Subgroups — #28316362

Order 13568.

750 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20291521	C₂	1	2	yes	maximal, complemented
2	Q₁₆×#88324	C₂	1	2	yes	maximal, complemented
3	#20290299	C₂	1	4	yes	maximal, complemented
4	#20290296	C₂	1	2	yes	maximal, complemented
5	#20290265	C₂	1	1	yes	maximal, characteristic, complemented
6	#20290009	C₂	1	2	yes	maximal, complemented
7	C₈×D₅₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#13133240	C₂	1	1	yes	maximal, characteristic
9	Q₈×#88324	C₂²	1	4	yes	—
10	#16811651	—	2	4	no	—
11	#16811651	C₂²	1	4	yes	—
12	#16811642	—	2	2	no	—
13	#16811642	C₂²	1	2	yes	—
14	C₄×D₅₃⋊C₈	—	2	2	no	—
15	C₄×D₅₃⋊C₈	C₂²	1	1	yes	characteristic
16	C₄×D₅₃⋊C₈	C₂²	1	1	yes	characteristic
17	#16811486	C₂²	1	2	yes	—
18	#16811485	C₂²	1	2	yes	—
19	Q₈⋊#88324	C₂²	1	4	yes	—
20	#16811471	C₂²	1	4	yes	—
21	C₈×#88324	C₂²	1	2	yes	—
22	C₈×#88324	C₂²	1	2	yes	—
23	C₂×#13133240	C₄	1	1	yes	characteristic
24	C₄×#11719949	C₂²	1	2	yes	—
25	Q₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
26	C₄×#11719924	C₄	1	2	yes	—
27	#16810708	C₂²	1	2	yes	—
28	C₄×D₄₂₄	C₄	1	1	yes	characteristic
29	#16810705	C₂²	1	1	yes	characteristic
30	#16810660	C₄	1	1	yes	characteristic
31	#16810618	C₄	1	2	yes	—
32	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
33	C₄×Q₁₆×C₅₃	C₄	1	1	yes	characteristic
34	C₂×D₅₃⋊C₈	—	2	1	no	—
35	C₂×D₅₃⋊C₈	—	2	4	no	—
36	C₂×D₅₃⋊C₈	D₄	1	2	yes	—
37	C₄⋊#88324	—	2	4	no	—
38	C₄⋊#88324	C₂³	1	4	yes	—
39	#13133274	—	2	2	no	—
40	#13133274	—	2	4	no	—
41	C₄×#88324	—	2	2	no	—
42	C₄×#88324	—	2	1	no	—
43	C₄×#88324	—	2	4	no	—
44	C₄×#88324	D₄	1	1	yes	characteristic
45	C₄×#88324	D₄	1	1	yes	characteristic
46	C₄×#88324	C₂³	1	2	yes	—
47	C₄×#88324	C₂³	1	2	yes	—
48	C₂×#11719949	C₂×C₄	1	2	yes	—
49	#13133240	C₈	1	8	yes	complemented
50	C₂×#11719924	C₂×C₄	1	2	yes	—
51	C₂×D₄₂₄	C₂×C₄	1	1	yes	characteristic
52	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
53	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
54	Q₈×Dic₅₃	C₂³	1	2	yes	—
55	C₄×D₂₁₂	—	2	2	no	—
56	C₄×D₂₁₂	C₂×C₄	1	2	yes	—
57	#13133179	—	2	2	no	—
58	#13133179	C₂³	1	2	yes	—
59	#13133175	—	2	2	no	—
60	#13133175	C₂×C₄	1	2	yes	—
61	C₄²×D₅₃	—	2	2	no	—
62	C₄²×D₅₃	C₂³	1	1	yes	characteristic
63	#13133151	C₂×C₄	1	1	yes	characteristic
64	#13133148	C₂×C₄	1	2	yes	—
65	#13133141	C₂×C₄	1	2	yes	—
66	#13133139	C₂×C₄	1	2	yes	—
67	#13133137	C₂×C₄	1	2	yes	—
68	C₄×#88322	C₂×C₄	1	1	yes	characteristic
69	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
70	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
71	C₂×Q₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
72	C₄×Q₈×C₅₃	C₂×C₄	1	2	yes	—
73	#107×C₅₃	C₂×C₄	1	1	yes	characteristic
74	#103×C₅₃	C₂×C₄	1	2	yes	—
75	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
76	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
77	#11719949	—	2	4	no	—
78	#11719949	C₂×C₈	1	8	yes	complemented
79	C₂×D₂₁₂	—	2	2	no	—
80	C₂×D₂₁₂	C₂²×C₄	1	2	yes	—
81	C₂×C₄×D₅₃	—	2	2	no	—
82	C₂×C₄×D₅₃	—	2	2	no	—
83	C₂×C₄×D₅₃	—	2	2	no	—
84	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₄×D₅₃	#41	1	1	yes	characteristic
87	C₂×#88324	—	2	2	no	—
88	C₂×#88324	—	2	4	no	—
89	C₂×#88324	C₂×D₄	1	2	yes	—
90	C₂×#88324	C₂×D₄	1	2	yes	—
91	D₅₃⋊C₈	—	2	4	no	—
92	D₅₃⋊C₈	—	2	4	no	—
93	D₅₃⋊C₈	—	4	8	no	—
94	Q₁₆×C₅₃	C₂×C₈	1	4	yes	complemented
95	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
97	C₄⋊C₄×C₅₃	—	2	2	no	—
98	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
99	C₄²×C₅₃	—	2	2	no	—
100	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
101	#11719924	C₂×C₈	1	8	yes	complemented
102	#11719921	—	2	2	no	—
103	#11719921	—	2	2	no	—
104	#11719921	—	2	2	no	—
105	C₄⋊Dic₅₃	—	2	2	no	—
106	C₄⋊Dic₅₃	C₂²×C₄	1	2	yes	—
107	C₄×Dic₅₃	—	2	2	no	—
108	C₄×Dic₅₃	—	2	2	no	—
109	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
110	C₄×Dic₅₃	#41	1	1	yes	characteristic
111	C₄×Dic₅₃	C₂⁴	1	1	yes	characteristic
112	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
113	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
114	D₄₂₄	C₂×C₈	1	4	yes	complemented
115	C₈×D₅₃	—	2	2	no	—
116	C₈×D₅₃	C₂×C₈	1	4	yes	complemented
117	C₂²×D₅₃	—	2	2	no	—
118	C₂²×D₅₃	C₄×D₄	1	1	yes	characteristic
119	Q₈×C₅₃	—	2	2	no	—
120	Q₈×C₅₃	C₂²×C₈	1	4	yes	—
121	C₂×C₄×C₅₃	—	2	2	no	—
122	C₂×C₄×C₅₃	—	2	2	no	—
123	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
125	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
126	C₂×Dic₅₃	—	2	2	no	—
127	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
128	C₂×Dic₅₃	C₂²×D₄	1	1	yes	characteristic
129	C₂×Dic₅₃	C₂×#41	1	1	yes	characteristic
130	D₂₁₂	—	2	8	no	—
131	D₂₁₂	—	2	2	no	—
132	D₂₁₂	C₂²×C₈	1	4	yes	—
133	C₄×D₅₃	—	2	2	no	—
134	C₄×D₅₃	—	2	2	no	—
135	C₄×D₅₃	—	2	8	no	—
136	C₄×D₅₃	—	4	4	no	—
137	C₄×D₅₃	—	4	4	no	—
138	C₄×D₅₃	C₂²×C₈	1	2	yes	—
139	C₄×D₅₃	#131	1	2	yes	—
140	#88324	—	2	4	no	—
141	#88324	—	2	2	no	—
142	#88324	—	4	4	no	—
143	#88324	#135	1	4	yes	complemented
144	C₈×C₅₃	—	2	1	no	—
145	C₈×C₅₃	C₂²×C₈	1	2	yes	—
146	#88322	—	2	1	no	—
147	#88322	C₂²×C₈	1	2	yes	—
148	C₈×#135	—	53	1	no	maximal, Sylow
149	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
150	C₂×D₅₃	—	2	1	no	—
151	C₂×D₅₃	—	2	1	no	—
152	C₂×D₅₃	—	2	4	no	—
153	C₂×D₅₃	—	4	2	no	—
154	C₂×D₅₃	—	4	2	no	—
155	C₂×D₅₃	C₈×D₄	1	2	yes	—
156	C₄×C₅₃	—	2	4	no	—
157	C₄×C₅₃	—	2	1	no	—
158	C₄×C₅₃	—	4	2	no	—
159	C₄×C₅₃	#443	1	2	yes	—
160	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
161	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
162	Dic₅₃	—	2	1	no	—
163	Dic₅₃	—	4	2	no	—
164	Dic₅₃	C₈×D₄	1	2	yes	—
165	Dic₅₃	C₂×#135	1	2	yes	—
166	C₈×#41	—	53	2	no	—
167	C₄×#135	—	53	1	no	—
168	C₈×Q₁₆	—	53	2	no	—
169	C₈×SD₁₆	—	53	4	no	—
170	C₈×D₈	—	53	2	no	—
171	#1564	—	53	1	no	—
172	#1476	—	53	2	no	—
173	C₂×C₈²	—	53	1	no	—
174	C₂×C₅₃	C₂×C₈×D₄	1	1	yes	characteristic
175	C₂×C₅₃	C₄×#135	1	1	yes	characteristic
176	C₂×C₅₃	C₂×#443	1	1	yes	characteristic
177	D₅₃	—	2	2	no	—
178	D₅₃	—	4	4	no	—
179	C₂×#135	—	53	1	no	—
180	C₄×#41	—	53	2	no	—
181	C₈×Q₈	—	53	4	no	—
182	C₄×Q₁₆	—	53	1	no	—
183	C₄×Q₁₆	#6157	1	1	yes	characteristic
184	C₄×SD₁₆	—	53	2	no	—
185	C₄×SD₁₆	—	53	2	no	—
186	C₄×D₈	—	53	1	no	—
187	C₄×D₈	—	53	1	no	—
188	C₈×D₄	—	53	4	no	—
189	C₈×D₄	—	106	4	no	—
190	#431	—	53	2	no	—
191	#431	—	106	2	no	—
192	#427	—	53	1	no	—
193	#416	—	53	2	no	—
194	C₂×C₄×C₈	—	53	1	no	—
195	C₂×C₄×C₈	—	53	1	no	—
196	C₂×C₄×C₈	—	53	1	no	—
197	C₂×C₄×C₈	—	106	2	no	—
198	#335	—	53	2	no	—
199	#334	—	53	2	no	—
200	Q₈⋊C₈	—	53	4	no	—
201	D₄⋊C₈	—	53	4	no	—
202	C₈²	—	53	2	no	—
203	C₈²	—	53	2	no	—
204	C₅₃	C₈×#135	1	1	yes	characteristic, complemented, Sylow
205	C₂×#41	—	53	2	no	—
206	#135	—	53	8	no	—
207	C₂×Q₁₆	C₂×#6157	1	1	yes	characteristic
208	C₂×SD₁₆	—	53	2	no	—
209	C₂×D₈	—	53	1	no	—
210	C₂²×C₈	—	53	1	no	—
211	C₂²×C₈	—	53	2	no	—
212	C₂²×C₈	—	53	1	no	—
213	C₂²×C₈	—	106	4	no	—
214	C₂²×C₈	—	106	1	no	—
215	C₄×Q₈	—	53	2	no	—
216	C₄×Q₈	C₂×#6157	1	2	yes	—
217	C₄×D₄	—	53	2	no	—
218	C₄×D₄	—	53	2	no	—
219	C₄×D₄	—	106	2	no	—
220	C₄×D₄	—	106	2	no	—
221	#117	—	53	2	no	—
222	#117	—	106	2	no	—
223	C₂×C₄²	—	53	1	no	—
224	C₂×C₄²	—	106	2	no	—
225	#107	—	53	1	no	—
226	#107	C₂×#6157	1	1	yes	characteristic
227	#106	—	53	2	no	—
228	C₄⋊C₈	—	53	4	no	—
229	C₄⋊C₈	—	106	4	no	—
230	#103	—	53	2	no	—
231	#103	C₂×#6157	1	2	yes	—
232	#102	—	53	2	no	—
233	#102	—	53	2	no	—
234	C₂²⋊C₈	—	106	4	no	—
235	C₂²⋊C₈	—	106	2	no	—
236	C₄×C₈	—	53	2	no	—
237	C₄×C₈	—	53	1	no	—
238	C₄×C₈	—	53	1	no	—
239	C₄×C₈	—	53	1	no	—
240	C₄×C₈	—	53	1	no	—
241	C₄×C₈	—	53	2	no	—
242	C₄×C₈	—	53	1	no	—
243	C₄×C₈	—	106	1	no	—
244	C₄×C₈	—	106	2	no	—
245	C₄×C₈	—	106	4	no	—
246	C₄×C₈	C₂×#6157	1	1	yes	characteristic
247	#41	—	53	8	no	—
248	#41	—	106	4	no	—
249	C₂×Q₈	C₂²×#6157	1	2	yes	—
250	C₂×D₄	—	53	2	no	—
251	C₂×D₄	—	106	2	no	—
252	C₂²×C₄	—	53	1	no	—
253	C₂²×C₄	—	53	1	no	—
254	C₂²×C₄	—	53	1	no	—
255	C₂²×C₄	—	106	2	no	—
256	C₂²×C₄	—	106	2	no	—
257	C₂²×C₄	—	106	2	no	—
258	Q₁₆	D₅₃⋊C₈	1	4	yes	complemented
259	SD₁₆	—	53	8	no	—
260	D₈	—	53	4	no	—
261	C₂×C₈	—	53	2	no	—
262	C₂×C₈	—	53	1	no	—
263	C₂×C₈	—	53	4	no	—
264	C₂×C₈	—	53	2	no	—
265	C₂×C₈	—	53	1	no	—
266	C₂×C₈	—	106	2	no	—
267	C₂×C₈	—	106	4	no	—
268	C₂×C₈	—	106	2	no	—
269	C₂×C₈	—	106	4	no	—
270	C₂×C₈	—	106	4	no	—
271	C₂×C₈	—	212	8	no	—
272	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
273	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	—	53	2	no	—
276	C₄⋊C₄	—	106	2	no	—
277	C₄⋊C₄	C₂²×#6157	1	2	yes	—
278	C₂²⋊C₄	—	106	2	no	—
279	C₂²⋊C₄	—	106	2	no	—
280	C₂²⋊C₄	—	106	2	no	—
281	C₄²	—	2	2	no	—
282	C₄²	—	53	1	no	—
283	C₄²	—	53	1	no	—
284	C₄²	—	53	1	no	—
285	C₄²	—	106	2	no	—
286	C₄²	—	106	2	no	—
287	C₄²	C₂²×#6157	1	1	yes	characteristic
288	C₂³	—	53	1	no	—
289	C₂³	—	106	2	no	—
290	Q₈	—	2	2	no	—
291	Q₈	C₂×D₅₃⋊C₈	1	4	yes	—
292	D₄	—	53	4	no	—
293	D₄	—	106	8	no	—
294	D₄	—	106	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	53	1	no	—
298	C₂×C₄	—	53	1	no	—
299	C₂×C₄	—	53	2	no	—
300	C₂×C₄	—	53	2	no	—
301	C₂×C₄	—	53	1	no	—
302	C₂×C₄	—	106	2	no	—
303	C₂×C₄	—	106	8	no	—
304	C₂×C₄	—	106	2	no	—
305	C₂×C₄	—	106	2	no	—
306	C₂×C₄	—	212	4	no	—
307	C₂×C₄	—	212	4	no	—
308	C₂×C₄	D₄×#6157	1	1	yes	characteristic
309	C₂×C₄	D₄×#6157	1	1	yes	characteristic
310	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
311	C₈	—	2	1	no	—
312	C₈	—	53	4	no	—
313	C₈	—	53	2	no	—
314	C₈	—	106	2	no	—
315	C₈	—	106	4	no	—
316	C₈	—	106	1	no	—
317	C₈	—	212	4	no	—
318	C₈	C₂×D₅₃⋊C₈	1	2	yes	—
319	C₂²	—	53	2	no	—
320	C₂²	—	106	1	no	—
321	C₂²	—	106	1	no	—
322	C₂²	—	106	4	no	—
323	C₂²	—	212	2	no	—
324	C₂²	—	212	2	no	—
325	C₂²	C₂×D₄×#6157	1	1	yes	characteristic
326	C₄	—	2	4	no	—
327	C₄	—	2	1	no	—
328	C₄	—	4	2	no	—
329	C₄	—	53	2	no	—
330	C₄	—	53	2	no	—
331	C₄	—	106	1	no	—
332	C₄	—	212	2	no	—
333	C₄	#16811667	1	2	yes	—
334	C₄	C₂²×D₅₃⋊C₈	1	1	yes	characteristic
335	C₄	D₅₃⋊#131	1	1	yes	characteristic
336	C₂	—	106	2	no	—
337	C₂	—	212	4	no	—
338	C₂	D₄×D₅₃⋊C₈	1	1	yes	characteristic
339	C₂	#135×#6157	1	1	yes	characteristic
340	C₂	C₂×#16811667	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×#6157
commutator	C₄×C₅₃	C₂³×C₈
Frattini	C₂×C₄	C₂³×#6157
Fitting	C₄×Q₁₆×C₅₃	C₄
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂×#13133240	C₄
cosocle	C₄×Dic₅₃	C₂⁴
hypercenter	C₄×Q₁₆	#6157
hypocenter	C₅₃	C₈×#135
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
1. #20291521	C₂	semidirect
2. Q₁₆×#88324	C₂	semidirect
3. #20290299	C₂	semidirect
4. #20290296	C₂	semidirect
5. #20290265	C₂	semidirect
6. #20290009	C₂	semidirect
7. C₈×D₅₃⋊C₈	C₂	semidirect
49. #13133240	C₈	semidirect
78. #11719949	C₂×C₈	semidirect
94. Q₁₆×C₅₃	C₂×C₈	semidirect
101. #11719924	C₂×C₈	semidirect
114. D₄₂₄	C₂×C₈	semidirect
116. C₈×D₅₃	C₂×C₈	semidirect
143. #88324	#135	semidirect
204. C₅₃	C₈×#135	semidirect
258. Q₁₆	D₅₃⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#20291521	2
Q₁₆×#88324	2
#20290299	2
#20290296	2
#20290265	2
#20290009	2
C₈×D₅₃⋊C₈	2
C₄×#13133240	2
C₈×#135	53