Subgroups — #27538751

Order 13568.

534 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₂₁₂	C₂	1	1	yes	maximal, characteristic
2	#20276876	C₂	1	4	yes	maximal, complemented
3	#20276864	C₂	1	2	yes	maximal, complemented
4	#20276854	C₂	1	2	yes	maximal, complemented
5	#20276504	C₂	1	2	yes	maximal
6	#20276471	C₂	1	1	yes	maximal, characteristic
7	#20276434	C₂	1	2	yes	maximal
8	C₅₃×#3160	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₂₁₂	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₁₀₆	—	2	1	no	—
11	C₂×#13133157	C₂²	1	2	yes	—
12	C₄×Dic₂₁₂	—	2	1	no	—
13	C₄×Dic₂₁₂	—	2	1	no	—
14	C₄×Dic₂₁₂	C₂²	1	2	yes	—
15	C₄×Dic₂₁₂	C₂²	1	2	yes	—
16	C₂×#13133151	C₂²	1	1	yes	characteristic
17	C₂×#13133142	—	2	1	no	—
18	#16810450	C₂²	1	4	yes	—
19	#16810448	C₂²	1	4	yes	—
20	C₄⋊#11719908	C₂²	1	2	yes	—
21	C₄⋊#11719908	C₂²	1	2	yes	—
22	#16810426	C₂²	1	4	yes	—
23	#16810426	C₂²	1	4	yes	—
24	C₅₃×#533	—	2	1	no	—
25	C₅₃×#500	C₂²	1	1	yes	characteristic
26	C₅₃×#500	C₂²	1	1	yes	characteristic
27	C₅₃×#499	—	2	1	no	—
28	C₅₃×#499	C₂²	1	2	yes	—
29	C₅₃×#498	—	2	1	no	—
30	C₅₃×#427	—	2	1	no	—
31	C₅₃×#427	C₂²	1	2	yes	—
32	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
33	C₂²×Dic₁₀₆	—	2	1	no	—
34	C₂×Dic₂₁₂	—	2	1	no	—
35	C₂×Dic₂₁₂	—	2	2	no	—
36	C₂×Dic₂₁₂	Q₈	1	4	yes	complemented
37	C₂×Dic₂₁₂	C₂³	1	2	yes	—
38	C₄×Dic₁₀₆	—	2	2	no	—
39	C₄×Dic₁₀₆	—	2	2	no	—
40	C₄×Dic₁₀₆	—	4	1	no	—
41	C₄×Dic₁₀₆	—	4	1	no	—
42	C₂×C₄⋊Dic₅₃	—	2	1	no	—
43	C₂×#11719919	—	4	1	no	—
44	C₂×C₄×Dic₅₃	—	4	1	no	—
45	#13133157	—	2	4	no	—
46	C₂×#11719908	C₂³	1	4	yes	—
47	#13133151	—	2	1	no	—
48	#13133151	C₂³	1	2	yes	—
49	#13133142	—	2	2	no	—
50	#13133142	—	2	2	no	—
51	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
52	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
53	C₄⋊Q₈×C₅₃	—	2	1	no	—
54	C₄⋊Q₈×C₅₃	—	2	1	no	—
55	#125×C₅₃	—	2	2	no	—
56	C₂²⋊Q₈×C₅₃	—	4	1	no	—
57	C₂²⋊Q₈×C₅₃	—	4	1	no	—
58	C₄×Q₈×C₅₃	—	4	1	no	—
59	C₄×Q₈×C₅₃	—	4	1	no	—
60	#117×C₅₃	—	2	2	no	—
61	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
62	#107×C₅₃	—	2	1	no	—
63	#107×C₅₃	—	2	1	no	—
64	#107×C₅₃	C₂³	1	2	yes	—
65	#107×C₅₃	C₂³	1	2	yes	—
66	#106×C₅₃	—	2	2	no	—
67	#106×C₅₃	—	2	2	no	—
68	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
69	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
70	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
71	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
72	C₂×Q₈×C₅₃	—	4	1	no	—
73	C₂×Q₈×C₅₃	—	4	1	no	—
74	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
75	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
76	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
77	C₂²×Dic₅₃	—	4	1	no	—
78	C₂²×Dic₅₃	—	4	1	no	—
79	C₂×Dic₁₀₆	—	2	4	no	—
80	C₂×Dic₁₀₆	—	2	2	no	—
81	C₂×Dic₁₀₆	—	4	1	no	—
82	C₂×Dic₁₀₆	—	4	2	no	—
83	C₂×C₈×C₅₃	—	2	2	no	—
84	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
88	C₂×C₈×C₅₃	#41	1	2	yes	—
89	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
90	C₄⋊C₄×C₅₃	—	2	2	no	—
91	C₄⋊C₄×C₅₃	—	2	2	no	—
92	C₄⋊C₄×C₅₃	—	4	2	no	—
93	C₄⋊C₄×C₅₃	—	4	2	no	—
94	C₄⋊C₄×C₅₃	—	4	1	no	—
95	C₄⋊C₄×C₅₃	—	4	1	no	—
96	C₂²⋊C₄×C₅₃	—	4	2	no	—
97	C₄²×C₅₃	—	4	2	no	—
98	C₄²×C₅₃	D₈	1	1	yes	characteristic
99	C₄²×C₅₃	D₈	1	1	yes	characteristic
100	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
101	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
102	C₄⋊Dic₅₃	—	2	2	no	—
103	C₄⋊Dic₅₃	—	4	1	no	—
104	#11719919	—	4	2	no	—
105	#11719919	—	4	2	no	—
106	C₄×Dic₅₃	—	4	2	no	—
107	C₄×Dic₅₃	—	4	2	no	—
108	Dic₂₁₂	—	2	4	no	—
109	Dic₂₁₂	—	2	2	no	—
110	Dic₂₁₂	C₂×Q₈	1	4	yes	—
111	#11719908	—	2	4	no	—
112	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
113	Q₈×C₅₃	—	8	2	no	—
114	Q₈×C₅₃	—	8	2	no	—
115	C₂×C₄×C₅₃	—	2	2	no	—
116	C₂×C₄×C₅₃	—	4	1	no	—
117	C₂×C₄×C₅₃	—	4	2	no	—
118	C₂×C₄×C₅₃	—	4	2	no	—
119	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
120	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
121	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₂×D₈	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	#137	1	2	yes	—
126	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
127	C₂×Dic₅₃	—	4	2	no	—
128	C₂×Dic₅₃	—	4	4	no	—
129	C₂×Dic₅₃	—	4	2	no	—
130	C₂×Dic₅₃	—	8	2	no	—
131	Dic₁₀₆	—	2	4	no	—
132	Dic₁₀₆	—	4	2	no	—
133	Dic₁₀₆	—	4	4	no	—
134	C₈×C₅₃	—	2	1	no	—
135	C₈×C₅₃	—	2	1	no	—
136	C₈×C₅₃	C₂²⋊Q₈	1	2	yes	—
137	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
138	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
139	#33628	—	53	1	no	maximal, Sylow
140	C₂²×C₅₃	—	2	1	no	—
141	C₂²×C₅₃	—	2	1	no	—
142	C₂²×C₅₃	#474	1	2	yes	—
143	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
144	C₄×C₅₃	—	2	1	no	—
145	C₄×C₅₃	—	2	1	no	—
146	C₄×C₅₃	—	2	1	no	—
147	C₄×C₅₃	—	8	2	no	—
148	C₄×C₅₃	—	8	2	no	—
149	C₄×C₅₃	#474	1	2	yes	—
150	C₄×C₅₃	#510	1	2	yes	—
151	C₄×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
152	C₄×C₅₃	C₂×#137	1	1	yes	characteristic
153	Dic₅₃	—	4	4	no	—
154	Dic₅₃	—	8	2	no	—
155	#3160	D₅₃	1	1	yes	characteristic
156	C₂×C₄×Q₁₆	—	53	1	no	—
157	#2229	—	53	4	no	—
158	#2227	—	53	2	no	—
159	#2225	—	53	2	no	—
160	#2159	—	53	2	no	—
161	#2150	—	53	1	no	—
162	#2141	—	53	2	no	—
163	C₂×C₅₃	—	2	2	no	—
164	C₂×C₅₃	C₂×#474	1	1	yes	characteristic
165	C₂×C₅₃	C₂×#510	1	1	yes	characteristic
166	C₂×C₅₃	#3414	1	1	yes	characteristic
167	C₂²×Q₁₆	—	53	1	no	—
168	#533	—	2	1	no	—
169	C₂×C₄×Q₈	—	106	1	no	—
170	C₂×M₃₂	—	53	2	no	—
171	#500	C₂×D₅₃	1	1	yes	characteristic
172	#500	C₂×D₅₃	1	1	yes	characteristic
173	#499	—	2	1	no	—
174	#499	C₂×D₅₃	1	2	yes	—
175	#498	—	2	1	no	—
176	C₄×Q₁₆	—	53	2	no	—
177	C₄×Q₁₆	—	53	2	no	—
178	C₄×Q₁₆	—	106	1	no	—
179	C₄×Q₁₆	—	106	1	no	—
180	#427	—	2	1	no	—
181	#427	C₂×D₅₃	1	2	yes	—
182	C₂×#107	—	53	1	no	—
183	C₂×#103	—	106	1	no	—
184	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
185	#367	—	53	4	no	—
186	#366	—	53	4	no	—
187	C₄⋊C₁₆	—	53	2	no	—
188	C₄⋊C₁₆	—	53	2	no	—
189	#358	—	53	4	no	—
190	#358	—	53	4	no	—
191	C₅₃	#33628	1	1	yes	characteristic, complemented, Sylow
192	C₂²×Q₈	—	106	1	no	—
193	C₂×Q₁₆	—	53	4	no	—
194	C₂×Q₁₆	—	53	2	no	—
195	C₂×Q₁₆	—	106	2	no	—
196	C₂×Q₁₆	—	106	1	no	—
197	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
198	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
199	C₄⋊Q₈	—	2	1	no	—
200	C₄⋊Q₈	—	2	1	no	—
201	#125	—	2	2	no	—
202	C₂²⋊Q₈	—	4	1	no	—
203	C₂²⋊Q₈	—	4	1	no	—
204	C₄×Q₈	—	4	1	no	—
205	C₄×Q₈	—	4	1	no	—
206	C₄×Q₈	—	106	2	no	—
207	C₄×Q₈	—	106	2	no	—
208	C₄×Q₈	—	212	1	no	—
209	C₄×Q₈	—	212	1	no	—
210	#117	—	2	2	no	—
211	C₂×C₄⋊C₄	—	106	1	no	—
212	C₂×C₄⋊C₄	—	212	1	no	—
213	C₂×C₄²	—	212	1	no	—
214	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
215	M₃₂	—	106	4	no	—
216	C₂×C₁₆	—	53	4	no	—
217	#107	—	2	1	no	—
218	#107	—	2	1	no	—
219	#107	—	53	2	no	—
220	#107	—	106	1	no	—
221	#107	C₂²×D₅₃	1	2	yes	—
222	#107	C₂²×D₅₃	1	2	yes	—
223	#106	—	2	2	no	—
224	#106	—	2	2	no	—
225	#103	—	106	2	no	—
226	#103	—	106	2	no	—
227	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
228	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
229	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
230	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
231	C₂×Q₈	—	4	1	no	—
232	C₂×Q₈	—	4	1	no	—
233	C₂×Q₈	—	106	4	no	—
234	C₂×Q₈	—	106	2	no	—
235	C₂×Q₈	—	212	1	no	—
236	C₂×Q₈	—	212	2	no	—
237	C₂²×C₄	—	212	1	no	—
238	C₂²×C₄	—	212	1	no	—
239	C₂²×C₄	#11719922	1	1	yes	characteristic
240	C₂²×C₄	#11719922	1	1	yes	characteristic
241	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
242	Q₁₆	—	53	4	no	—
243	Q₁₆	—	106	4	no	—
244	Q₁₆	—	106	2	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	Q₈×D₅₃	1	2	yes	—
247	C₂×C₈	#11719949	1	2	yes	—
248	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
249	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
250	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
251	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	4	2	no	—
255	C₄⋊C₄	—	4	2	no	—
256	C₄⋊C₄	—	4	1	no	—
257	C₄⋊C₄	—	4	1	no	—
258	C₄⋊C₄	—	106	2	no	—
259	C₄⋊C₄	—	212	1	no	—
260	C₄⋊C₄	—	212	2	no	—
261	C₄⋊C₄	—	212	2	no	—
262	C₂²⋊C₄	—	4	2	no	—
263	C₄²	—	4	2	no	—
264	C₄²	—	212	2	no	—
265	C₄²	—	212	2	no	—
266	C₄²	#11719922	1	1	yes	characteristic
267	C₄²	#11719922	1	1	yes	characteristic
268	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
269	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
270	C₁₆	—	106	4	no	—
271	C₂³	C₂×#11719922	1	1	yes	characteristic
272	Q₈	—	8	2	no	—
273	Q₈	—	8	2	no	—
274	Q₈	—	106	4	no	—
275	Q₈	—	212	4	no	—
276	Q₈	—	212	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	4	1	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	424	2	no	—
282	C₂×C₄	—	212	2	no	—
283	C₂×C₄	—	212	2	no	—
284	C₂×C₄	—	212	4	no	—
285	C₂×C₄	#13133193	1	2	yes	—
286	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
287	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
288	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
289	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
290	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
291	C₂×C₄	#13133248	1	2	yes	—
292	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
293	C₈	—	2	1	no	—
294	C₈	—	2	1	no	—
295	C₈	#13133193	1	2	yes	—
296	C₈	C₂×Q₈×D₅₃	1	1	yes	characteristic
297	C₈	C₂×#11719949	1	1	yes	characteristic
298	C₂²	—	2	1	no	—
299	C₂²	—	2	1	no	—
300	C₂²	#16810900	1	2	yes	—
301	C₂²	C₂²×#11719922	1	1	yes	characteristic
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	2	1	no	—
305	C₄	—	8	2	no	—
306	C₄	—	8	2	no	—
307	C₄	—	424	2	no	—
308	C₄	—	212	4	no	—
309	C₄	#16810900	1	2	yes	—
310	C₄	#16811063	1	2	yes	—
311	C₄	C₂×#13133193	1	1	yes	characteristic
312	C₄	C₂×#13133248	1	1	yes	characteristic
313	C₂	—	2	2	no	—
314	C₂	C₂×#16810900	1	1	yes	characteristic
315	C₂	C₂×#16811063	1	1	yes	characteristic
316	C₂	#20287849	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#11719922
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#3160	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂³×C₅₃	C₂×D₈
hypercenter	#3160	D₅₃
hypocenter	C₅₃	#33628
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3160

Complements

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

N	H ≅ G/N	type
2. #20276876	C₂	semidirect
3. #20276864	C₂	semidirect
4. #20276854	C₂	semidirect
36. C₂×Dic₂₁₂	Q₈	semidirect
191. C₅₃	#33628	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₂₁₂	2
#20276876	2
#20276864	2
#20276854	2
#20276504	2
#20276471	2
#20276434	2
C₅₃×#3160	2
#33628	53