Subgroups — #27537681

Order 13568.

618 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20276541	C₂	1	8	yes	maximal, complemented
2	#20276518	C₂	1	2	yes	maximal, complemented
3	#20276326	C₂	1	2	yes	maximal, complemented
4	C₂×C₂²⋊#11719908	C₂	1	2	yes	maximal, complemented
5	C₂×C₈×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133157	C₂²	1	4	yes	complemented
7	C₂²×#11719908	C₂²	1	4	yes	complemented
8	C₄⋊#11719908	C₂²	1	4	yes	complemented
9	C₂²⋊#11719908	C₂²	1	4	yes	—
10	C₂²⋊#11719908	C₂²	1	4	yes	complemented
11	C₂²⋊#11719908	C₂²	1	4	yes	complemented
12	#16810399	C₂²	1	4	yes	complemented
13	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₄×C₅₃	C₄	1	1	yes	characteristic
16	C₈×D₄×C₅₃	C₄	1	4	yes	—
17	C₈×D₄×C₅₃	C₂²	1	4	yes	—
18	C₂×C₄⋊C₈×C₅₃	C₄	1	1	yes	characteristic
19	C₂×C₂²⋊C₈×C₅₃	C₄	1	1	yes	characteristic
20	C₂×C₂²⋊C₈×C₅₃	C₄	1	1	yes	characteristic
21	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
22	#13133157	—	4	4	no	—
23	C₂×#11719908	—	4	4	no	—
24	C₂×#11719908	C₂³	1	8	yes	—
25	C₂²×D₄×C₅₃	C₈	1	1	yes	characteristic
26	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₅₃	—	4	2	no	—
29	C₂²×C₈×C₅₃	—	4	2	no	—
30	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
33	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
34	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
36	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
37	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
38	C₄×D₄×C₅₃	C₈	1	4	yes	—
39	C₄×D₄×C₅₃	C₂×C₄	1	4	yes	—
40	C₂×C₄⋊C₄×C₅₃	C₈	1	1	yes	characteristic
41	C₂×C₂²⋊C₄×C₅₃	C₈	1	1	yes	characteristic
42	C₂×C₂²⋊C₄×C₅₃	C₈	1	1	yes	characteristic
43	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
45	C₂²⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
46	C₂²⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
47	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
48	C₄×C₈×C₅₃	C₂³	1	2	yes	—
49	C₂⁴×C₅₃	C₂×C₈	1	1	yes	characteristic
50	C₂⁴×C₅₃	C₂×C₈	1	1	yes	characteristic
51	C₂×D₄×C₅₃	—	2	4	no	—
52	C₂×D₄×C₅₃	C₂×C₈	1	4	yes	—
53	C₂²×C₄×C₅₃	—	4	1	no	—
54	C₂²×C₄×C₅₃	—	4	1	no	—
55	C₂²×C₄×C₅₃	—	4	1	no	—
56	C₂²×C₄×C₅₃	—	4	1	no	—
57	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
62	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
63	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₈×C₅₃	—	2	2	no	—
67	C₂×C₈×C₅₃	—	2	4	no	—
68	C₂×C₈×C₅₃	—	2	4	no	—
69	C₂×C₈×C₅₃	—	4	2	no	—
70	C₂×C₈×C₅₃	—	4	2	no	—
71	C₂×C₈×C₅₃	—	4	1	no	—
72	C₂×C₈×C₅₃	—	4	1	no	—
73	C₂×C₈×C₅₃	—	4	1	no	—
74	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
77	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
79	C₄⋊C₄×C₅₃	C₂×C₈	1	4	yes	—
80	C₂²⋊C₄×C₅₃	C₂×C₈	1	4	yes	—
81	C₂²⋊C₄×C₅₃	C₂×C₈	1	4	yes	—
82	C₄²×C₅₃	C₂×C₈	1	2	yes	—
83	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
84	#11719908	—	2	8	no	—
85	C₂³×C₅₃	—	2	2	no	—
86	C₂³×C₅₃	—	4	1	no	—
87	C₂³×C₅₃	—	4	1	no	—
88	C₂³×C₅₃	—	4	1	no	—
89	C₂³×C₅₃	—	4	1	no	—
90	C₂³×C₅₃	M₃₂	1	4	yes	complemented
91	C₂³×C₅₃	C₂²×C₈	1	2	yes	—
92	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
93	C₂³×C₅₃	C₂²×C₈	1	2	yes	—
94	D₄×C₅₃	—	2	8	no	—
95	C₂×C₄×C₅₃	—	2	2	no	—
96	C₂×C₄×C₅₃	—	2	1	no	—
97	C₂×C₄×C₅₃	—	2	2	no	—
98	C₂×C₄×C₅₃	—	2	2	no	—
99	C₂×C₄×C₅₃	—	2	1	no	—
100	C₂×C₄×C₅₃	—	2	2	no	—
101	C₂×C₄×C₅₃	—	2	2	no	—
102	C₂×C₄×C₅₃	—	4	2	no	—
103	C₂×C₄×C₅₃	—	4	2	no	—
104	C₂×C₄×C₅₃	—	4	1	no	—
105	C₂×C₄×C₅₃	—	4	1	no	—
106	C₂×C₄×C₅₃	—	4	1	no	—
107	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
108	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
109	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
110	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
111	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
112	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
113	C₈×C₅₃	—	2	1	no	—
114	C₈×C₅₃	—	2	2	no	—
115	C₈×C₅₃	—	2	2	no	—
116	C₈×C₅₃	—	2	2	no	—
117	C₈×C₅₃	H(5,𝔽₂)	1	2	yes	—
118	#33487	—	53	1	no	maximal, Sylow
119	C₂²×C₅₃	—	2	1	no	—
120	C₂²×C₅₃	—	2	2	no	—
121	C₂²×C₅₃	—	2	1	no	—
122	C₂²×C₅₃	—	2	2	no	—
123	C₂²×C₅₃	—	2	2	no	—
124	C₂²×C₅₃	—	2	2	no	—
125	C₂²×C₅₃	—	2	2	no	—
126	C₂²×C₅₃	—	4	4	no	—
127	C₂²×C₅₃	—	4	4	no	—
128	C₂²×C₅₃	C₂×M₃₂	1	2	yes	—
129	C₂²×C₅₃	C₂×M₃₂	1	4	yes	—
130	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
131	C₄×C₅₃	—	2	2	no	—
132	C₄×C₅₃	—	2	2	no	—
133	C₄×C₅₃	—	2	1	no	—
134	C₄×C₅₃	—	2	2	no	—
135	C₄×C₅₃	—	2	2	no	—
136	C₄×C₅₃	#518	1	1	yes	characteristic
137	C₄×C₅₃	#568	1	1	yes	characteristic
138	C₂×C₈×D₄	D₅₃	1	1	yes	characteristic
139	#2168	—	53	8	no	—
140	#2163	—	53	2	no	—
141	#2112	—	53	2	no	—
142	C₂×C₂²⋊C₁₆	—	53	2	no	—
143	C₂×C₅₃	—	2	2	no	—
144	C₂×C₅₃	—	2	4	no	—
145	C₂×C₅₃	—	4	2	no	—
146	C₂×C₅₃	#2972	1	1	yes	characteristic
147	C₂×C₅₃	C₂²×M₃₂	1	1	yes	characteristic
148	C₂×C₅₃	#3407	1	1	yes	characteristic
149	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
150	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
151	C₂×C₄×D₄	Dic₅₃	1	1	yes	characteristic
152	C₂×M₃₂	—	53	4	no	—
153	C₂²×C₁₆	—	53	4	no	—
154	C₈×D₄	Dic₅₃	1	4	yes	—
155	C₈×D₄	C₂×D₅₃	1	4	yes	—
156	C₂×C₄⋊C₈	Dic₅₃	1	1	yes	characteristic
157	C₂×C₂²⋊C₈	Dic₅₃	1	1	yes	characteristic
158	C₂×C₂²⋊C₈	Dic₅₃	1	1	yes	characteristic
159	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
160	C₄⋊C₁₆	—	53	4	no	—
161	C₂²⋊C₁₆	—	53	4	no	—
162	C₂²⋊C₁₆	—	53	4	no	—
163	C₂²⋊C₁₆	—	53	4	no	—
164	#346	—	53	4	no	—
165	C₅₃	#33487	1	1	yes	characteristic, complemented, Sylow
166	C₂²×D₄	#88322	1	1	yes	characteristic
167	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
168	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
169	C₂²×C₈	—	4	2	no	—
170	C₂²×C₈	—	4	2	no	—
171	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
172	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
173	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
174	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
175	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
176	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
177	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
178	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
179	C₄×D₄	#88322	1	4	yes	—
180	C₄×D₄	C₂×Dic₅₃	1	4	yes	—
181	C₂×C₄⋊C₄	#88322	1	1	yes	characteristic
182	C₂×C₂²⋊C₄	#88322	1	1	yes	characteristic
183	C₂×C₂²⋊C₄	#88322	1	1	yes	characteristic
184	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
185	M₃₂	—	212	4	no	—
186	C₂×C₁₆	—	53	8	no	—
187	C₂×C₁₆	—	212	4	no	—
188	C₄⋊C₈	C₂×Dic₅₃	1	4	yes	—
189	C₂²⋊C₈	C₂×Dic₅₃	1	4	yes	—
190	C₂²⋊C₈	C₂×Dic₅₃	1	4	yes	—
191	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
192	C₄×C₈	C₂²×D₅₃	1	2	yes	—
193	C₂⁴	C₂×#88322	1	1	yes	characteristic
194	C₂⁴	C₂×#88322	1	1	yes	characteristic
195	C₂×D₄	—	2	4	no	—
196	C₂×D₄	C₂×#88322	1	4	yes	—
197	C₂²×C₄	—	4	1	no	—
198	C₂²×C₄	—	4	1	no	—
199	C₂²×C₄	—	4	1	no	—
200	C₂²×C₄	—	4	1	no	—
201	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
202	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
203	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
204	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
205	C₂²×C₄	C₂×#88322	1	2	yes	—
206	C₂²×C₄	C₂×#88322	1	2	yes	—
207	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
208	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
209	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
210	C₂×C₈	—	2	4	no	—
211	C₂×C₈	—	2	4	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	4	2	no	—
214	C₂×C₈	—	4	2	no	—
215	C₂×C₈	—	4	1	no	—
216	C₂×C₈	—	4	1	no	—
217	C₂×C₈	—	4	1	no	—
218	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
219	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
220	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
221	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
222	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
223	C₄⋊C₄	C₂×#88322	1	4	yes	—
224	C₂²⋊C₄	C₂×#88322	1	4	yes	—
225	C₂²⋊C₄	C₂×#88322	1	4	yes	—
226	C₄²	C₂×#88322	1	2	yes	—
227	C₄²	C₂²×Dic₅₃	1	2	yes	—
228	C₁₆	—	106	8	no	—
229	C₂³	—	2	2	no	—
230	C₂³	—	4	1	no	—
231	C₂³	—	4	1	no	—
232	C₂³	—	4	1	no	—
233	C₂³	—	4	1	no	—
234	C₂³	#13133157	1	4	yes	complemented
235	C₂³	C₂²×#88322	1	2	yes	—
236	C₂³	C₂²×#88322	1	2	yes	—
237	C₂³	C₂²×#88322	1	1	yes	characteristic
238	D₄	—	2	8	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	4	1	no	—
249	C₂×C₄	—	4	1	no	—
250	C₂×C₄	—	4	1	no	—
251	C₂×C₄	C₂²×#88322	1	2	yes	—
252	C₂×C₄	C₂²×#88322	1	2	yes	—
253	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
254	C₂×C₄	C₂²×#88322	1	2	yes	—
255	C₂×C₄	C₂²×#88322	1	2	yes	—
256	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
257	C₈	—	2	1	no	—
258	C₈	—	2	2	no	—
259	C₈	—	2	2	no	—
260	C₈	—	2	2	no	—
261	C₈	#13133262	1	2	yes	—
262	C₂²	—	2	1	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	2	1	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	2	2	no	—
267	C₂²	—	2	2	no	—
268	C₂²	—	2	2	no	—
269	C₂²	—	4	4	no	—
270	C₂²	—	4	4	no	—
271	C₂²	C₂×#13133157	1	2	yes	—
272	C₂²	C₂×#13133157	1	4	yes	—
273	C₂²	C₂³×#88322	1	1	yes	characteristic
274	C₄	—	2	2	no	—
275	C₄	—	2	2	no	—
276	C₄	—	2	1	no	—
277	C₄	—	2	2	no	—
278	C₄	—	2	2	no	—
279	C₄	#16811098	1	1	yes	characteristic
280	C₄	#16811390	1	1	yes	characteristic
281	C₂	—	2	2	no	—
282	C₂	—	2	4	no	—
283	C₂	—	4	2	no	—
284	C₂	#20283739	1	1	yes	characteristic
285	C₂	C₂²×#13133157	1	1	yes	characteristic
286	C₂	#20287818	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	C₂×C₈×D₄×C₅₃	C₂
prime radical	C₂²×D₄×C₅₃	C₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂×C₈×D₄	D₅₃
hypocenter	C₅₃	#33487
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₈×D₄

Complements

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

N	H ≅ G/N	type
1. #20276541	C₂	semidirect
2. #20276518	C₂	semidirect
3. #20276326	C₂	semidirect
4. C₂×C₂²⋊#11719908	C₂	semidirect
6. C₂×#13133157	C₂²	semidirect
7. C₂²×#11719908	C₂²	semidirect
8. C₄⋊#11719908	C₂²	semidirect
10. C₂²⋊#11719908	C₂²	semidirect
11. C₂²⋊#11719908	C₂²	semidirect
12. #16810399	C₂²	semidirect
90. C₂³×C₅₃	M₃₂	semidirect
165. C₅₃	#33487	semidirect
234. C₂³	#13133157	semidirect

Maximal subgroups

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

subgroup	index
#20276541	2
#20276518	2
#20276326	2
C₂×C₂²⋊#11719908	2
C₂×C₈×D₄×C₅₃	2
#33487	53