Subgroups — #27300244

Order 13568.

396 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133157	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810447	C₂	1	2	yes	maximal, complemented
3	#20274740	C₂	1	1	yes	maximal, characteristic
4	#20272815	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1810	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133157	—	2	4	no	—
7	C₂×#13133157	C₄	1	2	yes	complemented
8	C₂×#13133157	C₂²	1	1	yes	characteristic
9	C₂×#13133157	C₂²	1	1	yes	characteristic
10	C₂²×#11719908	C₄	1	2	yes	complemented
11	C₂×#13133149	C₂²	1	2	yes	—
12	#16810483	—	2	1	no	—
13	#16810447	—	2	4	no	—
14	#16810386	—	2	1	no	—
15	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×#108×C₅₃	C₂²	1	2	yes	—
17	C₅₃×#407	—	2	1	no	—
18	C₅₃×#341	—	2	1	no	—
19	C₂×C₄⋊Dic₅₃	—	2	2	no	—
20	C₂×C₂²⋊Dic₅₃	—	4	1	no	—
21	#13133157	—	2	2	no	—
22	#13133157	—	2	4	no	—
23	#13133157	D₄	1	4	yes	—
24	C₂×#11719908	—	2	4	no	—
25	C₂×#11719908	C₂×C₄	1	2	yes	—
26	C₂×#11719908	C₂×C₄	1	2	yes	—
27	#13133149	—	2	2	no	—
28	#13133149	—	2	2	no	—
29	#13133124	—	4	1	no	—
30	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
31	C₂×M₁₆×C₅₃	—	2	2	no	—
32	C₂²×C₈×C₅₃	—	2	2	no	—
33	C₂²×C₈×C₅₃	—	2	2	no	—
34	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
38	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
39	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
40	#108×C₅₃	—	2	2	no	—
41	#108×C₅₃	—	2	2	no	—
42	C₂²⋊C₈×C₅₃	—	4	1	no	—
43	C₂²⋊C₈×C₅₃	—	4	1	no	—
44	C₂⁴×C₅₃	SD₁₆	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	—	2	2	no	—
46	C₂²×C₄×C₅₃	—	2	2	no	—
47	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
53	C₂²×Dic₅₃	—	4	2	no	—
54	M₁₆×C₅₃	—	4	2	no	—
55	M₁₆×C₅₃	—	4	2	no	—
56	C₂×C₈×C₅₃	—	2	2	no	—
57	C₂×C₈×C₅₃	—	2	1	no	—
58	C₂×C₈×C₅₃	—	2	1	no	—
59	C₂×C₈×C₅₃	—	2	1	no	—
60	C₂×C₈×C₅₃	—	2	1	no	—
61	C₂×C₈×C₅₃	—	2	2	no	—
62	C₂×C₈×C₅₃	—	2	2	no	—
63	C₂×C₈×C₅₃	—	4	2	no	—
64	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
68	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
69	C₂×C₈×C₅₃	#41	1	2	yes	—
70	C₂²⋊Dic₅₃	—	8	2	no	—
71	C₄⋊Dic₅₃	—	4	2	no	—
72	C₄⋊Dic₅₃	—	4	2	no	—
73	#11719908	—	2	2	no	—
74	#11719908	C₂²⋊C₄	1	4	yes	complemented
75	C₂³×C₅₃	—	2	2	no	—
76	C₂³×C₅₃	—	2	2	no	—
77	C₂³×C₅₃	—	4	1	no	—
78	C₂³×C₅₃	#106	1	1	yes	characteristic
79	C₂³×C₅₃	#106	1	1	yes	characteristic
80	C₂³×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
81	C₂×C₄×C₅₃	—	2	2	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	2	no	—
87	C₂×C₄×C₅₃	—	2	2	no	—
88	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
89	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
90	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₅₃	#121	1	2	yes	—
92	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
93	C₂×C₄×C₅₃	#135	1	2	yes	—
94	C₂×Dic₅₃	—	8	2	no	—
95	C₂×Dic₅₃	—	8	2	no	—
96	C₂×Dic₅₃	—	8	2	no	—
97	C₈×C₅₃	—	2	1	no	—
98	C₈×C₅₃	—	2	1	no	—
99	C₈×C₅₃	—	8	2	no	—
100	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₈×C₅₃	#117	1	1	yes	characteristic
102	C₈×C₅₃	C₂²⋊Q₈	1	2	yes	—
103	#12831	—	53	1	no	maximal, Sylow
104	C₂²×C₅₃	—	2	1	no	—
105	C₂²×C₅₃	—	2	1	no	—
106	C₂²×C₅₃	—	2	1	no	—
107	C₂²×C₅₃	—	2	1	no	—
108	C₂²×C₅₃	—	2	1	no	—
109	C₂²×C₅₃	—	2	1	no	—
110	C₂²×C₅₃	—	2	1	no	—
111	C₂²×C₅₃	—	4	4	no	—
112	C₂²×C₅₃	#365	1	2	yes	complemented
113	C₂²×C₅₃	C₂×#106	1	1	yes	characteristic
114	C₂²×C₅₃	#465	1	2	yes	—
115	C₄×C₅₃	—	2	1	no	—
116	C₄×C₅₃	—	2	1	no	—
117	C₄×C₅₃	#380	1	1	yes	characteristic
118	C₄×C₅₃	#427	1	1	yes	characteristic
119	C₄×C₅₃	#465	1	2	yes	—
120	Dic₅₃	—	16	2	no	—
121	C₂²×M₃₂	—	53	1	no	—
122	C₂×#365	—	53	2	no	—
123	#1810	D₅₃	1	1	yes	characteristic
124	#1807	—	53	1	no	—
125	#1383	—	53	2	no	—
126	C₂×C₅₃	—	2	2	no	—
127	C₂×C₅₃	—	2	2	no	—
128	C₂×C₅₃	—	4	1	no	—
129	C₂×C₅₃	#1807	1	1	yes	characteristic
130	C₂×C₅₃	C₂×#365	1	1	yes	characteristic
131	C₂×C₅₃	#2155	1	1	yes	characteristic
132	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
133	C₂×M₃₂	—	53	1	no	—
134	C₂×M₃₂	—	53	2	no	—
135	C₂×M₃₂	—	53	1	no	—
136	C₂×M₃₂	—	106	4	no	—
137	C₂²×C₁₆	—	53	2	no	—
138	C₂×#108	C₂×D₅₃	1	2	yes	—
139	C₂×#106	—	53	2	no	—
140	#407	—	2	1	no	—
141	#380	—	106	1	no	—
142	#365	—	106	4	no	—
143	#341	—	2	1	no	—
144	#340	—	106	1	no	—
145	C₅₃	#12831	1	1	yes	characteristic, complemented, Sylow
146	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
147	C₂×M₁₆	—	2	2	no	—
148	C₂²×C₈	—	2	2	no	—
149	C₂²×C₈	—	2	2	no	—
150	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
151	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
152	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
153	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
154	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
155	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
156	C₂×C₄⋊C₄	—	106	2	no	—
157	C₂×C₂²⋊C₄	—	212	1	no	—
158	M₃₂	—	53	4	no	—
159	M₃₂	—	106	4	no	—
160	M₃₂	—	106	2	no	—
161	C₂×C₁₆	—	53	2	no	—
162	C₂×C₁₆	—	53	2	no	—
163	C₂×C₁₆	—	106	4	no	—
164	#108	—	2	2	no	—
165	#108	—	2	2	no	—
166	#106	—	106	2	no	—
167	#106	—	106	2	no	—
168	C₂²⋊C₈	—	4	1	no	—
169	C₂²⋊C₈	—	4	1	no	—
170	#95	—	212	1	no	—
171	C₂⁴	#11719924	1	1	yes	characteristic
172	C₂²×C₄	—	2	2	no	—
173	C₂²×C₄	—	2	2	no	—
174	C₂²×C₄	—	212	2	no	—
175	C₂²×C₄	#11719919	1	1	yes	characteristic
176	C₂²×C₄	#11719919	1	1	yes	characteristic
177	C₂²×C₄	#11719923	1	1	yes	characteristic
178	C₂²×C₄	#11719923	1	1	yes	characteristic
179	C₂²×C₄	#11719924	1	1	yes	characteristic
180	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
181	M₁₆	—	4	2	no	—
182	M₁₆	—	4	2	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	2	1	no	—
185	C₂×C₈	—	2	1	no	—
186	C₂×C₈	—	2	1	no	—
187	C₂×C₈	—	2	1	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	—	4	2	no	—
191	C₂×C₈	#11719919	1	1	yes	characteristic
192	C₂×C₈	#11719919	1	1	yes	characteristic
193	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
194	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
195	C₂×C₈	D₄×D₅₃	1	2	yes	—
196	C₂×C₈	#11719947	1	2	yes	—
197	C₄⋊C₄	—	212	2	no	—
198	C₄⋊C₄	—	212	2	no	—
199	C₂²⋊C₄	—	424	2	no	—
200	C₁₆	—	53	4	no	—
201	C₁₆	—	106	2	no	—
202	C₂³	—	2	2	no	—
203	C₂³	—	2	2	no	—
204	C₂³	—	4	1	no	—
205	C₂³	#13133148	1	1	yes	characteristic
206	C₂³	#13133148	1	1	yes	characteristic
207	C₂³	C₂×#11719924	1	1	yes	characteristic
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	424	2	no	—
216	C₂×C₄	—	424	2	no	—
217	C₂×C₄	—	424	2	no	—
218	C₂×C₄	#13133148	1	1	yes	characteristic
219	C₂×C₄	#13133148	1	1	yes	characteristic
220	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
221	C₂×C₄	#13133189	1	2	yes	—
222	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
223	C₂×C₄	#13133237	1	2	yes	—
224	C₈	—	2	1	no	—
225	C₈	—	2	1	no	—
226	C₈	—	8	2	no	—
227	C₈	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
228	C₈	#13133173	1	1	yes	characteristic
229	C₈	#13133194	1	2	yes	—
230	C₂²	—	2	1	no	—
231	C₂²	—	2	1	no	—
232	C₂²	—	2	1	no	—
233	C₂²	—	2	1	no	—
234	C₂²	—	2	1	no	—
235	C₂²	—	2	1	no	—
236	C₂²	—	2	1	no	—
237	C₂²	—	4	4	no	—
238	C₂²	#16810447	1	2	yes	complemented
239	C₂²	C₂×#13133148	1	1	yes	characteristic
240	C₂²	#16810854	1	2	yes	—
241	C₄	—	2	1	no	—
242	C₄	—	2	1	no	—
243	C₄	—	848	2	no	—
244	C₄	#16810481	1	1	yes	characteristic
245	C₄	#16810659	1	1	yes	characteristic
246	C₄	#16810850	1	2	yes	—
247	C₂	—	2	2	no	—
248	C₂	—	2	2	no	—
249	C₂	—	4	1	no	—
250	C₂	#20274738	1	1	yes	characteristic
251	C₂	C₂×#16810447	1	1	yes	characteristic
252	C₂	#20276489	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13133148
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1810	C₂
socle	C₂²×C₅₃	C₂×#106
prime radical	C₂⁴×C₅₃	SD₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1810	D₅₃
hypocenter	C₅₃	#12831
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ #1810

Complements

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

N	H ≅ G/N	type
2. C₂×#16810447	C₂	semidirect
4. #20272815	C₂	semidirect
7. C₂×#13133157	C₄	semidirect
10. C₂²×#11719908	C₄	semidirect
74. #11719908	C₂²⋊C₄	semidirect
112. C₂²×C₅₃	#365	semidirect
145. C₅₃	#12831	semidirect
238. C₂²	#16810447	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13133157	2
C₂×#16810447	2
#20274740	2
#20272815	2
C₅₃×#1810	2
#12831	53