Subgroups — #28317847

Order 13568.

572 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	#20291626	C₂	1	2	yes	maximal
2	#20290354	C₂	1	2	yes	maximal, complemented
3	#20290339	C₂	1	4	yes	maximal, complemented
4	#20290303	C₂	1	2	yes	maximal, complemented
5	#20290265	C₂	1	1	yes	maximal, characteristic
6	#20290039	C₂	1	2	yes	maximal
7	C₈×#11719936	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	#16811486	C₂²	1	1	yes	characteristic
18	#16811486	C₂²	1	1	yes	characteristic
19	#16811485	C₂²	1	2	yes	—
20	#16811473	C₂²	1	4	yes	—
21	#16811473	C₂²	1	4	yes	—
22	#88324⋊C₈	C₂²	1	2	yes	—
23	C₈×#88324	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	1	yes	characteristic
37	C₄⋊#88324	D₄	1	1	yes	characteristic
38	C₄⋊#88324	C₂³	1	2	yes	—
39	C₄⋊#88324	C₂³	1	2	yes	—
40	#13133274	—	2	2	no	—
41	#13133274	D₄	1	2	yes	—
42	#13133272	—	2	1	no	—
43	#13133272	—	2	4	no	—
44	C₄×#88324	—	2	2	no	—
45	C₄×#88324	—	2	4	no	—
46	C₄×#88324	C₂³	1	2	yes	—
47	C₄×#88324	C₂³	1	2	yes	—
48	C₂×Dic₂₁₂	C₂×C₄	1	2	yes	—
49	C₂×Dic₂₁₂	C₂×C₄	1	2	yes	—
50	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
51	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
52	#13133217	C₂×C₄	1	2	yes	—
53	#13133208	—	2	2	no	—
54	#13133197	C₂×C₄	1	4	yes	—
55	#13133193	—	2	2	no	—
56	Q₈×Dic₅₃	—	2	2	no	—
57	#13133182	C₂³	1	4	yes	—
58	#13133175	—	2	2	no	—
59	C₄²×D₅₃	C₂³	1	1	yes	characteristic
60	#13133140	C₂×C₄	1	4	yes	—
61	#13133140	C₂×C₄	1	4	yes	—
62	C₄×#88322	C₂×C₄	1	1	yes	characteristic
63	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
64	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
65	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
66	#107×C₅₃	C₂×C₄	1	2	yes	—
67	#107×C₅₃	C₂×C₄	1	2	yes	—
68	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
69	C₂×Q₈×C₅₃	—	2	2	no	—
70	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
72	C₂×C₄×D₅₃	#41	1	1	yes	characteristic
73	C₂×Dic₁₀₆	C₂²×C₄	1	4	yes	—
74	C₂×#88324	—	2	2	no	—
75	C₂×#88324	—	2	2	no	—
76	C₂×#88324	—	2	2	no	—
77	C₂×#88324	C₂×D₄	1	2	yes	—
78	C₂×#88324	C₂×D₄	1	2	yes	—
79	#11719936	—	2	4	no	—
80	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
82	C₄⋊C₄×C₅₃	—	2	2	no	—
83	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
84	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
85	#11719921	—	2	4	no	—
86	C₄⋊Dic₅₃	—	2	2	no	—
87	C₄⋊Dic₅₃	—	2	2	no	—
88	C₄⋊Dic₅₃	—	2	4	no	—
89	#11719919	—	2	4	no	—
90	C₄×Dic₅₃	—	2	4	no	—
91	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
92	C₄×Dic₅₃	#41	1	1	yes	characteristic
93	C₄×Dic₅₃	C₂⁴	1	1	yes	characteristic
94	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
95	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
96	Dic₂₁₂	—	2	4	no	—
97	Dic₂₁₂	—	4	2	no	—
98	C₈×D₅₃	—	2	2	no	—
99	C₈×D₅₃	—	4	1	no	—
100	C₂²×D₅₃	C₄×D₄	1	1	yes	characteristic
101	Q₈×C₅₃	—	4	4	no	—
102	C₂×C₄×C₅₃	—	2	4	no	—
103	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
104	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
106	C₂×Dic₅₃	—	2	4	no	—
107	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
108	C₂×Dic₅₃	C₂²×D₄	1	1	yes	characteristic
109	C₂×Dic₅₃	C₂×#41	1	1	yes	characteristic
110	C₄×D₅₃	—	2	1	no	—
111	C₄×D₅₃	—	2	1	no	—
112	C₄×D₅₃	—	2	2	no	—
113	C₄×D₅₃	—	2	1	no	—
114	C₄×D₅₃	—	2	1	no	—
115	Dic₁₀₆	—	2	4	no	—
116	Dic₁₀₆	—	2	4	no	—
117	#88324	—	2	4	no	—
118	#88324	—	2	2	no	—
119	#88324	—	2	2	no	—
120	#88324	—	4	2	no	—
121	#88324	—	4	2	no	—
122	C₈×C₅₃	—	2	1	no	—
123	C₈×C₅₃	#131	1	2	yes	—
124	#88322	—	2	1	no	—
125	#88322	—	2	1	no	—
126	#18086	—	53	1	no	maximal, Sylow
127	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
128	C₂×D₅₃	—	2	1	no	—
129	C₂×D₅₃	—	2	1	no	—
130	C₂×D₅₃	—	2	1	no	—
131	C₄×C₅₃	—	2	1	no	—
132	C₄×C₅₃	—	4	4	no	—
133	C₄×C₅₃	#443	1	2	yes	—
134	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
135	C₄×C₅₃	#568	1	1	yes	characteristic
136	Dic₅₃	—	2	1	no	—
137	Dic₅₃	—	2	1	no	—
138	Dic₅₃	—	4	4	no	—
139	Dic₅₃	#578	1	1	yes	characteristic
140	Dic₅₃	#578	1	1	yes	characteristic
141	#3170	—	53	1	no	—
142	#2986	—	53	2	no	—
143	#1591	—	53	2	no	—
144	#1587	—	53	4	no	—
145	C₈×Q₁₆	—	53	2	no	—
146	#1564	—	53	1	no	—
147	#1485	—	53	2	no	—
148	C₈×M₁₆	—	53	1	no	—
149	C₂×C₅₃	#2935	1	1	yes	characteristic
150	C₂×C₅₃	#2950	1	1	yes	characteristic
151	C₂×C₅₃	C₂×#443	1	1	yes	characteristic
152	D₅₃	—	4	1	no	—
153	#533	—	53	2	no	—
154	#500	#6157	1	1	yes	characteristic
155	#494	—	53	1	no	—
156	#487	—	53	2	no	—
157	#467	—	53	2	no	—
158	#467	—	53	2	no	—
159	#446	—	53	4	no	—
160	#446	—	106	2	no	—
161	C₈×Q₈	—	53	4	no	—
162	C₈×Q₈	—	106	2	no	—
163	C₄×Q₁₆	—	53	2	no	—
164	C₄×Q₁₆	—	53	2	no	—
165	#433	—	106	4	no	—
166	#431	—	53	1	no	—
167	#431	—	53	2	no	—
168	C₄×M₁₆	—	53	1	no	—
169	C₂×C₄×C₈	—	53	1	no	—
170	#335	—	53	1	no	—
171	#335	—	53	1	no	—
172	#334	—	53	2	no	—
173	Q₈⋊C₈	—	53	4	no	—
174	Q₈⋊C₈	—	53	4	no	—
175	#322	—	53	2	no	—
176	C₈²	—	53	2	no	—
177	C₅₃	#18086	1	1	yes	characteristic, complemented, Sylow
178	C₂×Q₁₆	—	53	2	no	—
179	C₂×Q₁₆	—	53	2	no	—
180	C₂×M₁₆	—	106	1	no	—
181	C₂²×C₈	—	53	1	no	—
182	C₂²×C₈	—	53	1	no	—
183	C₄⋊Q₈	—	53	2	no	—
184	C₄⋊Q₈	C₂×#6157	1	2	yes	—
185	#125	—	106	2	no	—
186	C₂²⋊Q₈	—	53	4	no	—
187	C₂²⋊Q₈	—	106	2	no	—
188	C₄×Q₈	—	53	4	no	—
189	C₄×Q₈	—	106	2	no	—
190	#117	—	106	2	no	—
191	C₂×C₄²	—	53	1	no	—
192	#107	C₂×#6157	1	2	yes	—
193	#107	C₂×#6157	1	2	yes	—
194	C₄⋊C₈	—	53	1	no	—
195	C₄⋊C₈	—	53	1	no	—
196	C₄⋊C₈	—	53	2	no	—
197	C₄⋊C₈	—	53	2	no	—
198	C₄⋊C₈	—	106	4	no	—
199	C₄⋊C₈	—	106	4	no	—
200	#103	—	53	4	no	—
201	#103	—	53	4	no	—
202	C₂²⋊C₈	—	53	2	no	—
203	C₂²⋊C₈	—	106	2	no	—
204	#97	—	106	1	no	—
205	#97	—	106	4	no	—
206	C₄×C₈	—	53	2	no	—
207	C₄×C₈	—	53	1	no	—
208	C₄×C₈	—	53	1	no	—
209	C₄×C₈	—	53	2	no	—
210	C₄×C₈	—	53	1	no	—
211	C₄×C₈	—	106	2	no	—
212	C₄×C₈	—	106	4	no	—
213	C₄×C₈	C₂×#6157	1	1	yes	characteristic
214	C₂×Q₈	—	2	2	no	—
215	C₂×Q₈	—	53	4	no	—
216	C₂²×C₄	—	53	1	no	—
217	C₂²×C₄	—	53	1	no	—
218	C₂²×C₄	—	53	1	no	—
219	Q₁₆	—	106	4	no	—
220	Q₁₆	—	212	2	no	—
221	M₁₆	—	106	4	no	—
222	C₂×C₈	—	53	2	no	—
223	C₂×C₈	—	53	2	no	—
224	C₂×C₈	—	53	1	no	—
225	C₂×C₈	—	53	1	no	—
226	C₂×C₈	—	106	2	no	—
227	C₂×C₈	—	106	2	no	—
228	C₂×C₈	—	106	2	no	—
229	C₂×C₈	—	106	2	no	—
230	C₂×C₈	—	212	1	no	—
231	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
232	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
233	C₄⋊C₄	—	2	2	no	—
234	C₄⋊C₄	—	106	4	no	—
235	C₄⋊C₄	—	106	4	no	—
236	C₄⋊C₄	—	106	2	no	—
237	C₄⋊C₄	—	106	2	no	—
238	C₄⋊C₄	C₂²×#6157	1	4	yes	—
239	C₂²⋊C₄	—	106	4	no	—
240	C₄²	—	53	1	no	—
241	C₄²	—	53	1	no	—
242	C₄²	—	53	1	no	—
243	C₄²	—	106	4	no	—
244	C₄²	C₂²×#6157	1	1	yes	characteristic
245	C₂³	—	53	1	no	—
246	Q₈	—	4	4	no	—
247	Q₈	—	106	4	no	—
248	Q₈	—	106	4	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	53	1	no	—
251	C₂×C₄	—	53	1	no	—
252	C₂×C₄	—	53	1	no	—
253	C₂×C₄	—	106	1	no	—
254	C₂×C₄	—	106	1	no	—
255	C₂×C₄	—	106	2	no	—
256	C₂×C₄	—	106	1	no	—
257	C₂×C₄	—	106	1	no	—
258	C₂×C₄	—	106	4	no	—
259	C₂×C₄	D₄×#6157	1	1	yes	characteristic
260	C₂×C₄	D₄×#6157	1	1	yes	characteristic
261	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
262	C₈	—	2	1	no	—
263	C₈	—	106	2	no	—
264	C₈	—	106	2	no	—
265	C₈	—	106	4	no	—
266	C₈	—	106	1	no	—
267	C₈	—	106	1	no	—
268	C₈	—	212	2	no	—
269	C₈	—	212	2	no	—
270	C₈	#13133305	1	2	yes	—
271	C₂²	—	106	1	no	—
272	C₂²	—	106	1	no	—
273	C₂²	—	106	1	no	—
274	C₂²	C₂×D₄×#6157	1	1	yes	characteristic
275	C₄	—	2	1	no	—
276	C₄	—	4	4	no	—
277	C₄	—	53	1	no	—
278	C₄	—	53	1	no	—
279	C₄	—	106	1	no	—
280	C₄	—	106	1	no	—
281	C₄	—	212	4	no	—
282	C₄	#16811665	1	2	yes	—
283	C₄	C₂×#13133305	1	1	yes	characteristic
284	C₄	#16811703	1	1	yes	characteristic
285	C₂	—	212	1	no	—
286	C₂	#20291461	1	1	yes	characteristic
287	C₂	#20291476	1	1	yes	characteristic
288	C₂	C₂×#16811665	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	#18086
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. #20290354	C₂	semidirect
3. #20290339	C₂	semidirect
4. #20290303	C₂	semidirect
177. C₅₃	#18086	semidirect

Maximal subgroups

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

subgroup	index
#20291626	2
#20290354	2
#20290339	2
#20290303	2
#20290265	2
#20290039	2
C₈×#11719936	2
#20285302	2
#18086	53