Subgroups — #27284218

Order 13568.

538 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274867	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic
3	#20272598	C₂	1	4	yes	maximal
4	C₅₃×#2868	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#88322	—	2	1	no	—
6	C₂×C₄⋊#88322	—	2	1	no	—
7	C₂×C₄⋊#88322	C₂²	1	2	yes	—
8	C₂×C₄⋊#88322	C₂²	1	2	yes	—
9	C₂×C₄×#88322	—	2	1	no	—
10	#16810378	—	2	1	no	—
11	C₂×C₄⋊Q₈×C₅₃	—	2	1	no	—
12	C₂×C₄⋊Q₈×C₅₃	—	2	1	no	—
13	C₂×C₄⋊Q₈×C₅₃	C₄	1	2	yes	—
14	C₅₃×#384	—	2	1	no	—
15	C₅₃×#384	—	2	1	no	—
16	C₅₃×#384	C₄	1	2	yes	—
17	C₅₃×#384	C₂²	1	2	yes	—
18	C₄³×C₅₃	C₂²	1	1	yes	characteristic
19	C₂²×#88322	—	2	2	no	—
20	C₂²×#88322	—	2	2	no	—
21	C₄⋊#88322	—	2	2	no	—
22	C₄⋊#88322	—	2	4	no	—
23	C₄⋊#88322	—	4	2	no	—
24	C₄⋊#88322	—	4	2	no	—
25	C₄⋊#88322	Q₈	1	4	yes	complemented
26	C₄×#88322	—	4	2	no	—
27	C₂²×Q₈×C₅₃	—	2	2	no	—
28	C₄⋊Q₈×C₅₃	—	4	2	no	—
29	C₄⋊Q₈×C₅₃	—	4	2	no	—
30	C₄⋊Q₈×C₅₃	—	4	2	no	—
31	C₄⋊Q₈×C₅₃	—	4	2	no	—
32	C₄⋊Q₈×C₅₃	—	4	4	no	—
33	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
34	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
35	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
36	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
37	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
38	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
39	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₅₃	C₂×C₄	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	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
46	C₂×Q₈×C₅₃	—	4	4	no	—
47	C₂×Q₈×C₅₃	—	4	4	no	—
48	C₂×Q₈×C₅₃	—	4	4	no	—
49	C₂²×C₄×C₅₃	—	2	4	no	—
50	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
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	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
57	C₄⋊C₄×C₅₃	—	4	4	no	—
58	C₄⋊C₄×C₅₃	—	4	4	no	—
59	C₄⋊C₄×C₅₃	—	4	4	no	—
60	C₄⋊C₄×C₅₃	—	4	4	no	—
61	C₄⋊C₄×C₅₃	—	4	2	no	—
62	C₄⋊C₄×C₅₃	—	4	2	no	—
63	C₄⋊C₄×C₅₃	—	4	2	no	—
64	C₄⋊C₄×C₅₃	—	4	2	no	—
65	C₄²×C₅₃	—	2	2	no	—
66	C₄²×C₅₃	—	2	2	no	—
67	C₄²×C₅₃	—	2	2	no	—
68	C₄²×C₅₃	D₈	1	2	yes	—
69	C₄²×C₅₃	D₈	1	2	yes	—
70	C₄²×C₅₃	SD₁₆	1	2	yes	—
71	C₄²×C₅₃	SD₁₆	1	2	yes	—
72	C₄²×C₅₃	SD₁₆	1	2	yes	—
73	C₄²×C₅₃	Q₁₆	1	2	yes	—
74	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
75	C₄²×C₅₃	#41	1	2	yes	—
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	C₂×#88322	—	4	2	no	—
81	C₂×#88322	—	4	2	no	—
82	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	Q₈×C₅₃	—	8	8	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	1	no	—
89	C₂×C₄×C₅₃	—	2	1	no	—
90	C₂×C₄×C₅₃	—	4	4	no	—
91	C₂×C₄×C₅₃	—	4	4	no	—
92	C₂×C₄×C₅₃	—	4	4	no	—
93	C₂×C₄×C₅₃	#100	1	2	yes	—
94	C₂×C₄×C₅₃	#101	1	2	yes	—
95	C₂×C₄×C₅₃	#102	1	2	yes	—
96	C₂×C₄×C₅₃	#102	1	2	yes	—
97	C₂×C₄×C₅₃	#102	1	2	yes	—
98	C₂×C₄×C₅₃	#102	1	2	yes	—
99	C₂×C₄×C₅₃	#103	1	2	yes	—
100	C₂×C₄×C₅₃	#103	1	2	yes	—
101	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
102	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
103	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
104	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
106	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
107	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
108	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
109	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
110	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
111	C₂×C₄×C₅₃	#137	1	2	yes	—
112	#88322	—	4	4	no	—
113	#88322	—	8	2	no	—
114	#9898	—	53	1	no	maximal, Sylow
115	C₂²×C₅₃	#391	1	1	yes	characteristic
116	C₂²×C₅₃	C₂×#100	1	1	yes	characteristic
117	C₂²×C₅₃	C₂×#101	1	1	yes	characteristic
118	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
119	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
120	C₂²×C₅₃	C₂×#103	1	1	yes	characteristic
121	C₂²×C₅₃	#419	1	1	yes	characteristic
122	C₄×C₅₃	—	2	2	no	—
123	C₄×C₅₃	—	2	2	no	—
124	C₄×C₅₃	—	2	2	no	—
125	C₄×C₅₃	—	8	4	no	—
126	C₄×C₅₃	#331	1	4	yes	—
127	C₄×C₅₃	#332	1	4	yes	—
128	C₄×C₅₃	#474	1	2	yes	—
129	C₄×C₅₃	#475	1	2	yes	—
130	C₄×C₅₃	#486	1	1	yes	characteristic
131	C₄×C₅₃	#487	1	1	yes	characteristic
132	C₄×C₅₃	#498	1	1	yes	characteristic
133	C₄×C₅₃	#500	1	1	yes	characteristic
134	#2868	D₅₃	1	1	yes	characteristic
135	#1842	—	53	1	no	—
136	C₄×C₄⋊C₈	—	53	1	no	—
137	#1296	—	53	4	no	—
138	C₂×C₅₃	C₂×#331	1	1	yes	characteristic
139	C₂×C₅₃	C₂×#332	1	1	yes	characteristic
140	C₂×C₅₃	#1546	1	1	yes	characteristic
141	C₂×C₅₃	#1548	1	1	yes	characteristic
142	C₂×C₅₃	#1988	1	1	yes	characteristic
143	C₂×C₅₃	#1990	1	1	yes	characteristic
144	C₂×C₅₃	#1997	1	1	yes	characteristic
145	C₂×C₄⋊Q₈	—	2	1	no	—
146	C₂×C₄⋊Q₈	—	2	1	no	—
147	C₂×C₄⋊Q₈	Dic₅₃	1	2	yes	—
148	C₂×C₄⋊C₈	—	53	2	no	—
149	C₂×C₄⋊C₈	—	53	2	no	—
150	C₂×C₄⋊C₈	—	106	1	no	—
151	C₂×C₄⋊C₈	—	106	1	no	—
152	C₂×C₄×C₈	—	106	1	no	—
153	#384	—	2	1	no	—
154	#384	—	2	1	no	—
155	#384	Dic₅₃	1	2	yes	—
156	#384	C₂×D₅₃	1	2	yes	—
157	C₄³	C₂×D₅₃	1	1	yes	characteristic
158	#336	—	106	1	no	—
159	C₅₃	#9898	1	1	yes	characteristic, complemented, Sylow
160	C₂²×Q₈	—	2	2	no	—
161	C₂²×C₈	—	106	2	no	—
162	C₂²×C₈	—	106	2	no	—
163	C₄⋊Q₈	—	4	4	no	—
164	C₄⋊Q₈	—	4	2	no	—
165	C₄⋊Q₈	—	4	2	no	—
166	C₄⋊Q₈	—	4	2	no	—
167	C₄⋊Q₈	—	4	2	no	—
168	C₂×C₄⋊C₄	—	2	2	no	—
169	C₂×C₄⋊C₄	—	2	2	no	—
170	C₂×C₄⋊C₄	—	2	2	no	—
171	C₂×C₄⋊C₄	—	2	2	no	—
172	C₂×C₄⋊C₄	—	2	2	no	—
173	C₂×C₄⋊C₄	C₂×Dic₅₃	1	4	yes	—
174	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
175	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
176	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
177	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
178	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
179	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
180	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
181	C₄⋊C₈	—	53	4	no	—
182	C₄⋊C₈	—	106	2	no	—
183	C₄⋊C₈	—	106	4	no	—
184	C₄⋊C₈	—	212	2	no	—
185	C₄⋊C₈	—	212	2	no	—
186	C₄×C₈	—	212	2	no	—
187	C₂×Q₈	—	4	4	no	—
188	C₂×Q₈	—	4	4	no	—
189	C₂×Q₈	—	4	4	no	—
190	C₂²×C₄	—	2	4	no	—
191	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
192	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
193	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
194	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
195	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
196	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
197	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
198	C₂×C₈	—	106	4	no	—
199	C₂×C₈	—	212	2	no	—
200	C₂×C₈	—	212	2	no	—
201	C₂×C₈	—	212	2	no	—
202	C₂×C₈	—	212	2	no	—
203	C₂×C₈	—	212	2	no	—
204	C₄⋊C₄	—	4	4	no	—
205	C₄⋊C₄	—	4	4	no	—
206	C₄⋊C₄	—	4	4	no	—
207	C₄⋊C₄	—	4	4	no	—
208	C₄⋊C₄	—	4	2	no	—
209	C₄⋊C₄	—	4	2	no	—
210	C₄⋊C₄	—	4	2	no	—
211	C₄⋊C₄	—	4	2	no	—
212	C₄²	—	2	2	no	—
213	C₄²	—	2	2	no	—
214	C₄²	—	2	2	no	—
215	C₄²	#11719922	1	2	yes	—
216	C₄²	#11719922	1	2	yes	—
217	C₄²	#11719923	1	2	yes	—
218	C₄²	#11719923	1	2	yes	—
219	C₄²	#11719924	1	2	yes	—
220	C₄²	#11719925	1	2	yes	—
221	C₄²	Q₈×D₅₃	1	2	yes	—
222	C₄²	#11719949	1	2	yes	—
223	C₂³	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
224	Q₈	—	8	8	no	—
225	C₂×C₄	—	2	1	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	—	4	4	no	—
232	C₂×C₄	—	4	4	no	—
233	C₂×C₄	—	4	4	no	—
234	C₂×C₄	#13133134	1	2	yes	—
235	C₂×C₄	#13133136	1	2	yes	—
236	C₂×C₄	#13133138	1	2	yes	—
237	C₂×C₄	#13133138	1	2	yes	—
238	C₂×C₄	#13133138	1	2	yes	—
239	C₂×C₄	#13133138	1	2	yes	—
240	C₂×C₄	#13133141	1	2	yes	—
241	C₂×C₄	#13133141	1	2	yes	—
242	C₂×C₄	Q₈×Dic₅₃	1	2	yes	—
243	C₂×C₄	#13133193	1	2	yes	—
244	C₂×C₄	#13133206	1	1	yes	characteristic
245	C₂×C₄	#13133215	1	1	yes	characteristic
246	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
247	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
248	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
249	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
250	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
251	C₂×C₄	C₂×#11719925	1	1	yes	characteristic
252	C₂×C₄	#13133248	1	2	yes	—
253	C₈	—	424	2	no	—
254	C₈	—	212	4	no	—
255	C₂²	#16810530	1	1	yes	characteristic
256	C₂²	C₂×#13133134	1	1	yes	characteristic
257	C₂²	C₂×#13133136	1	1	yes	characteristic
258	C₂²	C₂×#13133138	1	1	yes	characteristic
259	C₂²	C₂×#13133138	1	1	yes	characteristic
260	C₂²	C₂×#13133141	1	1	yes	characteristic
261	C₂²	#16810632	1	1	yes	characteristic
262	C₄	—	2	2	no	—
263	C₄	—	2	2	no	—
264	C₄	—	2	2	no	—
265	C₄	—	8	4	no	—
266	C₄	#16810367	1	4	yes	—
267	C₄	#16810369	1	4	yes	—
268	C₄	#16810900	1	2	yes	—
269	C₄	#16810905	1	2	yes	—
270	C₄	#16810971	1	1	yes	characteristic
271	C₄	#16810975	1	1	yes	characteristic
272	C₄	#16811019	1	1	yes	characteristic
273	C₄	#16811026	1	1	yes	characteristic
274	C₂	C₂×#16810367	1	1	yes	characteristic
275	C₂	C₂×#16810369	1	1	yes	characteristic
276	C₂	#20273387	1	1	yes	characteristic
277	C₂	#20273396	1	1	yes	characteristic
278	C₂	#20275668	1	1	yes	characteristic
279	C₂	#20275677	1	1	yes	characteristic
280	C₂	#20275705	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₅₃
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#2868	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#2868	D₅₃
hypocenter	C₅₃	#9898
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2868

Complements

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

N	H ≅ G/N	type
25. C₄⋊#88322	Q₈	semidirect
159. C₅₃	#9898	semidirect

Maximal subgroups

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

subgroup	index
#20274867	2
C₄×C₄⋊#88322	2
#20272598	2
C₅₃×#2868	2
#9898	53