Subgroups — #28301672

Order 13568.

530 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20290008	C₂	1	2	yes	maximal
2	#20289918	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₈×#88324	C₂	1	1	yes	maximal, characteristic, complemented
4	#20289548	C₂	1	2	yes	maximal, complemented
5	#20283137	C₂	1	1	yes	maximal, characteristic
6	#16811639	C₂²	1	2	yes	—
7	C₂×C₄×#88324	C₂²	1	2	yes	—
8	#16811473	C₄	1	4	yes	—
9	#16811471	C₄	1	4	yes	—
10	#16811464	—	2	1	no	—
11	C₈×#88324	—	2	1	no	—
12	C₈×#88324	—	2	1	no	—
13	C₈×#88324	—	2	1	no	—
14	C₂×#13133225	C₄	1	1	yes	characteristic
15	C₂×#13133223	C₄	1	1	yes	characteristic
16	#16811091	C₂²	1	1	yes	characteristic
17	#16810580	—	2	1	no	—
18	#16810579	C₄	1	1	yes	characteristic
19	#16810577	—	2	1	no	—
20	#16810575	—	2	1	no	—
21	M₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
22	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
23	C₄×C₈×D₅₃	—	2	1	no	—
24	C₅₃×#405	C₄	1	1	yes	characteristic
25	C₂²×#88324	—	2	2	no	—
26	C₄⋊#88324	C₂×C₄	1	4	yes	—
27	C₂²⋊#88324	—	2	2	no	—
28	C₄×#88324	—	2	2	no	—
29	C₄×#88324	—	2	2	no	—
30	C₄×#88324	C₂×C₄	1	4	yes	—
31	C₂×#11719945	C₂×C₄	1	1	yes	characteristic
32	#13133225	—	2	2	no	—
33	#13133225	C₈	1	4	yes	complemented
34	#13133223	—	2	2	no	—
35	#13133223	C₈	1	4	yes	complemented
36	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
37	C₂×#11719912	—	2	1	no	—
38	C₂×#11719912	—	2	1	no	—
39	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
40	C₂×C₈×D₅₃	—	2	1	no	—
41	C₂×C₈×D₅₃	—	2	1	no	—
42	#13133182	C₂×C₄	1	2	yes	—
43	#13133179	C₂×C₄	1	2	yes	—
44	#13133177	—	2	2	no	—
45	#13133174	—	2	1	no	—
46	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
47	C₄²×D₅₃	—	2	1	no	—
48	#13133128	—	2	1	no	—
49	#13133127	—	2	1	no	—
50	C₄×#88322	—	2	1	no	—
51	C₈×Dic₅₃	—	2	1	no	—
52	C₈×Dic₅₃	D₄	1	1	yes	characteristic
53	C₈×Dic₅₃	D₄	1	1	yes	characteristic
54	C₈×Dic₅₃	D₄	1	1	yes	characteristic
55	C₈×Dic₅₃	D₄	1	1	yes	characteristic
56	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
57	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
58	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
59	#97×C₅₃	—	2	1	no	—
60	C₄×C₈×C₅₃	—	2	1	no	—
61	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₂²⋊D₅₃	—	2	1	no	—
63	C₂²×Dic₅₃	#41	1	2	yes	—
64	#11719945	—	2	1	no	—
65	#11719945	—	2	1	no	—
66	#11719945	C₂×C₈	1	2	yes	—
67	#11719945	C₂×C₈	1	2	yes	—
68	C₂×D₂₁₂	C₄²	1	1	yes	characteristic
69	C₂×C₄×D₅₃	—	2	1	no	—
70	C₂×C₄×D₅₃	—	2	2	no	—
71	C₂×Dic₁₀₆	C₄²	1	1	yes	characteristic
72	C₂×#88324	—	2	8	no	—
73	C₂×#88324	—	2	4	no	—
74	C₂×#88324	—	2	4	no	—
75	M₁₆×C₅₃	—	2	1	no	—
76	M₁₆×C₅₃	C₂×C₈	1	2	yes	—
77	C₂×C₈×C₅₃	—	2	1	no	—
78	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
81	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
82	C₄⋊C₄×C₅₃	C₄²	1	2	yes	—
83	C₂²⋊C₄×C₅₃	—	2	1	no	—
84	C₄²×C₅₃	—	2	1	no	—
85	#11719921	—	2	2	no	—
86	#11719919	—	2	2	no	—
87	C₄×Dic₅₃	—	2	2	no	—
88	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
89	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
90	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
91	#11719917	—	2	1	no	—
92	#11719917	C₂×C₈	1	2	yes	—
93	C₂×#88322	—	2	1	no	—
94	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
95	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
96	C₂×#88322	C₂×C₈	1	2	yes	—
97	C₂×#88322	M₁₆	1	2	yes	—
98	#11719912	—	2	4	no	—
99	#11719912	—	2	4	no	—
100	C₈×D₅₃	—	2	4	no	—
101	C₈×D₅₃	—	2	4	no	—
102	C₂³×C₅₃	#117	1	1	yes	characteristic
103	C₂²×D₅₃	—	2	1	no	—
104	C₂×C₄×C₅₃	—	2	2	no	—
105	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
106	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
108	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
109	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
110	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
111	C₂²⋊D₅₃	—	2	2	no	—
112	C₂²⋊D₅₃	—	2	2	no	—
113	C₂×Dic₅₃	—	2	1	no	—
114	C₂×Dic₅₃	—	2	4	no	—
115	C₂×Dic₅₃	C₄×D₄	1	2	yes	—
116	C₂×Dic₅₃	C₄×D₄	1	2	yes	—
117	D₂₁₂	—	2	1	no	—
118	D₂₁₂	C₄×C₈	1	2	yes	—
119	C₄×D₅₃	—	2	2	no	—
120	C₄×D₅₃	—	2	2	no	—
121	C₄×D₅₃	—	4	4	no	—
122	Dic₁₀₆	—	2	1	no	—
123	Dic₁₀₆	C₄×C₈	1	2	yes	—
124	#88324	—	2	8	no	—
125	#88324	—	4	4	no	—
126	C₈×C₅₃	—	2	2	no	—
127	C₈×C₅₃	C₂²⋊C₈	1	2	yes	—
128	C₈×C₅₃	C₂²⋊C₈	1	2	yes	—
129	#88322	—	2	2	no	—
130	#88322	C₂²⋊C₈	1	2	yes	—
131	#88322	C₂²⋊C₈	1	2	yes	—
132	#8184	—	53	1	no	maximal, Sylow
133	C₂²×C₅₃	—	2	2	no	—
134	C₂²×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
135	C₂²×C₅₃	#431	1	1	yes	characteristic
136	C₂²×C₅₃	#431	1	1	yes	characteristic
137	C₂×D₅₃	—	2	2	no	—
138	C₂×D₅₃	—	4	2	no	—
139	C₄×C₅₃	—	4	2	no	—
140	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
141	C₄×C₅₃	C₄×M₁₆	1	1	yes	characteristic
142	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
143	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
144	Dic₅₃	—	2	2	no	—
145	Dic₅₃	—	2	2	no	—
146	Dic₅₃	#443	1	4	yes	—
147	C₄×#131	—	53	1	no	—
148	#1476	—	53	2	no	—
149	#1450	—	53	1	no	—
150	C₂×C₈²	—	53	1	no	—
151	#1278	—	53	2	no	—
152	C₂×C₅₃	—	2	1	no	—
153	C₂×C₅₃	—	2	1	no	—
154	C₂×C₅₃	C₄×C₂²⋊C₈	1	1	yes	characteristic
155	C₂×C₅₃	#1764	1	2	yes	—
156	D₅₃	—	4	2	no	—
157	C₂×#131	—	53	1	no	—
158	C₂×#131	—	53	1	no	—
159	C₄×#41	—	53	1	no	—
160	#431	—	53	2	no	—
161	#405	—	53	1	no	—
162	#405	—	106	1	no	—
163	#405	—	106	1	no	—
164	#405	#6157	1	1	yes	characteristic
165	C₄×M₁₆	—	53	1	no	—
166	C₄×M₁₆	—	106	1	no	—
167	C₂×C₄×C₈	—	53	1	no	—
168	C₂×C₄×C₈	—	53	2	no	—
169	C₂×C₄×C₈	—	106	1	no	—
170	Q₈⋊C₈	—	53	4	no	—
171	D₄⋊C₈	—	53	4	no	—
172	#322	—	106	1	no	—
173	C₈²	—	106	1	no	—
174	C₈²	—	106	1	no	—
175	C₈²	—	106	1	no	—
176	C₅₃	#8184	1	1	yes	characteristic, complemented, Sylow
177	C₂×#41	—	53	1	no	—
178	#131	—	53	4	no	—
179	#131	—	53	4	no	—
180	#131	—	106	2	no	—
181	#131	—	106	2	no	—
182	C₂×M₁₆	—	53	1	no	—
183	C₂×M₁₆	—	106	1	no	—
184	C₂×M₁₆	—	106	1	no	—
185	C₂×M₁₆	C₂×#6157	1	1	yes	characteristic
186	C₂²×C₈	—	53	1	no	—
187	C₂²×C₈	—	106	1	no	—
188	C₂²×C₈	—	106	2	no	—
189	C₂²×C₈	—	106	1	no	—
190	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
191	C₄×Q₈	—	53	2	no	—
192	C₄×D₄	—	53	2	no	—
193	C₄×D₄	—	106	2	no	—
194	#117	—	106	1	no	—
195	#117	C₂×#6157	1	1	yes	characteristic
196	C₂×C₄²	—	53	1	no	—
197	C₂×C₄²	—	106	1	no	—
198	C₄⋊C₈	—	53	4	no	—
199	C₂²⋊C₈	—	106	2	no	—
200	#97	—	2	1	no	—
201	#97	—	106	1	no	—
202	#97	—	106	1	no	—
203	C₄×C₈	—	2	1	no	—
204	C₄×C₈	—	53	1	no	—
205	C₄×C₈	—	53	1	no	—
206	C₄×C₈	—	53	1	no	—
207	C₄×C₈	—	53	1	no	—
208	C₄×C₈	—	53	4	no	—
209	C₄×C₈	—	106	1	no	—
210	C₄×C₈	—	106	1	no	—
211	C₄×C₈	—	106	2	no	—
212	C₄×C₈	—	106	2	no	—
213	#41	—	53	2	no	—
214	#41	—	53	2	no	—
215	#41	—	106	1	no	—
216	#41	—	106	1	no	—
217	C₂×Q₈	—	53	1	no	—
218	C₂×D₄	—	53	1	no	—
219	C₂×D₄	—	106	1	no	—
220	C₂²×C₄	—	53	2	no	—
221	C₂²×C₄	—	106	2	no	—
222	C₂²×C₄	—	106	1	no	—
223	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
224	M₁₆	—	2	1	no	—
225	M₁₆	—	53	2	no	—
226	M₁₆	—	106	4	no	—
227	M₁₆	—	106	1	no	—
228	M₁₆	—	106	4	no	—
229	M₁₆	D₅₃⋊C₈	1	2	yes	—
230	C₂×C₈	—	2	1	no	—
231	C₂×C₈	—	53	1	no	—
232	C₂×C₈	—	53	1	no	—
233	C₂×C₈	—	53	2	no	—
234	C₂×C₈	—	53	2	no	—
235	C₂×C₈	—	106	4	no	—
236	C₂×C₈	—	106	4	no	—
237	C₂×C₈	—	106	8	no	—
238	C₂×C₈	—	106	1	no	—
239	C₂×C₈	—	106	4	no	—
240	C₂×C₈	—	106	4	no	—
241	C₂×C₈	D₅₃⋊C₈	1	2	yes	—
242	C₂×C₈	#11719936	1	2	yes	—
243	C₂×C₈	C₂²⋊#6157	1	1	yes	characteristic
244	C₂×C₈	C₂²⋊#6157	1	1	yes	characteristic
245	C₄⋊C₄	—	106	2	no	—
246	C₄⋊C₄	C₄×#6157	1	2	yes	—
247	C₂²⋊C₄	—	2	1	no	—
248	C₂²⋊C₄	—	106	2	no	—
249	C₄²	—	2	1	no	—
250	C₄²	—	53	2	no	—
251	C₄²	—	53	1	no	—
252	C₄²	—	53	1	no	—
253	C₄²	—	106	2	no	—
254	C₂³	—	106	1	no	—
255	C₂³	#13133296	1	1	yes	characteristic
256	Q₈	—	53	2	no	—
257	Q₈	—	106	1	no	—
258	D₄	—	53	2	no	—
259	D₄	—	106	2	no	—
260	D₄	—	106	2	no	—
261	D₄	—	106	1	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	53	2	no	—
264	C₂×C₄	—	53	2	no	—
265	C₂×C₄	—	106	4	no	—
266	C₂×C₄	—	106	1	no	—
267	C₂×C₄	—	106	2	no	—
268	C₂×C₄	—	106	2	no	—
269	C₂×C₄	—	212	4	no	—
270	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
271	C₂×C₄	C₂×C₂²⋊#6157	1	1	yes	characteristic
272	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
273	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
274	C₂×C₄	D₅₃⋊M₁₆	1	1	yes	characteristic
275	C₂×C₄	C₂×#11719936	1	1	yes	characteristic
276	C₈	—	2	2	no	—
277	C₈	—	53	2	no	—
278	C₈	—	53	2	no	—
279	C₈	—	106	8	no	—
280	C₈	—	106	2	no	—
281	C₈	—	212	4	no	—
282	C₈	#13133274	1	2	yes	—
283	C₈	#13133274	1	2	yes	—
284	C₂²	—	2	2	no	—
285	C₂²	—	106	2	no	—
286	C₂²	—	212	2	no	—
287	C₂²	C₄×C₂²⋊#6157	1	1	yes	characteristic
288	C₂²	#16811641	1	1	yes	characteristic
289	C₂²	#16811642	1	1	yes	characteristic
290	C₄	—	4	2	no	—
291	C₄	—	53	4	no	—
292	C₄	—	106	2	no	—
293	C₄	—	106	2	no	—
294	C₄	C₄×D₅₃⋊C₈	1	1	yes	characteristic
295	C₄	C₄×#11719936	1	1	yes	characteristic
296	C₄	D₅₃⋊(C₂²⋊C₈)	1	1	yes	characteristic
297	C₄	C₂×#13133274	1	1	yes	characteristic
298	C₂	—	2	1	no	—
299	C₂	—	2	1	no	—
300	C₂	—	212	2	no	—
301	C₂	C₄×#13133274	1	1	yes	characteristic
302	C₂	#20290489	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂²⋊#6157
commutator	C₄×C₅₃	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×#6157
Fitting	C₅₃×#405	C₄
socle	C₂²×C₅₃	C₄×C₂²⋊C₄
prime radical	C₂×#11719945	C₂×C₄
cosocle	C₂×C₄×Dic₅₃	C₂³
hypercenter	#405	#6157
hypocenter	C₅₃	#8184
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #405

Complements

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

N	H ≅ G/N	type
2. #20289918	C₂	semidirect
3. C₂×C₈×#88324	C₂	semidirect
4. #20289548	C₂	semidirect
33. #13133225	C₈	semidirect
35. #13133223	C₈	semidirect
176. C₅₃	#8184	semidirect

Maximal subgroups

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

subgroup	index
#20290008	2
#20289918	2
C₂×C₈×#88324	2
#20289548	2
#20283137	2
#8184	53