Subgroups — #27537619

Order 13568.

558 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	#20276624	C₂	1	2	yes	maximal, complemented
2	#20276547	C₂	1	2	yes	maximal, complemented
3	#20276541	C₂	1	4	yes	maximal, complemented
4	#20276514	C₂	1	2	yes	maximal, complemented
5	#20276514	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#13133157	C₂	1	2	yes	maximal, complemented
7	C₂×C₄×#11719908	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈×#41×C₅₃	C₂	1	1	yes	maximal, characteristic
9	C₂×#13133157	C₂²	1	4	yes	—
10	C₂²×#11719908	C₂²	1	2	yes	—
11	C₄⋊#11719908	C₂²	1	4	yes	—
12	C₄⋊#11719908	C₂²	1	2	yes	—
13	C₂²⋊#11719908	C₂²	1	2	yes	—
14	C₂²⋊#11719908	C₂²	1	4	yes	—
15	#16810399	C₂²	1	4	yes	—
16	C₄×#11719908	C₂²	1	1	yes	characteristic
17	C₄×#11719908	C₂²	1	2	yes	—
18	C₄×#11719908	C₂²	1	2	yes	—
19	C₄×#11719908	C₂²	1	1	yes	characteristic
20	C₄×#41×C₅₃	C₄	1	1	yes	characteristic
21	C₈×Q₈×C₅₃	C₄	1	1	yes	characteristic
22	C₈×Q₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₈×D₄×C₅₃	C₄	1	1	yes	characteristic
24	C₈×D₄×C₅₃	C₄	1	2	yes	—
25	C₈×D₄×C₅₃	C₂²	1	2	yes	—
26	C₈×D₄×C₅₃	C₂²	1	1	yes	characteristic
27	C₅₃×#431	C₄	1	2	yes	—
28	C₅₃×#431	C₄	1	1	yes	characteristic
29	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
31	#13133157	—	2	8	no	—
32	C₂×#11719908	—	2	4	no	—
33	C₂×#11719908	C₂³	1	2	yes	—
34	C₂×#11719908	C₂³	1	4	yes	—
35	C₂×#11719908	C₂³	1	2	yes	—
36	C₂×#41×C₅₃	C₈	1	1	yes	characteristic
37	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
39	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
40	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
41	C₄×Q₈×C₅₃	C₈	1	1	yes	characteristic
42	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₄×D₄×C₅₃	C₈	1	1	yes	characteristic
44	C₄×D₄×C₅₃	C₈	1	2	yes	—
45	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
46	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
47	#117×C₅₃	C₈	1	2	yes	—
48	#117×C₅₃	C₈	1	1	yes	characteristic
49	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
51	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
52	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
53	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
54	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
55	C₂²⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
56	C₂²⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
57	C₂²⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
58	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
59	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
60	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
61	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
62	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
63	C₄×C₈×C₅₃	C₂³	1	2	yes	—
64	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
65	#41×C₅₃	—	2	4	no	—
66	C₂×Q₈×C₅₃	C₂×C₈	1	1	yes	characteristic
67	C₂×D₄×C₅₃	C₂×C₈	1	2	yes	—
68	C₂×D₄×C₅₃	C₂×C₈	1	1	yes	characteristic
69	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
70	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
71	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
72	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
73	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
74	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈×C₅₃	—	2	2	no	—
76	C₂×C₈×C₅₃	—	2	4	no	—
77	C₂×C₈×C₅₃	—	2	2	no	—
78	C₂×C₈×C₅₃	—	2	4	no	—
79	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
83	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
85	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
86	C₄⋊C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
87	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
88	C₄⋊C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
89	C₂²⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
90	C₂²⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
91	C₂²⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
92	C₄²×C₅₃	C₂×C₈	1	2	yes	—
93	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
94	C₄²×C₅₃	C₂×C₈	1	2	yes	—
95	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
96	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
97	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
98	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
99	#11719908	—	2	2	no	—
100	#11719908	—	2	4	no	—
101	#11719908	#41	1	4	yes	complemented
102	C₂³×C₅₃	C₂²×C₈	1	2	yes	—
103	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
104	Q₈×C₅₃	—	2	2	no	—
105	D₄×C₅₃	—	2	4	no	—
106	D₄×C₅₃	—	2	2	no	—
107	C₂×C₄×C₅₃	—	2	1	no	—
108	C₂×C₄×C₅₃	—	2	2	no	—
109	C₂×C₄×C₅₃	—	2	1	no	—
110	C₂×C₄×C₅₃	—	2	2	no	—
111	C₂×C₄×C₅₃	—	2	4	no	—
112	C₂×C₄×C₅₃	—	2	2	no	—
113	C₂×C₄×C₅₃	—	2	4	no	—
114	C₂×C₄×C₅₃	M₃₂	1	4	yes	complemented
115	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
116	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
117	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
118	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
119	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
120	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
121	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
123	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
124	C₈×C₅₃	—	2	1	no	—
125	C₈×C₅₃	—	2	2	no	—
126	C₈×C₅₃	—	2	1	no	—
127	C₈×C₅₃	—	2	2	no	—
128	C₈×C₅₃	#131	1	2	yes	—
129	C₈×C₅₃	C₂×#41	1	2	yes	—
130	#33479	—	53	1	no	maximal, Sylow
131	C₂²×C₅₃	—	2	1	no	—
132	C₂²×C₅₃	—	2	1	no	—
133	C₂²×C₅₃	—	2	2	no	—
134	C₂²×C₅₃	—	2	2	no	—
135	C₂²×C₅₃	—	2	2	no	—
136	C₂²×C₅₃	C₂×M₃₂	1	2	yes	—
137	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
138	C₄×C₅₃	—	2	2	no	—
139	C₄×C₅₃	—	2	1	no	—
140	C₄×C₅₃	—	2	2	no	—
141	C₄×C₅₃	—	2	1	no	—
142	C₄×C₅₃	—	2	2	no	—
143	C₄×C₅₃	C₂×M₃₂	1	2	yes	—
144	C₄×C₅₃	C₂×M₃₂	1	2	yes	—
145	C₄×C₅₃	#504	1	2	yes	—
146	C₄×C₅₃	C₄×#41	1	1	yes	characteristic
147	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
148	C₈×#41	D₅₃	1	1	yes	characteristic
149	#2183	—	53	2	no	—
150	#2169	—	53	2	no	—
151	#2168	—	53	4	no	—
152	#2162	—	53	2	no	—
153	#2162	—	53	1	no	—
154	C₄×M₃₂	—	53	2	no	—
155	C₂×C₄×C₁₆	—	53	1	no	—
156	C₂×C₅₃	—	2	2	no	—
157	C₂×C₅₃	—	4	2	no	—
158	C₂×C₅₃	C₈×#41	1	1	yes	characteristic
159	C₂×C₅₃	C₂²×M₃₂	1	1	yes	characteristic
160	C₂×C₅₃	C₂×#504	1	1	yes	characteristic
161	C₄×#41	Dic₅₃	1	1	yes	characteristic
162	C₂×M₃₂	—	53	4	no	—
163	C₂²×C₁₆	—	53	2	no	—
164	C₈×Q₈	Dic₅₃	1	1	yes	characteristic
165	C₈×Q₈	C₂×D₅₃	1	1	yes	characteristic
166	C₈×D₄	Dic₅₃	1	1	yes	characteristic
167	C₈×D₄	Dic₅₃	1	2	yes	—
168	C₈×D₄	C₂×D₅₃	1	2	yes	—
169	C₈×D₄	C₂×D₅₃	1	1	yes	characteristic
170	#431	Dic₅₃	1	2	yes	—
171	#431	Dic₅₃	1	1	yes	characteristic
172	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
173	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
174	C₄⋊C₁₆	—	53	4	no	—
175	C₄⋊C₁₆	—	53	2	no	—
176	C₂²⋊C₁₆	—	53	2	no	—
177	C₂²⋊C₁₆	—	53	4	no	—
178	#346	—	53	4	no	—
179	C₄×C₁₆	—	53	2	no	—
180	C₄×C₁₆	—	53	1	no	—
181	C₄×C₁₆	—	53	1	no	—
182	C₄×C₁₆	—	53	2	no	—
183	C₅₃	#33479	1	1	yes	characteristic, complemented, Sylow
184	C₂×#41	#88322	1	1	yes	characteristic
185	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
186	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
187	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
188	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
189	C₄×Q₈	#88322	1	1	yes	characteristic
190	C₄×Q₈	C₂×Dic₅₃	1	1	yes	characteristic
191	C₄×D₄	#88322	1	2	yes	—
192	C₄×D₄	#88322	1	1	yes	characteristic
193	C₄×D₄	C₂×Dic₅₃	1	2	yes	—
194	C₄×D₄	C₂×Dic₅₃	1	1	yes	characteristic
195	#117	#88322	1	2	yes	—
196	#117	#88322	1	1	yes	characteristic
197	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
198	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
199	M₃₂	—	106	8	no	—
200	C₂×C₁₆	—	53	2	no	—
201	C₂×C₁₆	—	53	2	no	—
202	C₂×C₁₆	—	53	4	no	—
203	C₂×C₁₆	—	106	4	no	—
204	C₄⋊C₈	C₂×Dic₅₃	1	2	yes	—
205	C₄⋊C₈	C₂×Dic₅₃	1	1	yes	characteristic
206	C₄⋊C₈	C₂×Dic₅₃	1	2	yes	—
207	C₄⋊C₈	C₂×Dic₅₃	1	1	yes	characteristic
208	C₂²⋊C₈	C₂×Dic₅₃	1	2	yes	—
209	C₂²⋊C₈	C₂×Dic₅₃	1	2	yes	—
210	C₂²⋊C₈	C₂×Dic₅₃	1	2	yes	—
211	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
212	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
213	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
214	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
215	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
216	C₄×C₈	C₂²×D₅₃	1	2	yes	—
217	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
218	#41	—	2	4	no	—
219	C₂×Q₈	C₂×#88322	1	1	yes	characteristic
220	C₂×D₄	C₂×#88322	1	2	yes	—
221	C₂×D₄	C₂×#88322	1	1	yes	characteristic
222	C₂²×C₄	C₂×#88322	1	2	yes	—
223	C₂²×C₄	C₂×#88322	1	2	yes	—
224	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
225	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
226	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
227	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	4	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
233	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
234	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
235	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
236	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
237	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
238	C₄⋊C₄	C₂×#88322	1	2	yes	—
239	C₄⋊C₄	C₂×#88322	1	1	yes	characteristic
240	C₄⋊C₄	C₂×#88322	1	2	yes	—
241	C₄⋊C₄	C₂×#88322	1	1	yes	characteristic
242	C₂²⋊C₄	C₂×#88322	1	2	yes	—
243	C₂²⋊C₄	C₂×#88322	1	2	yes	—
244	C₂²⋊C₄	C₂×#88322	1	2	yes	—
245	C₄²	C₂×#88322	1	2	yes	—
246	C₄²	C₂×#88322	1	1	yes	characteristic
247	C₄²	C₂×#88322	1	2	yes	—
248	C₄²	C₂×#88322	1	1	yes	characteristic
249	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
250	C₄²	C₂²×Dic₅₃	1	2	yes	—
251	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
252	C₁₆	—	53	4	no	—
253	C₁₆	—	106	2	no	—
254	C₁₆	—	106	4	no	—
255	C₂³	C₂²×#88322	1	2	yes	—
256	C₂³	C₂²×#88322	1	1	yes	characteristic
257	Q₈	—	2	2	no	—
258	D₄	—	2	4	no	—
259	D₄	—	2	2	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	#13133157	1	4	yes	complemented
268	C₂×C₄	C₂²×#88322	1	2	yes	—
269	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
270	C₂×C₄	C₂²×#88322	1	2	yes	—
271	C₂×C₄	C₂²×#88322	1	2	yes	—
272	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
273	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
274	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
275	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
276	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
277	C₈	—	2	1	no	—
278	C₈	—	2	2	no	—
279	C₈	—	2	1	no	—
280	C₈	—	2	2	no	—
281	C₈	#13133224	1	2	yes	—
282	C₈	#41×D₅₃	1	2	yes	—
283	C₂²	—	2	1	no	—
284	C₂²	—	2	1	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	2	2	no	—
288	C₂²	C₂×#13133157	1	2	yes	—
289	C₂²	C₂³×#88322	1	1	yes	characteristic
290	C₄	—	2	1	no	—
291	C₄	—	2	2	no	—
292	C₄	—	2	2	no	—
293	C₄	—	2	1	no	—
294	C₄	—	2	2	no	—
295	C₄	C₂×#13133157	1	2	yes	—
296	C₄	C₂×#13133157	1	2	yes	—
297	C₄	#16811040	1	2	yes	—
298	C₄	#41×Dic₅₃	1	1	yes	characteristic
299	C₄	C₂×#13133224	1	1	yes	characteristic
300	C₂	—	2	2	no	—
301	C₂	—	4	2	no	—
302	C₂	#41×#88322	1	1	yes	characteristic
303	C₂	C₂²×#13133157	1	1	yes	characteristic
304	C₂	C₂×#16811040	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₈	C₂²×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₈×#41×C₅₃	C₂
prime radical	C₂×#41×C₅₃	C₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₈×#41	D₅₃
hypocenter	C₅₃	#33479
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄×C₈ ◁ C₈×#41

Complements

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

N	H ≅ G/N	type
1. #20276624	C₂	semidirect
2. #20276547	C₂	semidirect
3. #20276541	C₂	semidirect
4. #20276514	C₂	semidirect
5. #20276514	C₂	semidirect
6. C₄×#13133157	C₂	semidirect
7. C₂×C₄×#11719908	C₂	semidirect
101. #11719908	#41	semidirect
114. C₂×C₄×C₅₃	M₃₂	semidirect
183. C₅₃	#33479	semidirect
267. C₂×C₄	#13133157	semidirect

Maximal subgroups

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

subgroup	index
#20276624	2
#20276547	2
#20276541	2
#20276514	2
C₄×#13133157	2
C₂×C₄×#11719908	2
C₈×#41×C₅₃	2
#33479	53