	—
265	C₂²	C₂²×#11719945	1	14	yes	—
266	C₂²	C₂²×#11719949	1	14	yes	—
267	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
268	C₄	—	2	8	no	—
269	C₄	—	2	8	no	—
270	C₄	—	2	8	no	—
271	C₄	—	106	8	no	—
272	C₄	—	106	8	no	—
273	C₂	—	106	16	no	—
274	C₂	—	212	8	no	—
275	C₂	C₂²×#13133202	1	28	yes	direct product factor
276	C₂	C₂³×D₄×D₅₃	1	1	yes	characteristic
277	C₂	C₂³×#11719945	1	1	yes	characteristic
278	C₂	C₂³×#11719949	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂²×D₅₃
commutator, cosocle	C₂²×C₅₃	C₂⁶
Frattini	C₂²	C₂⁵×D₅₃
Fitting	C₂³×C₄⋊C₄×C₅₃	C₂
socle	C₂⁵×C₅₃	C₂³
prime radical	C₂⁴×D₂₁₂	C₂
hypercenter	C₂³×C₄⋊C₄	D₅₃
hypocenter	C₅₃	C₂³×#123
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁵ ◁ C₂³×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂⁴×C₄×D₅₃	C₂	semidirect
3. C₂²×#13133202	C₂	direct (N × H)
4. C₂³×#11719919	C₂	semidirect
5. C₂³×#11719921	C₂	semidirect
6. C₂³×#11719921	C₂	semidirect
7. C₂³×#11719921	C₂	semidirect
8. C₂³×C₄⋊C₄×C₅₃	C₂	semidirect
16. C₂³×C₄×D₅₃	C₂²	semidirect
17. C₂×#13133202	C₂²	direct (N × H)
18. C₂²×#11719919	C₂²	semidirect
19. C₂²×#11719921	C₂²	semidirect
20. C₂²×#11719921	C₂²	semidirect
21. C₂²×#11719921	C₂²	semidirect
25. C₂²×C₄⋊C₄×C₅₃	C₂²	semidirect
41. C₂²×C₄×D₅₃	C₂³	semidirect
42. #13133202	C₂³	direct (N × H)
43. C₂×#11719919	C₂³	semidirect
44. C₂×#11719921	C₂³	semidirect
45. C₂×#11719921	C₂³	semidirect
46. C₂×#11719921	C₂³	semidirect
54. C₂×C₄⋊C₄×C₅₃	C₂³	semidirect
82. C₂×C₄×D₅₃	C₂⁴	semidirect
83. C₄⋊C₄×C₅₃	C₂⁴	semidirect
84. #11719921	C₂⁴	semidirect
85. #11719921	C₂⁴	semidirect
86. #11719921	C₂⁴	semidirect
87. #11719919	C₂⁴	semidirect
88. C₂³×C₅₃	#123	semidirect
116. C₂²×C₅₃	C₂×#123	semidirect
138. C₂³×C₄⋊C₄	D₅₃	semidirect
142. C₂×C₅₃	C₂²×#123	semidirect
161. C₂²×C₄⋊C₄	C₂×D₅₃	semidirect
165. C₅₃	C₂³×#123	semidirect
191. C₂×C₄⋊C₄	C₂²×D₅₃	semidirect
224. C₄⋊C₄	C₂³×D₅₃	semidirect
238. C₂³	#13133202	direct (N × H)
263. C₂²	C₂×#13133202	direct (N × H)
275. C₂	C₂²×#13133202	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
C₂⁴×C₄×D₅₃	2
C₂²×#13133202	2
C₂³×#11719919	2
C₂³×#11719921	2
C₂³×C₄⋊C₄×C₅₃	2
C₂³×#123	53

Subgroups — C23×#13133202 #28290742

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#13133202 #28290742