Subgroups — #27283024

Order 13568.

446 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20273463	C₂	1	1	yes	maximal, characteristic
2	#20273110	C₂	1	2	yes	maximal
3	#20273039	C₂	1	2	yes	maximal
4	#20272998	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×Q₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
6	#16810676	C₂²	1	2	yes	—
7	#16810676	C₂²	1	1	yes	characteristic
8	#16810676	C₂²	1	1	yes	characteristic
9	#16810376	—	2	1	no	—
10	#16810374	—	2	1	no	—
11	Q₈⋊#88322	—	2	4	no	—
12	#16810349	—	2	4	no	—
13	#16810344	—	2	1	no	—
14	C₈×#88322	—	2	1	no	—
15	C₂²×Q₁₆×C₅₃	C₄	1	1	yes	characteristic
16	C₂×C₄×Q₈×C₅₃	C₂²	1	2	yes	—
17	C₄×Q₁₆×C₅₃	—	2	1	no	—
18	C₄×Q₁₆×C₅₃	—	2	1	no	—
19	C₄×Q₁₆×C₅₃	—	2	1	no	—
20	C₄×Q₁₆×C₅₃	—	2	1	no	—
21	C₂×#107×C₅₃	C₄	1	1	yes	characteristic
22	C₂×#103×C₅₃	C₄	1	2	yes	—
23	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₄⋊#88322	—	2	1	no	—
25	C₄⋊#88322	—	2	2	no	—
26	C₄⋊#88322	D₄	1	2	yes	—
27	C₂²⋊#88322	—	2	2	no	—
28	C₂²⋊#88322	—	2	1	no	—
29	C₂²⋊#88322	D₄	1	2	yes	—
30	C₄×#88322	—	2	1	no	—
31	C₄×#88322	—	2	1	no	—
32	C₄×#88322	—	2	2	no	—
33	C₂²×Q₈×C₅₃	C₂×C₄	1	2	yes	—
34	C₂×Q₁₆×C₅₃	—	2	1	no	—
35	C₂×Q₁₆×C₅₃	—	2	1	no	—
36	C₂×Q₁₆×C₅₃	—	2	2	no	—
37	C₂×Q₁₆×C₅₃	C₈	1	4	yes	complemented
38	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₄×Q₈×C₅₃	—	2	2	no	—
41	C₄×Q₈×C₅₃	—	2	2	no	—
42	C₄×Q₈×C₅₃	—	2	2	no	—
43	C₄×Q₈×C₅₃	—	2	2	no	—
44	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
45	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₅₃	—	2	2	no	—
47	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
48	#107×C₅₃	—	2	1	no	—
49	#107×C₅₃	—	2	1	no	—
50	#103×C₅₃	—	2	2	no	—
51	#103×C₅₃	—	2	2	no	—
52	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
53	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
54	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
55	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
56	C₂×Q₈×C₅₃	—	2	2	no	—
57	C₂×Q₈×C₅₃	—	2	2	no	—
58	C₂×Q₈×C₅₃	—	2	2	no	—
59	C₂×Q₈×C₅₃	—	2	2	no	—
60	C₂×Q₈×C₅₃	—	2	2	no	—
61	C₂×Q₈×C₅₃	C₂×C₈	1	4	yes	—
62	C₂²×C₄×C₅₃	—	2	2	no	—
63	C₂²×C₄×C₅₃	—	2	2	no	—
64	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
67	Q₁₆×C₅₃	—	2	4	no	—
68	Q₁₆×C₅₃	—	2	4	no	—
69	C₂×C₈×C₅₃	—	2	1	no	—
70	C₂×C₈×C₅₃	—	2	1	no	—
71	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
76	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
77	C₄⋊C₄×C₅₃	—	2	2	no	—
78	C₄⋊C₄×C₅₃	—	2	2	no	—
79	C₄⋊C₄×C₅₃	—	4	2	no	—
80	C₄⋊C₄×C₅₃	—	4	2	no	—
81	C₄²×C₅₃	—	4	2	no	—
82	C₄²×C₅₃	—	4	2	no	—
83	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
85	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₅₃	#41	1	1	yes	characteristic
87	C₂×#88322	—	2	2	no	—
88	C₂×#88322	—	2	2	no	—
89	C₂×#88322	—	4	2	no	—
90	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
91	Q₈×C₅₃	—	2	4	no	—
92	Q₈×C₅₃	—	2	4	no	—
93	Q₈×C₅₃	—	4	2	no	—
94	Q₈×C₅₃	—	4	2	no	—
95	C₂×C₄×C₅₃	—	2	2	no	—
96	C₂×C₄×C₅₃	—	2	4	no	—
97	C₂×C₄×C₅₃	—	4	1	no	—
98	C₂×C₄×C₅₃	—	4	2	no	—
99	C₂×C₄×C₅₃	—	4	2	no	—
100	C₂×C₄×C₅₃	—	4	2	no	—
101	C₂×C₄×C₅₃	—	8	2	no	—
102	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
104	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
105	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
106	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
107	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
108	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
109	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
110	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
111	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
112	C₈×C₅₃	—	2	1	no	—
113	C₈×C₅₃	—	2	1	no	—
114	C₈×C₅₃	C₂²⋊C₈	1	2	yes	—
115	C₈×C₅₃	C₂²⋊C₈	1	2	yes	—
116	#88322	—	4	2	no	—
117	#88322	—	4	2	no	—
118	#88322	—	8	2	no	—
119	#9669	—	53	1	no	maximal, Sylow
120	C₂²×C₅₃	—	2	1	no	—
121	C₂²×C₅₃	—	2	1	no	—
122	C₂²×C₅₃	#390	1	1	yes	characteristic
123	C₂²×C₅₃	C₈×D₄	1	1	yes	characteristic
124	C₂²×C₅₃	#436	1	1	yes	characteristic
125	C₄×C₅₃	—	2	1	no	—
126	C₄×C₅₃	—	2	1	no	—
127	C₄×C₅₃	—	2	1	no	—
128	C₄×C₅₃	—	4	4	no	—
129	C₄×C₅₃	—	8	2	no	—
130	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
131	C₄×C₅₃	#408	1	1	yes	characteristic
132	C₄×C₅₃	C₈×D₄	1	1	yes	characteristic
133	C₄×C₅₃	#436	1	1	yes	characteristic
134	C₄×C₅₃	#473	1	2	yes	—
135	C₂×C₄×Q₁₆	D₅₃	1	1	yes	characteristic
136	#1564	—	53	1	no	—
137	#1479	—	53	2	no	—
138	#1461	—	53	2	no	—
139	#1450	—	53	1	no	—
140	C₂×C₅₃	—	2	1	no	—
141	C₂×C₅₃	—	2	1	no	—
142	C₂×C₅₃	#1954	1	1	yes	characteristic
143	C₂×C₅₃	#1976	1	1	yes	characteristic
144	C₂×C₅₃	#1979	1	1	yes	characteristic
145	C₂²×Q₁₆	Dic₅₃	1	1	yes	characteristic
146	C₂×C₄×Q₈	C₂×D₅₃	1	2	yes	—
147	C₄×Q₁₆	—	2	1	no	—
148	C₄×Q₁₆	—	2	1	no	—
149	C₄×Q₁₆	—	2	1	no	—
150	C₄×Q₁₆	—	2	1	no	—
151	#431	—	53	1	no	—
152	#431	—	53	1	no	—
153	#431	—	53	2	no	—
154	C₂×#107	Dic₅₃	1	1	yes	characteristic
155	C₂×#103	Dic₅₃	1	2	yes	—
156	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
157	#335	—	106	1	no	—
158	#334	—	106	1	no	—
159	Q₈⋊C₈	—	106	4	no	—
160	#324	—	106	4	no	—
161	#322	—	106	1	no	—
162	C₈²	—	106	1	no	—
163	C₅₃	#9669	1	1	yes	characteristic, complemented, Sylow
164	C₂²×Q₈	C₂×Dic₅₃	1	2	yes	—
165	C₂×Q₁₆	—	2	2	no	—
166	C₂×Q₁₆	—	2	1	no	—
167	C₂×Q₁₆	—	2	1	no	—
168	C₂×Q₁₆	#88322	1	4	yes	complemented
169	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
170	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
171	C₄×Q₈	—	2	2	no	—
172	C₄×Q₈	—	2	2	no	—
173	C₄×Q₈	—	2	2	no	—
174	C₄×Q₈	—	2	2	no	—
175	C₂×C₄⋊C₄	—	2	2	no	—
176	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
177	C₂×C₄²	—	2	2	no	—
178	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
179	#107	—	2	1	no	—
180	#107	—	2	1	no	—
181	C₄⋊C₈	—	53	2	no	—
182	C₄⋊C₈	—	106	1	no	—
183	C₄⋊C₈	—	106	2	no	—
184	#103	—	2	2	no	—
185	#103	—	2	2	no	—
186	C₂²⋊C₈	—	53	2	no	—
187	C₂²⋊C₈	—	106	1	no	—
188	C₂²⋊C₈	—	106	2	no	—
189	C₄×C₈	—	106	2	no	—
190	C₄×C₈	—	106	1	no	—
191	C₄×C₈	—	106	1	no	—
192	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
193	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
194	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
195	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
196	C₂×Q₈	—	2	2	no	—
197	C₂×Q₈	—	2	2	no	—
198	C₂×Q₈	—	2	2	no	—
199	C₂×Q₈	—	2	2	no	—
200	C₂×Q₈	—	2	2	no	—
201	C₂×Q₈	C₂×#88322	1	4	yes	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
205	C₂²×C₄	#11719947	1	1	yes	characteristic
206	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
207	Q₁₆	—	2	4	no	—
208	Q₁₆	—	2	4	no	—
209	C₂×C₈	—	2	1	no	—
210	C₂×C₈	—	2	1	no	—
211	C₂×C₈	—	106	2	no	—
212	C₂×C₈	—	106	2	no	—
213	C₂×C₈	—	212	2	no	—
214	C₂×C₈	C₂×#88322	1	2	yes	—
215	C₂×C₈	#11719917	1	2	yes	—
216	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
217	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
218	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
219	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
220	C₄⋊C₄	—	2	2	no	—
221	C₄⋊C₄	—	2	2	no	—
222	C₄⋊C₄	—	4	2	no	—
223	C₄⋊C₄	—	4	2	no	—
224	C₄²	—	4	2	no	—
225	C₄²	—	4	2	no	—
226	C₄²	D₄×D₅₃	1	1	yes	characteristic
227	C₄²	#11719947	1	1	yes	characteristic
228	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
229	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
230	C₂³	D₄×Dic₅₃	1	1	yes	characteristic
231	Q₈	—	2	4	no	—
232	Q₈	—	2	4	no	—
233	Q₈	—	4	2	no	—
234	Q₈	—	4	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	4	1	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	8	2	no	—
242	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
243	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
244	C₂×C₄	#13133188	1	1	yes	characteristic
245	C₂×C₄	#13133188	1	1	yes	characteristic
246	C₂×C₄	#13133204	1	1	yes	characteristic
247	C₂×C₄	#13133213	1	1	yes	characteristic
248	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
249	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
250	C₂×C₄	#13133224	1	1	yes	characteristic
251	C₂×C₄	#13133224	1	1	yes	characteristic
252	C₈	—	2	1	no	—
253	C₈	—	2	1	no	—
254	C₈	—	424	2	no	—
255	C₈	—	212	2	no	—
256	C₈	—	212	2	no	—
257	C₈	C₂²⋊#88322	1	2	yes	—
258	C₈	C₂²⋊#88322	1	2	yes	—
259	C₂²	—	2	1	no	—
260	C₂²	—	2	1	no	—
261	C₂²	#16810526	1	1	yes	characteristic
262	C₂²	D₄×#88322	1	1	yes	characteristic
263	C₂²	#16810701	1	1	yes	characteristic
264	C₄	—	2	1	no	—
265	C₄	—	2	1	no	—
266	C₄	—	2	1	no	—
267	C₄	—	4	4	no	—
268	C₄	—	8	2	no	—
269	C₄	C₂×C₂²⋊#88322	1	1	yes	characteristic
270	C₄	#16810590	1	1	yes	characteristic
271	C₄	D₄×#88322	1	1	yes	characteristic
272	C₄	#16810701	1	1	yes	characteristic
273	C₄	#16810895	1	2	yes	—
274	C₂	—	2	1	no	—
275	C₂	—	2	1	no	—
276	C₂	#20275510	1	1	yes	characteristic
277	C₂	#20275614	1	1	yes	characteristic
278	C₂	#20275631	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13133188
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₄×Q₁₆×C₅₃	C₂
socle	C₂²×C₅₃	#390
prime radical	C₂³×C₅₃	C₄×D₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×C₄×Q₁₆	D₅₃
hypocenter	C₅₃	#9669
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
37. C₂×Q₁₆×C₅₃	C₈	semidirect
163. C₅₃	#9669	semidirect
168. C₂×Q₁₆	#88322	semidirect

Maximal subgroups

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

subgroup	index
#20273463	2
#20273110	2
#20273039	2
#20272998	2
C₂×C₄×Q₁₆×C₅₃	2
#9669	53