Subgroups — #28313763

Order 13568.

922 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20291519	C₂	1	2	yes	maximal
2	#20289919	C₂	1	2	yes	maximal, complemented
3	#20289918	C₂	1	1	yes	maximal, characteristic, complemented
4	#20289913	C₂	1	4	yes	maximal, complemented
5	M₁₆×#88324	C₂	1	2	yes	maximal, complemented
6	C₈×D₅₃⋊C₈	C₂	1	2	yes	maximal, complemented
7	C₂×C₈×#88324	C₂	1	1	yes	maximal, characteristic, complemented
8	#20283137	C₂	1	1	yes	maximal, characteristic
9	#16811672	C₄	1	4	yes	—
10	#16811651	C₄	1	4	yes	—
11	#16811650	C₄	1	8	yes	—
12	#16811641	C₂²	1	4	yes	—
13	#16811639	C₂²	1	2	yes	—
14	C₄×D₅₃⋊C₈	C₂²	1	4	yes	—
15	C₂×C₄×#88324	C₂²	1	2	yes	—
16	#16811464	C₂²	1	4	yes	—
17	#16811464	C₂²	1	2	yes	—
18	C₈×#88324	C₂²	1	1	yes	characteristic
19	C₈×#88324	C₂²	1	2	yes	—
20	C₈×#88324	C₂²	1	1	yes	characteristic
21	C₈×#88324	C₂²	1	4	yes	—
22	C₈×#88324	C₂²	1	2	yes	—
23	C₂×#13133225	C₄	1	1	yes	characteristic
24	C₂×#13133223	C₄	1	1	yes	characteristic
25	#16811091	C₂²	1	1	yes	characteristic
26	#16810580	C₄	1	2	yes	—
27	#16810579	C₄	1	1	yes	characteristic
28	#16810577	C₄	1	2	yes	—
29	#16810575	C₂²	1	2	yes	—
30	M₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
31	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
32	C₄×C₈×D₅₃	C₂²	1	2	yes	—
33	C₅₃×#405	C₄	1	1	yes	characteristic
34	C₂×D₅₃⋊C₈	C₂×C₄	1	8	yes	—
35	C₂²×#88324	C₂×C₄	1	4	yes	—
36	#13133284	C₂×C₄	1	8	yes	—
37	C₄⋊#88324	C₂×C₄	1	4	yes	—
38	#13133274	C₂×C₄	1	8	yes	—
39	C₂²⋊#88324	C₂×C₄	1	4	yes	—
40	C₄×#88324	C₂×C₄	1	4	yes	—
41	C₄×#88324	C₂×C₄	1	8	yes	—
42	C₄×#88324	C₂³	1	2	yes	—
43	C₄×#88324	C₂³	1	2	yes	—
44	C₄×#88324	C₂³	1	4	yes	—
45	C₂×#11719945	C₂×C₄	1	1	yes	characteristic
46	#13133225	C₈	1	8	yes	complemented
47	#13133223	C₈	1	8	yes	complemented
48	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
49	C₂×#11719912	C₂×C₄	1	2	yes	—
50	C₂×#11719912	C₂×C₄	1	2	yes	—
51	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
52	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
53	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
54	#13133182	C₂×C₄	1	2	yes	—
55	#13133179	C₂×C₄	1	2	yes	—
56	#13133177	C₂×C₄	1	4	yes	—
57	#13133174	C₂×C₄	1	2	yes	—
58	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
59	C₄²×D₅₃	C₂³	1	2	yes	—
60	#13133128	C₂×C₄	1	2	yes	—
61	#13133127	C₂×C₄	1	2	yes	—
62	C₄×#88322	C₂×C₄	1	2	yes	—
63	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
64	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
65	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
66	C₈×Dic₅₃	C₂³	1	2	yes	—
67	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
68	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
69	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
70	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
71	#97×C₅₃	C₂×C₄	1	2	yes	—
72	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
73	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₂²⋊D₅₃	C₄²	1	2	yes	—
75	C₂²×Dic₅₃	C₂²×C₄	1	2	yes	—
76	#11719945	C₂×C₈	1	4	yes	—
77	#11719945	C₂×C₈	1	4	yes	—
78	C₂×D₂₁₂	C₄²	1	1	yes	characteristic
79	C₂×C₄×D₅₃	C₂²×C₄	1	4	yes	—
80	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
81	C₂×Dic₁₀₆	C₄²	1	1	yes	characteristic
82	C₂×#88324	—	2	8	no	—
83	C₂×#88324	C₂²×C₄	1	8	yes	—
84	C₂×#88324	C₂²×C₄	1	4	yes	—
85	C₂×#88324	C₂²×C₄	1	4	yes	—
86	D₅₃⋊C₈	—	2	16	no	—
87	M₁₆×C₅₃	C₂×C₈	1	4	yes	complemented
88	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	complemented
89	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
90	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
92	C₄⋊C₄×C₅₃	C₄²	1	2	yes	—
93	C₂²⋊C₄×C₅₃	C₄²	1	2	yes	—
94	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
95	#11719921	C₄²	1	4	yes	—
96	#11719919	C₄²	1	4	yes	—
97	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
98	C₄×Dic₅₃	C₂²×C₄	1	4	yes	—
99	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
100	C₄×Dic₅₃	C₂⁴	1	1	yes	characteristic
101	#11719917	C₂×C₈	1	4	yes	complemented
102	C₂×#88322	C₂×C₈	1	4	yes	complemented
103	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
104	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
105	C₂×#88322	C₂²×C₄	1	2	yes	—
106	#11719912	C₂×C₈	1	8	yes	complemented
107	#11719912	C₂×C₈	1	8	yes	complemented
108	C₈×D₅₃	C₂×C₈	1	8	yes	complemented
109	C₈×D₅₃	C₂×C₈	1	8	yes	complemented
110	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
111	C₂²×D₅₃	C₂×C₄²	1	2	yes	—
112	C₂×C₄×C₅₃	C₂×C₄²	1	4	yes	—
113	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
114	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
115	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
116	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
117	C₂²⋊D₅₃	C₄×C₈	1	8	yes	—
118	C₂×Dic₅₃	—	2	4	no	—
119	C₂×Dic₅₃	C₂×C₄²	1	2	yes	—
120	C₂×Dic₅₃	C₂×C₄²	1	2	yes	—
121	C₂×Dic₅₃	C₂³×C₄	1	2	yes	—
122	D₂₁₂	C₄×C₈	1	4	yes	—
123	C₄×D₅₃	—	2	8	no	—
124	C₄×D₅₃	C₂²×C₈	1	4	yes	—
125	C₄×D₅₃	C₂²×C₈	1	4	yes	—
126	Dic₁₀₆	C₄×C₈	1	4	yes	—
127	#88324	—	2	4	no	—
128	#88324	—	2	8	no	—
129	#88324	#131	1	8	yes	complemented
130	C₈×C₅₃	C₂²×C₈	1	2	yes	—
131	C₈×C₅₃	C₂²×C₈	1	4	yes	—
132	C₈×C₅₃	C₂²×C₈	1	2	yes	—
133	#88322	C₂²×C₈	1	2	yes	—
134	#88322	C₂²×C₈	1	4	yes	—
135	#88322	C₂²×C₈	1	2	yes	—
136	C₈×#131	—	53	1	no	maximal, Sylow
137	C₂²×C₅₃	—	2	2	no	—
138	C₂²×C₅₃	C₂×C₄×C₈	1	2	yes	—
139	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
140	C₂×D₅₃	—	2	4	no	—
141	C₂×D₅₃	C₂×C₄×C₈	1	4	yes	—
142	C₄×C₅₃	—	2	4	no	—
143	C₄×C₅₃	C₂×C₄×C₈	1	2	yes	—
144	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
145	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
146	Dic₅₃	—	2	2	no	—
147	Dic₅₃	C₂×C₄×C₈	1	4	yes	—
148	Dic₅₃	C₂×#131	1	4	yes	—
149	C₈×#41	—	53	2	no	—
150	C₄×#131	—	53	1	no	—
151	#1450	—	53	1	no	—
152	#1450	—	53	2	no	—
153	C₈×M₁₆	—	53	2	no	—
154	C₈×M₁₆	—	53	4	no	—
155	C₂×C₈²	—	53	1	no	—
156	C₂×C₈²	—	53	2	no	—
157	C₂×C₅₃	—	2	2	no	—
158	C₂×C₅₃	C₂²×C₄×C₈	1	1	yes	characteristic
159	C₂×C₅₃	C₄×#131	1	2	yes	—
160	D₅₃	—	2	4	no	—
161	C₂×#131	—	53	1	no	—
162	C₂×#131	—	53	1	no	—
163	C₄×#41	—	53	1	no	—
164	C₈×Q₈	—	53	4	no	—
165	C₈×D₄	—	53	8	no	—
166	C₈×D₄	—	53	4	no	—
167	#431	—	53	2	no	—
168	#431	—	53	4	no	—
169	#405	—	53	1	no	—
170	#405	—	53	2	no	—
171	#405	—	53	2	no	—
172	#405	#6157	1	1	yes	characteristic
173	C₄×M₁₆	—	53	1	no	—
174	C₄×M₁₆	—	53	2	no	—
175	C₂×C₄×C₈	—	53	2	no	—
176	C₂×C₄×C₈	—	53	4	no	—
177	C₂×C₄×C₈	—	53	1	no	—
178	C₂×C₄×C₈	—	53	2	no	—
179	#322	—	53	4	no	—
180	#322	—	53	2	no	—
181	C₈²	—	53	1	no	—
182	C₈²	—	53	2	no	—
183	C₈²	—	53	4	no	—
184	C₈²	—	53	1	no	—
185	C₈²	—	53	2	no	—
186	C₅₃	C₈×#131	1	1	yes	characteristic, complemented, Sylow
187	C₂×#41	—	53	1	no	—
188	#131	—	53	8	no	—
189	#131	—	53	8	no	—
190	C₂×M₁₆	—	53	1	no	—
191	C₂×M₁₆	—	53	2	no	—
192	C₂×M₁₆	—	53	2	no	—
193	C₂×M₁₆	C₂×#6157	1	1	yes	characteristic
194	C₂²×C₈	—	53	1	no	—
195	C₂²×C₈	—	53	2	no	—
196	C₂²×C₈	—	53	2	no	—
197	C₂²×C₈	—	53	4	no	—
198	C₂²×C₈	—	53	8	no	—
199	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
200	C₄×Q₈	—	53	2	no	—
201	C₄×D₄	—	53	2	no	—
202	C₄×D₄	—	53	4	no	—
203	#117	—	53	2	no	—
204	#117	C₂×#6157	1	1	yes	characteristic
205	C₂×C₄²	—	53	1	no	—
206	C₂×C₄²	—	53	2	no	—
207	C₄⋊C₈	—	53	4	no	—
208	C₄⋊C₈	—	53	8	no	—
209	C₂²⋊C₈	—	53	4	no	—
210	C₂²⋊C₈	—	53	8	no	—
211	#97	—	53	2	no	—
212	#97	—	53	2	no	—
213	#97	C₂×#6157	1	2	yes	—
214	C₄×C₈	—	53	2	no	—
215	C₄×C₈	—	53	2	no	—
216	C₄×C₈	—	53	4	no	—
217	C₄×C₈	—	53	4	no	—
218	C₄×C₈	—	53	8	no	—
219	C₄×C₈	—	53	1	no	—
220	C₄×C₈	—	53	2	no	—
221	C₄×C₈	—	53	1	no	—
222	C₄×C₈	—	53	1	no	—
223	C₄×C₈	—	53	1	no	—
224	C₄×C₈	—	53	2	no	—
225	C₄×C₈	C₂×#6157	1	2	yes	—
226	#41	—	53	4	no	—
227	#41	—	53	4	no	—
228	C₂×Q₈	—	53	1	no	—
229	C₂×D₄	—	53	2	no	—
230	C₂×D₄	—	53	1	no	—
231	C₂²×C₄	—	53	2	no	—
232	C₂²×C₄	—	53	2	no	—
233	C₂²×C₄	—	53	4	no	—
234	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
235	M₁₆	—	53	4	no	—
236	M₁₆	—	53	8	no	—
237	M₁₆	—	53	8	no	—
238	M₁₆	D₅₃⋊C₈	1	4	yes	complemented
239	C₂×C₈	—	53	4	no	—
240	C₂×C₈	—	53	4	no	—
241	C₂×C₈	—	53	1	no	—
242	C₂×C₈	—	53	1	no	—
243	C₂×C₈	—	53	2	no	—
244	C₂×C₈	—	53	8	no	—
245	C₂×C₈	—	53	8	no	—
246	C₂×C₈	—	53	8	no	—
247	C₂×C₈	—	53	4	no	—
248	C₂×C₈	—	106	8	no	—
249	C₂×C₈	—	106	16	no	—
250	C₂×C₈	D₅₃⋊C₈	1	4	yes	complemented
251	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
252	C₂×C₈	C₂²×#6157	1	2	yes	—
253	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
254	C₄⋊C₄	—	53	4	no	—
255	C₄⋊C₄	C₄×#6157	1	2	yes	—
256	C₂²⋊C₄	—	53	4	no	—
257	C₂²⋊C₄	C₄×#6157	1	2	yes	—
258	C₄²	—	53	4	no	—
259	C₄²	—	53	2	no	—
260	C₄²	—	53	1	no	—
261	C₄²	—	53	1	no	—
262	C₄²	C₂²×#6157	1	2	yes	—
263	C₂³	—	53	2	no	—
264	C₂³	C₂×C₄×#6157	1	1	yes	characteristic
265	Q₈	—	53	4	no	—
266	D₄	—	53	8	no	—
267	D₄	—	53	4	no	—
268	C₂×C₄	—	53	2	no	—
269	C₂×C₄	—	53	2	no	—
270	C₂×C₄	—	53	4	no	—
271	C₂×C₄	—	53	4	no	—
272	C₂×C₄	—	53	2	no	—
273	C₂×C₄	—	106	8	no	—
274	C₂×C₄	—	106	4	no	—
275	C₂×C₄	C₂×C₄×#6157	1	4	yes	—
276	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
277	C₂×C₄	C₂×D₅₃⋊C₈	1	2	yes	—
278	C₂×C₄	C₂×D₅₃⋊C₈	1	2	yes	—
279	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
280	C₈	—	53	8	no	—
281	C₈	—	53	2	no	—
282	C₈	—	53	4	no	—
283	C₈	—	53	2	no	—
284	C₈	—	106	4	no	—
285	C₈	—	106	8	no	—
286	C₈	C₂×D₅₃⋊C₈	1	2	yes	—
287	C₈	C₂×D₅₃⋊C₈	1	4	yes	—
288	C₈	C₂×D₅₃⋊C₈	1	2	yes	—
289	C₂²	—	2	2	no	—
290	C₂²	—	53	4	no	—
291	C₂²	—	106	4	no	—
292	C₂²	C₄×D₅₃⋊C₈	1	2	yes	—
293	C₂²	C₂²×C₄×#6157	1	1	yes	characteristic
294	C₄	—	2	4	no	—
295	C₄	—	53	4	no	—
296	C₄	—	53	4	no	—
297	C₄	—	106	2	no	—
298	C₄	C₄×D₅₃⋊C₈	1	2	yes	—
299	C₄	C₂²×D₅₃⋊C₈	1	1	yes	characteristic
300	C₄	C₂²×D₅₃⋊C₈	1	1	yes	characteristic
301	C₂	—	2	2	no	—
302	C₂	—	106	4	no	—
303	C₂	C₂×C₄×D₅₃⋊C₈	1	1	yes	characteristic
304	C₂	#131×#6157	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂²×#6157
commutator	C₂×C₅₃	C₂²×C₄×C₈
Frattini	C₂×C₄	C₂³×#6157
Fitting	C₅₃×#405	C₄
socle	C₂²×C₅₃	C₂²×C₄²
prime radical	C₂×#11719945	C₂×C₄
cosocle	C₄×Dic₅₃	C₂⁴
hypercenter	#405	#6157
hypocenter	C₅₃	C₈×#131
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ #405

Complements

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

N	H ≅ G/N	type
2. #20289919	C₂	semidirect
3. #20289918	C₂	semidirect
4. #20289913	C₂	semidirect
5. M₁₆×#88324	C₂	semidirect
6. C₈×D₅₃⋊C₈	C₂	semidirect
7. C₂×C₈×#88324	C₂	semidirect
46. #13133225	C₈	semidirect
47. #13133223	C₈	semidirect
87. M₁₆×C₅₃	C₂×C₈	semidirect
88. C₂×C₈×C₅₃	C₂×C₈	semidirect
101. #11719917	C₂×C₈	semidirect
102. C₂×#88322	C₂×C₈	semidirect
106. #11719912	C₂×C₈	semidirect
107. #11719912	C₂×C₈	semidirect
108. C₈×D₅₃	C₂×C₈	semidirect
109. C₈×D₅₃	C₂×C₈	semidirect
129. #88324	#131	semidirect
186. C₅₃	C₈×#131	semidirect
238. M₁₆	D₅₃⋊C₈	semidirect
250. C₂×C₈	D₅₃⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#20291519	2
#20289919	2
#20289918	2
#20289913	2
M₁₆×#88324	2
C₈×D₅₃⋊C₈	2
C₂×C₈×#88324	2
#20283137	2
C₈×#131	53