Subgroups — #28328260

Order 13568.

590 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20291082	C₂	1	4	yes	maximal, complemented
2	#20291078	C₂	1	2	yes	maximal, complemented
3	#20291074	C₂	1	2	yes	maximal, complemented
4	#20291049	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16811544	C₂	1	1	yes	maximal, characteristic, complemented
6	#20290947	C₂	1	2	yes	maximal
7	#20290924	C₂	1	2	yes	maximal, complemented
8	#20283137	C₂	1	1	yes	maximal, characteristic
9	C₂×#13133291	C₂²	1	2	yes	—
10	C₂²×#11719910	C₂²	1	2	yes	—
11	#16811544	C₂²	1	1	yes	characteristic
12	#16811544	C₂²	1	2	yes	—
13	#16811544	C₂²	1	1	yes	characteristic
14	#16811544	C₂²	1	2	yes	—
15	#16811544	C₂²	1	2	yes	—
16	#16811518	C₂²	1	4	yes	—
17	#16811518	C₂²	1	4	yes	—
18	#16811511	C₂²	1	4	yes	—
19	C₄×#11719910	C₂²	1	4	yes	—
20	C₂×#13133225	C₂²	1	1	yes	characteristic
21	C₂×#13133223	C₄	1	1	yes	characteristic
22	#16811091	C₄	1	1	yes	characteristic
23	#16810580	C₂²	1	2	yes	—
24	#16810579	C₂²	1	1	yes	characteristic
25	#16810577	C₄	1	2	yes	—
26	#16810575	C₄	1	2	yes	—
27	M₁₆×Dic₅₃	C₄	1	1	yes	characteristic
28	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
29	C₄×C₈×D₅₃	C₂²	1	2	yes	—
30	C₅₃×#405	C₄	1	1	yes	characteristic
31	#13133291	—	2	4	no	—
32	C₂×#11719910	—	2	4	no	—
33	C₂×#11719910	C₂³	1	2	yes	—
34	C₂×#11719910	C₂³	1	2	yes	—
35	C₂×#11719910	C₂³	1	4	yes	—
36	C₂×#11719945	C₂×C₄	1	1	yes	characteristic
37	#13133225	—	2	2	no	—
38	#13133225	—	2	2	no	—
39	#13133223	—	2	4	no	—
40	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
41	C₂×#11719912	C₂×C₄	1	2	yes	—
42	C₂×#11719912	C₂×C₄	1	2	yes	—
43	C₂²×#88322	C₂³	1	1	yes	characteristic
44	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
45	C₂×C₈×D₅₃	C₂³	1	2	yes	—
46	#13133182	C₈	1	1	yes	characteristic
47	#13133182	C₈	1	1	yes	characteristic
48	#13133179	C₈	1	1	yes	characteristic
49	#13133179	C₈	1	1	yes	characteristic
50	#13133177	C₈	1	2	yes	—
51	#13133177	C₈	1	2	yes	—
52	#13133174	C₂×C₄	1	2	yes	—
53	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
54	C₄²×D₅₃	C₂×C₄	1	2	yes	—
55	#13133128	C₂×C₄	1	2	yes	—
56	#13133127	C₂×C₄	1	2	yes	—
57	C₄×#88322	C₂³	1	2	yes	—
58	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
59	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
60	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
61	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
62	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
63	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
64	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
65	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
66	#97×C₅₃	C₂×C₄	1	2	yes	—
67	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
68	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₂²⋊D₅₃	C₂×C₈	1	2	yes	—
70	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
71	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
72	#11719945	—	2	2	no	—
73	#11719945	—	2	2	no	—
74	C₂×D₂₁₂	C₂×C₈	1	1	yes	characteristic
75	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
76	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
77	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
78	C₂×Dic₁₀₆	C₂×C₈	1	1	yes	characteristic
79	M₁₆×C₅₃	—	2	2	no	—
80	C₂×C₈×C₅₃	—	2	2	no	—
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	C₄⋊C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
85	C₄⋊C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
86	C₂²⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
87	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
88	#11719921	C₂×C₈	1	2	yes	—
89	#11719921	C₂×C₈	1	2	yes	—
90	#11719919	C₂×C₈	1	2	yes	—
91	#11719919	C₂×C₈	1	2	yes	—
92	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
93	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
94	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
95	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
96	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
97	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
98	#11719917	M₁₆	1	4	yes	—
99	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
100	C₂×#88322	C₂²×C₄	1	2	yes	—
101	C₂×#88322	#41	1	2	yes	—
102	C₂×#88322	#41	1	2	yes	—
103	C₂×#88322	C₂⁴	1	1	yes	characteristic
104	#11719912	—	2	4	no	—
105	#11719912	—	2	4	no	—
106	C₈×D₅₃	—	2	4	no	—
107	C₈×D₅₃	—	2	2	no	—
108	C₈×D₅₃	—	2	2	no	—
109	#11719910	—	2	2	no	—
110	#11719910	—	2	2	no	—
111	#11719910	—	2	4	no	—
112	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
113	C₂²×D₅₃	C₂²×C₈	1	2	yes	—
114	C₂×C₄×C₅₃	#117	1	2	yes	—
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	C₂×M₁₆	1	2	yes	—
119	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
120	C₂²⋊D₅₃	—	2	4	no	—
121	C₂×Dic₅₃	—	2	2	no	—
122	C₂×Dic₅₃	—	4	1	no	—
123	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
124	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
125	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
126	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
127	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
128	D₂₁₂	—	2	2	no	—
129	C₄×D₅₃	—	2	2	no	—
130	C₄×D₅₃	—	2	2	no	—
131	C₄×D₅₃	—	2	4	no	—
132	C₄×D₅₃	—	4	2	no	—
133	Dic₁₀₆	—	2	2	no	—
134	C₈×C₅₃	—	2	2	no	—
135	C₈×C₅₃	—	2	1	no	—
136	C₈×C₅₃	—	2	1	no	—
137	#88322	#117	1	2	yes	—
138	#88322	C₂×M₁₆	1	4	yes	—
139	#88322	C₂×#41	1	1	yes	characteristic
140	#88322	C₂×#41	1	1	yes	characteristic
141	#33423	—	53	1	no	maximal, Sylow
142	C₂²×C₅₃	—	2	1	no	—
143	C₂²×C₅₃	—	2	1	no	—
144	C₂²×C₅₃	#431	1	2	yes	—
145	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
146	C₂×D₅₃	—	2	2	no	—
147	C₂×D₅₃	—	2	2	no	—
148	C₂×D₅₃	—	2	2	no	—
149	C₄×C₅₃	—	2	2	no	—
150	C₄×C₅₃	—	2	2	no	—
151	C₄×C₅₃	#431	1	2	yes	—
152	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
153	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
154	Dic₅₃	—	2	1	no	—
155	Dic₅₃	—	2	1	no	—
156	Dic₅₃	—	2	1	no	—
157	Dic₅₃	—	2	2	no	—
158	Dic₅₃	#504	1	2	yes	—
159	C₄×#131	—	53	1	no	—
160	#2164	—	53	4	no	—
161	#2163	—	53	2	no	—
162	#2162	—	53	2	no	—
163	#2150	—	53	1	no	—
164	C₂×C₄⋊C₁₆	—	53	1	no	—
165	#2113	—	53	2	no	—
166	#2108	—	53	2	no	—
167	C₂×C₅₃	—	2	2	no	—
168	C₂×C₅₃	C₂×#431	1	1	yes	characteristic
169	C₂×C₅₃	C₂×#504	1	1	yes	characteristic
170	C₂×C₅₃	#3407	1	1	yes	characteristic
171	D₅₃	—	4	2	no	—
172	C₂×#131	—	53	1	no	—
173	C₂×#131	—	53	1	no	—
174	C₄×#41	—	53	1	no	—
175	C₂×M₃₂	—	53	2	no	—
176	C₂²×C₁₆	—	53	2	no	—
177	#405	—	53	1	no	—
178	#405	—	53	2	no	—
179	#405	—	53	2	no	—
180	#405	#6157	1	1	yes	characteristic
181	C₄×M₁₆	—	53	1	no	—
182	C₄×M₁₆	—	53	2	no	—
183	C₂×C₄×C₈	—	53	1	no	—
184	C₂×C₄×C₈	—	53	2	no	—
185	C₄⋊C₁₆	—	53	1	no	—
186	C₄⋊C₁₆	—	53	1	no	—
187	C₄⋊C₁₆	—	53	2	no	—
188	C₄⋊C₁₆	—	53	2	no	—
189	C₄⋊C₁₆	—	53	2	no	—
190	C₂²⋊C₁₆	—	53	4	no	—
191	C₂²⋊C₁₆	—	53	4	no	—
192	#346	—	53	4	no	—
193	C₄×C₁₆	—	53	4	no	—
194	C₅₃	#33423	1	1	yes	characteristic, complemented, Sylow
195	C₂×#41	—	53	1	no	—
196	#131	—	106	4	no	—
197	#131	—	106	2	no	—
198	#131	—	106	2	no	—
199	C₂×M₁₆	—	53	1	no	—
200	C₂×M₁₆	—	53	2	no	—
201	C₂×M₁₆	—	53	2	no	—
202	C₂×M₁₆	C₂×#6157	1	1	yes	characteristic
203	C₂²×C₈	—	53	1	no	—
204	C₂²×C₈	—	53	2	no	—
205	C₂²×C₈	—	53	2	no	—
206	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
207	C₄×Q₈	—	53	1	no	—
208	C₄×Q₈	—	53	1	no	—
209	C₄×D₄	—	53	1	no	—
210	C₄×D₄	—	53	1	no	—
211	C₄×D₄	—	53	2	no	—
212	C₄×D₄	—	53	2	no	—
213	#117	—	53	2	no	—
214	#117	C₂×#6157	1	1	yes	characteristic
215	C₂×C₄²	—	53	1	no	—
216	C₂×C₄²	—	53	2	no	—
217	M₃₂	—	106	4	no	—
218	C₂×C₁₆	—	53	2	no	—
219	C₂×C₁₆	—	53	4	no	—
220	C₂×C₁₆	—	53	2	no	—
221	C₂×C₁₆	—	106	4	no	—
222	#97	—	53	2	no	—
223	#97	—	53	2	no	—
224	#97	C₂×#6157	1	2	yes	—
225	C₄×C₈	—	53	1	no	—
226	C₄×C₈	—	53	2	no	—
227	C₄×C₈	—	53	1	no	—
228	C₄×C₈	—	53	1	no	—
229	C₄×C₈	—	53	1	no	—
230	C₄×C₈	—	53	2	no	—
231	C₄×C₈	C₂×#6157	1	2	yes	—
232	#41	—	106	2	no	—
233	#41	—	106	2	no	—
234	C₂×Q₈	—	53	1	no	—
235	C₂×D₄	—	53	2	no	—
236	C₂×D₄	—	53	1	no	—
237	C₂²×C₄	—	53	1	no	—
238	C₂²×C₄	—	53	1	no	—
239	C₂²×C₄	—	53	2	no	—
240	C₂²×C₄	—	53	2	no	—
241	C₂²×C₄	—	53	2	no	—
242	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
243	M₁₆	—	2	2	no	—
244	M₁₆	—	53	4	no	—
245	M₁₆	—	106	4	no	—
246	M₁₆	—	106	4	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	53	2	no	—
249	C₂×C₈	—	53	2	no	—
250	C₂×C₈	—	53	1	no	—
251	C₂×C₈	—	53	1	no	—
252	C₂×C₈	—	53	2	no	—
253	C₂×C₈	—	106	2	no	—
254	C₂×C₈	—	106	2	no	—
255	C₂×C₈	—	106	4	no	—
256	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
257	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
258	C₂×C₈	C₂²×#6157	1	2	yes	—
259	C₄⋊C₄	—	53	2	no	—
260	C₄⋊C₄	—	53	2	no	—
261	C₄⋊C₄	D₅₃⋊C₈	1	1	yes	characteristic
262	C₄⋊C₄	D₅₃⋊C₈	1	1	yes	characteristic
263	C₂²⋊C₄	—	53	2	no	—
264	C₂²⋊C₄	—	53	2	no	—
265	C₂²⋊C₄	D₅₃⋊C₈	1	2	yes	—
266	C₄²	—	53	1	no	—
267	C₄²	—	53	2	no	—
268	C₄²	—	53	1	no	—
269	C₄²	—	53	2	no	—
270	C₄²	—	53	1	no	—
271	C₄²	—	53	1	no	—
272	C₄²	C₂²×#6157	1	2	yes	—
273	C₁₆	—	106	2	no	—
274	C₁₆	—	106	2	no	—
275	C₁₆	—	106	4	no	—
276	C₂³	—	53	2	no	—
277	C₂³	C₂×D₅₃⋊C₈	1	1	yes	characteristic
278	Q₈	—	106	2	no	—
279	D₄	—	106	4	no	—
280	D₄	—	106	2	no	—
281	C₂×C₄	—	53	1	no	—
282	C₂×C₄	—	53	1	no	—
283	C₂×C₄	—	53	1	no	—
284	C₂×C₄	—	53	1	no	—
285	C₂×C₄	—	53	2	no	—
286	C₂×C₄	—	106	4	no	—
287	C₂×C₄	—	106	2	no	—
288	C₂×C₄	—	106	2	no	—
289	C₂×C₄	—	106	2	no	—
290	C₂×C₄	—	212	2	no	—
291	C₂×C₄	—	212	1	no	—
292	C₂×C₄	#13133296	1	2	yes	—
293	C₂×C₄	C₂×D₅₃⋊C₈	1	2	yes	—
294	C₂×C₄	C₂×D₅₃⋊C₈	1	2	yes	—
295	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
296	C₂×C₄	D₅₃⋊M₁₆	1	2	yes	—
297	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
298	C₈	—	2	2	no	—
299	C₈	—	2	1	no	—
300	C₈	—	2	1	no	—
301	C₈	—	53	2	no	—
302	C₈	—	53	1	no	—
303	C₈	—	53	1	no	—
304	C₈	—	53	4	no	—
305	C₂²	—	2	1	no	—
306	C₂²	—	2	1	no	—
307	C₂²	—	106	2	no	—
308	C₂²	—	106	2	no	—
309	C₂²	—	106	2	no	—
310	C₂²	#16811641	1	2	yes	—
311	C₂²	C₂²×D₅₃⋊C₈	1	1	yes	characteristic
312	C₄	—	2	2	no	—
313	C₄	—	2	2	no	—
314	C₄	—	53	2	no	—
315	C₄	—	106	2	no	—
316	C₄	—	106	1	no	—
317	C₄	—	106	1	no	—
318	C₄	—	106	1	no	—
319	C₄	#16811641	1	2	yes	—
320	C₄	C₂×#13133296	1	1	yes	characteristic
321	C₄	C₂×D₅₃⋊M₁₆	1	1	yes	characteristic
322	C₂	—	2	2	no	—
323	C₂	—	212	2	no	—
324	C₂	C₂×#16811641	1	1	yes	characteristic
325	C₂	D₅₃⋊#504	1	1	yes	characteristic
326	C₂	#20291683	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#6157
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	C₅₃×#405	C₄
prime radical	C₂×#11719945	C₂×C₄
cosocle	C₂×#88322	C₂⁴
hypercenter	#405	#6157
hypocenter	C₅₃	#33423
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #405

Complements

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

N	H ≅ G/N	type
1. #20291082	C₂	semidirect
2. #20291078	C₂	semidirect
3. #20291074	C₂	semidirect
4. #20291049	C₂	semidirect
5. C₂×#16811544	C₂	semidirect
7. #20290924	C₂	semidirect
194. C₅₃	#33423	semidirect

Maximal subgroups

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

subgroup	index
#20291082	2
#20291078	2
#20291074	2
#20291049	2
C₂×#16811544	2
#20290947	2
#20290924	2
#20283137	2
#33423	53