Subgroups — #27538843

Order 13568.

514 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20285246	C₂	1	1	yes	maximal, characteristic, complemented
2	#20283633	C₂	1	1	yes	maximal, characteristic
3	#20276892	C₂	1	4	yes	maximal, complemented
4	#20276887	C₂	1	4	yes	maximal, complemented
5	#20276488	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16810446	C₂	1	1	yes	maximal, characteristic, complemented
7	#20276452	C₂	1	1	yes	maximal, characteristic
8	C₂×#16810434	C₂	1	1	yes	maximal, characteristic
9	C₅₃×#2153	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13133239	C₂²	1	1	yes	characteristic
11	#16811389	—	2	1	no	—
12	#16811251	—	2	1	no	—
13	#16811032	—	2	2	no	—
14	#16811025	C₂²	1	2	yes	complemented
15	#16811021	C₂²	1	2	yes	complemented
16	#16810746	—	2	2	no	—
17	#16810741	C₂²	1	4	yes	—
18	C₂×#13133153	C₂²	1	1	yes	characteristic
19	#16810665	—	2	1	no	—
20	#16810661	C₂²	1	1	yes	characteristic
21	C₂×#13133149	C₂²	1	1	yes	characteristic
22	#16810641	—	2	1	no	—
23	#16810446	C₂²	1	4	yes	complemented
24	#16810446	C₂²	1	2	yes	complemented
25	#16810446	C₂²	1	2	yes	complemented
26	#16810434	C₂²	1	2	yes	—
27	#16810434	C₂²	1	4	yes	—
28	#16810434	C₂²	1	2	yes	—
29	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
30	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
31	C₅₃×#405	C₂²	1	1	yes	characteristic
32	C₅₃×#364	C₂²	1	4	yes	complemented
33	C₂×#11719945	—	2	1	no	—
34	#13133239	—	2	1	no	—
35	#13133239	—	2	1	no	—
36	#13133239	Q₈	1	2	yes	complemented
37	#13133239	C₂³	1	2	yes	—
38	C₂×Dic₂₁₂	Q₈	1	1	yes	characteristic, complemented
39	C₂×#11719913	—	2	1	no	—
40	C₂×D₄₂₄	Q₈	1	1	yes	characteristic, complemented
41	#13133225	—	4	2	no	—
42	#13133223	—	2	4	no	—
43	C₂×#11719917	—	2	1	no	—
44	M₁₆×D₅₃	—	4	2	no	—
45	#13133217	—	2	2	no	—
46	#13133216	—	4	1	no	—
47	#13133209	—	2	2	no	—
48	#13133208	—	4	1	no	—
49	#13133194	—	4	2	no	—
50	#13133176	—	2	1	no	—
51	C₂×C₄⋊Dic₅₃	—	2	1	no	—
52	#13133153	—	2	1	no	—
53	#13133153	C₂³	1	2	yes	—
54	#13133151	—	2	2	no	—
55	#13133151	—	2	1	no	—
56	#13133149	—	2	2	no	—
57	#13133149	—	2	1	no	—
58	#13133149	C₂³	1	1	yes	characteristic
59	#13133149	C₂³	1	2	yes	—
60	#13133149	C₂³	1	1	yes	characteristic
61	#13133145	—	2	4	no	—
62	C₂×M₁₆×C₅₃	D₄	1	1	yes	characteristic
63	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
64	#117×C₅₃	D₄	1	1	yes	characteristic
65	M₃₂×C₅₃	—	2	1	no	—
66	M₃₂×C₅₃	—	2	1	no	—
67	M₃₂×C₅₃	C₂³	1	2	yes	—
68	M₃₂×C₅₃	C₂³	1	2	yes	—
69	C₂×C₁₆×C₅₃	—	2	1	no	—
70	C₂×C₁₆×C₅₃	—	2	1	no	—
71	#97×C₅₃	D₄	1	2	yes	—
72	C₄×C₈×C₅₃	C₂³	1	2	yes	—
73	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
74	C₂×C₂²⋊D₅₃	—	4	1	no	—
75	C₂²×Dic₅₃	—	4	1	no	—
76	#11719945	—	2	2	no	—
77	#11719945	—	2	2	no	—
78	#11719945	—	4	1	no	—
79	#11719945	—	4	1	no	—
80	C₂×D₂₁₂	—	2	1	no	—
81	C₂×C₄×D₅₃	—	4	1	no	—
82	C₂×Dic₁₀₆	—	2	1	no	—
83	C₂×Dic₁₀₆	—	4	2	no	—
84	M₁₆×C₅₃	SD₁₆	1	2	yes	—
85	M₁₆×C₅₃	SD₁₆	1	2	yes	—
86	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
88	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
89	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
90	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
91	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
92	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
93	C₄⋊C₄×C₅₃	—	2	1	no	—
94	C₂²⋊C₄×C₅₃	—	2	1	no	—
95	C₄²×C₅₃	C₂×D₄	1	2	yes	—
96	C₂²⋊Dic₅₃	—	4	1	no	—
97	C₄⋊Dic₅₃	—	2	2	no	—
98	C₄⋊Dic₅₃	—	2	1	no	—
99	C₄⋊Dic₅₃	—	2	1	no	—
100	C₄⋊Dic₅₃	—	4	2	no	—
101	C₄⋊Dic₅₃	—	4	1	no	—
102	#11719919	—	4	2	no	—
103	#11719919	—	4	2	no	—
104	C₄×Dic₅₃	—	4	1	no	—
105	#11719917	—	2	2	no	—
106	#11719917	—	4	1	no	—
107	C₂×#88322	—	4	1	no	—
108	Dic₂₁₂	—	2	1	no	—
109	Dic₂₁₂	C₂×Q₈	1	2	yes	—
110	D₄₂₄	—	2	1	no	—
111	D₄₂₄	C₂×Q₈	1	2	yes	—
112	#11719913	—	2	2	no	—
113	#11719913	—	2	2	no	—
114	#11719912	—	4	4	no	—
115	C₈×D₅₃	—	4	4	no	—
116	C₁₆×C₅₃	—	2	2	no	—
117	C₁₆×C₅₃	—	2	2	no	—
118	C₂³×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
119	C₂²×D₅₃	—	4	1	no	—
120	C₂×C₄×C₅₃	—	2	2	no	—
121	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
123	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
124	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
125	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
126	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
127	C₂²⋊D₅₃	—	4	2	no	—
128	C₂²⋊D₅₃	—	4	2	no	—
129	C₂×Dic₅₃	—	4	1	no	—
130	C₂×Dic₅₃	—	4	1	no	—
131	C₂×Dic₅₃	—	4	2	no	—
132	C₂×Dic₅₃	—	4	1	no	—
133	C₂×Dic₅₃	—	16	1	no	—
134	D₂₁₂	—	2	2	no	—
135	D₂₁₂	—	4	1	no	—
136	C₄×D₅₃	—	4	2	no	—
137	C₄×D₅₃	—	4	2	no	—
138	Dic₁₀₆	—	2	2	no	—
139	Dic₁₀₆	—	4	1	no	—
140	Dic₁₀₆	—	16	2	no	—
141	C₈×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
142	C₈×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
143	C₈×C₅₃	C₂×SD₁₆	1	4	yes	—
144	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
145	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
146	#88322	—	4	2	no	—
147	#33638	—	53	1	no	maximal, Sylow
148	C₂²×C₅₃	—	4	1	no	—
149	C₂²×C₅₃	#476	1	1	yes	characteristic
150	C₂²×C₅₃	#476	1	1	yes	characteristic
151	C₂²×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
152	C₂×D₅₃	—	4	2	no	—
153	C₂×D₅₃	—	16	1	no	—
154	C₄×C₅₃	—	4	2	no	—
155	C₄×C₅₃	#476	1	1	yes	characteristic
156	C₄×C₅₃	#476	1	1	yes	characteristic
157	C₄×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
158	C₄×C₅₃	C₂²×SD₁₆	1	1	yes	characteristic
159	Dic₅₃	—	4	2	no	—
160	Dic₅₃	—	8	1	no	—
161	Dic₅₃	—	8	2	no	—
162	Dic₅₃	—	8	1	no	—
163	#3163	—	53	1	no	—
164	#2957	—	53	1	no	—
165	#2231	—	53	4	no	—
166	#2230	—	53	4	no	—
167	#2155	—	53	1	no	—
168	C₂×#365	—	53	1	no	—
169	#2153	D₅₃	1	1	yes	characteristic, complemented
170	#2145	—	53	1	no	—
171	C₂×#360	—	53	1	no	—
172	C₂×C₅₃	—	2	1	no	—
173	C₂×C₅₃	—	2	1	no	—
174	C₂×C₅₃	—	2	1	no	—
175	C₂×C₅₃	C₂×#476	1	1	yes	characteristic
176	D₅₃	—	8	2	no	—
177	C₂×#135	—	53	1	no	—
178	#568	—	106	1	no	—
179	#544	—	106	1	no	—
180	C₂×M₃₂	C₂×D₅₃	1	1	yes	characteristic
181	C₂×M₃₂	C₂×D₅₃	1	1	yes	characteristic
182	#501	—	106	2	no	—
183	#499	—	53	2	no	—
184	#498	—	53	2	no	—
185	#444	—	106	2	no	—
186	#443	—	53	4	no	—
187	C₂×#108	—	53	1	no	—
188	#428	—	106	1	no	—
189	#427	—	53	1	no	—
190	C₂×#106	—	53	1	no	—
191	#421	—	106	1	no	—
192	#405	C₂×D₅₃	1	1	yes	characteristic
193	#365	—	53	4	no	—
194	#365	—	53	2	no	—
195	#365	—	53	2	no	—
196	#364	C₂×D₅₃	1	4	yes	complemented
197	#360	—	53	4	no	—
198	#360	—	53	2	no	—
199	#360	—	53	2	no	—
200	C₅₃	#33638	1	1	yes	characteristic, complemented, Sylow
201	C₂×#41	—	106	1	no	—
202	#135	—	53	2	no	—
203	#135	—	53	2	no	—
204	#135	—	106	1	no	—
205	#135	—	106	1	no	—
206	C₂×Q₁₆	—	53	1	no	—
207	C₂×SD₁₆	—	106	1	no	—
208	C₂×D₈	—	53	1	no	—
209	#131	—	106	4	no	—
210	#131	—	212	2	no	—
211	C₂×M₁₆	—	106	1	no	—
212	C₂×M₁₆	—	212	2	no	—
213	C₂×M₁₆	D₂₁₂	1	1	yes	characteristic
214	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
215	C₄⋊Q₈	—	106	2	no	—
216	C₄⋊Q₈	—	212	1	no	—
217	#125	—	106	2	no	—
218	#125	—	212	1	no	—
219	C₂²⋊Q₈	—	212	2	no	—
220	#117	—	106	1	no	—
221	#117	D₂₁₂	1	1	yes	characteristic
222	C₂×C₄⋊C₄	—	106	1	no	—
223	M₃₂	—	2	1	no	—
224	M₃₂	—	2	1	no	—
225	M₃₂	C₂²×D₅₃	1	2	yes	—
226	M₃₂	C₂²×D₅₃	1	2	yes	—
227	C₂×C₁₆	—	2	1	no	—
228	C₂×C₁₆	—	2	1	no	—
229	#108	—	53	2	no	—
230	#108	—	106	1	no	—
231	#107	—	106	1	no	—
232	#107	—	106	2	no	—
233	#106	—	53	2	no	—
234	#106	—	53	1	no	—
235	#106	—	53	1	no	—
236	#106	—	106	1	no	—
237	#106	—	106	2	no	—
238	#104	—	106	4	no	—
239	#97	D₂₁₂	1	2	yes	—
240	C₄×C₈	C₂²×D₅₃	1	2	yes	—
241	#41	—	106	2	no	—
242	#41	—	106	2	no	—
243	#41	—	212	1	no	—
244	#41	—	212	1	no	—
245	C₂×Q₈	—	106	1	no	—
246	C₂×Q₈	—	212	2	no	—
247	C₂×D₄	—	106	1	no	—
248	C₂×D₄	—	212	1	no	—
249	C₂²×C₄	—	212	1	no	—
250	C₂²×C₄	—	212	1	no	—
251	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
252	Q₁₆	—	53	2	no	—
253	Q₁₆	—	106	1	no	—
254	SD₁₆	—	106	2	no	—
255	SD₁₆	—	106	2	no	—
256	D₈	—	53	2	no	—
257	D₈	—	106	1	no	—
258	M₁₆	—	106	2	no	—
259	M₁₆	—	212	1	no	—
260	M₁₆	—	212	4	no	—
261	M₁₆	#11719913	1	2	yes	—
262	M₁₆	#11719913	1	2	yes	—
263	C₂×C₈	—	212	1	no	—
264	C₂×C₈	—	212	4	no	—
265	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
266	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
267	C₂×C₈	#11719947	1	1	yes	characteristic
268	C₂×C₈	#11719947	1	1	yes	characteristic
269	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
270	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
271	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
272	C₄⋊C₄	—	2	1	no	—
273	C₄⋊C₄	—	106	2	no	—
274	C₄⋊C₄	—	106	1	no	—
275	C₄⋊C₄	—	106	1	no	—
276	C₄⋊C₄	—	212	2	no	—
277	C₄⋊C₄	—	212	2	no	—
278	C₄⋊C₄	—	212	2	no	—
279	C₄⋊C₄	—	212	1	no	—
280	C₂²⋊C₄	—	2	1	no	—
281	C₂²⋊C₄	—	212	1	no	—
282	C₄²	—	212	1	no	—
283	C₄²	C₂×D₂₁₂	1	2	yes	—
284	C₁₆	—	2	2	no	—
285	C₁₆	—	2	2	no	—
286	C₂³	—	212	1	no	—
287	C₂³	#13133245	1	1	yes	characteristic
288	Q₈	—	848	2	no	—
289	Q₈	—	106	2	no	—
290	Q₈	—	212	1	no	—
291	D₄	—	106	2	no	—
292	D₄	—	212	2	no	—
293	D₄	—	212	2	no	—
294	D₄	—	212	1	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	848	1	no	—
297	C₂×C₄	—	212	1	no	—
298	C₂×C₄	—	212	1	no	—
299	C₂×C₄	—	212	2	no	—
300	C₂×C₄	—	212	1	no	—
301	C₂×C₄	—	212	2	no	—
302	C₂×C₄	—	212	2	no	—
303	C₂×C₄	#13133197	1	1	yes	characteristic
304	C₂×C₄	#13133197	1	1	yes	characteristic
305	C₂×C₄	C₂×#11719913	1	1	yes	characteristic
306	C₂×C₄	C₂×#11719913	1	1	yes	characteristic
307	C₂×C₄	#13133245	1	1	yes	characteristic
308	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
309	C₈	—	212	2	no	—
310	C₈	#13133197	1	1	yes	characteristic
311	C₈	#13133197	1	1	yes	characteristic
312	C₈	C₂×#11719913	1	4	yes	—
313	C₈	C₂×Q₈×D₅₃	1	1	yes	characteristic
314	C₈	C₂×#11719947	1	1	yes	characteristic
315	C₂²	—	4	1	no	—
316	C₂²	—	848	1	no	—
317	C₂²	—	212	2	no	—
318	C₂²	#16810914	1	1	yes	characteristic
319	C₂²	#16810914	1	1	yes	characteristic
320	C₂²	C₂×#13133245	1	1	yes	characteristic
321	C₄	—	4	2	no	—
322	C₄	—	424	2	no	—
323	C₄	—	424	1	no	—
324	C₄	—	424	1	no	—
325	C₄	—	212	2	no	—
326	C₄	#16810914	1	1	yes	characteristic
327	C₄	#16810914	1	1	yes	characteristic
328	C₄	C₂×#13133197	1	1	yes	characteristic
329	C₄	C₂²×#11719913	1	1	yes	characteristic
330	C₂	—	2	1	no	—
331	C₂	—	2	1	no	—
332	C₂	—	2	1	no	—
333	C₂	—	424	2	no	—
334	C₂	C₂×#16810914	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#16810914
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2153	C₂
socle	C₂×C₅₃	C₂×#476
prime radical	C₂×#13133239	C₂²
hypercenter	#2153	D₅₃
hypocenter	C₅₃	#33638
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ #2153

Complements

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

N	H ≅ G/N	type
1. #20285246	C₂	semidirect
3. #20276892	C₂	semidirect
4. #20276887	C₂	semidirect
5. #20276488	C₂	semidirect
6. C₂×#16810446	C₂	semidirect
9. C₅₃×#2153	C₂	semidirect
14. #16811025	C₂²	semidirect
15. #16811021	C₂²	semidirect
23. #16810446	C₂²	semidirect
24. #16810446	C₂²	semidirect
25. #16810446	C₂²	semidirect
32. C₅₃×#364	C₂²	semidirect
36. #13133239	Q₈	semidirect
38. C₂×Dic₂₁₂	Q₈	semidirect
40. C₂×D₄₂₄	Q₈	semidirect
169. #2153	D₅₃	semidirect
196. #364	C₂×D₅₃	semidirect
200. C₅₃	#33638	semidirect

Maximal subgroups

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

subgroup	index
#20285246	2
#20283633	2
#20276892	2
#20276887	2
#20276488	2
C₂×#16810446	2
#20276452	2
C₂×#16810434	2
C₅₃×#2153	2
#33638	53