Subgroups — #27271048

Order 13568.

326 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊#11719908	C₂	1	1	yes	maximal, characteristic
2	#20274411	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₃×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11719917	C₄	1	1	yes	characteristic
5	C₂×#13133176	C₄	1	1	yes	characteristic
6	C₂²×#11719908	—	2	1	no	—
7	C₂²⋊#11719908	—	2	1	no	—
8	C₂²⋊#11719908	C₄	1	2	yes	complemented
9	#16810377	—	2	1	no	—
10	#16810377	—	2	1	no	—
11	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₂²×C₁₆×C₅₃	—	2	1	no	—
13	C₅₃×C₂²⋊C₁₆	—	2	1	no	—
14	C₅₃×C₂²⋊C₁₆	C₄	1	2	yes	complemented
15	C₂×#11719917	—	2	1	no	—
16	C₂×#11719917	—	4	1	no	—
17	C₂×#11719917	—	4	1	no	—
18	C₂×#11719917	C₈	1	2	yes	—
19	C₂²×#88322	—	2	1	no	—
20	#13133176	—	4	1	no	—
21	#13133176	—	4	1	no	—
22	C₂×C₄⋊Dic₅₃	C₈	1	2	yes	—
23	C₂×C₂²⋊Dic₅₃	—	2	1	no	—
24	C₂×C₄×Dic₅₃	—	2	1	no	—
25	C₂×#11719908	—	2	2	no	—
26	C₂×#11719908	—	4	1	no	—
27	C₂×#11719908	—	4	1	no	—
28	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₅₃	—	4	1	no	—
30	C₂²×C₈×C₅₃	—	4	1	no	—
31	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
37	C₂×C₁₆×C₅₃	—	2	2	no	—
38	C₂×C₁₆×C₅₃	—	4	1	no	—
39	C₂×C₁₆×C₅₃	—	4	1	no	—
40	C₂⁴×C₅₃	M₁₆	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	—	2	1	no	—
42	C₂²×C₄×C₅₃	—	2	1	no	—
43	C₂²×C₄×C₅₃	—	2	1	no	—
44	C₂²×C₄×C₅₃	—	2	1	no	—
45	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
51	C₂²×Dic₅₃	—	2	2	no	—
52	C₂×C₈×C₅₃	—	2	1	no	—
53	C₂×C₈×C₅₃	—	2	1	no	—
54	C₂×C₈×C₅₃	—	2	1	no	—
55	C₂×C₈×C₅₃	—	2	1	no	—
56	C₂×C₈×C₅₃	—	4	1	no	—
57	C₂×C₈×C₅₃	—	4	1	no	—
58	C₂×C₈×C₅₃	—	4	1	no	—
59	C₂×C₈×C₅₃	—	4	1	no	—
60	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
64	C₂²⋊Dic₅₃	—	4	2	no	—
65	C₄⋊Dic₅₃	—	4	2	no	—
66	C₄×Dic₅₃	—	4	1	no	—
67	C₄×Dic₅₃	—	4	1	no	—
68	#11719917	—	4	2	no	—
69	#11719917	—	4	2	no	—
70	C₂×#88322	—	2	2	no	—
71	C₂×#88322	—	4	1	no	—
72	C₂×#88322	—	4	1	no	—
73	C₁₆×C₅₃	—	4	2	no	—
74	#11719908	—	4	2	no	—
75	C₂³×C₅₃	—	2	1	no	—
76	C₂³×C₅₃	—	2	1	no	—
77	C₂³×C₅₃	—	2	2	no	—
78	C₂³×C₅₃	—	2	2	no	—
79	C₂³×C₅₃	#97	1	1	yes	characteristic
80	C₂³×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
81	C₂³×C₅₃	C₄⋊C₈	1	1	yes	characteristic
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	1	no	—
92	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
93	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
94	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
95	C₂×C₄×C₅₃	#99	1	1	yes	characteristic
96	C₂×C₄×C₅₃	#99	1	1	yes	characteristic
97	C₂×C₄×C₅₃	#100	1	1	yes	characteristic
98	C₂×C₄×C₅₃	#101	1	1	yes	characteristic
99	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
100	C₂×Dic₅₃	—	4	2	no	—
101	C₂×Dic₅₃	—	4	2	no	—
102	C₂×Dic₅₃	—	4	2	no	—
103	C₈×C₅₃	—	2	1	no	—
104	C₈×C₅₃	—	2	1	no	—
105	C₈×C₅₃	—	2	1	no	—
106	C₈×C₅₃	—	2	1	no	—
107	#88322	—	4	2	no	—
108	#7186	—	53	1	no	maximal, Sylow
109	C₂²×C₅₃	—	2	1	no	—
110	C₂²×C₅₃	—	2	1	no	—
111	C₂²×C₅₃	—	2	1	no	—
112	C₂²×C₅₃	—	2	1	no	—
113	C₂²×C₅₃	—	2	1	no	—
114	C₂²×C₅₃	—	2	1	no	—
115	C₂²×C₅₃	—	4	2	no	—
116	C₂²×C₅₃	—	4	2	no	—
117	C₂²×C₅₃	#323	1	1	yes	characteristic
118	C₂²×C₅₃	#324	1	1	yes	characteristic
119	C₂²×C₅₃	#336	1	1	yes	characteristic
120	C₂²×C₅₃	#350	1	2	yes	—
121	C₂²×C₅₃	#350	1	2	yes	—
122	C₄×C₅₃	—	2	1	no	—
123	C₄×C₅₃	—	2	1	no	—
124	C₄×C₅₃	#323	1	1	yes	characteristic
125	C₄×C₅₃	#324	1	1	yes	characteristic
126	C₄×C₅₃	#342	1	1	yes	characteristic
127	C₄×C₅₃	#343	1	1	yes	characteristic
128	Dic₅₃	—	8	2	no	—
129	C₂×C₂²⋊C₁₆	—	53	1	no	—
130	C₂×C₂²⋊C₁₆	D₅₃	1	1	yes	characteristic
131	#1729	—	53	1	no	—
132	C₂×C₅₃	—	4	1	no	—
133	C₂×C₅₃	—	4	1	no	—
134	C₂×C₅₃	#1280	1	1	yes	characteristic
135	C₂×C₅₃	#1315	1	2	yes	—
136	C₂×C₅₃	#1323	1	2	yes	—
137	C₂×C₅₃	#1379	1	1	yes	characteristic
138	C₂×C₅₃	#1379	1	1	yes	characteristic
139	C₂²×M₁₆	—	53	1	no	—
140	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
141	C₂×#117	—	53	1	no	—
142	C₂²×C₁₆	—	2	1	no	—
143	C₂²×C₁₆	—	106	1	no	—
144	C₂²⋊C₁₆	—	2	1	no	—
145	C₂²⋊C₁₆	—	53	2	no	—
146	C₂²⋊C₁₆	—	106	1	no	—
147	C₂²⋊C₁₆	Dic₅₃	1	2	yes	complemented
148	#336	—	106	1	no	—
149	#336	—	106	1	no	—
150	C₅₃	#7186	1	1	yes	characteristic, complemented, Sylow
151	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
152	C₂×M₁₆	—	53	2	no	—
153	C₂×M₁₆	—	106	1	no	—
154	C₂×M₁₆	—	212	1	no	—
155	C₂×M₁₆	—	212	1	no	—
156	C₂²×C₈	—	4	1	no	—
157	C₂²×C₈	—	4	1	no	—
158	C₂²×C₈	—	106	1	no	—
159	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
160	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
161	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
162	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
163	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
164	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
165	#117	—	212	1	no	—
166	#117	—	212	1	no	—
167	C₂×C₄⋊C₄	—	53	2	no	—
168	C₂×C₂²⋊C₄	—	106	1	no	—
169	C₂×C₄²	—	106	1	no	—
170	C₂×C₁₆	—	2	2	no	—
171	C₂×C₁₆	—	4	1	no	—
172	C₂×C₁₆	—	4	1	no	—
173	C₂×C₁₆	—	106	2	no	—
174	C₂×C₁₆	—	212	1	no	—
175	C₂×C₁₆	—	212	1	no	—
176	C₂⁴	#11719912	1	1	yes	characteristic
177	C₂²×C₄	—	2	1	no	—
178	C₂²×C₄	—	2	1	no	—
179	C₂²×C₄	—	2	1	no	—
180	C₂²×C₄	—	2	1	no	—
181	C₂²×C₄	—	106	2	no	—
182	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
183	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
184	C₂²×C₄	#11719912	1	1	yes	characteristic
185	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
186	C₂²×C₄	#11719919	1	1	yes	characteristic
187	C₂²×C₄	#11719921	1	1	yes	characteristic
188	M₁₆	—	212	2	no	—
189	M₁₆	—	212	2	no	—
190	C₂×C₈	—	2	1	no	—
191	C₂×C₈	—	2	1	no	—
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	—	2	1	no	—
194	C₂×C₈	—	4	1	no	—
195	C₂×C₈	—	4	1	no	—
196	C₂×C₈	—	4	1	no	—
197	C₂×C₈	—	4	1	no	—
198	C₂×C₈	—	106	2	no	—
199	C₂×C₈	—	212	1	no	—
200	C₂×C₈	—	212	1	no	—
201	C₂×C₈	#11719919	1	1	yes	characteristic
202	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
203	C₂×C₈	#11719921	1	1	yes	characteristic
204	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
205	C₄⋊C₄	—	212	2	no	—
206	C₂²⋊C₄	—	212	2	no	—
207	C₄²	—	212	1	no	—
208	C₄²	—	212	1	no	—
209	C₁₆	—	4	2	no	—
210	C₁₆	—	212	2	no	—
211	C₂³	—	2	1	no	—
212	C₂³	—	2	1	no	—
213	C₂³	—	2	2	no	—
214	C₂³	—	2	2	no	—
215	C₂³	#13133127	1	1	yes	characteristic
216	C₂³	#13133129	1	1	yes	characteristic
217	C₂³	#13133146	1	1	yes	characteristic
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	212	2	no	—
229	C₂×C₄	—	212	2	no	—
230	C₂×C₄	—	212	2	no	—
231	C₂×C₄	#13133124	1	1	yes	characteristic
232	C₂×C₄	C₈×Dic₅₃	1	1	yes	characteristic
233	C₂×C₄	#13133129	1	1	yes	characteristic
234	C₂×C₄	#13133131	1	1	yes	characteristic
235	C₂×C₄	#13133131	1	1	yes	characteristic
236	C₂×C₄	#13133133	1	1	yes	characteristic
237	C₂×C₄	#13133135	1	1	yes	characteristic
238	C₂×C₄	#13133146	1	1	yes	characteristic
239	C₈	—	2	1	no	—
240	C₈	—	2	1	no	—
241	C₈	—	2	1	no	—
242	C₈	—	2	1	no	—
243	C₈	—	212	2	no	—
244	C₂²	—	2	1	no	—
245	C₂²	—	2	1	no	—
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	—	2	1	no	—
249	C₂²	—	2	1	no	—
250	C₂²	—	4	2	no	—
251	C₂²	—	4	2	no	—
252	C₂²	#16810346	1	1	yes	characteristic
253	C₂²	#16810348	1	1	yes	characteristic
254	C₂²	#16810377	1	1	yes	characteristic
255	C₂²	#16810406	1	2	yes	—
256	C₂²	#16810408	1	2	yes	—
257	C₄	—	2	1	no	—
258	C₄	—	2	1	no	—
259	C₄	—	424	2	no	—
260	C₄	#16810346	1	1	yes	characteristic
261	C₄	#16810348	1	1	yes	characteristic
262	C₄	#16810389	1	1	yes	characteristic
263	C₄	#16810391	1	1	yes	characteristic
264	C₂	—	4	1	no	—
265	C₂	—	4	1	no	—
266	C₂	#20272557	1	1	yes	characteristic
267	C₂	#20272640	1	2	yes	—
268	C₂	#20272656	1	2	yes	—
269	C₂	#20272803	1	1	yes	characteristic
270	C₂	#20272805	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	#11719921
commutator	C₂×C₄×C₅₃	C₄×C₈
Frattini	C₂³×C₈	C₂×D₅₃
Fitting	C₂×C₅₃×C₂²⋊C₁₆	C₂
socle	C₂³×C₅₃	C₂²⋊C₈
prime radical	C₂⁴×C₅₃	M₁₆
cosocle	C₂³×C₈×C₅₃	C₂²
hypercenter	C₂×C₂²⋊C₁₆	D₅₃
hypocenter	C₅₃	#7186
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₈ ◁ C₂×C₂²⋊C₁₆

Complements

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

N	H ≅ G/N	type
8. C₂²⋊#11719908	C₄	semidirect
14. C₅₃×C₂²⋊C₁₆	C₄	semidirect
147. C₂²⋊C₁₆	Dic₅₃	semidirect
150. C₅₃	#7186	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂²⋊#11719908	2
#20274411	2
C₂×C₅₃×C₂²⋊C₁₆	2
#7186	53