Subgroups — #27402938

Order 13568.

750 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20277672	C₂	1	2	yes	maximal
2	#20275485	C₂	1	4	yes	maximal
3	#20275479	C₂	1	2	yes	maximal
4	#20274879	C₂	1	2	yes	maximal
5	#20274875	C₂	1	1	yes	maximal, characteristic
6	C₄×#13133151	C₂	1	2	yes	maximal
7	C₄×#13133151	C₂	1	1	yes	maximal, characteristic
8	C₂×C₈×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
9	C₂×#13133151	C₂²	1	2	yes	—
10	C₂×#13133151	C₂²	1	1	yes	characteristic
11	C₂×#13133151	C₂²	1	2	yes	—
12	C₂×#13133151	C₂²	1	1	yes	characteristic
13	C₂×#13133149	C₂²	1	4	yes	—
14	#16810530	—	2	2	no	—
15	#16810509	—	2	4	no	—
16	#16810493	C₂²	1	2	yes	—
17	#16810489	C₂²	1	4	yes	—
18	C₄⋊C₄×Dic₅₃	—	2	2	no	—
19	C₄×C₄⋊Dic₅₃	C₂²	1	2	yes	—
20	C₄×C₄⋊Dic₅₃	C₂²	1	4	yes	—
21	#16810386	C₂²	1	4	yes	—
22	#16810386	C₂²	1	2	yes	—
23	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₈×Q₈×C₅₃	C₄	1	8	yes	complemented
25	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
29	C₂×C₄⋊Dic₅₃	—	2	2	no	—
30	C₂×C₄⋊Dic₅₃	—	2	2	no	—
31	C₂×C₄⋊Dic₅₃	C₂³	1	2	yes	—
32	C₂×C₄⋊Dic₅₃	C₂³	1	4	yes	—
33	C₂×C₄⋊Dic₅₃	C₂³	1	2	yes	—
34	C₂×C₄×Dic₅₃	—	2	2	no	—
35	C₂×C₄×Dic₅₃	—	2	4	no	—
36	#13133151	—	2	4	no	—
37	#13133151	—	2	2	no	—
38	#13133151	—	4	2	no	—
39	#13133151	Q₈	1	4	yes	complemented
40	#13133149	—	4	4	no	—
41	#13133124	—	2	2	no	—
42	#13133124	—	2	4	no	—
43	#13133124	—	2	4	no	—
44	#13133124	—	2	4	no	—
45	C₂²×Q₈×C₅₃	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
47	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
48	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
49	C₄×Q₈×C₅₃	C₂×C₄	1	4	yes	—
50	C₄×Q₈×C₅₃	Q₈	1	4	yes	—
51	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
52	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
53	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
54	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
55	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
56	C₄⋊C₈×C₅₃	C₂×C₄	1	8	yes	—
57	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
58	C₄×C₈×C₅₃	C₂×C₄	1	8	yes	—
59	C₂×Q₈×C₅₃	—	2	2	no	—
60	C₂×Q₈×C₅₃	—	2	2	no	—
61	C₂×Q₈×C₅₃	C₄⋊C₄	1	4	yes	—
62	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
67	C₂²×Dic₅₃	—	2	2	no	—
68	C₂²×Dic₅₃	—	2	2	no	—
69	C₂²×Dic₅₃	—	2	4	no	—
70	C₂×C₈×C₅₃	—	2	2	no	—
71	C₂×C₈×C₅₃	—	2	1	no	—
72	C₂×C₈×C₅₃	—	2	2	no	—
73	C₂×C₈×C₅₃	—	2	1	no	—
74	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
75	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
77	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
78	C₂×C₈×C₅₃	#41	1	2	yes	—
79	C₄⋊C₄×C₅₃	C₄⋊C₄	1	8	yes	—
80	C₄⋊C₄×C₅₃	C₄⋊C₄	1	4	yes	—
81	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
82	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
83	C₄²×C₅₃	C₂×Q₈	1	4	yes	—
84	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
85	C₄⋊Dic₅₃	—	2	2	no	—
86	C₄⋊Dic₅₃	—	2	2	no	—
87	C₄⋊Dic₅₃	—	2	4	no	—
88	C₄⋊Dic₅₃	—	2	2	no	—
89	C₄⋊Dic₅₃	—	2	4	no	—
90	C₄⋊Dic₅₃	—	4	4	no	—
91	C₄⋊Dic₅₃	—	8	2	no	—
92	C₄⋊Dic₅₃	C₂×Q₈	1	4	yes	—
93	C₄×Dic₅₃	—	4	4	no	—
94	C₄×Dic₅₃	—	8	4	no	—
95	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
96	Q₈×C₅₃	—	2	8	no	—
97	C₂×C₄×C₅₃	—	2	2	no	—
98	C₂×C₄×C₅₃	—	2	2	no	—
99	C₂×C₄×C₅₃	—	2	2	no	—
100	C₂×C₄×C₅₃	—	2	1	no	—
101	C₂×C₄×C₅₃	—	2	2	no	—
102	C₂×C₄×C₅₃	—	2	1	no	—
103	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
104	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
105	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
106	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
107	C₂×C₄×C₅₃	#135	1	2	yes	—
108	C₂×C₄×C₅₃	#135	1	2	yes	—
109	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
110	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
111	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
112	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
113	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
114	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
115	C₂×Dic₅₃	—	2	4	no	—
116	C₂×Dic₅₃	—	4	2	no	—
117	C₂×Dic₅₃	—	4	2	no	—
118	C₂×Dic₅₃	—	4	2	no	—
119	C₂×Dic₅₃	—	4	4	no	—
120	C₂×Dic₅₃	—	4	2	no	—
121	C₂×Dic₅₃	—	4	4	no	—
122	C₂×Dic₅₃	—	4	4	no	—
123	C₂×Dic₅₃	—	4	2	no	—
124	C₈×C₅₃	—	2	4	no	—
125	C₈×C₅₃	—	2	2	no	—
126	C₈×C₅₃	C₄×Q₈	1	4	yes	—
127	#22727	—	53	1	no	maximal, Sylow
128	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
129	C₂²×C₅₃	D₄×Q₈	1	1	yes	characteristic
130	C₂²×C₅₃	#550	1	1	yes	characteristic
131	C₂²×C₅₃	C₂×#135	1	1	yes	characteristic
132	C₂²×C₅₃	C₂×#135	1	1	yes	characteristic
133	C₂²×C₅₃	#576	1	1	yes	characteristic
134	C₂²×C₅₃	#578	1	1	yes	characteristic
135	C₄×C₅₃	—	2	4	no	—
136	C₄×C₅₃	—	2	4	no	—
137	C₄×C₅₃	—	2	2	no	—
138	C₄×C₅₃	#427	1	4	yes	—
139	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
140	C₄×C₅₃	#520	1	1	yes	characteristic
141	C₄×C₅₃	#557	1	1	yes	characteristic
142	C₄×C₅₃	Q₈²	1	1	yes	characteristic
143	Dic₅₃	—	4	4	no	—
144	Dic₅₃	—	8	2	no	—
145	Dic₅₃	—	8	4	no	—
146	C₂×C₈×Q₈	D₅₃	1	1	yes	characteristic
147	#2355	—	53	2	no	—
148	#1949	—	53	4	no	—
149	#1948	—	53	2	no	—
150	#1846	—	53	2	no	—
151	#1845	—	53	1	no	—
152	C₄×#107	—	53	2	no	—
153	C₄×#107	—	53	1	no	—
154	C₂×C₅₃	Q₈×C₄⋊C₄	1	1	yes	characteristic
155	C₂×C₅₃	C₂×#427	1	1	yes	characteristic
156	C₂×C₅₃	#2913	1	1	yes	characteristic
157	C₂×C₅₃	#3380	1	1	yes	characteristic
158	C₂×C₅₃	#3383	1	1	yes	characteristic
159	C₂×C₅₃	#3393	1	1	yes	characteristic
160	C₂×C₅₃	#3396	1	1	yes	characteristic
161	C₂×C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
162	C₈×Q₈	Dic₅₃	1	8	yes	complemented
163	C₂×#107	—	53	1	no	—
164	C₂×#107	—	53	2	no	—
165	C₂×#107	—	53	1	no	—
166	C₂×#107	—	53	2	no	—
167	C₂×#106	—	53	4	no	—
168	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
169	C₂×C₄⋊C₈	C₂×D₅₃	1	2	yes	—
170	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
171	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
172	#391	—	106	2	no	—
173	#387	—	106	4	no	—
174	#384	—	53	2	no	—
175	#382	—	53	4	no	—
176	C₄×C₄⋊C₄	—	53	2	no	—
177	C₄×C₄⋊C₄	—	53	4	no	—
178	C₄×C₄⋊C₄	—	106	2	no	—
179	#340	—	53	4	no	—
180	#340	—	53	2	no	—
181	C₅₃	#22727	1	1	yes	characteristic, complemented, Sylow
182	C₂²×Q₈	D₂₁₂	1	1	yes	characteristic
183	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
184	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
185	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
186	C₄×Q₈	Dic₁₀₆	1	4	yes	—
187	C₄×Q₈	C₂×Dic₅₃	1	4	yes	—
188	C₂×C₄⋊C₄	—	53	2	no	—
189	C₂×C₄⋊C₄	—	53	2	no	—
190	C₂×C₄⋊C₄	—	53	4	no	—
191	C₂×C₄⋊C₄	—	106	2	no	—
192	C₂×C₄⋊C₄	—	106	2	no	—
193	C₂×C₄⋊C₄	D₂₁₂	1	1	yes	characteristic
194	C₂×C₄⋊C₄	D₂₁₂	1	2	yes	—
195	C₂×C₄²	—	106	4	no	—
196	C₂×C₄²	—	106	2	no	—
197	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
198	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
199	#107	—	53	4	no	—
200	#107	—	106	4	no	—
201	#107	—	106	2	no	—
202	#107	—	212	2	no	—
203	#106	—	212	4	no	—
204	C₄⋊C₈	C₂×Dic₅₃	1	4	yes	—
205	C₄⋊C₈	C₂×Dic₅₃	1	8	yes	—
206	C₄×C₈	C₂×Dic₅₃	1	4	yes	—
207	C₄×C₈	C₂×Dic₅₃	1	8	yes	—
208	#95	—	106	2	no	—
209	#95	—	106	4	no	—
210	#95	—	106	4	no	—
211	#95	—	106	4	no	—
212	C₂×Q₈	—	2	2	no	—
213	C₂×Q₈	—	2	2	no	—
214	C₂×Q₈	C₄⋊Dic₅₃	1	4	yes	—
215	C₂²×C₄	—	106	2	no	—
216	C₂²×C₄	—	106	2	no	—
217	C₂²×C₄	—	106	4	no	—
218	C₂²×C₄	C₂×D₂₁₂	1	2	yes	—
219	C₂²×C₄	C₂×D₂₁₂	1	2	yes	—
220	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
221	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
222	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	Q₈×D₅₃	1	2	yes	—
228	C₂×C₈	#11719949	1	2	yes	—
229	C₂×C₈	C₂²×Dic₅₃	1	4	yes	—
230	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
231	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
232	C₄⋊C₄	—	53	4	no	—
233	C₄⋊C₄	—	106	2	no	—
234	C₄⋊C₄	—	106	2	no	—
235	C₄⋊C₄	—	106	4	no	—
236	C₄⋊C₄	—	106	2	no	—
237	C₄⋊C₄	—	106	4	no	—
238	C₄⋊C₄	—	424	2	no	—
239	C₄⋊C₄	—	212	4	no	—
240	C₄⋊C₄	C₄⋊Dic₅₃	1	8	yes	—
241	C₄⋊C₄	C₄⋊Dic₅₃	1	4	yes	—
242	C₄²	—	424	4	no	—
243	C₄²	—	212	4	no	—
244	C₄²	C₂×Dic₁₀₆	1	4	yes	—
245	C₄²	C₂×Dic₁₀₆	1	2	yes	—
246	C₄²	C₂²×Dic₅₃	1	4	yes	—
247	C₄²	C₂²×Dic₅₃	1	2	yes	—
248	C₂³	C₂²×D₂₁₂	1	1	yes	characteristic
249	Q₈	—	2	8	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	106	4	no	—
257	C₂×C₄	—	212	2	no	—
258	C₂×C₄	—	212	4	no	—
259	C₂×C₄	—	212	2	no	—
260	C₂×C₄	—	212	4	no	—
261	C₂×C₄	—	212	4	no	—
262	C₂×C₄	—	212	2	no	—
263	C₂×C₄	—	212	2	no	—
264	C₂×C₄	—	212	2	no	—
265	C₂×C₄	C₂×C₄⋊Dic₅₃	1	4	yes	—
266	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
267	C₂×C₄	C₂×C₄⋊Dic₅₃	1	4	yes	—
268	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
269	C₂×C₄	#13133239	1	2	yes	—
270	C₂×C₄	#13133239	1	2	yes	—
271	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
272	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
273	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
274	C₂×C₄	C₂×#11719949	1	1	yes	characteristic
275	C₂×C₄	#13133263	1	1	yes	characteristic
276	C₂×C₄	#13133264	1	1	yes	characteristic
277	C₈	—	2	4	no	—
278	C₈	—	2	2	no	—
279	C₈	Q₈×Dic₅₃	1	4	yes	—
280	C₂²	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
281	C₂²	Q₈×D₂₁₂	1	1	yes	characteristic
282	C₂²	#16811295	1	1	yes	characteristic
283	C₂²	C₂×#13133239	1	1	yes	characteristic
284	C₂²	C₂×#13133239	1	1	yes	characteristic
285	C₂²	#16811428	1	1	yes	characteristic
286	C₂²	#16811440	1	1	yes	characteristic
287	C₄	—	2	4	no	—
288	C₄	—	2	4	no	—
289	C₄	—	2	2	no	—
290	C₄	—	424	2	no	—
291	C₄	—	424	4	no	—
292	C₄	—	212	4	no	—
293	C₄	#16810661	1	4	yes	—
294	C₄	C₂×Q₈×Dic₅₃	1	1	yes	characteristic
295	C₄	#16811106	1	1	yes	characteristic
296	C₄	#16811352	1	1	yes	characteristic
297	C₄	Q₈×Dic₁₀₆	1	1	yes	characteristic
298	C₂	Q₈×C₄⋊Dic₅₃	1	1	yes	characteristic
299	C₂	C₂×#16810661	1	1	yes	characteristic
300	C₂	#20283315	1	1	yes	characteristic
301	C₂	#20287583	1	1	yes	characteristic
302	C₂	#20287607	1	1	yes	characteristic
303	C₂	#20287704	1	1	yes	characteristic
304	C₂	#20287728	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₈×Q₈×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂²×D₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂×C₈×Q₈	D₅₃
hypocenter	C₅₃	#22727
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
24. C₈×Q₈×C₅₃	C₄	semidirect
39. #13133151	Q₈	semidirect
162. C₈×Q₈	Dic₅₃	semidirect
181. C₅₃	#22727	semidirect

Maximal subgroups

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

subgroup	index
#20277672	2
#20275485	2
#20275479	2
#20274879	2
#20274875	2
C₄×#13133151	2
C₂×C₈×Q₈×C₅₃	2
#22727	53