Subgroups — #27273577

Order 13568.

630 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810676	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic
3	#20272549	C₂	1	4	yes	maximal, complemented
4	C₂×C₅₃×#432	C₂	1	1	yes	maximal, characteristic
5	C₂³×#88322	—	2	1	no	—
6	#16810676	—	2	2	no	—
7	#16810676	—	2	2	no	—
8	C₂×C₄⋊#88322	—	2	2	no	—
9	C₂×C₄⋊#88322	—	2	1	no	—
10	C₂×C₄⋊#88322	C₄	1	2	yes	—
11	C₂×C₄⋊#88322	C₄	1	4	yes	—
12	C₂×C₄⋊#88322	C₂²	1	2	yes	—
13	C₂×C₂²⋊#88322	C₄	1	2	yes	—
14	C₂×C₄×#88322	C₂²	1	2	yes	—
15	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#432	—	2	2	no	—
17	C₅₃×#432	C₄	1	4	yes	—
18	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
19	C₂×C₂²⋊C₈×C₅₃	—	2	1	no	—
20	C₂×#97×C₅₃	—	2	1	no	—
21	C₂²×#88322	—	2	4	no	—
22	C₂²×#88322	—	2	2	no	—
23	C₂²×#88322	—	4	4	no	—
24	C₂²×#88322	C₂×C₄	1	4	yes	—
25	C₄⋊#88322	—	2	4	no	—
26	C₄⋊#88322	—	2	4	no	—
27	C₄⋊#88322	—	2	2	no	—
28	C₄⋊#88322	C₂×C₄	1	4	yes	—
29	C₂²⋊#88322	—	2	4	no	—
30	C₄×#88322	—	2	2	no	—
31	C₄×#88322	—	2	2	no	—
32	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
34	C₂³×C₄×C₅₃	Q₈	1	1	yes	characteristic
35	C₂²×C₈×C₅₃	—	2	2	no	—
36	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₅₃	D₄	1	2	yes	—
40	C₂×C₄²×C₅₃	D₄	1	2	yes	—
41	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
43	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
44	C₄⋊C₈×C₅₃	—	2	2	no	—
45	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
46	C₂²⋊C₈×C₅₃	—	2	4	no	—
47	#97×C₅₃	—	2	4	no	—
48	C₂⁴×C₅₃	C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	—	2	4	no	—
50	C₂²×C₄×C₅₃	—	2	4	no	—
51	C₂²×C₄×C₅₃	—	2	2	no	—
52	C₂²×C₄×C₅₃	—	2	2	no	—
53	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
54	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
55	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
56	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
57	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
61	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
62	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
65	C₂×C₈×C₅₃	—	2	4	no	—
66	C₂×C₈×C₅₃	—	4	2	no	—
67	C₂×C₈×C₅₃	—	4	2	no	—
68	C₄²×C₅₃	C₄²	1	2	yes	—
69	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
70	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
71	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
72	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
73	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
74	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
75	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
76	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
78	C₂×#88322	—	2	4	no	—
79	C₂×#88322	—	2	4	no	—
80	C₂×#88322	—	2	8	no	—
81	C₂×#88322	—	4	8	no	—
82	C₂×#88322	—	4	4	no	—
83	C₂³×C₅₃	—	2	2	no	—
84	C₂³×C₅₃	—	2	2	no	—
85	C₂³×C₅₃	#95	1	2	yes	—
86	C₂³×C₅₃	#108	1	4	yes	—
87	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₅₃	—	2	4	no	—
89	C₂×C₄×C₅₃	—	2	4	no	—
90	C₂×C₄×C₅₃	—	2	2	no	—
91	C₂×C₄×C₅₃	—	2	2	no	—
92	C₂×C₄×C₅₃	—	2	2	no	—
93	C₂×C₄×C₅₃	—	2	2	no	—
94	C₂×C₄×C₅₃	—	2	4	no	—
95	C₂×C₄×C₅₃	—	2	4	no	—
96	C₂×C₄×C₅₃	—	2	4	no	—
97	C₂×C₄×C₅₃	—	2	2	no	—
98	C₂×C₄×C₅₃	—	2	2	no	—
99	C₂×C₄×C₅₃	#95	1	2	yes	—
100	C₂×C₄×C₅₃	#95	1	2	yes	—
101	C₂×C₄×C₅₃	#95	1	2	yes	—
102	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
103	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₅₃	#131	1	4	yes	—
109	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
110	C₂×C₄×C₅₃	#137	1	2	yes	—
111	C₈×C₅₃	—	4	4	no	—
112	#88322	—	4	4	no	—
113	#88322	—	4	4	no	—
114	#7879	—	53	1	no	maximal, Sylow
115	C₂²×C₅₃	—	2	8	no	—
116	C₂²×C₅₃	—	2	2	no	—
117	C₂²×C₅₃	—	2	2	no	—
118	C₂²×C₅₃	#341	1	4	yes	—
119	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
120	C₂²×C₅₃	#424	1	2	yes	—
121	C₂²×C₅₃	#428	1	2	yes	—
122	C₂²×C₅₃	C₂×#108	1	2	yes	—
123	C₄×C₅₃	—	2	2	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	#405	1	2	yes	—
129	C₄×C₅₃	#408	1	2	yes	—
130	C₄×C₅₃	#417	1	2	yes	—
131	C₄×C₅₃	#418	1	1	yes	characteristic
132	C₄×C₅₃	#419	1	1	yes	characteristic
133	C₂×#432	D₅₃	1	1	yes	characteristic
134	C₂×#431	—	53	1	no	—
135	C₂²×C₄⋊C₈	—	53	1	no	—
136	#1277	—	53	4	no	—
137	C₂×C₅₃	—	2	4	no	—
138	C₂×C₅₃	#1573	1	4	yes	—
139	C₂×C₅₃	#1729	1	1	yes	characteristic
140	C₂×C₅₃	#1736	1	1	yes	characteristic
141	C₂×C₅₃	C₂×#341	1	1	yes	characteristic
142	C₂³×C₈	—	106	1	no	—
143	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
144	#432	—	2	2	no	—
145	#432	Dic₅₃	1	4	yes	—
146	#431	—	106	2	no	—
147	#431	—	106	2	no	—
148	C₂×C₄⋊C₈	—	53	4	no	—
149	C₂×C₄⋊C₈	—	53	2	no	—
150	C₂×C₄⋊C₈	—	53	2	no	—
151	C₂×C₄⋊C₈	—	106	2	no	—
152	C₂×C₄⋊C₈	—	106	1	no	—
153	C₂×C₄⋊C₈	C₂×D₅₃	1	2	yes	—
154	C₂×C₂²⋊C₈	—	2	1	no	—
155	C₂×C₂²⋊C₈	—	53	2	no	—
156	C₂×#97	—	2	1	no	—
157	C₂×C₄×C₈	—	53	2	no	—
158	C₅₃	#7879	1	1	yes	characteristic, complemented, Sylow
159	C₂³×C₄	Dic₁₀₆	1	1	yes	characteristic
160	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
161	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
162	C₂²×C₈	—	2	2	no	—
163	C₂²×C₈	—	53	4	no	—
164	C₂²×C₈	—	106	2	no	—
165	C₂²×C₈	—	106	4	no	—
166	C₂²×C₈	—	212	4	no	—
167	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
168	C₂×C₄²	C₄×D₅₃	1	2	yes	—
169	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
170	C₂×C₄²	D₂₁₂	1	2	yes	—
171	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
172	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
173	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
174	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
175	C₄⋊C₈	—	2	2	no	—
176	C₄⋊C₈	—	53	4	no	—
177	C₄⋊C₈	—	106	2	no	—
178	C₄⋊C₈	—	106	4	no	—
179	C₄⋊C₈	—	106	4	no	—
180	C₄⋊C₈	C₂×Dic₅₃	1	4	yes	—
181	C₂²⋊C₈	—	2	4	no	—
182	C₂²⋊C₈	—	106	4	no	—
183	#97	—	2	4	no	—
184	C₄×C₈	—	106	2	no	—
185	C₄×C₈	—	106	2	no	—
186	C₂⁴	#11719919	1	1	yes	characteristic
187	C₂²×C₄	—	2	4	no	—
188	C₂²×C₄	—	2	4	no	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	C₄×Dic₅₃	1	2	yes	—
192	C₂²×C₄	#11719919	1	1	yes	characteristic
193	C₂²×C₄	#11719919	1	1	yes	characteristic
194	C₂²×C₄	#11719919	1	1	yes	characteristic
195	C₂²×C₄	#11719919	1	2	yes	—
196	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
197	C₂²×C₄	#11719921	1	2	yes	—
198	C₂²×C₄	#11719921	1	2	yes	—
199	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
200	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
201	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
202	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
203	C₂×C₈	—	2	4	no	—
204	C₂×C₈	—	4	2	no	—
205	C₂×C₈	—	4	2	no	—
206	C₂×C₈	—	106	4	no	—
207	C₂×C₈	—	106	4	no	—
208	C₂×C₈	—	106	8	no	—
209	C₂×C₈	—	212	8	no	—
210	C₂×C₈	—	212	4	no	—
211	C₄²	C₄×Dic₅₃	1	2	yes	—
212	C₄²	#11719919	1	2	yes	—
213	C₄²	C₄⋊Dic₅₃	1	2	yes	—
214	C₄²	#11719921	1	2	yes	—
215	C₄²	#11719921	1	2	yes	—
216	C₄²	C₂²⋊Dic₅₃	1	2	yes	—
217	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
218	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
219	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
220	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
221	C₂³	—	2	2	no	—
222	C₂³	—	2	2	no	—
223	C₂³	#13133124	1	2	yes	—
224	C₂³	#13133152	1	4	yes	—
225	C₂³	C₂×#11719919	1	1	yes	characteristic
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	#13133124	1	2	yes	—
238	C₂×C₄	#13133124	1	2	yes	—
239	C₂×C₄	#13133124	1	2	yes	—
240	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
241	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
242	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
243	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
244	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
245	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
246	C₂×C₄	#13133225	1	4	yes	—
247	C₂×C₄	#13133242	1	2	yes	—
248	C₂×C₄	#13133247	1	2	yes	—
249	C₈	—	4	4	no	—
250	C₈	—	212	4	no	—
251	C₈	—	212	4	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	2	8	no	—
254	C₂²	—	2	2	no	—
255	C₂²	#16810387	1	4	yes	—
256	C₂²	C₂×#13133124	1	1	yes	characteristic
257	C₂²	#16810651	1	2	yes	—
258	C₂²	#16810666	1	2	yes	—
259	C₂²	C₂×#13133152	1	2	yes	—
260	C₄	—	2	2	no	—
261	C₄	—	2	2	no	—
262	C₄	—	2	1	no	—
263	C₄	—	2	1	no	—
264	C₄	—	2	2	no	—
265	C₄	#16810581	1	2	yes	—
266	C₄	#16810591	1	2	yes	—
267	C₄	#16810624	1	2	yes	—
268	C₄	#16810627	1	1	yes	characteristic
269	C₄	#16810631	1	1	yes	characteristic
270	C₂	—	2	4	no	—
271	C₂	#20273501	1	4	yes	—
272	C₂	#20274413	1	1	yes	characteristic
273	C₂	#20274436	1	1	yes	characteristic
274	C₂	C₂×#16810387	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₅₃×#432	C₂
socle	C₂³×C₅₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₅₃	C₄⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×#432	D₅₃
hypocenter	C₅₃	#7879
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#432

Complements

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

N	H ≅ G/N	type
3. #20272549	C₂	semidirect
158. C₅₃	#7879	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810676	2
C₂²×C₄⋊#88322	2
#20272549	2
C₂×C₅₃×#432	2
#7879	53