Subgroups — #27537742

Order 13568.

494 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic
2	#20276621	C₂	1	4	yes	maximal, complemented
3	#20276620	C₂	1	2	yes	maximal, complemented
4	#20276617	C₂	1	2	yes	maximal, complemented
5	#20276515	C₂	1	2	yes	maximal
6	#20276514	C₂	1	1	yes	maximal, characteristic
7	#20276511	C₂	1	2	yes	maximal
8	C₅₃×#2163	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Dic₁₀₆	C₄	1	1	yes	characteristic
10	C₈×Dic₁₀₆	C₄	1	2	yes	—
11	C₈×Dic₁₀₆	C₄	1	1	yes	characteristic
12	C₈×Dic₁₀₆	C₄	1	1	yes	characteristic
13	C₈×Dic₁₀₆	C₂²	1	1	yes	characteristic
14	C₈×Dic₁₀₆	C₂²	1	1	yes	characteristic
15	C₈×Dic₁₀₆	C₂²	1	2	yes	—
16	C₂×C₄⋊#88322	C₄	1	1	yes	characteristic
17	C₂×#13133146	C₄	1	2	yes	—
18	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
19	C₄⋊#11719908	C₂²	1	2	yes	—
20	#16810441	C₂²	1	4	yes	—
21	#16810441	C₂²	1	4	yes	—
22	C₂²⋊#11719908	C₂²	1	1	yes	characteristic
23	C₂²⋊#11719908	C₂²	1	1	yes	characteristic
24	#16810398	C₂²	1	4	yes	—
25	C₄×#11719908	C₂²	1	2	yes	—
26	C₁₆×Dic₅₃	C₂²	1	4	yes	—
27	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
28	C₅₃×C₄⋊C₁₆	C₂²	1	2	yes	—
29	C₅₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
30	C₅₃×#346	C₂²	1	2	yes	—
31	C₂²×Dic₁₀₆	C₈	1	1	yes	characteristic
32	C₂²×#88322	C₂×C₄	1	2	yes	—
33	C₄×Dic₁₀₆	C₈	1	1	yes	characteristic
34	C₄×Dic₁₀₆	C₈	1	2	yes	—
35	C₄×Dic₁₀₆	C₈	1	1	yes	characteristic
36	C₄×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
37	C₄×Dic₁₀₆	C₂×C₄	1	2	yes	—
38	C₄×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄⋊Dic₅₃	C₈	1	1	yes	characteristic
40	C₂×#11719919	C₈	1	2	yes	—
41	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
42	C₂×#11719908	C₂³	1	2	yes	—
43	C₂×#11719908	C₂³	1	2	yes	—
44	C₄⋊#88322	C₂×C₄	1	1	yes	characteristic
45	C₄⋊#88322	C₂×C₄	1	1	yes	characteristic
46	C₄⋊#88322	C₂×C₄	1	2	yes	—
47	#13133146	C₂×C₄	1	2	yes	—
48	#13133146	C₂×C₄	1	2	yes	—
49	#13133146	C₂×C₄	1	2	yes	—
50	#13133146	C₂×C₄	1	2	yes	—
51	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
52	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
53	C₈×Dic₅₃	C₂³	1	2	yes	—
54	C₈×Dic₅₃	C₂³	1	2	yes	—
55	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
56	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
57	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
58	C₂×C₁₆×C₅₃	C₂³	1	4	yes	—
59	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
60	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
61	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
62	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
66	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
67	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
68	C₂×Dic₁₀₆	—	2	4	no	—
69	C₂×Dic₁₀₆	C₂×C₈	1	1	yes	characteristic
70	C₂×Dic₁₀₆	C₂×C₈	1	2	yes	—
71	C₂×Dic₁₀₆	C₂×C₈	1	1	yes	characteristic
72	C₂×C₈×C₅₃	—	2	2	no	—
73	C₂×C₈×C₅₃	—	4	1	no	—
74	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
78	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
79	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
80	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
81	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
82	C₄⋊Dic₅₃	C₂×C₈	1	2	yes	—
83	C₄⋊Dic₅₃	C₂×C₈	1	1	yes	characteristic
84	C₄⋊Dic₅₃	C₂×C₈	1	1	yes	characteristic
85	#11719919	C₂×C₈	1	2	yes	—
86	#11719919	C₂×C₈	1	2	yes	—
87	#11719919	C₂×C₈	1	2	yes	—
88	#11719919	C₂×C₈	1	2	yes	—
89	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
90	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
91	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
92	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
93	C₂×#88322	—	4	2	no	—
94	C₂×#88322	C₂²×C₄	1	2	yes	—
95	C₂×#88322	C₂²×C₄	1	2	yes	—
96	C₁₆×C₅₃	—	2	4	no	—
97	#11719908	—	2	2	no	—
98	#11719908	—	2	2	no	—
99	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
100	C₂×C₄×C₅₃	—	2	2	no	—
101	C₂×C₄×C₅₃	—	2	1	no	—
102	C₂×C₄×C₅₃	—	2	1	no	—
103	C₂×C₄×C₅₃	—	2	2	no	—
104	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
106	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
107	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
108	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
109	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
110	C₂×Dic₅₃	—	2	4	no	—
111	C₂×Dic₅₃	—	2	4	no	—
112	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
113	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
114	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
115	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
116	Dic₁₀₆	—	2	2	no	—
117	Dic₁₀₆	—	2	4	no	—
118	Dic₁₀₆	M₃₂	1	4	yes	—
119	C₈×C₅₃	—	2	1	no	—
120	C₈×C₅₃	—	2	1	no	—
121	C₈×C₅₃	—	2	1	no	—
122	C₈×C₅₃	2₋¹⁺⁴	1	2	yes	—
123	#88322	—	2	2	no	—
124	#88322	—	2	2	no	—
125	#33498	—	53	1	no	maximal, Sylow
126	C₂²×C₅₃	—	2	1	no	—
127	C₂²×C₅₃	—	2	1	no	—
128	C₂²×C₅₃	#504	1	2	yes	—
129	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
130	C₄×C₅₃	—	2	1	no	—
131	C₄×C₅₃	—	2	1	no	—
132	C₄×C₅₃	—	2	1	no	—
133	C₄×C₅₃	—	2	1	no	—
134	C₄×C₅₃	C₂×M₃₂	1	2	yes	—
135	C₄×C₅₃	#519	1	1	yes	characteristic
136	C₄×C₅₃	#568	1	1	yes	characteristic
137	Dic₅₃	—	2	2	no	—
138	Dic₅₃	—	2	2	no	—
139	Dic₅₃	—	2	2	no	—
140	Dic₅₃	C₂×M₃₂	1	4	yes	—
141	C₂×C₈×Q₈	—	53	1	no	—
142	#2183	—	53	2	no	—
143	#2183	—	53	4	no	—
144	Q₈×C₁₆	—	53	2	no	—
145	#2163	—	53	2	no	—
146	#2163	D₅₃	1	1	yes	characteristic
147	#2162	—	53	1	no	—
148	#2162	—	53	2	no	—
149	C₂×C₅₃	—	2	2	no	—
150	C₂×C₅₃	#2982	1	1	yes	characteristic
151	C₂×C₅₃	C₂²×M₃₂	1	1	yes	characteristic
152	C₂×C₅₃	C₂×#504	1	1	yes	characteristic
153	C₂×C₄×Q₈	—	53	1	no	—
154	C₈×Q₈	—	53	2	no	—
155	C₈×Q₈	—	53	1	no	—
156	C₈×Q₈	—	53	1	no	—
157	C₈×Q₈	—	53	1	no	—
158	C₈×Q₈	—	53	2	no	—
159	C₈×Q₈	—	53	1	no	—
160	C₂×C₄⋊C₈	—	53	2	no	—
161	C₂×C₄⋊C₈	—	53	1	no	—
162	C₂×C₄×C₈	—	53	2	no	—
163	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
164	C₄⋊C₁₆	—	53	4	no	—
165	C₄⋊C₁₆	—	53	4	no	—
166	C₄⋊C₁₆	—	53	2	no	—
167	C₄⋊C₁₆	C₂×D₅₃	1	2	yes	—
168	C₂²⋊C₁₆	—	53	1	no	—
169	C₂²⋊C₁₆	—	53	1	no	—
170	C₂²⋊C₁₆	C₂×D₅₃	1	2	yes	—
171	#346	—	53	4	no	—
172	#346	C₂×D₅₃	1	2	yes	—
173	C₄×C₁₆	—	53	4	no	—
174	C₄×C₁₆	—	53	2	no	—
175	C₅₃	#33498	1	1	yes	characteristic, complemented, Sylow
176	C₂²×Q₈	—	53	1	no	—
177	C₂²×C₈	—	53	2	no	—
178	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
179	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
180	C₄×Q₈	—	53	1	no	—
181	C₄×Q₈	—	53	2	no	—
182	C₄×Q₈	—	53	1	no	—
183	C₄×Q₈	—	53	1	no	—
184	C₄×Q₈	—	53	2	no	—
185	C₄×Q₈	—	53	1	no	—
186	C₂×C₄⋊C₄	—	53	2	no	—
187	C₂×C₄⋊C₄	—	53	1	no	—
188	C₂×C₄²	—	53	2	no	—
189	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
190	C₂×C₁₆	—	53	2	no	—
191	C₂×C₁₆	—	53	2	no	—
192	C₂×C₁₆	C₂²×D₅₃	1	4	yes	—
193	C₄⋊C₈	—	53	2	no	—
194	C₄⋊C₈	—	53	2	no	—
195	C₄⋊C₈	—	53	2	no	—
196	C₄⋊C₈	—	53	2	no	—
197	C₄⋊C₈	—	53	1	no	—
198	C₄⋊C₈	—	53	2	no	—
199	C₄⋊C₈	—	53	1	no	—
200	C₄×C₈	—	53	2	no	—
201	C₄×C₈	—	53	2	no	—
202	C₄×C₈	—	53	2	no	—
203	C₄×C₈	—	53	2	no	—
204	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
205	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
206	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
207	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
208	C₂×Q₈	—	53	1	no	—
209	C₂×Q₈	—	53	2	no	—
210	C₂×Q₈	—	53	1	no	—
211	C₂×Q₈	—	106	4	no	—
212	C₂²×C₄	—	53	2	no	—
213	C₂²×C₄	—	53	2	no	—
214	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
215	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
216	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	4	1	no	—
219	C₂×C₈	—	53	2	no	—
220	C₂×C₈	—	53	2	no	—
221	C₂×C₈	—	212	2	no	—
222	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
223	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
224	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
225	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
226	C₄⋊C₄	—	53	2	no	—
227	C₄⋊C₄	—	53	2	no	—
228	C₄⋊C₄	—	53	2	no	—
229	C₄⋊C₄	—	53	2	no	—
230	C₄⋊C₄	—	53	2	no	—
231	C₄⋊C₄	—	53	1	no	—
232	C₄⋊C₄	—	53	1	no	—
233	C₄²	—	53	2	no	—
234	C₄²	—	53	2	no	—
235	C₄²	—	53	2	no	—
236	C₄²	—	53	2	no	—
237	C₄²	C₈×D₅₃	1	1	yes	characteristic
238	C₄²	C₈×D₅₃	1	1	yes	characteristic
239	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
240	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
241	C₁₆	—	2	4	no	—
242	C₁₆	—	106	2	no	—
243	C₁₆	—	106	2	no	—
244	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
245	Q₈	—	53	4	no	—
246	Q₈	—	106	4	no	—
247	Q₈	—	106	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	53	2	no	—
253	C₂×C₄	—	53	2	no	—
254	C₂×C₄	—	53	2	no	—
255	C₂×C₄	—	53	2	no	—
256	C₂×C₄	—	106	4	no	—
257	C₂×C₄	—	106	4	no	—
258	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
259	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
260	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
261	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
262	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
263	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
264	C₈	—	2	1	no	—
265	C₈	—	2	1	no	—
266	C₈	—	2	1	no	—
267	C₈	—	106	2	no	—
268	C₈	—	106	2	no	—
269	C₈	#13133264	1	2	yes	—
270	C₂²	—	2	1	no	—
271	C₂²	—	2	1	no	—
272	C₂²	#16811041	1	2	yes	—
273	C₂²	C₂²×C₈×D₅₃	1	1	yes	characteristic
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	53	4	no	—
279	C₄	—	106	2	no	—
280	C₄	—	106	2	no	—
281	C₄	—	106	2	no	—
282	C₄	M₃₂×D₅₃	1	2	yes	—
283	C₄	#16811100	1	1	yes	characteristic
284	C₄	#16811389	1	1	yes	characteristic
285	C₂	—	2	2	no	—
286	C₂	#20283795	1	1	yes	characteristic
287	C₂	C₂×M₃₂×D₅₃	1	1	yes	characteristic
288	C₂	C₂×#16811041	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	C₅₃×#2163	C₂
prime radical	C₂³×C₅₃	C₂²×C₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#2163	D₅₃
hypocenter	C₅₃	#33498
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ #2163

Complements

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

N	H ≅ G/N	type
2. #20276621	C₂	semidirect
3. #20276620	C₂	semidirect
4. #20276617	C₂	semidirect
175. C₅₃	#33498	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×Dic₁₀₆	2
#20276621	2
#20276620	2
#20276617	2
#20276515	2
#20276514	2
#20276511	2
C₅₃×#2163	2
#33498	53