Subgroups — #27283150

Order 13568.

498 conjugacy classes of subgroups.

The subgroup lattice has 250 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	C₂×#16810372	C₂	1	2	yes	maximal
4	C₂×#16810369	C₂	1	2	yes	maximal
5	C₅₃×#2862	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#88322	—	2	2	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	#16810372	—	2	4	no	—
12	#16810369	—	2	4	no	—
13	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×#391	—	2	1	no	—
16	C₅₃×#391	—	2	1	no	—
17	C₅₃×#391	C₄	1	4	yes	—
18	C₅₃×#382	—	2	2	no	—
19	C₄³×C₅₃	C₂²	1	1	yes	characteristic
20	C₂²×#88322	—	2	2	no	—
21	C₂²×#88322	—	2	2	no	—
22	C₄⋊#88322	—	2	2	no	—
23	C₄⋊#88322	—	2	2	no	—
24	C₄⋊#88322	—	2	2	no	—
25	C₄⋊#88322	—	4	4	no	—
26	C₄⋊#88322	D₄	1	4	yes	—
27	C₄×#88322	—	4	2	no	—
28	C₂²×Q₈×C₅₃	—	2	1	no	—
29	C₂²×Q₈×C₅₃	C₂×C₄	1	2	yes	—
30	C₄⋊Q₈×C₅₃	—	2	2	no	—
31	C₄⋊Q₈×C₅₃	—	2	2	no	—
32	C₄⋊Q₈×C₅₃	—	2	2	no	—
33	C₄⋊Q₈×C₅₃	—	2	2	no	—
34	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
35	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
36	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
37	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₅₃	D₄	1	2	yes	—
42	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
43	#95×C₅₃	—	2	4	no	—
44	#95×C₅₃	—	2	4	no	—
45	C₂×Q₈×C₅₃	—	2	4	no	—
46	C₂×Q₈×C₅₃	—	4	4	no	—
47	C₂×Q₈×C₅₃	—	4	2	no	—
48	C₂×Q₈×C₅₃	—	4	2	no	—
49	C₂×Q₈×C₅₃	—	4	1	no	—
50	C₂×Q₈×C₅₃	—	4	1	no	—
51	C₂²×C₄×C₅₃	—	2	2	no	—
52	C₂²×C₄×C₅₃	—	2	2	no	—
53	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
54	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
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	—	4	2	no	—
60	C₄⋊C₄×C₅₃	—	4	1	no	—
61	C₄⋊C₄×C₅₃	—	4	1	no	—
62	C₄⋊C₄×C₅₃	—	4	1	no	—
63	C₄⋊C₄×C₅₃	—	4	1	no	—
64	C₄²×C₅₃	—	2	2	no	—
65	C₄²×C₅₃	—	2	1	no	—
66	C₄²×C₅₃	—	2	1	no	—
67	C₄²×C₅₃	—	4	2	no	—
68	C₄²×C₅₃	SD₁₆	1	4	yes	—
69	C₄²×C₅₃	Q₁₆	1	4	yes	—
70	C₄²×C₅₃	C₂×D₄	1	2	yes	—
71	C₄²×C₅₃	#41	1	2	yes	—
72	C₂×#88322	—	2	4	no	—
73	C₂×#88322	—	4	2	no	—
74	C₂×#88322	—	4	2	no	—
75	C₂×#88322	—	4	2	no	—
76	C₂×#88322	—	4	4	no	—
77	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	Q₈×C₅₃	—	8	4	no	—
79	Q₈×C₅₃	—	8	2	no	—
80	Q₈×C₅₃	—	8	2	no	—
81	C₂×C₄×C₅₃	—	2	2	no	—
82	C₂×C₄×C₅₃	—	2	2	no	—
83	C₂×C₄×C₅₃	—	2	2	no	—
84	C₂×C₄×C₅₃	—	2	2	no	—
85	C₂×C₄×C₅₃	—	2	2	no	—
86	C₂×C₄×C₅₃	—	4	2	no	—
87	C₂×C₄×C₅₃	—	4	2	no	—
88	C₂×C₄×C₅₃	—	4	4	no	—
89	C₂×C₄×C₅₃	—	4	2	no	—
90	C₂×C₄×C₅₃	—	4	2	no	—
91	C₂×C₄×C₅₃	#103	1	4	yes	—
92	C₂×C₄×C₅₃	#103	1	4	yes	—
93	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
94	C₂×C₄×C₅₃	#121	1	2	yes	—
95	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
96	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
97	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
98	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
99	C₂×C₄×C₅₃	#135	1	2	yes	—
100	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
101	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
102	#88322	—	4	4	no	—
103	#88322	—	8	2	no	—
104	#9697	—	53	1	no	maximal, Sylow
105	C₂²×C₅₃	#390	1	1	yes	characteristic
106	C₂²×C₅₃	#413	1	2	yes	—
107	C₂²×C₅₃	C₂×#103	1	2	yes	—
108	C₂²×C₅₃	#416	1	1	yes	characteristic
109	C₂²×C₅₃	#417	1	1	yes	characteristic
110	C₄×C₅₃	—	2	4	no	—
111	C₄×C₅₃	—	2	4	no	—
112	C₄×C₅₃	—	2	2	no	—
113	C₄×C₅₃	—	8	2	no	—
114	C₄×C₅₃	—	8	2	no	—
115	C₄×C₅₃	#463	1	2	yes	—
116	C₄×C₅₃	#464	1	2	yes	—
117	C₄×C₅₃	#487	1	2	yes	—
118	C₄×C₅₃	#492	1	1	yes	characteristic
119	C₄×C₅₃	#494	1	1	yes	characteristic
120	#2862	D₅₃	1	1	yes	characteristic
121	#1842	—	53	1	no	—
122	C₄×C₄⋊C₈	—	53	1	no	—
123	C₂×#333	—	53	2	no	—
124	C₂×#332	—	53	2	no	—
125	C₂×C₅₃	#1542	1	2	yes	—
126	C₂×C₅₃	#1550	1	2	yes	—
127	C₂×C₅₃	#1956	1	1	yes	characteristic
128	C₂×C₅₃	#1967	1	1	yes	characteristic
129	C₂×C₅₃	#1983	1	1	yes	characteristic
130	C₂×C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
131	C₂×C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
132	C₂×C₄⋊C₈	—	53	2	no	—
133	C₂×C₄⋊C₈	—	53	2	no	—
134	C₂×C₄⋊C₈	—	106	2	no	—
135	C₂×C₄×C₈	—	106	1	no	—
136	#391	—	2	1	no	—
137	#391	—	2	1	no	—
138	#391	Dic₅₃	1	4	yes	—
139	#382	—	2	2	no	—
140	C₄³	C₂×D₅₃	1	1	yes	characteristic
141	#336	—	106	1	no	—
142	#333	—	106	4	no	—
143	#332	—	106	4	no	—
144	C₅₃	#9697	1	1	yes	characteristic, complemented, Sylow
145	C₂²×Q₈	—	2	1	no	—
146	C₂²×Q₈	C₂×Dic₅₃	1	2	yes	—
147	C₂²×C₈	—	106	2	no	—
148	C₂²×C₈	—	106	2	no	—
149	C₄⋊Q₈	—	2	2	no	—
150	C₄⋊Q₈	—	2	2	no	—
151	C₄⋊Q₈	—	2	2	no	—
152	C₄⋊Q₈	—	2	2	no	—
153	C₂×C₄⋊C₄	—	2	1	no	—
154	C₂×C₄⋊C₄	—	2	1	no	—
155	C₂×C₄⋊C₄	—	2	1	no	—
156	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
157	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
158	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
159	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
160	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
161	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
162	C₄⋊C₈	—	53	4	no	—
163	C₄⋊C₈	—	106	2	no	—
164	C₄⋊C₈	—	106	2	no	—
165	C₄⋊C₈	—	106	2	no	—
166	C₄⋊C₈	—	212	4	no	—
167	C₄×C₈	—	212	2	no	—
168	#95	—	2	4	no	—
169	#95	—	2	4	no	—
170	C₂×Q₈	—	2	4	no	—
171	C₂×Q₈	—	4	4	no	—
172	C₂×Q₈	—	4	2	no	—
173	C₂×Q₈	—	4	2	no	—
174	C₂×Q₈	—	4	1	no	—
175	C₂×Q₈	—	4	1	no	—
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	—	2	2	no	—
178	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
179	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
180	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
181	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
182	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
183	C₂×C₈	—	106	4	no	—
184	C₂×C₈	—	212	2	no	—
185	C₂×C₈	—	212	4	no	—
186	C₂×C₈	—	212	2	no	—
187	C₂×C₈	—	212	2	no	—
188	C₄⋊C₄	—	2	4	no	—
189	C₄⋊C₄	—	4	2	no	—
190	C₄⋊C₄	—	4	1	no	—
191	C₄⋊C₄	—	4	1	no	—
192	C₄⋊C₄	—	4	1	no	—
193	C₄⋊C₄	—	4	1	no	—
194	C₄²	—	2	2	no	—
195	C₄²	—	2	1	no	—
196	C₄²	—	2	1	no	—
197	C₄²	—	4	2	no	—
198	C₄²	#11719924	1	4	yes	—
199	C₄²	#11719925	1	4	yes	—
200	C₄²	D₄×D₅₃	1	2	yes	—
201	C₄²	#11719947	1	2	yes	—
202	C₂³	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
203	Q₈	—	8	4	no	—
204	Q₈	—	8	2	no	—
205	Q₈	—	8	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	4	4	no	—
214	C₂×C₄	—	4	2	no	—
215	C₂×C₄	—	4	2	no	—
216	C₂×C₄	#13133141	1	4	yes	—
217	C₂×C₄	#13133141	1	4	yes	—
218	C₂×C₄	D₄×Dic₅₃	1	2	yes	—
219	C₂×C₄	#13133188	1	2	yes	—
220	C₂×C₄	#13133204	1	1	yes	characteristic
221	C₂×C₄	#13133213	1	1	yes	characteristic
222	C₂×C₄	C₂×#11719924	1	2	yes	—
223	C₂×C₄	C₂×#11719925	1	2	yes	—
224	C₂×C₄	#13133237	1	2	yes	—
225	C₂×C₄	#13133242	1	1	yes	characteristic
226	C₂×C₄	#13133247	1	1	yes	characteristic
227	C₈	—	424	2	no	—
228	C₈	—	212	4	no	—
229	C₂²	#16810526	1	1	yes	characteristic
230	C₂²	#16810607	1	2	yes	—
231	C₂²	C₂×#13133141	1	2	yes	—
232	C₂²	#16810619	1	1	yes	characteristic
233	C₂²	#16810624	1	1	yes	characteristic
234	C₄	—	2	2	no	—
235	C₄	—	2	4	no	—
236	C₄	—	2	4	no	—
237	C₄	—	8	2	no	—
238	C₄	—	8	2	no	—
239	C₄	#16810835	1	2	yes	—
240	C₄	#16810842	1	2	yes	—
241	C₄	#16810973	1	2	yes	—
242	C₄	#16810997	1	1	yes	characteristic
243	C₄	#16811002	1	1	yes	characteristic
244	C₂	#20273372	1	2	yes	—
245	C₂	#20273405	1	2	yes	—
246	C₂	#20275518	1	1	yes	characteristic
247	C₂	#20275565	1	1	yes	characteristic
248	C₂	#20275648	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×#2862	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#2862	D₅₃
hypocenter	C₅₃	#9697
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2862

Complements

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

N	H ≅ G/N	type
144. C₅₃	#9697	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
C₂×#16810372	2
C₂×#16810369	2
C₅₃×#2862	2
#9697	53