Subgroups — #27273550

Order 13568.

718 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810676	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×#88322	C₂	1	1	yes	maximal, characteristic
3	#20272549	C₂	1	4	yes	maximal, complemented
4	C₂×C₅₃×#431	C₂	1	1	yes	maximal, characteristic
5	C₂³×#88322	C₄	1	2	yes	—
6	#16810676	—	2	2	no	—
7	#16810676	C₄	1	4	yes	—
8	C₂×C₄⋊#88322	C₂²	1	2	yes	—
9	C₂×C₂²⋊#88322	—	2	1	no	—
10	C₂×C₄×#88322	—	2	1	no	—
11	C₂×C₄×#88322	—	2	2	no	—
12	C₂×C₄×#88322	—	2	2	no	—
13	C₂×C₄×#88322	C₄	1	2	yes	—
14	C₂×C₄×#88322	C₂²	1	2	yes	—
15	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#431	—	2	2	no	—
17	C₅₃×#431	C₄	1	4	yes	—
18	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
19	C₂×C₂²⋊C₈×C₅₃	—	2	1	no	—
20	C₂×C₄×C₈×C₅₃	—	2	1	no	—
21	C₂²×#88322	—	2	4	no	—
22	C₂²×#88322	—	2	4	no	—
23	C₂²×#88322	—	2	2	no	—
24	C₂²×#88322	C₈	1	8	yes	complemented
25	C₂²×#88322	C₂×C₄	1	4	yes	—
26	C₄⋊#88322	—	2	2	no	—
27	C₄⋊#88322	C₂×C₄	1	4	yes	—
28	C₂²⋊#88322	—	2	4	no	—
29	C₄×#88322	—	2	4	no	—
30	C₄×#88322	—	2	4	no	—
31	C₄×#88322	—	2	2	no	—
32	C₄×#88322	—	2	2	no	—
33	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
35	C₂³×C₄×C₅₃	Q₈	1	1	yes	characteristic
36	C₂²×C₈×C₅₃	—	2	2	no	—
37	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₅₃	D₄	1	2	yes	—
41	C₂×C₄²×C₅₃	D₄	1	2	yes	—
42	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
44	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
45	C₄⋊C₈×C₅₃	—	2	2	no	—
46	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
47	C₂²⋊C₈×C₅₃	—	2	4	no	—
48	C₄×C₈×C₅₃	—	2	4	no	—
49	C₂⁴×C₅₃	C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	—	2	4	no	—
51	C₂²×C₄×C₅₃	—	2	4	no	—
52	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
53	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
54	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
55	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
56	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
58	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
59	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	C₂×C₈	1	4	yes	—
62	C₂²×C₄×C₅₃	M₁₆	1	4	yes	—
63	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
66	C₂×C₈×C₅₃	—	2	4	no	—
67	C₂×C₈×C₅₃	—	4	4	no	—
68	C₄²×C₅₃	C₄²	1	2	yes	—
69	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
70	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
71	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
72	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
73	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
74	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
75	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
76	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
78	C₂×#88322	—	2	16	no	—
79	C₂×#88322	—	2	4	no	—
80	C₂×#88322	—	2	4	no	—
81	C₂×#88322	—	2	4	no	—
82	C₂×#88322	—	4	4	no	—
83	C₂×#88322	C₂×C₈	1	8	yes	—
84	C₂³×C₅₃	—	2	2	no	—
85	C₂³×C₅₃	—	2	2	no	—
86	C₂³×C₅₃	#95	1	2	yes	—
87	C₂³×C₅₃	C₄⋊C₈	1	4	yes	—
88	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₅₃	—	2	2	no	—
90	C₂×C₄×C₅₃	—	2	2	no	—
91	C₂×C₄×C₅₃	—	2	4	no	—
92	C₂×C₄×C₅₃	—	2	4	no	—
93	C₂×C₄×C₅₃	—	2	4	no	—
94	C₂×C₄×C₅₃	—	2	4	no	—
95	C₂×C₄×C₅₃	#95	1	2	yes	—
96	C₂×C₄×C₅₃	#95	1	2	yes	—
97	C₂×C₄×C₅₃	#95	1	2	yes	—
98	C₂×C₄×C₅₃	C₄×C₈	1	4	yes	—
99	C₂×C₄×C₅₃	#97	1	4	yes	—
100	C₂×C₄×C₅₃	C₂²⋊C₈	1	8	yes	—
101	C₂×C₄×C₅₃	C₄⋊C₈	1	4	yes	—
102	C₂×C₄×C₅₃	#108	1	4	yes	—
103	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
104	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
110	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
111	C₂×C₄×C₅₃	#135	1	4	yes	—
112	C₈×C₅₃	—	4	4	no	—
113	#88322	—	2	8	no	—
114	#88322	—	4	4	no	—
115	#7872	—	53	1	no	maximal, Sylow
116	C₂²×C₅₃	—	2	8	no	—
117	C₂²×C₅₃	—	2	2	no	—
118	C₂²×C₅₃	—	2	2	no	—
119	C₂²×C₅₃	#336	1	4	yes	—
120	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
121	C₂²×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
122	C₂²×C₅₃	#427	1	2	yes	—
123	C₂²×C₅₃	C₂×#108	1	2	yes	—
124	C₄×C₅₃	—	2	2	no	—
125	C₄×C₅₃	—	2	1	no	—
126	C₄×C₅₃	—	2	1	no	—
127	C₄×C₅₃	#336	1	4	yes	—
128	C₄×C₅₃	#341	1	4	yes	—
129	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
130	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
131	C₄×C₅₃	C₂×C₂²⋊C₈	1	2	yes	—
132	C₄×C₅₃	#416	1	2	yes	—
133	C₄×C₅₃	#416	1	2	yes	—
134	C₂×#431	—	53	1	no	—
135	C₂×#431	D₅₃	1	1	yes	characteristic
136	C₂²×C₄×C₈	—	53	1	no	—
137	#1277	—	53	4	no	—
138	C₂×C₅₃	—	2	4	no	—
139	C₂×C₅₃	#1564	1	4	yes	—
140	C₂×C₅₃	C₂×#336	1	1	yes	characteristic
141	C₂×C₅₃	#1735	1	1	yes	characteristic
142	C₂×C₅₃	C₂×#341	1	1	yes	characteristic
143	C₂³×C₈	—	53	2	no	—
144	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
145	#431	—	2	2	no	—
146	#431	—	53	4	no	—
147	#431	—	106	2	no	—
148	#431	Dic₅₃	1	4	yes	—
149	C₂×C₄⋊C₈	—	53	2	no	—
150	C₂×C₄⋊C₈	C₂×D₅₃	1	2	yes	—
151	C₂×C₂²⋊C₈	—	2	1	no	—
152	C₂×C₂²⋊C₈	—	106	1	no	—
153	C₂×C₄×C₈	—	2	1	no	—
154	C₂×C₄×C₈	—	53	2	no	—
155	C₂×C₄×C₈	—	53	2	no	—
156	C₂×C₄×C₈	—	106	2	no	—
157	C₂×C₄×C₈	—	106	2	no	—
158	C₂×C₄×C₈	—	106	1	no	—
159	C₅₃	#7872	1	1	yes	characteristic, complemented, Sylow
160	C₂³×C₄	Dic₁₀₆	1	1	yes	characteristic
161	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
162	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
163	C₂²×C₈	—	2	2	no	—
164	C₂²×C₈	—	53	4	no	—
165	C₂²×C₈	—	53	8	no	—
166	C₂²×C₈	—	106	4	no	—
167	C₂²×C₈	—	106	4	no	—
168	C₂²×C₈	—	106	2	no	—
169	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
170	C₂×C₄²	C₄×D₅₃	1	2	yes	—
171	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
172	C₂×C₄²	D₂₁₂	1	2	yes	—
173	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
174	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
175	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
176	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
177	C₄⋊C₈	—	2	2	no	—
178	C₄⋊C₈	—	53	4	no	—
179	C₄⋊C₈	—	106	2	no	—
180	C₄⋊C₈	C₂×Dic₅₃	1	4	yes	—
181	C₂²⋊C₈	—	2	4	no	—
182	C₂²⋊C₈	—	106	4	no	—
183	C₄×C₈	—	2	4	no	—
184	C₄×C₈	—	106	2	no	—
185	C₄×C₈	—	106	2	no	—
186	C₄×C₈	—	106	4	no	—
187	C₄×C₈	—	106	4	no	—
188	C₂⁴	#11719919	1	1	yes	characteristic
189	C₂²×C₄	—	2	4	no	—
190	C₂²×C₄	—	2	4	no	—
191	C₂²×C₄	C₈×D₅₃	1	4	yes	—
192	C₂²×C₄	#11719912	1	4	yes	—
193	C₂²×C₄	C₄×Dic₅₃	1	2	yes	—
194	C₂²×C₄	#11719919	1	1	yes	characteristic
195	C₂²×C₄	#11719919	1	2	yes	—
196	C₂²×C₄	#11719919	1	1	yes	characteristic
197	C₂²×C₄	#11719919	1	1	yes	characteristic
198	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
199	C₂²×C₄	#11719921	1	2	yes	—
200	C₂²×C₄	#11719921	1	2	yes	—
201	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
202	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
203	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
204	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
205	C₂×C₈	—	2	4	no	—
206	C₂×C₈	—	4	4	no	—
207	C₂×C₈	—	53	8	no	—
208	C₂×C₈	—	106	16	no	—
209	C₂×C₈	—	106	4	no	—
210	C₂×C₈	—	106	4	no	—
211	C₂×C₈	—	106	4	no	—
212	C₂×C₈	—	212	4	no	—
213	C₄²	C₄×Dic₅₃	1	2	yes	—
214	C₄²	#11719919	1	2	yes	—
215	C₄²	C₄⋊Dic₅₃	1	2	yes	—
216	C₄²	#11719921	1	2	yes	—
217	C₄²	#11719921	1	2	yes	—
218	C₄²	C₂²⋊Dic₅₃	1	2	yes	—
219	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
220	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
221	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
222	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
223	C₂³	—	2	2	no	—
224	C₂³	—	2	2	no	—
225	C₂³	#13133124	1	2	yes	—
226	C₂³	#13133146	1	4	yes	—
227	C₂³	C₂×#11719919	1	1	yes	characteristic
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	#13133124	1	2	yes	—
235	C₂×C₄	#13133124	1	2	yes	—
236	C₂×C₄	#13133124	1	2	yes	—
237	C₂×C₄	C₈×Dic₅₃	1	4	yes	—
238	C₂×C₄	#13133127	1	4	yes	—
239	C₂×C₄	#13133129	1	8	yes	—
240	C₂×C₄	#13133146	1	4	yes	—
241	C₂×C₄	#13133152	1	4	yes	—
242	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
243	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
244	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
245	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
246	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
247	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
248	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
249	C₂×C₄	C₂×#11719912	1	2	yes	—
250	C₂×C₄	#13133237	1	4	yes	—
251	C₈	—	4	4	no	—
252	C₈	—	106	8	no	—
253	C₈	—	212	4	no	—
254	C₂²	—	2	2	no	—
255	C₂²	—	2	8	no	—
256	C₂²	—	2	2	no	—
257	C₂²	#16810377	1	4	yes	—
258	C₂²	C₂×#13133124	1	1	yes	characteristic
259	C₂²	C₂×#13133146	1	2	yes	—
260	C₂²	#16810659	1	2	yes	—
261	C₂²	C₂×#13133152	1	2	yes	—
262	C₄	—	2	1	no	—
263	C₄	—	2	1	no	—
264	C₄	—	2	2	no	—
265	C₄	#16810377	1	4	yes	—
266	C₄	#16810387	1	4	yes	—
267	C₄	C₂×C₈×Dic₅₃	1	1	yes	characteristic
268	C₄	C₂×#13133127	1	1	yes	characteristic
269	C₄	C₂×#13133129	1	2	yes	—
270	C₄	#16810617	1	2	yes	—
271	C₄	#16810619	1	2	yes	—
272	C₂	—	2	4	no	—
273	C₂	#20273461	1	4	yes	—
274	C₂	C₂×#16810377	1	1	yes	characteristic
275	C₂	#20274431	1	1	yes	characteristic
276	C₂	C₂×#16810387	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²	C₂²⋊D₅₃
commutator	C₄×C₅₃	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₅₃×#431	C₂
socle	C₂³×C₅₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₅₃	C₄⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×#431	D₅₃
hypocenter	C₅₃	#7872
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄² ◁ C₂²×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
3. #20272549	C₂	semidirect
24. C₂²×#88322	C₈	semidirect
159. C₅₃	#7872	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810676	2
C₂²×C₄×#88322	2
#20272549	2
C₂×C₅₃×#431	2
#7872	53