Subgroups — #27282995

Order 13568.

482 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20275652	C₂	1	2	yes	maximal
2	C₂×#16810374	C₂	1	1	yes	maximal, characteristic
3	C₂×Q₈⋊#88322	C₂	1	2	yes	maximal
4	C₂×#16810345	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×Q₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#88322	C₂²	1	2	yes	—
7	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
8	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
9	#16810378	—	2	4	no	—
10	#16810374	—	2	2	no	—
11	Q₈⋊#88322	—	2	4	no	—
12	#16810345	—	2	2	no	—
13	C₂²×Q₁₆×C₅₃	C₄	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₅₃	C₂²	1	2	yes	—
15	C₄×Q₁₆×C₅₃	—	2	2	no	—
16	C₄×Q₁₆×C₅₃	—	2	2	no	—
17	C₂×#107×C₅₃	C₄	1	1	yes	characteristic
18	C₂×#103×C₅₃	C₄	1	2	yes	—
19	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂²×#88322	—	2	2	no	—
21	C₂²×#88322	—	2	1	no	—
22	C₂²×#88322	D₄	1	2	yes	—
23	C₄⋊#88322	—	2	2	no	—
24	C₄⋊#88322	—	2	2	no	—
25	C₄×#88322	—	2	1	no	—
26	C₄×#88322	—	2	1	no	—
27	C₄×#88322	—	2	1	no	—
28	C₄×#88322	D₄	1	2	yes	—
29	C₂²×Q₈×C₅₃	C₂×C₄	1	2	yes	—
30	C₂×Q₁₆×C₅₃	—	2	2	no	—
31	C₂×Q₁₆×C₅₃	—	2	4	no	—
32	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₄×Q₈×C₅₃	—	2	2	no	—
35	C₄×Q₈×C₅₃	—	2	2	no	—
36	C₄×Q₈×C₅₃	—	2	2	no	—
37	C₄×Q₈×C₅₃	—	2	2	no	—
38	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
39	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₅₃	—	2	2	no	—
41	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
42	#107×C₅₃	—	2	2	no	—
43	#103×C₅₃	—	2	4	no	—
44	C₄×C₈×C₅₃	D₄	1	2	yes	—
45	C₄×C₈×C₅₃	D₄	1	2	yes	—
46	C₂×Q₈×C₅₃	—	2	2	no	—
47	C₂×Q₈×C₅₃	—	2	2	no	—
48	C₂×Q₈×C₅₃	—	2	2	no	—
49	C₂×Q₈×C₅₃	—	2	2	no	—
50	C₂×Q₈×C₅₃	—	2	2	no	—
51	C₂×Q₈×C₅₃	M₁₆	1	4	yes	—
52	C₂²×C₄×C₅₃	—	2	2	no	—
53	C₂²×C₄×C₅₃	—	2	2	no	—
54	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
57	Q₁₆×C₅₃	—	2	8	no	—
58	C₂×C₈×C₅₃	—	2	2	no	—
59	C₂×C₈×C₅₃	—	2	2	no	—
60	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
61	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
62	C₄⋊C₄×C₅₃	—	2	2	no	—
63	C₄⋊C₄×C₅₃	—	2	2	no	—
64	C₄⋊C₄×C₅₃	—	4	2	no	—
65	C₄⋊C₄×C₅₃	—	4	2	no	—
66	C₄²×C₅₃	—	4	2	no	—
67	C₄²×C₅₃	—	4	2	no	—
68	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
69	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
70	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
71	C₄²×C₅₃	#41	1	1	yes	characteristic
72	C₂×#88322	—	2	4	no	—
73	C₂×#88322	—	2	2	no	—
74	C₂×#88322	—	2	2	no	—
75	C₂×#88322	—	2	4	no	—
76	C₂×#88322	—	4	2	no	—
77	C₂×#88322	—	4	4	no	—
78	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
79	Q₈×C₅₃	—	2	4	no	—
80	Q₈×C₅₃	—	2	4	no	—
81	Q₈×C₅₃	—	4	2	no	—
82	Q₈×C₅₃	—	4	2	no	—
83	C₂×C₄×C₅₃	—	2	2	no	—
84	C₂×C₄×C₅₃	—	2	4	no	—
85	C₂×C₄×C₅₃	—	4	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	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
91	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
92	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
93	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
94	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
95	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
96	C₂×C₄×C₅₃	#131	1	2	yes	—
97	C₂×C₄×C₅₃	#137	1	4	yes	—
98	C₈×C₅₃	—	2	2	no	—
99	C₈×C₅₃	—	2	2	no	—
100	#88322	—	2	4	no	—
101	#88322	—	4	2	no	—
102	#88322	—	8	2	no	—
103	#9664	—	53	1	no	maximal, Sylow
104	C₂²×C₅₃	#390	1	1	yes	characteristic
105	C₂²×C₅₃	#435	1	2	yes	—
106	C₂²×C₅₃	#441	1	2	yes	—
107	C₂²×C₅₃	#443	1	2	yes	—
108	C₄×C₅₃	—	2	1	no	—
109	C₄×C₅₃	—	2	1	no	—
110	C₄×C₅₃	—	4	4	no	—
111	C₄×C₅₃	—	8	2	no	—
112	C₄×C₅₃	#407	1	1	yes	characteristic
113	C₄×C₅₃	#408	1	1	yes	characteristic
114	C₄×C₅₃	#435	1	2	yes	—
115	C₄×C₅₃	#470	1	2	yes	—
116	C₄×C₅₃	#497	1	2	yes	—
117	C₂×C₄×Q₁₆	D₅₃	1	1	yes	characteristic
118	#1984	—	53	2	no	—
119	C₂×#334	—	53	1	no	—
120	C₂×Q₈⋊C₈	—	53	2	no	—
121	C₂×#322	—	53	1	no	—
122	C₂×C₅₃	#1587	1	4	yes	—
123	C₂×C₅₃	#1955	1	1	yes	characteristic
124	C₂×C₅₃	#1971	1	1	yes	characteristic
125	C₂×C₅₃	#1974	1	1	yes	characteristic
126	C₂²×Q₁₆	Dic₅₃	1	1	yes	characteristic
127	C₂×C₄×Q₈	C₂×D₅₃	1	2	yes	—
128	C₄×Q₁₆	—	2	2	no	—
129	C₄×Q₁₆	—	2	2	no	—
130	C₂×#107	Dic₅₃	1	1	yes	characteristic
131	C₂×C₄⋊C₈	—	53	2	no	—
132	C₂×#103	Dic₅₃	1	2	yes	—
133	C₂×C₄×C₈	—	53	1	no	—
134	C₂×C₄×C₈	—	53	1	no	—
135	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
136	#336	—	106	4	no	—
137	#334	—	106	2	no	—
138	Q₈⋊C₈	—	106	4	no	—
139	#322	—	106	2	no	—
140	C₅₃	#9664	1	1	yes	characteristic, complemented, Sylow
141	C₂²×Q₈	C₂×Dic₅₃	1	2	yes	—
142	C₂×Q₁₆	—	2	4	no	—
143	C₂×Q₁₆	—	2	2	no	—
144	C₂²×C₈	—	53	2	no	—
145	C₂²×C₈	—	106	1	no	—
146	C₂²×C₈	—	106	2	no	—
147	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
148	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
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	2	no	—
154	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
155	C₂×C₄²	—	2	2	no	—
156	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
157	#107	—	2	2	no	—
158	C₄⋊C₈	—	106	2	no	—
159	C₄⋊C₈	—	106	2	no	—
160	#103	—	2	4	no	—
161	C₄×C₈	—	53	2	no	—
162	C₄×C₈	—	106	1	no	—
163	C₄×C₈	—	106	1	no	—
164	C₄×C₈	—	106	1	no	—
165	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
166	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
167	C₂×Q₈	—	2	2	no	—
168	C₂×Q₈	—	2	2	no	—
169	C₂×Q₈	—	2	2	no	—
170	C₂×Q₈	—	2	2	no	—
171	C₂×Q₈	—	2	2	no	—
172	C₂×Q₈	#11719917	1	4	yes	—
173	C₂²×C₄	—	2	2	no	—
174	C₂²×C₄	—	2	2	no	—
175	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
176	C₂²×C₄	#11719947	1	1	yes	characteristic
177	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
178	Q₁₆	—	2	8	no	—
179	C₂×C₈	—	2	2	no	—
180	C₂×C₈	—	2	2	no	—
181	C₂×C₈	—	106	4	no	—
182	C₂×C₈	—	106	2	no	—
183	C₂×C₈	—	106	2	no	—
184	C₂×C₈	—	106	4	no	—
185	C₂×C₈	—	212	2	no	—
186	C₂×C₈	—	212	4	no	—
187	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
188	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
189	C₄⋊C₄	—	2	2	no	—
190	C₄⋊C₄	—	2	2	no	—
191	C₄⋊C₄	—	4	2	no	—
192	C₄⋊C₄	—	4	2	no	—
193	C₄²	—	4	2	no	—
194	C₄²	—	4	2	no	—
195	C₄²	D₄×D₅₃	1	1	yes	characteristic
196	C₄²	#11719947	1	1	yes	characteristic
197	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
198	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
199	C₂³	D₄×Dic₅₃	1	1	yes	characteristic
200	Q₈	—	2	4	no	—
201	Q₈	—	2	4	no	—
202	Q₈	—	4	2	no	—
203	Q₈	—	4	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	4	2	no	—
207	C₂×C₄	—	4	2	no	—
208	C₂×C₄	—	4	2	no	—
209	C₂×C₄	—	4	4	no	—
210	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
211	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
212	C₂×C₄	#13133188	1	1	yes	characteristic
213	C₂×C₄	#13133188	1	1	yes	characteristic
214	C₂×C₄	#13133204	1	1	yes	characteristic
215	C₂×C₄	#13133213	1	1	yes	characteristic
216	C₂×C₄	C₂×#11719917	1	2	yes	—
217	C₂×C₄	#13133224	1	2	yes	—
218	C₂×C₄	#13133249	1	4	yes	—
219	C₈	—	2	2	no	—
220	C₈	—	2	2	no	—
221	C₈	—	106	4	no	—
222	C₈	—	424	2	no	—
223	C₈	—	212	2	no	—
224	C₂²	#16810526	1	1	yes	characteristic
225	C₂²	#16810695	1	2	yes	—
226	C₂²	#16810730	1	2	yes	—
227	C₂²	#16810740	1	2	yes	—
228	C₄	—	2	1	no	—
229	C₄	—	2	1	no	—
230	C₄	—	4	4	no	—
231	C₄	—	8	2	no	—
232	C₄	#16810587	1	1	yes	characteristic
233	C₄	#16810590	1	1	yes	characteristic
234	C₄	#16810695	1	2	yes	—
235	C₄	#16810878	1	2	yes	—
236	C₄	#16811016	1	2	yes	—
237	C₂	#20273587	1	4	yes	—
238	C₂	#20275514	1	1	yes	characteristic
239	C₂	#20275587	1	1	yes	characteristic
240	C₂	#20275604	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₅₃
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₄×Q₁₆×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₄×D₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×C₄×Q₁₆	D₅₃
hypocenter	C₅₃	#9664
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
140. C₅₃	#9664	semidirect

Maximal subgroups

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

subgroup	index
#20275652	2
C₂×#16810374	2
C₂×Q₈⋊#88322	2
C₂×#16810345	2
C₂×C₄×Q₁₆×C₅₃	2
#9664	53