Subgroups — #27542236

Order 13568.

466 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811055	C₂	1	1	yes	maximal, characteristic, complemented
2	#20277033	C₂	1	2	yes	maximal, complemented
3	#20277032	C₂	1	2	yes	maximal, complemented
4	#20277028	C₂	1	2	yes	maximal, complemented
5	#20277027	C₂	1	2	yes	maximal, complemented
6	#20277022	C₂	1	4	yes	maximal, complemented
7	C₂×#16810457	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅₃×#508	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13133239	—	2	1	no	—
10	#16811055	—	2	1	no	—
11	#16811055	—	2	1	no	—
12	#16811055	C₂²	1	2	yes	—
13	#16811055	C₂²	1	2	yes	complemented
14	C₂×Dic₄₂₄	C₂²	1	1	yes	characteristic, complemented
15	C₂×#13133162	—	2	1	no	—
16	C₂×D₈₄₈	C₂²	1	1	yes	characteristic, complemented
17	#16810463	C₂²	1	4	yes	complemented
18	#16810461	C₂²	1	4	yes	complemented
19	#16810460	C₂²	1	4	yes	complemented
20	#16810458	C₂²	1	4	yes	complemented
21	#16810457	C₂²	1	2	yes	complemented
22	#16810457	C₂²	1	2	yes	complemented
23	C₂×#13133123	C₂²	1	2	yes	complemented
24	C₂×#135×C₅₃	—	2	1	no	—
25	C₅₃×#508	—	2	1	no	—
26	C₅₃×#508	—	2	1	no	—
27	C₅₃×#508	C₂²	1	2	yes	—
28	C₅₃×#508	C₂²	1	2	yes	complemented
29	C₂×Q₃₂×C₅₃	C₂²	1	1	yes	characteristic, complemented
30	C₂×SD₃₂×C₅₃	—	2	1	no	—
31	C₂×D₁₆×C₅₃	C₂²	1	1	yes	characteristic, complemented
32	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
33	C₂×#11719945	—	4	1	no	—
34	#13133239	—	2	2	no	—
35	#13133239	—	2	2	no	—
36	#13133239	—	4	1	no	—
37	#13133239	—	4	1	no	—
38	C₂×Dic₂₁₂	—	2	1	no	—
39	C₂×#11719913	—	4	1	no	—
40	C₂×D₄₂₄	—	2	1	no	—
41	Dic₄₂₄	—	2	1	no	—
42	Dic₄₂₄	C₂³	1	2	yes	—
43	#13133162	—	2	2	no	—
44	#13133162	—	2	2	no	—
45	D₈₄₈	—	2	1	no	—
46	D₈₄₈	C₂³	1	2	yes	—
47	#13133123	C₂³	1	4	yes	—
48	C₂×#41×C₅₃	—	4	1	no	—
49	#135×C₅₃	—	2	2	no	—
50	#135×C₅₃	—	2	2	no	—
51	#135×C₅₃	—	4	1	no	—
52	#135×C₅₃	—	4	1	no	—
53	C₂×Q₁₆×C₅₃	—	2	1	no	—
54	C₂×SD₁₆×C₅₃	—	4	1	no	—
55	C₂×D₈×C₅₃	—	2	1	no	—
56	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
57	Q₃₂×C₅₃	—	2	1	no	—
58	Q₃₂×C₅₃	C₂³	1	2	yes	—
59	SD₃₂×C₅₃	—	2	2	no	—
60	SD₃₂×C₅₃	—	2	2	no	—
61	D₁₆×C₅₃	—	2	1	no	—
62	D₁₆×C₅₃	C₂³	1	2	yes	—
63	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
64	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
65	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
66	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
67	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
68	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
69	#41×C₅₃	—	4	2	no	—
70	#41×C₅₃	—	4	2	no	—
71	#41×C₅₃	—	8	1	no	—
72	#41×C₅₃	—	8	1	no	—
73	C₂×Q₈×C₅₃	—	4	1	no	—
74	C₂×D₄×C₅₃	—	4	1	no	—
75	C₂×D₄×C₅₃	—	8	1	no	—
76	C₂²×C₄×C₅₃	—	8	1	no	—
77	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
78	C₂×C₂²⋊D₅₃	—	8	1	no	—
79	#11719945	—	4	2	no	—
80	#11719945	—	4	2	no	—
81	#11719945	—	8	1	no	—
82	#11719945	—	8	1	no	—
83	C₂×D₂₁₂	—	4	1	no	—
84	C₂×C₄×D₅₃	—	8	1	no	—
85	C₂×Dic₁₀₆	—	4	1	no	—
86	Q₁₆×C₅₃	—	2	2	no	—
87	Q₁₆×C₅₃	—	4	1	no	—
88	SD₁₆×C₅₃	—	4	2	no	—
89	SD₁₆×C₅₃	—	4	2	no	—
90	D₈×C₅₃	—	2	2	no	—
91	D₈×C₅₃	—	4	1	no	—
92	C₂×C₈×C₅₃	D₈	1	1	yes	characteristic
93	C₂×C₈×C₅₃	D₈	1	1	yes	characteristic
94	C₂×C₈×C₅₃	D₈	1	1	yes	characteristic
95	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
98	Dic₂₁₂	—	2	2	no	—
99	Dic₂₁₂	—	4	1	no	—
100	D₄₂₄	—	2	2	no	—
101	D₄₂₄	—	4	1	no	—
102	#11719913	—	4	2	no	—
103	#11719913	—	4	2	no	—
104	C₁₆×C₅₃	C₂×D₄	1	1	yes	characteristic
105	C₁₆×C₅₃	C₂×D₄	1	1	yes	characteristic
106	C₁₆×C₅₃	C₂×D₄	1	1	yes	characteristic
107	C₁₆×C₅₃	C₂⁴	1	1	yes	characteristic
108	C₂³×C₅₃	—	8	1	no	—
109	C₂³×C₅₃	D₁₆	1	1	yes	characteristic
110	C₂²×D₅₃	—	8	1	no	—
111	Q₈×C₅₃	—	4	2	no	—
112	Q₈×C₅₃	—	8	1	no	—
113	D₄×C₅₃	—	4	2	no	—
114	D₄×C₅₃	—	8	2	no	—
115	D₄×C₅₃	—	8	2	no	—
116	D₄×C₅₃	—	8	1	no	—
117	C₂×C₄×C₅₃	—	8	1	no	—
118	C₂×C₄×C₅₃	—	8	2	no	—
119	C₂×C₄×C₅₃	—	8	2	no	—
120	C₂×C₄×C₅₃	D₁₆	1	1	yes	characteristic
121	C₂×C₄×C₅₃	D₁₆	1	1	yes	characteristic
122	C₂×C₄×C₅₃	D₁₆	1	1	yes	characteristic
123	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
124	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
125	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
126	C₂²⋊D₅₃	—	8	2	no	—
127	C₂²⋊D₅₃	—	8	2	no	—
128	C₂×Dic₅₃	—	8	1	no	—
129	D₂₁₂	—	4	2	no	—
130	D₂₁₂	—	8	1	no	—
131	C₄×D₅₃	—	8	2	no	—
132	C₄×D₅₃	—	8	2	no	—
133	Dic₁₀₆	—	4	2	no	—
134	Dic₁₀₆	—	8	1	no	—
135	C₈×C₅₃	C₂×D₈	1	1	yes	characteristic
136	C₈×C₅₃	C₂×D₈	1	1	yes	characteristic
137	C₈×C₅₃	C₂×D₈	1	1	yes	characteristic
138	C₈×C₅₃	C₂²×D₄	1	1	yes	characteristic
139	#33981	—	53	1	no	maximal, Sylow
140	C₂²×C₅₃	—	4	1	no	—
141	C₂²×C₅₃	—	8	2	no	—
142	C₂²×C₅₃	—	16	2	no	—
143	C₂²×C₅₃	C₂×D₁₆	1	1	yes	characteristic
144	C₂²×C₅₃	C₂×D₁₆	1	1	yes	characteristic
145	C₂²×C₅₃	C₂×D₁₆	1	1	yes	characteristic
146	C₂×D₅₃	—	8	2	no	—
147	C₂×D₅₃	—	16	2	no	—
148	C₄×C₅₃	—	8	2	no	—
149	C₄×C₅₃	C₂×D₁₆	1	1	yes	characteristic
150	C₄×C₅₃	C₂×D₁₆	1	1	yes	characteristic
151	C₄×C₅₃	C₂×D₁₆	1	1	yes	characteristic
152	C₄×C₅₃	C₂²×D₈	1	1	yes	characteristic
153	Dic₅₃	—	8	2	no	—
154	C₂×#508	—	53	1	no	—
155	C₂×#508	D₅₃	1	1	yes	characteristic, complemented
156	#2264	—	53	2	no	—
157	#2264	—	53	2	no	—
158	#2263	—	53	2	no	—
159	#2263	—	53	2	no	—
160	#2262	—	53	4	no	—
161	C₂×M₆₄	—	53	1	no	—
162	C₂×C₅₃	—	2	1	no	—
163	C₂×C₅₃	—	2	1	no	—
164	C₂×C₅₃	—	2	1	no	—
165	C₂×C₅₃	—	16	2	no	—
166	C₂×C₅₃	C₂²×D₁₆	1	1	yes	characteristic
167	D₅₃	—	16	2	no	—
168	C₂×#135	—	2	1	no	—
169	C₂×#135	—	106	1	no	—
170	#508	—	2	1	no	—
171	#508	—	2	1	no	—
172	#508	—	53	2	no	—
173	#508	—	53	2	no	—
174	#508	—	106	1	no	—
175	#508	—	106	1	no	—
176	#508	C₂×D₅₃	1	2	yes	—
177	#508	C₂×D₅₃	1	2	yes	complemented
178	C₂×Q₃₂	—	53	1	no	—
179	C₂×Q₃₂	C₂×D₅₃	1	1	yes	characteristic, complemented
180	C₂×SD₃₂	—	2	1	no	—
181	C₂×SD₃₂	—	106	1	no	—
182	C₂×D₁₆	—	53	1	no	—
183	C₂×D₁₆	C₂×D₅₃	1	1	yes	characteristic, complemented
184	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
185	Q₆₄	—	53	4	no	—
186	SD₆₄	—	53	4	no	—
187	SD₆₄	—	53	4	no	—
188	D₃₂	—	53	4	no	—
189	M₆₄	—	53	2	no	—
190	M₆₄	—	53	2	no	—
191	C₂×C₃₂	—	53	2	no	—
192	C₅₃	#33981	1	1	yes	characteristic, complemented, Sylow
193	C₂×#41	—	4	1	no	—
194	C₂×#41	—	212	1	no	—
195	#135	—	2	2	no	—
196	#135	—	2	2	no	—
197	#135	—	4	1	no	—
198	#135	—	4	1	no	—
199	#135	—	106	2	no	—
200	#135	—	106	2	no	—
201	#135	—	212	1	no	—
202	#135	—	212	1	no	—
203	C₂×Q₁₆	—	2	1	no	—
204	C₂×Q₁₆	—	106	1	no	—
205	C₂×SD₁₆	—	4	1	no	—
206	C₂×SD₁₆	—	212	1	no	—
207	C₂×D₈	—	2	1	no	—
208	C₂×D₈	—	106	1	no	—
209	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
210	Q₃₂	—	2	1	no	—
211	Q₃₂	—	53	2	no	—
212	Q₃₂	—	106	1	no	—
213	Q₃₂	C₂²×D₅₃	1	2	yes	—
214	SD₃₂	—	2	2	no	—
215	SD₃₂	—	2	2	no	—
216	SD₃₂	—	106	2	no	—
217	SD₃₂	—	106	2	no	—
218	D₁₆	—	2	1	no	—
219	D₁₆	—	53	2	no	—
220	D₁₆	—	106	1	no	—
221	D₁₆	C₂²×D₅₃	1	2	yes	—
222	C₂×C₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
223	C₂×C₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
224	C₂×C₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
225	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
226	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
227	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
228	C₃₂	—	53	4	no	—
229	#41	—	4	2	no	—
230	#41	—	4	2	no	—
231	#41	—	8	1	no	—
232	#41	—	8	1	no	—
233	#41	—	424	1	no	—
234	#41	—	424	1	no	—
235	#41	—	212	2	no	—
236	#41	—	212	2	no	—
237	C₂×Q₈	—	4	1	no	—
238	C₂×Q₈	—	212	1	no	—
239	C₂×D₄	—	4	1	no	—
240	C₂×D₄	—	8	1	no	—
241	C₂×D₄	—	424	1	no	—
242	C₂×D₄	—	212	1	no	—
243	C₂²×C₄	—	8	1	no	—
244	C₂²×C₄	—	424	1	no	—
245	C₂²×C₄	#11719922	1	1	yes	characteristic
246	Q₁₆	—	2	2	no	—
247	Q₁₆	—	4	1	no	—
248	Q₁₆	—	106	2	no	—
249	Q₁₆	—	212	1	no	—
250	SD₁₆	—	4	2	no	—
251	SD₁₆	—	4	2	no	—
252	SD₁₆	—	212	2	no	—
253	SD₁₆	—	212	2	no	—
254	D₈	—	2	2	no	—
255	D₈	—	4	1	no	—
256	D₈	—	106	2	no	—
257	D₈	—	212	1	no	—
258	C₂×C₈	#11719922	1	1	yes	characteristic
259	C₂×C₈	#11719922	1	1	yes	characteristic
260	C₂×C₈	#11719922	1	1	yes	characteristic
261	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
262	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
263	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
264	C₁₆	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
265	C₁₆	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
266	C₁₆	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
267	C₁₆	C₂³×D₅₃	1	1	yes	characteristic
268	C₂³	—	8	1	no	—
269	C₂³	—	424	1	no	—
270	C₂³	#13133158	1	1	yes	characteristic
271	Q₈	—	4	2	no	—
272	Q₈	—	8	1	no	—
273	Q₈	—	424	1	no	—
274	Q₈	—	212	2	no	—
275	D₄	—	4	2	no	—
276	D₄	—	8	2	no	—
277	D₄	—	8	2	no	—
278	D₄	—	8	1	no	—
279	D₄	—	424	2	no	—
280	D₄	—	424	2	no	—
281	D₄	—	424	1	no	—
282	D₄	—	212	2	no	—
283	C₂×C₄	—	8	1	no	—
284	C₂×C₄	—	8	2	no	—
285	C₂×C₄	—	8	2	no	—
286	C₂×C₄	—	424	1	no	—
287	C₂×C₄	—	424	2	no	—
288	C₂×C₄	—	424	2	no	—
289	C₂×C₄	#13133158	1	1	yes	characteristic
290	C₂×C₄	#13133158	1	1	yes	characteristic
291	C₂×C₄	#13133158	1	1	yes	characteristic
292	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
293	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
294	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
295	C₈	C₂×#11719922	1	1	yes	characteristic
296	C₈	C₂×#11719922	1	1	yes	characteristic
297	C₈	C₂×#11719922	1	1	yes	characteristic
298	C₈	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
299	C₂²	—	4	1	no	—
300	C₂²	—	8	2	no	—
301	C₂²	—	16	2	no	—
302	C₂²	—	848	2	no	—
303	C₂²	—	424	2	no	—
304	C₂²	C₂×#13133158	1	1	yes	characteristic
305	C₂²	C₂×#13133158	1	1	yes	characteristic
306	C₂²	C₂×#13133158	1	1	yes	characteristic
307	C₄	—	8	2	no	—
308	C₄	—	424	2	no	—
309	C₄	C₂×#13133158	1	1	yes	characteristic
310	C₄	C₂×#13133158	1	1	yes	characteristic
311	C₄	C₂×#13133158	1	1	yes	characteristic
312	C₄	C₂²×#11719922	1	1	yes	characteristic
313	C₂	—	2	1	no	—
314	C₂	—	2	1	no	—
315	C₂	—	2	1	no	—
316	C₂	—	16	2	no	—
317	C₂	—	848	2	no	—
318	C₂	C₂²×#13133158	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#13133158
commutator, cosocle	C₁₆×C₅₃	C₂⁴
Frattini	C₁₆	C₂³×D₅₃
Fitting	C₂×C₅₃×#508	C₂
socle	C₂×C₅₃	C₂²×D₁₆
hypercenter	C₂×#508	D₅₃
hypocenter	C₅₃	#33981
LCS	G ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ C₂×#508

Complements

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

N	H ≅ G/N	type
1. C₂×#16811055	C₂	semidirect
2. #20277033	C₂	semidirect
3. #20277032	C₂	semidirect
4. #20277028	C₂	semidirect
5. #20277027	C₂	semidirect
6. #20277022	C₂	semidirect
7. C₂×#16810457	C₂	semidirect
8. C₂×C₅₃×#508	C₂	semidirect
13. #16811055	C₂²	semidirect
14. C₂×Dic₄₂₄	C₂²	semidirect
16. C₂×D₈₄₈	C₂²	semidirect
17. #16810463	C₂²	semidirect
18. #16810461	C₂²	semidirect
19. #16810460	C₂²	semidirect
20. #16810458	C₂²	semidirect
21. #16810457	C₂²	semidirect
22. #16810457	C₂²	semidirect
23. C₂×#13133123	C₂²	semidirect
28. C₅₃×#508	C₂²	semidirect
29. C₂×Q₃₂×C₅₃	C₂²	semidirect
31. C₂×D₁₆×C₅₃	C₂²	semidirect
155. C₂×#508	D₅₃	semidirect
177. #508	C₂×D₅₃	semidirect
179. C₂×Q₃₂	C₂×D₅₃	semidirect
183. C₂×D₁₆	C₂×D₅₃	semidirect
192. C₅₃	#33981	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811055	2
#20277033	2
#20277032	2
#20277028	2
#20277027	2
#20277022	2
C₂×#16810457	2
C₂×C₅₃×#508	2
#33981	53