Subgroups — #28328469

Order 13568.

518 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20291130	C₂	1	2	yes	maximal, complemented
2	#20291098	C₂	1	2	yes	maximal, complemented
3	#20291098	C₂	1	2	yes	maximal, complemented
4	#20291094	C₂	1	2	yes	maximal, complemented
5	#20291083	C₂	1	2	yes	maximal, complemented
6	#20291075	C₂	1	1	yes	maximal, characteristic, complemented
7	#20290925	C₂	1	2	yes	maximal, complemented
8	#20290916	C₂	1	1	yes	maximal, characteristic, complemented
9	#20283689	C₂	1	1	yes	maximal, characteristic
10	C₂×#13133292	C₂²	1	2	yes	—
11	C₂×D₅₃⋊C₁₆	C₂²	1	2	yes	—
12	C₂×D₅₃⋊C₁₆	C₂²	1	2	yes	—
13	C₄⋊#11719910	C₂²	1	2	yes	—
14	C₄⋊#11719910	C₂²	1	2	yes	—
15	C₄⋊#11719910	C₂²	1	2	yes	—
16	#16811518	C₂²	1	1	yes	characteristic
17	#16811518	C₂²	1	1	yes	characteristic
18	#16811518	C₂²	1	2	yes	—
19	#16811518	C₂²	1	2	yes	—
20	#16811512	C₂²	1	1	yes	characteristic
21	#16811512	C₂²	1	1	yes	characteristic
22	#16811511	C₂²	1	2	yes	—
23	#16811511	C₂²	1	2	yes	—
24	#16811509	C₂²	1	2	yes	—
25	C₄×#11719910	C₂²	1	2	yes	—
26	#16811160	C₄	1	1	yes	characteristic
27	Q₈×#88322	C₂²	1	1	yes	characteristic
28	#16810702	C₄	1	1	yes	characteristic
29	#16810696	C₄	1	2	yes	—
30	#16810688	C₂²	1	2	yes	—
31	#16810688	C₂²	1	1	yes	characteristic
32	#16810683	C₄	1	2	yes	—
33	#16810679	C₄	1	1	yes	characteristic
34	#16810580	C₂²	1	2	yes	—
35	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
36	C₅₃×#446	C₄	1	1	yes	characteristic
37	#13133292	—	2	4	no	—
38	D₅₃⋊C₁₆	—	2	4	no	—
39	D₅₃⋊C₁₆	—	2	4	no	—
40	C₂×#11719910	C₂³	1	2	yes	—
41	C₂×#11719910	C₂³	1	2	yes	—
42	C₂×#11719910	C₂³	1	2	yes	—
43	C₂×#11719910	C₂³	1	2	yes	—
44	C₂×#11719912	C₂×C₄	1	2	yes	—
45	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
47	C₂×C₈×D₅₃	C₂³	1	2	yes	—
48	#13133212	C₈	1	2	yes	—
49	#13133208	C₈	1	1	yes	characteristic
50	#13133206	C₈	1	1	yes	characteristic
51	#13133202	C₈	1	2	yes	—
52	#13133193	C₈	1	1	yes	characteristic
53	#13133192	C₈	1	1	yes	characteristic
54	C₄×D₂₁₂	C₂×C₄	1	2	yes	—
55	C₄×D₂₁₂	C₂×C₄	1	1	yes	characteristic
56	#13133174	C₂×C₄	1	2	yes	—
57	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
58	C₄⋊#88322	C₂×C₄	1	1	yes	characteristic
59	C₄⋊#88322	C₂×C₄	1	2	yes	—
60	#13133129	C₂×C₄	1	2	yes	—
61	#13133129	C₂×C₄	1	2	yes	—
62	#13133129	C₂×C₄	1	2	yes	—
63	#13133127	C₂×C₄	1	2	yes	—
64	C₄×#88322	C₂³	1	1	yes	characteristic
65	C₄×#88322	C₂³	1	2	yes	—
66	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
67	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
68	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
69	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
70	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
71	#97×C₅₃	C₂×C₄	1	2	yes	—
72	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
73	C₂×Q₈×C₅₃	C₂×C₈	1	1	yes	characteristic
74	C₂×D₂₁₂	C₂×C₈	1	2	yes	—
75	C₂×D₂₁₂	C₂×C₈	1	1	yes	characteristic
76	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
77	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
78	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
83	C₄⋊C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
84	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
85	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
86	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
87	#11719921	C₂×C₈	1	2	yes	—
88	#11719921	C₂×C₈	1	2	yes	—
89	#11719921	C₂×C₈	1	2	yes	—
90	#11719921	C₂×C₈	1	2	yes	—
91	#11719921	C₂×C₈	1	2	yes	—
92	C₄⋊Dic₅₃	C₂×C₈	1	1	yes	characteristic
93	C₄⋊Dic₅₃	C₂×C₈	1	2	yes	—
94	#11719919	C₂×C₈	1	2	yes	—
95	#11719919	C₂×C₈	1	2	yes	—
96	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
97	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
98	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
99	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
100	C₂×#88322	C₂²×C₄	1	2	yes	—
101	C₂×#88322	C₂⁴	1	1	yes	characteristic
102	#11719912	—	2	4	no	—
103	C₈×D₅₃	—	2	2	no	—
104	C₈×D₅₃	—	2	2	no	—
105	C₈×D₅₃	—	2	2	no	—
106	C₈×D₅₃	—	2	2	no	—
107	#11719910	—	2	2	no	—
108	#11719910	—	2	2	no	—
109	#11719910	—	2	2	no	—
110	#11719910	#41	1	4	yes	—
111	C₂²×D₅₃	C₂²×C₈	1	2	yes	—
112	C₂²×D₅₃	C₂²×C₈	1	1	yes	characteristic
113	Q₈×C₅₃	—	4	1	no	—
114	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
115	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
116	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
117	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
118	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
119	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
120	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
121	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
122	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
123	D₂₁₂	—	2	4	no	—
124	D₂₁₂	—	4	1	no	—
125	C₄×D₅₃	—	2	2	no	—
126	C₄×D₅₃	—	2	1	no	—
127	C₄×D₅₃	—	2	2	no	—
128	C₄×D₅₃	—	2	1	no	—
129	C₄×D₅₃	—	2	2	no	—
130	C₄×D₅₃	—	4	1	no	—
131	C₈×C₅₃	—	2	1	no	—
132	C₈×C₅₃	—	2	2	no	—
133	C₈×C₅₃	#131	1	2	yes	—
134	#88322	—	2	1	no	—
135	#88322	—	2	2	no	—
136	#88322	C₂×#41	1	2	yes	—
137	#33482	—	53	1	no	maximal, Sylow
138	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
139	C₂×D₅₃	—	2	1	no	—
140	C₂×D₅₃	—	2	1	no	—
141	C₂×D₅₃	—	2	2	no	—
142	C₂×D₅₃	—	2	2	no	—
143	C₂×D₅₃	—	2	2	no	—
144	C₂×D₅₃	#504	1	2	yes	—
145	C₄×C₅₃	—	2	1	no	—
146	C₄×C₅₃	—	2	2	no	—
147	C₄×C₅₃	—	2	1	no	—
148	C₄×C₅₃	—	2	2	no	—
149	C₄×C₅₃	C₄×#41	1	1	yes	characteristic
150	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
151	Dic₅₃	—	2	1	no	—
152	Dic₅₃	—	2	1	no	—
153	Dic₅₃	—	2	1	no	—
154	Dic₅₃	—	2	2	no	—
155	#2966	—	53	1	no	—
156	#2183	—	53	2	no	—
157	#2168	—	53	2	no	—
158	#2168	—	53	2	no	—
159	D₄×C₁₆	—	53	2	no	—
160	#2164	—	53	2	no	—
161	#2162	—	53	1	no	—
162	#2108	—	53	2	no	—
163	C₂×#346	—	53	1	no	—
164	C₂×C₅₃	C₈×#41	1	1	yes	characteristic
165	C₂×C₅₃	C₂×#504	1	1	yes	characteristic
166	C₂×C₅₃	#3407	1	1	yes	characteristic
167	D₅₃	—	2	2	no	—
168	D₅₃	—	4	2	no	—
169	#529	—	53	1	no	—
170	C₂×M₃₂	—	53	2	no	—
171	C₂²×C₁₆	—	53	2	no	—
172	C₂²×C₁₆	—	53	2	no	—
173	#446	#6157	1	1	yes	characteristic
174	C₈×Q₈	—	53	1	no	—
175	#436	—	53	1	no	—
176	#435	—	53	2	no	—
177	C₈×D₄	—	53	2	no	—
178	C₈×D₄	—	53	1	no	—
179	#433	—	53	2	no	—
180	#432	—	53	1	no	—
181	#405	—	53	2	no	—
182	C₂×C₄×C₈	—	53	1	no	—
183	C₄⋊C₁₆	—	53	2	no	—
184	C₄⋊C₁₆	—	53	2	no	—
185	C₄⋊C₁₆	—	53	2	no	—
186	C₂²⋊C₁₆	—	53	1	no	—
187	C₂²⋊C₁₆	—	53	1	no	—
188	C₂²⋊C₁₆	—	53	2	no	—
189	C₂²⋊C₁₆	—	53	2	no	—
190	#346	—	53	1	no	—
191	#346	—	53	1	no	—
192	#346	—	53	2	no	—
193	#346	—	53	2	no	—
194	C₄×C₁₆	—	53	2	no	—
195	C₄×C₁₆	—	53	2	no	—
196	C₅₃	#33482	1	1	yes	characteristic, complemented, Sylow
197	C₂×M₁₆	—	53	2	no	—
198	C₂²×C₈	—	53	1	no	—
199	C₂²×C₈	—	53	2	no	—
200	C₂²×C₈	—	53	1	no	—
201	#126	—	53	2	no	—
202	#125	—	53	1	no	—
203	#124	—	53	1	no	—
204	#123	—	53	2	no	—
205	C₂²⋊Q₈	—	53	1	no	—
206	#121	—	53	1	no	—
207	C₄×Q₈	C₂×#6157	1	1	yes	characteristic
208	C₄×D₄	—	53	2	no	—
209	C₄×D₄	—	53	1	no	—
210	#117	—	53	2	no	—
211	C₂×C₄²	—	53	1	no	—
212	M₃₂	—	106	4	no	—
213	C₂×C₁₆	—	53	2	no	—
214	C₂×C₁₆	—	53	2	no	—
215	C₂×C₁₆	—	53	2	no	—
216	C₂×C₁₆	—	53	2	no	—
217	C₂×C₁₆	—	106	4	no	—
218	C₂×C₁₆	—	106	4	no	—
219	C₄⋊C₈	—	53	1	no	—
220	C₄⋊C₈	—	53	2	no	—
221	C₄⋊C₈	C₂×#6157	1	1	yes	characteristic
222	C₄⋊C₈	C₂×#6157	1	2	yes	—
223	C₂²⋊C₈	—	53	2	no	—
224	C₂²⋊C₈	—	53	2	no	—
225	C₂²⋊C₈	—	53	2	no	—
226	#97	—	53	2	no	—
227	#97	C₂×#6157	1	2	yes	—
228	C₄×C₈	—	53	1	no	—
229	C₄×C₈	—	53	1	no	—
230	C₄×C₈	—	53	2	no	—
231	C₄×C₈	—	53	1	no	—
232	C₄×C₈	C₂×#6157	1	1	yes	characteristic
233	C₂×Q₈	D₅₃⋊C₈	1	1	yes	characteristic
234	C₂×D₄	—	53	2	no	—
235	C₂×D₄	—	53	1	no	—
236	C₂²×C₄	—	53	1	no	—
237	C₂²×C₄	—	53	1	no	—
238	C₂²×C₄	—	53	2	no	—
239	C₂²×C₄	—	53	1	no	—
240	M₁₆	—	106	4	no	—
241	C₂×C₈	—	53	1	no	—
242	C₂×C₈	—	53	1	no	—
243	C₂×C₈	—	53	2	no	—
244	C₂×C₈	—	106	2	no	—
245	C₂×C₈	—	106	2	no	—
246	C₂×C₈	—	106	2	no	—
247	C₂×C₈	—	106	2	no	—
248	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
249	C₂×C₈	C₂²×#6157	1	2	yes	—
250	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
251	C₄⋊C₄	—	53	2	no	—
252	C₄⋊C₄	—	53	2	no	—
253	C₄⋊C₄	—	53	1	no	—
254	C₄⋊C₄	—	53	2	no	—
255	C₄⋊C₄	D₅₃⋊C₈	1	1	yes	characteristic
256	C₄⋊C₄	D₅₃⋊C₈	1	2	yes	—
257	C₂²⋊C₄	—	53	2	no	—
258	C₂²⋊C₄	—	53	2	no	—
259	C₂²⋊C₄	—	53	2	no	—
260	C₂²⋊C₄	—	53	2	no	—
261	C₂²⋊C₄	—	53	2	no	—
262	C₄²	—	53	1	no	—
263	C₄²	—	53	1	no	—
264	C₄²	—	53	1	no	—
265	C₄²	C₂²×#6157	1	1	yes	characteristic
266	C₄²	C₂²×#6157	1	2	yes	—
267	C₁₆	—	53	4	no	—
268	C₁₆	—	106	2	no	—
269	C₁₆	—	106	2	no	—
270	C₁₆	—	106	2	no	—
271	C₂³	—	53	1	no	—
272	C₂³	—	53	2	no	—
273	Q₈	—	4	1	no	—
274	D₄	—	106	4	no	—
275	D₄	—	212	1	no	—
276	C₂×C₄	—	53	1	no	—
277	C₂×C₄	—	53	1	no	—
278	C₂×C₄	—	53	1	no	—
279	C₂×C₄	—	53	2	no	—
280	C₂×C₄	—	106	2	no	—
281	C₂×C₄	—	106	1	no	—
282	C₂×C₄	—	106	2	no	—
283	C₂×C₄	—	106	1	no	—
284	C₂×C₄	—	106	2	no	—
285	C₂×C₄	—	212	1	no	—
286	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
287	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
288	C₂×C₄	C₂×D₅₃⋊C₈	1	2	yes	—
289	C₂×C₄	C₂×D₅₃⋊C₈	1	2	yes	—
290	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
291	C₈	—	2	1	no	—
292	C₈	—	2	2	no	—
293	C₈	—	53	2	no	—
294	C₈	—	106	1	no	—
295	C₈	—	106	2	no	—
296	C₈	#13133305	1	2	yes	—
297	C₂²	—	53	2	no	—
298	C₂²	—	106	1	no	—
299	C₂²	—	106	1	no	—
300	C₂²	—	106	2	no	—
301	C₂²	—	106	2	no	—
302	C₂²	—	106	2	no	—
303	C₂²	C₂²×D₅₃⋊C₈	1	1	yes	characteristic
304	C₄	—	2	1	no	—
305	C₄	—	2	2	no	—
306	C₄	—	2	1	no	—
307	C₄	—	2	2	no	—
308	C₄	—	106	1	no	—
309	C₄	—	106	1	no	—
310	C₄	—	106	1	no	—
311	C₄	—	106	2	no	—
312	C₄	#41×#6157	1	1	yes	characteristic
313	C₄	C₂×#13133305	1	1	yes	characteristic
314	C₂	—	106	2	no	—
315	C₂	—	212	2	no	—
316	C₂	#20291521	1	1	yes	characteristic
317	C₂	D₅₃⋊#504	1	1	yes	characteristic
318	C₂	#20291684	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#6157
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	C₅₃×#446	C₄
prime radical	#13133192	C₈
cosocle	C₂×#88322	C₂⁴
hypercenter	#446	#6157
hypocenter	C₅₃	#33482
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #446

Complements

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

N	H ≅ G/N	type
1. #20291130	C₂	semidirect
2. #20291098	C₂	semidirect
3. #20291098	C₂	semidirect
4. #20291094	C₂	semidirect
5. #20291083	C₂	semidirect
6. #20291075	C₂	semidirect
7. #20290925	C₂	semidirect
8. #20290916	C₂	semidirect
196. C₅₃	#33482	semidirect

Maximal subgroups

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

subgroup	index
#20291130	2
#20291098	2
#20291094	2
#20291083	2
#20291075	2
#20290925	2
#20290916	2
#20283689	2
#33482	53