Subgroups — #27283021

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	C₂×C₄×Dic₂₁₂	C₂	1	1	yes	maximal, characteristic
2	#20273111	C₂	1	2	yes	maximal
3	#20273038	C₂	1	2	yes	maximal
4	#20272998	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1564	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₁₂	C₄	1	1	yes	characteristic
7	C₂×C₄×Dic₁₀₆	C₂²	1	2	yes	—
8	C₄×Dic₂₁₂	—	2	1	no	—
9	C₄×Dic₂₁₂	—	2	1	no	—
10	C₄×Dic₂₁₂	—	2	1	no	—
11	C₄×Dic₂₁₂	—	2	1	no	—
12	#16810676	C₂²	1	1	yes	characteristic
13	#16810676	C₂²	1	1	yes	characteristic
14	C₂×#13133151	C₄	1	1	yes	characteristic
15	C₂×#13133142	C₄	1	2	yes	—
16	#16810355	—	2	4	no	—
17	#16810348	—	2	4	no	—
18	#16810344	—	2	1	no	—
19	C₈×#88322	—	2	1	no	—
20	C₅₃×#431	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₅₃×#335	—	2	1	no	—
23	C₅₃×#334	—	2	1	no	—
24	C₂²×Dic₁₀₆	C₂×C₄	1	2	yes	—
25	C₂×Dic₂₁₂	—	2	1	no	—
26	C₂×Dic₂₁₂	—	2	1	no	—
27	C₂×Dic₂₁₂	—	2	2	no	—
28	C₂×Dic₂₁₂	C₈	1	4	yes	complemented
29	C₄×Dic₁₀₆	—	2	2	no	—
30	C₄×Dic₁₀₆	—	2	2	no	—
31	C₄×Dic₁₀₆	—	2	2	no	—
32	C₄×Dic₁₀₆	—	2	2	no	—
33	C₂×C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
34	C₂×#11719919	—	2	2	no	—
35	C₂×C₄×Dic₅₃	—	2	2	no	—
36	#13133151	—	2	1	no	—
37	#13133151	—	2	1	no	—
38	C₄⋊#88322	—	2	1	no	—
39	C₄⋊#88322	D₄	1	2	yes	—
40	#13133142	—	2	2	no	—
41	#13133142	—	2	2	no	—
42	C₂²⋊#88322	—	2	1	no	—
43	C₂²⋊#88322	D₄	1	2	yes	—
44	C₄×#88322	—	2	1	no	—
45	C₄×#88322	—	2	1	no	—
46	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
49	C₄⋊C₈×C₅₃	—	2	2	no	—
50	C₂²⋊C₈×C₅₃	—	2	2	no	—
51	C₄×C₈×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	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
59	C₂²×Dic₅₃	—	2	2	no	—
60	C₂²×Dic₅₃	—	2	2	no	—
61	C₂×Dic₁₀₆	—	2	2	no	—
62	C₂×Dic₁₀₆	—	2	2	no	—
63	C₂×Dic₁₀₆	—	2	2	no	—
64	C₂×Dic₁₀₆	—	2	2	no	—
65	C₂×Dic₁₀₆	—	2	2	no	—
66	C₂×Dic₁₀₆	C₂×C₈	1	4	yes	—
67	C₂×C₈×C₅₃	—	2	1	no	—
68	C₂×C₈×C₅₃	—	2	1	no	—
69	C₂×C₈×C₅₃	—	4	2	no	—
70	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
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	C₂×C₈	1	2	yes	—
75	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
76	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₅₃	#41	1	1	yes	characteristic
80	C₄⋊Dic₅₃	—	2	2	no	—
81	C₄⋊Dic₅₃	—	2	2	no	—
82	#11719919	—	4	2	no	—
83	#11719919	—	4	2	no	—
84	C₄×Dic₅₃	—	4	2	no	—
85	C₄×Dic₅₃	—	4	2	no	—
86	C₂×#88322	—	2	2	no	—
87	C₂×#88322	—	2	2	no	—
88	Dic₂₁₂	—	2	4	no	—
89	Dic₂₁₂	—	2	4	no	—
90	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
91	C₂×C₄×C₅₃	—	2	2	no	—
92	C₂×C₄×C₅₃	—	4	1	no	—
93	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
95	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
96	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
97	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
98	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
99	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
100	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
101	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
102	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
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	—	8	2	no	—
108	Dic₁₀₆	—	2	4	no	—
109	Dic₁₀₆	—	2	4	no	—
110	Dic₁₀₆	—	4	2	no	—
111	Dic₁₀₆	—	4	2	no	—
112	C₈×C₅₃	—	2	1	no	—
113	C₈×C₅₃	—	2	1	no	—
114	C₈×C₅₃	—	8	2	no	—
115	C₈×C₅₃	C₂²⋊C₈	1	2	yes	—
116	C₈×C₅₃	C₂²⋊C₈	1	2	yes	—
117	#88322	—	4	2	no	—
118	#88322	—	4	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	C₂×C₂²⋊C₈	1	1	yes	characteristic
129	C₄×C₅₃	#408	1	1	yes	characteristic
130	C₄×C₅₃	C₈×D₄	1	1	yes	characteristic
131	C₄×C₅₃	#436	1	1	yes	characteristic
132	C₄×C₅₃	#473	1	2	yes	—
133	Dic₅₃	—	4	4	no	—
134	Dic₅₃	—	8	2	no	—
135	C₂×C₄×Q₁₆	—	53	1	no	—
136	#1564	D₅₃	1	1	yes	characteristic
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	—	53	1	no	—
146	C₂×C₄×Q₈	—	53	2	no	—
147	C₄×Q₁₆	—	106	1	no	—
148	C₄×Q₁₆	—	106	1	no	—
149	C₄×Q₁₆	—	106	1	no	—
150	C₄×Q₁₆	—	106	1	no	—
151	#431	—	53	1	no	—
152	#431	—	53	1	no	—
153	#431	C₂×D₅₃	1	2	yes	—
154	C₂×#107	—	53	1	no	—
155	C₂×#103	—	53	2	no	—
156	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
157	#335	—	2	1	no	—
158	#334	—	2	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	—	53	2	no	—
165	C₂×Q₁₆	—	53	4	no	—
166	C₂×Q₁₆	—	106	2	no	—
167	C₂×Q₁₆	—	106	1	no	—
168	C₂×Q₁₆	—	106	1	no	—
169	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
170	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
171	C₄×Q₈	—	106	2	no	—
172	C₄×Q₈	—	106	2	no	—
173	C₄×Q₈	—	106	2	no	—
174	C₄×Q₈	—	106	2	no	—
175	C₂×C₄⋊C₄	—	53	2	no	—
176	C₂×C₄⋊C₄	—	106	2	no	—
177	C₂×C₄²	—	106	2	no	—
178	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
179	#107	—	106	1	no	—
180	#107	—	106	1	no	—
181	C₄⋊C₈	—	2	2	no	—
182	C₄⋊C₈	—	53	2	no	—
183	C₄⋊C₈	—	106	1	no	—
184	#103	—	106	2	no	—
185	#103	—	106	2	no	—
186	C₂²⋊C₈	—	2	2	no	—
187	C₂²⋊C₈	—	53	2	no	—
188	C₂²⋊C₈	—	106	1	no	—
189	C₄×C₈	—	2	2	no	—
190	C₄×C₈	—	106	1	no	—
191	C₄×C₈	—	106	1	no	—
192	C₄×C₈	D₂₁₂	1	1	yes	characteristic
193	C₄×C₈	D₂₁₂	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	—	53	4	no	—
197	C₂×Q₈	—	106	2	no	—
198	C₂×Q₈	—	106	2	no	—
199	C₂×Q₈	—	106	2	no	—
200	C₂×Q₈	—	106	2	no	—
201	C₂×Q₈	—	106	2	no	—
202	C₂²×C₄	—	106	2	no	—
203	C₂²×C₄	—	106	2	no	—
204	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
205	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
206	C₂²×C₄	#11719949	1	1	yes	characteristic
207	Q₁₆	—	106	4	no	—
208	Q₁₆	—	106	4	no	—
209	C₂×C₈	—	2	1	no	—
210	C₂×C₈	—	2	1	no	—
211	C₂×C₈	—	4	2	no	—
212	C₂×C₈	—	106	2	no	—
213	C₂×C₈	—	106	2	no	—
214	C₂×C₈	C₈×D₅₃	1	2	yes	—
215	C₂×C₈	#11719912	1	2	yes	—
216	C₂×C₈	#11719921	1	1	yes	characteristic
217	C₂×C₈	#11719921	1	1	yes	characteristic
218	C₂×C₈	#11719921	1	1	yes	characteristic
219	C₂×C₈	#11719921	1	1	yes	characteristic
220	C₄⋊C₄	—	106	2	no	—
221	C₄⋊C₄	—	106	2	no	—
222	C₄⋊C₄	—	212	2	no	—
223	C₄⋊C₄	—	212	2	no	—
224	C₄²	—	212	2	no	—
225	C₄²	—	212	2	no	—
226	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
227	C₄²	D₄×D₅₃	1	1	yes	characteristic
228	C₄²	#11719949	1	1	yes	characteristic
229	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
230	C₂³	#13133179	1	1	yes	characteristic
231	Q₈	—	106	4	no	—
232	Q₈	—	106	4	no	—
233	Q₈	—	212	2	no	—
234	Q₈	—	212	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	4	1	no	—
237	C₂×C₄	—	106	4	no	—
238	C₂×C₄	—	424	2	no	—
239	C₂×C₄	—	212	2	no	—
240	C₂×C₄	—	212	2	no	—
241	C₂×C₄	—	212	2	no	—
242	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
243	C₂×C₄	#13133179	1	1	yes	characteristic
244	C₂×C₄	#13133192	1	1	yes	characteristic
245	C₂×C₄	#13133192	1	1	yes	characteristic
246	C₂×C₄	#13133206	1	1	yes	characteristic
247	C₂×C₄	#13133213	1	1	yes	characteristic
248	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
249	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
250	C₂×C₄	#13133225	1	1	yes	characteristic
251	C₂×C₄	#13133225	1	1	yes	characteristic
252	C₈	—	2	1	no	—
253	C₈	—	2	1	no	—
254	C₈	—	8	2	no	—
255	C₈	—	212	2	no	—
256	C₈	—	212	2	no	—
257	C₈	#13133129	1	2	yes	—
258	C₈	#13133129	1	2	yes	—
259	C₂²	—	2	1	no	—
260	C₂²	—	2	1	no	—
261	C₂²	#16810524	1	1	yes	characteristic
262	C₂²	#16810688	1	1	yes	characteristic
263	C₂²	#16810700	1	1	yes	characteristic
264	C₄	—	2	1	no	—
265	C₄	—	2	1	no	—
266	C₄	—	2	1	no	—
267	C₄	—	424	2	no	—
268	C₄	—	212	4	no	—
269	C₄	C₂×#13133129	1	1	yes	characteristic
270	C₄	#16810591	1	1	yes	characteristic
271	C₄	#16810688	1	1	yes	characteristic
272	C₄	#16810700	1	1	yes	characteristic
273	C₄	#16810899	1	2	yes	—
274	C₂	—	2	1	no	—
275	C₂	—	2	1	no	—
276	C₂	#20275508	1	1	yes	characteristic
277	C₂	#20275611	1	1	yes	characteristic
278	C₂	#20275628	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13133192
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1564	C₂
socle	C₂²×C₅₃	#390
prime radical	C₂³×C₅₃	C₄×D₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1564	D₅₃
hypocenter	C₅₃	#9669
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
28. C₂×Dic₂₁₂	C₈	semidirect
163. C₅₃	#9669	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₂₁₂	2
#20273111	2
#20273038	2
#20272998	2
C₅₃×#1564	2
#9669	53