Subgroups — #27356085

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	#20285301	C₂	1	1	yes	maximal, characteristic
2	#20283816	C₂	1	2	yes	maximal
3	#20273613	C₂	1	2	yes	maximal, complemented
4	#20273588	C₂	1	4	yes	maximal, complemented
5	C₈×Dic₂₁₂	C₂	1	2	yes	maximal, complemented
6	#20273463	C₂	1	1	yes	maximal, characteristic
7	#20273137	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	#16810750	—	2	2	no	—
17	C₈×Dic₁₀₆	C₂²	1	4	yes	—
18	#16810747	—	2	2	no	—
19	C₄×Dic₂₁₂	C₂²	1	2	yes	—
20	C₄×Dic₂₁₂	C₂²	1	2	yes	—
21	#16810681	—	2	4	no	—
22	#16810676	C₂²	1	2	yes	—
23	#16810376	C₂²	1	1	yes	characteristic
24	#16810376	C₂²	1	1	yes	characteristic
25	#16810374	C₂²	1	2	yes	—
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	#13133217	C₂×C₄	1	2	yes	—
36	#13133217	C₂×C₄	1	2	yes	—
37	#13133209	—	2	2	no	—
38	#13133196	—	2	2	no	—
39	#13133196	C₂×C₄	1	4	yes	—
40	C₄×Dic₁₀₆	—	2	2	no	—
41	C₄×Dic₁₀₆	C₂³	1	4	yes	—
42	#13133176	—	2	2	no	—
43	#13133151	C₂×C₄	1	2	yes	—
44	#13133151	C₂×C₄	1	2	yes	—
45	C₄⋊#88322	C₂³	1	2	yes	—
46	C₄⋊#88322	C₂³	1	2	yes	—
47	#13133146	—	2	4	no	—
48	#13133146	—	2	4	no	—
49	#13133142	C₂×C₄	1	4	yes	—
50	#13133142	C₂×C₄	1	4	yes	—
51	C₂²⋊#88322	—	2	2	no	—
52	#13133127	—	2	4	no	—
53	C₄×#88322	—	2	2	no	—
54	C₈×Dic₅₃	—	2	4	no	—
55	C₂×M₁₆×C₅₃	—	2	1	no	—
56	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
57	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
59	C₄⋊C₈×C₅₃	D₄	1	1	yes	characteristic
60	C₄⋊C₈×C₅₃	D₄	1	1	yes	characteristic
61	C₂²⋊C₈×C₅₃	D₄	1	2	yes	—
62	#97×C₅₃	—	2	1	no	—
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	4	no	—
75	C₂×C₈×C₅₃	—	2	2	no	—
76	C₂×C₈×C₅₃	—	2	2	no	—
77	C₂×C₈×C₅₃	—	4	1	no	—
78	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
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	C₂×D₄	1	2	yes	—
83	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
84	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
85	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₅₃	#41	1	1	yes	characteristic
87	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
88	C₂²⋊Dic₅₃	—	2	4	no	—
89	C₄⋊Dic₅₃	—	2	2	no	—
90	C₄⋊Dic₅₃	—	2	2	no	—
91	C₄⋊Dic₅₃	—	2	2	no	—
92	C₄⋊Dic₅₃	C₂²×C₄	1	4	yes	—
93	#11719919	—	2	4	no	—
94	#11719919	—	2	4	no	—
95	C₄×Dic₅₃	—	2	4	no	—
96	C₂×#88322	—	2	2	no	—
97	C₂×#88322	—	2	2	no	—
98	Dic₂₁₂	—	2	4	no	—
99	Dic₂₁₂	—	4	2	no	—
100	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
101	C₂×C₄×C₅₃	—	2	1	no	—
102	C₂×C₄×C₅₃	—	2	1	no	—
103	C₂×C₄×C₅₃	—	2	1	no	—
104	C₂×C₄×C₅₃	—	2	1	no	—
105	C₂×C₄×C₅₃	—	2	2	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	C₂³×C₄	1	1	yes	characteristic
110	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
111	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
112	C₂×Dic₅₃	—	2	4	no	—
113	C₂×Dic₅₃	—	2	4	no	—
114	Dic₁₀₆	—	2	4	no	—
115	Dic₁₀₆	—	2	4	no	—
116	Dic₁₀₆	—	4	4	no	—
117	C₈×C₅₃	—	2	4	no	—
118	C₈×C₅₃	—	2	1	no	—
119	C₈×C₅₃	—	2	1	no	—
120	C₈×C₅₃	—	2	1	no	—
121	C₈×C₅₃	—	2	2	no	—
122	C₈×C₅₃	—	2	2	no	—
123	C₈×C₅₃	#131	1	2	yes	—
124	#88322	—	4	2	no	—
125	#88322	—	4	2	no	—
126	#18086	—	53	1	no	maximal, Sylow
127	C₂²×C₅₃	—	2	1	no	—
128	C₂²×C₅₃	—	2	1	no	—
129	C₂²×C₅₃	—	2	1	no	—
130	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
131	C₄×C₅₃	—	2	1	no	—
132	C₄×C₅₃	—	2	1	no	—
133	C₄×C₅₃	—	2	1	no	—
134	C₄×C₅₃	#443	1	2	yes	—
135	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
136	C₄×C₅₃	#568	1	1	yes	characteristic
137	C₄×C₅₃	#578	1	1	yes	characteristic
138	C₄×C₅₃	#578	1	1	yes	characteristic
139	Dic₅₃	—	4	4	no	—
140	Dic₅₃	—	4	4	no	—
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	D₅₃	1	1	yes	characteristic
149	C₂×C₅₃	—	4	1	no	—
150	C₂×C₅₃	#2935	1	1	yes	characteristic
151	C₂×C₅₃	#2950	1	1	yes	characteristic
152	C₂×C₅₃	C₂×#443	1	1	yes	characteristic
153	#533	—	53	2	no	—
154	#500	—	53	1	no	—
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	2	no	—
167	#431	C₂×D₅₃	1	1	yes	characteristic
168	C₄×M₁₆	C₂×D₅₃	1	1	yes	characteristic
169	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
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	C₂×D₅₃	1	2	yes	—
176	C₈²	C₂×D₅₃	1	2	yes	—
177	C₅₃	#18086	1	1	yes	characteristic, complemented, Sylow
178	C₂×Q₁₆	—	53	2	no	—
179	C₂×Q₁₆	—	53	2	no	—
180	C₂×M₁₆	—	2	1	no	—
181	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
182	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
183	C₄⋊Q₈	—	53	2	no	—
184	C₄⋊Q₈	—	53	2	no	—
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	C₂²×D₅₃	1	1	yes	characteristic
192	#107	—	53	2	no	—
193	#107	—	53	2	no	—
194	C₄⋊C₈	—	53	2	no	—
195	C₄⋊C₈	—	53	2	no	—
196	C₄⋊C₈	—	106	4	no	—
197	C₄⋊C₈	—	106	4	no	—
198	C₄⋊C₈	D₂₁₂	1	1	yes	characteristic
199	C₄⋊C₈	D₂₁₂	1	1	yes	characteristic
200	#103	—	53	4	no	—
201	#103	—	53	4	no	—
202	C₂²⋊C₈	—	106	2	no	—
203	C₂²⋊C₈	D₂₁₂	1	2	yes	—
204	#97	—	2	1	no	—
205	#97	—	106	4	no	—
206	C₄×C₈	—	106	4	no	—
207	C₄×C₈	—	106	2	no	—
208	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
209	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
210	C₄×C₈	C₂²×D₅₃	1	2	yes	—
211	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
212	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
213	C₄×C₈	C₂²×D₅₃	1	2	yes	—
214	C₂×Q₈	—	53	4	no	—
215	C₂×Q₈	—	106	2	no	—
216	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
217	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
218	C₂²×C₄	#11719945	1	1	yes	characteristic
219	Q₁₆	—	106	4	no	—
220	Q₁₆	—	212	2	no	—
221	M₁₆	—	2	4	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	4	1	no	—
225	C₂×C₈	—	106	2	no	—
226	C₂×C₈	—	106	2	no	—
227	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
228	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
229	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
230	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
231	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
232	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
233	C₄⋊C₄	—	53	4	no	—
234	C₄⋊C₄	—	106	2	no	—
235	C₄⋊C₄	—	106	2	no	—
236	C₄⋊C₄	—	106	4	no	—
237	C₄⋊C₄	—	106	2	no	—
238	C₄⋊C₄	—	106	4	no	—
239	C₂²⋊C₄	—	106	4	no	—
240	C₄²	—	106	4	no	—
241	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
242	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
243	C₄²	#11719945	1	1	yes	characteristic
244	C₄²	C₂³×D₅₃	1	1	yes	characteristic
245	C₂³	C₄×D₂₁₂	1	1	yes	characteristic
246	Q₈	—	106	4	no	—
247	Q₈	—	106	4	no	—
248	Q₈	—	212	4	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	106	4	no	—
255	C₂×C₄	—	106	4	no	—
256	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
257	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
258	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
259	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
260	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
261	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
262	C₈	—	2	4	no	—
263	C₈	—	2	1	no	—
264	C₈	—	2	1	no	—
265	C₈	—	2	1	no	—
266	C₈	—	2	2	no	—
267	C₈	—	2	2	no	—
268	C₈	—	212	2	no	—
269	C₈	—	212	2	no	—
270	C₈	#13133225	1	2	yes	—
271	C₂²	—	2	1	no	—
272	C₂²	—	2	1	no	—
273	C₂²	—	2	1	no	—
274	C₂²	C₂×C₄×D₂₁₂	1	1	yes	characteristic
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	212	4	no	—
279	C₄	—	212	4	no	—
280	C₄	#16810741	1	2	yes	—
281	C₄	C₂×#13133225	1	1	yes	characteristic
282	C₄	#16811389	1	1	yes	characteristic
283	C₄	#16811440	1	1	yes	characteristic
284	C₄	#16811440	1	1	yes	characteristic
285	C₂	—	4	1	no	—
286	C₂	#20283461	1	1	yes	characteristic
287	C₂	#20283584	1	1	yes	characteristic
288	C₂	C₂×#16810741	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	#18086
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
3. #20273613	C₂	semidirect
4. #20273588	C₂	semidirect
5. C₈×Dic₂₁₂	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
#20285301	2
#20283816	2
#20273613	2
#20273588	2
C₈×Dic₂₁₂	2
#20273463	2
#20273137	2
C₈×M₁₆×C₅₃	2
#18086	53