Subgroups — #27355871

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	#20285301	C₂	1	1	yes	maximal, characteristic
2	#20283662	C₂	1	2	yes	maximal
3	#20273613	C₂	1	2	yes	maximal, complemented
4	#20273567	C₂	1	4	yes	maximal, complemented
5	C₈×Dic₂₁₂	C₂	1	2	yes	maximal, complemented
6	C₂×#16810376	C₂	1	1	yes	maximal, characteristic
7	#20273119	C₂	1	2	yes	maximal
8	C₈×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
9	#16811180	C₂²	1	2	yes	—
10	#16811028	C₄	1	1	yes	characteristic
11	#16811003	C₄	1	1	yes	characteristic
12	#16810974	C₄	1	2	yes	—
13	#16810864	C₄	1	2	yes	—
14	#16810864	C₄	1	2	yes	—
15	#16810753	C₂²	1	4	yes	—
16	#16810751	—	2	4	no	—
17	C₈×Dic₁₀₆	C₂²	1	4	yes	—
18	C₄×Dic₂₁₂	C₂²	1	2	yes	—
19	C₄×Dic₂₁₂	C₂²	1	2	yes	—
20	#16810651	—	2	2	no	—
21	C₂×C₄⋊#88322	C₂²	1	2	yes	—
22	#16810581	—	2	2	no	—
23	#16810376	C₂²	1	1	yes	characteristic
24	#16810376	C₂²	1	2	yes	—
25	#16810376	C₂²	1	1	yes	characteristic
26	#16810355	C₂²	1	4	yes	—
27	#16810355	C₂²	1	4	yes	—
28	C₅₃×#431	C₂²	1	1	yes	characteristic
29	C₄×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
31	C₅₃×#322	C₂²	1	2	yes	—
32	C₈²×C₅₃	C₂²	1	2	yes	—
33	C₂×Dic₂₁₂	C₂×C₄	1	2	yes	—
34	C₂×Dic₂₁₂	C₂×C₄	1	2	yes	—
35	C₂²×#88322	—	2	2	no	—
36	#13133217	C₂×C₄	1	2	yes	—
37	#13133217	C₂×C₄	1	2	yes	—
38	#13133209	—	2	2	no	—
39	#13133196	—	2	2	no	—
40	#13133196	C₂×C₄	1	4	yes	—
41	C₄×Dic₁₀₆	—	2	2	no	—
42	C₄×Dic₁₀₆	C₂³	1	4	yes	—
43	#13133176	—	2	2	no	—
44	#13133151	C₂×C₄	1	2	yes	—
45	#13133151	C₂×C₄	1	2	yes	—
46	C₄⋊#88322	—	2	2	no	—
47	C₄⋊#88322	C₂³	1	2	yes	—
48	C₄⋊#88322	C₂³	1	2	yes	—
49	#13133146	—	2	4	no	—
50	#13133146	—	2	4	no	—
51	#13133142	C₂×C₄	1	4	yes	—
52	#13133142	C₂×C₄	1	4	yes	—
53	#13133127	—	2	4	no	—
54	C₈×Dic₅₃	—	2	4	no	—
55	C₂×M₁₆×C₅₃	D₄	1	1	yes	characteristic
56	C₂×M₁₆×C₅₃	D₄	1	1	yes	characteristic
57	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
58	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
60	C₄⋊C₈×C₅₃	—	2	1	no	—
61	C₂²⋊C₈×C₅₃	—	2	1	no	—
62	#97×C₅₃	D₄	1	2	yes	—
63	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
64	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
65	C₄×C₈×C₅₃	C₂³	1	2	yes	—
66	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
68	C₄×C₈×C₅₃	C₂³	1	2	yes	—
69	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
72	C₂×Dic₁₀₆	—	2	2	no	—
73	C₂×Dic₁₀₆	C₂²×C₄	1	4	yes	—
74	M₁₆×C₅₃	—	2	2	no	—
75	M₁₆×C₅₃	Q₁₆	1	4	yes	complemented
76	C₂×C₈×C₅₃	—	2	2	no	—
77	C₂×C₈×C₅₃	—	2	2	no	—
78	C₂×C₈×C₅₃	—	4	1	no	—
79	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
84	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
85	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
86	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₅₃	#41	1	1	yes	characteristic
88	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
89	C₂²⋊Dic₅₃	—	2	4	no	—
90	C₄⋊Dic₅₃	—	2	2	no	—
91	C₄⋊Dic₅₃	—	2	2	no	—
92	C₄⋊Dic₅₃	—	2	2	no	—
93	C₄⋊Dic₅₃	C₂²×C₄	1	4	yes	—
94	#11719919	—	2	4	no	—
95	#11719919	—	2	4	no	—
96	C₄×Dic₅₃	—	2	4	no	—
97	C₂×#88322	—	2	2	no	—
98	C₂×#88322	—	2	2	no	—
99	C₂×#88322	—	4	4	no	—
100	Dic₂₁₂	—	2	4	no	—
101	Dic₂₁₂	—	4	2	no	—
102	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
103	C₂×C₄×C₅₃	—	2	1	no	—
104	C₂×C₄×C₅₃	—	2	1	no	—
105	C₂×C₄×C₅₃	—	4	1	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	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
110	C₂×C₄×C₅₃	#135	1	2	yes	—
111	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
112	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
114	C₂×Dic₅₃	—	2	4	no	—
115	C₂×Dic₅₃	—	2	4	no	—
116	Dic₁₀₆	—	2	4	no	—
117	Dic₁₀₆	—	2	4	no	—
118	Dic₁₀₆	—	8	2	no	—
119	C₈×C₅₃	—	2	1	no	—
120	C₈×C₅₃	—	2	1	no	—
121	C₈×C₅₃	—	2	1	no	—
122	C₈×C₅₃	—	2	2	no	—
123	C₈×C₅₃	—	4	2	no	—
124	C₈×C₅₃	#131	1	2	yes	—
125	C₈×C₅₃	C₂×Q₁₆	1	4	yes	—
126	#88322	—	4	2	no	—
127	#88322	—	4	2	no	—
128	#18068	—	53	1	no	maximal, Sylow
129	C₂²×C₅₃	—	2	1	no	—
130	C₂²×C₅₃	—	2	1	no	—
131	C₂²×C₅₃	C₄×Q₁₆	1	2	yes	—
132	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
133	C₄×C₅₃	—	2	1	no	—
134	C₄×C₅₃	—	2	1	no	—
135	C₄×C₅₃	—	2	1	no	—
136	C₄×C₅₃	C₄×Q₁₆	1	2	yes	—
137	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
138	C₄×C₅₃	#568	1	1	yes	characteristic
139	C₄×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
140	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
141	Dic₅₃	—	4	4	no	—
142	Dic₅₃	—	4	4	no	—
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	D₅₃	1	1	yes	characteristic
151	C₂×C₅₃	—	2	2	no	—
152	C₂×C₅₃	#2935	1	1	yes	characteristic
153	C₂×C₅₃	C₂×C₄×Q₁₆	1	1	yes	characteristic
154	C₂×C₅₃	#2957	1	1	yes	characteristic
155	#533	—	53	2	no	—
156	#500	—	53	1	no	—
157	#494	—	53	1	no	—
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	C₂×D₅₃	1	1	yes	characteristic
167	#424	—	106	2	no	—
168	C₂×C₄⋊C₈	—	53	2	no	—
169	#405	—	106	2	no	—
170	C₄×M₁₆	C₂×D₅₃	1	1	yes	characteristic
171	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
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	C₂×D₅₃	1	2	yes	—
178	C₈²	C₂×D₅₃	1	2	yes	—
179	C₅₃	#18068	1	1	yes	characteristic, complemented, Sylow
180	C₂×Q₁₆	—	53	2	no	—
181	C₂×Q₁₆	—	53	2	no	—
182	C₂×M₁₆	D₂₁₂	1	1	yes	characteristic
183	C₂×M₁₆	D₂₁₂	1	1	yes	characteristic
184	C₂²×C₈	—	106	2	no	—
185	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
186	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
187	C₄⋊Q₈	—	53	2	no	—
188	C₄⋊Q₈	—	53	2	no	—
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	C₂²×D₅₃	1	1	yes	characteristic
196	#107	—	53	2	no	—
197	#107	—	53	2	no	—
198	C₄⋊C₈	—	2	1	no	—
199	C₄⋊C₈	—	53	2	no	—
200	C₄⋊C₈	—	53	2	no	—
201	C₄⋊C₈	—	106	4	no	—
202	C₄⋊C₈	—	106	4	no	—
203	C₄⋊C₈	—	106	2	no	—
204	#103	—	53	4	no	—
205	#103	—	53	4	no	—
206	C₂²⋊C₈	—	2	1	no	—
207	#97	—	106	4	no	—
208	#97	D₂₁₂	1	2	yes	—
209	C₄×C₈	—	106	4	no	—
210	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
211	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
212	C₄×C₈	C₂²×D₅₃	1	2	yes	—
213	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
214	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
215	C₄×C₈	C₂²×D₅₃	1	2	yes	—
216	C₂×Q₈	—	53	4	no	—
217	C₂×Q₈	—	106	2	no	—
218	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
219	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
220	C₂²×C₄	#11719945	1	1	yes	characteristic
221	Q₁₆	—	106	4	no	—
222	Q₁₆	—	212	2	no	—
223	M₁₆	—	2	2	no	—
224	M₁₆	Dic₂₁₂	1	4	yes	complemented
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	4	1	no	—
228	C₂×C₈	—	106	2	no	—
229	C₂×C₈	—	106	2	no	—
230	C₂×C₈	—	212	4	no	—
231	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
232	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
233	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
234	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
235	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
236	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
237	C₄⋊C₄	—	53	4	no	—
238	C₄⋊C₄	—	106	2	no	—
239	C₄⋊C₄	—	106	2	no	—
240	C₄⋊C₄	—	106	4	no	—
241	C₄⋊C₄	—	106	2	no	—
242	C₄⋊C₄	—	106	4	no	—
243	C₂²⋊C₄	—	106	4	no	—
244	C₄²	—	106	4	no	—
245	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
246	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
247	C₄²	#11719945	1	1	yes	characteristic
248	C₄²	C₂³×D₅₃	1	1	yes	characteristic
249	C₂³	C₄×D₂₁₂	1	1	yes	characteristic
250	Q₈	—	106	4	no	—
251	Q₈	—	106	4	no	—
252	Q₈	—	424	2	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	4	1	no	—
256	C₂×C₄	—	106	4	no	—
257	C₂×C₄	—	106	4	no	—
258	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
259	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
260	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
261	C₂×C₄	C₂×Dic₂₁₂	1	2	yes	—
262	C₂×C₄	#13133239	1	2	yes	—
263	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
264	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
265	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
266	C₈	—	2	1	no	—
267	C₈	—	2	1	no	—
268	C₈	—	2	1	no	—
269	C₈	—	2	2	no	—
270	C₈	—	4	2	no	—
271	C₈	—	212	2	no	—
272	C₈	—	212	2	no	—
273	C₈	#13133225	1	2	yes	—
274	C₈	C₂×Dic₂₁₂	1	4	yes	—
275	C₂²	—	2	1	no	—
276	C₂²	—	2	1	no	—
277	C₂²	C₄×Dic₂₁₂	1	2	yes	—
278	C₂²	C₂×C₄×D₂₁₂	1	1	yes	characteristic
279	C₄	—	2	1	no	—
280	C₄	—	2	1	no	—
281	C₄	—	2	1	no	—
282	C₄	—	212	4	no	—
283	C₄	—	212	4	no	—
284	C₄	C₄×Dic₂₁₂	1	2	yes	—
285	C₄	C₂×#13133225	1	1	yes	characteristic
286	C₄	#16811389	1	1	yes	characteristic
287	C₄	C₂²×Dic₂₁₂	1	1	yes	characteristic
288	C₄	C₂×#13133239	1	1	yes	characteristic
289	C₂	—	2	2	no	—
290	C₂	#20283461	1	1	yes	characteristic
291	C₂	C₂×C₄×Dic₂₁₂	1	1	yes	characteristic
292	C₂	#20283633	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₈×M₁₆×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂³×C₅₃	C₄×D₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	C₈×M₁₆	D₅₃
hypocenter	C₅₃	#18068
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
3. #20273613	C₂	semidirect
4. #20273567	C₂	semidirect
5. C₈×Dic₂₁₂	C₂	semidirect
75. M₁₆×C₅₃	Q₁₆	semidirect
179. C₅₃	#18068	semidirect
224. M₁₆	Dic₂₁₂	semidirect

Maximal subgroups

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

subgroup	index
#20285301	2
#20283662	2
#20273613	2
#20273567	2
C₈×Dic₂₁₂	2
C₂×#16810376	2
#20273119	2
C₈×M₁₆×C₅₃	2
#18068	53