Subgroups — #28326750

Order 13568.

638 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20291509	C₂	1	2	yes	maximal
2	#20290891	C₂	1	2	yes	maximal
3	#20290823	C₂	1	2	yes	maximal
4	#20290780	C₂	1	4	yes	maximal
5	C₄×#13133288	C₂	1	1	yes	maximal, characteristic
6	#20290519	C₂	1	2	yes	maximal
7	C₈×C₄⋊#6157	C₂	1	1	yes	maximal, characteristic
8	#20285226	C₂	1	1	yes	maximal, characteristic
9	#16811675	—	2	4	no	—
10	#16811674	—	2	2	no	—
11	Q₈×#88324	—	2	2	no	—
12	C₂×#13133288	C₂²	1	1	yes	characteristic
13	C₂×#13133288	C₂²	1	1	yes	characteristic
14	#16811630	C₂²	1	4	yes	—
15	D₅₃⋊(C₄⋊C₈)	C₂²	1	1	yes	characteristic
16	#16811588	C₂²	1	4	yes	—
17	C₄×#11719936	C₂²	1	2	yes	—
18	C₂×C₈×#6157	C₂²	1	2	yes	—
19	C₄×C₄⋊#6157	C₂²	1	1	yes	characteristic
20	#16811499	C₂²	1	2	yes	—
21	#16811493	C₂²	1	4	yes	—
22	#16811493	C₂²	1	4	yes	—
23	(C₂×#6157)⋊C₈	C₂²	1	2	yes	—
24	#16811178	C₂²	1	2	yes	—
25	#16811028	C₄	1	1	yes	characteristic, complemented
26	#16811024	C₄	1	2	yes	complemented
27	#16811024	C₄	1	2	yes	complemented
28	#16811019	C₄	1	2	yes	complemented
29	#16810660	C₂²	1	2	yes	—
30	#16810660	C₂²	1	2	yes	—
31	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
32	C₅₃×#500	C₄	1	1	yes	characteristic, complemented
33	C₂×#11719936	C₂³	1	2	yes	—
34	C₂×#11719936	C₂³	1	2	yes	—
35	C₂×D₅₃⋊C₈	C₂³	1	2	yes	—
36	C₂×C₄⋊#6157	—	2	1	no	—
37	C₂×C₄×#6157	C₂³	1	2	yes	—
38	#13133288	—	2	2	no	—
39	#13133288	Q₈	1	4	yes	complemented
40	#13133284	—	2	4	no	—
41	#13133284	D₄	1	2	yes	—
42	C₄⋊#88324	—	2	4	no	—
43	C₄⋊#88324	D₄	1	1	yes	characteristic
44	C₄⋊#88324	D₄	1	1	yes	characteristic
45	#13133272	—	2	2	no	—
46	#13133272	—	2	4	no	—
47	C₄×#88324	—	2	2	no	—
48	C₄×#88324	—	2	4	no	—
49	C₈×#6157	—	2	4	no	—
50	(C₂×#6157)⋊C₄	—	2	1	no	—
51	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
52	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
53	#13133217	C₂×C₄	1	2	yes	—
54	#13133208	C₂×C₄	1	4	yes	—
55	#13133197	—	2	2	no	—
56	#13133193	—	2	2	no	—
57	Q₈×Dic₅₃	—	2	2	no	—
58	#13133182	—	2	2	no	—
59	#13133175	C₂³	1	4	yes	—
60	C₄²×D₅₃	C₂³	1	1	yes	characteristic
61	#13133151	C₂×C₄	1	2	yes	—
62	#13133151	C₂×C₄	1	2	yes	—
63	#13133148	C₂×C₄	1	4	yes	—
64	#13133148	C₂×C₄	1	4	yes	—
65	C₄×#88322	C₂×C₄	1	1	yes	characteristic
66	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
67	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
68	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
69	#107×C₅₃	C₂×C₄	1	2	yes	—
70	#107×C₅₃	C₂×C₄	1	2	yes	—
71	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
72	C₂²×#6157	—	2	2	no	—
73	C₂×Q₈×C₅₃	—	2	2	no	—
74	C₂×C₄×D₅₃	#41	1	1	yes	characteristic
75	C₂×C₄×D₅₃	#41	1	1	yes	characteristic
76	C₂×C₄×D₅₃	C₂⁴	1	1	yes	characteristic
77	C₂×Dic₁₀₆	—	2	2	no	—
78	C₂×#88324	—	2	2	no	—
79	C₂×#88324	—	2	2	no	—
80	C₂×#88324	C₂×D₄	1	2	yes	—
81	C₂×#88324	C₂×D₄	1	2	yes	—
82	C₄⋊#6157	—	2	4	no	—
83	C₄×#6157	—	2	2	no	—
84	C₄×#6157	—	2	2	no	—
85	#11719936	—	2	2	no	—
86	#11719936	—	2	2	no	—
87	#11719936	—	2	2	no	—
88	#11719936	C₂×Q₈	1	4	yes	—
89	D₅₃⋊C₈	—	2	2	no	—
90	D₅₃⋊C₈	—	2	2	no	—
91	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
93	C₄⋊C₄×C₅₃	—	2	2	no	—
94	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
95	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
96	#11719921	—	2	4	no	—
97	C₄⋊Dic₅₃	—	2	2	no	—
98	C₄⋊Dic₅₃	—	2	4	no	—
99	C₄⋊Dic₅₃	C₂²×C₄	1	4	yes	—
100	#11719919	—	2	4	no	—
101	C₄×Dic₅₃	—	2	4	no	—
102	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
103	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
104	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
105	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
106	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
107	C₈×D₅₃	—	2	1	no	—
108	C₈×D₅₃	—	2	1	no	—
109	C₈×D₅₃	C₂×Q₈	1	2	yes	—
110	C₈×D₅₃	#41	1	2	yes	—
111	C₂²×D₅₃	C₂×#41	1	1	yes	characteristic
112	C₂×#6157	—	2	4	no	—
113	C₂×#6157	—	4	2	no	—
114	C₂×#6157	—	4	2	no	—
115	Q₈×C₅₃	—	4	4	no	—
116	C₂×C₄×C₅₃	—	2	4	no	—
117	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
118	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
119	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
120	C₂×Dic₅₃	—	2	4	no	—
121	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
122	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
123	C₂×Dic₅₃	C₂²×D₄	1	1	yes	characteristic
124	C₄×D₅₃	—	2	2	no	—
125	C₄×D₅₃	—	4	1	no	—
126	C₄×D₅₃	C₂²×Q₈	1	1	yes	characteristic
127	C₄×D₅₃	C₂×#41	1	1	yes	characteristic
128	C₄×D₅₃	2₋¹⁺⁴	1	1	yes	characteristic
129	C₄×D₅₃	2₋¹⁺⁴	1	1	yes	characteristic
130	Dic₁₀₆	—	8	2	no	—
131	#88324	—	2	2	no	—
132	#88324	—	2	2	no	—
133	#88324	—	2	4	no	—
134	#88324	—	4	2	no	—
135	C₈×C₅₃	—	2	1	no	—
136	C₈×C₅₃	C₄×Q₈	1	2	yes	—
137	#88322	—	2	1	no	—
138	#88322	C₄×Q₈	1	2	yes	—
139	#23170	—	53	1	no	maximal, Sylow
140	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
141	C₂×D₅₃	—	2	1	no	—
142	C₂×D₅₃	—	2	1	no	—
143	C₂×D₅₃	#556	1	1	yes	characteristic
144	C₂×D₅₃	#557	1	1	yes	characteristic
145	#6157	—	4	4	no	—
146	C₄×C₅₃	—	2	1	no	—
147	C₄×C₅₃	—	4	4	no	—
148	C₄×C₅₃	#443	1	2	yes	—
149	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
150	C₄×C₅₃	#519	1	1	yes	characteristic
151	Dic₅₃	—	2	1	no	—
152	Dic₅₃	—	2	1	no	—
153	Dic₅₃	—	4	4	no	—
154	Dic₅₃	#547	1	1	yes	characteristic
155	Dic₅₃	D₄×Q₈	1	1	yes	characteristic
156	#3160	—	53	1	no	—
157	#2962	—	53	2	no	—
158	#1986	—	53	2	no	—
159	#1945	—	53	2	no	—
160	#1922	—	53	4	no	—
161	C₄×#108	—	53	1	no	—
162	#1772	—	53	2	no	—
163	C₈×C₄⋊C₄	—	53	1	no	—
164	C₂×C₅₃	#2365	1	1	yes	characteristic
165	C₂×C₅₃	C₂×#443	1	1	yes	characteristic
166	C₂×C₅₃	#2957	1	1	yes	characteristic
167	D₅₃	—	2	1	no	—
168	D₅₃	—	2	1	no	—
169	#533	—	53	2	no	—
170	#500	—	53	1	no	—
171	#500	#6157	1	1	yes	characteristic, complemented
172	#499	—	53	2	no	—
173	#499	—	53	2	no	—
174	#498	—	53	2	no	—
175	#446	—	106	2	no	—
176	#446	—	106	4	no	—
177	C₈×Q₈	—	106	2	no	—
178	C₂×#108	—	53	1	no	—
179	C₂×#108	—	53	1	no	—
180	#427	—	53	2	no	—
181	#427	—	53	2	no	—
182	#424	—	53	4	no	—
183	C₂×C₄⋊C₈	—	53	1	no	—
184	#405	—	53	4	no	—
185	C₄×M₁₆	—	53	2	no	—
186	C₂×C₄×C₈	—	53	1	no	—
187	C₂×C₄×C₈	—	53	2	no	—
188	C₄×C₄⋊C₄	—	53	1	no	—
189	#341	—	53	2	no	—
190	#339	—	53	4	no	—
191	#339	—	53	4	no	—
192	#336	—	53	2	no	—
193	C₅₃	#23170	1	1	yes	characteristic, complemented, Sylow
194	C₂×M₁₆	—	53	2	no	—
195	C₂×M₁₆	—	53	2	no	—
196	C₂²×C₈	—	53	1	no	—
197	C₂²×C₈	—	53	2	no	—
198	C₂²×C₈	—	53	1	no	—
199	C₄⋊Q₈	—	53	2	no	—
200	C₄⋊Q₈	C₂×#6157	1	2	yes	—
201	#125	—	53	4	no	—
202	C₂²⋊Q₈	—	106	2	no	—
203	C₂²⋊Q₈	—	106	2	no	—
204	C₄×Q₈	—	106	2	no	—
205	C₄×Q₈	—	106	2	no	—
206	#117	—	53	4	no	—
207	C₂×C₄⋊C₄	—	106	1	no	—
208	C₂×C₄²	—	53	1	no	—
209	C₂×C₄²	—	53	2	no	—
210	#108	—	53	4	no	—
211	#108	—	106	2	no	—
212	#107	—	53	2	no	—
213	#107	—	53	2	no	—
214	#107	C₂×#6157	1	2	yes	—
215	#107	C₂×#6157	1	2	yes	—
216	#106	—	53	4	no	—
217	#106	—	53	4	no	—
218	C₄⋊C₈	—	53	1	no	—
219	C₄⋊C₈	—	53	1	no	—
220	C₄⋊C₈	—	53	2	no	—
221	C₄⋊C₈	—	106	4	no	—
222	C₄⋊C₈	—	106	4	no	—
223	#97	—	106	2	no	—
224	#97	—	106	4	no	—
225	C₄×C₈	—	53	1	no	—
226	C₄×C₈	—	53	1	no	—
227	C₄×C₈	—	53	1	no	—
228	C₄×C₈	—	106	2	no	—
229	C₄×C₈	—	106	4	no	—
230	C₄×C₈	—	106	4	no	—
231	C₄×C₈	C₂×#6157	1	1	yes	characteristic
232	#95	—	106	1	no	—
233	C₂×Q₈	—	2	2	no	—
234	C₂×Q₈	—	106	2	no	—
235	C₂²×C₄	—	53	1	no	—
236	C₂²×C₄	—	53	1	no	—
237	C₂²×C₄	—	53	1	no	—
238	C₂²×C₄	—	106	2	no	—
239	M₁₆	—	53	4	no	—
240	M₁₆	—	106	2	no	—
241	M₁₆	—	106	2	no	—
242	M₁₆	—	106	2	no	—
243	C₂×C₈	—	53	1	no	—
244	C₂×C₈	—	53	2	no	—
245	C₂×C₈	—	53	2	no	—
246	C₂×C₈	—	53	2	no	—
247	C₂×C₈	—	53	2	no	—
248	C₂×C₈	—	53	1	no	—
249	C₂×C₈	—	106	1	no	—
250	C₂×C₈	—	106	1	no	—
251	C₂×C₈	—	106	2	no	—
252	C₂×C₈	—	106	2	no	—
253	C₂×C₈	—	106	2	no	—
254	C₂×C₈	—	106	2	no	—
255	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
256	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	53	4	no	—
259	C₄⋊C₄	—	106	4	no	—
260	C₄⋊C₄	—	106	4	no	—
261	C₄⋊C₄	—	106	4	no	—
262	C₄⋊C₄	—	106	2	no	—
263	C₄⋊C₄	C₂²×#6157	1	4	yes	—
264	C₂²⋊C₄	—	106	4	no	—
265	C₄²	—	53	1	no	—
266	C₄²	—	53	1	no	—
267	C₄²	—	53	1	no	—
268	C₄²	—	106	4	no	—
269	C₄²	—	106	2	no	—
270	C₄²	—	106	2	no	—
271	C₄²	C₂²×#6157	1	1	yes	characteristic
272	C₂³	—	53	1	no	—
273	Q₈	—	4	4	no	—
274	Q₈	—	424	2	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	53	1	no	—
277	C₂×C₄	—	53	1	no	—
278	C₂×C₄	—	53	1	no	—
279	C₂×C₄	—	53	1	no	—
280	C₂×C₄	—	53	1	no	—
281	C₂×C₄	—	53	1	no	—
282	C₂×C₄	—	53	1	no	—
283	C₂×C₄	—	106	2	no	—
284	C₂×C₄	—	106	4	no	—
285	C₂×C₄	—	106	4	no	—
286	C₂×C₄	—	212	1	no	—
287	C₂×C₄	—	212	2	no	—
288	C₂×C₄	—	212	2	no	—
289	C₂×C₄	D₄×#6157	1	1	yes	characteristic
290	C₂×C₄	D₄×#6157	1	1	yes	characteristic
291	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
292	C₈	—	2	1	no	—
293	C₈	—	53	2	no	—
294	C₈	—	106	2	no	—
295	C₈	—	106	2	no	—
296	C₈	—	106	4	no	—
297	C₈	—	106	1	no	—
298	C₈	—	212	2	no	—
299	C₈	Q₈×#6157	1	2	yes	—
300	C₂²	—	53	1	no	—
301	C₂²	—	53	1	no	—
302	C₂²	—	106	1	no	—
303	C₂²	—	106	1	no	—
304	C₂²	C₂×D₄×#6157	1	1	yes	characteristic
305	C₄	—	2	1	no	—
306	C₄	—	4	4	no	—
307	C₄	—	53	1	no	—
308	C₄	—	53	1	no	—
309	C₄	—	106	1	no	—
310	C₄	—	106	1	no	—
311	C₄	—	212	4	no	—
312	C₄	—	212	4	no	—
313	C₄	#16811665	1	2	yes	—
314	C₄	C₂×Q₈×#6157	1	1	yes	characteristic
315	C₄	#16811691	1	1	yes	characteristic
316	C₂	—	106	1	no	—
317	C₂	—	106	1	no	—
318	C₂	#20291244	1	1	yes	characteristic
319	C₂	C₂×#16811665	1	1	yes	characteristic
320	C₂	#20291495	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×#6157
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#6157
Fitting	C₅₃×#500	C₄
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂²×D₅₃	C₂×#41
cosocle	C₂×C₄×D₅₃	C₂⁴
hypercenter	#500	#6157
hypocenter	C₅₃	#23170
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

Complements

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

N	H ≅ G/N	type
25. #16811028	C₄	semidirect
26. #16811024	C₄	semidirect
27. #16811024	C₄	semidirect
28. #16811019	C₄	semidirect
32. C₅₃×#500	C₄	semidirect
39. #13133288	Q₈	semidirect
171. #500	#6157	semidirect
193. C₅₃	#23170	semidirect

Maximal subgroups

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

subgroup	index
#20291509	2
#20290891	2
#20290823	2
#20290780	2
C₄×#13133288	2
#20290519	2
C₈×C₄⋊#6157	2
#20285226	2
#23170	53