Subgroups — #27384714

Order 13568.

718 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	#20283352	C₂	1	2	yes	maximal
2	C₂×#16810577	C₂	1	1	yes	maximal, characteristic
3	#20274887	C₂	1	1	yes	maximal, characteristic, complemented
4	#20274855	C₂	1	2	yes	maximal, complemented
5	#20274754	C₂	1	2	yes	maximal
6	#20274750	C₂	1	2	yes	maximal
7	#20274623	C₂	1	2	yes	maximal, complemented
8	#108×Dic₅₃	C₂	1	2	yes	maximal, complemented
9	C₅₃×#1848	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#11719912	C₂²	1	1	yes	characteristic
11	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
12	C₂×#13133174	C₂²	1	1	yes	characteristic
13	#16810683	—	2	8	no	—
14	C₂×#13133153	C₂²	1	2	yes	—
15	C₂×#13133153	C₂²	1	2	yes	—
16	#16810647	C₂²	1	2	yes	—
17	#16810586	C₂²	1	4	yes	—
18	#16810577	C₄	1	8	yes	—
19	M₁₆×Dic₅₃	C₂²	1	4	yes	—
20	C₂×#13133128	C₂²	1	1	yes	characteristic
21	C₂×#13133127	C₂²	1	1	yes	characteristic
22	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
23	#16810387	C₂²	1	4	yes	—
24	#16810387	C₂²	1	4	yes	—
25	C₂×#108×C₅₃	C₂²	1	2	yes	—
26	C₂×#108×C₅₃	C₂²	1	2	yes	—
27	C₅₃×#423	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
29	C₂²×C₄×D₅₃	C₂³	1	1	yes	characteristic
30	C₂×#11719917	C₂³	1	4	yes	—
31	C₂×#11719912	—	2	2	no	—
32	C₂×#11719912	—	2	2	no	—
33	C₂×#11719912	C₂×C₄	1	4	yes	—
34	C₂²×#88322	C₂³	1	1	yes	characteristic
35	C₂²×#88322	C₂³	1	1	yes	characteristic
36	C₂×C₈×D₅₃	—	2	2	no	—
37	C₂×C₈×D₅₃	—	2	2	no	—
38	C₂×C₈×D₅₃	C₂×C₄	1	4	yes	—
39	#13133174	C₂×C₄	1	4	yes	—
40	#13133174	D₄	1	4	yes	—
41	C₂×#11719919	Q₈	1	2	yes	—
42	C₂×#11719921	Q₈	1	2	yes	—
43	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
44	#13133153	—	2	4	no	—
45	#13133153	—	2	4	no	—
46	C₄⋊#88322	—	2	4	no	—
47	#13133129	—	2	8	no	—
48	#13133128	C₂×C₄	1	4	yes	—
49	#13133127	—	2	4	no	—
50	#13133127	C₂×C₄	1	4	yes	—
51	C₈×Dic₅₃	—	2	4	no	—
52	C₈×Dic₅₃	C₂×C₄	1	4	yes	—
53	C₂×M₁₆×C₅₃	C₂³	1	4	yes	—
54	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
55	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
57	#108×C₅₃	—	2	4	no	—
58	#108×C₅₃	—	2	4	no	—
59	C₄⋊C₈×C₅₃	—	2	4	no	—
60	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
61	C₂³×D₅₃	C₂×Q₈	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
63	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
64	C₂²×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
65	C₂²×Dic₅₃	C₂×Q₈	1	2	yes	—
66	C₂×C₄×D₅₃	—	2	1	no	—
67	C₂×C₄×D₅₃	—	2	1	no	—
68	C₂×C₄×D₅₃	—	2	1	no	—
69	C₂×C₄×D₅₃	—	2	1	no	—
70	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
71	C₂×C₄×D₅₃	C₂×D₄	1	2	yes	—
72	M₁₆×C₅₃	—	2	4	no	—
73	M₁₆×C₅₃	—	2	4	no	—
74	C₂×C₈×C₅₃	—	2	4	no	—
75	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
77	C₂×C₈×C₅₃	#41	1	2	yes	—
78	C₂×C₈×C₅₃	#41	1	2	yes	—
79	C₂×C₈×C₅₃	#41	1	2	yes	—
80	C₂×C₈×C₅₃	#41	1	2	yes	—
81	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
82	C₄²×C₅₃	C₂×D₄	1	2	yes	—
83	#11719921	C₄⋊C₄	1	8	yes	—
84	#11719919	C₄⋊C₄	1	8	yes	—
85	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
86	C₄×Dic₅₃	C₂×D₄	1	2	yes	—
87	#11719917	—	2	4	no	—
88	#11719917	—	2	4	no	—
89	C₂×#88322	—	2	1	no	—
90	C₂×#88322	—	2	1	no	—
91	C₂×#88322	—	2	1	no	—
92	C₂×#88322	—	2	1	no	—
93	C₂×#88322	—	2	4	no	—
94	C₂×#88322	C₂²×C₄	1	2	yes	—
95	C₂×#88322	C₂²×C₄	1	2	yes	—
96	#11719912	—	2	8	no	—
97	C₈×D₅₃	—	2	8	no	—
98	C₂³×C₅₃	C₂²×Q₈	1	1	yes	characteristic
99	C₂²×D₅₃	—	2	1	no	—
100	C₂²×D₅₃	—	2	1	no	—
101	C₂²×D₅₃	—	4	2	no	—
102	C₂²×D₅₃	C₂×C₄⋊C₄	1	2	yes	—
103	C₂×C₄×C₅₃	—	4	2	no	—
104	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
105	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
106	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
107	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
108	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
109	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
110	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
111	C₂×Dic₅₃	—	2	1	no	—
112	C₂×Dic₅₃	—	2	1	no	—
113	C₂×Dic₅₃	—	4	2	no	—
114	C₂×Dic₅₃	C₂×C₄⋊C₄	1	2	yes	—
115	C₂×Dic₅₃	C₂×C₄⋊C₄	1	4	yes	—
116	C₄×D₅₃	—	2	2	no	—
117	C₄×D₅₃	—	2	2	no	—
118	C₄×D₅₃	—	2	2	no	—
119	C₄×D₅₃	—	2	2	no	—
120	C₈×C₅₃	—	4	4	no	—
121	C₈×C₅₃	#117	1	4	yes	—
122	C₈×C₅₃	#117	1	4	yes	—
123	#88322	—	2	2	no	—
124	#88322	—	2	2	no	—
125	#88322	—	4	4	no	—
126	#20828	—	53	1	no	maximal, Sylow
127	C₂²×C₅₃	—	2	2	no	—
128	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
129	C₂²×C₅₃	#533	1	1	yes	characteristic
130	C₂²×C₅₃	#533	1	1	yes	characteristic
131	C₂×D₅₃	—	2	4	no	—
132	C₂×D₅₃	—	2	2	no	—
133	C₂×D₅₃	—	4	2	no	—
134	C₂×D₅₃	—	4	2	no	—
135	C₄×C₅₃	—	2	4	no	—
136	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
137	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
138	C₄×C₅₃	#532	1	1	yes	characteristic
139	C₄×C₅₃	#532	1	1	yes	characteristic
140	Dic₅₃	—	2	4	no	—
141	Dic₅₃	—	2	2	no	—
142	#2921	—	53	2	no	—
143	C₂×#405	—	53	1	no	—
144	#1848	—	53	1	no	—
145	#1848	D₅₃	1	1	yes	characteristic, complemented
146	#1838	—	53	2	no	—
147	#1811	—	53	2	no	—
148	#1810	—	53	2	no	—
149	#1778	—	53	2	no	—
150	C₄×#108	—	53	2	no	—
151	C₂×C₅₃	—	2	1	no	—
152	C₂×C₅₃	—	2	1	no	—
153	C₂×C₅₃	#2285	1	1	yes	characteristic
154	C₂×C₅₃	#2916	1	2	yes	—
155	D₅₃	—	4	2	no	—
156	C₂²×M₁₆	—	53	1	no	—
157	C₂³×C₈	—	53	1	no	—
158	C₂×#117	—	53	1	no	—
159	#433	—	106	8	no	—
160	C₂×#108	—	53	2	no	—
161	C₂×#108	—	53	2	no	—
162	C₂×#108	C₂×D₅₃	1	2	yes	—
163	C₂×#108	C₂×D₅₃	1	2	yes	—
164	#423	—	53	2	no	—
165	#423	C₂×D₅₃	1	2	yes	—
166	#407	—	53	4	no	—
167	#405	—	53	8	no	—
168	C₄×M₁₆	—	53	4	no	—
169	C₂×#97	—	53	1	no	—
170	C₂×#97	—	53	1	no	—
171	C₂×C₄×C₈	—	53	1	no	—
172	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
173	#341	—	53	4	no	—
174	#341	—	53	4	no	—
175	C₅₃	#20828	1	1	yes	characteristic, complemented, Sylow
176	C₂³×C₄	—	53	1	no	—
177	C₂×M₁₆	—	53	4	no	—
178	C₂×M₁₆	—	53	4	no	—
179	C₂×M₁₆	—	106	2	no	—
180	C₂×M₁₆	—	106	2	no	—
181	C₂×M₁₆	C₂²×D₅₃	1	4	yes	—
182	C₂²×C₈	—	53	1	no	—
183	C₂²×C₈	—	53	1	no	—
184	C₂²×C₈	—	53	4	no	—
185	C₂²×C₈	—	106	2	no	—
186	C₂²×C₈	—	106	2	no	—
187	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
188	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
189	#117	—	53	4	no	—
190	#117	—	53	4	no	—
191	C₂×C₄⋊C₄	—	53	2	no	—
192	C₂×C₂²⋊C₄	—	53	2	no	—
193	C₂×C₄²	—	53	1	no	—
194	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
195	#108	—	2	4	no	—
196	#108	—	2	4	no	—
197	#108	—	106	4	no	—
198	#108	—	106	4	no	—
199	C₄⋊C₈	—	2	4	no	—
200	C₄⋊C₈	—	106	4	no	—
201	C₂²⋊C₈	—	106	8	no	—
202	#97	—	53	4	no	—
203	#97	—	53	4	no	—
204	#97	—	106	4	no	—
205	C₄×C₈	—	53	4	no	—
206	C₄×C₈	—	106	4	no	—
207	C₄×C₈	C₄×D₅₃	1	4	yes	—
208	C₂⁴	—	53	1	no	—
209	C₂²×C₄	—	53	1	no	—
210	C₂²×C₄	—	53	2	no	—
211	C₂²×C₄	—	53	2	no	—
212	C₂²×C₄	—	53	2	no	—
213	C₂²×C₄	—	106	1	no	—
214	C₂²×C₄	—	106	1	no	—
215	C₂²×C₄	—	106	1	no	—
216	C₂²×C₄	—	106	1	no	—
217	C₂²×C₄	Q₈×D₅₃	1	2	yes	—
218	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
219	M₁₆	—	2	4	no	—
220	M₁₆	—	2	4	no	—
221	M₁₆	—	106	8	no	—
222	M₁₆	—	106	4	no	—
223	M₁₆	—	106	4	no	—
224	C₂×C₈	—	2	4	no	—
225	C₂×C₈	—	53	2	no	—
226	C₂×C₈	—	53	2	no	—
227	C₂×C₈	—	106	1	no	—
228	C₂×C₈	—	106	1	no	—
229	C₂×C₈	—	106	1	no	—
230	C₂×C₈	—	106	1	no	—
231	C₂×C₈	—	106	8	no	—
232	C₂×C₈	—	106	4	no	—
233	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
234	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
235	C₂×C₈	#11719947	1	2	yes	—
236	C₂×C₈	#11719947	1	2	yes	—
237	C₂×C₈	#11719949	1	2	yes	—
238	C₂×C₈	#11719949	1	2	yes	—
239	C₄⋊C₄	—	53	8	no	—
240	C₂²⋊C₄	—	53	8	no	—
241	C₄²	—	53	2	no	—
242	C₄²	—	53	2	no	—
243	C₄²	C₂×C₄×D₅₃	1	2	yes	—
244	C₄²	D₄×D₅₃	1	2	yes	—
245	C₂³	—	53	2	no	—
246	C₂³	—	106	1	no	—
247	C₂³	—	106	1	no	—
248	C₂³	—	212	2	no	—
249	C₂³	C₂×Q₈×D₅₃	1	1	yes	characteristic
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	53	4	no	—
252	C₂×C₄	—	53	2	no	—
253	C₂×C₄	—	106	1	no	—
254	C₂×C₄	—	106	1	no	—
255	C₂×C₄	—	106	2	no	—
256	C₂×C₄	—	106	2	no	—
257	C₂×C₄	—	106	2	no	—
258	C₂×C₄	—	106	2	no	—
259	C₂×C₄	—	212	2	no	—
260	C₂×C₄	C₄⋊C₄×D₅₃	1	4	yes	—
261	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
262	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
263	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
264	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
265	C₂×C₄	C₂×#11719949	1	1	yes	characteristic
266	C₂×C₄	C₂×#11719949	1	1	yes	characteristic
267	C₈	—	4	4	no	—
268	C₈	—	106	2	no	—
269	C₈	—	106	2	no	—
270	C₈	—	212	4	no	—
271	C₈	#13133175	1	4	yes	—
272	C₈	#13133175	1	4	yes	—
273	C₂²	—	2	2	no	—
274	C₂²	—	106	2	no	—
275	C₂²	—	106	4	no	—
276	C₂²	—	212	2	no	—
277	C₂²	—	212	2	no	—
278	C₂²	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
279	C₂²	#16811178	1	1	yes	characteristic
280	C₂²	#16811178	1	1	yes	characteristic
281	C₄	—	2	4	no	—
282	C₄	—	106	2	no	—
283	C₄	—	106	4	no	—
284	C₄	C₂×#13133175	1	1	yes	characteristic
285	C₄	C₂×#13133175	1	1	yes	characteristic
286	C₄	#16811171	1	1	yes	characteristic
287	C₄	#16811174	1	1	yes	characteristic
288	C₂	—	2	1	no	—
289	C₂	—	2	1	no	—
290	C₂	—	212	2	no	—
291	C₂	#20277120	1	1	yes	characteristic
292	C₂	#20283323	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1848	C₂
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂³×D₅₃	C₂×Q₈
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1848	D₅₃
hypocenter	C₅₃	#20828
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1848

Complements

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

N	H ≅ G/N	type
3. #20274887	C₂	semidirect
4. #20274855	C₂	semidirect
7. #20274623	C₂	semidirect
8. #108×Dic₅₃	C₂	semidirect
9. C₅₃×#1848	C₂	semidirect
145. #1848	D₅₃	semidirect
175. C₅₃	#20828	semidirect

Maximal subgroups

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

subgroup	index
#20283352	2
C₂×#16810577	2
#20274887	2
#20274855	2
#20274754	2
#20274750	2
#20274623	2
#108×Dic₅₃	2
C₅₃×#1848	2
#20828	53