Subgroups — #28317870

Order 13568.

564 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	#20291626	C₂	1	2	yes	maximal
2	#20290361	C₂	1	2	yes	maximal, complemented
3	#20290327	C₂	1	4	yes	maximal, complemented
4	#20290310	C₂	1	2	yes	maximal, complemented
5	#20290263	C₂	1	1	yes	maximal, characteristic
6	#20290039	C₂	1	2	yes	maximal
7	#20289925	C₂	1	1	yes	maximal, characteristic
8	#20285302	C₂	1	1	yes	maximal, characteristic
9	#16811674	—	2	2	no	—
10	#16811673	C₂²	1	4	yes	—
11	#16811672	C₂²	1	4	yes	—
12	Q₈×#88324	—	2	2	no	—
13	#16811648	—	2	4	no	—
14	#16811642	C₂²	1	2	yes	—
15	#16811642	C₂²	1	1	yes	characteristic
16	C₄×#11719936	C₂²	1	1	yes	characteristic
17	#16811487	C₂²	1	2	yes	—
18	#16811484	C₂²	1	1	yes	characteristic
19	#16811484	C₂²	1	1	yes	characteristic
20	#16811473	C₂²	1	4	yes	—
21	#16811473	C₂²	1	4	yes	—
22	#88324⋊C₈	C₂²	1	2	yes	—
23	#16811464	C₂²	1	2	yes	—
24	#16811178	C₂²	1	2	yes	—
25	#16811003	C₄	1	1	yes	characteristic
26	#16810975	C₄	1	2	yes	—
27	#16810865	C₄	1	2	yes	—
28	#16810865	C₄	1	2	yes	—
29	#16810717	C₂²	1	2	yes	—
30	#16810717	C₂²	1	2	yes	—
31	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
32	C₅₃×#500	C₄	1	1	yes	characteristic
33	C₂×#11719936	—	2	1	no	—
34	#13133284	—	2	4	no	—
35	C₄⋊#88324	—	2	4	no	—
36	C₄⋊#88324	D₄	1	2	yes	—
37	C₄⋊#88324	C₂³	1	2	yes	—
38	C₄⋊#88324	C₂³	1	2	yes	—
39	#13133274	—	2	2	no	—
40	#13133274	D₄	1	2	yes	—
41	#13133272	—	2	1	no	—
42	#13133272	—	2	4	no	—
43	C₄×#88324	—	2	2	no	—
44	C₄×#88324	—	2	4	no	—
45	C₄×#88324	C₂³	1	2	yes	—
46	C₄×#88324	C₂³	1	2	yes	—
47	C₂×Dic₂₁₂	C₂×C₄	1	2	yes	—
48	C₂×Dic₂₁₂	C₂×C₄	1	2	yes	—
49	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
51	#13133217	C₂×C₄	1	2	yes	—
52	#13133208	—	2	2	no	—
53	#13133197	C₂×C₄	1	4	yes	—
54	#13133193	—	2	2	no	—
55	Q₈×Dic₅₃	—	2	2	no	—
56	#13133182	C₂³	1	4	yes	—
57	#13133175	—	2	2	no	—
58	C₄²×D₅₃	C₂³	1	1	yes	characteristic
59	#13133140	C₂×C₄	1	4	yes	—
60	#13133140	C₂×C₄	1	4	yes	—
61	C₄×#88322	C₂×C₄	1	1	yes	characteristic
62	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
63	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
64	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
65	#107×C₅₃	C₂×C₄	1	2	yes	—
66	#107×C₅₃	C₂×C₄	1	2	yes	—
67	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
68	C₂×Q₈×C₅₃	—	2	2	no	—
69	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
71	C₂×C₄×D₅₃	#41	1	1	yes	characteristic
72	C₂×Dic₁₀₆	C₂²×C₄	1	4	yes	—
73	C₂×#88324	—	2	2	no	—
74	C₂×#88324	—	2	2	no	—
75	C₂×#88324	—	2	2	no	—
76	C₂×#88324	C₂×D₄	1	4	yes	—
77	#11719936	—	4	2	no	—
78	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
80	C₄⋊C₄×C₅₃	—	2	2	no	—
81	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
82	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
83	#11719921	—	2	4	no	—
84	C₄⋊Dic₅₃	—	2	2	no	—
85	C₄⋊Dic₅₃	—	2	2	no	—
86	C₄⋊Dic₅₃	—	2	4	no	—
87	#11719919	—	2	4	no	—
88	C₄×Dic₅₃	—	2	4	no	—
89	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
90	C₄×Dic₅₃	#41	1	1	yes	characteristic
91	C₄×Dic₅₃	C₂⁴	1	1	yes	characteristic
92	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
93	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
94	Dic₂₁₂	—	2	4	no	—
95	Dic₂₁₂	—	4	2	no	—
96	C₈×D₅₃	—	2	2	no	—
97	C₈×D₅₃	—	4	1	no	—
98	C₂²×D₅₃	C₄×D₄	1	1	yes	characteristic
99	Q₈×C₅₃	—	4	4	no	—
100	C₂×C₄×C₅₃	—	2	4	no	—
101	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
102	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
103	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
104	C₂×Dic₅₃	—	2	4	no	—
105	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
106	C₂×Dic₅₃	C₂²×D₄	1	1	yes	characteristic
107	C₂×Dic₅₃	C₂×#41	1	1	yes	characteristic
108	C₄×D₅₃	—	2	1	no	—
109	C₄×D₅₃	—	2	1	no	—
110	C₄×D₅₃	—	2	2	no	—
111	C₄×D₅₃	—	2	2	no	—
112	Dic₁₀₆	—	2	4	no	—
113	Dic₁₀₆	—	2	4	no	—
114	#88324	—	2	4	no	—
115	#88324	—	4	2	no	—
116	#88324	—	4	2	no	—
117	#88324	—	4	2	no	—
118	C₈×C₅₃	—	2	1	no	—
119	C₈×C₅₃	#131	1	2	yes	—
120	#88322	—	2	1	no	—
121	#88322	—	2	1	no	—
122	#18090	—	53	1	no	maximal, Sylow
123	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
124	C₂×D₅₃	—	2	1	no	—
125	C₂×D₅₃	—	2	1	no	—
126	C₂×D₅₃	—	2	1	no	—
127	C₄×C₅₃	—	2	1	no	—
128	C₄×C₅₃	—	4	4	no	—
129	C₄×C₅₃	#444	1	2	yes	—
130	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
131	C₄×C₅₃	#568	1	1	yes	characteristic
132	Dic₅₃	—	2	1	no	—
133	Dic₅₃	—	2	1	no	—
134	Dic₅₃	—	4	4	no	—
135	Dic₅₃	#578	1	2	yes	—
136	#3170	—	53	1	no	—
137	#2986	—	53	2	no	—
138	#1593	—	53	2	no	—
139	#1584	—	53	4	no	—
140	#1579	—	53	2	no	—
141	#1564	—	53	1	no	—
142	#1485	—	53	2	no	—
143	#1451	—	53	1	no	—
144	C₂×C₅₃	#2935	1	1	yes	characteristic
145	C₂×C₅₃	#2950	1	1	yes	characteristic
146	C₂×C₅₃	C₂×#444	1	1	yes	characteristic
147	D₅₃	—	4	1	no	—
148	#533	—	53	2	no	—
149	#500	#6157	1	1	yes	characteristic
150	#494	—	53	1	no	—
151	#487	—	53	2	no	—
152	#467	—	53	2	no	—
153	#467	—	53	2	no	—
154	#446	—	53	4	no	—
155	#446	—	106	2	no	—
156	C₈×Q₈	—	53	4	no	—
157	C₈×Q₈	—	106	2	no	—
158	C₄×Q₁₆	—	53	2	no	—
159	C₄×Q₁₆	—	53	2	no	—
160	#433	—	106	4	no	—
161	#431	—	53	1	no	—
162	#431	—	53	2	no	—
163	C₄×M₁₆	—	53	1	no	—
164	C₂×C₄×C₈	—	53	1	no	—
165	#335	—	53	2	no	—
166	#334	—	53	1	no	—
167	#334	—	53	1	no	—
168	Q₈⋊C₈	—	53	4	no	—
169	Q₈⋊C₈	—	53	4	no	—
170	#322	—	53	2	no	—
171	#322	—	53	2	no	—
172	C₅₃	#18090	1	1	yes	characteristic, complemented, Sylow
173	C₂×Q₁₆	—	53	2	no	—
174	C₂×Q₁₆	—	53	2	no	—
175	C₂×M₁₆	—	106	1	no	—
176	C₂²×C₈	—	53	1	no	—
177	C₂²×C₈	—	53	1	no	—
178	C₄⋊Q₈	—	53	2	no	—
179	C₄⋊Q₈	C₂×#6157	1	2	yes	—
180	#125	—	106	2	no	—
181	C₂²⋊Q₈	—	53	4	no	—
182	C₂²⋊Q₈	—	106	2	no	—
183	C₄×Q₈	—	53	4	no	—
184	C₄×Q₈	—	106	2	no	—
185	#117	—	106	2	no	—
186	C₂×C₄²	—	53	1	no	—
187	#107	C₂×#6157	1	2	yes	—
188	#107	C₂×#6157	1	2	yes	—
189	C₄⋊C₈	—	53	2	no	—
190	C₄⋊C₈	—	53	2	no	—
191	C₄⋊C₈	—	53	2	no	—
192	C₄⋊C₈	—	106	4	no	—
193	C₄⋊C₈	—	106	4	no	—
194	#103	—	53	4	no	—
195	#103	—	53	4	no	—
196	C₂²⋊C₈	—	53	2	no	—
197	C₂²⋊C₈	—	106	2	no	—
198	#97	—	106	1	no	—
199	#97	—	106	4	no	—
200	C₄×C₈	—	53	2	no	—
201	C₄×C₈	—	53	1	no	—
202	C₄×C₈	—	53	1	no	—
203	C₄×C₈	—	53	2	no	—
204	C₄×C₈	—	53	1	no	—
205	C₄×C₈	—	106	2	no	—
206	C₄×C₈	—	106	4	no	—
207	C₄×C₈	C₂×#6157	1	1	yes	characteristic
208	C₂×Q₈	—	2	2	no	—
209	C₂×Q₈	—	53	4	no	—
210	C₂²×C₄	—	53	1	no	—
211	C₂²×C₄	—	53	1	no	—
212	C₂²×C₄	—	53	1	no	—
213	Q₁₆	—	106	4	no	—
214	Q₁₆	—	212	2	no	—
215	M₁₆	—	212	2	no	—
216	C₂×C₈	—	53	4	no	—
217	C₂×C₈	—	53	1	no	—
218	C₂×C₈	—	53	1	no	—
219	C₂×C₈	—	106	2	no	—
220	C₂×C₈	—	106	2	no	—
221	C₂×C₈	—	106	2	no	—
222	C₂×C₈	—	106	2	no	—
223	C₂×C₈	—	212	1	no	—
224	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
225	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
226	C₄⋊C₄	—	2	2	no	—
227	C₄⋊C₄	—	106	4	no	—
228	C₄⋊C₄	—	106	4	no	—
229	C₄⋊C₄	—	106	2	no	—
230	C₄⋊C₄	—	106	2	no	—
231	C₄⋊C₄	C₂²×#6157	1	4	yes	—
232	C₂²⋊C₄	—	106	4	no	—
233	C₄²	—	53	1	no	—
234	C₄²	—	53	1	no	—
235	C₄²	—	53	1	no	—
236	C₄²	—	106	4	no	—
237	C₄²	C₂²×#6157	1	1	yes	characteristic
238	C₂³	—	53	1	no	—
239	Q₈	—	4	4	no	—
240	Q₈	—	106	4	no	—
241	Q₈	—	106	4	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	53	1	no	—
244	C₂×C₄	—	53	1	no	—
245	C₂×C₄	—	53	1	no	—
246	C₂×C₄	—	106	2	no	—
247	C₂×C₄	—	106	2	no	—
248	C₂×C₄	—	106	1	no	—
249	C₂×C₄	—	106	1	no	—
250	C₂×C₄	—	106	4	no	—
251	C₂×C₄	D₄×#6157	1	1	yes	characteristic
252	C₂×C₄	D₄×#6157	1	1	yes	characteristic
253	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
254	C₈	—	2	1	no	—
255	C₈	—	106	4	no	—
256	C₈	—	106	1	no	—
257	C₈	—	106	1	no	—
258	C₈	—	212	2	no	—
259	C₈	—	212	2	no	—
260	C₈	—	212	2	no	—
261	C₈	#13133305	1	2	yes	—
262	C₂²	—	106	1	no	—
263	C₂²	—	106	1	no	—
264	C₂²	—	106	1	no	—
265	C₂²	C₂×D₄×#6157	1	1	yes	characteristic
266	C₄	—	2	1	no	—
267	C₄	—	4	4	no	—
268	C₄	—	53	2	no	—
269	C₄	—	106	1	no	—
270	C₄	—	106	1	no	—
271	C₄	—	212	4	no	—
272	C₄	#16811670	1	2	yes	—
273	C₄	C₂×#13133305	1	1	yes	characteristic
274	C₄	#16811703	1	1	yes	characteristic
275	C₂	—	212	1	no	—
276	C₂	#20291461	1	1	yes	characteristic
277	C₂	#20291476	1	1	yes	characteristic
278	C₂	C₂×#16811670	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×#6157
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#6157
Fitting	C₅₃×#500	C₄
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂²×D₅₃	C₄×D₄
cosocle	C₄×Dic₅₃	C₂⁴
hypercenter	#500	#6157
hypocenter	C₅₃	#18090
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

Complements

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

N	H ≅ G/N	type
2. #20290361	C₂	semidirect
3. #20290327	C₂	semidirect
4. #20290310	C₂	semidirect
172. C₅₃	#18090	semidirect

Maximal subgroups

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

subgroup	index
#20291626	2
#20290361	2
#20290327	2
#20290310	2
#20290263	2
#20290039	2
#20289925	2
#20285302	2
#18090	53