Subgroups — #27300977

Order 13568.

488 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20287818	C₂	1	1	yes	maximal, characteristic, complemented
2	#20276336	C₂	1	1	yes	maximal, characteristic, complemented
3	#20276326	C₂	1	2	yes	maximal, complemented
4	#20272801	C₂	1	2	yes	maximal, complemented
5	C₅₃×#2929	C₂	1	1	yes	maximal, characteristic
6	#16811040	—	2	4	no	—
7	C₂×#13133157	—	2	2	no	—
8	C₂×#13133157	C₂²	1	2	yes	—
9	C₂×#13133157	C₂²	1	2	yes	—
10	#16810405	—	2	2	no	—
11	C₂²⋊#11719908	—	2	4	no	—
12	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
13	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×#525	C₄	1	1	yes	characteristic
15	C₅₃×#435	—	2	2	no	—
16	C₈×D₄×C₅₃	—	2	2	no	—
17	C₅₃×#424	C₄	1	1	yes	characteristic
18	C₅₃×#408	C₄	1	1	yes	characteristic
19	C₅₃×#407	C₄	1	1	yes	characteristic
20	C₅₃×#405	C₂²	1	1	yes	characteristic
21	#13133157	—	2	4	no	—
22	#13133157	—	2	2	no	—
23	#13133157	—	2	2	no	—
24	#13133157	—	2	2	no	—
25	#13133157	—	2	2	no	—
26	C₂×#11719908	—	2	4	no	—
27	C₂×#11719908	—	2	2	no	—
28	C₂×#11719908	D₄	1	4	yes	complemented
29	C₂×#41×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
31	#131×C₅₃	—	2	2	no	—
32	#131×C₅₃	—	2	2	no	—
33	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂×M₁₆×C₅₃	D₄	1	2	yes	—
36	C₂²×C₈×C₅₃	—	2	2	no	—
37	C₂²×C₈×C₅₃	—	2	1	no	—
38	C₂²×C₈×C₅₃	—	2	1	no	—
39	C₂²×C₈×C₅₃	—	2	2	no	—
40	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₅₃	D₄	1	2	yes	—
42	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
43	#123×C₅₃	C₈	1	2	yes	—
44	C₂²⋊Q₈×C₅₃	—	2	1	no	—
45	#121×C₅₃	—	2	1	no	—
46	#120×C₅₃	C₈	1	2	yes	—
47	C₄×D₄×C₅₃	—	2	2	no	—
48	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
49	C₄⋊C₈×C₅₃	—	2	2	no	—
50	C₂²⋊C₈×C₅₃	—	2	2	no	—
51	C₂²⋊C₈×C₅₃	—	2	2	no	—
52	#97×C₅₃	—	2	1	no	—
53	C₄×C₈×C₅₃	—	2	1	no	—
54	C₂⁴×C₅₃	C₂×C₈	1	1	yes	characteristic
55	#41×C₅₃	—	2	2	no	—
56	#41×C₅₃	—	2	2	no	—
57	C₂×Q₈×C₅₃	M₁₆	1	1	yes	characteristic
58	C₂×D₄×C₅₃	—	2	2	no	—
59	C₂×D₄×C₅₃	C₂×C₈	1	2	yes	—
60	C₂×D₄×C₅₃	M₁₆	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	—	2	1	no	—
62	C₂²×C₄×C₅₃	—	2	2	no	—
63	C₂²×C₄×C₅₃	—	2	1	no	—
64	C₂²×C₄×C₅₃	—	2	2	no	—
65	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
66	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
68	M₁₆×C₅₃	—	2	2	no	—
69	M₁₆×C₅₃	—	2	4	no	—
70	M₁₆×C₅₃	—	2	2	no	—
71	C₂×C₈×C₅₃	—	2	2	no	—
72	C₂×C₈×C₅₃	—	2	2	no	—
73	C₂×C₈×C₅₃	—	2	1	no	—
74	C₂×C₈×C₅₃	—	2	1	no	—
75	C₂×C₈×C₅₃	—	2	2	no	—
76	C₂×C₈×C₅₃	—	4	1	no	—
77	C₂×C₈×C₅₃	—	4	2	no	—
78	C₂×C₈×C₅₃	—	4	1	no	—
79	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
80	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
81	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₅₃	#41	1	2	yes	—
84	C₄⋊C₄×C₅₃	—	2	2	no	—
85	C₄⋊C₄×C₅₃	—	2	1	no	—
86	C₂²⋊C₄×C₅₃	—	2	2	no	—
87	C₂²⋊C₄×C₅₃	—	2	2	no	—
88	C₂²⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
89	C₄²×C₅₃	—	2	1	no	—
90	#11719908	—	2	4	no	—
91	#11719908	—	4	2	no	—
92	C₂³×C₅₃	—	2	1	no	—
93	C₂³×C₅₃	—	2	2	no	—
94	C₂³×C₅₃	—	4	2	no	—
95	C₂³×C₅₃	C₂²⋊C₈	1	2	yes	—
96	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
97	Q₈×C₅₃	—	2	2	no	—
98	D₄×C₅₃	—	2	4	no	—
99	D₄×C₅₃	—	2	2	no	—
100	D₄×C₅₃	—	4	4	no	—
101	C₂×C₄×C₅₃	—	2	2	no	—
102	C₂×C₄×C₅₃	—	2	2	no	—
103	C₂×C₄×C₅₃	—	2	1	no	—
104	C₂×C₄×C₅₃	—	2	2	no	—
105	C₂×C₄×C₅₃	—	2	2	no	—
106	C₂×C₄×C₅₃	—	4	1	no	—
107	C₂×C₄×C₅₃	—	4	1	no	—
108	C₂×C₄×C₅₃	—	4	2	no	—
109	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
110	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
112	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
113	C₂×C₄×C₅₃	#131	1	2	yes	—
114	C₈×C₅₃	—	2	2	no	—
115	C₈×C₅₃	—	4	1	no	—
116	C₈×C₅₃	—	4	1	no	—
117	C₈×C₅₃	#120	1	1	yes	characteristic
118	C₈×C₅₃	#121	1	2	yes	—
119	C₈×C₅₃	#123	1	1	yes	characteristic
120	#12981	—	53	1	no	maximal, Sylow
121	C₂²×C₅₃	—	2	2	no	—
122	C₂²×C₅₃	—	2	4	no	—
123	C₂²×C₅₃	—	2	2	no	—
124	C₂²×C₅₃	—	4	1	no	—
125	C₂²×C₅₃	—	4	2	no	—
126	C₂²×C₅₃	—	4	1	no	—
127	C₂²×C₅₃	—	4	2	no	—
128	C₂²×C₅₃	—	4	1	no	—
129	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
130	C₂²×C₅₃	C₈×D₄	1	2	yes	—
131	C₄×C₅₃	—	2	2	no	—
132	C₄×C₅₃	—	4	2	no	—
133	C₄×C₅₃	—	4	1	no	—
134	C₄×C₅₃	#386	1	1	yes	characteristic
135	C₄×C₅₃	#408	1	1	yes	characteristic
136	C₄×C₅₃	#435	1	2	yes	—
137	#3407	—	53	1	no	—
138	#2929	D₅₃	1	1	yes	characteristic
139	#2115	—	53	1	no	—
140	#2112	—	53	2	no	—
141	#1378	—	53	2	no	—
142	C₂×C₅₃	—	2	1	no	—
143	C₂×C₅₃	—	2	2	no	—
144	C₂×C₅₃	—	4	2	no	—
145	C₂×C₅₃	—	4	2	no	—
146	C₂×C₅₃	#1869	1	1	yes	characteristic
147	C₂×#131	C₂×D₅₃	1	1	yes	characteristic
148	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
149	#525	Dic₅₃	1	1	yes	characteristic
150	#504	—	106	4	no	—
151	C₂×M₃₂	—	53	2	no	—
152	C₂×M₃₂	—	53	2	no	—
153	C₂×M₃₂	—	106	2	no	—
154	#435	—	2	2	no	—
155	C₈×D₄	—	2	2	no	—
156	#424	Dic₅₃	1	1	yes	characteristic
157	#408	Dic₅₃	1	1	yes	characteristic
158	#407	Dic₅₃	1	1	yes	characteristic
159	#405	C₂×D₅₃	1	1	yes	characteristic
160	#349	—	106	2	no	—
161	C₂²⋊C₁₆	—	106	4	no	—
162	C₅₃	#12981	1	1	yes	characteristic, complemented, Sylow
163	C₂×#41	C₂×Dic₅₃	1	1	yes	characteristic
164	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
165	#131	—	2	2	no	—
166	#131	—	2	2	no	—
167	C₂×M₁₆	C₂×Dic₅₃	1	1	yes	characteristic
168	C₂×M₁₆	C₂×Dic₅₃	1	1	yes	characteristic
169	C₂×M₁₆	C₂²⋊D₅₃	1	2	yes	—
170	C₂²×C₈	—	2	2	no	—
171	C₂²×C₈	—	2	2	no	—
172	C₂²×C₈	—	2	1	no	—
173	C₂²×C₈	—	2	1	no	—
174	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
175	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
176	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
177	#123	#88322	1	2	yes	—
178	C₂²⋊Q₈	—	2	1	no	—
179	#121	—	2	1	no	—
180	#120	#88322	1	2	yes	—
181	C₄×D₄	—	2	2	no	—
182	#117	C₂×Dic₅₃	1	1	yes	characteristic
183	M₃₂	—	106	4	no	—
184	M₃₂	—	106	2	no	—
185	M₃₂	—	106	2	no	—
186	M₃₂	—	106	2	no	—
187	M₃₂	—	106	2	no	—
188	C₂×C₁₆	—	53	4	no	—
189	C₂×C₁₆	—	106	4	no	—
190	C₂×C₁₆	—	106	2	no	—
191	C₄⋊C₈	—	2	2	no	—
192	C₂²⋊C₈	—	2	2	no	—
193	C₂²⋊C₈	—	2	2	no	—
194	#97	—	2	1	no	—
195	C₄×C₈	—	2	1	no	—
196	C₂⁴	C₂×#88322	1	1	yes	characteristic
197	#41	—	2	2	no	—
198	#41	—	2	2	no	—
199	C₂×Q₈	#11719917	1	1	yes	characteristic
200	C₂×D₄	—	2	2	no	—
201	C₂×D₄	C₂×#88322	1	2	yes	—
202	C₂×D₄	#11719917	1	1	yes	characteristic
203	C₂²×C₄	—	2	1	no	—
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	2	1	no	—
206	C₂²×C₄	—	2	2	no	—
207	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
208	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
209	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
210	M₁₆	—	2	2	no	—
211	M₁₆	—	2	4	no	—
212	M₁₆	—	2	2	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	1	no	—
216	C₂×C₈	—	2	1	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	4	1	no	—
219	C₂×C₈	—	4	2	no	—
220	C₂×C₈	—	4	1	no	—
221	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
222	C₂×C₈	D₄×D₅₃	1	2	yes	—
223	C₂×C₈	#11719947	1	2	yes	—
224	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
225	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
226	C₄⋊C₄	—	2	2	no	—
227	C₄⋊C₄	—	2	1	no	—
228	C₂²⋊C₄	—	2	2	no	—
229	C₂²⋊C₄	—	2	2	no	—
230	C₂²⋊C₄	C₂×#88322	1	2	yes	—
231	C₄²	—	2	1	no	—
232	C₁₆	—	106	4	no	—
233	C₁₆	—	212	2	no	—
234	C₂³	—	2	1	no	—
235	C₂³	—	2	2	no	—
236	C₂³	—	4	2	no	—
237	C₂³	C₂²⋊#88322	1	2	yes	—
238	C₂³	C₂²×#88322	1	1	yes	characteristic
239	Q₈	—	2	2	no	—
240	D₄	—	2	4	no	—
241	D₄	—	2	2	no	—
242	D₄	—	4	4	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	4	1	no	—
249	C₂×C₄	—	4	1	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	C₂²⋊#88322	1	2	yes	—
252	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
253	C₂×C₄	D₄×Dic₅₃	1	2	yes	—
254	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
255	C₂×C₄	#13133224	1	2	yes	—
256	C₈	—	2	2	no	—
257	C₈	—	4	1	no	—
258	C₈	—	4	1	no	—
259	C₈	#13133185	1	1	yes	characteristic
260	C₈	#13133189	1	2	yes	—
261	C₈	#13133198	1	1	yes	characteristic
262	C₂²	—	2	2	no	—
263	C₂²	—	2	4	no	—
264	C₂²	—	2	2	no	—
265	C₂²	—	4	1	no	—
266	C₂²	—	4	2	no	—
267	C₂²	—	4	1	no	—
268	C₂²	—	4	2	no	—
269	C₂²	—	4	1	no	—
270	C₂²	C₂×C₂²⋊#88322	1	1	yes	characteristic
271	C₂²	D₄×#88322	1	2	yes	—
272	C₄	—	2	2	no	—
273	C₄	—	4	2	no	—
274	C₄	—	4	1	no	—
275	C₄	#16810503	1	1	yes	characteristic
276	C₄	#16810590	1	1	yes	characteristic
277	C₄	#16810695	1	2	yes	—
278	C₂	—	2	1	no	—
279	C₂	—	2	2	no	—
280	C₂	—	4	2	no	—
281	C₂	—	4	2	no	—
282	C₂	#20275008	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#13133185
commutator	C₂³×C₅₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#2929	C₂
socle	C₂×C₅₃	#1869
prime radical	C₅₃×#525	C₄
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#2929	D₅₃
hypocenter	C₅₃	#12981
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2929

Complements

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

N	H ≅ G/N	type
1. #20287818	C₂	semidirect
2. #20276336	C₂	semidirect
3. #20276326	C₂	semidirect
4. #20272801	C₂	semidirect
28. C₂×#11719908	D₄	semidirect
162. C₅₃	#12981	semidirect

Maximal subgroups

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

subgroup	index
#20287818	2
#20276336	2
#20276326	2
#20272801	2
C₅₃×#2929	2
#12981	53