Subgroups — #27542129

Order 13568.

498 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20287704	C₂	1	1	yes	maximal, characteristic
2	#20276984	C₂	1	2	yes	maximal
3	#20276967	C₂	1	1	yes	maximal, characteristic, complemented
4	#20276777	C₂	1	2	yes	maximal
5	#20276765	C₂	1	1	yes	maximal, characteristic, complemented
6	#20276762	C₂	1	2	yes	maximal, complemented
7	#20276757	C₂	1	1	yes	maximal, characteristic
8	#20276624	C₂	1	1	yes	maximal, characteristic, complemented
9	#20276584	C₂	1	2	yes	maximal, complemented
10	C₄×#13133160	C₂	1	1	yes	maximal, characteristic
11	C₅₃×#3395	C₂	1	1	yes	maximal, characteristic, complemented
12	#16811346	—	2	1	no	—
13	C₂×#13133163	C₂²	1	2	yes	—
14	C₂×#13133163	C₂²	1	2	yes	—
15	C₂×#13133163	C₂²	1	2	yes	—
16	C₂×#13133160	C₂²	1	1	yes	characteristic
17	C₂×#13133160	C₂²	1	2	yes	—
18	#16811007	C₂²	1	2	yes	—
19	#16811003	C₂²	1	1	yes	characteristic
20	#16810846	—	2	2	no	—
21	#16810829	—	2	1	no	—
22	C₄×Dic₂₁₂	C₂²	1	2	yes	—
23	C₄×D₄₂₄	C₂²	1	1	yes	characteristic
24	#16810450	C₂²	1	1	yes	characteristic
25	C₄⋊#11719908	C₂²	1	1	yes	characteristic
26	C₄⋊#11719908	C₂²	1	2	yes	—
27	#16810427	C₂²	1	2	yes	—
28	#16810427	C₂²	1	1	yes	characteristic
29	#16810426	C₂²	1	2	yes	—
30	#16810423	C₂²	1	1	yes	characteristic
31	#16810399	C₂²	1	2	yes	—
32	C₄×#11719908	C₂²	1	1	yes	characteristic
33	C₅₃×#557	—	2	1	no	—
34	C₅₃×#494	C₂²	1	1	yes	characteristic
35	C₅₃×#494	C₂²	1	2	yes	—
36	C₅₃×#462	—	2	1	no	—
37	C₅₃×#462	—	2	2	no	—
38	C₈×Q₈×C₅₃	C₂²	1	1	yes	characteristic
39	C₄×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
40	C₄×Q₁₆×C₅₃	C₂²	1	2	yes	—
41	C₂×Dic₂₁₂	—	2	1	no	—
42	C₂×Dic₂₁₂	C₂³	1	2	yes	—
43	C₂×#11719913	—	2	2	no	—
44	C₂×D₄₂₄	C₂³	1	1	yes	characteristic
45	#13133217	—	2	1	no	—
46	#13133212	—	4	2	no	—
47	#13133205	—	2	2	no	—
48	#13133193	—	4	1	no	—
49	C₄×Dic₁₀₆	—	2	2	no	—
50	C₄×D₂₁₂	—	2	1	no	—
51	#13133175	—	4	1	no	—
52	#13133163	—	2	4	no	—
53	#13133163	—	4	2	no	—
54	#13133163	—	4	2	no	—
55	#13133160	—	2	2	no	—
56	#13133160	—	4	2	no	—
57	C₂×#11719908	C₂³	1	2	yes	—
58	C₂×#11719908	C₂³	1	1	yes	characteristic
59	C₂×#11719908	C₂³	1	1	yes	characteristic
60	#13133151	C₂³	1	1	yes	characteristic
61	#13133142	—	2	2	no	—
62	#13133139	—	2	1	no	—
63	C₂×Q₁₆×C₅₃	—	2	1	no	—
64	C₂×Q₁₆×C₅₃	—	2	2	no	—
65	C₂×Q₁₆×C₅₃	C₂³	1	1	yes	characteristic
66	C₂×Q₁₆×C₅₃	C₂³	1	2	yes	—
67	C₄⋊Q₈×C₅₃	—	2	1	no	—
68	C₄⋊Q₈×C₅₃	—	2	2	no	—
69	#125×C₅₃	—	4	1	no	—
70	#125×C₅₃	—	4	2	no	—
71	C₄×Q₈×C₅₃	—	2	1	no	—
72	C₄×Q₈×C₅₃	—	2	2	no	—
73	C₄×Q₈×C₅₃	—	4	1	no	—
74	C₄×Q₈×C₅₃	D₄	1	1	yes	characteristic
75	#107×C₅₃	C₂³	1	1	yes	characteristic
76	C₄⋊C₈×C₅₃	D₄	1	1	yes	characteristic
77	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
78	#103×C₅₃	—	2	1	no	—
79	#103×C₅₃	—	2	2	no	—
80	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
81	C₄×C₈×C₅₃	C₂³	1	2	yes	—
82	C₂×Q₈×C₅₃	—	2	1	no	—
83	C₂×Q₈×C₅₃	—	2	2	no	—
84	C₂×Q₈×C₅₃	D₈	1	1	yes	characteristic
85	C₂×D₂₁₂	—	2	1	no	—
86	C₂×C₄×D₅₃	—	4	1	no	—
87	C₂×Dic₁₀₆	—	2	2	no	—
88	Q₁₆×C₅₃	—	2	1	no	—
89	Q₁₆×C₅₃	—	2	2	no	—
90	Q₁₆×C₅₃	—	2	2	no	—
91	Q₁₆×C₅₃	—	2	1	no	—
92	Q₁₆×C₅₃	—	8	2	no	—
93	Q₁₆×C₅₃	—	8	1	no	—
94	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
96	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
97	C₄⋊C₄×C₅₃	—	2	1	no	—
98	C₄⋊C₄×C₅₃	—	4	2	no	—
99	C₄⋊C₄×C₅₃	—	4	1	no	—
100	C₄⋊C₄×C₅₃	—	4	2	no	—
101	C₄⋊C₄×C₅₃	—	4	1	no	—
102	C₄⋊C₄×C₅₃	—	4	2	no	—
103	C₄⋊C₄×C₅₃	—	4	1	no	—
104	C₄⋊C₄×C₅₃	D₈	1	1	yes	characteristic
105	C₄⋊C₄×C₅₃	D₈	1	2	yes	—
106	C₄²×C₅₃	—	4	2	no	—
107	C₄²×C₅₃	—	4	1	no	—
108	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₅₃	C₂×D₄	1	2	yes	—
110	#11719921	—	4	2	no	—
111	#11719921	—	4	1	no	—
112	#11719921	—	4	2	no	—
113	C₄⋊Dic₅₃	—	2	1	no	—
114	C₄⋊Dic₅₃	—	4	1	no	—
115	C₄⋊Dic₅₃	—	4	1	no	—
116	#11719919	—	4	2	no	—
117	C₄×Dic₅₃	—	4	2	no	—
118	Dic₂₁₂	—	2	2	no	—
119	Dic₂₁₂	—	2	2	no	—
120	Dic₂₁₂	—	8	1	no	—
121	D₄₂₄	—	2	1	no	—
122	D₄₂₄	—	2	1	no	—
123	#11719913	—	8	2	no	—
124	#11719908	—	2	2	no	—
125	#11719908	—	2	1	no	—
126	#11719908	#41	1	2	yes	—
127	C₂²×D₅₃	—	4	1	no	—
128	Q₈×C₅₃	—	2	2	no	—
129	Q₈×C₅₃	—	4	1	no	—
130	Q₈×C₅₃	—	4	2	no	—
131	Q₈×C₅₃	—	4	1	no	—
132	Q₈×C₅₃	—	4	2	no	—
133	C₂×C₄×C₅₃	—	4	2	no	—
134	C₂×C₄×C₅₃	—	4	1	no	—
135	C₂×C₄×C₅₃	—	4	1	no	—
136	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
137	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
138	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
139	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
140	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
141	C₂×Dic₅₃	—	4	2	no	—
142	C₂×Dic₅₃	—	4	1	no	—
143	D₂₁₂	—	4	1	no	—
144	D₂₁₂	—	4	1	no	—
145	C₄×D₅₃	—	8	2	no	—
146	Dic₁₀₆	—	4	2	no	—
147	Dic₁₀₆	—	4	2	no	—
148	C₈×C₅₃	—	2	2	no	—
149	C₈×C₅₃	—	2	1	no	—
150	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
151	C₈×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
152	#33965	—	53	1	no	maximal, Sylow
153	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
154	C₂×D₅₃	—	8	1	no	—
155	C₂×D₅₃	—	8	1	no	—
156	C₂×D₅₃	—	8	1	no	—
157	C₄×C₅₃	—	2	2	no	—
158	C₄×C₅₃	—	2	2	no	—
159	C₄×C₅₃	—	2	1	no	—
160	C₄×C₅₃	—	8	1	no	—
161	C₄×C₅₃	—	8	2	no	—
162	C₄×C₅₃	—	8	1	no	—
163	C₄×C₅₃	#510	1	2	yes	—
164	C₄×C₅₃	#550	1	1	yes	characteristic
165	C₄×C₅₃	#576	1	1	yes	characteristic
166	Dic₅₃	—	8	2	no	—
167	Dic₅₃	—	8	1	no	—
168	#3395	D₅₃	1	1	yes	characteristic, complemented
169	#3393	—	53	1	no	—
170	#2251	—	53	2	no	—
171	#2247	—	53	1	no	—
172	#2211	—	53	2	no	—
173	#2209	—	53	2	no	—
174	#2209	—	53	1	no	—
175	#2208	—	53	1	no	—
176	#2183	—	53	1	no	—
177	#2176	—	53	2	no	—
178	C₄×SD₃₂	—	53	1	no	—
179	C₂×C₅₃	#3391	1	1	yes	characteristic
180	C₂×C₅₃	C₂×#510	1	1	yes	characteristic
181	C₂×C₅₃	#3417	1	1	yes	characteristic
182	D₅₃	—	16	1	no	—
183	#557	—	2	1	no	—
184	#556	—	106	1	no	—
185	C₂×Q₃₂	—	53	2	no	—
186	C₂×Q₃₂	—	53	2	no	—
187	C₂×Q₃₂	—	53	2	no	—
188	C₂×SD₃₂	—	53	1	no	—
189	C₂×SD₃₂	—	53	2	no	—
190	#495	—	53	2	no	—
191	#494	—	53	1	no	—
192	#494	C₂×D₅₃	1	1	yes	characteristic
193	#494	C₂×D₅₃	1	2	yes	—
194	#464	—	106	2	no	—
195	#462	—	2	1	no	—
196	#462	—	2	2	no	—
197	#461	—	106	1	no	—
198	C₈×Q₈	C₂×D₅₃	1	1	yes	characteristic
199	C₄×Q₁₆	—	53	2	no	—
200	C₄×Q₁₆	C₂×D₅₃	1	1	yes	characteristic
201	C₄×Q₁₆	C₂×D₅₃	1	2	yes	—
202	C₄×D₈	—	53	1	no	—
203	#367	—	53	1	no	—
204	C₄⋊C₁₆	—	53	1	no	—
205	C₄⋊C₁₆	—	53	2	no	—
206	#358	—	53	1	no	—
207	#358	—	53	2	no	—
208	#358	—	53	2	no	—
209	#357	—	53	1	no	—
210	#346	—	53	2	no	—
211	C₄×C₁₆	—	53	1	no	—
212	C₅₃	#33965	1	1	yes	characteristic, complemented, Sylow
213	C₂×Q₁₆	—	2	1	no	—
214	C₂×Q₁₆	—	2	2	no	—
215	C₂×Q₁₆	—	53	2	no	—
216	C₂×Q₁₆	—	106	1	no	—
217	C₂×Q₁₆	C₂²×D₅₃	1	1	yes	characteristic
218	C₂×Q₁₆	C₂²×D₅₃	1	2	yes	—
219	C₂×SD₁₆	—	106	2	no	—
220	C₂×D₈	—	53	1	no	—
221	C₄⋊Q₈	—	2	1	no	—
222	C₄⋊Q₈	—	2	2	no	—
223	C₄⋊Q₈	—	106	1	no	—
224	#126	—	212	2	no	—
225	#125	—	4	1	no	—
226	#125	—	4	2	no	—
227	#124	—	106	2	no	—
228	C₂²⋊Q₈	—	212	1	no	—
229	C₄×Q₈	—	2	1	no	—
230	C₄×Q₈	—	2	2	no	—
231	C₄×Q₈	—	4	1	no	—
232	C₄×Q₈	—	106	2	no	—
233	C₄×Q₈	C₂²⋊D₅₃	1	1	yes	characteristic
234	C₄×D₄	—	106	1	no	—
235	#117	—	212	1	no	—
236	Q₃₂	—	106	4	no	—
237	Q₃₂	—	212	2	no	—
238	Q₃₂	—	212	2	no	—
239	SD₃₂	—	106	2	no	—
240	SD₃₂	—	212	2	no	—
241	C₂×C₁₆	—	53	1	no	—
242	C₂×C₁₆	—	53	1	no	—
243	C₂×C₁₆	—	53	2	no	—
244	#107	—	53	1	no	—
245	#107	C₂²×D₅₃	1	1	yes	characteristic
246	C₄⋊C₈	C₂²⋊D₅₃	1	1	yes	characteristic
247	C₄⋊C₈	C₂²⋊D₅₃	1	2	yes	—
248	#103	—	2	1	no	—
249	#103	—	2	2	no	—
250	#103	—	106	2	no	—
251	#102	—	106	1	no	—
252	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
253	C₄×C₈	C₂²×D₅₃	1	2	yes	—
254	C₂×Q₈	—	2	1	no	—
255	C₂×Q₈	—	2	2	no	—
256	C₂×Q₈	—	106	2	no	—
257	C₂×Q₈	#11719922	1	1	yes	characteristic
258	C₂×D₄	—	106	1	no	—
259	C₂²×C₄	—	212	1	no	—
260	Q₁₆	—	2	1	no	—
261	Q₁₆	—	2	2	no	—
262	Q₁₆	—	2	2	no	—
263	Q₁₆	—	2	1	no	—
264	Q₁₆	—	8	2	no	—
265	Q₁₆	—	8	1	no	—
266	Q₁₆	—	106	2	no	—
267	Q₁₆	—	106	2	no	—
268	Q₁₆	—	424	1	no	—
269	SD₁₆	—	424	2	no	—
270	D₈	—	106	1	no	—
271	D₈	—	106	1	no	—
272	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
273	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
274	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
275	C₄⋊C₄	—	2	1	no	—
276	C₄⋊C₄	—	4	2	no	—
277	C₄⋊C₄	—	4	1	no	—
278	C₄⋊C₄	—	4	2	no	—
279	C₄⋊C₄	—	4	1	no	—
280	C₄⋊C₄	—	4	2	no	—
281	C₄⋊C₄	—	4	1	no	—
282	C₄⋊C₄	—	106	1	no	—
283	C₄⋊C₄	—	212	1	no	—
284	C₄⋊C₄	—	212	2	no	—
285	C₄⋊C₄	—	212	1	no	—
286	C₄⋊C₄	#11719922	1	1	yes	characteristic
287	C₄⋊C₄	#11719922	1	2	yes	—
288	C₂²⋊C₄	—	212	2	no	—
289	C₂²⋊C₄	—	212	1	no	—
290	C₂²⋊C₄	—	212	2	no	—
291	C₄²	—	4	2	no	—
292	C₄²	—	4	1	no	—
293	C₄²	—	212	2	no	—
294	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
295	C₄²	C₂×C₂²⋊D₅₃	1	2	yes	—
296	C₁₆	—	53	2	no	—
297	C₁₆	—	106	2	no	—
298	C₁₆	—	106	1	no	—
299	C₂³	—	212	1	no	—
300	Q₈	—	2	2	no	—
301	Q₈	—	4	1	no	—
302	Q₈	—	4	2	no	—
303	Q₈	—	4	1	no	—
304	Q₈	—	4	2	no	—
305	Q₈	—	212	2	no	—
306	Q₈	—	212	2	no	—
307	D₄	—	212	1	no	—
308	D₄	—	212	1	no	—
309	C₂×C₄	—	4	1	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	4	1	no	—
312	C₂×C₄	—	424	2	no	—
313	C₂×C₄	—	212	1	no	—
314	C₂×C₄	—	212	2	no	—
315	C₂×C₄	C₂×#11719922	1	2	yes	—
316	C₂×C₄	C₂×#11719922	1	2	yes	—
317	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
318	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
319	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
320	C₈	—	2	2	no	—
321	C₈	—	2	1	no	—
322	C₈	#41×D₅₃	1	1	yes	characteristic
323	C₈	#13133263	1	1	yes	characteristic
324	C₂²	—	424	1	no	—
325	C₂²	—	424	1	no	—
326	C₂²	—	424	1	no	—
327	C₂²	C₂²×#11719922	1	1	yes	characteristic
328	C₄	—	2	2	no	—
329	C₄	—	2	2	no	—
330	C₄	—	2	1	no	—
331	C₄	—	8	1	no	—
332	C₄	—	8	1	no	—
333	C₄	—	8	2	no	—
334	C₄	—	424	1	no	—
335	C₄	—	424	2	no	—
336	C₄	#16811064	1	2	yes	—
337	C₄	#16811292	1	1	yes	characteristic
338	C₄	#16811426	1	1	yes	characteristic
339	C₂	—	848	1	no	—
340	C₂	#20287687	1	1	yes	characteristic
341	C₂	C₂×#16811064	1	1	yes	characteristic
342	C₂	#20287866	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#11719922
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#3395	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂×D₄₂₄	C₂³
hypercenter	#3395	D₅₃
hypocenter	C₅₃	#33965
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ #3395

Complements

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

N	H ≅ G/N	type
3. #20276967	C₂	semidirect
5. #20276765	C₂	semidirect
6. #20276762	C₂	semidirect
8. #20276624	C₂	semidirect
9. #20276584	C₂	semidirect
11. C₅₃×#3395	C₂	semidirect
168. #3395	D₅₃	semidirect
212. C₅₃	#33965	semidirect

Maximal subgroups

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

subgroup	index
#20287704	2
#20276984	2
#20276967	2
#20276777	2
#20276765	2
#20276762	2
#20276757	2
#20276624	2
#20276584	2
C₄×#13133160	2
C₅₃×#3395	2
#33965	53