Subgroups — #27392895

Order 13568.

750 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811025	C₂	1	1	yes	maximal, characteristic
2	C₄×#13133209	C₂	1	2	yes	maximal
3	#20275680	C₂	1	2	yes	maximal
4	#20275458	C₂	1	2	yes	maximal
5	#20275453	C₂	1	2	yes	maximal
6	#20274879	C₂	1	1	yes	maximal, characteristic
7	#20274612	C₂	1	2	yes	maximal
8	#20274612	C₂	1	2	yes	maximal
9	C₄×#97×C₅₃	C₂	1	1	yes	maximal, characteristic
10	C₂×#13133209	C₂²	1	2	yes	—
11	#16811025	C₄	1	8	yes	complemented
12	C₂×#13133151	C₂²	1	2	yes	—
13	C₂×#13133151	C₂²	1	2	yes	—
14	C₂×#13133149	C₂²	1	2	yes	—
15	C₂×#13133149	C₂²	1	2	yes	—
16	#16810521	C₂²	1	4	yes	—
17	#16810507	—	2	4	no	—
18	#16810489	C₂²	1	2	yes	—
19	C₄×#11719919	—	2	4	no	—
20	C₄×C₄⋊Dic₅₃	C₂²	1	4	yes	—
21	#16810386	C₂²	1	4	yes	—
22	#16810386	C₂²	1	4	yes	—
23	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
25	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
26	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
29	C₄³×C₅₃	C₂²	1	1	yes	characteristic
30	#13133209	—	2	4	no	—
31	#13133209	C₂×C₄	1	8	yes	—
32	C₂×C₄⋊Dic₅₃	C₂³	1	4	yes	—
33	C₂×C₄⋊Dic₅₃	C₂³	1	4	yes	—
34	C₂×#11719919	—	2	4	no	—
35	C₂×#11719919	—	2	4	no	—
36	C₂×C₄×Dic₅₃	—	2	4	no	—
37	#13133151	—	2	4	no	—
38	#13133151	C₂×C₄	1	8	yes	—
39	#13133149	—	2	4	no	—
40	#13133149	C₂×C₄	1	8	yes	—
41	#13133124	—	2	4	no	—
42	#13133124	—	2	4	no	—
43	#13133124	—	2	4	no	—
44	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
45	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
46	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₅₃	D₄	1	4	yes	—
49	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
52	#97×C₅₃	—	2	2	no	—
53	#97×C₅₃	—	2	2	no	—
54	#97×C₅₃	Q₈	1	4	yes	—
55	#97×C₅₃	Q₈	1	4	yes	—
56	C₄×C₈×C₅₃	—	2	2	no	—
57	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
58	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
59	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
60	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
62	C₂²×Dic₅₃	—	2	4	no	—
63	C₂²×Dic₅₃	—	2	4	no	—
64	C₂×C₈×C₅₃	—	2	1	no	—
65	C₂×C₈×C₅₃	—	2	1	no	—
66	C₂×C₈×C₅₃	—	2	1	no	—
67	C₂×C₈×C₅₃	—	2	1	no	—
68	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
69	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
70	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
71	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
72	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
73	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
74	C₂×C₈×C₅₃	#41	1	2	yes	—
75	C₂×C₈×C₅₃	#41	1	2	yes	—
76	C₄²×C₅₃	—	2	1	no	—
77	C₄²×C₅₃	—	2	1	no	—
78	C₄²×C₅₃	—	4	4	no	—
79	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
80	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
81	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
82	C₄²×C₅₃	#41	1	2	yes	—
83	C₄⋊Dic₅₃	—	2	4	no	—
84	C₄⋊Dic₅₃	—	2	4	no	—
85	C₄⋊Dic₅₃	—	2	4	no	—
86	C₄⋊Dic₅₃	C₂²×C₄	1	8	yes	—
87	#11719919	—	2	16	no	—
88	#11719919	—	4	8	no	—
89	C₄×Dic₅₃	—	4	8	no	—
90	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
91	C₂×C₄×C₅₃	—	2	4	no	—
92	C₂×C₄×C₅₃	—	2	4	no	—
93	C₂×C₄×C₅₃	—	2	2	no	—
94	C₂×C₄×C₅₃	—	2	4	no	—
95	C₂×C₄×C₅₃	C₄×D₄	1	4	yes	—
96	C₂×C₄×C₅₃	C₄⋊Q₈	1	4	yes	—
97	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
98	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
99	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
100	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
101	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
102	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
103	C₂×Dic₅₃	—	2	8	no	—
104	C₂×Dic₅₃	—	4	4	no	—
105	C₂×Dic₅₃	—	4	4	no	—
106	C₂×Dic₅₃	—	4	4	no	—
107	C₂×Dic₅₃	—	4	4	no	—
108	C₂×Dic₅₃	—	4	4	no	—
109	C₈×C₅₃	—	2	2	no	—
110	C₈×C₅₃	—	2	2	no	—
111	C₈×C₅₃	C₄×Q₈	1	4	yes	—
112	C₈×C₅₃	C₄×Q₈	1	4	yes	—
113	#21728	—	53	1	no	maximal, Sylow
114	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
115	C₂²×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
116	C₂²×C₅₃	#532	1	1	yes	characteristic
117	C₂²×C₅₃	#575	1	4	yes	—
118	C₄×C₅₃	—	2	4	no	—
119	C₄×C₅₃	—	2	4	no	—
120	C₄×C₅₃	—	4	2	no	—
121	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
122	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
123	C₄×C₅₃	#533	1	1	yes	characteristic
124	C₄×C₅₃	#533	1	1	yes	characteristic
125	Dic₅₃	—	4	8	no	—
126	Dic₅₃	—	8	4	no	—
127	C₂×#499	—	53	1	no	—
128	C₄×#125	—	53	2	no	—
129	#1991	—	53	2	no	—
130	#1944	—	53	2	no	—
131	#1943	—	53	2	no	—
132	#1846	—	53	1	no	—
133	#1776	—	53	2	no	—
134	#1776	—	53	2	no	—
135	C₄×#97	D₅₃	1	1	yes	characteristic
136	C₂×C₅₃	C₄×C₄⋊Q₈	1	1	yes	characteristic
137	C₂×C₅₃	#2943	1	2	yes	—
138	C₂×C₅₃	#3166	1	2	yes	—
139	C₂×C₅₃	#3183	1	2	yes	—
140	C₂×#125	—	53	2	no	—
141	#499	—	53	8	no	—
142	C₂×#107	—	53	2	no	—
143	C₂×#107	—	53	2	no	—
144	C₂×#106	—	53	2	no	—
145	C₂×#106	—	53	2	no	—
146	C₂×#97	C₂×D₅₃	1	1	yes	characteristic
147	C₂×#97	C₂×D₅₃	1	1	yes	characteristic
148	C₂×#97	C₂×D₅₃	1	1	yes	characteristic
149	C₂×#97	C₂×D₅₃	1	1	yes	characteristic
150	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
151	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
152	#389	—	53	4	no	—
153	#387	—	106	4	no	—
154	#382	—	53	2	no	—
155	C₄×C₄⋊C₄	—	53	4	no	—
156	C₄×C₄⋊C₄	—	106	4	no	—
157	C₄³	C₂×D₅₃	1	1	yes	characteristic
158	#340	—	53	4	no	—
159	#340	—	53	4	no	—
160	C₅₃	#21728	1	1	yes	characteristic, complemented, Sylow
161	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
162	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
163	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
164	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
165	#125	—	53	8	no	—
166	#125	—	106	4	no	—
167	C₂×C₄⋊C₄	—	53	4	no	—
168	C₂×C₄⋊C₄	—	53	4	no	—
169	C₂×C₄⋊C₄	—	106	4	no	—
170	C₂×C₄⋊C₄	—	106	4	no	—
171	C₂×C₄²	—	106	4	no	—
172	C₂×C₄²	D₂₁₂	1	4	yes	—
173	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
174	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
175	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
176	#107	—	53	8	no	—
177	#107	—	106	4	no	—
178	#106	—	53	8	no	—
179	#106	—	106	4	no	—
180	#97	—	2	2	no	—
181	#97	—	2	2	no	—
182	#97	Dic₁₀₆	1	4	yes	—
183	#97	Dic₁₀₆	1	4	yes	—
184	C₄×C₈	—	2	2	no	—
185	C₄×C₈	C₄×D₅₃	1	4	yes	—
186	#95	—	106	4	no	—
187	#95	—	106	4	no	—
188	#95	—	106	4	no	—
189	C₂²×C₄	—	106	4	no	—
190	C₂²×C₄	—	106	4	no	—
191	C₂²×C₄	C₂×D₂₁₂	1	2	yes	—
192	C₂²×C₄	C₂×D₂₁₂	1	2	yes	—
193	C₂²×C₄	C₂×D₂₁₂	1	2	yes	—
194	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	—	2	1	no	—
197	C₂×C₈	—	2	1	no	—
198	C₂×C₈	—	2	1	no	—
199	C₂×C₈	C₂×Dic₁₀₆	1	2	yes	—
200	C₂×C₈	C₂×Dic₁₀₆	1	2	yes	—
201	C₂×C₈	C₂×Dic₁₀₆	1	2	yes	—
202	C₂×C₈	C₂×Dic₁₀₆	1	2	yes	—
203	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
204	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
205	C₂×C₈	#11719945	1	2	yes	—
206	C₂×C₈	#11719945	1	2	yes	—
207	C₄⋊C₄	—	53	8	no	—
208	C₄⋊C₄	—	106	16	no	—
209	C₄⋊C₄	—	106	4	no	—
210	C₄⋊C₄	—	106	4	no	—
211	C₄⋊C₄	—	106	4	no	—
212	C₄⋊C₄	—	212	8	no	—
213	C₄²	—	2	1	no	—
214	C₄²	—	2	1	no	—
215	C₄²	—	4	4	no	—
216	C₄²	—	212	8	no	—
217	C₄²	C₂×Dic₁₀₆	1	2	yes	—
218	C₄²	C₂×Dic₁₀₆	1	2	yes	—
219	C₄²	C₂×C₄×D₅₃	1	2	yes	—
220	C₄²	#11719945	1	2	yes	—
221	C₂³	C₂²×D₂₁₂	1	1	yes	characteristic
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	106	8	no	—
227	C₂×C₄	—	212	4	no	—
228	C₂×C₄	—	212	4	no	—
229	C₂×C₄	—	212	4	no	—
230	C₂×C₄	—	212	4	no	—
231	C₂×C₄	—	212	4	no	—
232	C₂×C₄	C₄×D₂₁₂	1	4	yes	—
233	C₂×C₄	#13133217	1	4	yes	—
234	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
235	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
236	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
237	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
238	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
239	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
240	C₈	—	2	2	no	—
241	C₈	—	2	2	no	—
242	C₈	C₄×Dic₁₀₆	1	4	yes	—
243	C₈	C₄×Dic₁₀₆	1	4	yes	—
244	C₂²	C₂×C₄×D₂₁₂	1	1	yes	characteristic
245	C₂²	C₂×#13133217	1	1	yes	characteristic
246	C₂²	#16811173	1	1	yes	characteristic
247	C₂²	#16811423	1	4	yes	—
248	C₄	—	2	4	no	—
249	C₄	—	2	4	no	—
250	C₄	—	4	2	no	—
251	C₄	—	424	4	no	—
252	C₄	—	212	8	no	—
253	C₄	C₂×C₄×Dic₁₀₆	1	1	yes	characteristic
254	C₄	C₂×C₄×Dic₁₀₆	1	1	yes	characteristic
255	C₄	#16811180	1	1	yes	characteristic
256	C₄	#16811180	1	1	yes	characteristic
257	C₂	C₄×#13133217	1	1	yes	characteristic
258	C₂	#20283517	1	2	yes	—
259	C₂	#20285267	1	2	yes	—
260	C₂	#20285444	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₄×#97×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂²×D₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₄×#97	D₅₃
hypocenter	C₅₃	#21728
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×#97

Complements

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

N	H ≅ G/N	type
11. #16811025	C₄	semidirect
160. C₅₃	#21728	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811025	2
C₄×#13133209	2
#20275680	2
#20275458	2
#20275453	2
#20274879	2
#20274612	2
C₄×#97×C₅₃	2
#21728	53