Subgroups — #27270659

Order 13568.

352 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1842	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊#88322	—	2	1	no	—
5	C₂×C₄⋊#88322	C₄	1	2	yes	—
6	C₂×C₄×#88322	—	2	1	no	—
7	C₂×C₄×#88322	—	2	1	no	—
8	#16810378	—	2	1	no	—
9	#16810378	C₄	1	2	yes	—
10	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
11	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
12	C₂×C₄⋊C₈×C₅₃	C₄	1	2	yes	—
13	C₄³×C₅₃	C₂²	1	1	yes	characteristic
14	C₂²×#88322	—	2	2	no	—
15	C₂²×#88322	—	2	2	no	—
16	C₄⋊#88322	—	2	4	no	—
17	C₄⋊#88322	—	2	2	no	—
18	C₄⋊#88322	C₈	1	4	yes	complemented
19	C₄×#88322	—	4	1	no	—
20	C₄×#88322	—	4	1	no	—
21	C₄×#88322	—	4	1	no	—
22	C₄×#88322	—	4	1	no	—
23	C₂²×C₈×C₅₃	—	2	2	no	—
24	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
28	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
31	C₄⋊C₈×C₅₃	—	4	2	no	—
32	C₄⋊C₈×C₅₃	—	4	1	no	—
33	C₄⋊C₈×C₅₃	—	4	1	no	—
34	C₄⋊C₈×C₅₃	—	4	1	no	—
35	C₄⋊C₈×C₅₃	—	4	1	no	—
36	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
37	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
38	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
39	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₅₃	—	4	2	no	—
44	C₂×C₈×C₅₃	—	4	2	no	—
45	C₂×C₈×C₅₃	—	4	2	no	—
46	C₄²×C₅₃	—	2	1	no	—
47	C₄²×C₅₃	—	2	1	no	—
48	C₄²×C₅₃	—	2	2	no	—
49	C₄²×C₅₃	—	2	2	no	—
50	C₄²×C₅₃	—	4	1	no	—
51	C₄²×C₅₃	C₂×C₈	1	2	yes	—
52	C₄²×C₅₃	M₁₆	1	2	yes	—
53	C₄²×C₅₃	D₈	1	1	yes	characteristic
54	C₄²×C₅₃	D₈	1	1	yes	characteristic
55	C₄²×C₅₃	SD₁₆	1	1	yes	characteristic
56	C₄²×C₅₃	SD₁₆	1	1	yes	characteristic
57	C₄²×C₅₃	SD₁₆	1	1	yes	characteristic
58	C₄²×C₅₃	SD₁₆	1	1	yes	characteristic
59	C₄²×C₅₃	Q₁₆	1	1	yes	characteristic
60	C₄²×C₅₃	Q₁₆	1	1	yes	characteristic
61	C₂×#88322	—	2	4	no	—
62	C₂×#88322	—	2	4	no	—
63	C₂×#88322	—	4	2	no	—
64	C₂×#88322	—	4	2	no	—
65	C₂×#88322	—	4	2	no	—
66	C₂×#88322	—	4	2	no	—
67	C₂³×C₅₃	#95	1	1	yes	characteristic
68	C₂×C₄×C₅₃	—	2	1	no	—
69	C₂×C₄×C₅₃	—	2	1	no	—
70	C₂×C₄×C₅₃	—	2	1	no	—
71	C₂×C₄×C₅₃	—	2	1	no	—
72	C₂×C₄×C₅₃	—	2	1	no	—
73	C₂×C₄×C₅₃	—	2	1	no	—
74	C₂×C₄×C₅₃	—	2	1	no	—
75	C₂×C₄×C₅₃	—	2	1	no	—
76	C₂×C₄×C₅₃	—	2	1	no	—
77	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
78	C₂×C₄×C₅₃	#97	1	1	yes	characteristic
79	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
80	C₂×C₄×C₅₃	#100	1	1	yes	characteristic
81	C₂×C₄×C₅₃	#101	1	1	yes	characteristic
82	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
83	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
84	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
85	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
86	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
87	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
88	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
89	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
90	C₂×C₄×C₅₃	#104	1	2	yes	—
91	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
92	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
93	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
94	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
95	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
96	C₂×C₄×C₅₃	#108	1	2	yes	—
97	C₈×C₅₃	—	8	2	no	—
98	#88322	—	4	4	no	—
99	#88322	—	4	4	no	—
100	#7031	—	53	1	no	maximal, Sylow
101	C₂²×C₅₃	#336	1	1	yes	characteristic
102	C₂²×C₅₃	#339	1	1	yes	characteristic
103	C₂²×C₅₃	#340	1	1	yes	characteristic
104	C₂²×C₅₃	#340	1	1	yes	characteristic
105	C₂²×C₅₃	#341	1	1	yes	characteristic
106	C₂²×C₅₃	#343	1	1	yes	characteristic
107	C₂²×C₅₃	#344	1	1	yes	characteristic
108	C₄×C₅₃	—	2	1	no	—
109	C₄×C₅₃	—	2	1	no	—
110	C₄×C₅₃	—	2	1	no	—
111	C₄×C₅₃	—	2	1	no	—
112	C₄×C₅₃	—	2	1	no	—
113	C₄×C₅₃	—	2	1	no	—
114	C₄×C₅₃	—	4	1	no	—
115	C₄×C₅₃	D₄⋊C₈	1	2	yes	—
116	C₄×C₅₃	Q₈⋊C₈	1	2	yes	—
117	C₄×C₅₃	#331	1	1	yes	characteristic
118	C₄×C₅₃	#332	1	1	yes	characteristic
119	C₄×C₅₃	#332	1	1	yes	characteristic
120	C₄×C₅₃	#333	1	1	yes	characteristic
121	C₄×C₅₃	#334	1	2	yes	—
122	C₄×C₅₃	#335	1	2	yes	—
123	#1842	D₅₃	1	1	yes	characteristic
124	C₄×C₄⋊C₈	—	53	1	no	—
125	C₄×C₄⋊C₈	—	53	1	no	—
126	C₂×C₅₃	#1277	1	1	yes	characteristic
127	C₂×C₅₃	#1279	1	1	yes	characteristic
128	C₂×C₅₃	#1283	1	1	yes	characteristic
129	C₂×C₅₃	#1296	1	1	yes	characteristic
130	C₂×C₅₃	#1299	1	1	yes	characteristic
131	C₂×C₅₃	#1301	1	1	yes	characteristic
132	C₂×C₅₃	#1303	1	1	yes	characteristic
133	C₂×C₄⋊C₈	—	2	1	no	—
134	C₂×C₄⋊C₈	—	2	1	no	—
135	C₂×C₄⋊C₈	—	53	2	no	—
136	C₂×C₄⋊C₈	—	106	1	no	—
137	C₂×C₄⋊C₈	Dic₅₃	1	2	yes	—
138	C₂×C₄×C₈	—	106	1	no	—
139	C₂×C₄×C₈	—	106	1	no	—
140	C₄³	C₂×D₅₃	1	1	yes	characteristic
141	#336	—	53	2	no	—
142	#336	—	106	1	no	—
143	C₅₃	#7031	1	1	yes	characteristic, complemented, Sylow
144	C₂²×C₈	—	2	2	no	—
145	C₂²×C₈	—	106	2	no	—
146	C₂²×C₈	—	106	2	no	—
147	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
148	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
149	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
150	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
151	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
152	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
153	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
154	C₄⋊C₈	—	4	2	no	—
155	C₄⋊C₈	—	4	1	no	—
156	C₄⋊C₈	—	4	1	no	—
157	C₄⋊C₈	—	4	1	no	—
158	C₄⋊C₈	—	4	1	no	—
159	C₄⋊C₈	—	53	4	no	—
160	C₄⋊C₈	—	106	4	no	—
161	C₄⋊C₈	—	106	2	no	—
162	C₄×C₈	—	212	1	no	—
163	C₄×C₈	—	212	1	no	—
164	C₄×C₈	—	212	1	no	—
165	C₄×C₈	—	212	1	no	—
166	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
167	C₂²×C₄	#11719919	1	1	yes	characteristic
168	C₂²×C₄	#11719919	1	1	yes	characteristic
169	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
170	C₂²×C₄	#11719921	1	1	yes	characteristic
171	C₂²×C₄	#11719921	1	1	yes	characteristic
172	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
173	C₂×C₈	—	4	2	no	—
174	C₂×C₈	—	4	2	no	—
175	C₂×C₈	—	4	2	no	—
176	C₂×C₈	—	106	4	no	—
177	C₂×C₈	—	106	4	no	—
178	C₂×C₈	—	212	2	no	—
179	C₂×C₈	—	212	2	no	—
180	C₂×C₈	—	212	2	no	—
181	C₂×C₈	—	212	2	no	—
182	C₄²	—	2	1	no	—
183	C₄²	—	2	1	no	—
184	C₄²	—	2	2	no	—
185	C₄²	—	2	2	no	—
186	C₄²	—	4	1	no	—
187	C₄²	C₈×D₅₃	1	2	yes	—
188	C₄²	#11719912	1	2	yes	—
189	C₄²	#11719922	1	1	yes	characteristic
190	C₄²	#11719922	1	1	yes	characteristic
191	C₄²	#11719923	1	1	yes	characteristic
192	C₄²	#11719923	1	1	yes	characteristic
193	C₄²	#11719924	1	1	yes	characteristic
194	C₄²	#11719924	1	1	yes	characteristic
195	C₄²	#11719925	1	1	yes	characteristic
196	C₄²	#11719925	1	1	yes	characteristic
197	C₂³	#13133124	1	1	yes	characteristic
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	C₈×Dic₅₃	1	1	yes	characteristic
208	C₂×C₄	#13133127	1	1	yes	characteristic
209	C₂×C₄	#13133129	1	2	yes	—
210	C₂×C₄	#13133134	1	1	yes	characteristic
211	C₂×C₄	#13133136	1	1	yes	characteristic
212	C₂×C₄	#13133137	1	1	yes	characteristic
213	C₂×C₄	#13133137	1	1	yes	characteristic
214	C₂×C₄	#13133138	1	1	yes	characteristic
215	C₂×C₄	#13133138	1	1	yes	characteristic
216	C₂×C₄	#13133140	1	1	yes	characteristic
217	C₂×C₄	#13133140	1	1	yes	characteristic
218	C₂×C₄	#13133141	1	1	yes	characteristic
219	C₂×C₄	#13133141	1	1	yes	characteristic
220	C₂×C₄	#13133143	1	2	yes	—
221	C₂×C₄	#13133146	1	2	yes	—
222	C₂×C₄	#13133148	1	1	yes	characteristic
223	C₂×C₄	#13133148	1	1	yes	characteristic
224	C₂×C₄	#13133150	1	1	yes	characteristic
225	C₂×C₄	#13133150	1	1	yes	characteristic
226	C₂×C₄	#13133152	1	2	yes	—
227	C₈	—	8	2	no	—
228	C₈	—	212	4	no	—
229	C₈	—	212	4	no	—
230	C₂²	#16810377	1	1	yes	characteristic
231	C₂²	#16810384	1	1	yes	characteristic
232	C₂²	#16810385	1	1	yes	characteristic
233	C₂²	#16810385	1	1	yes	characteristic
234	C₂²	#16810387	1	1	yes	characteristic
235	C₂²	#16810392	1	1	yes	characteristic
236	C₂²	#16810394	1	1	yes	characteristic
237	C₄	—	2	1	no	—
238	C₄	—	2	1	no	—
239	C₄	—	2	1	no	—
240	C₄	—	2	1	no	—
241	C₄	—	2	1	no	—
242	C₄	—	2	1	no	—
243	C₄	—	4	1	no	—
244	C₄	#16810350	1	2	yes	—
245	C₄	#16810353	1	2	yes	—
246	C₄	#16810367	1	1	yes	characteristic
247	C₄	#16810369	1	1	yes	characteristic
248	C₄	#16810369	1	1	yes	characteristic
249	C₄	#16810372	1	1	yes	characteristic
250	C₄	#16810373	1	2	yes	—
251	C₄	#16810375	1	2	yes	—
252	C₂	#20272549	1	1	yes	characteristic
253	C₂	#20272554	1	1	yes	characteristic
254	C₂	#20272565	1	1	yes	characteristic
255	C₂	#20272598	1	1	yes	characteristic
256	C₂	#20272604	1	1	yes	characteristic
257	C₂	#20272610	1	1	yes	characteristic
258	C₂	#20272616	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²⋊Dic₅₃
commutator	C₂×C₄×C₅₃	C₄×C₈
Frattini	C₄³	C₂×D₅₃
Fitting	C₅₃×#1842	C₂
socle, prime radical	C₂³×C₅₃	#95
cosocle	C₄³×C₅₃	C₂²
hypercenter	#1842	D₅₃
hypocenter	C₅₃	#7031
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1842

Complements

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

N	H ≅ G/N	type
18. C₄⋊#88322	C₈	semidirect
143. C₅₃	#7031	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊#88322	2
C₅₃×#1842	2
#7031	53