Subgroups — #28253734

Order 13568.

1770 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20288120	C₂	1	2	yes	maximal
2	#20285254	C₂	1	8	yes	maximal, complemented
3	#20285246	C₂	1	8	yes	maximal, complemented
4	#20285239	C₂	1	4	yes	maximal, complemented
5	#20285229	C₂	1	4	yes	maximal, complemented
6	#20283305	C₂	1	2	yes	maximal
7	C₂×#16810661	C₂	1	2	yes	maximal
8	C₂×C₅₃×#405	C₂	1	1	yes	maximal, characteristic
9	#16811254	—	2	4	no	—
10	#16811251	C₂²	1	16	yes	—
11	#16811249	—	2	4	no	—
12	#16811180	C₂²	1	8	yes	—
13	#16811176	—	2	8	no	—
14	C₂×#13133176	C₂²	1	4	yes	—
15	#16811032	C₂²	1	32	yes	complemented
16	#16811028	C₂²	1	8	yes	complemented
17	#16811025	C₂²	1	8	yes	complemented
18	#16811025	C₂²	1	8	yes	complemented
19	#16811021	C₂²	1	8	yes	complemented
20	#16810665	C₂²	1	16	yes	—
21	#16810661	C₂²	1	4	yes	—
22	#16810661	C₂²	1	8	yes	—
23	#16810661	C₂²	1	4	yes	—
24	#16810661	C₂²	1	4	yes	—
25	#16810661	C₂²	1	4	yes	—
26	C₂×#13133151	C₂²	1	4	yes	—
27	C₂×#13133149	C₂²	1	4	yes	—
28	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
29	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
30	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
31	C₅₃×#405	C₂²	1	4	yes	—
32	C₅₃×#405	C₂²	1	4	yes	—
33	C₂×#97×C₅₃	C₂²	1	2	yes	—
34	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
35	#13133217	C₂³	1	16	yes	—
36	#13133216	—	2	16	no	—
37	#13133209	C₂³	1	16	yes	—
38	#13133208	—	2	16	no	—
39	#13133196	—	2	16	no	—
40	#13133194	—	2	32	no	—
41	C₄×Dic₁₀₆	—	2	16	no	—
42	#13133176	—	2	4	no	—
43	#13133176	—	2	4	no	—
44	#13133176	C₂³	1	8	yes	—
45	#13133176	C₂³	1	8	yes	—
46	C₂×C₄⋊Dic₅₃	C₂³	1	8	yes	—
47	C₂×C₂²⋊Dic₅₃	—	2	4	no	—
48	C₂×C₄×Dic₅₃	—	2	4	no	—
49	#13133151	C₂³	1	8	yes	—
50	#13133151	C₂³	1	16	yes	—
51	#13133151	C₂³	1	8	yes	—
52	#13133149	C₂³	1	8	yes	—
53	#13133149	C₂³	1	16	yes	—
54	#13133149	C₂³	1	8	yes	—
55	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
56	C₂×M₁₆×C₅₃	Q₈	1	8	yes	complemented
57	C₂×M₁₆×C₅₃	C₂³	1	4	yes	—
58	C₂²×C₈×C₅₃	Q₈	1	8	yes	complemented
59	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
60	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
61	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
62	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
63	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
64	#117×C₅₃	D₄	1	4	yes	—
65	#117×C₅₃	C₂³	1	4	yes	—
66	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
67	C₂×C₂²⋊C₄×C₅₃	D₄	1	2	yes	—
68	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
69	#97×C₅₃	C₂³	1	8	yes	—
70	C₄×C₈×C₅₃	C₂³	1	4	yes	—
71	C₄×C₈×C₅₃	C₂³	1	4	yes	—
72	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
74	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
76	C₂²×C₄×C₅₃	C₂×Q₈	1	4	yes	—
77	C₂²×C₄×C₅₃	C₂×Q₈	1	4	yes	—
78	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
79	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
80	C₂²×Dic₅₃	—	2	8	no	—
81	C₂×Dic₁₀₆	—	2	16	no	—
82	M₁₆×C₅₃	C₂×Q₈	1	16	yes	—
83	C₂×C₈×C₅₃	C₂×Q₈	1	8	yes	—
84	C₂×C₈×C₅₃	C₂×Q₈	1	8	yes	—
85	C₂×C₈×C₅₃	C₂×Q₈	1	4	yes	—
86	C₂×C₈×C₅₃	C₂×Q₈	1	4	yes	—
87	C₂×C₈×C₅₃	C₂×Q₈	1	8	yes	—
88	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
89	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
90	C₂×C₈×C₅₃	C₂⁴	1	4	yes	—
91	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
92	C₄⋊C₄×C₅₃	C₂×D₄	1	8	yes	—
93	C₂²⋊C₄×C₅₃	C₂×D₄	1	8	yes	—
94	C₄²×C₅₃	C₂×D₄	1	4	yes	—
95	C₄²×C₅₃	C₂⁴	1	4	yes	—
96	C₂²⋊Dic₅₃	—	2	8	no	—
97	C₂²⋊Dic₅₃	—	2	8	no	—
98	C₄⋊Dic₅₃	—	2	8	no	—
99	C₄⋊Dic₅₃	—	2	16	no	—
100	C₄⋊Dic₅₃	C₂⁴	1	16	yes	—
101	#11719919	—	2	32	no	—
102	C₄×Dic₅₃	—	2	8	no	—
103	C₄×Dic₅₃	—	2	8	no	—
104	C₂³×C₅₃	—	4	2	no	—
105	C₂³×C₅₃	C₄⋊Q₈	1	4	yes	—
106	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
107	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
108	C₂×C₄×C₅₃	—	4	4	no	—
109	C₂×C₄×C₅₃	C₄⋊Q₈	1	4	yes	—
110	C₂×C₄×C₅₃	C₄⋊Q₈	1	8	yes	—
111	C₂×C₄×C₅₃	C₂²×D₄	1	8	yes	—
112	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
113	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₄×C₅₃	C₂²×Q₈	1	2	yes	—
115	C₂×C₄×C₅₃	C₂²×Q₈	1	4	yes	—
116	C₂×C₄×C₅₃	C₂²×Q₈	1	2	yes	—
117	C₂×C₄×C₅₃	C₂²×Q₈	1	4	yes	—
118	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
119	C₂×Dic₅₃	—	2	16	no	—
120	C₂×Dic₅₃	—	8	8	no	—
121	Dic₁₀₆	—	8	16	no	—
122	C₈×C₅₃	C₂²×Q₈	1	2	yes	—
123	C₈×C₅₃	C₂²×Q₈	1	2	yes	—
124	C₈×C₅₃	C₂²×Q₈	1	4	yes	—
125	C₈×C₅₃	C₂²×Q₈	1	8	yes	—
126	#60660	—	53	1	no	maximal, Sylow
127	C₂²×C₅₃	—	2	4	no	—
128	C₂²×C₅₃	—	2	2	no	—
129	C₂²×C₅₃	—	4	4	no	—
130	C₂²×C₅₃	C₂×C₄⋊Q₈	1	2	yes	—
131	C₂²×C₅₃	C₂×C₄⋊Q₈	1	4	yes	—
132	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
133	C₄×C₅₃	—	2	8	no	—
134	C₄×C₅₃	C₂×C₄⋊Q₈	1	4	yes	—
135	C₄×C₅₃	C₂×C₄⋊Q₈	1	2	yes	—
136	C₄×C₅₃	C₂³×Q₈	1	1	yes	characteristic
137	C₄×C₅₃	C₂³×Q₈	1	1	yes	characteristic
138	Dic₅₃	—	4	16	no	—
139	#3458	—	53	2	no	—
140	#3164	—	53	8	no	—
141	#3163	—	53	8	no	—
142	#3162	—	53	4	no	—
143	#3160	—	53	4	no	—
144	#2911	—	53	2	no	—
145	C₂×#427	—	53	2	no	—
146	C₂×#405	D₅₃	1	1	yes	characteristic
147	C₂×C₅₃	—	2	4	no	—
148	C₂×C₅₃	—	2	2	no	—
149	C₂×C₅₃	C₂²×C₄⋊Q₈	1	1	yes	characteristic
150	C₂×C₅₃	#3584	1	2	yes	—
151	C₂²×M₁₆	C₂×D₅₃	1	1	yes	characteristic
152	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
153	#544	—	53	16	no	—
154	#544	—	106	4	no	—
155	#543	—	106	4	no	—
156	#533	—	53	8	no	—
157	#533	—	106	8	no	—
158	C₂×#117	—	53	4	no	—
159	C₂×#117	C₂×D₅₃	1	1	yes	characteristic
160	#501	—	53	32	no	—
161	#500	—	53	8	no	—
162	#499	—	53	8	no	—
163	#499	—	53	8	no	—
164	#498	—	53	8	no	—
165	#428	—	53	16	no	—
166	#427	—	53	4	no	—
167	#427	—	53	8	no	—
168	#427	—	53	4	no	—
169	#427	—	53	4	no	—
170	#427	—	53	4	no	—
171	C₂×#107	—	53	4	no	—
172	C₂×#106	—	53	4	no	—
173	#405	C₂×D₅₃	1	4	yes	—
174	#405	C₂×D₅₃	1	4	yes	—
175	C₂×#97	C₂×D₅₃	1	2	yes	—
176	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
177	C₅₃	#60660	1	1	yes	characteristic, complemented, Sylow
178	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
179	C₂×M₁₆	Dic₁₀₆	1	8	yes	complemented
180	C₂×M₁₆	C₂²×D₅₃	1	4	yes	—
181	C₂²×C₈	Dic₁₀₆	1	8	yes	complemented
182	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
183	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
184	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
185	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
186	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
187	C₄⋊Q₈	—	53	16	no	—
188	C₄⋊Q₈	—	106	16	no	—
189	#125	—	53	16	no	—
190	#125	—	106	16	no	—
191	C₂²⋊Q₈	—	106	16	no	—
192	C₂²⋊Q₈	—	106	32	no	—
193	C₄×Q₈	—	106	16	no	—
194	#117	—	53	8	no	—
195	#117	—	53	8	no	—
196	#117	—	106	4	no	—
197	#117	—	106	4	no	—
198	#117	D₂₁₂	1	4	yes	—
199	#117	C₂²×D₅₃	1	4	yes	—
200	C₂×C₄⋊C₄	—	53	8	no	—
201	C₂×C₄⋊C₄	D₂₁₂	1	2	yes	—
202	C₂×C₂²⋊C₄	—	106	4	no	—
203	C₂×C₂²⋊C₄	D₂₁₂	1	2	yes	—
204	C₂×C₄²	—	106	4	no	—
205	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
206	#107	—	53	8	no	—
207	#107	—	53	8	no	—
208	#107	—	53	16	no	—
209	#106	—	53	8	no	—
210	#106	—	53	8	no	—
211	#106	—	53	16	no	—
212	#97	C₂²×D₅₃	1	8	yes	—
213	C₄×C₈	C₂²×D₅₃	1	4	yes	—
214	C₄×C₈	C₂²×D₅₃	1	4	yes	—
215	C₂⁴	C₂×D₂₁₂	1	1	yes	characteristic
216	C₂×Q₈	—	106	16	no	—
217	C₂²×C₄	—	106	8	no	—
218	C₂²×C₄	C₂×Dic₁₀₆	1	4	yes	—
219	C₂²×C₄	C₂×Dic₁₀₆	1	4	yes	—
220	C₂²×C₄	C₂×D₂₁₂	1	4	yes	—
221	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
222	C₂²×C₄	C₂×D₂₁₂	1	2	yes	—
223	C₂²×C₄	C₂³×D₅₃	1	2	yes	—
224	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
225	M₁₆	C₂×Dic₁₀₆	1	16	yes	—
226	C₂×C₈	C₂×Dic₁₀₆	1	4	yes	—
227	C₂×C₈	C₂×Dic₁₀₆	1	4	yes	—
228	C₂×C₈	C₂×Dic₁₀₆	1	8	yes	—
229	C₂×C₈	C₂×Dic₁₀₆	1	8	yes	—
230	C₂×C₈	C₂×Dic₁₀₆	1	8	yes	—
231	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
232	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
233	C₂×C₈	C₂³×D₅₃	1	4	yes	—
234	C₂×C₈	C₂³×D₅₃	1	2	yes	—
235	C₄⋊C₄	—	53	16	no	—
236	C₄⋊C₄	—	106	8	no	—
237	C₄⋊C₄	—	106	32	no	—
238	C₄⋊C₄	—	106	16	no	—
239	C₄⋊C₄	C₂×D₂₁₂	1	8	yes	—
240	C₂²⋊C₄	—	106	8	no	—
241	C₂²⋊C₄	—	106	8	no	—
242	C₂²⋊C₄	C₂×D₂₁₂	1	8	yes	—
243	C₄²	—	106	8	no	—
244	C₄²	—	106	8	no	—
245	C₄²	C₂×D₂₁₂	1	4	yes	—
246	C₄²	C₂³×D₅₃	1	4	yes	—
247	C₂³	—	4	2	no	—
248	C₂³	#13133217	1	4	yes	—
249	C₂³	C₂²×D₂₁₂	1	1	yes	characteristic
250	C₂³	C₂²×D₂₁₂	1	2	yes	—
251	Q₈	—	424	16	no	—
252	C₂×C₄	—	4	4	no	—
253	C₂×C₄	—	106	16	no	—
254	C₂×C₄	—	424	8	no	—
255	C₂×C₄	#13133217	1	4	yes	—
256	C₂×C₄	#13133217	1	8	yes	—
257	C₂×C₄	C₂²×D₂₁₂	1	8	yes	—
258	C₂×C₄	C₂²×D₂₁₂	1	2	yes	—
259	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
260	C₂×C₄	C₂²×Dic₁₀₆	1	4	yes	—
261	C₂×C₄	C₂²×Dic₁₀₆	1	2	yes	—
262	C₂×C₄	C₂²×Dic₁₀₆	1	2	yes	—
263	C₂×C₄	C₂²×Dic₁₀₆	1	4	yes	—
264	C₂×C₄	C₂⁴×D₅₃	1	1	yes	characteristic
265	C₈	C₂²×Dic₁₀₆	1	2	yes	—
266	C₈	C₂²×Dic₁₀₆	1	2	yes	—
267	C₈	C₂²×Dic₁₀₆	1	4	yes	—
268	C₈	C₂²×Dic₁₀₆	1	8	yes	—
269	C₂²	—	2	2	no	—
270	C₂²	—	2	4	no	—
271	C₂²	—	4	4	no	—
272	C₂²	C₂×#13133217	1	2	yes	—
273	C₂²	C₂×#13133217	1	4	yes	—
274	C₂²	C₂³×D₂₁₂	1	1	yes	characteristic
275	C₄	—	2	8	no	—
276	C₄	—	212	16	no	—
277	C₄	C₂×#13133217	1	4	yes	—
278	C₄	C₂×#13133217	1	2	yes	—
279	C₄	C₂³×Dic₁₀₆	1	1	yes	characteristic
280	C₄	C₂³×Dic₁₀₆	1	1	yes	characteristic
281	C₂	—	2	2	no	—
282	C₂	—	2	4	no	—
283	C₂	C₂²×#13133217	1	1	yes	characteristic
284	C₂	#20289489	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₁₂
commutator, cosocle	C₂×C₄×C₅₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₅₃
Fitting	C₂×C₅₃×#405	C₂
socle	C₂²×C₅₃	C₂³×D₄
prime radical	C₂⁴×C₅₃	C₂×D₄
hypercenter	C₂×#405	D₅₃
hypocenter	C₅₃	#60660
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117 ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
2. #20285254	C₂	semidirect
3. #20285246	C₂	semidirect
4. #20285239	C₂	semidirect
5. #20285229	C₂	semidirect
15. #16811032	C₂²	semidirect
16. #16811028	C₂²	semidirect
17. #16811025	C₂²	semidirect
18. #16811025	C₂²	semidirect
19. #16811021	C₂²	semidirect
56. C₂×M₁₆×C₅₃	Q₈	semidirect
58. C₂²×C₈×C₅₃	Q₈	semidirect
177. C₅₃	#60660	semidirect
179. C₂×M₁₆	Dic₁₀₆	semidirect
181. C₂²×C₈	Dic₁₀₆	semidirect

Maximal subgroups

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

subgroup	index
#20288120	2
#20285254	2
#20285246	2
#20285239	2
#20285229	2
#20283305	2
C₂×#16810661	2
C₂×C₅₃×#405	2
#60660	53