Subgroups — #27273621

Order 13568.

830 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810676	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×#88322	C₂	1	1	yes	maximal, characteristic
3	#20272556	C₂	1	4	yes	maximal, complemented
4	C₄×#117×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂³×#88322	—	2	1	no	—
6	#16810676	—	2	2	no	—
7	#16810676	C₄	1	4	yes	complemented
8	C₂×C₄⋊#88322	C₂²	1	2	yes	—
9	C₂×C₂²⋊#88322	—	2	1	no	—
10	C₂×C₄×#88322	—	2	1	no	—
11	C₂×C₄×#88322	—	2	2	no	—
12	C₂×C₄×#88322	—	2	1	no	—
13	C₂×C₄×#88322	C₄	1	4	yes	complemented
14	C₂×C₄×#88322	C₂²	1	2	yes	—
15	C₂×#117×C₅₃	C₄	1	2	yes	—
16	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
17	C₄×C₄⋊C₄×C₅₃	C₄	1	2	yes	—
18	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
19	C₄×C₂²⋊C₄×C₅₃	—	2	1	no	—
20	C₄×C₂²⋊C₄×C₅₃	—	2	1	no	—
21	C₅₃×#376	—	2	1	no	—
22	C₄³×C₅₃	—	2	1	no	—
23	C₂²×#88322	—	2	8	no	—
24	C₂²×#88322	—	2	4	no	—
25	C₂²×#88322	—	2	2	no	—
26	C₂²×#88322	—	2	2	no	—
27	C₄⋊#88322	—	2	2	no	—
28	C₄⋊#88322	C₂×C₄	1	4	yes	—
29	C₂²⋊#88322	—	2	4	no	—
30	C₄×#88322	—	2	4	no	—
31	C₄×#88322	—	2	4	no	—
32	C₄×#88322	—	2	2	no	—
33	C₄×#88322	C₂×C₄	1	4	yes	—
34	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
36	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
37	#117×C₅₃	—	2	4	no	—
38	#117×C₅₃	C₈	1	8	yes	complemented
39	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
40	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
41	C₂×C₄²×C₅₃	—	2	2	no	—
42	C₂×C₄²×C₅₃	—	2	2	no	—
43	C₂×C₄²×C₅₃	—	2	2	no	—
44	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₅₃	D₄	1	2	yes	—
48	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
50	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
51	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
52	#95×C₅₃	—	2	2	no	—
53	#95×C₅₃	—	2	2	no	—
54	C₂⁴×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	—	2	4	no	—
56	C₂²×C₄×C₅₃	—	2	4	no	—
57	C₂²×C₄×C₅₃	—	2	4	no	—
58	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
59	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
61	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
62	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
65	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
66	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
67	C₂²×C₄×C₅₃	C₂×C₈	1	4	yes	—
68	C₂²×C₄×C₅₃	M₁₆	1	4	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	C₂×D₄	1	1	yes	characteristic
72	C₄⋊C₄×C₅₃	—	2	4	no	—
73	C₄⋊C₄×C₅₃	C₂×C₈	1	8	yes	complemented
74	C₂²⋊C₄×C₅₃	—	2	8	no	—
75	C₄²×C₅₃	—	2	8	no	—
76	C₄²×C₅₃	—	4	4	no	—
77	C₄²×C₅₃	—	4	4	no	—
78	C₄²×C₅₃	C₄²	1	2	yes	—
79	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
80	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
81	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
82	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
83	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
84	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
85	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
86	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
88	C₂×#88322	—	2	8	no	—
89	C₂×#88322	—	2	16	no	—
90	C₂×#88322	—	2	4	no	—
91	C₂×#88322	—	2	4	no	—
92	C₂×#88322	—	4	4	no	—
93	C₂³×C₅₃	—	2	2	no	—
94	C₂³×C₅₃	—	2	2	no	—
95	C₂³×C₅₃	#95	1	2	yes	—
96	C₂³×C₅₃	C₂²⋊C₈	1	4	yes	—
97	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₅₃	—	2	2	no	—
99	C₂×C₄×C₅₃	—	2	2	no	—
100	C₂×C₄×C₅₃	—	2	4	no	—
101	C₂×C₄×C₅₃	—	2	4	no	—
102	C₂×C₄×C₅₃	—	2	4	no	—
103	C₂×C₄×C₅₃	—	2	4	no	—
104	C₂×C₄×C₅₃	—	2	8	no	—
105	C₂×C₄×C₅₃	—	4	4	no	—
106	C₂×C₄×C₅₃	—	4	4	no	—
107	C₂×C₄×C₅₃	#95	1	2	yes	—
108	C₂×C₄×C₅₃	#95	1	2	yes	—
109	C₂×C₄×C₅₃	#95	1	2	yes	—
110	C₂×C₄×C₅₃	C₄×C₈	1	4	yes	—
111	C₂×C₄×C₅₃	#97	1	4	yes	—
112	C₂×C₄×C₅₃	C₂²⋊C₈	1	4	yes	—
113	C₂×C₄×C₅₃	#104	1	4	yes	—
114	C₂×C₄×C₅₃	C₄⋊C₈	1	8	yes	—
115	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
116	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
122	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
123	C₂×C₄×C₅₃	#135	1	4	yes	—
124	#88322	—	2	8	no	—
125	#88322	—	4	4	no	—
126	#7890	—	53	1	no	maximal, Sylow
127	C₂²×C₅₃	—	2	8	no	—
128	C₂²×C₅₃	—	2	2	no	—
129	C₂²×C₅₃	—	2	2	no	—
130	C₂²×C₅₃	#336	1	4	yes	—
131	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
132	C₂²×C₅₃	C₂×C₂²⋊C₈	1	2	yes	—
133	C₂²×C₅₃	#416	1	2	yes	—
134	C₂²×C₅₃	C₂×#104	1	2	yes	—
135	C₄×C₅₃	—	2	1	no	—
136	C₄×C₅₃	—	2	1	no	—
137	C₄×C₅₃	—	2	2	no	—
138	C₄×C₅₃	—	4	8	no	—
139	C₄×C₅₃	#336	1	4	yes	—
140	C₄×C₅₃	#339	1	4	yes	—
141	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
142	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
143	C₄×C₅₃	#416	1	2	yes	—
144	C₄×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
145	C₄×C₅₃	#427	1	2	yes	—
146	C₂×#431	—	53	1	no	—
147	C₂²×C₄×C₈	—	53	1	no	—
148	C₄×#117	D₅₃	1	1	yes	characteristic
149	#1279	—	53	4	no	—
150	C₂×C₅₃	—	2	4	no	—
151	C₂×C₅₃	#1476	1	4	yes	—
152	C₂×C₅₃	C₂×#336	1	1	yes	characteristic
153	C₂×C₅₃	C₂×#339	1	1	yes	characteristic
154	C₂×C₅₃	#1735	1	1	yes	characteristic
155	C₂³×C₈	—	106	1	no	—
156	C₂×#117	Dic₅₃	1	2	yes	—
157	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
158	#431	—	53	4	no	—
159	#431	—	106	2	no	—
160	C₂×C₄⋊C₈	—	53	2	no	—
161	C₂×C₂²⋊C₈	—	106	1	no	—
162	C₂×C₄×C₈	—	53	4	no	—
163	C₂×C₄×C₈	—	53	2	no	—
164	C₂×C₄×C₈	—	106	2	no	—
165	C₂×C₄×C₈	—	106	1	no	—
166	C₂×C₄×C₈	—	106	1	no	—
167	C₄×C₄⋊C₄	Dic₅₃	1	2	yes	—
168	C₄×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
169	C₄×C₂²⋊C₄	—	2	1	no	—
170	C₄×C₂²⋊C₄	—	2	1	no	—
171	#376	—	2	1	no	—
172	C₄³	—	2	1	no	—
173	C₅₃	#7890	1	1	yes	characteristic, complemented, Sylow
174	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
175	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
176	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
177	C₂²×C₈	—	106	8	no	—
178	C₂²×C₈	—	106	2	no	—
179	C₂²×C₈	—	106	4	no	—
180	C₂²×C₈	—	106	2	no	—
181	#117	—	2	4	no	—
182	#117	#88322	1	8	yes	complemented
183	C₂×C₄⋊C₄	C₂×Dic₅₃	1	4	yes	—
184	C₂×C₂²⋊C₄	—	2	2	no	—
185	C₂×C₄²	—	2	2	no	—
186	C₂×C₄²	—	2	2	no	—
187	C₂×C₄²	—	2	2	no	—
188	C₂×C₄²	Dic₁₀₆	1	2	yes	—
189	C₂×C₄²	C₄×D₅₃	1	2	yes	—
190	C₂×C₄²	C₄×D₅₃	1	2	yes	—
191	C₂×C₄²	D₂₁₂	1	2	yes	—
192	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
193	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
194	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
195	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
196	C₄⋊C₈	—	53	4	no	—
197	C₄⋊C₈	—	106	2	no	—
198	C₂²⋊C₈	—	106	4	no	—
199	C₄×C₈	—	53	4	no	—
200	C₄×C₈	—	106	2	no	—
201	C₄×C₈	—	106	4	no	—
202	C₄×C₈	—	106	4	no	—
203	#95	—	2	2	no	—
204	#95	—	2	2	no	—
205	C₂⁴	C₂²⋊Dic₅₃	1	1	yes	characteristic
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	2	4	no	—
208	C₂²×C₄	—	2	4	no	—
209	C₂²×C₄	C₂×#88322	1	4	yes	—
210	C₂²×C₄	#11719917	1	4	yes	—
211	C₂²×C₄	C₄×Dic₅₃	1	2	yes	—
212	C₂²×C₄	#11719919	1	2	yes	—
213	C₂²×C₄	#11719919	1	2	yes	—
214	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
215	C₂²×C₄	#11719921	1	2	yes	—
216	C₂²×C₄	#11719921	1	2	yes	—
217	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
218	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
219	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
220	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
221	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
222	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
223	C₂×C₈	—	106	8	no	—
224	C₂×C₈	—	106	16	no	—
225	C₂×C₈	—	106	4	no	—
226	C₂×C₈	—	106	4	no	—
227	C₂×C₈	—	212	4	no	—
228	C₄⋊C₄	—	2	4	no	—
229	C₄⋊C₄	C₂×#88322	1	8	yes	complemented
230	C₂²⋊C₄	—	2	8	no	—
231	C₄²	—	2	8	no	—
232	C₄²	—	4	4	no	—
233	C₄²	—	4	4	no	—
234	C₄²	C₄×Dic₅₃	1	2	yes	—
235	C₄²	#11719919	1	2	yes	—
236	C₄²	#11719919	1	2	yes	—
237	C₄²	C₄⋊Dic₅₃	1	2	yes	—
238	C₄²	#11719921	1	2	yes	—
239	C₄²	#11719921	1	2	yes	—
240	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
241	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
242	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
243	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
244	C₂³	—	2	2	no	—
245	C₂³	—	2	2	no	—
246	C₂³	#13133124	1	2	yes	—
247	C₂³	C₂²⋊#88322	1	4	yes	—
248	C₂³	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	2	8	no	—
256	C₂×C₄	—	4	4	no	—
257	C₂×C₄	—	4	4	no	—
258	C₂×C₄	#13133124	1	2	yes	—
259	C₂×C₄	#13133124	1	2	yes	—
260	C₂×C₄	#13133124	1	2	yes	—
261	C₂×C₄	C₄×#88322	1	4	yes	—
262	C₂×C₄	#13133128	1	4	yes	—
263	C₂×C₄	C₂²⋊#88322	1	4	yes	—
264	C₂×C₄	#13133144	1	4	yes	—
265	C₂×C₄	C₄⋊#88322	1	8	yes	—
266	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
267	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
268	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
269	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
270	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
271	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
272	C₂×C₄	C₂²×#88322	1	2	yes	—
273	C₂×C₄	C₂×#11719917	1	2	yes	—
274	C₂×C₄	#13133237	1	4	yes	—
275	C₈	—	106	8	no	—
276	C₈	—	212	4	no	—
277	C₂²	—	2	8	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	2	2	no	—
280	C₂²	#16810378	1	4	yes	—
281	C₂²	C₂×#13133124	1	1	yes	characteristic
282	C₂²	C₂×C₂²⋊#88322	1	2	yes	—
283	C₂²	#16810619	1	2	yes	—
284	C₂²	C₂×#13133144	1	2	yes	—
285	C₄	—	2	1	no	—
286	C₄	—	2	1	no	—
287	C₄	—	2	2	no	—
288	C₄	—	4	8	no	—
289	C₄	#16810378	1	4	yes	—
290	C₄	#16810382	1	4	yes	—
291	C₄	C₂×C₄×#88322	1	1	yes	characteristic
292	C₄	C₂×#13133128	1	1	yes	characteristic
293	C₄	#16810617	1	2	yes	—
294	C₄	C₂×C₄⋊#88322	1	2	yes	—
295	C₄	#16810659	1	2	yes	—
296	C₂	—	2	4	no	—
297	C₂	#20273097	1	4	yes	—
298	C₂	C₂×#16810378	1	1	yes	characteristic
299	C₂	C₂×#16810382	1	1	yes	characteristic
300	C₂	#20274431	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²	C₂²⋊D₅₃
commutator	C₄×C₅₃	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₄×#117×C₅₃	C₂
socle	C₂³×C₅₃	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₅₃	C₂²⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₄×#117	D₅₃
hypocenter	C₅₃	#7890
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄² ◁ C₂²×C₄² ◁ C₄×#117

Complements

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

N	H ≅ G/N	type
3. #20272556	C₂	semidirect
7. #16810676	C₄	semidirect
13. C₂×C₄×#88322	C₄	semidirect
38. #117×C₅₃	C₈	semidirect
73. C₄⋊C₄×C₅₃	C₂×C₈	semidirect
173. C₅₃	#7890	semidirect
182. #117	#88322	semidirect
229. C₄⋊C₄	C₂×#88322	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810676	2
C₂²×C₄×#88322	2
#20272556	2
C₄×#117×C₅₃	2
#7890	53