Subgroups — #27388491

Order 13568.

746 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	#20283359	C₂	1	2	yes	maximal, complemented
2	#20275609	C₂	1	2	yes	maximal
3	#20275469	C₂	1	2	yes	maximal, complemented
4	#20275428	C₂	1	4	yes	maximal, complemented
5	C₄×#13133153	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#13133144	C₂	1	2	yes	maximal, complemented
7	C₄×#13133127	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#135×C₅₃	C₂	1	1	yes	maximal, characteristic
9	#16810684	—	2	4	no	—
10	#16810676	—	2	4	no	—
11	C₂×#13133153	C₂²	1	2	yes	—
12	#16810650	C₂²	1	4	yes	—
13	C₂×#13133144	C₂²	1	4	yes	—
14	#16810590	C₂²	1	4	yes	—
15	C₄×#11719917	C₂²	1	2	yes	—
16	C₂×#13133127	C₂²	1	2	yes	—
17	C₂×#13133127	C₂²	1	2	yes	—
18	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
19	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
20	#16810388	C₂²	1	2	yes	—
21	#16810382	C₂²	1	4	yes	—
22	C₂×#135×C₅₃	C₂²	1	1	yes	characteristic
23	C₄×#41×C₅₃	C₂²	1	2	yes	—
24	C₄×Q₁₆×C₅₃	C₄	1	2	yes	complemented
25	C₄×SD₁₆×C₅₃	C₄	1	4	yes	complemented
26	C₄×D₈×C₅₃	C₄	1	2	yes	complemented
27	C₅₃×#427	C₂²	1	1	yes	characteristic
28	C₅₃×#416	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
30	C₂×#11719917	C₂³	1	4	yes	—
31	C₂²×#88322	C₂³	1	2	yes	—
32	C₂×C₄×Dic₅₃	—	2	2	no	—
33	C₂×C₄×Dic₅₃	C₂³	1	2	yes	—
34	#13133153	—	2	4	no	—
35	C₄⋊#88322	—	2	8	no	—
36	#13133144	—	2	8	no	—
37	C₂²⋊#88322	—	2	8	no	—
38	#13133128	—	2	2	no	—
39	#13133127	—	2	4	no	—
40	#13133127	—	2	4	no	—
41	C₄×#88322	—	2	2	no	—
42	C₄×#88322	D₄	1	2	yes	—
43	C₄×#88322	D₄	1	2	yes	—
44	C₂×#41×C₅₃	C₂³	1	2	yes	—
45	#135×C₅₃	—	2	4	no	—
46	C₂×Q₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
47	C₂×SD₁₆×C₅₃	C₂×C₄	1	2	yes	—
48	C₂×D₈×C₅₃	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
51	C₄×Q₈×C₅₃	C₂×C₄	1	4	yes	—
52	C₄×D₄×C₅₃	—	2	4	no	—
53	C₄×D₄×C₅₃	C₂×C₄	1	4	yes	—
54	#117×C₅₃	—	2	2	no	—
55	#117×C₅₃	C₂³	1	2	yes	—
56	C₂×C₄²×C₅₃	—	2	2	no	—
57	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
58	#107×C₅₃	C₂×C₄	1	2	yes	—
59	#106×C₅₃	C₂×C₄	1	2	yes	—
60	#103×C₅₃	C₂×C₄	1	4	yes	—
61	#102×C₅₃	C₂×C₄	1	4	yes	—
62	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
63	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
64	#41×C₅₃	—	2	4	no	—
65	#41×C₅₃	—	2	4	no	—
66	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
67	C₂×D₄×C₅₃	—	2	2	no	—
68	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₅₃	—	2	4	no	—
70	C₂²×C₄×C₅₃	—	2	2	no	—
71	C₂²×C₄×C₅₃	#41	1	2	yes	—
72	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
73	C₂²×Dic₅₃	—	2	2	no	—
74	Q₁₆×C₅₃	—	2	2	no	—
75	SD₁₆×C₅₃	—	2	4	no	—
76	D₈×C₅₃	—	2	2	no	—
77	C₂×C₈×C₅₃	—	2	2	no	—
78	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₅₃	#41	1	4	yes	—
82	C₄⋊C₄×C₅₃	—	2	4	no	—
83	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
84	C₂²⋊C₄×C₅₃	—	2	2	no	—
85	C₂²⋊C₄×C₅₃	—	2	4	no	—
86	C₄²×C₅₃	—	2	4	no	—
87	C₄²×C₅₃	—	2	2	no	—
88	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
89	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
91	C₄×Dic₅₃	—	2	2	no	—
92	C₄×Dic₅₃	—	4	4	no	—
93	C₄×Dic₅₃	#41	1	4	yes	—
94	#11719917	—	2	4	no	—
95	#11719917	—	2	4	no	—
96	C₂×#88322	—	2	2	no	—
97	C₂×#88322	—	2	2	no	—
98	C₂×#88322	—	2	4	no	—
99	C₂×#88322	C₂×D₄	1	4	yes	—
100	C₂³×C₅₃	—	2	2	no	—
101	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
102	Q₈×C₅₃	—	2	2	no	—
103	Q₈×C₅₃	—	2	2	no	—
104	D₄×C₅₃	—	2	2	no	—
105	D₄×C₅₃	—	2	2	no	—
106	D₄×C₅₃	—	4	4	no	—
107	C₂×C₄×C₅₃	—	2	4	no	—
108	C₂×C₄×C₅₃	—	2	2	no	—
109	C₂×C₄×C₅₃	—	2	4	no	—
110	C₂×C₄×C₅₃	—	4	4	no	—
111	C₂×C₄×C₅₃	—	4	8	no	—
112	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
114	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
115	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
116	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
117	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
118	C₂×Dic₅₃	—	2	4	no	—
119	C₂×Dic₅₃	—	4	4	no	—
120	C₈×C₅₃	—	2	2	no	—
121	C₈×C₅₃	#117	1	2	yes	—
122	C₈×C₅₃	#117	1	2	yes	—
123	#88322	—	2	4	no	—
124	#88322	—	4	4	no	—
125	#21293	—	53	1	no	maximal, Sylow
126	C₂²×C₅₃	—	2	2	no	—
127	C₂²×C₅₃	—	4	4	no	—
128	C₂²×C₅₃	—	4	2	no	—
129	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
130	C₂²×C₅₃	#529	1	2	yes	—
131	C₄×C₅₃	—	2	2	no	—
132	C₄×C₅₃	—	4	4	no	—
133	C₄×C₅₃	—	4	2	no	—
134	C₄×C₅₃	#444	1	4	yes	—
135	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
136	C₄×C₅₃	C₄×#41	1	1	yes	characteristic
137	C₄×C₅₃	#532	1	2	yes	—
138	Dic₅₃	—	4	4	no	—
139	C₄×#135	D₅₃	1	1	yes	characteristic
140	#2922	—	53	2	no	—
141	#1975	—	53	2	no	—
142	#1946	—	53	2	no	—
143	#1938	—	53	4	no	—
144	C₄×#108	—	53	1	no	—
145	C₄×#104	—	53	2	no	—
146	C₄×#97	—	53	1	no	—
147	C₂×C₅₃	—	2	2	no	—
148	C₂×C₅₃	—	8	2	no	—
149	C₂×C₅₃	#2293	1	1	yes	characteristic
150	C₂×C₅₃	C₂×#444	1	2	yes	—
151	C₂×#135	C₂×D₅₃	1	1	yes	characteristic
152	C₄×#41	C₂×D₅₃	1	2	yes	—
153	C₄×Q₁₆	Dic₅₃	1	2	yes	complemented
154	C₄×SD₁₆	Dic₅₃	1	4	yes	complemented
155	C₄×D₈	Dic₅₃	1	2	yes	complemented
156	#433	—	106	4	no	—
157	#431	—	106	4	no	—
158	C₂×#108	—	53	2	no	—
159	#427	C₂×D₅₃	1	1	yes	characteristic
160	#424	—	53	4	no	—
161	C₂×#104	—	53	4	no	—
162	#416	C₂×D₅₃	1	2	yes	—
163	#408	—	53	4	no	—
164	C₄×M₁₆	—	53	2	no	—
165	C₂×#97	—	53	2	no	—
166	C₂×#97	—	53	2	no	—
167	C₂×C₄×C₈	—	53	1	no	—
168	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
169	C₄³	—	53	1	no	—
170	#341	—	53	2	no	—
171	#339	—	53	4	no	—
172	C₅₃	#21293	1	1	yes	characteristic, complemented, Sylow
173	C₂×#41	C₂²×D₅₃	1	2	yes	—
174	#135	—	2	4	no	—
175	C₂×Q₁₆	C₂×Dic₅₃	1	1	yes	characteristic
176	C₂×SD₁₆	C₂×Dic₅₃	1	2	yes	—
177	C₂×D₈	C₂×Dic₅₃	1	1	yes	characteristic
178	C₂×M₁₆	—	53	4	no	—
179	C₂²×C₈	—	53	2	no	—
180	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
181	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
182	C₄×Q₈	C₂×Dic₅₃	1	4	yes	—
183	C₄×D₄	—	2	4	no	—
184	C₄×D₄	C₂×Dic₅₃	1	4	yes	—
185	#117	—	2	2	no	—
186	#117	C₂²×D₅₃	1	2	yes	—
187	C₂×C₄²	—	2	2	no	—
188	C₂×C₄²	—	53	2	no	—
189	C₂×C₄²	—	106	2	no	—
190	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
191	#108	—	106	4	no	—
192	#107	C₂×Dic₅₃	1	2	yes	—
193	#106	C₂×Dic₅₃	1	2	yes	—
194	C₄⋊C₈	—	106	8	no	—
195	#104	—	106	8	no	—
196	#103	C₂×Dic₅₃	1	4	yes	—
197	#102	C₂×Dic₅₃	1	4	yes	—
198	C₂²⋊C₈	—	106	8	no	—
199	#97	—	106	4	no	—
200	#97	—	106	4	no	—
201	#97	—	106	2	no	—
202	C₄×C₈	—	53	2	no	—
203	C₄×C₈	—	53	2	no	—
204	C₄×C₈	—	106	2	no	—
205	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
206	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
207	#41	—	2	4	no	—
208	#41	—	2	4	no	—
209	C₂×Q₈	C₂²×Dic₅₃	1	2	yes	—
210	C₂×D₄	—	2	2	no	—
211	C₂×D₄	C₂²×Dic₅₃	1	2	yes	—
212	C₂²×C₄	—	2	4	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	106	2	no	—
215	C₂²×C₄	#11719947	1	2	yes	—
216	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
217	Q₁₆	—	2	2	no	—
218	SD₁₆	—	2	4	no	—
219	D₈	—	2	2	no	—
220	M₁₆	—	106	4	no	—
221	M₁₆	—	106	4	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	53	4	no	—
224	C₂×C₈	—	106	2	no	—
225	C₂×C₈	—	106	2	no	—
226	C₂×C₈	—	106	4	no	—
227	C₂×C₈	#11719945	1	4	yes	—
228	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
229	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
230	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
231	C₄⋊C₄	—	2	4	no	—
232	C₄⋊C₄	C₂²×Dic₅₃	1	4	yes	—
233	C₂²⋊C₄	—	2	2	no	—
234	C₂²⋊C₄	—	2	4	no	—
235	C₄²	—	2	4	no	—
236	C₄²	—	2	2	no	—
237	C₄²	—	53	4	no	—
238	C₄²	—	106	2	no	—
239	C₄²	—	212	4	no	—
240	C₄²	D₄×D₅₃	1	1	yes	characteristic
241	C₄²	D₄×D₅₃	1	1	yes	characteristic
242	C₄²	C₂²×Dic₅₃	1	2	yes	—
243	C₂³	—	2	2	no	—
244	C₂³	C₂×#11719947	1	1	yes	characteristic
245	Q₈	—	2	2	no	—
246	Q₈	—	2	2	no	—
247	D₄	—	2	2	no	—
248	D₄	—	2	2	no	—
249	D₄	—	4	4	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	4	8	no	—
255	C₂×C₄	—	106	4	no	—
256	C₂×C₄	—	212	4	no	—
257	C₂×C₄	D₄×Dic₅₃	1	2	yes	—
258	C₂×C₄	D₄×Dic₅₃	1	2	yes	—
259	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
260	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
261	C₂×C₄	#41×D₅₃	1	2	yes	—
262	C₂×C₄	C₂×#11719945	1	2	yes	—
263	C₈	—	2	2	no	—
264	C₈	—	106	4	no	—
265	C₈	—	212	4	no	—
266	C₈	#13133176	1	2	yes	—
267	C₈	#13133176	1	2	yes	—
268	C₂²	—	2	2	no	—
269	C₂²	—	4	2	no	—
270	C₂²	—	4	4	no	—
271	C₂²	C₂×D₄×Dic₅₃	1	1	yes	characteristic
272	C₂²	#16811155	1	2	yes	—
273	C₄	—	2	2	no	—
274	C₄	—	4	2	no	—
275	C₄	—	4	4	no	—
276	C₄	—	212	4	no	—
277	C₄	#16810745	1	4	yes	—
278	C₄	C₂×#13133176	1	1	yes	characteristic
279	C₄	#41×Dic₅₃	1	1	yes	characteristic
280	C₄	#16811170	1	2	yes	—
281	C₂	—	2	2	no	—
282	C₂	—	8	2	no	—
283	C₂	#20277180	1	1	yes	characteristic
284	C₂	C₂×#16810745	1	2	yes	—

	subgroup	G/H
center	C₄²	D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₄×#135×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂×#135×C₅₃	C₂²
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₄×#135	D₅₃
hypocenter	C₅₃	#21293
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₄×#135

Complements

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

N	H ≅ G/N	type
1. #20283359	C₂	semidirect
3. #20275469	C₂	semidirect
4. #20275428	C₂	semidirect
5. C₄×#13133153	C₂	semidirect
6. C₄×#13133144	C₂	semidirect
7. C₄×#13133127	C₂	semidirect
24. C₄×Q₁₆×C₅₃	C₄	semidirect
25. C₄×SD₁₆×C₅₃	C₄	semidirect
26. C₄×D₈×C₅₃	C₄	semidirect
153. C₄×Q₁₆	Dic₅₃	semidirect
154. C₄×SD₁₆	Dic₅₃	semidirect
155. C₄×D₈	Dic₅₃	semidirect
172. C₅₃	#21293	semidirect

Maximal subgroups

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

subgroup	index
#20283359	2
#20275609	2
#20275469	2
#20275428	2
C₄×#13133153	2
C₄×#13133144	2
C₄×#13133127	2
C₄×#135×C₅₃	2
#21293	53