Subgroups — #27273474

Order 13568.

798 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	C₂×#16810676	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#11719917	C₂	1	1	yes	maximal, characteristic
3	#20272542	C₂	1	4	yes	maximal, complemented
4	C₂²×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719917	C₄	1	2	yes	—
6	#16810676	—	2	2	no	—
7	#16810676	—	2	2	no	—
8	C₂×C₄⋊#88322	C₄	1	2	yes	—
9	C₂×C₂²⋊#88322	C₄	1	2	yes	—
10	C₄×#11719917	—	2	2	no	—
11	C₄×#11719917	—	2	2	no	—
12	C₂×#13133128	C₄	1	2	yes	—
13	C₂×C₄×#88322	C₂²	1	2	yes	—
14	C₂×C₄×#88322	C₂²	1	2	yes	—
15	C₂³×C₈×C₅₃	C₄	1	2	yes	—
16	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₅₃	—	2	2	no	—
18	C₂×C₄×C₈×C₅₃	—	2	2	no	—
19	C₂×C₄×C₈×C₅₃	C₄	1	2	yes	—
20	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
21	C₂×#11719917	—	2	4	no	—
22	C₂×#11719917	—	2	4	no	—
23	C₂×#11719917	C₈	1	8	yes	complemented
24	C₂²×#88322	C₂×C₄	1	4	yes	—
25	C₂²×#88322	C₂×C₄	1	4	yes	—
26	C₄⋊#88322	—	2	4	no	—
27	C₂²⋊#88322	—	2	4	no	—
28	#13133128	—	2	4	no	—
29	C₄×#88322	—	2	2	no	—
30	C₄×#88322	—	2	2	no	—
31	C₄×#88322	—	2	2	no	—
32	C₄×#88322	—	2	2	no	—
33	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₅₃	—	2	8	no	—
37	C₂²×C₈×C₅₃	—	2	4	no	—
38	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
39	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₅₃	D₄	1	2	yes	—
43	C₂×C₄²×C₅₃	D₄	1	2	yes	—
44	C₂×C₄²×C₅₃	D₄	1	2	yes	—
45	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
46	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
47	C₄×C₈×C₅₃	—	2	4	no	—
48	C₄×C₈×C₅₃	—	2	2	no	—
49	C₄×C₈×C₅₃	—	2	2	no	—
50	C₂⁴×C₅₃	C₄²	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	—	2	4	no	—
52	C₂²×C₄×C₅₃	—	2	4	no	—
53	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
57	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
58	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
59	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
60	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
61	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
62	C₂²×C₄×C₅₃	C₂×C₈	1	4	yes	—
63	C₂²×C₄×C₅₃	M₁₆	1	4	yes	—
64	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₈×C₅₃	—	2	16	no	—
68	C₂×C₈×C₅₃	—	2	4	no	—
69	C₂×C₈×C₅₃	—	2	4	no	—
70	C₂×C₈×C₅₃	M₁₆	1	8	yes	complemented
71	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
72	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
73	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
74	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
75	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
76	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
77	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
81	#11719917	—	2	16	no	—
82	C₂×#88322	—	2	4	no	—
83	C₂×#88322	—	2	4	no	—
84	C₂×#88322	—	2	4	no	—
85	C₂×#88322	—	2	8	no	—
86	C₂×#88322	C₂×C₈	1	8	yes	complemented
87	C₂³×C₅₃	—	2	2	no	—
88	C₂³×C₅₃	—	2	2	no	—
89	C₂³×C₅₃	#95	1	2	yes	—
90	C₂³×C₅₃	C₄×C₈	1	2	yes	—
91	C₂³×C₅₃	#97	1	2	yes	—
92	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
93	C₂×C₄×C₅₃	—	2	2	no	—
94	C₂×C₄×C₅₃	—	2	2	no	—
95	C₂×C₄×C₅₃	—	2	4	no	—
96	C₂×C₄×C₅₃	—	2	4	no	—
97	C₂×C₄×C₅₃	—	2	4	no	—
98	C₂×C₄×C₅₃	—	2	4	no	—
99	C₂×C₄×C₅₃	#95	1	2	yes	—
100	C₂×C₄×C₅₃	#95	1	2	yes	—
101	C₂×C₄×C₅₃	#95	1	2	yes	—
102	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
103	C₂×C₄×C₅₃	#97	1	2	yes	—
104	C₂×C₄×C₅₃	C₂²⋊C₈	1	8	yes	—
105	C₂×C₄×C₅₃	C₄⋊C₈	1	8	yes	—
106	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
113	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
114	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
115	C₂×C₄×C₅₃	#131	1	4	yes	—
116	C₈×C₅₃	—	2	8	no	—
117	#88322	—	2	8	no	—
118	#88322	—	4	4	no	—
119	#7848	—	53	1	no	maximal, Sylow
120	C₂²×C₅₃	—	2	4	no	—
121	C₂²×C₅₃	—	2	4	no	—
122	C₂²×C₅₃	—	2	2	no	—
123	C₂²×C₅₃	—	2	2	no	—
124	C₂²×C₅₃	#336	1	4	yes	—
125	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
126	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
127	C₂²×C₅₃	C₂×#97	1	1	yes	characteristic
128	C₂²×C₅₃	C₄×M₁₆	1	2	yes	—
129	C₂²×C₅₃	#405	1	2	yes	—
130	C₄×C₅₃	—	2	2	no	—
131	C₄×C₅₃	—	2	2	no	—
132	C₄×C₅₃	#336	1	4	yes	—
133	C₄×C₅₃	C₂×C₂²⋊C₈	1	2	yes	—
134	C₄×C₅₃	#407	1	2	yes	—
135	C₄×C₅₃	#408	1	2	yes	—
136	C₄×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
137	C₄×C₅₃	#423	1	2	yes	—
138	C₄×C₅₃	#424	1	2	yes	—
139	C₂×#431	—	53	1	no	—
140	C₂×C₄×M₁₆	—	53	1	no	—
141	C₂²×C₄×C₈	D₅₃	1	1	yes	characteristic
142	#1273	—	53	4	no	—
143	C₂×C₅₃	—	2	4	no	—
144	C₂×C₅₃	C₈×M₁₆	1	2	yes	—
145	C₂×C₅₃	#1451	1	2	yes	—
146	C₂×C₅₃	C₂×#336	1	1	yes	characteristic
147	C₂×C₅₃	#1728	1	1	yes	characteristic
148	C₂×C₅₃	#1729	1	1	yes	characteristic
149	C₂²×M₁₆	—	53	2	no	—
150	C₂³×C₈	Dic₅₃	1	2	yes	—
151	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
152	#431	—	106	2	no	—
153	#431	—	106	2	no	—
154	C₂×C₄⋊C₈	—	53	2	no	—
155	C₂×C₂²⋊C₈	—	53	2	no	—
156	C₄×M₁₆	—	106	2	no	—
157	C₄×M₁₆	—	106	2	no	—
158	C₂×#97	—	53	2	no	—
159	C₂×C₄×C₈	—	2	2	no	—
160	C₂×C₄×C₈	—	2	2	no	—
161	C₂×C₄×C₈	—	53	2	no	—
162	C₂×C₄×C₈	—	53	2	no	—
163	C₂×C₄×C₈	Dic₅₃	1	2	yes	—
164	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
165	C₅₃	#7848	1	1	yes	characteristic, complemented, Sylow
166	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
167	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
168	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
169	C₂×M₁₆	—	53	8	no	—
170	C₂×M₁₆	—	106	4	no	—
171	C₂×M₁₆	—	106	4	no	—
172	C₂²×C₈	—	2	4	no	—
173	C₂²×C₈	—	2	8	no	—
174	C₂²×C₈	—	53	4	no	—
175	C₂²×C₈	—	53	4	no	—
176	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
177	C₂×C₄²	Dic₁₀₆	1	2	yes	—
178	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
179	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
180	C₂×C₄²	D₂₁₂	1	2	yes	—
181	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
182	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
183	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
184	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
185	C₄⋊C₈	—	106	4	no	—
186	C₂²⋊C₈	—	106	4	no	—
187	#97	—	106	4	no	—
188	C₄×C₈	—	2	2	no	—
189	C₄×C₈	—	2	2	no	—
190	C₄×C₈	—	2	4	no	—
191	C₄×C₈	—	106	2	no	—
192	C₄×C₈	—	106	2	no	—
193	C₄×C₈	—	106	2	no	—
194	C₄×C₈	—	106	2	no	—
195	C₂⁴	C₄×Dic₅₃	1	1	yes	characteristic
196	C₂²×C₄	—	2	4	no	—
197	C₂²×C₄	—	2	4	no	—
198	C₂²×C₄	C₈×D₅₃	1	4	yes	—
199	C₂²×C₄	#11719912	1	4	yes	—
200	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
201	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
202	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
203	C₂²×C₄	#11719919	1	2	yes	—
204	C₂²×C₄	#11719919	1	2	yes	—
205	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
206	C₂²×C₄	#11719921	1	2	yes	—
207	C₂²×C₄	#11719921	1	2	yes	—
208	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
209	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
210	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
211	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
212	M₁₆	—	106	16	no	—
213	C₂×C₈	—	2	16	no	—
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	2	4	no	—
216	C₂×C₈	—	53	8	no	—
217	C₂×C₈	—	106	4	no	—
218	C₂×C₈	—	106	4	no	—
219	C₂×C₈	—	106	4	no	—
220	C₂×C₈	—	106	8	no	—
221	C₂×C₈	#11719917	1	8	yes	complemented
222	C₄²	#11719919	1	2	yes	—
223	C₄²	#11719919	1	2	yes	—
224	C₄²	C₄⋊Dic₅₃	1	2	yes	—
225	C₄²	#11719921	1	2	yes	—
226	C₄²	#11719921	1	2	yes	—
227	C₄²	C₂²⋊Dic₅₃	1	2	yes	—
228	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
229	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
230	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
231	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
232	C₂³	—	2	2	no	—
233	C₂³	—	2	2	no	—
234	C₂³	#13133124	1	2	yes	—
235	C₂³	C₈×Dic₅₃	1	2	yes	—
236	C₂³	#13133127	1	2	yes	—
237	C₂³	C₂×C₄×Dic₅₃	1	1	yes	characteristic
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	#13133124	1	2	yes	—
245	C₂×C₄	#13133124	1	2	yes	—
246	C₂×C₄	#13133124	1	2	yes	—
247	C₂×C₄	C₈×Dic₅₃	1	2	yes	—
248	C₂×C₄	#13133127	1	2	yes	—
249	C₂×C₄	#13133129	1	8	yes	—
250	C₂×C₄	#13133146	1	8	yes	—
251	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
252	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
253	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
254	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
255	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
256	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
257	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
258	C₂×C₄	C₂×#11719912	1	2	yes	—
259	C₂×C₄	C₂×#11719917	1	4	yes	—
260	C₂×C₄	#13133223	1	4	yes	—
261	C₈	—	2	8	no	—
262	C₈	—	106	8	no	—
263	C₈	—	212	4	no	—
264	C₂²	—	2	4	no	—
265	C₂²	—	2	4	no	—
266	C₂²	—	2	2	no	—
267	C₂²	—	2	2	no	—
268	C₂²	#16810377	1	4	yes	—
269	C₂²	C₂×#13133124	1	1	yes	characteristic
270	C₂²	C₂×C₈×Dic₅₃	1	1	yes	characteristic
271	C₂²	C₂×#13133127	1	1	yes	characteristic
272	C₂²	C₄×#11719917	1	2	yes	—
273	C₂²	#16810578	1	2	yes	—
274	C₄	—	2	2	no	—
275	C₄	—	2	2	no	—
276	C₄	#16810377	1	4	yes	—
277	C₄	C₂×#13133129	1	2	yes	—
278	C₄	#16810587	1	2	yes	—
279	C₄	#16810589	1	2	yes	—
280	C₄	C₂×#13133146	1	2	yes	—
281	C₄	#16810647	1	2	yes	—
282	C₄	#16810649	1	2	yes	—
283	C₂	—	2	4	no	—
284	C₂	C₈×#11719917	1	2	yes	—
285	C₂	#20273004	1	2	yes	—
286	C₂	C₂×#16810377	1	1	yes	characteristic
287	C₂	#20274409	1	1	yes	characteristic
288	C₂	#20274411	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂×C₄×C₈
Fitting	C₂²×C₄×C₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂×C₄²
prime radical	C₂⁴×C₅₃	C₄²
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂²×C₄×C₈	D₅₃
hypocenter	C₅₃	#7848
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. #20272542	C₂	semidirect
23. C₂×#11719917	C₈	semidirect
70. C₂×C₈×C₅₃	M₁₆	semidirect
86. C₂×#88322	C₂×C₈	semidirect
165. C₅₃	#7848	semidirect
221. C₂×C₈	#11719917	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
C₂×C₄×#11719917	2
#20272542	2
C₂²×C₄×C₈×C₅₃	2
#7848	53