Subgroups — #28328372

Order 13568.

718 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20291474	C₂	1	2	yes	maximal, complemented
2	#20291113	C₂	1	4	yes	maximal, complemented
3	D₁₆×#6157	C₂	1	4	yes	maximal, complemented
4	#20291066	C₂	1	1	yes	maximal, characteristic, complemented
5	#20291024	C₂	1	2	yes	maximal, complemented
6	#20290923	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×D₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic
8	#16811690	—	2	2	no	—
9	D₅₃⋊M₃₂	C₂²	1	1	yes	characteristic
10	#16811664	—	2	4	no	—
11	D₈×#6157	C₂²	1	8	yes	complemented
12	#16811633	C₂²	1	2	yes	—
13	#16811615	—	2	2	no	—
14	#16811586	C₂²	1	1	yes	characteristic
15	#16811550	C₂²	1	2	yes	complemented
16	#16811549	C₂²	1	2	yes	complemented
17	#16811535	C₂²	1	8	yes	complemented
18	#16811510	C₂²	1	2	yes	complemented
19	C₁₆×#6157	C₂²	1	2	yes	complemented
20	C₂×D₈×D₅₃	C₂²	1	2	yes	—
21	C₂×D₈₄₈	C₄	1	1	yes	characteristic, complemented
22	C₂×#13133158	C₄	1	2	yes	complemented
23	D₁₆×D₅₃	C₄	1	2	yes	complemented
24	D₁₆×D₅₃	C₄	1	2	yes	complemented
25	D₁₆×D₅₃	C₂²	1	2	yes	—
26	D₁₆×D₅₃	C₂²	1	2	yes	—
27	C₂×C₁₆×D₅₃	C₂²	1	1	yes	characteristic
28	C₂×D₁₆×C₅₃	C₄	1	1	yes	characteristic, complemented
29	D₅₃⋊M₁₆	D₄	1	1	yes	characteristic
30	D₄×#6157	—	2	8	no	—
31	D₄×#6157	—	4	4	no	—
32	#13133296	—	2	2	no	—
33	#13133296	D₄	1	1	yes	characteristic
34	C₂×C₂²⋊#6157	—	4	4	no	—
35	#13133292	—	2	1	no	—
36	#13133291	—	2	1	no	—
37	D₅₃⋊C₁₆	C₂³	1	2	yes	—
38	#13133286	C₂³	1	4	yes	—
39	#13133285	—	2	2	no	—
40	#13133279	—	2	8	no	—
41	#13133271	D₄	1	2	yes	—
42	C₈×#6157	C₂³	1	2	yes	—
43	C₂×D₄×D₅₃	—	2	2	no	—
44	C₂×D₄₂₄	C₂×C₄	1	2	yes	complemented
45	C₂×#11719922	—	2	2	no	—
46	D₈×D₅₃	—	2	2	no	—
47	D₈×D₅₃	—	2	2	no	—
48	D₈×D₅₃	C₂×C₄	1	4	yes	complemented
49	D₈×D₅₃	C₂³	1	4	yes	—
50	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
51	D₈₄₈	C₂×C₄	1	2	yes	complemented
52	D₈₄₈	C₂×C₄	1	2	yes	complemented
53	#13133158	C₂×C₄	1	4	yes	complemented
54	#13133158	C₂×C₄	1	4	yes	complemented
55	C₂×#11719908	C₂×C₄	1	1	yes	characteristic, complemented
56	C₁₆×D₅₃	C₂×C₄	1	1	yes	characteristic, complemented
57	C₁₆×D₅₃	C₂×C₄	1	1	yes	characteristic, complemented
58	C₁₆×D₅₃	C₂³	1	1	yes	characteristic
59	C₁₆×D₅₃	C₂³	1	1	yes	characteristic
60	C₂×D₈×C₅₃	C₂×C₄	1	2	yes	complemented
61	D₁₆×C₅₃	C₂×C₄	1	2	yes	complemented
62	D₁₆×C₅₃	C₂×C₄	1	2	yes	complemented
63	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic, complemented
64	C₂³×D₅₃	—	4	2	no	—
65	C₂²×#6157	—	4	8	no	—
66	C₂×D₄×C₅₃	—	2	2	no	—
67	C₂×C₂²⋊D₅₃	—	4	2	no	—
68	D₄×D₅₃	—	2	4	no	—
69	D₄×D₅₃	—	2	4	no	—
70	D₄×D₅₃	—	4	2	no	—
71	D₄×D₅₃	—	4	2	no	—
72	C₂×D₂₁₂	—	2	2	no	—
73	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
74	C₂²⋊#6157	—	4	2	no	—
75	C₂²⋊#6157	—	4	8	no	—
76	C₂²⋊#6157	D₈	1	2	yes	—
77	#11719940	—	2	1	no	—
78	C₄⋊#6157	—	2	4	no	—
79	C₄⋊#6157	D₈	1	2	yes	—
80	C₄×#6157	—	4	2	no	—
81	C₄×#6157	C₂×D₄	1	2	yes	—
82	#11719936	—	2	1	no	—
83	D₅₃⋊C₈	C₂×D₄	1	2	yes	—
84	D₈×C₅₃	—	2	2	no	—
85	D₈×C₅₃	C₂²×C₄	1	4	yes	—
86	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
87	#11719922	—	2	4	no	—
88	#11719922	—	2	4	no	—
89	C₂×#88322	#41	1	1	yes	characteristic
90	D₄₂₄	—	2	2	no	—
91	D₄₂₄	C₂²×C₄	1	4	yes	—
92	C₈×D₅₃	C₂²×C₄	1	1	yes	characteristic
93	C₈×D₅₃	C₂×D₄	1	1	yes	characteristic
94	C₈×D₅₃	#41	1	1	yes	characteristic
95	C₈×D₅₃	C₂⁴	1	1	yes	characteristic
96	#11719910	—	2	2	no	—
97	C₁₆×C₅₃	C₂²×C₄	1	1	yes	characteristic
98	C₁₆×C₅₃	C₂²×C₄	1	1	yes	characteristic
99	#11719908	C₂²×C₄	1	1	yes	characteristic
100	#11719908	C₂²×C₄	1	1	yes	characteristic
101	C₂³×C₅₃	—	4	2	no	—
102	C₂²×D₅₃	—	4	2	no	—
103	C₂²×D₅₃	—	4	4	no	—
104	C₂²×D₅₃	—	4	4	no	—
105	C₂²×D₅₃	—	16	4	no	—
106	C₂²×D₅₃	C₂×D₈	1	1	yes	characteristic
107	C₂×#6157	—	4	4	no	—
108	C₂×#6157	—	16	8	no	—
109	C₂×#6157	C₂×D₈	1	4	yes	—
110	D₄×C₅₃	—	2	4	no	—
111	D₄×C₅₃	—	4	2	no	—
112	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
113	C₂²⋊D₅₃	—	4	4	no	—
114	C₂²⋊D₅₃	—	4	4	no	—
115	C₂×Dic₅₃	#135	1	1	yes	characteristic
116	D₂₁₂	—	2	4	no	—
117	D₂₁₂	—	4	2	no	—
118	C₄×D₅₃	C₄×D₄	1	1	yes	characteristic
119	C₄×D₅₃	C₂×D₈	1	1	yes	characteristic
120	C₄×D₅₃	#135	1	1	yes	characteristic
121	C₄×D₅₃	C₂²×D₄	1	1	yes	characteristic
122	#88324	—	2	2	no	—
123	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
124	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
125	#88322	C₄×D₄	1	1	yes	characteristic
126	#88322	C₂×#41	1	1	yes	characteristic
127	#33456	—	53	1	no	maximal, Sylow
128	C₂²×C₅₃	—	4	4	no	—
129	C₂²×C₅₃	—	16	2	no	—
130	C₂²×C₅₃	C₄×D₈	1	1	yes	characteristic
131	C₂×D₅₃	—	2	2	no	—
132	C₂×D₅₃	—	4	4	no	—
133	C₂×D₅₃	—	8	8	no	—
134	C₂×D₅₃	—	16	4	no	—
135	C₂×D₅₃	—	16	2	no	—
136	C₂×D₅₃	C₄×D₈	1	1	yes	characteristic
137	C₂×D₅₃	C₂²×D₈	1	1	yes	characteristic
138	#6157	—	2	4	no	—
139	#6157	—	8	4	no	—
140	C₄×C₅₃	C₄×D₈	1	1	yes	characteristic
141	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
142	Dic₅₃	C₄×D₈	1	1	yes	characteristic
143	Dic₅₃	C₂×#135	1	1	yes	characteristic
144	C₂²×D₁₆	—	53	1	no	—
145	#2948	—	53	2	no	—
146	#2177	—	53	4	no	—
147	C₄×D₁₆	—	53	4	no	—
148	#2158	—	53	1	no	—
149	#2140	—	53	2	no	—
150	#2108	—	53	1	no	—
151	C₂×C₅₃	—	2	1	no	—
152	C₂×C₅₃	—	8	4	no	—
153	C₂×C₅₃	C₂×C₄×D₈	1	1	yes	characteristic
154	D₅₃	—	2	1	no	—
155	D₅₃	—	8	4	no	—
156	D₅₃	#3415	1	2	yes	—
157	C₂²×D₈	—	53	2	no	—
158	#518	—	106	2	no	—
159	C₂×D₁₆	—	53	2	no	—
160	C₂×D₁₆	—	53	2	no	—
161	C₂×D₁₆	—	53	2	no	—
162	C₂×D₁₆	—	53	2	no	—
163	C₂×D₁₆	—	53	2	no	—
164	C₂×D₁₆	—	53	1	no	—
165	C₂×D₁₆	#6157	1	1	yes	characteristic, complemented
166	C₂×M₃₂	—	53	1	no	—
167	C₂²×C₁₆	—	53	1	no	—
168	#442	—	106	4	no	—
169	C₄×D₈	—	53	8	no	—
170	#427	—	53	2	no	—
171	#418	—	106	2	no	—
172	#405	—	53	1	no	—
173	#367	—	53	2	no	—
174	#366	—	53	2	no	—
175	#357	—	53	8	no	—
176	#346	—	53	2	no	—
177	C₄×C₁₆	—	53	2	no	—
178	C₅₃	#33456	1	1	yes	characteristic, complemented, Sylow
179	C₂²×D₄	—	106	2	no	—
180	C₂×D₈	—	53	4	no	—
181	C₂×D₈	—	53	4	no	—
182	C₂×D₈	—	53	2	no	—
183	C₂×D₈	—	106	2	no	—
184	C₂×D₈	—	106	2	no	—
185	C₂×D₈	—	106	2	no	—
186	C₂×D₈	C₂×#6157	1	2	yes	complemented
187	C₂×M₁₆	—	53	1	no	—
188	C₂²×C₈	—	53	1	no	—
189	C₄×D₄	—	106	8	no	—
190	C₄×D₄	—	212	4	no	—
191	#117	—	53	1	no	—
192	#117	—	106	2	no	—
193	C₂×C₂²⋊C₄	—	212	4	no	—
194	D₁₆	—	53	4	no	—
195	D₁₆	—	53	4	no	—
196	D₁₆	—	53	2	no	—
197	D₁₆	—	53	2	no	—
198	D₁₆	C₂×#6157	1	2	yes	complemented
199	D₁₆	C₂×#6157	1	2	yes	complemented
200	M₃₂	—	106	1	no	—
201	M₃₂	—	106	1	no	—
202	C₂×C₁₆	—	53	1	no	—
203	C₂×C₁₆	—	53	1	no	—
204	C₂×C₁₆	—	53	1	no	—
205	C₂×C₁₆	—	53	1	no	—
206	C₂×C₁₆	—	53	1	no	—
207	C₂×C₁₆	—	53	2	no	—
208	C₂×C₁₆	C₂×#6157	1	1	yes	characteristic, complemented
209	#107	—	53	4	no	—
210	#106	—	106	2	no	—
211	#102	—	106	8	no	—
212	#97	—	53	2	no	—
213	C₄×C₈	—	53	2	no	—
214	C₂⁴	—	212	2	no	—
215	C₂×D₄	—	2	2	no	—
216	C₂×D₄	—	106	4	no	—
217	C₂×D₄	—	106	4	no	—
218	C₂×D₄	—	106	2	no	—
219	C₂×D₄	—	212	2	no	—
220	C₂×D₄	—	212	2	no	—
221	C₂×D₄	—	212	2	no	—
222	C₂²×C₄	—	53	1	no	—
223	C₂²×C₄	—	212	8	no	—
224	D₈	—	2	2	no	—
225	D₈	—	53	4	no	—
226	D₈	—	106	4	no	—
227	D₈	—	106	4	no	—
228	D₈	—	106	2	no	—
229	D₈	C₂²×#6157	1	4	yes	—
230	M₁₆	—	106	1	no	—
231	M₁₆	—	106	1	no	—
232	C₂×C₈	—	53	1	no	—
233	C₂×C₈	—	53	1	no	—
234	C₂×C₈	—	53	1	no	—
235	C₂×C₈	—	53	1	no	—
236	C₂×C₈	—	53	1	no	—
237	C₂×C₈	—	53	2	no	—
238	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
239	C₄⋊C₄	—	53	2	no	—
240	C₄⋊C₄	—	106	4	no	—
241	C₂²⋊C₄	—	53	2	no	—
242	C₂²⋊C₄	—	212	8	no	—
243	C₂²⋊C₄	—	212	2	no	—
244	C₄²	—	53	2	no	—
245	C₄²	—	212	2	no	—
246	C₁₆	—	53	1	no	—
247	C₁₆	—	53	1	no	—
248	C₁₆	—	106	2	no	—
249	C₁₆	C₂²×#6157	1	1	yes	characteristic
250	C₁₆	C₂²×#6157	1	1	yes	characteristic
251	C₂³	—	4	2	no	—
252	C₂³	—	53	1	no	—
253	C₂³	—	848	4	no	—
254	C₂³	—	212	4	no	—
255	C₂³	—	212	4	no	—
256	C₂³	—	212	2	no	—
257	D₄	—	2	4	no	—
258	D₄	—	4	2	no	—
259	D₄	—	106	4	no	—
260	D₄	—	212	4	no	—
261	D₄	—	212	4	no	—
262	D₄	—	212	2	no	—
263	C₂×C₄	—	53	1	no	—
264	C₂×C₄	—	53	1	no	—
265	C₂×C₄	—	53	1	no	—
266	C₂×C₄	—	53	1	no	—
267	C₂×C₄	—	53	1	no	—
268	C₂×C₄	—	53	4	no	—
269	C₂×C₄	—	848	8	no	—
270	C₂×C₄	—	212	4	no	—
271	C₂×C₄	D₄×#6157	1	1	yes	characteristic
272	C₈	—	53	1	no	—
273	C₈	—	53	1	no	—
274	C₈	—	106	2	no	—
275	C₈	D₄×#6157	1	1	yes	characteristic
276	C₈	C₂³×#6157	1	1	yes	characteristic
277	C₂²	—	4	4	no	—
278	C₂²	—	16	2	no	—
279	C₂²	—	53	1	no	—
280	C₂²	—	53	1	no	—
281	C₂²	—	848	2	no	—
282	C₂²	—	848	4	no	—
283	C₂²	—	106	2	no	—
284	C₂²	—	424	8	no	—
285	C₂²	—	212	4	no	—
286	C₂²	D₈×#6157	1	1	yes	characteristic
287	C₄	—	53	1	no	—
288	C₄	—	53	1	no	—
289	C₄	—	106	4	no	—
290	C₄	—	424	4	no	—
291	C₄	D₈×#6157	1	1	yes	characteristic
292	C₄	C₂×D₄×#6157	1	1	yes	characteristic
293	C₂	—	2	1	no	—
294	C₂	—	8	4	no	—
295	C₂	—	53	2	no	—
296	C₂	—	106	1	no	—
297	C₂	—	424	4	no	—
298	C₂	C₂×D₈×#6157	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₈×#6157
commutator	C₈×C₅₃	C₂³×C₄
Frattini	C₈	C₂³×#6157
Fitting	C₂×D₁₆×C₅₃	C₄
socle	C₂×C₅₃	C₂×C₄×D₈
prime radical	C₂×D₁₆×D₅₃	C₂
cosocle	C₈×D₅₃	C₂⁴
hypercenter	C₂×D₁₆	#6157
hypocenter	C₅₃	#33456
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×D₁₆

Complements

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

N	H ≅ G/N	type
1. #20291474	C₂	semidirect
2. #20291113	C₂	semidirect
3. D₁₆×#6157	C₂	semidirect
4. #20291066	C₂	semidirect
5. #20291024	C₂	semidirect
6. #20290923	C₂	semidirect
11. D₈×#6157	C₂²	semidirect
15. #16811550	C₂²	semidirect
16. #16811549	C₂²	semidirect
17. #16811535	C₂²	semidirect
18. #16811510	C₂²	semidirect
19. C₁₆×#6157	C₂²	semidirect
21. C₂×D₈₄₈	C₄	semidirect
22. C₂×#13133158	C₄	semidirect
23. D₁₆×D₅₃	C₄	semidirect
24. D₁₆×D₅₃	C₄	semidirect
28. C₂×D₁₆×C₅₃	C₄	semidirect
44. C₂×D₄₂₄	C₂×C₄	semidirect
48. D₈×D₅₃	C₂×C₄	semidirect
51. D₈₄₈	C₂×C₄	semidirect
52. D₈₄₈	C₂×C₄	semidirect
53. #13133158	C₂×C₄	semidirect
54. #13133158	C₂×C₄	semidirect
55. C₂×#11719908	C₂×C₄	semidirect
56. C₁₆×D₅₃	C₂×C₄	semidirect
57. C₁₆×D₅₃	C₂×C₄	semidirect
60. C₂×D₈×C₅₃	C₂×C₄	semidirect
61. D₁₆×C₅₃	C₂×C₄	semidirect
62. D₁₆×C₅₃	C₂×C₄	semidirect
63. C₂×C₁₆×C₅₃	C₂×C₄	semidirect
165. C₂×D₁₆	#6157	semidirect
178. C₅₃	#33456	semidirect
186. C₂×D₈	C₂×#6157	semidirect
198. D₁₆	C₂×#6157	semidirect
199. D₁₆	C₂×#6157	semidirect
208. 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
#20291474	2
#20291113	2
D₁₆×#6157	2
#20291066	2
#20291024	2
#20290923	2
C₂×D₁₆×D₅₃	2
#33456	53