Subgroups — #27284128

Order 13568.

518 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274832	C₂	1	1	yes	maximal, characteristic
2	C₄×#13133128	C₂	1	1	yes	maximal, characteristic
3	#20272556	C₂	1	4	yes	maximal
4	C₄×C₄⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#88322	C₂²	1	2	yes	—
6	C₂×#13133128	—	2	1	no	—
7	C₂×#13133128	—	2	1	no	—
8	C₂×C₄×#88322	C₂²	1	2	yes	—
9	#16810378	—	2	1	no	—
10	#16810378	—	2	1	no	—
11	C₂×C₄⋊Q₈×C₅₃	C₄	1	1	yes	characteristic
12	C₂×C₄×Q₈×C₅₃	—	2	1	no	—
13	C₅₃×#391	—	2	1	no	—
14	C₅₃×#389	—	2	1	no	—
15	C₅₃×#389	C₄	1	2	yes	—
16	C₅₃×#384	C₄	1	1	yes	characteristic
17	C₄×C₄⋊C₄×C₅₃	—	2	1	no	—
18	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
19	C₄³×C₅₃	C₂²	1	1	yes	characteristic
20	C₂²×#88322	—	2	2	no	—
21	C₂²×#88322	—	2	2	no	—
22	C₄⋊#88322	—	2	4	no	—
23	#13133128	—	2	4	no	—
24	#13133128	—	2	4	no	—
25	C₄×#88322	—	2	2	no	—
26	C₄×#88322	Q₈	1	4	yes	complemented
27	C₂²×Q₈×C₅₃	—	2	1	no	—
28	C₄⋊Q₈×C₅₃	—	2	4	no	—
29	C₄×Q₈×C₅₃	—	4	4	no	—
30	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
31	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
32	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
33	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
34	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
35	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₅₃	—	2	2	no	—
37	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
44	#95×C₅₃	—	2	2	no	—
45	#95×C₅₃	—	2	2	no	—
46	C₂×Q₈×C₅₃	—	2	4	no	—
47	C₂×Q₈×C₅₃	—	4	2	no	—
48	C₂×Q₈×C₅₃	—	4	2	no	—
49	C₂²×C₄×C₅₃	—	2	2	no	—
50	C₂²×C₄×C₅₃	—	2	2	no	—
51	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₄⋊C₄×C₅₃	—	2	4	no	—
59	C₄⋊C₄×C₅₃	—	2	2	no	—
60	C₄⋊C₄×C₅₃	—	2	4	no	—
61	C₄⋊C₄×C₅₃	—	4	2	no	—
62	C₄⋊C₄×C₅₃	—	4	4	no	—
63	C₄⋊C₄×C₅₃	—	4	2	no	—
64	C₄⋊C₄×C₅₃	M₁₆	1	4	yes	—
65	C₄²×C₅₃	—	2	2	no	—
66	C₄²×C₅₃	—	2	2	no	—
67	C₄²×C₅₃	—	2	2	no	—
68	C₄²×C₅₃	—	2	2	no	—
69	C₄²×C₅₃	—	2	2	no	—
70	C₄²×C₅₃	—	2	2	no	—
71	C₄²×C₅₃	—	4	4	no	—
72	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
73	C₄²×C₅₃	#41	1	2	yes	—
74	C₂×#88322	—	2	4	no	—
75	C₂×#88322	—	2	4	no	—
76	C₂×#88322	—	2	4	no	—
77	C₂×#88322	—	4	2	no	—
78	C₂×#88322	—	4	2	no	—
79	C₂×#88322	—	4	2	no	—
80	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
81	Q₈×C₅₃	—	4	8	no	—
82	C₂×C₄×C₅₃	—	2	1	no	—
83	C₂×C₄×C₅₃	—	2	1	no	—
84	C₂×C₄×C₅₃	—	2	1	no	—
85	C₂×C₄×C₅₃	—	2	1	no	—
86	C₂×C₄×C₅₃	—	2	1	no	—
87	C₂×C₄×C₅₃	—	2	1	no	—
88	C₂×C₄×C₅₃	—	2	4	no	—
89	C₂×C₄×C₅₃	—	2	1	no	—
90	C₂×C₄×C₅₃	—	2	1	no	—
91	C₂×C₄×C₅₃	—	2	1	no	—
92	C₂×C₄×C₅₃	—	2	1	no	—
93	C₂×C₄×C₅₃	—	2	1	no	—
94	C₂×C₄×C₅₃	—	2	1	no	—
95	C₂×C₄×C₅₃	—	4	2	no	—
96	C₂×C₄×C₅₃	—	4	2	no	—
97	C₂×C₄×C₅₃	—	4	2	no	—
98	C₂×C₄×C₅₃	—	4	2	no	—
99	C₂×C₄×C₅₃	—	4	2	no	—
100	C₂×C₄×C₅₃	#104	1	2	yes	—
101	C₂×C₄×C₅₃	#104	1	2	yes	—
102	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
103	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
104	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
106	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
107	C₂×C₄×C₅₃	#131	1	2	yes	—
108	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
109	C₂×C₄×C₅₃	#137	1	2	yes	—
110	#88322	—	2	8	no	—
111	#88322	—	8	2	no	—
112	#9877	—	53	1	no	maximal, Sylow
113	C₂²×C₅₃	#391	1	1	yes	characteristic
114	C₂²×C₅₃	#407	1	1	yes	characteristic
115	C₂²×C₅₃	#408	1	1	yes	characteristic
116	C₂²×C₅₃	#418	1	1	yes	characteristic
117	C₂²×C₅₃	#419	1	1	yes	characteristic
118	C₂²×C₅₃	C₂×#104	1	1	yes	characteristic
119	C₂²×C₅₃	C₂×#104	1	1	yes	characteristic
120	C₄×C₅₃	—	2	2	no	—
121	C₄×C₅₃	—	2	2	no	—
122	C₄×C₅₃	—	2	2	no	—
123	C₄×C₅₃	—	4	1	no	—
124	C₄×C₅₃	—	4	1	no	—
125	C₄×C₅₃	—	4	4	no	—
126	C₄×C₅₃	—	8	2	no	—
127	C₄×C₅₃	#446	1	4	yes	—
128	C₄×C₅₃	#489	1	2	yes	—
129	C₄×C₅₃	#501	1	2	yes	—
130	C₄×C₄⋊Q₈	D₅₃	1	1	yes	characteristic
131	#1831	—	53	1	no	—
132	C₄×#97	—	53	1	no	—
133	#1279	—	53	4	no	—
134	C₂×C₅₃	#1484	1	1	yes	characteristic
135	C₂×C₅₃	#1485	1	1	yes	characteristic
136	C₂×C₅₃	#1490	1	1	yes	characteristic
137	C₂×C₅₃	#1491	1	1	yes	characteristic
138	C₂×C₅₃	#1985	1	1	yes	characteristic
139	C₂×C₅₃	#1993	1	1	yes	characteristic
140	C₂×C₅₃	#1994	1	1	yes	characteristic
141	C₂×C₄⋊Q₈	Dic₅₃	1	1	yes	characteristic
142	C₂×C₄×Q₈	—	2	1	no	—
143	C₂×C₄⋊C₈	—	53	2	no	—
144	C₂×#97	—	106	1	no	—
145	C₂×#97	—	106	1	no	—
146	C₂×C₄×C₈	—	53	2	no	—
147	#391	—	2	1	no	—
148	#389	—	2	1	no	—
149	#389	Dic₅₃	1	2	yes	—
150	#384	Dic₅₃	1	1	yes	characteristic
151	C₄×C₄⋊C₄	—	2	1	no	—
152	C₄×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
153	C₄³	C₂×D₅₃	1	1	yes	characteristic
154	#336	—	106	1	no	—
155	#336	—	106	1	no	—
156	C₅₃	#9877	1	1	yes	characteristic, complemented, Sylow
157	C₂²×Q₈	—	2	1	no	—
158	C₂²×C₈	—	106	2	no	—
159	C₂²×C₈	—	106	2	no	—
160	C₄⋊Q₈	—	2	4	no	—
161	C₄×Q₈	—	4	4	no	—
162	C₂×C₄⋊C₄	—	2	1	no	—
163	C₂×C₄⋊C₄	—	2	1	no	—
164	C₂×C₄⋊C₄	—	2	2	no	—
165	C₂×C₄⋊C₄	—	2	1	no	—
166	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
167	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
168	C₂×C₄²	—	2	2	no	—
169	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
170	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
171	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
172	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
173	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
174	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
175	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
176	C₄⋊C₈	—	106	4	no	—
177	#97	—	106	4	no	—
178	#97	—	106	4	no	—
179	C₄×C₈	—	53	4	no	—
180	C₄×C₈	—	106	2	no	—
181	#95	—	2	2	no	—
182	#95	—	2	2	no	—
183	C₂×Q₈	—	2	4	no	—
184	C₂×Q₈	—	4	2	no	—
185	C₂×Q₈	—	4	2	no	—
186	C₂²×C₄	—	2	2	no	—
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
189	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
190	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
191	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
192	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
193	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
194	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
195	C₂×C₈	—	106	4	no	—
196	C₂×C₈	—	106	4	no	—
197	C₂×C₈	—	106	4	no	—
198	C₂×C₈	—	212	2	no	—
199	C₂×C₈	—	212	2	no	—
200	C₂×C₈	—	212	2	no	—
201	C₄⋊C₄	—	2	4	no	—
202	C₄⋊C₄	—	2	2	no	—
203	C₄⋊C₄	—	2	4	no	—
204	C₄⋊C₄	—	4	2	no	—
205	C₄⋊C₄	—	4	4	no	—
206	C₄⋊C₄	—	4	2	no	—
207	C₄⋊C₄	#11719917	1	4	yes	—
208	C₄²	—	2	2	no	—
209	C₄²	—	2	2	no	—
210	C₄²	—	2	2	no	—
211	C₄²	—	2	2	no	—
212	C₄²	—	2	2	no	—
213	C₄²	—	2	2	no	—
214	C₄²	—	4	4	no	—
215	C₄²	Q₈×D₅₃	1	2	yes	—
216	C₄²	#11719949	1	2	yes	—
217	C₂³	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
218	Q₈	—	4	8	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	#13133144	1	2	yes	—
238	C₂×C₄	#13133144	1	2	yes	—
239	C₂×C₄	Q₈×Dic₅₃	1	2	yes	—
240	C₂×C₄	#13133193	1	2	yes	—
241	C₂×C₄	#13133206	1	1	yes	characteristic
242	C₂×C₄	#13133215	1	1	yes	characteristic
243	C₂×C₄	C₂×#11719917	1	2	yes	—
244	C₂×C₄	#13133224	1	2	yes	—
245	C₂×C₄	#13133243	1	2	yes	—
246	C₂×C₄	#13133248	1	2	yes	—
247	C₈	—	106	8	no	—
248	C₈	—	424	2	no	—
249	C₂²	#16810530	1	1	yes	characteristic
250	C₂²	#16810587	1	1	yes	characteristic
251	C₂²	#16810590	1	1	yes	characteristic
252	C₂²	#16810628	1	1	yes	characteristic
253	C₂²	#16810632	1	1	yes	characteristic
254	C₂²	C₂×#13133144	1	1	yes	characteristic
255	C₂²	C₂×#13133144	1	1	yes	characteristic
256	C₄	—	2	2	no	—
257	C₄	—	2	2	no	—
258	C₄	—	2	2	no	—
259	C₄	—	4	4	no	—
260	C₄	—	4	1	no	—
261	C₄	—	4	1	no	—
262	C₄	—	8	2	no	—
263	C₄	#16810754	1	4	yes	—
264	C₄	#16810985	1	2	yes	—
265	C₄	#16811029	1	2	yes	—
266	C₂	#20273132	1	1	yes	characteristic
267	C₂	#20273136	1	1	yes	characteristic
268	C₂	#20273158	1	1	yes	characteristic
269	C₂	#20273162	1	1	yes	characteristic
270	C₂	#20275656	1	1	yes	characteristic
271	C₂	#20275689	1	1	yes	characteristic
272	C₂	#20275693	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₅₃
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₄×C₄⋊Q₈×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₄×C₄⋊Q₈	D₅₃
hypocenter	C₅₃	#9877
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
26. C₄×#88322	Q₈	semidirect
156. C₅₃	#9877	semidirect

Maximal subgroups

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

subgroup	index
#20274832	2
C₄×#13133128	2
#20272556	2
C₄×C₄⋊Q₈×C₅₃	2
#9877	53