Subgroups — #27272160

Order 13568.

478 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	C₄×#13133217	C₂	1	1	yes	maximal, characteristic
2	#20274832	C₂	1	1	yes	maximal, characteristic
3	#20272540	C₂	1	4	yes	maximal
4	C₅₃×#1744	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133217	C₄	1	1	yes	characteristic
6	C₂×C₄×Dic₁₀₆	C₄	1	2	yes	—
7	C₂×C₄⋊#88322	—	2	1	no	—
8	#16810529	C₂²	1	2	yes	—
9	#16810521	—	2	1	no	—
10	#16810521	—	2	1	no	—
11	#16810493	C₄	1	1	yes	characteristic
12	C₄×C₄⋊Dic₅₃	—	2	1	no	—
13	C₄×C₄⋊Dic₅₃	—	2	1	no	—
14	#16810378	—	2	1	no	—
15	#16810378	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₅₃	—	2	1	no	—
17	C₄³×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×#336	—	2	1	no	—
19	C₅₃×#336	C₂²	1	2	yes	—
20	C₂²×Dic₁₀₆	C₂×C₄	1	2	yes	—
21	C₂²×#88322	—	2	2	no	—
22	#13133217	—	2	2	no	—
23	#13133217	C₈	1	4	yes	complemented
24	C₄×Dic₁₀₆	—	4	4	no	—
25	C₂×C₄⋊Dic₅₃	—	2	1	no	—
26	C₂×C₄⋊Dic₅₃	—	2	1	no	—
27	C₂×C₄⋊Dic₅₃	—	2	1	no	—
28	C₂×C₄⋊Dic₅₃	—	2	1	no	—
29	C₂×C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
30	C₂×#11719919	—	2	2	no	—
31	C₂×C₄×Dic₅₃	—	2	2	no	—
32	C₄⋊#88322	—	4	2	no	—
33	#13133124	—	2	2	no	—
34	#13133124	—	2	2	no	—
35	C₂²×C₈×C₅₃	—	2	2	no	—
36	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
43	C₄×C₈×C₅₃	—	2	4	no	—
44	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
51	C₂²×Dic₅₃	—	2	2	no	—
52	C₂²×Dic₅₃	—	2	2	no	—
53	C₂×Dic₁₀₆	—	2	2	no	—
54	C₂×Dic₁₀₆	—	4	2	no	—
55	C₂×Dic₁₀₆	—	4	2	no	—
56	C₂×Dic₁₀₆	C₂×C₈	1	4	yes	—
57	C₂×C₈×C₅₃	—	2	4	no	—
58	C₂×C₈×C₅₃	—	4	2	no	—
59	C₂×C₈×C₅₃	—	4	2	no	—
60	C₄²×C₅₃	—	2	2	no	—
61	C₄²×C₅₃	—	2	2	no	—
62	C₄²×C₅₃	—	2	2	no	—
63	C₄²×C₅₃	—	4	1	no	—
64	C₄²×C₅₃	—	4	1	no	—
65	C₄²×C₅₃	C₂×C₈	1	2	yes	—
66	C₄²×C₅₃	M₁₆	1	2	yes	—
67	C₄²×C₅₃	SD₁₆	1	2	yes	—
68	C₄²×C₅₃	Q₁₆	1	2	yes	—
69	C₄⋊Dic₅₃	—	2	4	no	—
70	C₄⋊Dic₅₃	—	2	4	no	—
71	C₄⋊Dic₅₃	—	4	2	no	—
72	C₄⋊Dic₅₃	—	4	2	no	—
73	C₄⋊Dic₅₃	—	4	2	no	—
74	#11719919	—	4	4	no	—
75	C₄×Dic₅₃	—	4	4	no	—
76	C₂×#88322	—	4	2	no	—
77	C₂×#88322	—	4	2	no	—
78	C₂×#88322	—	4	2	no	—
79	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₅₃	—	2	1	no	—
81	C₂×C₄×C₅₃	—	2	1	no	—
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	C₂²⋊C₈	1	2	yes	—
93	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
94	C₂×C₄×C₅₃	#100	1	2	yes	—
95	C₂×C₄×C₅₃	#103	1	2	yes	—
96	C₂×C₄×C₅₃	#103	1	2	yes	—
97	C₂×C₄×C₅₃	#104	1	2	yes	—
98	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
99	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
100	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
101	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
102	C₂×C₄×C₅₃	#137	1	2	yes	—
103	C₂×Dic₅₃	—	2	4	no	—
104	C₂×Dic₅₃	—	4	2	no	—
105	C₂×Dic₅₃	—	4	2	no	—
106	C₂×Dic₅₃	—	4	2	no	—
107	C₂×Dic₅₃	—	4	2	no	—
108	C₂×Dic₅₃	—	4	2	no	—
109	Dic₁₀₆	—	4	4	no	—
110	Dic₁₀₆	—	4	4	no	—
111	C₈×C₅₃	—	4	4	no	—
112	#88322	—	8	2	no	—
113	#7590	—	53	1	no	maximal, Sylow
114	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
115	C₂²×C₅₃	#410	1	1	yes	characteristic
116	C₂²×C₅₃	C₂×#100	1	1	yes	characteristic
117	C₂²×C₅₃	C₂×#103	1	1	yes	characteristic
118	C₂²×C₅₃	#419	1	1	yes	characteristic
119	C₂²×C₅₃	C₂×#104	1	1	yes	characteristic
120	C₂²×C₅₃	#421	1	1	yes	characteristic
121	C₄×C₅₃	—	2	2	no	—
122	C₄×C₅₃	—	2	1	no	—
123	C₄×C₅₃	—	2	1	no	—
124	C₄×C₅₃	—	2	2	no	—
125	C₄×C₅₃	—	2	2	no	—
126	C₄×C₅₃	—	2	1	no	—
127	C₄×C₅₃	—	2	1	no	—
128	C₄×C₅₃	—	2	2	no	—
129	C₄×C₅₃	Q₈⋊C₈	1	4	yes	—
130	Dic₅₃	—	4	4	no	—
131	Dic₅₃	—	8	2	no	—
132	C₄×C₄⋊Q₈	—	53	1	no	—
133	#1831	—	53	1	no	—
134	#1744	D₅₃	1	1	yes	characteristic
135	#1272	—	53	4	no	—
136	C₂×C₅₃	#1460	1	1	yes	characteristic
137	C₂×C₅₃	C₂×Q₈⋊C₈	1	1	yes	characteristic
138	C₂×C₅₃	#1479	1	1	yes	characteristic
139	C₂×C₅₃	#1506	1	1	yes	characteristic
140	C₂×C₅₃	#1518	1	1	yes	characteristic
141	C₂×C₅₃	#1530	1	1	yes	characteristic
142	C₂×C₅₃	#1548	1	1	yes	characteristic
143	C₂×C₄⋊Q₈	—	53	1	no	—
144	C₂×C₄×Q₈	—	53	2	no	—
145	C₂×C₄⋊C₈	—	106	1	no	—
146	C₂×C₄×C₈	—	2	1	no	—
147	#391	—	53	2	no	—
148	#389	—	106	1	no	—
149	#389	—	106	1	no	—
150	#384	—	53	1	no	—
151	C₄×C₄⋊C₄	—	106	1	no	—
152	C₄×C₄⋊C₄	—	106	1	no	—
153	C₄³	C₂×D₅₃	1	1	yes	characteristic
154	#336	—	2	1	no	—
155	#336	—	53	2	no	—
156	#336	—	106	1	no	—
157	#336	C₂×D₅₃	1	2	yes	—
158	C₅₃	#7590	1	1	yes	characteristic, complemented, Sylow
159	C₂²×Q₈	—	53	2	no	—
160	C₂²×C₈	—	2	2	no	—
161	C₂²×C₈	—	106	2	no	—
162	C₄⋊Q₈	—	53	4	no	—
163	C₄⋊Q₈	—	106	2	no	—
164	C₄×Q₈	—	212	4	no	—
165	C₂×C₄⋊C₄	—	53	2	no	—
166	C₂×C₄⋊C₄	—	106	1	no	—
167	C₂×C₄⋊C₄	—	106	1	no	—
168	C₂×C₄⋊C₄	—	106	2	no	—
169	C₂×C₄⋊C₄	—	106	1	no	—
170	C₂×C₄⋊C₄	—	106	1	no	—
171	C₂×C₄²	—	106	2	no	—
172	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
173	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
174	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
175	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
176	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
177	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
178	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
179	C₄⋊C₈	—	212	2	no	—
180	C₄×C₈	—	2	4	no	—
181	#95	—	106	2	no	—
182	#95	—	106	2	no	—
183	C₂×Q₈	—	53	4	no	—
184	C₂×Q₈	—	106	2	no	—
185	C₂×Q₈	—	212	2	no	—
186	C₂×Q₈	—	212	2	no	—
187	C₂²×C₄	—	106	2	no	—
188	C₂²×C₄	—	106	2	no	—
189	C₂²×C₄	#11719921	1	1	yes	characteristic
190	C₂²×C₄	#11719921	1	1	yes	characteristic
191	C₂²×C₄	#11719921	1	1	yes	characteristic
192	C₂²×C₄	#11719921	1	1	yes	characteristic
193	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
194	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
195	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
196	C₂×C₈	—	2	4	no	—
197	C₂×C₈	—	4	2	no	—
198	C₂×C₈	—	4	2	no	—
199	C₂×C₈	—	212	2	no	—
200	C₂×C₈	—	212	2	no	—
201	C₂×C₈	—	212	2	no	—
202	C₄⋊C₄	—	106	4	no	—
203	C₄⋊C₄	—	106	4	no	—
204	C₄⋊C₄	—	212	2	no	—
205	C₄⋊C₄	—	212	2	no	—
206	C₄⋊C₄	—	212	4	no	—
207	C₄⋊C₄	—	212	2	no	—
208	C₄²	—	2	2	no	—
209	C₄²	—	2	2	no	—
210	C₄²	—	2	2	no	—
211	C₄²	—	4	1	no	—
212	C₄²	—	4	1	no	—
213	C₄²	—	212	4	no	—
214	C₄²	C₈×D₅₃	1	2	yes	—
215	C₄²	#11719912	1	2	yes	—
216	C₄²	#11719913	1	2	yes	—
217	C₄²	Dic₂₁₂	1	2	yes	—
218	C₂³	C₂×#11719921	1	1	yes	characteristic
219	Q₈	—	212	4	no	—
220	Q₈	—	212	4	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	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	106	4	no	—
234	C₂×C₄	—	212	2	no	—
235	C₂×C₄	—	212	2	no	—
236	C₂×C₄	—	212	2	no	—
237	C₂×C₄	—	212	2	no	—
238	C₂×C₄	—	212	2	no	—
239	C₂×C₄	#13133129	1	2	yes	—
240	C₂×C₄	#13133129	1	2	yes	—
241	C₂×C₄	#13133133	1	2	yes	—
242	C₂×C₄	#13133142	1	2	yes	—
243	C₂×C₄	#13133142	1	2	yes	—
244	C₂×C₄	#13133145	1	2	yes	—
245	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
246	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
247	C₂×C₄	C₂×#11719913	1	1	yes	characteristic
248	C₂×C₄	C₂×Dic₂₁₂	1	1	yes	characteristic
249	C₂×C₄	#13133247	1	2	yes	—
250	C₈	—	4	4	no	—
251	C₈	—	424	2	no	—
252	C₂²	C₂×#13133129	1	1	yes	characteristic
253	C₂²	#16810596	1	1	yes	characteristic
254	C₂²	C₂×#13133133	1	1	yes	characteristic
255	C₂²	C₂×#13133142	1	1	yes	characteristic
256	C₂²	#16810631	1	1	yes	characteristic
257	C₂²	C₂×#13133145	1	1	yes	characteristic
258	C₂²	#16810639	1	1	yes	characteristic
259	C₄	—	2	2	no	—
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	2	1	no	—
263	C₄	—	2	1	no	—
264	C₄	—	2	2	no	—
265	C₄	—	2	2	no	—
266	C₄	—	2	2	no	—
267	C₄	—	424	2	no	—
268	C₄	—	212	4	no	—
269	C₄	#16810355	1	4	yes	—
270	C₂	#20273035	1	1	yes	characteristic
271	C₂	C₂×#16810355	1	1	yes	characteristic
272	C₂	#20273109	1	1	yes	characteristic
273	C₂	#20273226	1	1	yes	characteristic
274	C₂	#20273280	1	1	yes	characteristic
275	C₂	#20273325	1	1	yes	characteristic
276	C₂	#20273395	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11719921
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1744	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1744	D₅₃
hypocenter	C₅₃	#7590
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1744

Complements

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

N	H ≅ G/N	type
23. #13133217	C₈	semidirect
158. C₅₃	#7590	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13133217	2
#20274832	2
#20272540	2
C₅₃×#1744	2
#7590	53