Subgroups — #28310290

Order 13568.

614 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₅₃⋊#108	C₂	1	1	yes	maximal, characteristic
2	C₂²×#13133285	C₂	1	1	yes	maximal, characteristic
3	#20289780	C₂	1	4	yes	maximal, complemented
4	C₂×M₃₂×D₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×D₅₃⋊M₁₆	—	2	1	no	—
6	C₂²×C₄⋊#6157	—	2	1	no	—
7	C₂×#13133288	—	2	1	no	—
8	D₅₃⋊#108	—	2	2	no	—
9	D₅₃⋊#108	C₄	1	4	yes	complemented
10	C₂×#13133287	C₂²	1	2	yes	—
11	C₂×#13133285	—	2	2	no	—
12	C₂×#13133285	—	2	1	no	—
13	C₂×#13133285	C₄	1	4	yes	—
14	C₂×#13133285	C₂²	1	2	yes	—
15	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
16	C₂×#13133157	C₄	1	1	yes	characteristic, complemented
17	C₂×#13133156	C₂²	1	2	yes	—
18	M₃₂×D₅₃	—	2	2	no	—
19	M₃₂×D₅₃	—	2	2	no	—
20	C₂×C₁₆×D₅₃	C₄	1	2	yes	complemented
21	C₂×M₃₂×C₅₃	C₄	1	1	yes	characteristic, complemented
22	C₂³×#6157	—	4	1	no	—
23	C₂×#11719936	—	2	2	no	—
24	D₅₃⋊M₁₆	—	2	4	no	—
25	D₅₃⋊M₁₆	—	4	2	no	—
26	C₂×#11719940	—	4	1	no	—
27	C₂×D₅₃⋊C₈	—	4	1	no	—
28	C₂×C₄⋊#6157	—	2	4	no	—
29	C₂×C₄⋊#6157	—	2	2	no	—
30	C₂×C₄⋊#6157	—	4	2	no	—
31	C₂×C₄⋊#6157	—	4	1	no	—
32	#13133288	—	2	4	no	—
33	#13133287	—	2	2	no	—
34	#13133287	C₂×C₄	1	4	yes	complemented
35	#13133285	—	2	4	no	—
36	#13133285	—	2	2	no	—
37	#13133285	C₂×C₄	1	4	yes	—
38	C₂²×C₄×D₅₃	D₄	1	1	yes	characteristic
39	C₂²×#88322	Q₈	1	1	yes	characteristic
40	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
41	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
42	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₈×D₅₃	D₄	1	2	yes	—
44	C₂×C₈×D₅₃	D₄	1	2	yes	—
45	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
46	C₂×C₈×D₅₃	Q₈	1	1	yes	characteristic
47	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
48	#13133157	—	2	2	no	—
49	#13133156	—	2	2	no	—
50	#13133156	—	2	2	no	—
51	C₂×#11719908	C₂×C₄	1	2	yes	complemented
52	C₁₆×D₅₃	—	2	4	no	—
53	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
54	M₃₂×C₅₃	—	2	2	no	—
55	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	complemented
56	C₂³×D₅₃	SD₁₆	1	1	yes	characteristic
57	C₂²×#6157	—	4	4	no	—
58	C₂²×#6157	—	4	2	no	—
59	C₂²×#6157	—	8	4	no	—
60	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
61	C₂²×Dic₅₃	SD₁₆	1	1	yes	characteristic
62	C₂×C₄×D₅₃	C₂²⋊C₄	1	2	yes	—
63	C₂×C₄×D₅₃	C₂²⋊C₄	1	2	yes	—
64	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×D₅₃	D₈	1	2	yes	—
66	C₂×C₄×D₅₃	SD₁₆	1	1	yes	characteristic
67	C₂×C₄×D₅₃	SD₁₆	1	1	yes	characteristic
68	C₂×C₄×D₅₃	Q₁₆	1	2	yes	—
69	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
70	#11719940	—	4	4	no	—
71	C₂×#88324	—	4	2	no	—
72	C₄⋊#6157	—	2	4	no	—
73	C₄⋊#6157	—	4	4	no	—
74	C₄⋊#6157	—	4	2	no	—
75	#11719936	—	2	4	no	—
76	#11719936	—	4	2	no	—
77	D₅₃⋊C₈	—	4	4	no	—
78	C₂×C₈×C₅₃	C₄²	1	2	yes	—
79	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
80	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
81	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
82	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
83	C₂×#88322	C₄⋊C₄	1	2	yes	—
84	C₂×#88322	C₄⋊C₄	1	2	yes	—
85	C₂×#88322	C₂×Q₈	1	1	yes	characteristic
86	C₈×D₅₃	C₄²	1	2	yes	—
87	C₈×D₅₃	C₂²⋊C₄	1	2	yes	—
88	C₈×D₅₃	C₂²⋊C₄	1	2	yes	—
89	C₈×D₅₃	C₂²⋊C₄	1	2	yes	—
90	C₈×D₅₃	C₄⋊C₄	1	2	yes	—
91	C₈×D₅₃	C₄⋊C₄	1	2	yes	—
92	C₈×D₅₃	C₂²×C₄	1	1	yes	characteristic
93	C₈×D₅₃	C₂²×C₄	1	1	yes	characteristic
94	C₈×D₅₃	C₂×D₄	1	1	yes	characteristic
95	C₈×D₅₃	C₂×D₄	1	1	yes	characteristic
96	C₁₆×C₅₃	—	2	2	no	—
97	#11719908	—	2	2	no	—
98	C₂³×C₅₃	#106	1	1	yes	characteristic
99	C₂²×D₅₃	—	2	4	no	—
100	C₂²×D₅₃	#102	1	2	yes	—
101	C₂²×D₅₃	#103	1	2	yes	—
102	C₂²×D₅₃	#106	1	1	yes	characteristic
103	C₂²×D₅₃	C₂×SD₁₆	1	1	yes	characteristic
104	C₂×#6157	—	4	4	no	—
105	C₂×#6157	—	8	4	no	—
106	C₂×#6157	—	8	8	no	—
107	C₂×C₄×C₅₃	#95	1	2	yes	—
108	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
109	C₂×C₄×C₅₃	#107	1	2	yes	—
110	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂×Dic₅₃	#102	1	2	yes	—
112	C₂×Dic₅₃	#103	1	2	yes	—
113	C₂×Dic₅₃	#106	1	1	yes	characteristic
114	C₂×Dic₅₃	C₂×SD₁₆	1	1	yes	characteristic
115	C₄×D₅₃	#95	1	2	yes	—
116	C₄×D₅₃	#102	1	2	yes	—
117	C₄×D₅₃	#102	1	2	yes	—
118	C₄×D₅₃	#103	1	2	yes	—
119	C₄×D₅₃	#103	1	2	yes	—
120	C₄×D₅₃	#107	1	2	yes	—
121	C₄×D₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₄×D₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₄×D₅₃	C₂×D₈	1	1	yes	characteristic
124	C₄×D₅₃	C₂×Q₁₆	1	1	yes	characteristic
125	#88324	—	4	4	no	—
126	C₈×C₅₃	#95	1	2	yes	—
127	C₈×C₅₃	C₂×C₄²	1	1	yes	characteristic
128	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
129	#88322	#95	1	2	yes	—
130	#88322	C₂×C₄⋊C₄	1	1	yes	characteristic
131	#88322	C₂×C₄⋊C₄	1	1	yes	characteristic
132	C₂×#1383	—	53	1	no	maximal, Sylow
133	C₂²×C₅₃	—	2	2	no	—
134	C₂²×C₅₃	#340	1	2	yes	—
135	C₂²×C₅₃	C₂×#106	1	1	yes	characteristic
136	C₂×D₅₃	—	2	2	no	—
137	C₂×D₅₃	—	2	4	no	—
138	C₂×D₅₃	—	2	4	no	—
139	C₂×D₅₃	#340	1	2	yes	—
140	C₂×D₅₃	#365	1	4	yes	—
141	C₂×D₅₃	C₂×#102	1	1	yes	characteristic
142	C₂×D₅₃	C₂×#103	1	1	yes	characteristic
143	#6157	—	8	4	no	—
144	C₄×C₅₃	#340	1	2	yes	—
145	C₄×C₅₃	C₂×#95	1	1	yes	characteristic
146	C₄×C₅₃	C₂×#107	1	1	yes	characteristic
147	Dic₅₃	#340	1	2	yes	—
148	Dic₅₃	C₂×#102	1	1	yes	characteristic
149	Dic₅₃	C₂×#103	1	1	yes	characteristic
150	C₂²×M₃₂	—	53	1	no	—
151	C₂²×#108	—	53	1	no	—
152	C₂²×#106	—	53	1	no	—
153	#1383	—	53	4	no	—
154	C₂×C₅₃	—	2	2	no	—
155	C₂×C₅₃	C₂×#340	1	1	yes	characteristic
156	C₂×C₅₃	C₂×#365	1	2	yes	—
157	D₅₃	—	2	1	no	—
158	D₅₃	—	2	1	no	—
159	D₅₃	#1383	1	4	yes	complemented
160	C₂²×M₁₆	—	106	1	no	—
161	C₂³×C₈	—	53	1	no	—
162	C₂²×C₄⋊C₄	—	106	1	no	—
163	C₂×M₃₂	—	53	1	no	—
164	C₂×M₃₂	—	53	2	no	—
165	C₂×M₃₂	—	106	2	no	—
166	C₂×M₃₂	—	106	2	no	—
167	C₂×M₃₂	#6157	1	1	yes	characteristic, complemented
168	C₂²×C₁₆	—	53	2	no	—
169	C₂×#108	—	53	2	no	—
170	C₂×#108	—	53	4	no	—
171	C₂×#108	—	106	2	no	—
172	C₂×#108	—	106	1	no	—
173	C₂×#106	—	53	2	no	—
174	C₂×#106	—	53	4	no	—
175	C₂×#106	—	106	1	no	—
176	C₂×#106	—	106	2	no	—
177	C₅₃	C₂×#1383	1	1	yes	characteristic, complemented, Sylow
178	C₂³×C₄	—	53	1	no	—
179	C₂³×C₄	—	212	1	no	—
180	C₂×M₁₆	—	106	4	no	—
181	C₂×M₁₆	—	106	2	no	—
182	C₂×M₁₆	—	212	1	no	—
183	C₂×M₁₆	—	212	2	no	—
184	C₂²×C₈	—	53	1	no	—
185	C₂²×C₈	—	53	2	no	—
186	C₂²×C₈	—	53	2	no	—
187	C₂²×C₈	—	53	1	no	—
188	C₂²×C₈	—	53	1	no	—
189	C₂²×C₈	—	53	2	no	—
190	C₂²×C₈	—	53	2	no	—
191	C₂²×C₈	—	53	1	no	—
192	C₂²×C₈	—	53	1	no	—
193	C₂²×C₈	—	212	1	no	—
194	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
195	C₂×C₄⋊C₄	—	106	4	no	—
196	C₂×C₄⋊C₄	—	106	2	no	—
197	C₂×C₄⋊C₄	—	212	1	no	—
198	C₂×C₄⋊C₄	—	212	2	no	—
199	M₃₂	—	2	2	no	—
200	M₃₂	—	106	2	no	—
201	M₃₂	—	106	2	no	—
202	M₃₂	—	106	2	no	—
203	C₂×C₁₆	—	53	2	no	—
204	C₂×C₁₆	—	106	4	no	—
205	C₂×C₁₆	C₂×#6157	1	2	yes	complemented
206	#108	—	53	4	no	—
207	#108	—	106	2	no	—
208	#108	—	106	4	no	—
209	#106	—	53	4	no	—
210	#106	—	106	4	no	—
211	#106	—	106	2	no	—
212	C₂⁴	—	53	1	no	—
213	C₂²×C₄	—	53	1	no	—
214	C₂²×C₄	—	53	2	no	—
215	C₂²×C₄	—	53	2	no	—
216	C₂²×C₄	—	53	1	no	—
217	C₂²×C₄	—	53	1	no	—
218	C₂²×C₄	—	53	2	no	—
219	C₂²×C₄	—	53	2	no	—
220	C₂²×C₄	—	53	1	no	—
221	C₂²×C₄	—	53	1	no	—
222	C₂²×C₄	—	424	4	no	—
223	C₂²×C₄	—	212	2	no	—
224	C₂²×C₄	—	212	4	no	—
225	C₂²×C₄	C₄⋊#6157	1	1	yes	characteristic
226	M₁₆	—	106	4	no	—
227	M₁₆	—	212	2	no	—
228	M₁₆	—	212	4	no	—
229	C₂×C₈	—	53	2	no	—
230	C₂×C₈	—	53	2	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	—	53	2	no	—
236	C₂×C₈	—	53	2	no	—
237	C₂×C₈	—	53	1	no	—
238	C₂×C₈	—	53	1	no	—
239	C₂×C₈	—	53	2	no	—
240	C₂×C₈	—	53	1	no	—
241	C₂×C₈	—	53	1	no	—
242	C₂×C₈	—	53	2	no	—
243	C₂×C₈	—	212	2	no	—
244	C₂×C₈	—	212	4	no	—
245	C₂×C₈	C₄×#6157	1	2	yes	—
246	C₂×C₈	C₄⋊#6157	1	1	yes	characteristic
247	C₂×C₈	C₂²⋊#6157	1	2	yes	—
248	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
249	C₄⋊C₄	—	106	4	no	—
250	C₄⋊C₄	—	212	2	no	—
251	C₄⋊C₄	—	212	4	no	—
252	C₁₆	—	2	2	no	—
253	C₁₆	—	106	2	no	—
254	C₂³	—	53	1	no	—
255	C₂³	—	53	1	no	—
256	C₂³	—	53	2	no	—
257	C₂³	—	53	2	no	—
258	C₂³	—	106	4	no	—
259	C₂³	#13133285	1	1	yes	characteristic
260	C₂×C₄	—	53	1	no	—
261	C₂×C₄	—	53	1	no	—
262	C₂×C₄	—	53	2	no	—
263	C₂×C₄	—	53	2	no	—
264	C₂×C₄	—	53	2	no	—
265	C₂×C₄	—	53	1	no	—
266	C₂×C₄	—	53	1	no	—
267	C₂×C₄	—	53	2	no	—
268	C₂×C₄	—	53	1	no	—
269	C₂×C₄	—	53	1	no	—
270	C₂×C₄	—	53	2	no	—
271	C₂×C₄	—	53	2	no	—
272	C₂×C₄	—	53	2	no	—
273	C₂×C₄	—	53	2	no	—
274	C₂×C₄	—	424	8	no	—
275	C₂×C₄	—	424	4	no	—
276	C₂×C₄	—	212	4	no	—
277	C₂×C₄	(C₂×#6157)⋊C₄	1	2	yes	—
278	C₂×C₄	#13133285	1	1	yes	characteristic
279	C₂×C₄	#13133286	1	2	yes	—
280	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
281	C₈	—	53	2	no	—
282	C₈	—	53	1	no	—
283	C₈	—	53	1	no	—
284	C₈	—	212	4	no	—
285	C₈	(C₂×#6157)⋊C₄	1	2	yes	—
286	C₈	C₂×C₄×#6157	1	1	yes	characteristic
287	C₈	C₂×C₂²⋊#6157	1	1	yes	characteristic
288	C₂²	—	2	2	no	—
289	C₂²	—	53	2	no	—
290	C₂²	—	53	4	no	—
291	C₂²	—	53	1	no	—
292	C₂²	—	53	1	no	—
293	C₂²	—	106	4	no	—
294	C₂²	—	106	4	no	—
295	C₂²	—	106	2	no	—
296	C₂²	#16811497	1	2	yes	—
297	C₂²	C₂×#13133285	1	1	yes	characteristic
298	C₄	—	53	1	no	—
299	C₄	—	53	2	no	—
300	C₄	—	53	1	no	—
301	C₄	—	424	4	no	—
302	C₄	#16811497	1	2	yes	—
303	C₄	C₂×(C₂×#6157)⋊C₄	1	1	yes	characteristic
304	C₄	C₂×#13133286	1	1	yes	characteristic
305	C₂	—	2	2	no	—
306	C₂	—	53	4	no	—
307	C₂	—	106	1	no	—
308	C₂	—	106	1	no	—
309	C₂	C₂×#16811497	1	1	yes	characteristic
310	C₂	D₅₃⋊#365	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#13133285
commutator	C₈×C₅₃	C₂×C₄²
Frattini	C₂×C₈	C₂²×#6157
Fitting	C₂×M₃₂×C₅₃	C₄
socle	C₂²×C₅₃	C₂×#106
prime radical	C₂³×D₅₃	SD₁₆
cosocle	C₂×C₈×D₅₃	C₂³
hypercenter	C₂×M₃₂	#6157
hypocenter	C₅₃	C₂×#1383
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

Complements

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

N	H ≅ G/N	type
3. #20289780	C₂	semidirect
9. D₅₃⋊#108	C₄	semidirect
16. C₂×#13133157	C₄	semidirect
20. C₂×C₁₆×D₅₃	C₄	semidirect
21. C₂×M₃₂×C₅₃	C₄	semidirect
34. #13133287	C₂×C₄	semidirect
51. C₂×#11719908	C₂×C₄	semidirect
55. C₂×C₁₆×C₅₃	C₂×C₄	semidirect
159. D₅₃	#1383	semidirect
167. C₂×M₃₂	#6157	semidirect
177. C₅₃	C₂×#1383	semidirect
205. C₂×C₁₆	C₂×#6157	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₅₃⋊#108	2
C₂²×#13133285	2
#20289780	2
C₂×M₃₂×D₅₃	2
C₂×#1383	53