Subgroups — C₂³×#13133249 #28297891

Order 13568.

11166 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#11719949	C₂	1	1	yes	maximal, characteristic
2	C₂³×Q₈×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#13133249	C₂	1	56	yes	maximal, direct product factor
4	C₂³×#11719925	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×#11719913	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×#11719924	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×#11719912	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×Q₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#11719949	—	2	14	no	—
10	C₂²×#11719949	C₂²	1	28	yes	—
11	C₂³×Dic₁₀₆	—	2	1	no	—
12	C₂³×Dic₁₀₆	C₂²	1	1	yes	characteristic
13	C₂²×Q₈×D₅₃	—	2	14	no	—
14	C₂²×Q₈×D₅₃	C₂²	1	28	yes	complemented
15	C₂³×D₂₁₂	—	2	1	no	—
16	C₂³×D₂₁₂	C₂²	1	1	yes	characteristic
17	C₂³×C₄×D₅₃	—	2	1	no	—
18	C₂³×C₄×D₅₃	—	2	1	no	—
19	C₂³×C₄×D₅₃	C₂²	1	1	yes	characteristic
20	C₂×#13133249	C₂²	1	448	yes	direct product factor
21	C₂²×#11719925	C₂²	1	28	yes	complemented
22	C₂²×#11719913	C₂²	1	28	yes	complemented
23	C₂²×#11719924	C₂²	1	28	yes	complemented
24	C₂²×#11719912	C₂²	1	28	yes	complemented
25	C₂³×#88322	C₂²	1	1	yes	characteristic
26	C₂³×Q₈×C₅₃	C₂²	1	1	yes	characteristic
27	C₂³×Q₈×C₅₃	C₂²	1	1	yes	characteristic
28	C₂²×Q₁₆×C₅₃	C₂²	1	28	yes	complemented
29	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
30	C₂⁴×D₅₃	—	2	1	no	—
31	C₂⁴×D₅₃	D₄	1	1	yes	characteristic
32	C₂×#11719949	—	2	168	no	—
33	C₂×#11719949	C₂³	1	112	yes	—
34	C₂²×Dic₁₀₆	—	2	7	no	—
35	C₂²×Dic₁₀₆	—	2	28	no	—
36	C₂²×Dic₁₀₆	C₂³	1	14	yes	—
37	C₂×Q₈×D₅₃	—	2	168	no	—
38	C₂×Q₈×D₅₃	C₂³	1	112	yes	complemented
39	C₂²×D₂₁₂	—	2	28	no	—
40	C₂²×D₂₁₂	—	2	7	no	—
41	C₂²×D₂₁₂	C₂³	1	14	yes	—
42	C₂³×Dic₅₃	—	2	1	no	—
43	C₂³×Dic₅₃	D₄	1	1	yes	characteristic
44	C₂²×C₄×D₅₃	—	2	28	no	—
45	C₂²×C₄×D₅₃	—	2	28	no	—
46	C₂²×C₄×D₅₃	D₄	1	14	yes	—
47	C₂²×C₄×D₅₃	C₂³	1	14	yes	—
48	#13133249	C₂³	1	512	yes	direct product factor
49	C₂×#11719925	C₂³	1	112	yes	complemented
50	C₂×#11719913	C₂³	1	112	yes	complemented
51	C₂×#11719924	C₂³	1	112	yes	complemented
52	C₂×#11719912	C₂³	1	112	yes	complemented
53	C₂²×#88322	C₂³	1	14	yes	—
54	C₂²×Q₈×C₅₃	—	2	7	no	—
55	C₂²×Q₈×C₅₃	—	2	7	no	—
56	C₂²×Q₈×C₅₃	C₂³	1	14	yes	—
57	C₂²×Q₈×C₅₃	C₂³	1	14	yes	—
58	C₂³×C₄×C₅₃	—	2	1	no	—
59	C₂³×C₄×C₅₃	—	2	1	no	—
60	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
61	C₂×Q₁₆×C₅₃	C₂³	1	112	yes	complemented
62	C₂²×C₈×C₅₃	C₂³	1	14	yes	—
63	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
64	C₂³×D₅₃	—	2	8	no	—
65	C₂³×D₅₃	—	2	14	no	—
66	C₂³×D₅₃	—	4	8	no	—
67	C₂³×D₅₃	C₂×D₄	1	14	yes	—
68	C₂×Q₈×C₅₃	—	2	42	no	—
69	C₂×Q₈×C₅₃	—	2	42	no	—
70	C₂×Q₈×C₅₃	C₂⁴	1	28	yes	—
71	C₂×Q₈×C₅₃	C₂⁴	1	28	yes	—
72	C₂²×C₄×C₅₃	—	2	14	no	—
73	C₂²×C₄×C₅₃	—	2	14	no	—
74	C₂²×C₄×C₅₃	C₂×D₄	1	7	yes	—
75	C₂²×C₄×C₅₃	C₂⁴	1	7	yes	—
76	C₂²×Dic₅₃	—	2	14	no	—
77	C₂²×Dic₅₃	C₂×D₄	1	14	yes	—
78	#11719949	—	2	224	no	—
79	#11719949	C₂⁴	1	64	yes	—
80	Q₈×D₅₃	—	2	224	no	—
81	Q₈×D₅₃	C₂⁴	1	64	yes	complemented
82	C₂×D₂₁₂	—	2	112	no	—
83	C₂×D₂₁₂	—	2	42	no	—
84	C₂×D₂₁₂	C₂⁴	1	28	yes	—
85	C₂×C₄×D₅₃	—	2	112	no	—
86	C₂×C₄×D₅₃	—	2	112	no	—
87	C₂×C₄×D₅₃	C₂×D₄	1	84	yes	—
88	C₂×C₄×D₅₃	C₂⁴	1	28	yes	—
89	C₂×Dic₁₀₆	—	2	42	no	—
90	C₂×Dic₁₀₆	—	2	112	no	—
91	C₂×Dic₁₀₆	C₂⁴	1	28	yes	—
92	Q₁₆×C₅₃	C₂⁴	1	64	yes	complemented
93	C₂×C₈×C₅₃	C₂⁴	1	28	yes	—
94	#11719925	C₂⁴	1	64	yes	complemented
95	#11719924	C₂⁴	1	64	yes	complemented
96	C₂×#88322	C₂⁴	1	28	yes	—
97	#11719913	C₂⁴	1	64	yes	complemented
98	#11719912	C₂⁴	1	64	yes	complemented
99	C₂³×C₅₃	#137	1	8	yes	complemented
100	C₂³×C₅₃	C₂²×D₄	1	7	yes	—
101	C₂²×D₅₃	—	2	56	no	—
102	C₂²×D₅₃	—	2	28	no	—
103	C₂²×D₅₃	—	4	56	no	—
104	C₂²×D₅₃	C₂²×D₄	1	28	yes	—
105	Q₈×C₅₃	—	2	28	no	—
106	Q₈×C₅₃	—	2	28	no	—
107	Q₈×C₅₃	C₂⁵	1	8	yes	—
108	Q₈×C₅₃	C₂⁵	1	8	yes	—
109	C₂×C₄×C₅₃	—	2	28	no	—
110	C₂×C₄×C₅₃	—	2	28	no	—
111	C₂×C₄×C₅₃	C₂²×D₄	1	21	yes	—
112	C₂×C₄×C₅₃	C₂⁵	1	7	yes	—
113	C₂×Dic₅₃	—	2	28	no	—
114	C₂×Dic₅₃	C₂²×D₄	1	28	yes	—
115	D₂₁₂	—	2	64	no	—
116	D₂₁₂	—	2	28	no	—
117	D₂₁₂	C₂⁵	1	8	yes	—
118	C₄×D₅₃	—	2	64	no	—
119	C₄×D₅₃	—	2	64	no	—
120	C₄×D₅₃	C₂²×D₄	1	56	yes	—
121	C₄×D₅₃	C₂⁵	1	8	yes	—
122	Dic₁₀₆	—	2	28	no	—
123	Dic₁₀₆	—	2	64	no	—
124	Dic₁₀₆	C₂⁵	1	8	yes	—
125	C₈×C₅₃	C₂⁵	1	8	yes	—
126	#88322	C₂⁵	1	8	yes	—
127	C₂³×#137	—	53	1	no	maximal, Sylow
128	C₂²×C₅₃	C₂×#137	1	28	yes	complemented
129	C₂²×C₅₃	C₂³×D₄	1	7	yes	—
130	C₂×D₅₃	—	2	56	no	—
131	C₂×D₅₃	—	2	8	no	—
132	C₂×D₅₃	—	4	56	no	—
133	C₂×D₅₃	C₂³×D₄	1	8	yes	—
134	C₄×C₅₃	—	2	8	no	—
135	C₄×C₅₃	—	2	8	no	—
136	C₄×C₅₃	C₂³×D₄	1	7	yes	—
137	C₄×C₅₃	C₂⁶	1	1	yes	characteristic
138	Dic₅₃	—	2	8	no	—
139	Dic₅₃	C₂³×D₄	1	8	yes	—
140	C₂³×#41	—	53	1	no	—
141	C₂⁴×Q₈	—	53	1	no	—
142	C₂²×#137	—	53	56	no	—
143	C₂³×Q₁₆	—	53	1	no	—
144	C₂³×Q₁₆	D₅₃	1	1	yes	characteristic, complemented
145	C₂³×SD₁₆	—	53	1	no	—
146	C₂³×SD₁₆	—	53	1	no	—
147	C₂³×M₁₆	—	53	1	no	—
148	C₂×C₅₃	C₂²×#137	1	14	yes	complemented
149	C₂×C₅₃	C₂⁴×D₄	1	1	yes	characteristic
150	D₅₃	—	2	8	no	—
151	D₅₃	—	4	8	no	—
152	C₂²×#41	—	53	28	no	—
153	C₂²×#41	—	106	14	no	—
154	C₂³×Q₈	—	53	28	no	—
155	C₂³×Q₈	—	53	1	no	—
156	C₂³×Q₈	—	106	1	no	—
157	C₂³×Q₈	—	106	14	no	—
158	C₂³×Q₈	C₂×D₅₃	1	1	yes	characteristic
159	C₂³×Q₈	C₂×D₅₃	1	1	yes	characteristic
160	C₂³×D₄	—	53	1	no	—
161	C₂³×D₄	—	106	1	no	—
162	C₂⁴×C₄	—	53	1	no	—
163	C₂⁴×C₄	—	106	1	no	—
164	C₂⁴×C₄	—	106	1	no	—
165	C₂×#137	—	53	448	no	—
166	C₂²×Q₁₆	—	53	28	no	—
167	C₂²×Q₁₆	C₂×D₅₃	1	28	yes	complemented
168	C₂²×SD₁₆	—	53	28	no	—
169	C₂²×SD₁₆	—	53	28	no	—
170	C₂²×M₁₆	—	53	28	no	—
171	C₂³×C₈	—	53	1	no	—
172	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
173	C₅₃	C₂³×#137	1	1	yes	characteristic, complemented, Sylow
174	C₂⁵	—	53	1	no	—
175	C₂⁵	—	106	1	no	—
176	C₂×#41	—	53	112	no	—
177	C₂×#41	—	106	168	no	—
178	C₂²×Q₈	—	2	7	no	—
179	C₂²×Q₈	—	2	7	no	—
180	C₂²×Q₈	—	53	112	no	—
181	C₂²×Q₈	—	53	14	no	—
182	C₂²×Q₈	—	106	28	no	—
183	C₂²×Q₈	—	106	168	no	—
184	C₂²×Q₈	—	106	7	no	—
185	C₂²×Q₈	C₂²×D₅₃	1	14	yes	—
186	C₂²×Q₈	C₂²×D₅₃	1	14	yes	—
187	C₂²×D₄	—	53	14	no	—
188	C₂²×D₄	—	106	28	no	—
189	C₂²×D₄	—	106	7	no	—
190	C₂³×C₄	—	2	1	no	—
191	C₂³×C₄	—	2	1	no	—
192	C₂³×C₄	—	53	1	no	—
193	C₂³×C₄	—	53	14	no	—
194	C₂³×C₄	—	53	14	no	—
195	C₂³×C₄	—	106	28	no	—
196	C₂³×C₄	—	106	28	no	—
197	C₂³×C₄	—	106	1	no	—
198	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
199	#137	—	53	512	no	—
200	C₂×Q₁₆	—	53	112	no	—
201	C₂×Q₁₆	C₂²×D₅₃	1	112	yes	complemented
202	C₂×SD₁₆	—	53	112	no	—
203	C₂×SD₁₆	—	53	112	no	—
204	C₂×M₁₆	—	53	112	no	—
205	C₂²×C₈	—	53	14	no	—
206	C₂²×C₈	C₂²×D₅₃	1	14	yes	—
207	C₂⁴	—	53	14	no	—
208	C₂⁴	—	106	8	no	—
209	C₂⁴	—	106	14	no	—
210	C₂⁴	—	212	8	no	—
211	C₂⁴	D₄×D₅₃	1	1	yes	characteristic
212	#41	—	53	64	no	—
213	#41	—	106	224	no	—
214	C₂×Q₈	—	2	42	no	—
215	C₂×Q₈	—	2	42	no	—
216	C₂×Q₈	—	53	64	no	—
217	C₂×Q₈	—	53	28	no	—
218	C₂×Q₈	—	106	112	no	—
219	C₂×Q₈	—	106	224	no	—
220	C₂×Q₈	—	106	42	no	—
221	C₂×Q₈	C₂³×D₅₃	1	28	yes	—
222	C₂×Q₈	C₂³×D₅₃	1	28	yes	—
223	C₂×D₄	—	53	28	no	—
224	C₂×D₄	—	106	112	no	—
225	C₂×D₄	—	106	42	no	—
226	C₂²×C₄	—	2	14	no	—
227	C₂²×C₄	—	2	14	no	—
228	C₂²×C₄	—	53	14	no	—
229	C₂²×C₄	—	53	84	no	—
230	C₂²×C₄	—	53	28	no	—
231	C₂²×C₄	—	106	112	no	—
232	C₂²×C₄	—	106	112	no	—
233	C₂²×C₄	—	106	14	no	—
234	C₂²×C₄	D₄×D₅₃	1	7	yes	—
235	C₂²×C₄	C₂³×D₅₃	1	7	yes	—
236	Q₁₆	—	53	64	no	—
237	Q₁₆	C₂³×D₅₃	1	64	yes	complemented
238	SD₁₆	—	53	64	no	—
239	SD₁₆	—	53	64	no	—
240	M₁₆	—	53	64	no	—
241	C₂×C₈	—	53	28	no	—
242	C₂×C₈	C₂³×D₅₃	1	28	yes	—
243	C₂³	—	53	28	no	—
244	C₂³	—	106	56	no	—
245	C₂³	—	106	28	no	—
246	C₂³	—	212	56	no	—
247	C₂³	#13133249	1	8	yes	direct product factor
248	C₂³	C₂×D₄×D₅₃	1	7	yes	—
249	Q₈	—	2	28	no	—
250	Q₈	—	2	28	no	—
251	Q₈	—	53	8	no	—
252	Q₈	—	106	64	no	—
253	Q₈	—	106	28	no	—
254	Q₈	C₂⁴×D₅₃	1	8	yes	—
255	Q₈	C₂⁴×D₅₃	1	8	yes	—
256	D₄	—	53	8	no	—
257	D₄	—	106	64	no	—
258	D₄	—	106	28	no	—
259	C₂×C₄	—	2	28	no	—
260	C₂×C₄	—	2	28	no	—
261	C₂×C₄	—	53	28	no	—
262	C₂×C₄	—	53	56	no	—
263	C₂×C₄	—	53	8	no	—
264	C₂×C₄	—	106	64	no	—
265	C₂×C₄	—	106	64	no	—
266	C₂×C₄	—	106	28	no	—
267	C₂×C₄	C₂×D₄×D₅₃	1	21	yes	—
268	C₂×C₄	C₂⁴×D₅₃	1	7	yes	—
269	C₈	—	53	8	no	—
270	C₈	C₂⁴×D₅₃	1	8	yes	—
271	C₂²	—	53	8	no	—
272	C₂²	—	106	56	no	—
273	C₂²	—	106	8	no	—
274	C₂²	—	212	56	no	—
275	C₂²	C₂×#13133249	1	28	yes	direct product factor
276	C₂²	C₂²×D₄×D₅₃	1	7	yes	—
277	C₄	—	2	8	no	—
278	C₄	—	2	8	no	—
279	C₄	—	53	8	no	—
280	C₄	—	106	8	no	—
281	C₄	C₂²×D₄×D₅₃	1	7	yes	—
282	C₄	C₂⁵×D₅₃	1	1	yes	characteristic
283	C₂	—	106	8	no	—
284	C₂	—	212	8	no	—
285	C₂	C₂²×#13133249	1	14	yes	direct product factor
286	C₂	C₂³×D₄×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	D₄×D₅₃
commutator, cosocle	C₄×C₅₃	C₂⁶
Frattini	C₄	C₂⁵×D₅₃
Fitting	C₂³×Q₁₆×C₅₃	C₂
socle	C₂⁴×C₅₃	C₂×D₄
prime radical	C₂³×#11719949	C₂
hypercenter	C₂³×Q₁₆	D₅₃
hypocenter	C₅₃	C₂³×#137
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂³×Q₁₆

Complements

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

N	H ≅ G/N	type
2. C₂³×Q₈×D₅₃	C₂	semidirect
3. C₂²×#13133249	C₂	direct (N × H)
4. C₂³×#11719925	C₂	semidirect
5. C₂³×#11719913	C₂	semidirect
6. C₂³×#11719924	C₂	semidirect
7. C₂³×#11719912	C₂	semidirect
8. C₂³×Q₁₆×C₅₃	C₂	semidirect
14. C₂²×Q₈×D₅₃	C₂²	semidirect
20. C₂×#13133249	C₂²	direct (N × H)
21. C₂²×#11719925	C₂²	semidirect
22. C₂²×#11719913	C₂²	semidirect
23. C₂²×#11719924	C₂²	semidirect
24. C₂²×#11719912	C₂²	semidirect
28. C₂²×Q₁₆×C₅₃	C₂²	semidirect
38. C₂×Q₈×D₅₃	C₂³	semidirect
48. #13133249	C₂³	direct (N × H)
49. C₂×#11719925	C₂³	semidirect
50. C₂×#11719913	C₂³	semidirect
51. C₂×#11719924	C₂³	semidirect
52. C₂×#11719912	C₂³	semidirect
61. C₂×Q₁₆×C₅₃	C₂³	semidirect
81. Q₈×D₅₃	C₂⁴	semidirect
92. Q₁₆×C₅₃	C₂⁴	semidirect
94. #11719925	C₂⁴	semidirect
95. #11719924	C₂⁴	semidirect
97. #11719913	C₂⁴	semidirect
98. #11719912	C₂⁴	semidirect
99. C₂³×C₅₃	#137	semidirect
128. C₂²×C₅₃	C₂×#137	semidirect
144. C₂³×Q₁₆	D₅₃	semidirect
148. C₂×C₅₃	C₂²×#137	semidirect
167. C₂²×Q₁₆	C₂×D₅₃	semidirect
173. C₅₃	C₂³×#137	semidirect
201. C₂×Q₁₆	C₂²×D₅₃	semidirect
237. Q₁₆	C₂³×D₅₃	semidirect
247. C₂³	#13133249	direct (N × H)
275. C₂²	C₂×#13133249	direct (N × H)
285. C₂	C₂²×#13133249	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×#11719949	2
C₂³×Q₈×D₅₃	2
C₂²×#13133249	2
C₂³×#11719925	2
C₂³×#11719913	2
C₂³×#11719924	2
C₂³×#11719912	2
C₂³×Q₁₆×C₅₃	2
C₂³×#137	53

Subgroups — C23×#13133249 #28297891

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#13133249 #28297891