Subgroups — #27382111

Order 13568.

1018 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810672	C₂	1	1	yes	maximal, characteristic
2	#20283305	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810578	C₂	1	2	yes	maximal
4	#20274599	C₂	1	4	yes	maximal, complemented
5	#20274594	C₂	1	4	yes	maximal, complemented
6	#20274425	C₂	1	2	yes	maximal
7	C₂×C₅₃×#405	C₂	1	1	yes	maximal, characteristic
8	C₂²×#11719917	C₂²	1	2	yes	—
9	C₂×#13133176	C₂²	1	2	yes	—
10	#16810672	C₄	1	8	yes	complemented
11	C₂×#13133153	C₂²	1	4	yes	—
12	#16810665	C₄	1	8	yes	complemented
13	#16810661	C₂²	1	4	yes	—
14	#16810578	—	2	2	no	—
15	#16810578	—	2	2	no	—
16	#16810578	C₂²	1	4	yes	—
17	#16810578	C₂²	1	4	yes	—
18	C₂×#13133127	—	2	2	no	—
19	C₂×C₈×Dic₅₃	—	2	2	no	—
20	#16810383	C₂²	1	8	yes	—
21	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
22	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
24	C₅₃×#405	C₄	1	8	yes	complemented
25	C₂×#97×C₅₃	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
27	C₂×#11719917	—	2	2	no	—
28	C₂×#11719917	—	2	4	no	—
29	C₂×#11719917	C₂×C₄	1	8	yes	—
30	C₂×#11719917	C₂³	1	4	yes	—
31	C₂²×#88322	—	2	2	no	—
32	#13133176	—	2	2	no	—
33	#13133176	—	2	2	no	—
34	#13133176	C₂×C₄	1	4	yes	—
35	#13133176	C₂³	1	4	yes	—
36	C₂×C₄⋊Dic₅₃	C₂×C₄	1	4	yes	—
37	C₂×C₂²⋊Dic₅₃	—	2	2	no	—
38	C₂×C₄×Dic₅₃	—	2	2	no	—
39	#13133153	C₂×C₄	1	16	yes	complemented
40	#13133151	C₂×C₄	1	8	yes	—
41	#13133149	C₂×C₄	1	8	yes	—
42	#13133127	—	2	4	no	—
43	#13133127	—	2	4	no	—
44	C₈×Dic₅₃	—	2	4	no	—
45	C₈×Dic₅₃	—	2	4	no	—
46	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
47	C₂×M₁₆×C₅₃	C₂×C₄	1	4	yes	—
48	C₂×M₁₆×C₅₃	C₂×C₄	1	4	yes	—
49	C₂×M₁₆×C₅₃	C₂×C₄	1	4	yes	—
50	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
51	C₂²×C₈×C₅₃	D₄	1	2	yes	—
52	C₂²×C₈×C₅₃	D₄	1	2	yes	—
53	C₂²×C₈×C₅₃	Q₈	1	4	yes	—
54	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
55	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
56	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
57	#117×C₅₃	—	2	2	no	—
58	#117×C₅₃	C₂×C₄	1	4	yes	—
59	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
60	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
61	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
62	#97×C₅₃	C₂×C₄	1	8	yes	complemented
63	C₄×C₈×C₅₃	C₂×C₄	1	8	yes	complemented
64	C₂⁴×C₅₃	#41	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	—	2	2	no	—
66	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
71	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
72	C₂²×C₄×C₅₃	#41	1	2	yes	—
73	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
74	C₂²×Dic₅₃	—	2	4	no	—
75	M₁₆×C₅₃	C₄²	1	16	yes	—
76	C₂×C₈×C₅₃	C₄⋊C₄	1	8	yes	—
77	C₂×C₈×C₅₃	C₄⋊C₄	1	8	yes	—
78	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
79	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
80	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
81	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
83	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
84	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
87	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
88	C₄⋊C₄×C₅₃	—	2	4	no	—
89	C₂²⋊C₄×C₅₃	—	2	4	no	—
90	C₄²×C₅₃	—	2	2	no	—
91	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
92	C₂²⋊Dic₅₃	—	2	4	no	—
93	C₂²⋊Dic₅₃	—	2	4	no	—
94	C₄⋊Dic₅₃	—	2	4	no	—
95	C₄⋊Dic₅₃	C₂²×C₄	1	8	yes	—
96	C₄×Dic₅₃	—	2	4	no	—
97	C₄×Dic₅₃	—	2	4	no	—
98	#11719917	—	2	8	no	—
99	#11719917	—	2	4	no	—
100	#11719917	C₂²×C₄	1	8	yes	—
101	C₂×#88322	—	2	8	no	—
102	C₂×#88322	—	2	4	no	—
103	C₂³×C₅₃	—	4	2	no	—
104	C₂³×C₅₃	#117	1	2	yes	—
105	C₂³×C₅₃	C₄×D₄	1	2	yes	—
106	C₂³×C₅₃	C₄×Q₈	1	2	yes	—
107	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
108	C₂×C₄×C₅₃	—	2	4	no	—
109	C₂×C₄×C₅₃	—	4	4	no	—
110	C₂×C₄×C₅₃	C₂×C₄²	1	4	yes	—
111	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
112	C₂×C₄×C₅₃	#117	1	2	yes	—
113	C₂×C₄×C₅₃	C₄×D₄	1	4	yes	—
114	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
115	C₂×C₄×C₅₃	C₄×Q₈	1	4	yes	—
116	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
117	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
118	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
119	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
120	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
121	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
123	C₂×Dic₅₃	—	2	8	no	—
124	C₂×Dic₅₃	—	4	8	no	—
125	C₈×C₅₃	C₂×C₄²	1	8	yes	—
126	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
127	C₈×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
128	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
129	#88322	—	2	8	no	—
130	#20444	—	53	1	no	maximal, Sylow
131	C₂²×C₅₃	—	2	4	no	—
132	C₂²×C₅₃	—	2	2	no	—
133	C₂²×C₅₃	—	4	2	no	—
134	C₂²×C₅₃	—	4	2	no	—
135	C₂²×C₅₃	C₄×C₄⋊C₄	1	4	yes	—
136	C₂²×C₅₃	C₂×#117	1	1	yes	characteristic
137	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
138	C₂²×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
139	C₄×C₅₃	—	4	4	no	—
140	C₄×C₅₃	C₄×C₄⋊C₄	1	4	yes	—
141	C₄×C₅₃	C₂²×C₄²	1	1	yes	characteristic
142	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
143	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
144	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
145	Dic₅₃	—	4	8	no	—
146	C₂×#430	—	53	1	no	—
147	#2911	—	53	1	no	—
148	C₂×#405	—	53	2	no	—
149	C₂×#405	D₅₃	1	1	yes	characteristic
150	#1773	—	53	4	no	—
151	#1772	—	53	4	no	—
152	#1733	—	53	2	no	—
153	C₂×C₅₃	—	2	1	no	—
154	C₂×C₅₃	—	2	1	no	—
155	C₂×C₅₃	—	2	4	no	—
156	C₂×C₅₃	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
157	C₂×C₅₃	#2955	1	2	yes	—
158	C₂²×M₁₆	—	53	2	no	—
159	C₂²×M₁₆	C₂×D₅₃	1	1	yes	characteristic
160	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
161	C₂×#117	—	53	2	no	—
162	C₂×#117	C₂×D₅₃	1	1	yes	characteristic
163	#430	—	53	8	no	—
164	C₂×#108	—	53	4	no	—
165	#428	—	53	8	no	—
166	#427	—	53	4	no	—
167	#405	—	53	4	no	—
168	#405	—	53	4	no	—
169	#405	—	106	2	no	—
170	#405	—	106	2	no	—
171	#405	Dic₅₃	1	8	yes	complemented
172	C₂×#97	—	106	2	no	—
173	C₂×#97	C₂×D₅₃	1	2	yes	—
174	C₂×C₄×C₈	—	106	2	no	—
175	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
176	#339	—	53	8	no	—
177	C₅₃	#20444	1	1	yes	characteristic, complemented, Sylow
178	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
179	C₂×M₁₆	—	53	8	no	—
180	C₂×M₁₆	—	53	4	no	—
181	C₂×M₁₆	—	106	2	no	—
182	C₂×M₁₆	—	106	4	no	—
183	C₂×M₁₆	C₄×D₅₃	1	4	yes	—
184	C₂×M₁₆	C₄×D₅₃	1	4	yes	—
185	C₂×M₁₆	C₂×Dic₅₃	1	4	yes	—
186	C₂²×C₈	—	106	2	no	—
187	C₂²×C₈	Dic₁₀₆	1	4	yes	—
188	C₂²×C₈	D₂₁₂	1	2	yes	—
189	C₂²×C₈	D₂₁₂	1	2	yes	—
190	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
191	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
192	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
193	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
194	#117	—	2	2	no	—
195	#117	—	53	4	no	—
196	#117	—	53	4	no	—
197	#117	—	106	2	no	—
198	#117	—	106	2	no	—
199	#117	C₂×Dic₅₃	1	4	yes	—
200	C₂×C₄⋊C₄	—	2	1	no	—
201	C₂×C₄⋊C₄	—	53	4	no	—
202	C₂×C₂²⋊C₄	—	2	1	no	—
203	C₂×C₂²⋊C₄	—	106	2	no	—
204	C₂×C₄²	—	106	2	no	—
205	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
206	#108	—	53	16	no	—
207	#107	—	53	8	no	—
208	#106	—	53	8	no	—
209	#97	—	106	4	no	—
210	#97	—	106	4	no	—
211	#97	C₂×Dic₅₃	1	8	yes	complemented
212	C₄×C₈	—	106	4	no	—
213	C₄×C₈	—	106	4	no	—
214	C₄×C₈	C₂×Dic₅₃	1	8	yes	complemented
215	C₂⁴	#11719945	1	1	yes	characteristic
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	—	106	4	no	—
218	C₂²×C₄	C₂×Dic₁₀₆	1	2	yes	—
219	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
220	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
221	C₂²×C₄	C₂×D₂₁₂	1	2	yes	—
222	C₂²×C₄	#11719945	1	1	yes	characteristic
223	C₂²×C₄	#11719945	1	2	yes	—
224	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
225	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
226	M₁₆	—	53	8	no	—
227	M₁₆	—	106	8	no	—
228	M₁₆	—	106	4	no	—
229	M₁₆	C₄×Dic₅₃	1	16	yes	—
230	C₂×C₈	—	106	8	no	—
231	C₂×C₈	—	106	4	no	—
232	C₂×C₈	C₄⋊Dic₅₃	1	8	yes	—
233	C₂×C₈	C₄⋊Dic₅₃	1	8	yes	—
234	C₂×C₈	C₂×Dic₁₀₆	1	2	yes	—
235	C₂×C₈	C₂×Dic₁₀₆	1	2	yes	—
236	C₂×C₈	C₂×C₄×D₅₃	1	4	yes	—
237	C₂×C₈	C₂×C₄×D₅₃	1	4	yes	—
238	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
239	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
240	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
241	C₂×C₈	C₂²×Dic₅₃	1	4	yes	—
242	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
243	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
244	C₄⋊C₄	—	2	4	no	—
245	C₄⋊C₄	—	53	8	no	—
246	C₄⋊C₄	—	106	4	no	—
247	C₂²⋊C₄	—	2	4	no	—
248	C₂²⋊C₄	—	106	4	no	—
249	C₂²⋊C₄	—	106	4	no	—
250	C₄²	—	2	2	no	—
251	C₄²	—	106	4	no	—
252	C₄²	—	106	4	no	—
253	C₄²	C₂²×Dic₅₃	1	4	yes	—
254	C₂³	—	4	2	no	—
255	C₂³	#13133176	1	2	yes	—
256	C₂³	C₄×D₂₁₂	1	2	yes	—
257	C₂³	C₄×Dic₁₀₆	1	2	yes	—
258	C₂³	C₂×#11719945	1	1	yes	characteristic
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	4	4	no	—
261	C₂×C₄	—	106	8	no	—
262	C₂×C₄	—	212	8	no	—
263	C₂×C₄	C₂×C₄×Dic₅₃	1	4	yes	—
264	C₂×C₄	C₂×C₄⋊Dic₅₃	1	4	yes	—
265	C₂×C₄	#13133176	1	2	yes	—
266	C₂×C₄	C₄×D₂₁₂	1	4	yes	—
267	C₂×C₄	C₄×D₂₁₂	1	2	yes	—
268	C₂×C₄	C₄×Dic₁₀₆	1	4	yes	—
269	C₂×C₄	C₄×Dic₁₀₆	1	2	yes	—
270	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
271	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
272	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
273	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
274	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
275	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
276	C₈	—	106	8	no	—
277	C₈	C₂×C₄×Dic₅₃	1	8	yes	—
278	C₈	C₂×C₄⋊Dic₅₃	1	4	yes	—
279	C₈	C₂×C₄⋊Dic₅₃	1	2	yes	—
280	C₈	C₂×C₄⋊Dic₅₃	1	2	yes	—
281	C₂²	—	2	2	no	—
282	C₂²	—	2	4	no	—
283	C₂²	—	4	2	no	—
284	C₂²	—	4	2	no	—
285	C₂²	C₄×C₄⋊Dic₅₃	1	4	yes	—
286	C₂²	C₂×#13133176	1	1	yes	characteristic
287	C₂²	C₂×C₄×D₂₁₂	1	1	yes	characteristic
288	C₂²	C₂×C₄×Dic₁₀₆	1	1	yes	characteristic
289	C₄	—	4	4	no	—
290	C₄	—	212	8	no	—
291	C₄	C₄×C₄⋊Dic₅₃	1	4	yes	—
292	C₄	C₂²×C₄×Dic₅₃	1	1	yes	characteristic
293	C₄	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
294	C₄	C₂×C₄×D₂₁₂	1	1	yes	characteristic
295	C₄	C₂×C₄×Dic₁₀₆	1	1	yes	characteristic
296	C₂	—	2	1	no	—
297	C₂	—	2	1	no	—
298	C₂	—	2	4	no	—
299	C₂	C₂×C₄×C₄⋊Dic₅₃	1	1	yes	characteristic
300	C₂	#20283619	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂×D₂₁₂
commutator	C₄×C₅₃	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₅₃×#405	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂⁴×C₅₃	#41
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂×#405	D₅₃
hypocenter	C₅₃	#20444
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
4. #20274599	C₂	semidirect
5. #20274594	C₂	semidirect
10. #16810672	C₄	semidirect
12. #16810665	C₄	semidirect
24. C₅₃×#405	C₄	semidirect
39. #13133153	C₂×C₄	semidirect
62. #97×C₅₃	C₂×C₄	semidirect
63. C₄×C₈×C₅₃	C₂×C₄	semidirect
171. #405	Dic₅₃	semidirect
177. C₅₃	#20444	semidirect
211. #97	C₂×Dic₅₃	semidirect
214. C₄×C₈	C₂×Dic₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810672	2
#20283305	2
C₂×#16810578	2
#20274599	2
#20274594	2
#20274425	2
C₂×C₅₃×#405	2
#20444	53