Subgroups — #27537687

Order 13568.

626 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20276547	C₂	1	2	yes	maximal, complemented
2	#20276541	C₂	1	4	yes	maximal, complemented
3	D₄×#11719908	C₂	1	2	yes	maximal, complemented
4	#20276518	C₂	1	1	yes	maximal, characteristic, complemented
5	#20276514	C₂	1	1	yes	maximal, characteristic, complemented
6	#20276326	C₂	1	2	yes	maximal, complemented
7	C₂×C₂²⋊#11719908	C₂	1	2	yes	maximal, complemented
8	C₂×C₈×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic
9	C₂×#13133157	C₂²	1	4	yes	complemented
10	C₂²×#11719908	C₂²	1	4	yes	complemented
11	C₄⋊#11719908	C₂²	1	2	yes	complemented
12	C₄⋊#11719908	C₂²	1	2	yes	complemented
13	C₂²⋊#11719908	C₂²	1	1	yes	characteristic
14	C₂²⋊#11719908	C₂²	1	1	yes	characteristic
15	C₂²⋊#11719908	C₂²	1	2	yes	—
16	C₂²⋊#11719908	C₂²	1	4	yes	complemented
17	C₂²⋊#11719908	C₂²	1	4	yes	complemented
18	#16810399	C₂²	1	2	yes	complemented
19	C₄×#11719908	C₂²	1	2	yes	complemented
20	C₂³×C₈×C₅₃	C₂²	1	2	yes	—
21	C₂×C₄×D₄×C₅₃	C₄	1	1	yes	characteristic
22	C₈×D₄×C₅₃	C₄	1	2	yes	—
23	C₈×D₄×C₅₃	C₄	1	1	yes	characteristic
24	C₈×D₄×C₅₃	C₄	1	1	yes	characteristic
25	C₈×D₄×C₅₃	C₂²	1	1	yes	characteristic
26	C₈×D₄×C₅₃	C₂²	1	2	yes	—
27	C₈×D₄×C₅₃	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊C₈×C₅₃	C₄	1	1	yes	characteristic
29	C₂×C₂²⋊C₈×C₅₃	C₄	1	2	yes	—
30	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
31	#13133157	—	4	4	no	—
32	C₂×#11719908	—	4	4	no	—
33	C₂×#11719908	C₂³	1	2	yes	—
34	C₂×#11719908	C₂³	1	4	yes	—
35	C₂×#11719908	C₂³	1	2	yes	—
36	C₂²×D₄×C₅₃	C₈	1	1	yes	characteristic
37	C₂³×C₄×C₅₃	C₂×C₄	1	2	yes	—
38	C₂²×C₈×C₅₃	—	4	4	no	—
39	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
40	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
41	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
42	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
44	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
45	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
46	C₄×D₄×C₅₃	C₈	1	1	yes	characteristic
47	C₄×D₄×C₅₃	C₈	1	2	yes	—
48	C₄×D₄×C₅₃	C₈	1	1	yes	characteristic
49	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
50	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
51	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄⋊C₄×C₅₃	C₈	1	1	yes	characteristic
53	C₂×C₂²⋊C₄×C₅₃	C₈	1	2	yes	—
54	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
55	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
56	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
57	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
58	C₂²⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
59	C₂²⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
60	C₂²⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
61	C₂²⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
62	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
63	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
64	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
65	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
66	C₂⁴×C₅₃	C₂×C₈	1	2	yes	—
67	C₂×D₄×C₅₃	—	2	4	no	—
68	C₂×D₄×C₅₃	C₂×C₈	1	2	yes	—
69	C₂×D₄×C₅₃	C₂×C₈	1	1	yes	characteristic
70	C₂×D₄×C₅₃	C₂×C₈	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	—	4	2	no	—
72	C₂²×C₄×C₅₃	—	4	2	no	—
73	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
74	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
75	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
76	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
77	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
78	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₅₃	—	2	2	no	—
82	C₂×C₈×C₅₃	—	2	4	no	—
83	C₂×C₈×C₅₃	—	2	4	no	—
84	C₂×C₈×C₅₃	—	4	2	no	—
85	C₂×C₈×C₅₃	—	4	2	no	—
86	C₂×C₈×C₅₃	—	4	2	no	—
87	C₂×C₈×C₅₃	—	4	1	no	—
88	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
92	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
93	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
94	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
95	C₄⋊C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
96	C₄⋊C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
97	C₂²⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
98	C₂²⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
99	C₂²⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
100	C₂²⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
101	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
102	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
103	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
104	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
105	#11719908	—	2	2	no	—
106	#11719908	—	2	4	no	—
107	#11719908	—	2	2	no	—
108	C₂³×C₅₃	—	2	4	no	—
109	C₂³×C₅₃	—	2	4	no	—
110	C₂³×C₅₃	—	4	2	no	—
111	C₂³×C₅₃	C₂²×C₈	1	2	yes	—
112	C₂³×C₅₃	C₂²×C₈	1	2	yes	—
113	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
114	D₄×C₅₃	—	2	4	no	—
115	D₄×C₅₃	—	2	2	no	—
116	D₄×C₅₃	M₃₂	1	4	yes	complemented
117	C₂×C₄×C₅₃	—	2	2	no	—
118	C₂×C₄×C₅₃	—	2	1	no	—
119	C₂×C₄×C₅₃	—	2	2	no	—
120	C₂×C₄×C₅₃	—	2	2	no	—
121	C₂×C₄×C₅₃	—	2	1	no	—
122	C₂×C₄×C₅₃	—	2	2	no	—
123	C₂×C₄×C₅₃	—	2	2	no	—
124	C₂×C₄×C₅₃	—	2	2	no	—
125	C₂×C₄×C₅₃	—	4	2	no	—
126	C₂×C₄×C₅₃	—	4	2	no	—
127	C₂×C₄×C₅₃	—	4	2	no	—
128	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
129	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
130	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
131	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
132	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
133	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
134	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
135	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
136	C₈×C₅₃	—	2	1	no	—
137	C₈×C₅₃	—	2	2	no	—
138	C₈×C₅₃	—	2	2	no	—
139	C₈×C₅₃	—	2	1	no	—
140	C₈×C₅₃	—	2	1	no	—
141	C₈×C₅₃	H(5,𝔽₂)	1	2	yes	—
142	#33488	—	53	1	no	maximal, Sylow
143	C₂²×C₅₃	—	2	1	no	—
144	C₂²×C₅₃	—	2	2	no	—
145	C₂²×C₅₃	—	2	1	no	—
146	C₂²×C₅₃	—	2	2	no	—
147	C₂²×C₅₃	—	2	2	no	—
148	C₂²×C₅₃	—	2	2	no	—
149	C₂²×C₅₃	—	2	2	no	—
150	C₂²×C₅₃	—	4	8	no	—
151	C₂²×C₅₃	C₂×M₃₂	1	4	yes	—
152	C₂²×C₅₃	#504	1	2	yes	—
153	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
154	C₄×C₅₃	—	2	1	no	—
155	C₄×C₅₃	—	2	1	no	—
156	C₄×C₅₃	—	2	1	no	—
157	C₄×C₅₃	—	2	1	no	—
158	C₄×C₅₃	—	2	2	no	—
159	C₄×C₅₃	—	2	2	no	—
160	C₄×C₅₃	C₂×M₃₂	1	2	yes	—
161	C₄×C₅₃	#518	1	1	yes	characteristic
162	C₄×C₅₃	#568	1	1	yes	characteristic
163	C₂×C₈×D₄	D₅₃	1	1	yes	characteristic
164	#2169	—	53	2	no	—
165	#2168	—	53	4	no	—
166	D₄×C₁₆	—	53	2	no	—
167	#2163	—	53	1	no	—
168	#2162	—	53	1	no	—
169	#2112	—	53	2	no	—
170	C₂×C₂²⋊C₁₆	—	53	2	no	—
171	C₂×C₅₃	—	2	2	no	—
172	C₂×C₅₃	—	2	4	no	—
173	C₂×C₅₃	—	4	2	no	—
174	C₂×C₅₃	#2972	1	1	yes	characteristic
175	C₂×C₅₃	C₂²×M₃₂	1	1	yes	characteristic
176	C₂×C₅₃	C₂×#504	1	1	yes	characteristic
177	C₂³×C₈	C₂×D₅₃	1	2	yes	—
178	C₂×C₄×D₄	Dic₅₃	1	1	yes	characteristic
179	C₂×M₃₂	—	53	4	no	—
180	C₂²×C₁₆	—	53	4	no	—
181	C₈×D₄	Dic₅₃	1	2	yes	—
182	C₈×D₄	Dic₅₃	1	1	yes	characteristic
183	C₈×D₄	Dic₅₃	1	1	yes	characteristic
184	C₈×D₄	C₂×D₅₃	1	1	yes	characteristic
185	C₈×D₄	C₂×D₅₃	1	2	yes	—
186	C₈×D₄	C₂×D₅₃	1	1	yes	characteristic
187	C₂×C₄⋊C₈	Dic₅₃	1	1	yes	characteristic
188	C₂×C₂²⋊C₈	Dic₅₃	1	2	yes	—
189	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
190	C₄⋊C₁₆	—	53	2	no	—
191	C₄⋊C₁₆	—	53	2	no	—
192	C₂²⋊C₁₆	—	53	1	no	—
193	C₂²⋊C₁₆	—	53	2	no	—
194	C₂²⋊C₁₆	—	53	4	no	—
195	C₂²⋊C₁₆	—	53	1	no	—
196	C₂²⋊C₁₆	—	53	4	no	—
197	#346	—	53	2	no	—
198	C₄×C₁₆	—	53	2	no	—
199	C₅₃	#33488	1	1	yes	characteristic, complemented, Sylow
200	C₂²×D₄	#88322	1	1	yes	characteristic
201	C₂³×C₄	C₂×Dic₅₃	1	2	yes	—
202	C₂²×C₈	—	4	4	no	—
203	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
204	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
205	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
206	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
207	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
208	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
209	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
210	C₄×D₄	#88322	1	2	yes	—
211	C₄×D₄	#88322	1	1	yes	characteristic
212	C₄×D₄	#88322	1	1	yes	characteristic
213	C₄×D₄	C₂×Dic₅₃	1	2	yes	—
214	C₄×D₄	C₂×Dic₅₃	1	1	yes	characteristic
215	C₄×D₄	C₂×Dic₅₃	1	1	yes	characteristic
216	C₂×C₄⋊C₄	#88322	1	1	yes	characteristic
217	C₂×C₂²⋊C₄	#88322	1	2	yes	—
218	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
219	M₃₂	—	212	4	no	—
220	C₂×C₁₆	—	53	2	no	—
221	C₂×C₁₆	—	53	2	no	—
222	C₂×C₁₆	—	53	4	no	—
223	C₂×C₁₆	—	212	4	no	—
224	C₄⋊C₈	C₂×Dic₅₃	1	1	yes	characteristic
225	C₄⋊C₈	C₂×Dic₅₃	1	2	yes	—
226	C₄⋊C₈	C₂×Dic₅₃	1	1	yes	characteristic
227	C₂²⋊C₈	C₂×Dic₅₃	1	2	yes	—
228	C₂²⋊C₈	C₂×Dic₅₃	1	2	yes	—
229	C₂²⋊C₈	C₂×Dic₅₃	1	2	yes	—
230	C₂²⋊C₈	C₂×Dic₅₃	1	2	yes	—
231	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
232	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
233	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
234	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
235	C₂⁴	C₂×#88322	1	2	yes	—
236	C₂×D₄	—	2	4	no	—
237	C₂×D₄	C₂×#88322	1	2	yes	—
238	C₂×D₄	C₂×#88322	1	1	yes	characteristic
239	C₂×D₄	C₂×#88322	1	1	yes	characteristic
240	C₂²×C₄	—	4	2	no	—
241	C₂²×C₄	—	4	2	no	—
242	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
243	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
244	C₂²×C₄	C₂×#88322	1	2	yes	—
245	C₂²×C₄	C₂×#88322	1	2	yes	—
246	C₂²×C₄	C₂×#88322	1	2	yes	—
247	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
248	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
249	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
250	C₂×C₈	—	2	4	no	—
251	C₂×C₈	—	2	4	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	4	2	no	—
254	C₂×C₈	—	4	2	no	—
255	C₂×C₈	—	4	2	no	—
256	C₂×C₈	—	4	1	no	—
257	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
258	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
259	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
260	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
261	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
262	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
263	C₄⋊C₄	C₂×#88322	1	2	yes	—
264	C₄⋊C₄	C₂×#88322	1	1	yes	characteristic
265	C₄⋊C₄	C₂×#88322	1	1	yes	characteristic
266	C₂²⋊C₄	C₂×#88322	1	2	yes	—
267	C₂²⋊C₄	C₂×#88322	1	2	yes	—
268	C₂²⋊C₄	C₂×#88322	1	2	yes	—
269	C₂²⋊C₄	C₂×#88322	1	2	yes	—
270	C₄²	C₂×#88322	1	1	yes	characteristic
271	C₄²	C₂×#88322	1	1	yes	characteristic
272	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
273	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
274	C₁₆	—	106	2	no	—
275	C₁₆	—	106	4	no	—
276	C₁₆	—	106	2	no	—
277	C₂³	—	2	4	no	—
278	C₂³	—	2	4	no	—
279	C₂³	—	4	2	no	—
280	C₂³	C₂²×#88322	1	2	yes	—
281	C₂³	C₂²×#88322	1	2	yes	—
282	C₂³	C₂²×#88322	1	1	yes	characteristic
283	D₄	—	2	4	no	—
284	D₄	—	2	2	no	—
285	D₄	#13133157	1	4	yes	complemented
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
298	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
299	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
300	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
301	C₂×C₄	C₂²×#88322	1	2	yes	—
302	C₂×C₄	C₂²×#88322	1	2	yes	—
303	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
304	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
305	C₈	—	2	1	no	—
306	C₈	—	2	2	no	—
307	C₈	—	2	2	no	—
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	#13133262	1	2	yes	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	2	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	2	no	—
315	C₂²	—	2	2	no	—
316	C₂²	—	2	2	no	—
317	C₂²	—	2	2	no	—
318	C₂²	—	4	8	no	—
319	C₂²	C₂×#13133157	1	4	yes	—
320	C₂²	#16811040	1	2	yes	—
321	C₂²	C₂³×#88322	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	2	2	no	—
327	C₄	—	2	2	no	—
328	C₄	C₂×#13133157	1	2	yes	—
329	C₄	#16811098	1	1	yes	characteristic
330	C₄	#16811390	1	1	yes	characteristic
331	C₂	—	2	2	no	—
332	C₂	—	2	4	no	—
333	C₂	—	4	2	no	—
334	C₂	#20283739	1	1	yes	characteristic
335	C₂	C₂²×#13133157	1	1	yes	characteristic
336	C₂	C₂×#16811040	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	C₂×C₈×D₄×C₅₃	C₂
prime radical	C₂²×D₄×C₅₃	C₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂×C₈×D₄	D₅₃
hypocenter	C₅₃	#33488
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₈×D₄

Complements

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

N	H ≅ G/N	type
1. #20276547	C₂	semidirect
2. #20276541	C₂	semidirect
3. D₄×#11719908	C₂	semidirect
4. #20276518	C₂	semidirect
5. #20276514	C₂	semidirect
6. #20276326	C₂	semidirect
7. C₂×C₂²⋊#11719908	C₂	semidirect
9. C₂×#13133157	C₂²	semidirect
10. C₂²×#11719908	C₂²	semidirect
11. C₄⋊#11719908	C₂²	semidirect
12. C₄⋊#11719908	C₂²	semidirect
16. C₂²⋊#11719908	C₂²	semidirect
17. C₂²⋊#11719908	C₂²	semidirect
18. #16810399	C₂²	semidirect
19. C₄×#11719908	C₂²	semidirect
116. D₄×C₅₃	M₃₂	semidirect
199. C₅₃	#33488	semidirect
285. D₄	#13133157	semidirect

Maximal subgroups

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

subgroup	index
#20276547	2
#20276541	2
D₄×#11719908	2
#20276518	2
#20276514	2
#20276326	2
C₂×C₂²⋊#11719908	2
C₂×C₈×D₄×C₅₃	2
#33488	53