Subgroups — #28301458

Order 13568.

478 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13133272	C₂	1	1	yes	maximal, characteristic
2	C₄²×#88324	C₂	1	1	yes	maximal, characteristic
3	#20289537	C₂	1	2	yes	maximal
4	#20289537	C₂	1	2	yes	maximal
5	#20274459	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133272	C₄	1	2	yes	—
7	C₂×#13133272	C₄	1	2	yes	—
8	C₂×C₄×#88324	C₄	1	2	yes	—
9	C₂×C₄×#88324	C₄	1	2	yes	—
10	C₂×C₄×#88324	C₂²	1	1	yes	characteristic
11	C₂×C₄×#88324	C₂²	1	1	yes	characteristic
12	C₂×C₄×#88324	C₂²	1	2	yes	—
13	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
14	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
15	#16810378	C₄	1	1	yes	characteristic
16	#16810377	C₄	1	2	yes	—
17	C₅₃×#336	C₄	1	1	yes	characteristic
18	C₂²×#88324	C₂×C₄	1	4	yes	—
19	C₂²×#88324	C₂×C₄	1	2	yes	—
20	C₂²×#88324	C₂×C₄	1	2	yes	—
21	#13133272	—	2	4	no	—
22	#13133272	C₈	1	8	yes	complemented
23	C₄×#88324	—	2	2	no	—
24	C₄×#88324	—	2	2	no	—
25	C₄×#88324	—	2	2	no	—
26	C₄×#88324	—	2	2	no	—
27	C₄×#88324	—	2	4	no	—
28	C₄×#88324	—	2	4	no	—
29	C₂²×#88322	C₂×C₄	1	2	yes	—
30	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
36	C₈×Dic₅₃	—	2	2	no	—
37	C₈×Dic₅₃	—	2	2	no	—
38	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
43	C₂²×Dic₅₃	C₄²	1	1	yes	characteristic
44	C₂²×Dic₅₃	C₄²	1	1	yes	characteristic
45	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
46	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
47	C₂×#88324	—	2	4	no	—
48	C₂×#88324	—	2	2	no	—
49	C₂×#88324	—	2	2	no	—
50	C₂×#88324	—	2	2	no	—
51	C₂×#88324	—	2	2	no	—
52	C₂×#88324	—	2	4	no	—
53	C₂×#88324	C₂×C₈	1	8	yes	—
54	C₂×#88324	M₁₆	1	4	yes	—
55	C₂×#88324	M₁₆	1	4	yes	—
56	C₂×C₈×C₅₃	—	2	2	no	—
57	C₂×C₈×C₅₃	—	2	2	no	—
58	C₂×C₈×C₅₃	—	2	2	no	—
59	C₄²×C₅₃	—	2	2	no	—
60	C₄×Dic₅₃	—	2	1	no	—
61	C₄×Dic₅₃	—	2	1	no	—
62	C₄×Dic₅₃	—	2	2	no	—
63	C₄×Dic₅₃	—	2	2	no	—
64	C₄×Dic₅₃	C₂×C₈	1	4	yes	—
65	C₄×Dic₅₃	M₁₆	1	4	yes	—
66	C₄×Dic₅₃	#41	1	2	yes	—
67	C₄×Dic₅₃	#41	1	2	yes	—
68	C₂×#88322	—	2	2	no	—
69	C₂×#88322	—	2	2	no	—
70	C₂×#88322	—	2	2	no	—
71	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
72	C₂×C₄×C₅₃	—	2	1	no	—
73	C₂×C₄×C₅₃	—	2	1	no	—
74	C₂×C₄×C₅₃	—	2	1	no	—
75	C₂×C₄×C₅₃	—	2	1	no	—
76	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
77	C₂×C₄×C₅₃	#97	1	2	yes	—
78	C₂×C₄×C₅₃	#117	1	2	yes	—
79	C₂×C₄×C₅₃	#131	1	2	yes	—
80	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
81	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
82	C₂×Dic₅₃	—	2	1	no	—
83	C₂×Dic₅₃	—	2	1	no	—
84	C₂×Dic₅₃	—	2	1	no	—
85	C₂×Dic₅₃	—	2	1	no	—
86	C₂×Dic₅₃	C₄×C₈	1	2	yes	—
87	C₂×Dic₅₃	#97	1	2	yes	—
88	C₂×Dic₅₃	#117	1	2	yes	—
89	C₂×Dic₅₃	#117	1	1	yes	characteristic
90	C₂×Dic₅₃	#117	1	1	yes	characteristic
91	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
92	C₂×Dic₅₃	C₂×M₁₆	1	2	yes	—
93	C₂×Dic₅₃	C₂×M₁₆	1	2	yes	—
94	C₂×Dic₅₃	C₂×M₁₆	1	2	yes	—
95	#88324	—	2	8	no	—
96	#88324	—	2	4	no	—
97	#88324	—	2	4	no	—
98	C₈×C₅₃	—	4	2	no	—
99	#88322	—	4	2	no	—
100	#8132	—	53	1	no	maximal, Sylow
101	C₂²×C₅₃	#376	1	1	yes	characteristic
102	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
103	C₂²×C₅₃	C₂×#97	1	1	yes	characteristic
104	C₂²×C₅₃	C₄×M₁₆	1	1	yes	characteristic
105	C₂²×C₅₃	C₄×M₁₆	1	1	yes	characteristic
106	C₂²×C₅₃	#405	1	1	yes	characteristic
107	C₂²×C₅₃	#405	1	1	yes	characteristic
108	C₄×C₅₃	—	2	2	no	—
109	C₄×C₅₃	—	2	2	no	—
110	C₄×C₅₃	#433	1	2	yes	—
111	C₄×C₅₃	#433	1	2	yes	—
112	Dic₅₃	—	2	2	no	—
113	Dic₅₃	—	2	2	no	—
114	Dic₅₃	#431	1	4	yes	—
115	Dic₅₃	#432	1	2	yes	—
116	Dic₅₃	#432	1	2	yes	—
117	#1744	—	53	1	no	—
118	C₄×#97	—	53	1	no	—
119	C₄²×C₈	—	53	1	no	—
120	#1273	—	53	2	no	—
121	#1273	—	53	2	no	—
122	C₂×C₅₃	C₈×M₁₆	1	1	yes	characteristic
123	C₂×C₅₃	C₈×M₁₆	1	1	yes	characteristic
124	C₂×C₅₃	#1451	1	1	yes	characteristic
125	C₂×C₅₃	#1451	1	1	yes	characteristic
126	C₂×C₅₃	#1744	1	1	yes	characteristic
127	C₂×C₅₃	#1745	1	1	yes	characteristic
128	C₂×C₅₃	#1746	1	1	yes	characteristic
129	C₂×#97	—	53	2	no	—
130	C₂×#97	—	53	2	no	—
131	C₂×C₄×C₈	—	53	2	no	—
132	C₂×C₄×C₈	—	53	2	no	—
133	C₂×C₄×C₈	—	53	1	no	—
134	C₂×C₄×C₈	—	53	1	no	—
135	C₂×C₄×C₈	—	53	2	no	—
136	C₂×C₄×C₈	—	53	2	no	—
137	C₄³	—	53	1	no	—
138	#336	—	53	2	no	—
139	#336	—	53	1	no	—
140	#336	#6157	1	1	yes	characteristic
141	C₅₃	#8132	1	1	yes	characteristic, complemented, Sylow
142	C₂²×C₈	—	53	2	no	—
143	C₂²×C₈	—	53	2	no	—
144	C₂²×C₈	—	53	4	no	—
145	C₂²×C₈	—	53	2	no	—
146	C₂²×C₈	C₂×#6157	1	2	yes	—
147	C₂×C₄²	—	53	1	no	—
148	C₂×C₄²	—	53	1	no	—
149	C₂×C₄²	—	53	1	no	—
150	C₂×C₄²	—	53	1	no	—
151	C₂×C₄²	—	53	1	no	—
152	C₂×C₄²	—	53	1	no	—
153	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
154	#97	—	53	8	no	—
155	#97	—	106	4	no	—
156	C₄×C₈	—	106	2	no	—
157	C₄×C₈	—	106	2	no	—
158	C₄×C₈	—	106	2	no	—
159	C₄×C₈	—	106	2	no	—
160	C₄×C₈	—	106	2	no	—
161	C₄×C₈	—	106	2	no	—
162	C₄×C₈	—	106	4	no	—
163	C₄×C₈	—	106	4	no	—
164	C₂²×C₄	—	53	1	no	—
165	C₂²×C₄	—	53	1	no	—
166	C₂²×C₄	—	53	1	no	—
167	C₂²×C₄	—	53	1	no	—
168	C₂²×C₄	C₄×#6157	1	1	yes	characteristic
169	C₂²×C₄	C₄×#6157	1	1	yes	characteristic
170	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
171	C₂×C₈	—	2	2	no	—
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	—	53	8	no	—
175	C₂×C₈	—	53	4	no	—
176	C₂×C₈	—	53	4	no	—
177	C₂×C₈	—	106	2	no	—
178	C₂×C₈	—	106	2	no	—
179	C₂×C₈	—	106	4	no	—
180	C₂×C₈	—	106	2	no	—
181	C₂×C₈	—	106	2	no	—
182	C₂×C₈	—	106	4	no	—
183	C₂×C₈	—	106	2	no	—
184	C₂×C₈	—	106	2	no	—
185	C₂×C₈	—	106	2	no	—
186	C₄²	—	2	2	no	—
187	C₄²	—	53	4	no	—
188	C₄²	—	53	4	no	—
189	C₄²	—	53	2	no	—
190	C₄²	—	53	2	no	—
191	C₄²	—	106	2	no	—
192	C₄²	—	106	2	no	—
193	C₄²	—	106	1	no	—
194	C₄²	—	106	1	no	—
195	C₂³	C₂×C₄×#6157	1	1	yes	characteristic
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	53	2	no	—
201	C₂×C₄	—	53	1	no	—
202	C₂×C₄	—	53	1	no	—
203	C₂×C₄	—	53	2	no	—
204	C₂×C₄	—	53	2	no	—
205	C₂×C₄	—	53	2	no	—
206	C₂×C₄	—	53	2	no	—
207	C₂×C₄	—	53	2	no	—
208	C₂×C₄	—	53	2	no	—
209	C₂×C₄	—	106	1	no	—
210	C₂×C₄	—	106	1	no	—
211	C₂×C₄	—	106	1	no	—
212	C₂×C₄	—	106	1	no	—
213	C₂×C₄	C₈×#6157	1	2	yes	—
214	C₂×C₄	#13133271	1	2	yes	—
215	C₂×C₄	#13133296	1	2	yes	—
216	C₂×C₄	#13133304	1	1	yes	characteristic
217	C₂×C₄	#13133304	1	2	yes	—
218	C₂×C₄	#13133305	1	1	yes	characteristic
219	C₈	—	4	2	no	—
220	C₈	—	106	8	no	—
221	C₈	—	106	4	no	—
222	C₈	—	106	4	no	—
223	C₈	—	212	2	no	—
224	C₂²	#16811559	1	1	yes	characteristic
225	C₂²	C₂×C₈×#6157	1	1	yes	characteristic
226	C₂²	C₂×#13133271	1	1	yes	characteristic
227	C₂²	M₁₆×#6157	1	1	yes	characteristic
228	C₂²	M₁₆×#6157	1	1	yes	characteristic
229	C₂²	#16811587	1	1	yes	characteristic
230	C₂²	#16811588	1	1	yes	characteristic
231	C₄	—	2	2	no	—
232	C₄	—	2	2	no	—
233	C₄	—	53	4	no	—
234	C₄	—	53	2	no	—
235	C₄	—	53	2	no	—
236	C₄	—	106	2	no	—
237	C₄	—	106	2	no	—
238	C₄	#16811647	1	2	yes	—
239	C₄	#16811648	1	2	yes	—
240	C₂	#20289916	1	1	yes	characteristic
241	C₂	#20289917	1	1	yes	characteristic
242	C₂	#20289926	1	1	yes	characteristic
243	C₂	#20289927	1	1	yes	characteristic
244	C₂	#20290428	1	1	yes	characteristic
245	C₂	#20290431	1	1	yes	characteristic
246	C₂	#20290434	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#6157
commutator	C₂²×C₅₃	C₂×C₄×C₈
Fitting	C₅₃×#336	C₄
socle, prime radical	C₂³×C₅₃	C₂×C₄²
cosocle	C₂×C₄×Dic₅₃	C₂³
hypercenter	#336	#6157
hypocenter	C₅₃	#8132
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ #336

Complements

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

N	H ≅ G/N	type
22. #13133272	C₈	semidirect
141. C₅₃	#8132	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13133272	2
C₄²×#88324	2
#20289537	2
#20274459	2
#8132	53