Subgroups — #27537736

Order 13568.

486 conjugacy classes of subgroups.

The subgroup lattice has 286 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	#20276622	C₂	1	4	yes	maximal, complemented
3	#20276621	C₂	1	4	yes	maximal, complemented
4	#20276518	C₂	1	1	yes	maximal, characteristic
5	#20276515	C₂	1	2	yes	maximal
6	#20276511	C₂	1	2	yes	maximal
7	C₅₃×#2163	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×Dic₁₀₆	C₄	1	1	yes	characteristic
9	C₈×Dic₁₀₆	C₄	1	2	yes	—
10	C₈×Dic₁₀₆	C₄	1	2	yes	—
11	C₈×Dic₁₀₆	C₂²	1	2	yes	—
12	C₈×Dic₁₀₆	C₂²	1	2	yes	—
13	C₂×C₄⋊#88322	C₄	1	1	yes	characteristic
14	C₂×#13133146	C₄	1	1	yes	characteristic
15	C₂×#13133146	C₄	1	1	yes	characteristic
16	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
17	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
18	C₄⋊#11719908	C₂²	1	2	yes	—
19	#16810441	C₂²	1	4	yes	—
20	#16810441	C₂²	1	4	yes	—
21	C₂²⋊#11719908	C₂²	1	2	yes	—
22	#16810399	C₂²	1	2	yes	—
23	#16810398	C₂²	1	4	yes	—
24	C₁₆×Dic₅₃	C₂²	1	4	yes	—
25	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
26	C₅₃×C₄⋊C₁₆	C₂²	1	2	yes	—
27	C₅₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
28	C₅₃×#346	C₂²	1	2	yes	—
29	C₂²×Dic₁₀₆	C₈	1	1	yes	characteristic
30	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
31	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
32	C₄×Dic₁₀₆	C₈	1	2	yes	—
33	C₄×Dic₁₀₆	C₈	1	2	yes	—
34	C₄×Dic₁₀₆	C₂×C₄	1	2	yes	—
35	C₄×Dic₁₀₆	C₂×C₄	1	2	yes	—
36	C₂×C₄⋊Dic₅₃	C₈	1	1	yes	characteristic
37	C₂×#11719919	C₈	1	1	yes	characteristic
38	C₂×#11719919	C₈	1	1	yes	characteristic
39	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
41	C₂×#11719908	C₂³	1	4	yes	—
42	C₄⋊#88322	C₂×C₄	1	2	yes	—
43	C₄⋊#88322	C₂×C₄	1	2	yes	—
44	#13133146	C₂×C₄	1	2	yes	—
45	#13133146	C₂×C₄	1	2	yes	—
46	#13133146	C₂×C₄	1	2	yes	—
47	#13133146	C₂×C₄	1	2	yes	—
48	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
49	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
50	C₈×Dic₅₃	C₂³	1	2	yes	—
51	C₈×Dic₅₃	C₂³	1	2	yes	—
52	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
55	C₂×C₁₆×C₅₃	C₂³	1	4	yes	—
56	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
57	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
58	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
59	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
63	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
64	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
65	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
66	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
67	C₂×Dic₁₀₆	—	2	4	no	—
68	C₂×Dic₁₀₆	C₂×C₈	1	2	yes	—
69	C₂×Dic₁₀₆	C₂×C₈	1	2	yes	—
70	C₂×C₈×C₅₃	—	2	2	no	—
71	C₂×C₈×C₅₃	—	4	1	no	—
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	C₂⁴	1	1	yes	characteristic
76	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
77	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
78	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
79	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
80	C₄⋊Dic₅₃	C₂×C₈	1	2	yes	—
81	C₄⋊Dic₅₃	C₂×C₈	1	2	yes	—
82	#11719919	C₂×C₈	1	2	yes	—
83	#11719919	C₂×C₈	1	2	yes	—
84	#11719919	C₂×C₈	1	2	yes	—
85	#11719919	C₂×C₈	1	2	yes	—
86	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
87	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
88	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
89	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
90	C₂×#88322	—	4	1	no	—
91	C₂×#88322	—	4	1	no	—
92	C₂×#88322	C₂²×C₄	1	2	yes	—
93	C₂×#88322	C₂²×C₄	1	2	yes	—
94	C₁₆×C₅₃	—	2	4	no	—
95	#11719908	—	2	4	no	—
96	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
97	C₂×C₄×C₅₃	—	2	2	no	—
98	C₂×C₄×C₅₃	—	2	1	no	—
99	C₂×C₄×C₅₃	—	2	1	no	—
100	C₂×C₄×C₅₃	—	4	1	no	—
101	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
102	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
103	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
104	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
106	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
107	C₂×Dic₅₃	—	2	2	no	—
108	C₂×Dic₅₃	—	4	1	no	—
109	C₂×Dic₅₃	—	4	1	no	—
110	C₂×Dic₅₃	M₃₂	1	4	yes	—
111	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
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	Dic₁₀₆	—	2	4	no	—
116	Dic₁₀₆	—	2	4	no	—
117	C₈×C₅₃	—	2	1	no	—
118	C₈×C₅₃	—	2	1	no	—
119	C₈×C₅₃	—	2	1	no	—
120	C₈×C₅₃	2₋¹⁺⁴	1	2	yes	—
121	#88322	—	2	2	no	—
122	#88322	—	2	2	no	—
123	#33497	—	53	1	no	maximal, Sylow
124	C₂²×C₅₃	—	2	1	no	—
125	C₂²×C₅₃	—	2	1	no	—
126	C₂²×C₅₃	C₂×M₃₂	1	2	yes	—
127	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
128	C₄×C₅₃	—	2	1	no	—
129	C₄×C₅₃	—	2	1	no	—
130	C₄×C₅₃	—	2	1	no	—
131	C₄×C₅₃	—	2	1	no	—
132	C₄×C₅₃	—	2	1	no	—
133	C₄×C₅₃	#519	1	1	yes	characteristic
134	C₄×C₅₃	#568	1	1	yes	characteristic
135	Dic₅₃	—	2	2	no	—
136	Dic₅₃	—	2	2	no	—
137	Dic₅₃	—	2	2	no	—
138	Dic₅₃	C₂×M₃₂	1	4	yes	—
139	C₂×C₈×Q₈	—	53	1	no	—
140	#2183	—	53	4	no	—
141	#2183	—	53	4	no	—
142	#2163	—	53	1	no	—
143	#2163	—	53	2	no	—
144	#2163	D₅₃	1	1	yes	characteristic
145	#2162	—	53	2	no	—
146	C₂×C₅₃	—	2	2	no	—
147	C₂×C₅₃	#2982	1	1	yes	characteristic
148	C₂×C₅₃	C₂²×M₃₂	1	1	yes	characteristic
149	C₂×C₅₃	#3407	1	1	yes	characteristic
150	C₂×C₄×Q₈	—	53	1	no	—
151	C₈×Q₈	—	53	2	no	—
152	C₈×Q₈	—	53	2	no	—
153	C₈×Q₈	—	53	2	no	—
154	C₈×Q₈	—	53	2	no	—
155	C₂×C₄⋊C₈	—	53	1	no	—
156	C₂×C₄⋊C₈	—	53	1	no	—
157	C₂×C₄⋊C₈	—	53	1	no	—
158	C₂×C₄×C₈	—	53	1	no	—
159	C₂×C₄×C₈	—	53	1	no	—
160	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
161	C₄⋊C₁₆	—	53	4	no	—
162	C₄⋊C₁₆	—	53	4	no	—
163	C₄⋊C₁₆	—	53	2	no	—
164	C₄⋊C₁₆	C₂×D₅₃	1	2	yes	—
165	C₂²⋊C₁₆	—	53	2	no	—
166	C₂²⋊C₁₆	C₂×D₅₃	1	2	yes	—
167	#346	—	53	4	no	—
168	#346	—	53	2	no	—
169	#346	C₂×D₅₃	1	2	yes	—
170	C₄×C₁₆	—	53	4	no	—
171	C₅₃	#33497	1	1	yes	characteristic, complemented, Sylow
172	C₂²×Q₈	—	53	1	no	—
173	C₂²×C₈	—	53	1	no	—
174	C₂²×C₈	—	53	1	no	—
175	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
176	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
177	C₄×Q₈	—	53	2	no	—
178	C₄×Q₈	—	53	2	no	—
179	C₄×Q₈	—	53	2	no	—
180	C₄×Q₈	—	53	2	no	—
181	C₂×C₄⋊C₄	—	53	1	no	—
182	C₂×C₄⋊C₄	—	53	1	no	—
183	C₂×C₄⋊C₄	—	53	1	no	—
184	C₂×C₄²	—	53	1	no	—
185	C₂×C₄²	—	53	1	no	—
186	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
187	C₂×C₁₆	—	53	4	no	—
188	C₂×C₁₆	C₂²×D₅₃	1	4	yes	—
189	C₄⋊C₈	—	53	2	no	—
190	C₄⋊C₈	—	53	2	no	—
191	C₄⋊C₈	—	53	2	no	—
192	C₄⋊C₈	—	53	2	no	—
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	2	no	—
198	C₄×C₈	—	53	2	no	—
199	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
200	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
201	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
202	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
203	C₂×Q₈	—	53	2	no	—
204	C₂×Q₈	—	53	2	no	—
205	C₂×Q₈	—	106	4	no	—
206	C₂²×C₄	—	53	1	no	—
207	C₂²×C₄	—	53	1	no	—
208	C₂²×C₄	—	53	1	no	—
209	C₂²×C₄	—	53	1	no	—
210	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
211	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
212	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	4	1	no	—
215	C₂×C₈	—	53	2	no	—
216	C₂×C₈	—	53	2	no	—
217	C₂×C₈	—	212	1	no	—
218	C₂×C₈	—	212	1	no	—
219	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
220	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
221	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
222	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
223	C₄⋊C₄	—	53	2	no	—
224	C₄⋊C₄	—	53	2	no	—
225	C₄⋊C₄	—	53	2	no	—
226	C₄⋊C₄	—	53	2	no	—
227	C₄⋊C₄	—	53	2	no	—
228	C₄⋊C₄	—	53	2	no	—
229	C₄²	—	53	2	no	—
230	C₄²	—	53	2	no	—
231	C₄²	—	53	2	no	—
232	C₄²	—	53	2	no	—
233	C₄²	C₈×D₅₃	1	1	yes	characteristic
234	C₄²	C₈×D₅₃	1	1	yes	characteristic
235	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
236	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
237	C₁₆	—	2	4	no	—
238	C₁₆	—	106	4	no	—
239	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
240	Q₈	—	106	4	no	—
241	Q₈	—	106	4	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	4	1	no	—
246	C₂×C₄	—	53	4	no	—
247	C₂×C₄	—	53	2	no	—
248	C₂×C₄	—	53	2	no	—
249	C₂×C₄	—	53	2	no	—
250	C₂×C₄	—	53	2	no	—
251	C₂×C₄	—	106	2	no	—
252	C₂×C₄	—	212	1	no	—
253	C₂×C₄	—	212	1	no	—
254	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
255	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
256	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
257	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
258	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
259	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
260	C₈	—	2	1	no	—
261	C₈	—	2	1	no	—
262	C₈	—	2	1	no	—
263	C₈	—	106	2	no	—
264	C₈	—	106	2	no	—
265	C₈	#13133264	1	2	yes	—
266	C₂²	—	2	1	no	—
267	C₂²	—	2	1	no	—
268	C₂²	M₃₂×D₅₃	1	2	yes	—
269	C₂²	C₂²×C₈×D₅₃	1	1	yes	characteristic
270	C₄	—	2	1	no	—
271	C₄	—	2	1	no	—
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	53	4	no	—
276	C₄	—	106	2	no	—
277	C₄	—	106	2	no	—
278	C₄	—	106	2	no	—
279	C₄	#16811100	1	1	yes	characteristic
280	C₄	#16811389	1	1	yes	characteristic
281	C₂	—	2	2	no	—
282	C₂	#20283795	1	1	yes	characteristic
283	C₂	C₂×M₃₂×D₅₃	1	1	yes	characteristic
284	C₂	#20287817	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	#33497
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. #20276622	C₂	semidirect
3. #20276621	C₂	semidirect
171. C₅₃	#33497	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
#20276622	2
#20276621	2
#20276518	2
#20276515	2
#20276511	2
C₅₃×#2163	2
#33497	53