Subgroups — #27539523

Order 13568.

918 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811058	C₂	1	4	yes	maximal, complemented
2	C₂×#16811000	C₂	1	1	yes	maximal, characteristic, complemented
3	#20276982	C₂	1	4	yes	maximal, complemented
4	#20276972	C₂	1	2	yes	maximal, complemented
5	#20276964	C₂	1	2	yes	maximal, complemented
6	C₄×#13133157	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#3147	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₄₂₄	—	2	1	no	—
9	C₂²×D₄₂₄	C₂²	1	2	yes	—
10	C₂×#13133214	—	2	1	no	—
11	#16811058	—	2	16	no	—
12	C₂×#13133160	C₂²	1	8	yes	complemented
13	C₂×#13133158	C₂²	1	8	yes	complemented
14	C₂×#13133157	C₂²	1	2	yes	—
15	#16811000	—	2	1	no	—
16	#16811000	—	2	1	no	—
17	#16811000	C₂²	1	2	yes	—
18	#16811000	C₂²	1	2	yes	—
19	#16810399	C₂²	1	2	yes	complemented
20	C₄×#11719908	C₂²	1	2	yes	complemented
21	C₂×#135×C₅₃	—	2	1	no	—
22	C₂×#135×C₅₃	C₂²	1	2	yes	—
23	C₅₃×#532	—	2	1	no	—
24	C₅₃×#495	—	2	1	no	—
25	C₅₃×#495	C₂²	1	2	yes	complemented
26	C₅₃×#494	C₂²	1	1	yes	characteristic, complemented
27	C₅₃×#493	C₂²	1	1	yes	characteristic, complemented
28	C₅₃×#492	—	2	1	no	—
29	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
30	C₂²×D₂₁₂	—	2	2	no	—
31	C₂²×D₂₁₂	—	4	2	no	—
32	C₂×D₄₂₄	—	2	2	no	—
33	C₂×D₄₂₄	—	2	2	no	—
34	C₂×D₄₂₄	—	2	2	no	—
35	C₂×D₄₂₄	—	2	8	no	—
36	C₂×D₄₂₄	—	4	4	no	—
37	C₂×D₄₂₄	C₂³	1	4	yes	—
38	#13133214	—	2	2	no	—
39	#13133214	—	2	2	no	—
40	#13133214	—	4	2	no	—
41	#13133160	—	2	16	no	—
42	#13133158	—	2	16	no	—
43	#13133157	D₄	1	8	yes	complemented
44	C₂×#11719908	C₂³	1	4	yes	—
45	C₂×#41×C₅₃	—	2	2	no	—
46	#135×C₅₃	—	2	8	no	—
47	#135×C₅₃	—	4	4	no	—
48	C₂×Q₁₆×C₅₃	—	2	1	no	—
49	C₂×Q₁₆×C₅₃	C₂³	1	2	yes	—
50	C₂×SD₁₆×C₅₃	—	2	2	no	—
51	C₂×SD₁₆×C₅₃	—	2	2	no	—
52	C₂×D₈×C₅₃	—	2	1	no	—
53	C₂×D₈×C₅₃	C₂³	1	2	yes	—
54	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
55	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
56	C₄⋊Q₈×C₅₃	—	2	1	no	—
57	#127×C₅₃	—	2	1	no	—
58	#124×C₅₃	—	2	2	no	—
59	#121×C₅₃	—	4	2	no	—
60	C₄×D₄×C₅₃	—	4	2	no	—
61	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
62	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
63	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
64	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
65	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
66	C₂³×D₅₃	—	4	2	no	—
67	#41×C₅₃	—	4	4	no	—
68	#41×C₅₃	—	4	4	no	—
69	C₂×Q₈×C₅₃	—	2	2	no	—
70	C₂×D₄×C₅₃	—	2	2	no	—
71	C₂×D₄×C₅₃	—	4	2	no	—
72	C₂×D₄×C₅₃	—	4	2	no	—
73	C₂²×C₄×C₅₃	—	4	2	no	—
74	C₂²×C₄×C₅₃	D₈	1	2	yes	—
75	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
76	C₂×D₂₁₂	—	2	4	no	—
77	C₂×D₂₁₂	—	4	4	no	—
78	C₂×D₂₁₂	—	4	4	no	—
79	C₂×D₂₁₂	—	4	4	no	—
80	C₂×D₂₁₂	—	4	4	no	—
81	C₂×D₂₁₂	—	4	2	no	—
82	C₂×D₂₁₂	—	8	8	no	—
83	Q₁₆×C₅₃	—	2	4	no	—
84	Q₁₆×C₅₃	—	4	2	no	—
85	SD₁₆×C₅₃	—	4	4	no	—
86	SD₁₆×C₅₃	—	4	4	no	—
87	D₈×C₅₃	—	2	4	no	—
88	D₈×C₅₃	—	4	2	no	—
89	C₂×C₈×C₅₃	D₈	1	4	yes	—
90	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
92	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
95	C₄⋊C₄×C₅₃	—	4	2	no	—
96	C₂²⋊C₄×C₅₃	—	4	4	no	—
97	C₄²×C₅₃	D₈	1	2	yes	—
98	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
100	D₄₂₄	—	2	8	no	—
101	D₄₂₄	—	4	4	no	—
102	D₄₂₄	—	4	4	no	—
103	D₄₂₄	—	4	4	no	—
104	#11719908	C₂×D₄	1	8	yes	complemented
105	C₂³×C₅₃	—	4	2	no	—
106	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
107	C₂²×D₅₃	—	4	4	no	—
108	C₂²×D₅₃	—	8	8	no	—
109	C₂²×D₅₃	—	8	4	no	—
110	Q₈×C₅₃	—	4	2	no	—
111	Q₈×C₅₃	—	4	2	no	—
112	D₄×C₅₃	—	4	2	no	—
113	D₄×C₅₃	—	4	2	no	—
114	D₄×C₅₃	—	8	4	no	—
115	D₄×C₅₃	—	8	4	no	—
116	C₂×C₄×C₅₃	—	2	4	no	—
117	C₂×C₄×C₅₃	—	4	2	no	—
118	C₂×C₄×C₅₃	—	8	4	no	—
119	C₂×C₄×C₅₃	#127	1	2	yes	—
120	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
121	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
122	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
123	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
124	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
125	D₂₁₂	—	4	4	no	—
126	D₂₁₂	—	4	4	no	—
127	D₂₁₂	—	8	8	no	—
128	D₂₁₂	—	8	4	no	—
129	D₂₁₂	—	8	8	no	—
130	C₈×C₅₃	#127	1	2	yes	—
131	C₈×C₅₃	C₂×D₈	1	2	yes	—
132	C₈×C₅₃	C₂×D₈	1	2	yes	—
133	C₈×C₅₃	C₂²×D₄	1	2	yes	—
134	#33717	—	53	1	no	maximal, Sylow
135	C₂²×C₅₃	—	2	2	no	—
136	C₂²×C₅₃	—	8	2	no	—
137	C₂²×C₅₃	—	8	4	no	—
138	C₂²×C₅₃	#493	1	2	yes	—
139	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
140	C₂×D₅₃	—	8	4	no	—
141	C₂×D₅₃	—	8	8	no	—
142	C₂×D₅₃	—	16	8	no	—
143	C₄×C₅₃	—	2	2	no	—
144	C₄×C₅₃	—	8	2	no	—
145	C₄×C₅₃	#493	1	2	yes	—
146	C₄×C₅₃	#509	1	4	yes	complemented
147	C₄×C₅₃	C₂×#127	1	1	yes	characteristic
148	C₄×C₅₃	C₂²×D₈	1	1	yes	characteristic
149	C₂×#509	—	53	4	no	—
150	#3147	D₅₃	1	1	yes	characteristic, complemented
151	C₂×#493	—	53	1	no	—
152	#2250	—	53	4	no	—
153	#2248	—	53	2	no	—
154	#2246	—	53	2	no	—
155	C₄×M₃₂	—	53	1	no	—
156	C₂×C₅₃	—	2	2	no	—
157	C₂×C₅₃	—	16	2	no	—
158	C₂×C₅₃	C₂×#493	1	1	yes	characteristic
159	C₂×C₅₃	C₂×#509	1	2	yes	—
160	D₅₃	—	16	4	no	—
161	C₂×#135	—	2	1	no	—
162	C₂×#135	C₂×D₅₃	1	2	yes	—
163	C₂²×D₈	—	53	2	no	—
164	C₂²×D₈	—	106	1	no	—
165	#532	—	2	1	no	—
166	C₂×#127	—	106	1	no	—
167	#509	—	106	16	no	—
168	C₂×SD₃₂	—	53	8	no	—
169	C₂×D₁₆	—	53	8	no	—
170	C₂×M₃₂	—	53	2	no	—
171	#495	—	2	1	no	—
172	#495	C₂×D₅₃	1	2	yes	complemented
173	#494	C₂×D₅₃	1	1	yes	characteristic, complemented
174	#493	—	53	2	no	—
175	#493	—	53	2	no	—
176	#493	—	106	1	no	—
177	#493	—	106	1	no	—
178	#493	C₂×D₅₃	1	1	yes	characteristic, complemented
179	#492	—	2	1	no	—
180	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
181	#346	—	53	2	no	—
182	C₄×C₁₆	—	53	2	no	—
183	C₅₃	#33717	1	1	yes	characteristic, complemented, Sylow
184	C₂×#41	—	2	2	no	—
185	C₂²×D₄	—	106	2	no	—
186	C₂²×D₄	—	212	2	no	—
187	#135	—	2	8	no	—
188	#135	—	4	4	no	—
189	C₂×Q₁₆	—	2	1	no	—
190	C₂×Q₁₆	C₂²×D₅₃	1	2	yes	—
191	C₂×SD₁₆	—	2	2	no	—
192	C₂×SD₁₆	—	2	2	no	—
193	C₂×D₈	—	2	1	no	—
194	C₂×D₈	—	53	4	no	—
195	C₂×D₈	—	106	8	no	—
196	C₂×D₈	—	106	2	no	—
197	C₂×D₈	—	106	2	no	—
198	C₂×D₈	—	106	2	no	—
199	C₂×D₈	—	212	4	no	—
200	C₂×D₈	C₂²×D₅₃	1	2	yes	—
201	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
202	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
203	C₄⋊Q₈	—	2	1	no	—
204	#127	—	2	1	no	—
205	#127	—	106	2	no	—
206	#127	—	106	2	no	—
207	#127	—	212	2	no	—
208	#124	—	2	2	no	—
209	#121	—	4	2	no	—
210	C₄×D₄	—	4	2	no	—
211	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
212	SD₃₂	—	106	16	no	—
213	D₁₆	—	106	16	no	—
214	M₃₂	—	53	8	no	—
215	C₂×C₁₆	—	53	4	no	—
216	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
217	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
218	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
219	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
220	C₂⁴	—	212	2	no	—
221	#41	—	4	4	no	—
222	#41	—	4	4	no	—
223	C₂×Q₈	—	2	2	no	—
224	C₂×D₄	—	2	2	no	—
225	C₂×D₄	—	4	2	no	—
226	C₂×D₄	—	4	2	no	—
227	C₂×D₄	—	106	4	no	—
228	C₂×D₄	—	424	8	no	—
229	C₂×D₄	—	212	4	no	—
230	C₂×D₄	—	212	4	no	—
231	C₂×D₄	—	212	4	no	—
232	C₂×D₄	—	212	2	no	—
233	C₂×D₄	—	212	4	no	—
234	C₂²×C₄	—	4	2	no	—
235	C₂²×C₄	#11719922	1	2	yes	—
236	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
237	Q₁₆	—	2	4	no	—
238	Q₁₆	—	4	2	no	—
239	SD₁₆	—	4	4	no	—
240	SD₁₆	—	4	4	no	—
241	D₈	—	2	4	no	—
242	D₈	—	4	2	no	—
243	D₈	—	106	8	no	—
244	D₈	—	212	4	no	—
245	D₈	—	212	4	no	—
246	D₈	—	212	4	no	—
247	C₂×C₈	#11719922	1	4	yes	—
248	C₂×C₈	D₄×D₅₃	1	4	yes	—
249	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
250	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
251	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
252	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
253	C₄⋊C₄	—	4	2	no	—
254	C₂²⋊C₄	—	4	4	no	—
255	C₄²	#11719922	1	2	yes	—
256	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
257	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
258	C₁₆	—	53	8	no	—
259	C₂³	—	4	2	no	—
260	C₂³	—	424	8	no	—
261	C₂³	—	424	4	no	—
262	C₂³	—	212	4	no	—
263	C₂³	C₂×#11719922	1	1	yes	characteristic
264	Q₈	—	4	2	no	—
265	Q₈	—	4	2	no	—
266	D₄	—	4	2	no	—
267	D₄	—	4	2	no	—
268	D₄	—	8	4	no	—
269	D₄	—	8	4	no	—
270	D₄	—	424	8	no	—
271	D₄	—	424	8	no	—
272	D₄	—	424	4	no	—
273	D₄	—	212	4	no	—
274	D₄	—	212	4	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	8	4	no	—
278	C₂×C₄	#13133213	1	2	yes	—
279	C₂×C₄	C₂×#11719922	1	2	yes	—
280	C₂×C₄	C₂×#11719922	1	2	yes	—
281	C₂×C₄	C₂×#11719922	1	2	yes	—
282	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
283	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
284	C₈	#13133213	1	2	yes	—
285	C₈	C₂×#11719922	1	2	yes	—
286	C₈	C₂×#11719922	1	2	yes	—
287	C₈	C₂×D₄×D₅₃	1	2	yes	—
288	C₂²	—	2	2	no	—
289	C₂²	—	8	4	no	—
290	C₂²	—	8	2	no	—
291	C₂²	—	848	8	no	—
292	C₂²	—	424	8	no	—
293	C₂²	—	424	4	no	—
294	C₂²	#16810999	1	2	yes	—
295	C₂²	C₂²×#11719922	1	1	yes	characteristic
296	C₄	—	2	2	no	—
297	C₄	—	8	2	no	—
298	C₄	#16810999	1	2	yes	—
299	C₄	#16811058	1	4	yes	complemented
300	C₄	C₂×#13133213	1	1	yes	characteristic
301	C₄	C₂²×#11719922	1	1	yes	characteristic
302	C₂	—	2	2	no	—
303	C₂	—	16	2	no	—
304	C₂	—	848	4	no	—
305	C₂	C₂×#16810999	1	1	yes	characteristic
306	C₂	C₂×#16811058	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×#11719922
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#3147	C₂
socle	C₂²×C₅₃	C₂²×D₈
hypercenter	#3147	D₅₃
hypocenter	C₅₃	#33717
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3147

Complements

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

N	H ≅ G/N	type
1. C₂×#16811058	C₂	semidirect
2. C₂×#16811000	C₂	semidirect
3. #20276982	C₂	semidirect
4. #20276972	C₂	semidirect
5. #20276964	C₂	semidirect
6. C₄×#13133157	C₂	semidirect
7. C₅₃×#3147	C₂	semidirect
12. C₂×#13133160	C₂²	semidirect
13. C₂×#13133158	C₂²	semidirect
19. #16810399	C₂²	semidirect
20. C₄×#11719908	C₂²	semidirect
25. C₅₃×#495	C₂²	semidirect
26. C₅₃×#494	C₂²	semidirect
27. C₅₃×#493	C₂²	semidirect
43. #13133157	D₄	semidirect
104. #11719908	C₂×D₄	semidirect
146. C₄×C₅₃	#509	semidirect
150. #3147	D₅₃	semidirect
172. #495	C₂×D₅₃	semidirect
173. #494	C₂×D₅₃	semidirect
178. #493	C₂×D₅₃	semidirect
183. C₅₃	#33717	semidirect
299. C₄	#16811058	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811058	2
C₂×#16811000	2
#20276982	2
#20276972	2
#20276964	2
C₄×#13133157	2
C₅₃×#3147	2
#33717	53