Subgroups — #27278119

Order 13568.

420 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283279	C₂	1	1	yes	maximal, characteristic
2	#20274699	C₂	1	1	yes	maximal, characteristic
3	#20273486	C₂	1	2	yes	maximal, complemented
4	#20272611	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1458	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719917	—	2	1	no	—
7	#16810650	—	2	4	no	—
8	#16810647	—	2	1	no	—
9	#16810647	C₂²	1	2	yes	—
10	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊#88322	—	2	1	no	—
13	#16810378	—	2	1	no	—
14	#16810374	—	2	4	no	—
15	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#431	C₂²	1	2	yes	—
17	C₅₃×#324	C₄	1	2	yes	—
18	C₅₃×#323	C₄	1	2	yes	—
19	C₂×#11719917	—	2	2	no	—
20	C₂×#11719917	—	2	2	no	—
21	C₂×#11719917	—	4	2	no	—
22	C₂×#11719917	—	4	2	no	—
23	C₂²×#88322	—	2	2	no	—
24	C₂²×#88322	—	2	1	no	—
25	C₂²×#88322	—	2	1	no	—
26	C₄⋊#88322	—	2	4	no	—
27	C₄⋊#88322	—	2	2	no	—
28	C₄⋊#88322	—	2	2	no	—
29	C₄⋊#88322	—	2	2	no	—
30	C₄⋊#88322	D₄	1	4	yes	—
31	C₂²⋊#88322	—	4	2	no	—
32	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
33	#117×C₅₃	—	2	2	no	—
34	#117×C₅₃	—	2	2	no	—
35	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄⋊C₄×C₅₃	Q₈	1	1	yes	characteristic
37	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₂²⋊C₄×C₅₃	Q₈	1	1	yes	characteristic
39	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
41	C₄⋊C₈×C₅₃	—	2	2	no	—
42	C₂²⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
43	C₄×C₈×C₅₃	—	2	2	no	—
44	C₂⁴×C₅₃	C₄⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	—	2	1	no	—
46	C₂²×C₄×C₅₃	—	2	1	no	—
47	C₂²×C₄×C₅₃	—	2	1	no	—
48	C₂²×C₄×C₅₃	—	2	1	no	—
49	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	SD₁₆	1	2	yes	—
53	C₂²×C₄×C₅₃	SD₁₆	1	2	yes	—
54	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₈×C₅₃	—	2	4	no	—
58	C₄⋊C₄×C₅₃	—	2	2	no	—
59	C₄⋊C₄×C₅₃	—	2	2	no	—
60	C₂²⋊C₄×C₅₃	—	2	2	no	—
61	C₂²⋊C₄×C₅₃	—	2	2	no	—
62	C₄²×C₅₃	—	2	1	no	—
63	C₄²×C₅₃	—	2	1	no	—
64	C₄²×C₅₃	C₂×D₄	1	2	yes	—
65	C₄²×C₅₃	#41	1	2	yes	—
66	#11719917	—	4	4	no	—
67	#11719917	—	4	2	no	—
68	#11719917	—	4	2	no	—
69	C₂×#88322	—	2	4	no	—
70	C₂×#88322	—	4	4	no	—
71	C₂×#88322	—	4	1	no	—
72	C₂×#88322	—	4	1	no	—
73	C₂×#88322	—	4	1	no	—
74	C₂×#88322	—	4	1	no	—
75	C₂×#88322	—	4	1	no	—
76	C₂×#88322	—	4	1	no	—
77	C₂³×C₅₃	—	2	1	no	—
78	C₂³×C₅₃	—	2	1	no	—
79	C₂³×C₅₃	—	4	1	no	—
80	C₂³×C₅₃	—	4	1	no	—
81	C₂³×C₅₃	#106	1	2	yes	—
82	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₅₃	—	2	1	no	—
84	C₂×C₄×C₅₃	—	2	1	no	—
85	C₂×C₄×C₅₃	—	2	2	no	—
86	C₂×C₄×C₅₃	—	2	2	no	—
87	C₂×C₄×C₅₃	—	2	2	no	—
88	C₂×C₄×C₅₃	—	2	2	no	—
89	C₂×C₄×C₅₃	—	2	1	no	—
90	C₂×C₄×C₅₃	—	2	1	no	—
91	C₂×C₄×C₅₃	—	4	1	no	—
92	C₂×C₄×C₅₃	—	4	1	no	—
93	C₂×C₄×C₅₃	—	4	1	no	—
94	C₂×C₄×C₅₃	—	4	1	no	—
95	C₂×C₄×C₅₃	#106	1	2	yes	—
96	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
97	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
98	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
99	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
100	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
101	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
102	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
103	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
104	C₈×C₅₃	—	4	4	no	—
105	#88322	—	4	4	no	—
106	#88322	—	8	1	no	—
107	#88322	—	8	1	no	—
108	#8862	—	53	1	no	maximal, Sylow
109	C₂²×C₅₃	—	2	1	no	—
110	C₂²×C₅₃	—	2	1	no	—
111	C₂²×C₅₃	—	2	2	no	—
112	C₂²×C₅₃	—	4	2	no	—
113	C₂²×C₅₃	—	4	1	no	—
114	C₂²×C₅₃	—	4	1	no	—
115	C₂²×C₅₃	—	8	1	no	—
116	C₂²×C₅₃	#385	1	1	yes	characteristic
117	C₂²×C₅₃	C₂×#106	1	1	yes	characteristic
118	C₂²×C₅₃	#428	1	1	yes	characteristic
119	C₄×C₅₃	—	2	2	no	—
120	C₄×C₅₃	—	2	2	no	—
121	C₄×C₅₃	—	2	2	no	—
122	C₄×C₅₃	—	4	1	no	—
123	C₄×C₅₃	—	4	1	no	—
124	C₄×C₅₃	#448	1	1	yes	characteristic
125	C₄×C₅₃	#450	1	1	yes	characteristic
126	C₄×C₅₃	#481	1	1	yes	characteristic
127	C₄×C₅₃	#483	1	1	yes	characteristic
128	#2905	—	53	1	no	—
129	#1798	—	53	1	no	—
130	#1570	—	53	2	no	—
131	#1458	D₅₃	1	1	yes	characteristic
132	#1301	—	53	2	no	—
133	C₂×C₅₃	—	2	1	no	—
134	C₂×C₅₃	—	2	1	no	—
135	C₂×C₅₃	—	4	2	no	—
136	C₂×C₅₃	#1853	1	1	yes	characteristic
137	C₂×C₅₃	#1858	1	1	yes	characteristic
138	C₂×C₅₃	#1868	1	1	yes	characteristic
139	C₂²×M₁₆	—	106	1	no	—
140	C₂×#117	C₂×D₅₃	1	1	yes	characteristic
141	#431	C₂×D₅₃	1	2	yes	—
142	#424	—	106	4	no	—
143	#423	—	53	2	no	—
144	#423	—	106	1	no	—
145	C₂×C₄⋊C₈	—	53	1	no	—
146	C₂×C₄⋊C₈	—	53	1	no	—
147	C₂×C₂²⋊C₈	—	106	1	no	—
148	#336	—	106	1	no	—
149	#334	—	106	4	no	—
150	#324	Dic₅₃	1	2	yes	—
151	#323	Dic₅₃	1	2	yes	—
152	C₅₃	#8862	1	1	yes	characteristic, complemented, Sylow
153	C₂³×C₄	D₂₁₂	1	1	yes	characteristic
154	C₂×M₁₆	—	106	2	no	—
155	C₂×M₁₆	—	106	2	no	—
156	C₂×M₁₆	—	212	2	no	—
157	C₂×M₁₆	—	212	2	no	—
158	C₂²×C₈	—	106	2	no	—
159	C₂²×C₈	—	106	1	no	—
160	C₂²×C₈	—	106	1	no	—
161	#117	—	2	2	no	—
162	#117	—	2	2	no	—
163	C₂×C₄⋊C₄	Dic₁₀₆	1	1	yes	characteristic
164	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
165	C₂×C₂²⋊C₄	Dic₁₀₆	1	1	yes	characteristic
166	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
167	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
168	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
169	C₄⋊C₈	—	2	2	no	—
170	C₄⋊C₈	—	53	4	no	—
171	C₄⋊C₈	—	106	4	no	—
172	C₄⋊C₈	—	106	2	no	—
173	C₄⋊C₈	—	106	2	no	—
174	C₄⋊C₈	—	106	2	no	—
175	C₂²⋊C₈	—	212	2	no	—
176	C₂²⋊C₈	C₂×Dic₅₃	1	4	yes	—
177	C₄×C₈	—	2	2	no	—
178	C₂⁴	C₄⋊Dic₅₃	1	1	yes	characteristic
179	C₂²×C₄	—	2	1	no	—
180	C₂²×C₄	—	2	1	no	—
181	C₂²×C₄	—	2	1	no	—
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	#11719913	1	2	yes	—
184	C₂²×C₄	#11719913	1	2	yes	—
185	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
186	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
187	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
188	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
189	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
190	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
191	M₁₆	—	212	4	no	—
192	M₁₆	—	212	2	no	—
193	M₁₆	—	212	2	no	—
194	C₂×C₈	—	2	4	no	—
195	C₂×C₈	—	106	4	no	—
196	C₂×C₈	—	212	1	no	—
197	C₂×C₈	—	212	1	no	—
198	C₂×C₈	—	212	1	no	—
199	C₂×C₈	—	212	1	no	—
200	C₂×C₈	—	212	1	no	—
201	C₂×C₈	—	212	1	no	—
202	C₂×C₈	—	212	4	no	—
203	C₄⋊C₄	—	2	2	no	—
204	C₄⋊C₄	—	2	2	no	—
205	C₂²⋊C₄	—	2	2	no	—
206	C₂²⋊C₄	—	2	2	no	—
207	C₄²	—	2	1	no	—
208	C₄²	—	2	1	no	—
209	C₄²	D₄×D₅₃	1	2	yes	—
210	C₄²	#11719947	1	2	yes	—
211	C₂³	—	2	1	no	—
212	C₂³	—	2	1	no	—
213	C₂³	—	4	1	no	—
214	C₂³	—	4	1	no	—
215	C₂³	#13133149	1	2	yes	—
216	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	4	1	no	—
226	C₂×C₄	—	4	1	no	—
227	C₂×C₄	—	4	1	no	—
228	C₂×C₄	—	4	1	no	—
229	C₂×C₄	#13133149	1	2	yes	—
230	C₂×C₄	D₄×Dic₅₃	1	2	yes	—
231	C₂×C₄	#13133187	1	1	yes	characteristic
232	C₂×C₄	#13133194	1	2	yes	—
233	C₂×C₄	#13133201	1	1	yes	characteristic
234	C₂×C₄	C₂×#11719913	1	1	yes	characteristic
235	C₂×C₄	C₂×#11719913	1	1	yes	characteristic
236	C₂×C₄	#13133245	1	1	yes	characteristic
237	C₂×C₄	#13133250	1	1	yes	characteristic
238	C₈	—	4	4	no	—
239	C₈	—	424	1	no	—
240	C₈	—	424	1	no	—
241	C₈	—	212	4	no	—
242	C₂²	—	2	1	no	—
243	C₂²	—	2	1	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	4	2	no	—
246	C₂²	—	4	1	no	—
247	C₂²	—	4	1	no	—
248	C₂²	—	8	1	no	—
249	C₂²	#16810498	1	1	yes	characteristic
250	C₂²	C₂×#13133149	1	1	yes	characteristic
251	C₂²	#16810665	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	2	2	no	—
254	C₄	—	2	2	no	—
255	C₄	—	4	1	no	—
256	C₄	—	4	1	no	—
257	C₄	#16810764	1	1	yes	characteristic
258	C₄	#16810776	1	1	yes	characteristic
259	C₄	#16810942	1	1	yes	characteristic
260	C₄	#16810954	1	1	yes	characteristic
261	C₂	—	2	1	no	—
262	C₂	—	2	1	no	—
263	C₂	—	4	2	no	—
264	C₂	#20274910	1	1	yes	characteristic
265	C₂	#20274937	1	1	yes	characteristic
266	C₂	#20275003	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13133187
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1458	C₂
socle	C₂²×C₅₃	#385
prime radical	C₂⁴×C₅₃	C₄⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1458	D₅₃
hypocenter	C₅₃	#8862
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1458

Complements

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

N	H ≅ G/N	type
3. #20273486	C₂	semidirect
4. #20272611	C₂	semidirect
152. C₅₃	#8862	semidirect

Maximal subgroups

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

subgroup	index
#20283279	2
#20274699	2
#20273486	2
#20272611	2
C₅₃×#1458	2
#8862	53