Subgroups — #27270651

Order 13568.

560 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274695	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂²⋊#88322	C₂	1	1	yes	maximal, characteristic
3	C₄×#95×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊#88322	C₄	1	2	yes	complemented
5	C₂×C₂²⋊#88322	—	2	1	no	—
6	C₂×C₂²⋊#88322	—	2	1	no	—
7	C₂×C₄×#88322	—	2	1	no	—
8	#16810378	—	2	1	no	—
9	#16810378	C₄	1	2	yes	complemented
10	C₂²×C₄²×C₅₃	—	2	1	no	—
11	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×#95×C₅₃	—	2	1	no	—
13	C₂×#95×C₅₃	C₄	1	2	yes	—
14	C₂²×#88322	—	2	2	no	—
15	C₂²×#88322	—	2	2	no	—
16	C₄⋊#88322	—	4	2	no	—
17	C₂²⋊#88322	—	2	4	no	—
18	C₂²⋊#88322	—	4	1	no	—
19	C₂²⋊#88322	—	4	1	no	—
20	C₄×#88322	—	4	2	no	—
21	C₂³×C₄×C₅₃	—	2	2	no	—
22	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
24	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
25	C₂×C₄²×C₅₃	—	2	4	no	—
26	C₂×C₄²×C₅₃	—	4	2	no	—
27	C₂×C₄²×C₅₃	—	4	2	no	—
28	C₂×C₄²×C₅₃	—	4	1	no	—
29	C₂×C₄²×C₅₃	—	4	1	no	—
30	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
34	#95×C₅₃	—	2	4	no	—
35	#95×C₅₃	—	2	2	no	—
36	#95×C₅₃	C₈	1	4	yes	complemented
37	C₂⁴×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
38	C₂²×C₄×C₅₃	—	2	2	no	—
39	C₂²×C₄×C₅₃	—	2	2	no	—
40	C₂²×C₄×C₅₃	—	2	4	no	—
41	C₂²×C₄×C₅₃	—	2	4	no	—
42	C₂²×C₄×C₅₃	—	2	4	no	—
43	C₂²×C₄×C₅₃	—	2	2	no	—
44	C₂²×C₄×C₅₃	—	2	2	no	—
45	C₂²×C₄×C₅₃	—	4	1	no	—
46	C₂²×C₄×C₅₃	—	4	1	no	—
47	C₂²×C₄×C₅₃	—	4	1	no	—
48	C₂²×C₄×C₅₃	—	4	2	no	—
49	C₂²×C₄×C₅₃	—	4	2	no	—
50	C₂²×C₄×C₅₃	—	4	2	no	—
51	C₂²×C₄×C₅₃	—	4	2	no	—
52	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
59	C₂²×C₄×C₅₃	M₁₆	1	2	yes	—
60	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	Q₁₆	1	1	yes	characteristic
64	C₄²×C₅₃	—	2	2	no	—
65	C₄²×C₅₃	—	2	2	no	—
66	C₄²×C₅₃	—	4	1	no	—
67	C₄²×C₅₃	—	4	1	no	—
68	C₄²×C₅₃	—	4	4	no	—
69	C₄²×C₅₃	—	4	4	no	—
70	C₂×#88322	—	2	4	no	—
71	C₂×#88322	—	4	2	no	—
72	C₂×#88322	—	4	2	no	—
73	C₂×#88322	—	4	2	no	—
74	C₂×#88322	—	4	2	no	—
75	C₂×#88322	—	4	2	no	—
76	C₂³×C₅₃	—	2	1	no	—
77	C₂³×C₅₃	—	2	1	no	—
78	C₂³×C₅₃	—	2	1	no	—
79	C₂³×C₅₃	—	2	1	no	—
80	C₂³×C₅₃	#95	1	1	yes	characteristic
81	C₂³×C₅₃	C₂²⋊C₈	1	2	yes	—
82	C₂³×C₅₃	#102	1	1	yes	characteristic
83	C₂³×C₅₃	#103	1	1	yes	characteristic
84	C₂³×C₅₃	#104	1	2	yes	—
85	C₂×C₄×C₅₃	—	2	1	no	—
86	C₂×C₄×C₅₃	—	2	1	no	—
87	C₂×C₄×C₅₃	—	2	1	no	—
88	C₂×C₄×C₅₃	—	2	1	no	—
89	C₂×C₄×C₅₃	—	2	1	no	—
90	C₂×C₄×C₅₃	—	2	1	no	—
91	C₂×C₄×C₅₃	—	2	1	no	—
92	C₂×C₄×C₅₃	—	2	1	no	—
93	C₂×C₄×C₅₃	—	2	2	no	—
94	C₂×C₄×C₅₃	—	2	2	no	—
95	C₂×C₄×C₅₃	—	2	2	no	—
96	C₂×C₄×C₅₃	—	2	2	no	—
97	C₂×C₄×C₅₃	—	2	8	no	—
98	C₂×C₄×C₅₃	—	2	1	no	—
99	C₂×C₄×C₅₃	—	2	1	no	—
100	C₂×C₄×C₅₃	—	2	1	no	—
101	C₂×C₄×C₅₃	—	4	2	no	—
102	C₂×C₄×C₅₃	—	4	2	no	—
103	C₂×C₄×C₅₃	—	4	4	no	—
104	C₂×C₄×C₅₃	—	4	4	no	—
105	C₂×C₄×C₅₃	—	4	4	no	—
106	C₂×C₄×C₅₃	—	4	4	no	—
107	C₂×C₄×C₅₃	—	4	4	no	—
108	C₂×C₄×C₅₃	—	4	4	no	—
109	C₂×C₄×C₅₃	—	4	4	no	—
110	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
111	C₂×C₄×C₅₃	#97	1	1	yes	characteristic
112	C₂×C₄×C₅₃	#99	1	2	yes	—
113	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
114	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
115	C₂×C₄×C₅₃	#104	1	2	yes	—
116	C₂×C₄×C₅₃	#104	1	2	yes	—
117	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
118	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
119	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
120	#88322	—	4	4	no	—
121	#88322	—	8	2	no	—
122	#7028	—	53	1	no	maximal, Sylow
123	C₂²×C₅₃	—	2	2	no	—
124	C₂²×C₅₃	—	2	1	no	—
125	C₂²×C₅₃	—	2	1	no	—
126	C₂²×C₅₃	—	2	2	no	—
127	C₂²×C₅₃	—	2	2	no	—
128	C₂²×C₅₃	—	2	1	no	—
129	C₂²×C₅₃	—	2	1	no	—
130	C₂²×C₅₃	—	2	2	no	—
131	C₂²×C₅₃	D₄⋊C₈	1	2	yes	—
132	C₂²×C₅₃	Q₈⋊C₈	1	2	yes	—
133	C₂²×C₅₃	#336	1	1	yes	characteristic
134	C₂²×C₅₃	#339	1	1	yes	characteristic
135	C₂²×C₅₃	#339	1	1	yes	characteristic
136	C₂²×C₅₃	#339	1	1	yes	characteristic
137	C₂²×C₅₃	#340	1	1	yes	characteristic
138	C₂²×C₅₃	#342	1	1	yes	characteristic
139	C₂²×C₅₃	#344	1	1	yes	characteristic
140	C₄×C₅₃	—	2	1	no	—
141	C₄×C₅₃	—	2	1	no	—
142	C₄×C₅₃	—	2	2	no	—
143	C₄×C₅₃	—	2	1	no	—
144	C₄×C₅₃	—	2	1	no	—
145	C₄×C₅₃	—	2	2	no	—
146	C₄×C₅₃	—	4	8	no	—
147	C₄×C₅₃	—	4	1	no	—
148	C₄×C₅₃	#327	1	2	yes	—
149	C₄×C₅₃	#328	1	2	yes	—
150	#1797	—	53	1	no	—
151	C₄×C₂²⋊C₈	—	53	1	no	—
152	C₄×#95	D₅₃	1	1	yes	characteristic
153	C₂×C₅₃	—	2	2	no	—
154	C₂×C₅₃	—	2	2	no	—
155	C₂×C₅₃	#1276	1	1	yes	characteristic
156	C₂×C₅₃	#1279	1	1	yes	characteristic
157	C₂×C₅₃	#1282	1	1	yes	characteristic
158	C₂×C₅₃	#1294	1	1	yes	characteristic
159	C₂×C₅₃	#1297	1	1	yes	characteristic
160	C₂×C₅₃	#1300	1	1	yes	characteristic
161	C₂×C₅₃	#1303	1	1	yes	characteristic
162	C₂²×C₄²	—	2	1	no	—
163	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
164	C₂×C₄⋊C₈	—	53	2	no	—
165	C₂×C₂²⋊C₈	—	106	1	no	—
166	C₂×C₂²⋊C₈	—	106	1	no	—
167	C₂×C₄×C₈	—	106	1	no	—
168	C₂×#95	—	2	1	no	—
169	C₂×#95	Dic₅₃	1	2	yes	—
170	#336	—	53	2	no	—
171	#336	—	106	1	no	—
172	C₅₃	#7028	1	1	yes	characteristic, complemented, Sylow
173	C₂³×C₄	—	2	2	no	—
174	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
175	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
176	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
177	C₂²×C₈	—	106	2	no	—
178	C₂²×C₈	—	106	2	no	—
179	C₂×C₄²	—	2	4	no	—
180	C₂×C₄²	—	4	2	no	—
181	C₂×C₄²	—	4	2	no	—
182	C₂×C₄²	—	4	1	no	—
183	C₂×C₄²	—	4	1	no	—
184	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
185	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
186	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
187	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
188	C₄⋊C₈	—	212	2	no	—
189	C₂²⋊C₈	—	106	4	no	—
190	C₂²⋊C₈	—	212	1	no	—
191	C₂²⋊C₈	—	212	1	no	—
192	C₄×C₈	—	212	2	no	—
193	#95	—	2	4	no	—
194	#95	—	2	2	no	—
195	#95	#88322	1	4	yes	complemented
196	C₂⁴	C₂²⋊Dic₅₃	1	1	yes	characteristic
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	—	2	4	no	—
200	C₂²×C₄	—	2	4	no	—
201	C₂²×C₄	—	2	4	no	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	—	4	1	no	—
205	C₂²×C₄	—	4	1	no	—
206	C₂²×C₄	—	4	1	no	—
207	C₂²×C₄	—	4	2	no	—
208	C₂²×C₄	—	4	2	no	—
209	C₂²×C₄	—	4	2	no	—
210	C₂²×C₄	—	4	2	no	—
211	C₂²×C₄	C₂×#88322	1	2	yes	—
212	C₂²×C₄	#11719917	1	2	yes	—
213	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
214	C₂²×C₄	#11719919	1	1	yes	characteristic
215	C₂²×C₄	#11719919	1	1	yes	characteristic
216	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
217	C₂²×C₄	#11719921	1	1	yes	characteristic
218	C₂²×C₄	#11719921	1	1	yes	characteristic
219	C₂²×C₄	#11719922	1	1	yes	characteristic
220	C₂²×C₄	#11719923	1	1	yes	characteristic
221	C₂²×C₄	#11719924	1	1	yes	characteristic
222	C₂²×C₄	#11719925	1	1	yes	characteristic
223	C₂×C₈	—	106	4	no	—
224	C₂×C₈	—	212	2	no	—
225	C₂×C₈	—	212	2	no	—
226	C₂×C₈	—	212	2	no	—
227	C₂×C₈	—	212	2	no	—
228	C₂×C₈	—	212	2	no	—
229	C₄²	—	2	2	no	—
230	C₄²	—	2	2	no	—
231	C₄²	—	4	1	no	—
232	C₄²	—	4	1	no	—
233	C₄²	—	4	4	no	—
234	C₄²	—	4	4	no	—
235	C₂³	—	2	1	no	—
236	C₂³	—	2	1	no	—
237	C₂³	—	2	1	no	—
238	C₂³	—	2	1	no	—
239	C₂³	#13133124	1	1	yes	characteristic
240	C₂³	C₂²⋊#88322	1	2	yes	—
241	C₂³	#13133138	1	1	yes	characteristic
242	C₂³	#13133141	1	1	yes	characteristic
243	C₂³	#13133144	1	2	yes	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	8	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	4	no	—
263	C₂×C₄	—	4	4	no	—
264	C₂×C₄	—	4	4	no	—
265	C₂×C₄	—	4	4	no	—
266	C₂×C₄	—	4	4	no	—
267	C₂×C₄	—	4	4	no	—
268	C₂×C₄	—	4	4	no	—
269	C₂×C₄	C₄×#88322	1	1	yes	characteristic
270	C₂×C₄	#13133128	1	1	yes	characteristic
271	C₂×C₄	#13133131	1	2	yes	—
272	C₂×C₄	#13133137	1	1	yes	characteristic
273	C₂×C₄	#13133140	1	1	yes	characteristic
274	C₂×C₄	#13133145	1	2	yes	—
275	C₂×C₄	#13133145	1	2	yes	—
276	C₂×C₄	C₄⋊#88322	1	2	yes	—
277	C₂×C₄	#13133148	1	1	yes	characteristic
278	C₂×C₄	#13133150	1	1	yes	characteristic
279	C₈	—	424	2	no	—
280	C₈	—	212	4	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	2	2	no	—
283	C₂²	—	2	1	no	—
284	C₂²	—	2	1	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	2	1	no	—
288	C₂²	—	2	1	no	—
289	C₂²	D₄⋊#88322	1	2	yes	—
290	C₂²	Q₈⋊#88322	1	2	yes	—
291	C₂²	#16810378	1	1	yes	characteristic
292	C₂²	#16810382	1	1	yes	characteristic
293	C₂²	#16810383	1	1	yes	characteristic
294	C₂²	#16810383	1	1	yes	characteristic
295	C₂²	#16810385	1	1	yes	characteristic
296	C₂²	#16810389	1	1	yes	characteristic
297	C₂²	#16810393	1	1	yes	characteristic
298	C₄	—	2	1	no	—
299	C₄	—	2	1	no	—
300	C₄	—	2	2	no	—
301	C₄	—	2	1	no	—
302	C₄	—	2	1	no	—
303	C₄	—	2	2	no	—
304	C₄	—	4	8	no	—
305	C₄	—	4	1	no	—
306	C₄	#16810356	1	2	yes	—
307	C₄	#16810359	1	2	yes	—
308	C₂	—	2	2	no	—
309	C₂	—	2	2	no	—
310	C₂	#20272548	1	1	yes	characteristic
311	C₂	#20272556	1	1	yes	characteristic
312	C₂	#20272563	1	1	yes	characteristic
313	C₂	#20272594	1	1	yes	characteristic
314	C₂	#20272601	1	1	yes	characteristic
315	C₂	#20272608	1	1	yes	characteristic
316	C₂	#20272618	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	#11719921
commutator	C₂×C₄×C₅₃	C₄×C₈
Frattini	C₂²×C₄²	C₂×D₅₃
Fitting	C₄×#95×C₅₃	C₂
socle	C₂³×C₅₃	#95
prime radical	C₂⁴×C₅₃	C₂²⋊C₄
cosocle	C₂²×C₄²×C₅₃	C₂²
hypercenter	C₄×#95	D₅₃
hypocenter	C₅₃	#7028
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₄×#95

Complements

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

N	H ≅ G/N	type
4. C₂×C₄⋊#88322	C₄	semidirect
9. #16810378	C₄	semidirect
36. #95×C₅₃	C₈	semidirect
172. C₅₃	#7028	semidirect
195. #95	#88322	semidirect

Maximal subgroups

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

subgroup	index
#20274695	2
C₄×C₂²⋊#88322	2
C₄×#95×C₅₃	2
#7028	53