Subgroups — #28317724

Order 13568.

586 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20291509	C₂	1	2	yes	maximal
2	#20290355	C₂	1	2	yes	maximal, complemented
3	#20290328	C₂	1	4	yes	maximal, complemented
4	Q₁₆×#88324	C₂	1	2	yes	maximal, complemented
5	D₅₃⋊#335	C₂	1	1	yes	maximal, characteristic
6	#20290024	C₂	1	2	yes	maximal
7	C₈×#11719936	C₂	1	1	yes	maximal, characteristic
8	#20285299	C₂	1	1	yes	maximal, characteristic
9	#16811675	—	2	4	no	—
10	#16811674	C₂²	1	4	yes	—
11	Q₈×#88324	C₂²	1	4	yes	—
12	#16811642	C₂²	1	1	yes	characteristic
13	#16811630	—	2	2	no	—
14	D₅₃⋊(C₄⋊C₈)	C₂²	1	2	yes	—
15	#16811588	—	2	2	no	—
16	C₄×#11719936	C₂²	1	1	yes	characteristic
17	#16811487	C₂²	1	2	yes	—
18	#16811486	C₂²	1	1	yes	characteristic
19	#16811486	C₂²	1	1	yes	characteristic
20	Q₈⋊#88324	C₂²	1	4	yes	—
21	Q₈⋊#88324	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	#16811028	C₄	1	1	yes	characteristic
26	#16810973	C₄	1	2	yes	—
27	#16810861	C₄	1	2	yes	—
28	#16810861	C₄	1	2	yes	—
29	Q₁₆×Dic₅₃	C₂²	1	2	yes	—
30	Q₁₆×Dic₅₃	C₂²	1	2	yes	—
31	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
32	C₅₃×#494	C₄	1	1	yes	characteristic
33	C₂×#11719936	D₄	1	1	yes	characteristic
34	C₂×#11719936	D₄	1	1	yes	characteristic
35	C₂×D₅₃⋊C₈	—	2	2	no	—
36	#13133284	—	2	2	no	—
37	#13133284	—	2	4	no	—
38	C₄⋊#88324	—	2	1	no	—
39	C₄⋊#88324	—	2	4	no	—
40	C₄⋊#88324	C₂³	1	2	yes	—
41	C₄⋊#88324	C₂³	1	2	yes	—
42	#13133274	—	2	1	no	—
43	#13133272	—	2	4	no	—
44	#13133272	D₄	1	2	yes	—
45	C₄×#88324	—	2	4	no	—
46	C₄×#88324	C₂³	1	2	yes	—
47	C₄×#88324	C₂³	1	2	yes	—
48	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
50	#13133217	C₂×C₄	1	2	yes	—
51	#13133208	—	2	2	no	—
52	#13133197	—	2	2	no	—
53	#13133193	C₂×C₄	1	4	yes	—
54	Q₈×Dic₅₃	C₂³	1	4	yes	—
55	#13133182	—	2	2	no	—
56	#13133175	—	2	2	no	—
57	C₄²×D₅₃	C₂³	1	1	yes	characteristic
58	#13133151	C₂×C₄	1	2	yes	—
59	#13133151	C₂×C₄	1	2	yes	—
60	#13133141	C₂×C₄	1	4	yes	—
61	#13133141	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	C₂×Q₁₆×C₅₃	C₂×C₄	1	2	yes	—
66	C₂×Q₁₆×C₅₃	C₂×C₄	1	2	yes	—
67	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
68	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
69	C₂×Q₈×C₅₃	C₂²×C₄	1	4	yes	—
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	—	2	2	no	—
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	2	no	—
80	#11719936	Q₁₆	1	4	yes	complemented
81	D₅₃⋊C₈	—	4	4	no	—
82	Q₁₆×C₅₃	—	2	4	no	—
83	Q₁₆×C₅₃	—	4	2	no	—
84	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
86	C₄⋊C₄×C₅₃	—	2	2	no	—
87	C₄⋊C₄×C₅₃	—	2	2	no	—
88	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
89	#11719921	—	2	4	no	—
90	C₄⋊Dic₅₃	—	2	2	no	—
91	C₄⋊Dic₅₃	—	2	4	no	—
92	C₄⋊Dic₅₃	C₂²×C₄	1	4	yes	—
93	#11719919	—	2	4	no	—
94	C₄×Dic₅₃	—	2	4	no	—
95	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
96	C₄×Dic₅₃	#41	1	1	yes	characteristic
97	C₄×Dic₅₃	C₂⁴	1	1	yes	characteristic
98	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
99	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
100	C₈×D₅₃	—	2	2	no	—
101	C₈×D₅₃	—	4	1	no	—
102	C₂²×D₅₃	C₄×D₄	1	1	yes	characteristic
103	Q₈×C₅₃	—	2	4	no	—
104	Q₈×C₅₃	—	2	4	no	—
105	C₂×C₄×C₅₃	—	2	4	no	—
106	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
108	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
109	C₂×Dic₅₃	—	2	4	no	—
110	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
111	C₂×Dic₅₃	C₂²×D₄	1	1	yes	characteristic
112	C₂×Dic₅₃	C₂×#41	1	1	yes	characteristic
113	C₄×D₅₃	—	2	1	no	—
114	C₄×D₅₃	—	2	1	no	—
115	C₄×D₅₃	—	4	1	no	—
116	C₄×D₅₃	C₂×Q₁₆	1	2	yes	—
117	C₄×D₅₃	#135	1	2	yes	—
118	Dic₁₀₆	—	8	2	no	—
119	#88324	—	2	2	no	—
120	#88324	—	4	2	no	—
121	#88324	—	4	2	no	—
122	#88324	—	4	2	no	—
123	#88324	C₂×Q₁₆	1	4	yes	—
124	C₈×C₅₃	—	2	1	no	—
125	C₈×C₅₃	#131	1	2	yes	—
126	#88322	—	2	1	no	—
127	#88322	—	2	1	no	—
128	#18068	—	53	1	no	maximal, Sylow
129	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
130	C₂×D₅₃	—	2	1	no	—
131	C₂×D₅₃	—	2	1	no	—
132	C₂×D₅₃	C₄×Q₁₆	1	2	yes	—
133	C₄×C₅₃	—	2	1	no	—
134	C₄×C₅₃	—	4	4	no	—
135	C₄×C₅₃	C₄×Q₁₆	1	2	yes	—
136	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
137	C₄×C₅₃	#568	1	1	yes	characteristic
138	Dic₅₃	—	2	1	no	—
139	Dic₅₃	—	2	1	no	—
140	Dic₅₃	—	4	4	no	—
141	Dic₅₃	C₂²×Q₁₆	1	1	yes	characteristic
142	Dic₅₃	C₂×#135	1	1	yes	characteristic
143	#3170	—	53	1	no	—
144	#2962	—	53	2	no	—
145	#1591	—	53	2	no	—
146	#1584	—	53	4	no	—
147	C₈×Q₁₆	—	53	2	no	—
148	C₂×#335	—	53	1	no	—
149	#1481	—	53	2	no	—
150	C₈×M₁₆	—	53	1	no	—
151	C₂×C₅₃	#2935	1	1	yes	characteristic
152	C₂×C₅₃	C₂×C₄×Q₁₆	1	1	yes	characteristic
153	C₂×C₅₃	#2957	1	1	yes	characteristic
154	D₅₃	—	2	2	no	—
155	#533	—	53	2	no	—
156	#500	—	53	1	no	—
157	#494	#6157	1	1	yes	characteristic
158	#487	—	53	2	no	—
159	#467	—	53	2	no	—
160	#467	—	53	2	no	—
161	#446	—	53	4	no	—
162	#446	—	106	4	no	—
163	C₈×Q₈	—	53	4	no	—
164	C₄×Q₁₆	—	53	2	no	—
165	C₄×Q₁₆	—	53	2	no	—
166	#431	—	53	1	no	—
167	#424	—	106	2	no	—
168	C₂×C₄⋊C₈	—	53	2	no	—
169	#405	—	106	2	no	—
170	C₄×M₁₆	—	53	1	no	—
171	C₂×C₄×C₈	—	53	1	no	—
172	#335	—	53	1	no	—
173	#335	—	53	2	no	—
174	#335	—	53	1	no	—
175	Q₈⋊C₈	—	53	4	no	—
176	Q₈⋊C₈	—	53	4	no	—
177	#322	—	53	2	no	—
178	C₈²	—	53	2	no	—
179	C₅₃	#18068	1	1	yes	characteristic, complemented, Sylow
180	C₂×Q₁₆	C₂×#6157	1	2	yes	—
181	C₂×Q₁₆	C₂×#6157	1	2	yes	—
182	C₂×M₁₆	—	53	1	no	—
183	C₂×M₁₆	—	53	1	no	—
184	C₂²×C₈	—	53	1	no	—
185	C₂²×C₈	—	53	1	no	—
186	C₂²×C₈	—	106	2	no	—
187	C₄⋊Q₈	—	53	2	no	—
188	C₄⋊Q₈	C₂×#6157	1	2	yes	—
189	#125	—	106	2	no	—
190	C₂²⋊Q₈	—	53	4	no	—
191	C₂²⋊Q₈	—	106	2	no	—
192	C₄×Q₈	—	53	4	no	—
193	C₄×Q₈	—	106	2	no	—
194	#117	—	106	2	no	—
195	C₂×C₄²	—	53	1	no	—
196	#107	—	53	2	no	—
197	#107	—	53	2	no	—
198	C₄⋊C₈	—	53	2	no	—
199	C₄⋊C₈	—	53	2	no	—
200	C₄⋊C₈	—	106	2	no	—
201	C₄⋊C₈	—	106	1	no	—
202	C₄⋊C₈	—	106	4	no	—
203	C₄⋊C₈	—	106	4	no	—
204	#103	—	53	4	no	—
205	#103	—	53	4	no	—
206	C₂²⋊C₈	—	106	1	no	—
207	#97	—	53	2	no	—
208	#97	—	106	4	no	—
209	C₄×C₈	—	53	2	no	—
210	C₄×C₈	—	53	1	no	—
211	C₄×C₈	—	53	1	no	—
212	C₄×C₈	—	53	2	no	—
213	C₄×C₈	—	53	1	no	—
214	C₄×C₈	—	106	4	no	—
215	C₄×C₈	C₂×#6157	1	1	yes	characteristic
216	C₂×Q₈	—	106	2	no	—
217	C₂×Q₈	C₂²×#6157	1	4	yes	—
218	C₂²×C₄	—	53	1	no	—
219	C₂²×C₄	—	53	1	no	—
220	C₂²×C₄	—	53	1	no	—
221	Q₁₆	—	2	4	no	—
222	Q₁₆	—	4	2	no	—
223	M₁₆	—	53	4	no	—
224	M₁₆	—	106	2	no	—
225	C₂×C₈	—	53	2	no	—
226	C₂×C₈	—	53	2	no	—
227	C₂×C₈	—	53	1	no	—
228	C₂×C₈	—	53	1	no	—
229	C₂×C₈	—	106	2	no	—
230	C₂×C₈	—	106	2	no	—
231	C₂×C₈	—	106	2	no	—
232	C₂×C₈	—	106	2	no	—
233	C₂×C₈	—	212	1	no	—
234	C₂×C₈	—	212	4	no	—
235	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
236	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
237	C₄⋊C₄	—	2	2	no	—
238	C₄⋊C₄	—	2	2	no	—
239	C₄⋊C₄	—	53	4	no	—
240	C₄⋊C₄	—	106	4	no	—
241	C₄⋊C₄	—	106	4	no	—
242	C₄⋊C₄	—	106	2	no	—
243	C₂²⋊C₄	—	106	4	no	—
244	C₄²	—	53	1	no	—
245	C₄²	—	53	1	no	—
246	C₄²	—	53	1	no	—
247	C₄²	—	106	4	no	—
248	C₄²	C₂²×#6157	1	1	yes	characteristic
249	C₂³	—	53	1	no	—
250	Q₈	—	2	4	no	—
251	Q₈	—	2	4	no	—
252	Q₈	—	424	2	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	53	1	no	—
255	C₂×C₄	—	53	2	no	—
256	C₂×C₄	—	53	2	no	—
257	C₂×C₄	—	53	1	no	—
258	C₂×C₄	—	53	1	no	—
259	C₂×C₄	—	106	1	no	—
260	C₂×C₄	—	106	1	no	—
261	C₂×C₄	—	106	4	no	—
262	C₂×C₄	—	212	1	no	—
263	C₂×C₄	D₄×#6157	1	1	yes	characteristic
264	C₂×C₄	D₄×#6157	1	1	yes	characteristic
265	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
266	C₈	—	2	1	no	—
267	C₈	—	53	4	no	—
268	C₈	—	106	2	no	—
269	C₈	—	106	1	no	—
270	C₈	—	106	1	no	—
271	C₈	—	212	2	no	—
272	C₈	—	212	2	no	—
273	C₈	—	212	2	no	—
274	C₈	#13133304	1	2	yes	—
275	C₂²	—	53	2	no	—
276	C₂²	—	106	1	no	—
277	C₂²	—	106	1	no	—
278	C₂²	C₂×D₄×#6157	1	1	yes	characteristic
279	C₄	—	2	1	no	—
280	C₄	—	4	4	no	—
281	C₄	—	53	1	no	—
282	C₄	—	53	1	no	—
283	C₄	—	106	1	no	—
284	C₄	—	106	1	no	—
285	C₄	—	212	4	no	—
286	C₄	Q₁₆×#6157	1	2	yes	—
287	C₄	C₂×#13133304	1	1	yes	characteristic
288	C₄	#16811701	1	1	yes	characteristic
289	C₂	—	106	2	no	—
290	C₂	#20291458	1	1	yes	characteristic
291	C₂	C₂×Q₁₆×#6157	1	1	yes	characteristic
292	C₂	#20291495	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×#494	C₄
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂²×D₅₃	C₄×D₄
cosocle	C₄×Dic₅₃	C₂⁴
hypercenter	#494	#6157
hypocenter	C₅₃	#18068
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

Complements

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

N	H ≅ G/N	type
2. #20290355	C₂	semidirect
3. #20290328	C₂	semidirect
4. Q₁₆×#88324	C₂	semidirect
80. #11719936	Q₁₆	semidirect
179. C₅₃	#18068	semidirect

Maximal subgroups

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

subgroup	index
#20291509	2
#20290355	2
#20290328	2
Q₁₆×#88324	2
D₅₃⋊#335	2
#20290024	2
C₈×#11719936	2
#20285299	2
#18068	53