Subgroups — #27270621

Order 13568.

474 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₂²⋊#88322	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810378	C₂	1	1	yes	maximal, characteristic
3	C₄×C₂²⋊C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂³×#88322	C₄	1	2	yes	—
5	C₂×C₂²⋊#88322	C₄	1	2	yes	—
6	C₂×C₄×#88322	—	2	1	no	—
7	#16810378	—	2	1	no	—
8	#16810378	—	2	1	no	—
9	#16810378	—	2	1	no	—
10	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
11	C₂×C₂²⋊C₈×C₅₃	C₄	1	2	yes	—
12	C₂×C₄×C₈×C₅₃	—	2	1	no	—
13	C₅₃×#336	—	2	1	no	—
14	C₂²×#88322	—	2	2	no	—
15	C₂²×#88322	—	2	2	no	—
16	C₂²×#88322	—	2	2	no	—
17	C₂²×#88322	—	4	2	no	—
18	C₂²×#88322	—	4	2	no	—
19	C₂²×#88322	C₈	1	4	yes	complemented
20	C₂²⋊#88322	—	2	2	no	—
21	C₂²⋊#88322	C₈	1	4	yes	complemented
22	C₄×#88322	—	4	2	no	—
23	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₅₃	—	2	2	no	—
27	C₂×C₄²×C₅₃	—	4	2	no	—
28	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
32	C₂²⋊C₈×C₅₃	—	2	2	no	—
33	C₂²⋊C₈×C₅₃	C₈	1	4	yes	complemented
34	C₄×C₈×C₅₃	—	4	2	no	—
35	C₂⁴×C₅₃	C₄²	1	1	yes	characteristic
36	C₂²×C₄×C₅₃	—	2	2	no	—
37	C₂²×C₄×C₅₃	—	2	1	no	—
38	C₂²×C₄×C₅₃	—	2	1	no	—
39	C₂²×C₄×C₅₃	—	4	1	no	—
40	C₂²×C₄×C₅₃	—	4	1	no	—
41	C₂²×C₄×C₅₃	—	4	1	no	—
42	C₂²×C₄×C₅₃	—	4	1	no	—
43	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
50	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
51	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
52	C₂²×C₄×C₅₃	M₁₆	1	2	yes	—
53	C₂²×C₄×C₅₃	M₁₆	1	2	yes	—
54	C₂²×C₄×C₅₃	M₁₆	1	2	yes	—
55	C₂×C₈×C₅₃	—	2	4	no	—
56	C₂×C₈×C₅₃	—	4	2	no	—
57	C₂×C₈×C₅₃	—	4	2	no	—
58	C₄²×C₅₃	—	2	2	no	—
59	C₄²×C₅₃	—	2	2	no	—
60	C₄²×C₅₃	—	4	1	no	—
61	C₄²×C₅₃	—	4	1	no	—
62	C₂×#88322	—	2	4	no	—
63	C₂×#88322	—	2	4	no	—
64	C₂×#88322	—	4	2	no	—
65	C₂×#88322	—	4	2	no	—
66	C₂×#88322	—	4	4	no	—
67	C₂×#88322	—	4	2	no	—
68	C₂×#88322	—	4	2	no	—
69	C₂×#88322	—	4	4	no	—
70	C₂³×C₅₃	—	2	1	no	—
71	C₂³×C₅₃	—	2	1	no	—
72	C₂³×C₅₃	—	2	1	no	—
73	C₂³×C₅₃	—	2	1	no	—
74	C₂³×C₅₃	#95	1	1	yes	characteristic
75	C₂³×C₅₃	C₄×C₈	1	1	yes	characteristic
76	C₂³×C₅₃	C₄×C₈	1	1	yes	characteristic
77	C₂³×C₅₃	C₄×C₈	1	1	yes	characteristic
78	C₂³×C₅₃	#97	1	1	yes	characteristic
79	C₂³×C₅₃	#97	1	1	yes	characteristic
80	C₂³×C₅₃	#97	1	1	yes	characteristic
81	C₂×C₄×C₅₃	—	2	1	no	—
82	C₂×C₄×C₅₃	—	2	1	no	—
83	C₂×C₄×C₅₃	—	2	1	no	—
84	C₂×C₄×C₅₃	—	2	1	no	—
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	2	no	—
92	C₂×C₄×C₅₃	—	2	2	no	—
93	C₂×C₄×C₅₃	—	2	2	no	—
94	C₂×C₄×C₅₃	—	2	2	no	—
95	C₂×C₄×C₅₃	—	4	2	no	—
96	C₂×C₄×C₅₃	—	4	2	no	—
97	C₂×C₄×C₅₃	—	4	2	no	—
98	C₂×C₄×C₅₃	—	4	2	no	—
99	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
100	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
101	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
102	C₂×C₄×C₅₃	#99	1	2	yes	—
103	C₂×C₄×C₅₃	#100	1	1	yes	characteristic
104	C₂×C₄×C₅₃	#101	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
106	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
107	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
108	C₈×C₅₃	—	4	4	no	—
109	#88322	—	4	4	no	—
110	#88322	—	4	4	no	—
111	#7016	—	53	1	no	maximal, Sylow
112	C₂²×C₅₃	—	2	1	no	—
113	C₂²×C₅₃	—	2	1	no	—
114	C₂²×C₅₃	—	2	1	no	—
115	C₂²×C₅₃	—	2	1	no	—
116	C₂²×C₅₃	—	2	1	no	—
117	C₂²×C₅₃	—	2	1	no	—
118	C₂²×C₅₃	—	2	1	no	—
119	C₂²×C₅₃	—	2	1	no	—
120	C₂²×C₅₃	—	2	1	no	—
121	C₂²×C₅₃	—	2	1	no	—
122	C₂²×C₅₃	—	2	1	no	—
123	C₂²×C₅₃	—	2	1	no	—
124	C₂²×C₅₃	C₈²	1	1	yes	characteristic
125	C₂²×C₅₃	#322	1	1	yes	characteristic
126	C₂²×C₅₃	#322	1	1	yes	characteristic
127	C₂²×C₅₃	#322	1	1	yes	characteristic
128	C₂²×C₅₃	#336	1	1	yes	characteristic
129	C₂²×C₅₃	#336	1	1	yes	characteristic
130	C₂²×C₅₃	#336	1	1	yes	characteristic
131	C₂²×C₅₃	#342	1	1	yes	characteristic
132	C₂²×C₅₃	#343	1	1	yes	characteristic
133	C₂²×C₅₃	#344	1	1	yes	characteristic
134	C₂²×C₅₃	#344	1	1	yes	characteristic
135	C₄×C₅₃	—	2	2	no	—
136	C₄×C₅₃	—	2	2	no	—
137	C₄×C₅₃	—	2	2	no	—
138	C₄×C₅₃	—	2	2	no	—
139	C₄×C₅₃	—	2	2	no	—
140	C₄×C₅₃	#323	1	2	yes	—
141	C₄×C₅₃	#324	1	2	yes	—
142	C₄×C₂²⋊C₈	—	53	1	no	—
143	C₄×C₂²⋊C₈	D₅₃	1	1	yes	characteristic
144	C₂×#336	—	53	1	no	—
145	C₂×C₅₃	—	2	1	no	—
146	C₂×C₅₃	—	2	1	no	—
147	C₂×C₅₃	—	2	1	no	—
148	C₂×C₅₃	—	2	1	no	—
149	C₂×C₅₃	#1273	1	1	yes	characteristic
150	C₂×C₅₃	#1280	1	1	yes	characteristic
151	C₂×C₅₃	#1281	1	1	yes	characteristic
152	C₂×C₅₃	#1282	1	1	yes	characteristic
153	C₂×C₅₃	#1282	1	1	yes	characteristic
154	C₂×C₅₃	#1283	1	1	yes	characteristic
155	C₂×C₅₃	#1283	1	1	yes	characteristic
156	C₂³×C₈	—	53	2	no	—
157	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
158	C₂×C₂²⋊C₈	—	53	2	no	—
159	C₂×C₂²⋊C₈	Dic₅₃	1	2	yes	—
160	C₂×C₄×C₈	—	2	1	no	—
161	C₂×C₄×C₈	—	106	1	no	—
162	#336	—	2	1	no	—
163	#336	—	106	1	no	—
164	#336	—	106	1	no	—
165	#336	—	106	1	no	—
166	C₅₃	#7016	1	1	yes	characteristic, complemented, Sylow
167	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
168	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
169	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
170	C₂²×C₈	—	2	2	no	—
171	C₂²×C₈	—	53	4	no	—
172	C₂²×C₈	—	106	2	no	—
173	C₂²×C₈	—	106	2	no	—
174	C₂²×C₈	—	106	2	no	—
175	C₂²×C₈	—	212	2	no	—
176	C₂²×C₈	—	212	2	no	—
177	C₂×C₄²	—	4	2	no	—
178	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
179	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
180	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
181	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
182	C₂²⋊C₈	—	2	2	no	—
183	C₂²⋊C₈	—	53	4	no	—
184	C₂²⋊C₈	—	106	2	no	—
185	C₂²⋊C₈	#88322	1	4	yes	complemented
186	C₄×C₈	—	4	2	no	—
187	C₄×C₈	—	212	2	no	—
188	C₂⁴	C₄×Dic₅₃	1	1	yes	characteristic
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	—	2	1	no	—
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	—	4	1	no	—
193	C₂²×C₄	—	4	1	no	—
194	C₂²×C₄	—	4	1	no	—
195	C₂²×C₄	—	4	1	no	—
196	C₂²×C₄	C₈×D₅₃	1	2	yes	—
197	C₂²×C₄	C₈×D₅₃	1	2	yes	—
198	C₂²×C₄	#11719912	1	2	yes	—
199	C₂²×C₄	#11719912	1	2	yes	—
200	C₂²×C₄	C₂×#88322	1	2	yes	—
201	C₂²×C₄	#11719917	1	2	yes	—
202	C₂²×C₄	#11719919	1	1	yes	characteristic
203	C₂²×C₄	#11719919	1	1	yes	characteristic
204	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
205	C₂²×C₄	#11719921	1	1	yes	characteristic
206	C₂²×C₄	#11719921	1	1	yes	characteristic
207	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
208	C₂×C₈	—	2	4	no	—
209	C₂×C₈	—	4	2	no	—
210	C₂×C₈	—	4	2	no	—
211	C₂×C₈	—	106	4	no	—
212	C₂×C₈	—	106	4	no	—
213	C₂×C₈	—	212	2	no	—
214	C₂×C₈	—	212	4	no	—
215	C₂×C₈	—	212	2	no	—
216	C₂×C₈	—	212	2	no	—
217	C₂×C₈	—	212	2	no	—
218	C₂×C₈	—	212	4	no	—
219	C₄²	—	2	2	no	—
220	C₄²	—	2	2	no	—
221	C₄²	—	4	1	no	—
222	C₄²	—	4	1	no	—
223	C₂³	—	2	1	no	—
224	C₂³	—	2	1	no	—
225	C₂³	—	2	1	no	—
226	C₂³	—	2	1	no	—
227	C₂³	#13133124	1	1	yes	characteristic
228	C₂³	C₈×Dic₅₃	1	1	yes	characteristic
229	C₂³	C₈×Dic₅₃	1	1	yes	characteristic
230	C₂³	C₄×#88322	1	1	yes	characteristic
231	C₂³	#13133127	1	1	yes	characteristic
232	C₂³	#13133127	1	1	yes	characteristic
233	C₂³	#13133128	1	1	yes	characteristic
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	#13133129	1	2	yes	—
253	C₂×C₄	#13133129	1	2	yes	—
254	C₂×C₄	C₂²⋊#88322	1	2	yes	—
255	C₂×C₄	#13133131	1	2	yes	—
256	C₂×C₄	#13133133	1	1	yes	characteristic
257	C₂×C₄	#13133135	1	1	yes	characteristic
258	C₂×C₄	#13133146	1	2	yes	—
259	C₂×C₄	#13133146	1	2	yes	—
260	C₂×C₄	C₄⋊#88322	1	2	yes	—
261	C₈	—	4	4	no	—
262	C₈	—	212	4	no	—
263	C₈	—	212	4	no	—
264	C₂²	—	2	1	no	—
265	C₂²	—	2	1	no	—
266	C₂²	—	2	1	no	—
267	C₂²	—	2	1	no	—
268	C₂²	—	2	1	no	—
269	C₂²	—	2	1	no	—
270	C₂²	—	2	1	no	—
271	C₂²	—	2	1	no	—
272	C₂²	—	2	1	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	2	1	no	—
275	C₂²	—	2	1	no	—
276	C₂²	C₈×#88322	1	1	yes	characteristic
277	C₂²	#16810344	1	1	yes	characteristic
278	C₂²	#16810345	1	1	yes	characteristic
279	C₂²	#16810345	1	1	yes	characteristic
280	C₂²	#16810377	1	1	yes	characteristic
281	C₂²	#16810377	1	1	yes	characteristic
282	C₂²	#16810378	1	1	yes	characteristic
283	C₂²	#16810389	1	1	yes	characteristic
284	C₂²	#16810391	1	1	yes	characteristic
285	C₂²	#16810393	1	1	yes	characteristic
286	C₂²	#16810395	1	1	yes	characteristic
287	C₄	—	2	2	no	—
288	C₄	—	2	2	no	—
289	C₄	—	2	2	no	—
290	C₄	—	2	2	no	—
291	C₄	—	2	2	no	—
292	C₄	#16810346	1	2	yes	—
293	C₄	#16810348	1	2	yes	—
294	C₂	—	2	1	no	—
295	C₂	—	2	1	no	—
296	C₂	—	2	1	no	—
297	C₂	—	2	1	no	—
298	C₂	#20272542	1	1	yes	characteristic
299	C₂	#20272557	1	1	yes	characteristic
300	C₂	#20272559	1	1	yes	characteristic
301	C₂	#20272561	1	1	yes	characteristic
302	C₂	#20272563	1	1	yes	characteristic
303	C₂	#20272564	1	1	yes	characteristic
304	C₂	#20272566	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	#11719921
commutator	C₂²×C₅₃	C₈²
Frattini	C₂²×C₄²	C₂×D₅₃
Fitting	C₄×C₂²⋊C₈×C₅₃	C₂
socle	C₂³×C₅₃	#95
prime radical	C₂⁴×C₅₃	C₄²
cosocle	C₂²×C₄²×C₅₃	C₂²
hypercenter	C₄×C₂²⋊C₈	D₅₃
hypocenter	C₅₃	#7016
LCS	G ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₄×C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
19. C₂²×#88322	C₈	semidirect
21. C₂²⋊#88322	C₈	semidirect
33. C₂²⋊C₈×C₅₃	C₈	semidirect
166. C₅₃	#7016	semidirect
185. C₂²⋊C₈	#88322	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₂²⋊#88322	2
C₂×#16810378	2
C₄×C₂²⋊C₈×C₅₃	2
#7016	53