Subgroups — #28328459

Order 13568.

508 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	#20291132	C₂	1	2	yes	maximal, complemented
2	#20291100	C₂	1	2	yes	maximal, complemented
3	#20291097	C₂	1	2	yes	maximal, complemented
4	#20291095	C₂	1	2	yes	maximal, complemented
5	#20291081	C₂	1	2	yes	maximal, complemented
6	#20291079	C₂	1	1	yes	maximal, characteristic, complemented
7	#20290926	C₂	1	2	yes	maximal, complemented
8	#20290922	C₂	1	1	yes	maximal, characteristic, complemented
9	#20283693	C₂	1	1	yes	maximal, characteristic
10	C₂×#13133292	C₂²	1	2	yes	—
11	C₂×#13133291	C₂²	1	2	yes	—
12	C₂²×#11719910	C₂²	1	2	yes	—
13	#16811544	C₂²	1	2	yes	—
14	#16811544	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	C₂²⋊#11719910	C₂²	1	2	yes	—
19	C₂²⋊#11719910	C₂²	1	2	yes	—
20	#16811512	C₂²	1	2	yes	—
21	#16811512	C₂²	1	2	yes	—
22	#16811511	C₂²	1	2	yes	—
23	#16811509	C₂²	1	1	yes	characteristic
24	#16811509	C₂²	1	2	yes	—
25	#16811509	C₂²	1	1	yes	characteristic
26	#16811155	C₄	1	1	yes	characteristic
27	#16810753	C₄	1	1	yes	characteristic
28	#16810747	C₂²	1	1	yes	characteristic
29	#16810692	C₄	1	2	yes	—
30	D₄×#88322	C₂²	1	1	yes	characteristic
31	#16810686	C₂²	1	2	yes	—
32	#16810681	C₄	1	2	yes	—
33	#16810679	C₄	1	1	yes	characteristic
34	#16810581	C₂²	1	2	yes	—
35	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
36	C₅₃×#436	C₄	1	1	yes	characteristic
37	#13133292	—	2	4	no	—
38	#13133291	—	2	4	no	—
39	C₂×#11719910	—	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	C₂²×#88322	C₂³	1	2	yes	—
45	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
47	#13133210	C₈	1	2	yes	—
48	#13133208	C₈	1	1	yes	characteristic
49	#13133204	C₈	1	1	yes	characteristic
50	#13133199	C₈	1	2	yes	—
51	#13133197	C₈	1	1	yes	characteristic
52	#13133188	C₈	1	1	yes	characteristic
53	C₄×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
54	C₄×C₂²⋊D₅₃	C₂×C₄	1	2	yes	—
55	#13133176	C₂×C₄	1	2	yes	—
56	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
57	C₄⋊#88322	C₂×C₄	1	1	yes	characteristic
58	#13133146	C₂×C₄	1	2	yes	—
59	#13133146	C₂×C₄	1	2	yes	—
60	C₂²⋊#88322	C₂×C₄	1	2	yes	—
61	#13133129	C₂×C₄	1	2	yes	—
62	#13133127	C₂×C₄	1	2	yes	—
63	#13133127	C₂×C₄	1	2	yes	—
64	C₄×#88322	C₂³	1	1	yes	characteristic
65	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
66	C₈×Dic₅₃	C₂³	1	2	yes	—
67	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
68	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
69	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
70	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
71	C₂²⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
72	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
73	C₂×D₄×C₅₃	C₂×C₈	1	1	yes	characteristic
74	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
75	C₂×C₂²⋊D₅₃	C₂×C₈	1	2	yes	—
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×Dic₁₀₆	C₂×C₈	1	1	yes	characteristic
80	M₁₆×C₅₃	—	2	4	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	C₂²⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
86	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
87	C₂²⋊Dic₅₃	C₂×C₈	1	2	yes	—
88	C₂²⋊Dic₅₃	C₂×C₈	1	2	yes	—
89	C₂²⋊Dic₅₃	C₂×C₈	1	2	yes	—
90	#11719921	C₂×C₈	1	2	yes	—
91	C₄⋊Dic₅₃	C₂×C₈	1	2	yes	—
92	C₄⋊Dic₅₃	C₂×C₈	1	2	yes	—
93	C₄⋊Dic₅₃	C₂×C₈	1	1	yes	characteristic
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	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
100	C₂×#88322	—	2	2	no	—
101	C₂×#88322	—	2	2	no	—
102	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
103	C₂×#88322	C₂²×C₄	1	2	yes	—
104	C₂×#88322	C₂⁴	1	1	yes	characteristic
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	C₂³×C₅₃	C₂²×C₈	1	2	yes	—
112	C₂²×D₅₃	C₂²×C₈	1	1	yes	characteristic
113	D₄×C₅₃	—	4	1	no	—
114	C₂×C₄×C₅₃	—	2	2	no	—
115	C₂×C₄×C₅₃	—	2	2	no	—
116	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
117	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
118	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
119	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
120	C₂²⋊D₅₃	—	4	2	no	—
121	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
122	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
123	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
124	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
125	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
126	C₄×D₅₃	—	2	1	no	—
127	C₄×D₅₃	—	2	1	no	—
128	C₄×D₅₃	—	4	1	no	—
129	C₄×D₅₃	—	4	1	no	—
130	Dic₁₀₆	—	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	2	no	—
135	#88322	—	2	1	no	—
136	#88322	C₂×#41	1	2	yes	—
137	#33481	—	53	1	no	maximal, Sylow
138	C₂²×C₅₃	—	2	2	no	—
139	C₂²×C₅₃	—	2	2	no	—
140	C₂²×C₅₃	—	2	2	no	—
141	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
142	C₂×D₅₃	—	2	1	no	—
143	C₂×D₅₃	—	2	1	no	—
144	C₂×D₅₃	—	2	1	no	—
145	C₄×C₅₃	—	2	1	no	—
146	C₄×C₅₃	—	2	1	no	—
147	C₄×C₅₃	—	2	2	no	—
148	C₄×C₅₃	C₄×#41	1	1	yes	characteristic
149	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
150	Dic₅₃	—	2	2	no	—
151	Dic₅₃	—	2	1	no	—
152	Dic₅₃	—	2	1	no	—
153	Dic₅₃	—	2	2	no	—
154	Dic₅₃	—	2	1	no	—
155	#2966	—	53	1	no	—
156	#2183	—	53	2	no	—
157	#2169	—	53	2	no	—
158	#2168	—	53	2	no	—
159	#2168	—	53	2	no	—
160	#2164	—	53	2	no	—
161	#2163	—	53	1	no	—
162	#2108	—	53	2	no	—
163	C₄×M₃₂	—	53	1	no	—
164	C₂×C₅₃	—	4	2	no	—
165	C₂×C₅₃	C₈×#41	1	1	yes	characteristic
166	C₂×C₅₃	#3407	1	1	yes	characteristic
167	C₂×C₅₃	#3407	1	1	yes	characteristic
168	D₅₃	—	4	1	no	—
169	#529	—	53	1	no	—
170	C₂×M₃₂	—	53	2	no	—
171	C₂×M₃₂	—	53	2	no	—
172	C₂²×C₁₆	—	53	2	no	—
173	#446	—	53	1	no	—
174	C₈×Q₈	—	53	1	no	—
175	#436	#6157	1	1	yes	characteristic
176	#435	—	53	2	no	—
177	C₈×D₄	—	53	1	no	—
178	C₈×D₄	—	53	2	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	2	no	—
191	#346	—	53	2	no	—
192	#346	—	53	2	no	—
193	C₄×C₁₆	—	53	1	no	—
194	C₄×C₁₆	—	53	1	no	—
195	C₄×C₁₆	—	53	2	no	—
196	C₅₃	#33481	1	1	yes	characteristic, complemented, Sylow
197	C₂×M₁₆	C₂×#6157	1	2	yes	—
198	C₂²×C₈	—	53	2	no	—
199	C₂²×C₈	—	53	1	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	—	53	1	no	—
208	C₄×D₄	—	53	2	no	—
209	C₄×D₄	C₂×#6157	1	1	yes	characteristic
210	#117	—	53	2	no	—
211	C₂×C₄²	—	53	1	no	—
212	M₃₂	—	106	4	no	—
213	M₃₂	—	106	4	no	—
214	C₂×C₁₆	—	53	2	no	—
215	C₂×C₁₆	—	53	2	no	—
216	C₂×C₁₆	—	53	2	no	—
217	C₂×C₁₆	—	53	2	no	—
218	C₂×C₁₆	—	106	4	no	—
219	C₄⋊C₈	—	53	2	no	—
220	C₄⋊C₈	—	53	2	no	—
221	C₄⋊C₈	—	53	1	no	—
222	C₄⋊C₈	C₂×#6157	1	1	yes	characteristic
223	C₂²⋊C₈	—	53	2	no	—
224	C₂²⋊C₈	—	53	2	no	—
225	C₂²⋊C₈	C₂×#6157	1	2	yes	—
226	#97	—	53	2	no	—
227	#97	—	53	2	no	—
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	—	53	1	no	—
234	C₂×D₄	—	53	2	no	—
235	C₂×D₄	D₅₃⋊C₈	1	1	yes	characteristic
236	C₂²×C₄	—	53	1	no	—
237	C₂²×C₄	—	53	1	no	—
238	C₂²×C₄	—	53	1	no	—
239	C₂²×C₄	C₂²×#6157	1	2	yes	—
240	M₁₆	—	2	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	2	no	—
254	C₄⋊C₄	—	53	1	no	—
255	C₄⋊C₄	—	53	2	no	—
256	C₄⋊C₄	D₅₃⋊C₈	1	1	yes	characteristic
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	D₅₃⋊C₈	1	2	yes	—
262	C₄²	—	53	1	no	—
263	C₄²	—	53	1	no	—
264	C₄²	—	53	1	no	—
265	C₄²	—	53	2	no	—
266	C₄²	C₂²×#6157	1	1	yes	characteristic
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	C₂×D₅₃⋊C₈	1	2	yes	—
273	Q₈	—	212	1	no	—
274	D₄	—	4	1	no	—
275	D₄	—	212	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	53	1	no	—
279	C₂×C₄	—	53	1	no	—
280	C₂×C₄	—	53	1	no	—
281	C₂×C₄	—	53	2	no	—
282	C₂×C₄	—	53	2	no	—
283	C₂×C₄	—	106	1	no	—
284	C₂×C₄	—	106	1	no	—
285	C₂×C₄	—	212	1	no	—
286	C₂×C₄	—	212	1	no	—
287	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
288	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
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	2	no	—
295	C₈	—	106	1	no	—
296	C₈	#13133304	1	2	yes	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	106	1	no	—
301	C₂²	—	106	1	no	—
302	C₂²	—	106	1	no	—
303	C₂²	C₂²×D₅₃⋊C₈	1	1	yes	characteristic
304	C₄	—	2	1	no	—
305	C₄	—	2	1	no	—
306	C₄	—	2	2	no	—
307	C₄	—	106	1	no	—
308	C₄	—	106	2	no	—
309	C₄	—	106	1	no	—
310	C₄	—	106	2	no	—
311	C₄	—	106	1	no	—
312	C₄	#41×#6157	1	1	yes	characteristic
313	C₄	C₂×#13133304	1	1	yes	characteristic
314	C₂	—	4	2	no	—
315	C₂	—	212	1	no	—
316	C₂	#20291520	1	1	yes	characteristic
317	C₂	#20291683	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×#436	C₄
prime radical	#13133188	C₈
cosocle	C₂×#88322	C₂⁴
hypercenter	#436	#6157
hypocenter	C₅₃	#33481
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #436

Complements

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

N	H ≅ G/N	type
1. #20291132	C₂	semidirect
2. #20291100	C₂	semidirect
3. #20291097	C₂	semidirect
4. #20291095	C₂	semidirect
5. #20291081	C₂	semidirect
6. #20291079	C₂	semidirect
7. #20290926	C₂	semidirect
8. #20290922	C₂	semidirect
196. C₅₃	#33481	semidirect

Maximal subgroups

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

subgroup	index
#20291132	2
#20291100	2
#20291097	2
#20291095	2
#20291081	2
#20291079	2
#20290926	2
#20290922	2
#20283693	2
#33481	53