Subgroups — #27353550

Order 13568.

782 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	#20283739	C₂	1	2	yes	maximal, complemented
2	#20273539	C₂	1	4	yes	maximal, complemented
3	D₈×#88322	C₂	1	4	yes	maximal, complemented
4	#20273463	C₂	1	1	yes	maximal, characteristic, complemented
5	#20273106	C₂	1	2	yes	maximal, complemented
6	#20272998	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄×D₈×C₅₃	C₂	1	1	yes	maximal, characteristic
8	D₄×#88322	—	2	4	no	—
9	D₄×#88322	C₂²	1	8	yes	complemented
10	#16810676	C₂²	1	2	yes	—
11	#16810676	C₂²	1	1	yes	characteristic
12	#16810676	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊#88322	—	2	4	no	—
14	#16810376	C₂²	1	2	yes	complemented
15	#16810374	C₂²	1	2	yes	complemented
16	D₄⋊#88322	C₂²	1	8	yes	complemented
17	#16810344	C₂²	1	2	yes	complemented
18	C₈×#88322	C₂²	1	2	yes	complemented
19	C₂²×D₈×C₅₃	C₄	1	1	yes	characteristic
20	C₂×C₄×D₄×C₅₃	C₂²	1	2	yes	—
21	C₄×D₈×C₅₃	C₄	1	2	yes	—
22	C₄×D₈×C₅₃	C₄	1	2	yes	—
23	C₄×D₈×C₅₃	C₂²	1	2	yes	—
24	C₄×D₈×C₅₃	C₂²	1	2	yes	—
25	C₂×#107×C₅₃	C₄	1	1	yes	characteristic
26	C₂×#102×C₅₃	C₄	1	2	yes	—
27	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
28	C₂²×#88322	—	2	8	no	—
29	C₄⋊#88322	—	2	1	no	—
30	C₄⋊#88322	D₄	1	2	yes	—
31	C₄⋊#88322	C₂³	1	4	yes	—
32	C₂²⋊#88322	—	2	8	no	—
33	C₂²⋊#88322	—	2	2	no	—
34	C₂²⋊#88322	—	2	1	no	—
35	C₂²⋊#88322	D₄	1	2	yes	—
36	C₄×#88322	—	2	2	no	—
37	C₄×#88322	C₂³	1	2	yes	—
38	C₄×#88322	C₂³	1	2	yes	—
39	C₂²×D₄×C₅₃	C₂×C₄	1	2	yes	—
40	C₂³×C₄×C₅₃	—	2	2	no	—
41	C₂×D₈×C₅₃	C₈	1	8	yes	complemented
42	C₂×D₈×C₅₃	C₂×C₄	1	2	yes	—
43	C₂×D₈×C₅₃	C₂×C₄	1	2	yes	—
44	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
46	C₄×D₄×C₅₃	—	2	2	no	—
47	C₄×D₄×C₅₃	—	2	2	no	—
48	C₄×D₄×C₅₃	C₂×C₄	1	4	yes	—
49	C₄×D₄×C₅₃	C₂³	1	4	yes	—
50	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
51	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
52	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
53	#107×C₅₃	C₂×C₄	1	2	yes	—
54	#107×C₅₃	C₂×C₄	1	2	yes	—
55	#102×C₅₃	C₂×C₄	1	4	yes	—
56	#102×C₅₃	C₂×C₄	1	4	yes	—
57	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
58	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
59	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
60	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
61	C₂⁴×C₅₃	—	2	2	no	—
62	C₂×D₄×C₅₃	—	2	2	no	—
63	C₂×D₄×C₅₃	—	2	4	no	—
64	C₂×D₄×C₅₃	C₂×C₈	1	8	yes	complemented
65	C₂×D₄×C₅₃	C₂²×C₄	1	4	yes	—
66	C₂²×C₄×C₅₃	—	2	2	no	—
67	C₂²×C₄×C₅₃	—	2	4	no	—
68	C₂²×C₄×C₅₃	—	2	4	no	—
69	C₂²×C₄×C₅₃	—	8	4	no	—
70	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
73	D₈×C₅₃	C₂×C₈	1	8	yes	complemented
74	D₈×C₅₃	C₂×C₈	1	8	yes	complemented
75	C₂×C₈×C₅₃	—	2	2	no	—
76	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	complemented
77	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
81	C₄⋊C₄×C₅₃	—	2	2	no	—
82	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
83	C₂²⋊C₄×C₅₃	—	2	4	no	—
84	C₂²⋊C₄×C₅₃	—	2	4	no	—
85	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
86	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₅₃	#41	1	1	yes	characteristic
88	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
89	C₂×#88322	—	2	4	no	—
90	C₂×#88322	—	2	2	no	—
91	C₂×#88322	—	8	8	no	—
92	C₂×#88322	C₂×D₄	1	4	yes	—
93	C₂³×C₅₃	—	2	4	no	—
94	C₂³×C₅₃	—	2	8	no	—
95	C₂³×C₅₃	—	8	2	no	—
96	C₂³×C₅₃	—	8	2	no	—
97	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
98	D₄×C₅₃	—	2	4	no	—
99	D₄×C₅₃	—	2	8	no	—
100	D₄×C₅₃	C₂²×C₈	1	8	yes	—
101	C₂×C₄×C₅₃	—	2	2	no	—
102	C₂×C₄×C₅₃	—	2	4	no	—
103	C₂×C₄×C₅₃	—	4	1	no	—
104	C₂×C₄×C₅₃	—	4	8	no	—
105	C₂×C₄×C₅₃	—	8	4	no	—
106	C₂×C₄×C₅₃	—	8	2	no	—
107	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
108	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
110	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
111	C₂×C₄×C₅₃	#131	1	2	yes	—
112	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
113	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
115	C₈×C₅₃	—	2	1	no	—
116	C₈×C₅₃	—	2	1	no	—
117	C₈×C₅₃	C₂²×C₈	1	2	yes	—
118	C₈×C₅₃	C₂²×C₈	1	2	yes	—
119	#88322	—	2	4	no	—
120	#88322	—	4	2	no	—
121	#88322	—	4	4	no	—
122	#17854	—	53	1	no	maximal, Sylow
123	C₂²×C₅₃	—	2	1	no	—
124	C₂²×C₅₃	—	2	1	no	—
125	C₂²×C₅₃	—	2	8	no	—
126	C₂²×C₅₃	—	4	4	no	—
127	C₂²×C₅₃	—	4	4	no	—
128	C₂²×C₅₃	—	8	8	no	—
129	C₂²×C₅₃	C₈×D₄	1	2	yes	—
130	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
131	C₄×C₅₃	—	2	1	no	—
132	C₄×C₅₃	—	2	1	no	—
133	C₄×C₅₃	—	2	1	no	—
134	C₄×C₅₃	—	4	4	no	—
135	C₄×C₅₃	C₈×D₄	1	2	yes	—
136	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
137	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
138	C₄×C₅₃	#576	1	2	yes	—
139	#2972	—	53	2	no	—
140	C₂×C₄×D₈	D₅₃	1	1	yes	characteristic
141	#1580	—	53	4	no	—
142	C₈×D₈	—	53	4	no	—
143	#1564	—	53	1	no	—
144	#1478	—	53	2	no	—
145	#1450	—	53	1	no	—
146	C₂×C₅₃	—	2	2	no	—
147	C₂×C₅₃	—	4	8	no	—
148	C₂×C₅₃	C₂×C₈×D₄	1	1	yes	characteristic
149	C₂×C₅₃	#2948	1	1	yes	characteristic
150	C₂×C₅₃	#2957	1	1	yes	characteristic
151	C₂²×D₈	Dic₅₃	1	1	yes	characteristic
152	C₂×C₄×D₄	C₂×D₅₃	1	2	yes	—
153	C₄×D₈	Dic₅₃	1	2	yes	—
154	C₄×D₈	Dic₅₃	1	2	yes	—
155	C₄×D₈	C₂×D₅₃	1	2	yes	—
156	C₄×D₈	C₂×D₅₃	1	2	yes	—
157	C₈×D₄	—	53	8	no	—
158	C₈×D₄	—	106	4	no	—
159	#431	—	53	1	no	—
160	#431	—	53	1	no	—
161	#431	—	53	2	no	—
162	C₂×#107	Dic₅₃	1	1	yes	characteristic
163	C₂×#102	Dic₅₃	1	2	yes	—
164	C₂×C₂²⋊C₈	—	106	4	no	—
165	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
166	#335	—	53	2	no	—
167	#334	—	53	2	no	—
168	D₄⋊C₈	—	53	8	no	—
169	#322	—	53	2	no	—
170	C₈²	—	53	2	no	—
171	C₅₃	#17854	1	1	yes	characteristic, complemented, Sylow
172	C₂²×D₄	C₂×Dic₅₃	1	2	yes	—
173	C₂³×C₄	—	2	2	no	—
174	C₂×D₈	#88322	1	8	yes	complemented
175	C₂×D₈	C₂×Dic₅₃	1	2	yes	—
176	C₂×D₈	C₂×Dic₅₃	1	2	yes	—
177	C₂²×C₈	—	106	8	no	—
178	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
179	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
180	C₄×D₄	—	2	2	no	—
181	C₄×D₄	—	2	2	no	—
182	C₄×D₄	C₂×Dic₅₃	1	4	yes	—
183	C₄×D₄	C₂²×D₅₃	1	4	yes	—
184	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
185	C₂×C₂²⋊C₄	—	2	2	no	—
186	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
187	#107	C₂×Dic₅₃	1	2	yes	—
188	#107	C₂×Dic₅₃	1	2	yes	—
189	C₄⋊C₈	—	53	2	no	—
190	C₄⋊C₈	—	53	4	no	—
191	C₄⋊C₈	—	106	1	no	—
192	#102	C₂×Dic₅₃	1	4	yes	—
193	#102	C₂×Dic₅₃	1	4	yes	—
194	C₂²⋊C₈	—	53	2	no	—
195	C₂²⋊C₈	—	106	8	no	—
196	C₂²⋊C₈	—	106	2	no	—
197	C₂²⋊C₈	—	106	1	no	—
198	C₄×C₈	—	53	2	no	—
199	C₄×C₈	—	53	2	no	—
200	C₄×C₈	—	106	2	no	—
201	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
202	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
203	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
204	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
205	C₂⁴	—	2	2	no	—
206	C₂×D₄	—	2	2	no	—
207	C₂×D₄	—	2	4	no	—
208	C₂×D₄	C₂×#88322	1	8	yes	complemented
209	C₂×D₄	C₂²×Dic₅₃	1	4	yes	—
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	—	2	4	no	—
212	C₂²×C₄	—	2	4	no	—
213	C₂²×C₄	—	8	4	no	—
214	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
215	C₂²×C₄	#11719947	1	1	yes	characteristic
216	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
217	D₈	C₂×#88322	1	8	yes	complemented
218	D₈	C₂×#88322	1	8	yes	complemented
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	53	4	no	—
221	C₂×C₈	—	106	4	no	—
222	C₂×C₈	—	106	2	no	—
223	C₂×C₈	—	424	8	no	—
224	C₂×C₈	C₂×#88322	1	4	yes	complemented
225	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
226	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
227	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
228	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
229	C₄⋊C₄	—	2	2	no	—
230	C₄⋊C₄	C₂²×Dic₅₃	1	4	yes	—
231	C₂²⋊C₄	—	2	4	no	—
232	C₂²⋊C₄	—	2	4	no	—
233	C₄²	D₄×D₅₃	1	1	yes	characteristic
234	C₄²	#11719947	1	1	yes	characteristic
235	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
236	C₄²	C₂³×D₅₃	1	1	yes	characteristic
237	C₂³	—	2	4	no	—
238	C₂³	—	2	8	no	—
239	C₂³	—	8	2	no	—
240	C₂³	—	8	2	no	—
241	C₂³	D₄×Dic₅₃	1	1	yes	characteristic
242	D₄	—	2	4	no	—
243	D₄	—	2	8	no	—
244	D₄	C₂²×#88322	1	8	yes	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	4	1	no	—
248	C₂×C₄	—	4	8	no	—
249	C₂×C₄	—	8	4	no	—
250	C₂×C₄	—	8	2	no	—
251	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
252	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
253	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
254	C₂×C₄	C₂²×#88322	1	2	yes	—
255	C₂×C₄	#13133224	1	2	yes	—
256	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
257	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
258	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
259	C₈	—	2	1	no	—
260	C₈	—	2	1	no	—
261	C₈	—	106	4	no	—
262	C₈	—	212	2	no	—
263	C₈	—	212	4	no	—
264	C₈	C₂²×#88322	1	2	yes	—
265	C₈	C₂²×#88322	1	2	yes	—
266	C₂²	—	2	1	no	—
267	C₂²	—	2	1	no	—
268	C₂²	—	2	8	no	—
269	C₂²	—	4	4	no	—
270	C₂²	—	4	4	no	—
271	C₂²	—	8	8	no	—
272	C₂²	D₄×#88322	1	2	yes	—
273	C₂²	C₂×D₄×Dic₅₃	1	1	yes	characteristic
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	4	4	no	—
278	C₄	D₄×#88322	1	2	yes	—
279	C₄	C₂³×#88322	1	1	yes	characteristic
280	C₄	C₂×#13133224	1	1	yes	characteristic
281	C₄	#16811424	1	2	yes	—
282	C₂	—	2	2	no	—
283	C₂	—	4	8	no	—
284	C₂	C₂×D₄×#88322	1	1	yes	characteristic
285	C₂	#20283567	1	1	yes	characteristic
286	C₂	#20283632	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₄×C₅₃	C₂³×C₈
Frattini	C₄²	C₂³×D₅₃
Fitting	C₂×C₄×D₈×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂²×D₈×C₅₃	C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	C₂×C₄×D₈	D₅₃
hypocenter	C₅₃	#17854
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×D₈

Complements

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

N	H ≅ G/N	type
1. #20283739	C₂	semidirect
2. #20273539	C₂	semidirect
3. D₈×#88322	C₂	semidirect
4. #20273463	C₂	semidirect
5. #20273106	C₂	semidirect
6. #20272998	C₂	semidirect
9. D₄×#88322	C₂²	semidirect
14. #16810376	C₂²	semidirect
15. #16810374	C₂²	semidirect
16. D₄⋊#88322	C₂²	semidirect
17. #16810344	C₂²	semidirect
18. C₈×#88322	C₂²	semidirect
41. C₂×D₈×C₅₃	C₈	semidirect
64. C₂×D₄×C₅₃	C₂×C₈	semidirect
73. D₈×C₅₃	C₂×C₈	semidirect
74. D₈×C₅₃	C₂×C₈	semidirect
76. C₂×C₈×C₅₃	C₂×C₈	semidirect
171. C₅₃	#17854	semidirect
174. C₂×D₈	#88322	semidirect
208. C₂×D₄	C₂×#88322	semidirect
217. D₈	C₂×#88322	semidirect
218. D₈	C₂×#88322	semidirect
224. C₂×C₈	C₂×#88322	semidirect

Maximal subgroups

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

subgroup	index
#20283739	2
#20273539	2
D₈×#88322	2
#20273463	2
#20273106	2
#20272998	2
C₂×C₄×D₈×C₅₃	2
#17854	53