Subgroups — #27402995

Order 13568.

876 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×#88322	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	#20275433	C₂	1	3	yes	maximal
4	#20275431	C₂	1	3	yes	maximal
5	#20274831	C₂	1	3	yes	maximal
6	C₄×#13133146	C₂	1	3	yes	maximal
7	C₂×C₈×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
8	C₂×Q₈×Dic₅₃	C₄	1	1	yes	characteristic
9	C₂×Q₈×Dic₅₃	C₄	1	1	yes	characteristic
10	Q₈×#88322	—	2	4	no	—
11	C₂×C₄⋊#88322	C₂²	1	3	yes	—
12	C₂×#13133146	C₂²	1	3	yes	—
13	C₂×#13133146	C₂²	1	3	yes	—
14	C₂×C₄×#88322	C₂²	1	3	yes	—
15	C₄⋊C₄×Dic₅₃	C₄	1	3	yes	—
16	C₄⋊C₄×Dic₅₃	C₄	1	3	yes	—
17	C₄×C₄⋊Dic₅₃	C₂²	1	3	yes	—
18	C₄²×Dic₅₃	C₂²	1	3	yes	—
19	#16810377	C₂²	1	1	yes	characteristic
20	#16810377	C₂²	1	3	yes	—
21	#16810377	C₂²	1	6	yes	—
22	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₈×Q₈×C₅₃	—	2	4	no	—
24	C₂×C₄⋊C₈×C₅₃	C₂²	1	3	yes	—
25	C₂×C₄×C₈×C₅₃	C₂²	1	3	yes	—
26	C₂²×#88322	C₂³	1	3	yes	—
27	C₂²×#88322	C₂³	1	1	yes	characteristic
28	Q₈×Dic₅₃	—	2	4	no	—
29	Q₈×Dic₅₃	C₈	1	8	yes	complemented
30	C₂×C₄⋊Dic₅₃	C₂×C₄	1	3	yes	—
31	C₂×C₄⋊Dic₅₃	C₂×C₄	1	3	yes	—
32	C₂×C₄×Dic₅₃	C₂×C₄	1	3	yes	—
33	C₂×C₄×Dic₅₃	C₂×C₄	1	3	yes	—
34	C₂×C₄×Dic₅₃	C₂×C₄	1	3	yes	—
35	C₂×C₄×Dic₅₃	C₂×C₄	1	3	yes	—
36	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
37	C₂×C₄×Dic₅₃	C₂³	1	3	yes	—
38	C₄⋊#88322	—	2	6	no	—
39	#13133146	—	2	6	no	—
40	#13133146	—	2	6	no	—
41	C₄×#88322	—	2	6	no	—
42	#13133124	C₂×C₄	1	6	yes	—
43	C₂²×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₅₃	C₂³	1	3	yes	—
45	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
46	C₄×Q₈×C₅₃	D₄	1	4	yes	—
47	C₄×Q₈×C₅₃	Q₈	1	4	yes	—
48	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	3	yes	—
49	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
50	C₄⋊C₈×C₅₃	—	2	6	no	—
51	C₄×C₈×C₅₃	—	2	6	no	—
52	C₂×Q₈×C₅₃	C₄⋊C₄	1	4	yes	—
53	C₂×Q₈×C₅₃	C₂×C₈	1	4	yes	—
54	C₂×Q₈×C₅₃	M₁₆	1	4	yes	—
55	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
56	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
57	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
58	C₂²×Dic₅₃	C₂²×C₄	1	3	yes	—
59	C₂²×Dic₅₃	C₂²×C₄	1	3	yes	—
60	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
61	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₈×C₅₃	—	2	1	no	—
63	C₂×C₈×C₅₃	—	2	1	no	—
64	C₂×C₈×C₅₃	—	2	3	no	—
65	C₂×C₈×C₅₃	—	2	3	no	—
66	C₂×C₈×C₅₃	—	2	3	no	—
67	C₂×C₈×C₅₃	#41	1	2	yes	—
68	C₄⋊C₄×C₅₃	C₄⋊C₄	1	12	yes	—
69	C₄²×C₅₃	C₂×D₄	1	6	yes	—
70	C₄²×C₅₃	C₂×Q₈	1	6	yes	—
71	C₄⋊Dic₅₃	—	2	6	no	—
72	C₄⋊Dic₅₃	C₂×C₈	1	12	yes	—
73	C₄×Dic₅₃	—	2	6	no	—
74	C₄×Dic₅₃	—	2	3	no	—
75	C₄×Dic₅₃	—	2	3	no	—
76	C₄×Dic₅₃	—	2	1	no	—
77	C₄×Dic₅₃	—	2	1	no	—
78	C₄×Dic₅₃	—	2	6	no	—
79	C₄×Dic₅₃	—	4	3	no	—
80	C₄×Dic₅₃	C₂×C₈	1	12	yes	—
81	C₂×#88322	—	2	1	no	—
82	C₂×#88322	—	2	1	no	—
83	C₂×#88322	—	2	3	no	—
84	C₂×#88322	—	2	3	no	—
85	C₂×#88322	—	2	3	no	—
86	C₂×#88322	#41	1	2	yes	—
87	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
88	Q₈×C₅₃	C₄⋊C₈	1	16	yes	complemented
89	C₂×C₄×C₅₃	—	2	3	no	—
90	C₂×C₄×C₅₃	—	2	3	no	—
91	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	6	yes	—
92	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	6	yes	—
93	C₂×C₄×C₅₃	C₂²×C₈	1	6	yes	—
94	C₂×C₄×C₅₃	C₂×M₁₆	1	6	yes	—
95	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
96	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
97	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
98	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
99	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
100	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
101	C₂×Dic₅₃	—	2	3	no	—
102	C₂×Dic₅₃	—	2	1	no	—
103	C₂×Dic₅₃	—	2	1	no	—
104	C₂×Dic₅₃	—	2	1	no	—
105	C₂×Dic₅₃	—	2	3	no	—
106	C₂×Dic₅₃	—	2	1	no	—
107	C₂×Dic₅₃	—	2	3	no	—
108	C₂×Dic₅₃	—	2	3	no	—
109	C₂×Dic₅₃	—	2	3	no	—
110	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
111	C₂×Dic₅₃	C₂²×C₈	1	6	yes	—
112	C₂×Dic₅₃	#131	1	2	yes	—
113	C₈×C₅₃	—	2	2	no	—
114	C₈×C₅₃	—	4	3	no	—
115	#88322	—	2	2	no	—
116	#88322	—	4	3	no	—
117	#22734	—	53	1	no	maximal, Sylow
118	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
119	C₂²×C₅₃	C₄×#41	1	1	yes	characteristic
120	C₂²×C₅₃	#519	1	1	yes	characteristic
121	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
122	C₂²×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
123	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
124	C₂²×C₅₃	#568	1	1	yes	characteristic
125	C₄×C₅₃	—	2	6	no	—
126	C₄×C₅₃	C₂×C₄⋊C₈	1	12	yes	—
127	C₄×C₅₃	#548	1	1	yes	characteristic
128	C₄×C₅₃	#548	1	1	yes	characteristic
129	C₄×C₅₃	#557	1	1	yes	characteristic
130	C₄×C₅₃	#557	1	1	yes	characteristic
131	Dic₅₃	—	2	6	no	—
132	Dic₅₃	—	2	2	no	—
133	Dic₅₃	—	2	2	no	—
134	Dic₅₃	—	4	3	no	—
135	C₂×C₈×Q₈	—	53	1	no	—
136	C₂×C₈×Q₈	D₅₃	1	1	yes	characteristic
137	C₄²×Q₈	—	53	1	no	—
138	#1939	—	53	3	no	—
139	#1939	—	53	3	no	—
140	#1831	—	53	3	no	—
141	C₄×C₄⋊C₈	—	53	3	no	—
142	C₂×C₅₃	#2351	1	1	yes	characteristic
143	C₂×C₅₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
144	C₂×C₅₃	#2906	1	1	yes	characteristic
145	C₂×C₅₃	C₈×#41	1	1	yes	characteristic
146	C₂×C₅₃	#2971	1	1	yes	characteristic
147	C₂×C₅₃	#2982	1	1	yes	characteristic
148	C₂×C₅₃	#2984	1	1	yes	characteristic
149	C₂×C₄×Q₈	—	53	1	no	—
150	C₂×C₄×Q₈	—	53	1	no	—
151	C₂×C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
152	C₈×Q₈	—	2	4	no	—
153	C₈×Q₈	—	106	4	no	—
154	C₂×C₄⋊C₈	—	53	3	no	—
155	C₂×C₄⋊C₈	—	53	3	no	—
156	C₂×C₄⋊C₈	—	53	3	no	—
157	C₂×C₄⋊C₈	C₂×D₅₃	1	3	yes	—
158	C₂×C₄×C₈	—	53	3	no	—
159	C₂×C₄×C₈	C₂×D₅₃	1	3	yes	—
160	C₄×C₄⋊C₄	—	53	3	no	—
161	C₄×C₄⋊C₄	—	53	3	no	—
162	C₄×C₄⋊C₄	—	53	3	no	—
163	C₄³	—	53	3	no	—
164	#336	—	53	3	no	—
165	#336	—	53	6	no	—
166	#336	—	53	1	no	—
167	C₅₃	#22734	1	1	yes	characteristic, complemented, Sylow
168	C₂²×Q₈	C₄×D₅₃	1	1	yes	characteristic
169	C₂²×C₈	—	53	1	no	—
170	C₂²×C₈	—	53	3	no	—
171	C₂²×C₈	C₂²×D₅₃	1	3	yes	—
172	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
173	C₄×Q₈	—	53	8	no	—
174	C₄×Q₈	—	106	4	no	—
175	C₄×Q₈	Dic₁₀₆	1	4	yes	—
176	C₄×Q₈	C₂²⋊D₅₃	1	4	yes	—
177	C₂×C₄⋊C₄	—	53	3	no	—
178	C₂×C₄⋊C₄	—	53	3	no	—
179	C₂×C₄⋊C₄	C₄×D₅₃	1	3	yes	—
180	C₂×C₄²	—	53	3	no	—
181	C₂×C₄²	—	53	3	no	—
182	C₂×C₄²	—	53	3	no	—
183	C₂×C₄²	—	53	1	no	—
184	C₂×C₄²	—	53	3	no	—
185	C₂×C₄²	—	53	3	no	—
186	C₂×C₄²	C₂²×D₅₃	1	3	yes	—
187	C₄⋊C₈	—	2	6	no	—
188	C₄⋊C₈	—	106	6	no	—
189	C₄⋊C₈	—	106	6	no	—
190	C₄⋊C₈	—	106	6	no	—
191	C₄×C₈	—	2	6	no	—
192	C₄×C₈	—	106	6	no	—
193	#95	—	53	6	no	—
194	C₂×Q₈	C₈×D₅₃	1	4	yes	—
195	C₂×Q₈	#11719912	1	4	yes	—
196	C₂×Q₈	#11719919	1	4	yes	—
197	C₂²×C₄	—	53	3	no	—
198	C₂²×C₄	—	53	1	no	—
199	C₂²×C₄	—	53	1	no	—
200	C₂²×C₄	—	53	3	no	—
201	C₂²×C₄	C₂×C₄×D₅₃	1	3	yes	—
202	C₂²×C₄	C₂×C₄×D₅₃	1	3	yes	—
203	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
204	C₂×C₈	—	2	1	no	—
205	C₂×C₈	—	2	1	no	—
206	C₂×C₈	—	2	3	no	—
207	C₂×C₈	—	2	3	no	—
208	C₂×C₈	—	2	3	no	—
209	C₂×C₈	—	53	2	no	—
210	C₂×C₈	—	106	1	no	—
211	C₂×C₈	—	106	1	no	—
212	C₂×C₈	—	106	3	no	—
213	C₂×C₈	—	106	3	no	—
214	C₂×C₈	—	106	3	no	—
215	C₂×C₈	#11719949	1	2	yes	—
216	C₄⋊C₄	—	53	12	no	—
217	C₄⋊C₄	—	106	6	no	—
218	C₄⋊C₄	#11719919	1	12	yes	—
219	C₄²	—	53	12	no	—
220	C₄²	—	106	6	no	—
221	C₄²	—	106	6	no	—
222	C₄²	—	106	3	no	—
223	C₄²	—	106	3	no	—
224	C₄²	—	106	1	no	—
225	C₄²	—	106	1	no	—
226	C₄²	—	212	3	no	—
227	C₄²	C₂×Dic₁₀₆	1	6	yes	—
228	C₄²	C₂×C₂²⋊D₅₃	1	6	yes	—
229	C₂³	C₂²×C₄×D₅₃	1	1	yes	characteristic
230	Q₈	#13133146	1	16	yes	complemented
231	C₂×C₄	—	2	3	no	—
232	C₂×C₄	—	2	3	no	—
233	C₂×C₄	—	53	2	no	—
234	C₂×C₄	—	53	6	no	—
235	C₂×C₄	—	53	2	no	—
236	C₂×C₄	—	106	3	no	—
237	C₂×C₄	—	106	1	no	—
238	C₂×C₄	—	106	1	no	—
239	C₂×C₄	—	106	1	no	—
240	C₂×C₄	—	106	3	no	—
241	C₂×C₄	—	106	1	no	—
242	C₂×C₄	—	106	3	no	—
243	C₂×C₄	—	106	3	no	—
244	C₂×C₄	—	106	3	no	—
245	C₂×C₄	C₂×#11719919	1	6	yes	—
246	C₂×C₄	C₂×#11719919	1	6	yes	—
247	C₂×C₄	C₂×C₈×D₅₃	1	6	yes	—
248	C₂×C₄	C₂×#11719912	1	6	yes	—
249	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
250	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
251	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
252	C₂×C₄	C₂×#11719949	1	1	yes	characteristic
253	C₂×C₄	#13133264	1	1	yes	characteristic
254	C₂×C₄	#13133265	1	1	yes	characteristic
255	C₈	—	2	2	no	—
256	C₈	—	4	3	no	—
257	C₈	—	106	2	no	—
258	C₈	—	212	3	no	—
259	C₂²	C₂²×#11719919	1	1	yes	characteristic
260	C₂²	C₄×#11719949	1	1	yes	characteristic
261	C₂²	#16811099	1	1	yes	characteristic
262	C₂²	C₂²×C₈×D₅₃	1	1	yes	characteristic
263	C₂²	C₂²×#11719912	1	1	yes	characteristic
264	C₂²	#131×D₅₃	1	1	yes	characteristic
265	C₂²	#16811388	1	1	yes	characteristic
266	C₄	—	2	6	no	—
267	C₄	—	106	6	no	—
268	C₄	—	106	2	no	—
269	C₄	—	106	2	no	—
270	C₄	—	212	3	no	—
271	C₄	C₂×#13133146	1	12	yes	—
272	C₄	#16811280	1	1	yes	characteristic
273	C₄	#16811284	1	1	yes	characteristic
274	C₄	#16811349	1	1	yes	characteristic
275	C₄	#16811352	1	1	yes	characteristic
276	C₂	#20277633	1	1	yes	characteristic
277	C₂	C₂²×#13133146	1	1	yes	characteristic
278	C₂	#20283281	1	1	yes	characteristic
279	C₂	C₈×#11719949	1	1	yes	characteristic
280	C₂	#20283734	1	1	yes	characteristic
281	C₂	#20283794	1	1	yes	characteristic
282	C₂	#20283803	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₅₃
commutator	C₂²×C₅₃	C₂³×C₈
Fitting	C₂×C₈×Q₈×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂³×C₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂×C₈×Q₈	D₅₃
hypocenter	C₅₃	#22734
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
29. Q₈×Dic₅₃	C₈	semidirect
88. Q₈×C₅₃	C₄⋊C₈	semidirect
167. C₅₃	#22734	semidirect
230. Q₈	#13133146	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×#88322	2
C₄×Q₈×Dic₅₃	2
#20275433	2
#20275431	2
#20274831	2
C₄×#13133146	2
C₂×C₈×Q₈×C₅₃	2
#22734	53