Subgroups — #27402522

Order 13568.

926 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₂₁₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810717	C₂	1	2	yes	maximal
3	#20277627	C₂	1	2	yes	maximal
4	#20275446	C₂	1	2	yes	maximal
5	#20274966	C₂	1	4	yes	maximal
6	#20274717	C₂	1	2	yes	maximal
7	C₈×C₄⋊Dic₅₃	C₂	1	1	yes	maximal, characteristic
8	C₅₃×#1845	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₂₁₂	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₁₀₆	C₂²	1	2	yes	—
11	C₂×#13133182	C₂²	1	4	yes	—
12	C₄×Dic₂₁₂	C₄	1	8	yes	complemented
13	#16810717	—	2	8	no	—
14	C₂×#13133151	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
16	C₂×#13133142	C₂²	1	2	yes	—
17	C₂×#13133140	C₂²	1	4	yes	—
18	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
19	#16810519	C₂²	1	4	yes	—
20	#16810518	—	2	4	no	—
21	#16810492	—	2	2	no	—
22	C₄⋊C₄×Dic₅₃	—	2	2	no	—
23	C₄×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
24	#16810385	C₂²	1	4	yes	—
25	#16810377	C₂²	1	2	yes	—
26	C₂×#107×C₅₃	C₂²	1	2	yes	—
27	C₂×#107×C₅₃	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
29	C₅₃×#384	C₂²	1	2	yes	—
30	C₂²×Dic₁₀₆	C₂³	1	2	yes	—
31	C₂×Dic₂₁₂	C₂×C₄	1	4	yes	—
32	C₂×Dic₂₁₂	D₄	1	4	yes	—
33	C₂×Dic₂₁₂	Q₈	1	4	yes	—
34	C₂²×#88322	C₂³	1	2	yes	—
35	C₄×Dic₁₀₆	—	2	4	no	—
36	C₄×Dic₁₀₆	C₂×C₄	1	8	yes	—
37	#13133182	—	2	8	no	—
38	#13133182	—	4	4	no	—
39	C₂×C₄⋊Dic₅₃	D₄	1	1	yes	characteristic
40	C₂×C₄⋊Dic₅₃	D₄	1	1	yes	characteristic
41	C₂×C₄⋊Dic₅₃	C₂³	1	2	yes	—
42	C₂×#11719919	—	2	2	no	—
43	C₂×#11719919	—	2	4	no	—
44	C₂×#11719919	—	2	4	no	—
45	C₂×C₄×Dic₅₃	—	2	2	no	—
46	C₂×C₄×Dic₅₃	—	2	4	no	—
47	C₂×C₄×Dic₅₃	C₂³	1	2	yes	—
48	#13133151	C₂×C₄	1	4	yes	—
49	C₄⋊#88322	—	2	2	no	—
50	#13133142	C₂×C₄	1	8	yes	—
51	#13133140	—	2	8	no	—
52	C₈×Dic₅₃	—	2	4	no	—
53	#13133124	—	2	4	no	—
54	#13133124	D₄	1	2	yes	—
55	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
56	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
57	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
58	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
59	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
60	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
61	#107×C₅₃	—	2	4	no	—
62	#107×C₅₃	—	4	2	no	—
63	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
64	C₂²×C₄×C₅₃	—	2	4	no	—
65	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
68	C₂²×Dic₅₃	—	2	2	no	—
69	C₂²×Dic₅₃	—	2	2	no	—
70	C₂²×Dic₅₃	C₂×D₄	1	2	yes	—
71	C₂²×Dic₅₃	C₂×D₄	1	2	yes	—
72	C₂×Dic₁₀₆	—	2	2	no	—
73	C₂×Dic₁₀₆	—	2	2	no	—
74	C₂×Dic₁₀₆	—	2	2	no	—
75	C₂×Dic₁₀₆	C₂²×C₄	1	4	yes	—
76	C₂×Dic₁₀₆	C₂×D₄	1	4	yes	—
77	C₂×Dic₁₀₆	C₂×Q₈	1	4	yes	—
78	C₂×C₈×C₅₃	—	2	1	no	—
79	C₂×C₈×C₅₃	—	2	1	no	—
80	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
83	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
84	C₄⋊C₄×C₅₃	—	2	4	no	—
85	C₄⋊C₄×C₅₃	—	2	4	no	—
86	C₄⋊C₄×C₅₃	—	4	4	no	—
87	C₄⋊C₄×C₅₃	—	8	2	no	—
88	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
89	C₄²×C₅₃	#41	1	2	yes	—
90	C₄⋊Dic₅₃	—	2	2	no	—
91	C₄⋊Dic₅₃	—	2	2	no	—
92	C₄⋊Dic₅₃	Q₁₆	1	4	yes	complemented
93	C₄⋊Dic₅₃	C₂²×C₄	1	4	yes	—
94	#11719919	—	2	8	no	—
95	#11719919	—	4	8	no	—
96	#11719919	—	8	4	no	—
97	C₄×Dic₅₃	—	2	8	no	—
98	C₄×Dic₅₃	—	2	2	no	—
99	C₄×Dic₅₃	—	2	2	no	—
100	C₄×Dic₅₃	—	4	8	no	—
101	C₂×#88322	—	2	2	no	—
102	C₂×#88322	—	2	2	no	—
103	C₂×#88322	—	2	2	no	—
104	Dic₂₁₂	C₄⋊C₄	1	16	yes	complemented
105	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
106	C₂×C₄×C₅₃	—	2	1	no	—
107	C₂×C₄×C₅₃	—	2	1	no	—
108	C₂×C₄×C₅₃	—	4	4	no	—
109	C₂×C₄×C₅₃	—	4	4	no	—
110	C₂×C₄×C₅₃	—	4	4	no	—
111	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
112	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
114	C₂×C₄×C₅₃	#135	1	2	yes	—
115	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
116	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
117	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
118	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
119	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
120	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
121	C₂×Dic₅₃	—	2	4	no	—
122	C₂×Dic₅₃	—	2	4	no	—
123	C₂×Dic₅₃	—	2	2	no	—
124	C₂×Dic₅₃	—	2	4	no	—
125	C₂×Dic₅₃	—	2	2	no	—
126	C₂×Dic₅₃	—	2	4	no	—
127	C₂×Dic₅₃	—	2	2	no	—
128	C₂×Dic₅₃	—	2	2	no	—
129	C₂×Dic₅₃	—	2	2	no	—
130	C₂×Dic₅₃	—	4	2	no	—
131	C₂×Dic₅₃	—	4	2	no	—
132	C₂×Dic₅₃	C₂×Q₁₆	1	4	yes	—
133	Dic₁₀₆	—	2	4	no	—
134	Dic₁₀₆	—	2	4	no	—
135	Dic₁₀₆	—	2	4	no	—
136	Dic₁₀₆	C₂×C₄⋊C₄	1	8	yes	—
137	C₈×C₅₃	—	2	2	no	—
138	C₈×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
139	#88322	—	4	2	no	—
140	C₄⋊C₄×Q₁₆	—	53	1	no	maximal, Sylow
141	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
142	C₂²×C₅₃	D₄²	1	1	yes	characteristic
143	C₂²×C₅₃	D₄×Q₈	1	1	yes	characteristic
144	C₂²×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
145	C₂²×C₅₃	C₂×#135	1	1	yes	characteristic
146	C₂²×C₅₃	#576	1	1	yes	characteristic
147	C₂²×C₅₃	#578	1	1	yes	characteristic
148	C₄×C₅₃	—	2	2	no	—
149	C₄×C₅₃	—	8	4	no	—
150	C₄×C₅₃	C₄×Q₁₆	1	4	yes	complemented
151	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
152	C₄×C₅₃	#520	1	1	yes	characteristic
153	C₄×C₅₃	#547	1	1	yes	characteristic
154	C₄×C₅₃	#554	1	1	yes	characteristic
155	Dic₅₃	—	2	8	no	—
156	Dic₅₃	—	2	4	no	—
157	Dic₅₃	—	4	4	no	—
158	Dic₅₃	—	4	2	no	—
159	C₂×C₄×Q₁₆	—	53	2	no	—
160	C₂×C₄×Q₁₆	—	53	1	no	—
161	Q₈×C₄⋊C₄	—	53	2	no	—
162	#1942	—	53	2	no	—
163	#1863	—	53	4	no	—
164	#1845	D₅₃	1	1	yes	characteristic
165	#1803	—	53	2	no	—
166	C₈×C₄⋊C₄	—	53	1	no	—
167	C₂×C₅₃	D₄×C₄⋊C₄	1	1	yes	characteristic
168	C₂×C₅₃	C₂×C₄×Q₁₆	1	1	yes	characteristic
169	C₂×C₅₃	#2951	1	1	yes	characteristic
170	C₂×C₅₃	D₄×Q₁₆	1	1	yes	characteristic
171	C₂×C₅₃	#3287	1	1	yes	characteristic
172	C₂×C₅₃	Q₈×Q₁₆	1	1	yes	characteristic
173	C₂×C₅₃	#3383	1	1	yes	characteristic
174	C₂²×Q₁₆	—	53	1	no	—
175	C₂×C₄×Q₈	—	53	4	no	—
176	C₂×C₄×Q₈	—	53	2	no	—
177	C₄×Q₁₆	—	53	8	no	—
178	C₄×Q₁₆	—	106	8	no	—
179	C₂×#107	—	53	1	no	—
180	C₂×#107	C₂×D₅₃	1	2	yes	—
181	C₂×#107	C₂×D₅₃	1	2	yes	—
182	C₂×C₄⋊C₈	—	53	1	no	—
183	C₂×#103	—	53	2	no	—
184	C₂×#103	—	53	4	no	—
185	C₂×C₄×C₈	—	53	2	no	—
186	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
187	#389	—	53	4	no	—
188	#389	—	106	4	no	—
189	#384	—	106	2	no	—
190	#384	C₂×D₅₃	1	2	yes	—
191	C₄×C₄⋊C₄	—	53	1	no	—
192	C₄×C₄⋊C₄	—	106	2	no	—
193	#340	—	53	4	no	—
194	#336	—	53	2	no	—
195	C₅₃	C₄⋊C₄×Q₁₆	1	1	yes	characteristic, complemented, Sylow
196	C₂²×Q₈	—	53	2	no	—
197	C₂×Q₁₆	—	53	4	no	—
198	C₂×Q₁₆	—	53	4	no	—
199	C₂×Q₁₆	—	53	4	no	—
200	C₂²×C₈	—	53	2	no	—
201	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
202	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
203	C₄×Q₈	—	53	8	no	—
204	C₄×Q₈	—	106	4	no	—
205	C₄×Q₈	—	106	8	no	—
206	C₄×Q₈	—	212	4	no	—
207	C₂×C₄⋊C₄	—	2	2	no	—
208	C₂×C₄⋊C₄	—	2	2	no	—
209	C₂×C₄⋊C₄	—	53	1	no	—
210	C₂×C₄⋊C₄	—	53	2	no	—
211	C₂×C₄⋊C₄	—	53	1	no	—
212	C₂×C₄⋊C₄	—	106	2	no	—
213	C₂×C₄⋊C₄	—	106	4	no	—
214	C₂×C₄⋊C₄	—	106	4	no	—
215	C₂×C₄⋊C₄	C₂²×D₅₃	1	4	yes	—
216	C₂×C₄²	—	53	2	no	—
217	C₂×C₄²	—	106	2	no	—
218	C₂×C₄²	—	106	4	no	—
219	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
220	#107	—	2	4	no	—
221	#107	—	4	2	no	—
222	#107	—	53	4	no	—
223	C₄⋊C₈	—	106	2	no	—
224	#103	—	53	8	no	—
225	#103	—	106	8	no	—
226	C₄×C₈	—	106	4	no	—
227	C₄×C₈	C₄×D₅₃	1	4	yes	—
228	#95	—	53	2	no	—
229	#95	—	106	4	no	—
230	C₂×Q₈	—	53	4	no	—
231	C₂×Q₈	—	53	4	no	—
232	C₂×Q₈	—	53	4	no	—
233	C₂×Q₈	—	106	2	no	—
234	C₂×Q₈	—	106	2	no	—
235	C₂×Q₈	—	106	2	no	—
236	C₂²×C₄	—	2	4	no	—
237	C₂²×C₄	—	53	2	no	—
238	C₂²×C₄	—	53	2	no	—
239	C₂²×C₄	—	106	2	no	—
240	C₂²×C₄	—	106	2	no	—
241	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
242	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
243	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
244	Q₁₆	—	53	16	no	—
245	C₂×C₈	—	2	1	no	—
246	C₂×C₈	—	2	1	no	—
247	C₂×C₈	—	106	2	no	—
248	C₂×C₈	—	106	2	no	—
249	C₂×C₈	—	106	2	no	—
250	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
251	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
252	C₂×C₈	D₄×D₅₃	1	2	yes	—
253	C₂×C₈	Q₈×D₅₃	1	2	yes	—
254	C₄⋊C₄	—	2	4	no	—
255	C₄⋊C₄	—	2	4	no	—
256	C₄⋊C₄	—	4	4	no	—
257	C₄⋊C₄	—	8	2	no	—
258	C₄⋊C₄	—	53	4	no	—
259	C₄⋊C₄	—	53	4	no	—
260	C₄⋊C₄	—	106	8	no	—
261	C₄⋊C₄	—	106	2	no	—
262	C₄⋊C₄	—	106	2	no	—
263	C₄⋊C₄	—	424	4	no	—
264	C₄⋊C₄	—	212	8	no	—
265	C₄²	—	106	8	no	—
266	C₄²	—	106	2	no	—
267	C₄²	—	106	2	no	—
268	C₄²	—	212	8	no	—
269	C₄²	C₂×C₄×D₅₃	1	2	yes	—
270	C₄²	#11719949	1	2	yes	—
271	C₂³	C₂×D₄×D₅₃	1	1	yes	characteristic
272	Q₈	—	53	8	no	—
273	Q₈	—	106	4	no	—
274	Q₈	—	106	4	no	—
275	Q₈	—	106	4	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	4	4	no	—
279	C₂×C₄	—	4	4	no	—
280	C₂×C₄	—	4	4	no	—
281	C₂×C₄	—	53	4	no	—
282	C₂×C₄	—	106	4	no	—
283	C₂×C₄	—	106	4	no	—
284	C₂×C₄	—	106	2	no	—
285	C₂×C₄	—	106	2	no	—
286	C₂×C₄	—	106	4	no	—
287	C₂×C₄	—	106	4	no	—
288	C₂×C₄	—	106	2	no	—
289	C₂×C₄	—	106	2	no	—
290	C₂×C₄	—	106	2	no	—
291	C₂×C₄	—	212	2	no	—
292	C₂×C₄	—	212	2	no	—
293	C₂×C₄	#13133179	1	2	yes	—
294	C₂×C₄	#13133179	1	2	yes	—
295	C₂×C₄	Q₁₆×D₅₃	1	2	yes	—
296	C₂×C₄	#13133236	1	2	yes	—
297	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
298	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
299	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
300	C₂×C₄	C₂×#11719949	1	1	yes	characteristic
301	C₂×C₄	#13133262	1	1	yes	characteristic
302	C₂×C₄	#13133265	1	1	yes	characteristic
303	C₈	—	2	2	no	—
304	C₈	—	212	2	no	—
305	C₈	C₄⋊C₄×D₅₃	1	4	yes	—
306	C₂²	C₂×#13133179	1	1	yes	characteristic
307	C₂²	#16811258	1	1	yes	characteristic
308	C₂²	#16811288	1	1	yes	characteristic
309	C₂²	C₂×Q₁₆×D₅₃	1	1	yes	characteristic
310	C₂²	C₂×#13133236	1	1	yes	characteristic
311	C₂²	#16811424	1	1	yes	characteristic
312	C₂²	#16811436	1	1	yes	characteristic
313	C₄	—	2	2	no	—
314	C₄	—	8	4	no	—
315	C₄	—	106	4	no	—
316	C₄	—	106	8	no	—
317	C₄	—	212	2	no	—
318	C₄	—	212	4	no	—
319	C₄	#16810717	1	4	yes	complemented
320	C₄	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
321	C₄	#16811104	1	1	yes	characteristic
322	C₄	#16811276	1	1	yes	characteristic
323	C₄	#16811334	1	1	yes	characteristic
324	C₂	#20277615	1	1	yes	characteristic
325	C₂	C₂×#16810717	1	1	yes	characteristic
326	C₂	#20283589	1	1	yes	characteristic
327	C₂	#20286465	1	1	yes	characteristic
328	C₂	#20286473	1	1	yes	characteristic
329	C₂	#20287598	1	1	yes	characteristic
330	C₂	#20287608	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1845	C₂
socle, prime radical	C₂³×C₅₃	C₂²×D₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1845	D₅₃
hypocenter	C₅₃	C₄⋊C₄×Q₁₆
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

Complements

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

N	H ≅ G/N	type
12. C₄×Dic₂₁₂	C₄	semidirect
92. C₄⋊Dic₅₃	Q₁₆	semidirect
104. Dic₂₁₂	C₄⋊C₄	semidirect
150. C₄×C₅₃	C₄×Q₁₆	semidirect
195. C₅₃	C₄⋊C₄×Q₁₆	semidirect
319. C₄	#16810717	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₂₁₂	2
C₂×#16810717	2
#20277627	2
#20275446	2
#20274966	2
#20274717	2
C₈×C₄⋊Dic₅₃	2
C₅₃×#1845	2
C₄⋊C₄×Q₁₆	53