Subgroups — #27354935

Order 13568.

624 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283791	C₂	1	2	yes	maximal, complemented
2	#20273580	C₂	1	4	yes	maximal, complemented
3	#20273545	C₂	1	4	yes	maximal, complemented
4	#20273478	C₂	1	1	yes	maximal, characteristic, complemented
5	#20273184	C₂	1	2	yes	maximal, complemented
6	#20273007	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#3169	C₂	1	1	yes	maximal, characteristic
8	#16810701	C₂²	1	4	yes	—
9	#16810695	—	2	4	no	—
10	D₄×#88322	C₂²	1	4	yes	—
11	#16810680	C₂²	1	2	yes	—
12	#16810676	C₂²	1	1	yes	characteristic
13	#16810676	C₂²	1	1	yes	characteristic
14	#16810590	—	2	4	no	—
15	#16810376	C₂²	1	2	yes	—
16	#16810374	C₂²	1	2	yes	—
17	D₄⋊#88322	C₂²	1	8	yes	—
18	#16810345	C₂²	1	2	yes	—
19	#16810345	C₂²	1	2	yes	—
20	C₅₃×#532	C₂²	1	2	yes	—
21	C₅₃×#500	C₄	1	1	yes	characteristic
22	C₅₃×#493	C₄	1	1	yes	characteristic
23	C₅₃×#486	C₄	1	2	yes	—
24	C₅₃×#466	C₄	1	2	yes	—
25	C₅₃×#466	C₄	1	2	yes	—
26	C₄×D₈×C₅₃	C₂²	1	2	yes	—
27	C₄×D₈×C₅₃	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
29	C₂×#11719917	—	2	4	no	—
30	C₂²×#88322	—	2	4	no	—
31	C₄⋊#88322	—	2	1	no	—
32	C₄⋊#88322	D₄	1	2	yes	—
33	C₄⋊#88322	C₂³	1	4	yes	—
34	C₂²⋊#88322	—	2	4	no	—
35	C₂²⋊#88322	—	2	4	no	—
36	C₂²⋊#88322	—	2	2	no	—
37	C₂²⋊#88322	—	2	1	no	—
38	C₂²⋊#88322	D₄	1	2	yes	—
39	#13133128	—	2	2	no	—
40	C₄×#88322	C₂³	1	2	yes	—
41	C₄×#88322	C₂³	1	2	yes	—
42	C₂×#41×C₅₃	—	2	2	no	—
43	C₂×D₈×C₅₃	C₂×C₄	1	2	yes	—
44	C₂×D₈×C₅₃	C₂×C₄	1	2	yes	—
45	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
47	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
48	#127×C₅₃	C₂×C₄	1	2	yes	—
49	#124×C₅₃	—	2	2	no	—
50	#121×C₅₃	—	2	2	no	—
51	#121×C₅₃	C₂×C₄	1	4	yes	—
52	C₄×D₄×C₅₃	—	2	2	no	—
53	C₄×D₄×C₅₃	C₂³	1	4	yes	—
54	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
55	#107×C₅₃	C₂×C₄	1	2	yes	—
56	#107×C₅₃	C₂×C₄	1	2	yes	—
57	#102×C₅₃	C₂×C₄	1	4	yes	—
58	#102×C₅₃	C₂×C₄	1	4	yes	—
59	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
60	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
61	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
62	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
63	#41×C₅₃	—	8	4	no	—
64	C₂×Q₈×C₅₃	—	2	2	no	—
65	C₂×D₄×C₅₃	—	2	4	no	—
66	C₂×D₄×C₅₃	—	2	2	no	—
67	C₂×D₄×C₅₃	—	2	2	no	—
68	C₂×D₄×C₅₃	C₂²×C₄	1	4	yes	—
69	C₂²×C₄×C₅₃	—	2	4	no	—
70	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
73	D₈×C₅₃	—	4	2	no	—
74	D₈×C₅₃	—	4	2	no	—
75	C₂×C₈×C₅₃	—	4	1	no	—
76	C₂×C₈×C₅₃	—	4	1	no	—
77	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
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	C₄⋊C₄×C₅₃	—	2	2	no	—
82	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
83	C₂²⋊C₄×C₅₃	—	2	4	no	—
84	C₂²⋊C₄×C₅₃	—	2	4	no	—
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	#11719917	—	8	4	no	—
90	C₂×#88322	—	2	4	no	—
91	C₂×#88322	—	2	2	no	—
92	C₂×#88322	—	8	4	no	—
93	C₂×#88322	C₂×D₄	1	4	yes	—
94	C₂³×C₅₃	—	2	4	no	—
95	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
96	Q₈×C₅₃	—	8	2	no	—
97	D₄×C₅₃	—	2	4	no	—
98	D₄×C₅₃	—	2	4	no	—
99	D₄×C₅₃	—	8	2	no	—
100	D₄×C₅₃	—	8	2	no	—
101	D₄×C₅₃	—	8	4	no	—
102	C₂×C₄×C₅₃	—	2	1	no	—
103	C₂×C₄×C₅₃	—	2	1	no	—
104	C₂×C₄×C₅₃	—	2	2	no	—
105	C₂×C₄×C₅₃	—	2	4	no	—
106	C₂×C₄×C₅₃	—	4	1	no	—
107	C₂×C₄×C₅₃	—	4	4	no	—
108	C₂×C₄×C₅₃	—	4	4	no	—
109	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
110	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
113	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
115	C₈×C₅₃	—	2	1	no	—
116	C₈×C₅₃	—	2	1	no	—
117	C₈×C₅₃	—	2	1	no	—
118	C₈×C₅₃	—	2	1	no	—
119	#88322	—	2	4	no	—
120	#88322	—	4	2	no	—
121	#88322	—	4	4	no	—
122	#17989	—	53	1	no	maximal, Sylow
123	C₂²×C₅₃	—	2	1	no	—
124	C₂²×C₅₃	—	2	1	no	—
125	C₂²×C₅₃	—	2	1	no	—
126	C₂²×C₅₃	—	4	4	no	—
127	C₂²×C₅₃	—	4	4	no	—
128	C₂²×C₅₃	—	4	4	no	—
129	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
130	C₄×C₅₃	—	2	1	no	—
131	C₄×C₅₃	—	2	1	no	—
132	C₄×C₅₃	—	2	1	no	—
133	C₄×C₅₃	—	2	1	no	—
134	C₄×C₅₃	—	4	4	no	—
135	C₄×C₅₃	#568	1	1	yes	characteristic
136	C₄×C₅₃	#568	1	1	yes	characteristic
137	C₄×C₅₃	#576	1	2	yes	—
138	#3169	D₅₃	1	1	yes	characteristic
139	#2981	—	53	2	no	—
140	#1586	—	53	4	no	—
141	#1581	—	53	4	no	—
142	#1568	—	53	1	no	—
143	#1496	—	53	2	no	—
144	#1452	—	53	1	no	—
145	C₂×C₅₃	—	4	1	no	—
146	C₂×C₅₃	—	8	4	no	—
147	C₂×C₅₃	#2932	1	1	yes	characteristic
148	C₂×C₅₃	#2948	1	1	yes	characteristic
149	C₂×C₅₃	#2957	1	1	yes	characteristic
150	#532	C₂×D₅₃	1	2	yes	—
151	#500	Dic₅₃	1	1	yes	characteristic
152	#493	Dic₅₃	1	1	yes	characteristic
153	#486	Dic₅₃	1	2	yes	—
154	#466	Dic₅₃	1	2	yes	—
155	#466	Dic₅₃	1	2	yes	—
156	C₄×D₈	C₂×D₅₃	1	2	yes	—
157	C₄×D₈	C₂×D₅₃	1	2	yes	—
158	#436	—	53	4	no	—
159	#435	—	106	4	no	—
160	C₈×D₄	—	53	4	no	—
161	#432	—	53	2	no	—
162	#431	—	53	1	no	—
163	#431	—	53	1	no	—
164	#408	—	106	4	no	—
165	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
166	#335	—	53	2	no	—
167	#334	—	53	2	no	—
168	D₄⋊C₈	—	53	8	no	—
169	#322	—	53	2	no	—
170	#322	—	53	2	no	—
171	C₅₃	#17989	1	1	yes	characteristic, complemented, Sylow
172	C₂×#41	—	2	2	no	—
173	C₂×D₈	C₂×Dic₅₃	1	2	yes	—
174	C₂×D₈	C₂×Dic₅₃	1	2	yes	—
175	C₂×M₁₆	—	106	4	no	—
176	C₂²×C₈	—	106	4	no	—
177	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
178	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
179	C₄⋊Q₈	C₂×Dic₅₃	1	2	yes	—
180	#127	C₂×Dic₅₃	1	2	yes	—
181	#124	—	2	2	no	—
182	#121	—	2	2	no	—
183	#121	C₂×Dic₅₃	1	4	yes	—
184	C₄×D₄	—	2	2	no	—
185	C₄×D₄	C₂²×D₅₃	1	4	yes	—
186	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
187	#107	C₂×Dic₅₃	1	2	yes	—
188	#107	C₂×Dic₅₃	1	2	yes	—
189	C₄⋊C₈	—	53	2	no	—
190	C₄⋊C₈	—	53	4	no	—
191	C₄⋊C₈	—	106	1	no	—
192	#102	C₂×Dic₅₃	1	4	yes	—
193	#102	C₂×Dic₅₃	1	4	yes	—
194	C₂²⋊C₈	—	53	2	no	—
195	C₂²⋊C₈	—	106	4	no	—
196	C₂²⋊C₈	—	106	4	no	—
197	C₂²⋊C₈	—	106	2	no	—
198	C₂²⋊C₈	—	106	1	no	—
199	#97	—	106	2	no	—
200	C₄×C₈	—	53	2	no	—
201	C₄×C₈	—	53	2	no	—
202	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
203	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
204	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
205	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
206	#41	—	8	4	no	—
207	C₂×Q₈	—	2	2	no	—
208	C₂×D₄	—	2	4	no	—
209	C₂×D₄	—	2	2	no	—
210	C₂×D₄	—	2	2	no	—
211	C₂×D₄	C₂²×Dic₅₃	1	4	yes	—
212	C₂²×C₄	—	2	4	no	—
213	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
214	C₂²×C₄	#11719947	1	1	yes	characteristic
215	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
216	D₈	—	4	2	no	—
217	D₈	—	4	2	no	—
218	M₁₆	—	424	4	no	—
219	C₂×C₈	—	4	1	no	—
220	C₂×C₈	—	4	1	no	—
221	C₂×C₈	—	53	4	no	—
222	C₂×C₈	—	106	4	no	—
223	C₂×C₈	—	106	2	no	—
224	C₂×C₈	—	424	4	no	—
225	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
226	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
227	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
228	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
229	C₄⋊C₄	—	2	2	no	—
230	C₄⋊C₄	C₂²×Dic₅₃	1	4	yes	—
231	C₂²⋊C₄	—	2	4	no	—
232	C₂²⋊C₄	—	2	4	no	—
233	C₄²	D₄×D₅₃	1	1	yes	characteristic
234	C₄²	#11719947	1	1	yes	characteristic
235	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
236	C₄²	C₂³×D₅₃	1	1	yes	characteristic
237	C₂³	—	2	4	no	—
238	C₂³	D₄×Dic₅₃	1	1	yes	characteristic
239	Q₈	—	8	2	no	—
240	D₄	—	2	4	no	—
241	D₄	—	2	4	no	—
242	D₄	—	8	2	no	—
243	D₄	—	8	2	no	—
244	D₄	—	8	4	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	4	1	no	—
250	C₂×C₄	—	4	4	no	—
251	C₂×C₄	—	4	4	no	—
252	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
253	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
254	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
255	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
256	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
257	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
258	C₈	—	2	1	no	—
259	C₈	—	2	1	no	—
260	C₈	—	2	1	no	—
261	C₈	—	2	1	no	—
262	C₈	—	106	4	no	—
263	C₈	—	212	2	no	—
264	C₈	—	212	4	no	—
265	C₂²	—	2	1	no	—
266	C₂²	—	2	1	no	—
267	C₂²	—	2	1	no	—
268	C₂²	—	4	4	no	—
269	C₂²	—	4	4	no	—
270	C₂²	—	4	4	no	—
271	C₂²	C₂×D₄×Dic₅₃	1	1	yes	characteristic
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	4	4	no	—
277	C₄	#16811390	1	1	yes	characteristic
278	C₄	#16811390	1	1	yes	characteristic
279	C₄	#16811424	1	2	yes	—
280	C₂	—	4	1	no	—
281	C₂	—	8	4	no	—
282	C₂	#20283433	1	1	yes	characteristic
283	C₂	#20283567	1	1	yes	characteristic
284	C₂	#20283632	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting, prime radical	C₅₃×#3169	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#3169	D₅₃
hypocenter	C₅₃	#17989
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3169

Complements

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

N	H ≅ G/N	type
1. #20283791	C₂	semidirect
2. #20273580	C₂	semidirect
3. #20273545	C₂	semidirect
4. #20273478	C₂	semidirect
5. #20273184	C₂	semidirect
6. #20273007	C₂	semidirect
171. C₅₃	#17989	semidirect

Maximal subgroups

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

subgroup	index
#20283791	2
#20273580	2
#20273545	2
#20273478	2
#20273184	2
#20273007	2
C₅₃×#3169	2
#17989	53