Subgroups — #28301461

Order 13568.

446 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20290430	C₂	1	1	yes	maximal, characteristic
2	C₄²×#88324	C₂	1	1	yes	maximal, characteristic
3	#20289537	C₂	1	4	yes	maximal
4	#20274459	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×#88324	C₄	1	2	yes	—
6	C₂×C₄×#88324	C₄	1	2	yes	—
7	C₂×C₄×#88324	C₂²	1	2	yes	—
8	C₂×C₄×#88324	C₂²	1	2	yes	—
9	#16811489	C₄	1	2	yes	—
10	#16811489	C₄	1	2	yes	—
11	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
12	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
13	#16810378	C₄	1	1	yes	characteristic
14	#16810377	C₄	1	2	yes	—
15	C₅₃×#336	C₄	1	1	yes	characteristic
16	C₂²×#88324	C₂×C₄	1	4	yes	—
17	C₂²×#88324	C₂×C₄	1	4	yes	—
18	C₄×#88324	—	2	4	no	—
19	C₄×#88324	—	2	4	no	—
20	C₄×#88324	—	2	4	no	—
21	C₄×#88324	C₈	1	8	yes	complemented
22	C₂²×#88322	C₂×C₄	1	2	yes	—
23	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
29	C₈×Dic₅₃	—	2	4	no	—
30	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
33	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
34	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
35	C₂²×Dic₅₃	C₄²	1	1	yes	characteristic
36	C₂²×Dic₅₃	C₄²	1	1	yes	characteristic
37	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
38	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
39	C₂×#88324	—	2	4	no	—
40	C₂×#88324	—	2	4	no	—
41	C₂×#88324	—	2	4	no	—
42	C₂×#88324	—	2	4	no	—
43	C₂×#88324	—	2	4	no	—
44	C₂×#88324	C₂×C₈	1	8	yes	—
45	C₂×C₈×C₅₃	—	2	2	no	—
46	C₂×C₈×C₅₃	—	2	2	no	—
47	C₂×C₈×C₅₃	—	2	2	no	—
48	C₄²×C₅₃	—	2	2	no	—
49	C₄×Dic₅₃	—	2	1	no	—
50	C₄×Dic₅₃	—	2	1	no	—
51	C₄×Dic₅₃	—	2	2	no	—
52	C₄×Dic₅₃	—	2	2	no	—
53	C₄×Dic₅₃	C₂×C₈	1	4	yes	—
54	C₄×Dic₅₃	M₁₆	1	4	yes	—
55	C₄×Dic₅₃	#41	1	4	yes	—
56	C₂×#88322	—	2	2	no	—
57	C₂×#88322	—	2	2	no	—
58	C₂×#88322	—	2	2	no	—
59	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
60	C₂×C₄×C₅₃	—	2	1	no	—
61	C₂×C₄×C₅₃	—	2	1	no	—
62	C₂×C₄×C₅₃	—	2	1	no	—
63	C₂×C₄×C₅₃	—	2	1	no	—
64	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
65	C₂×C₄×C₅₃	#97	1	2	yes	—
66	C₂×C₄×C₅₃	#117	1	2	yes	—
67	C₂×C₄×C₅₃	#131	1	2	yes	—
68	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
69	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
70	C₂×Dic₅₃	—	2	1	no	—
71	C₂×Dic₅₃	—	2	1	no	—
72	C₂×Dic₅₃	—	2	1	no	—
73	C₂×Dic₅₃	—	2	1	no	—
74	C₂×Dic₅₃	C₄×C₈	1	2	yes	—
75	C₂×Dic₅₃	#97	1	2	yes	—
76	C₂×Dic₅₃	#117	1	2	yes	—
77	C₂×Dic₅₃	#117	1	2	yes	—
78	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
79	C₂×Dic₅₃	C₂×M₁₆	1	2	yes	—
80	C₂×Dic₅₃	#131	1	2	yes	—
81	C₂×Dic₅₃	#131	1	2	yes	—
82	#88324	—	2	8	no	—
83	#88324	—	4	4	no	—
84	C₈×C₅₃	—	4	2	no	—
85	#88322	—	4	2	no	—
86	#8133	—	53	1	no	maximal, Sylow
87	C₂²×C₅₃	#376	1	1	yes	characteristic
88	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
89	C₂²×C₅₃	C₂×#97	1	1	yes	characteristic
90	C₂²×C₅₃	#405	1	1	yes	characteristic
91	C₂²×C₅₃	#405	1	1	yes	characteristic
92	C₂²×C₅₃	#405	1	1	yes	characteristic
93	C₂²×C₅₃	#405	1	1	yes	characteristic
94	C₄×C₅₃	—	2	2	no	—
95	C₄×C₅₃	—	2	2	no	—
96	C₄×C₅₃	#433	1	2	yes	—
97	C₄×C₅₃	#433	1	2	yes	—
98	Dic₅₃	—	2	2	no	—
99	Dic₅₃	—	2	2	no	—
100	Dic₅₃	#431	1	4	yes	—
101	Dic₅₃	#433	1	4	yes	—
102	#1744	—	53	1	no	—
103	#1744	—	53	1	no	—
104	C₄²×C₈	—	53	1	no	—
105	#1273	—	53	4	no	—
106	C₂×C₅₃	#1450	1	1	yes	characteristic
107	C₂×C₅₃	#1450	1	1	yes	characteristic
108	C₂×C₅₃	#1452	1	1	yes	characteristic
109	C₂×C₅₃	#1452	1	1	yes	characteristic
110	C₂×C₅₃	#1744	1	1	yes	characteristic
111	C₂×C₅₃	#1746	1	1	yes	characteristic
112	C₂×C₅₃	#1746	1	1	yes	characteristic
113	C₂×C₄×C₈	—	53	2	no	—
114	C₂×C₄×C₈	—	53	2	no	—
115	C₂×C₄×C₈	—	53	2	no	—
116	C₂×C₄×C₈	—	53	2	no	—
117	C₂×C₄×C₈	—	53	2	no	—
118	C₄³	—	53	1	no	—
119	#336	—	53	2	no	—
120	#336	—	53	2	no	—
121	#336	—	53	2	no	—
122	#336	—	53	1	no	—
123	#336	#6157	1	1	yes	characteristic
124	C₅₃	#8133	1	1	yes	characteristic, complemented, Sylow
125	C₂²×C₈	—	53	4	no	—
126	C₂²×C₈	—	53	4	no	—
127	C₂²×C₈	—	53	2	no	—
128	C₂²×C₈	C₂×#6157	1	2	yes	—
129	C₂×C₄²	—	53	1	no	—
130	C₂×C₄²	—	53	1	no	—
131	C₂×C₄²	—	53	1	no	—
132	C₂×C₄²	—	53	1	no	—
133	C₂×C₄²	—	53	1	no	—
134	C₂×C₄²	—	53	1	no	—
135	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
136	C₄×C₈	—	53	8	no	—
137	C₄×C₈	—	106	4	no	—
138	C₄×C₈	—	106	4	no	—
139	C₄×C₈	—	106	4	no	—
140	C₄×C₈	—	106	4	no	—
141	C₂²×C₄	—	53	1	no	—
142	C₂²×C₄	—	53	1	no	—
143	C₂²×C₄	—	53	1	no	—
144	C₂²×C₄	—	53	1	no	—
145	C₂²×C₄	C₄×#6157	1	1	yes	characteristic
146	C₂²×C₄	C₄×#6157	1	1	yes	characteristic
147	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
148	C₂×C₈	—	2	2	no	—
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	53	8	no	—
152	C₂×C₈	—	106	4	no	—
153	C₂×C₈	—	106	4	no	—
154	C₂×C₈	—	106	4	no	—
155	C₂×C₈	—	106	4	no	—
156	C₂×C₈	—	106	4	no	—
157	C₂×C₈	—	106	2	no	—
158	C₂×C₈	—	106	2	no	—
159	C₂×C₈	—	106	2	no	—
160	C₄²	—	2	2	no	—
161	C₄²	—	53	4	no	—
162	C₄²	—	53	4	no	—
163	C₄²	—	53	4	no	—
164	C₄²	—	106	2	no	—
165	C₄²	—	106	2	no	—
166	C₄²	—	106	1	no	—
167	C₄²	—	106	1	no	—
168	C₂³	C₂×C₄×#6157	1	1	yes	characteristic
169	C₂×C₄	—	2	1	no	—
170	C₂×C₄	—	2	1	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	53	2	no	—
174	C₂×C₄	—	53	2	no	—
175	C₂×C₄	—	53	2	no	—
176	C₂×C₄	—	53	2	no	—
177	C₂×C₄	—	53	2	no	—
178	C₂×C₄	—	53	2	no	—
179	C₂×C₄	—	53	2	no	—
180	C₂×C₄	—	53	2	no	—
181	C₂×C₄	—	106	1	no	—
182	C₂×C₄	—	106	1	no	—
183	C₂×C₄	—	106	1	no	—
184	C₂×C₄	—	106	1	no	—
185	C₂×C₄	C₈×#6157	1	2	yes	—
186	C₂×C₄	#13133271	1	2	yes	—
187	C₂×C₄	#13133296	1	2	yes	—
188	C₂×C₄	#13133304	1	1	yes	characteristic
189	C₂×C₄	#13133304	1	2	yes	—
190	C₂×C₄	#13133305	1	1	yes	characteristic
191	C₈	—	4	2	no	—
192	C₈	—	106	8	no	—
193	C₈	—	212	4	no	—
194	C₈	—	212	2	no	—
195	C₂²	#16811559	1	1	yes	characteristic
196	C₂²	C₂×C₈×#6157	1	1	yes	characteristic
197	C₂²	C₂×#13133271	1	1	yes	characteristic
198	C₂²	#16811587	1	1	yes	characteristic
199	C₂²	#16811588	1	1	yes	characteristic
200	C₂²	#16811589	1	1	yes	characteristic
201	C₂²	#16811589	1	1	yes	characteristic
202	C₄	—	2	2	no	—
203	C₄	—	2	2	no	—
204	C₄	—	53	4	no	—
205	C₄	—	53	4	no	—
206	C₄	—	106	2	no	—
207	C₄	—	106	2	no	—
208	C₄	#16811647	1	2	yes	—
209	C₄	#16811648	1	2	yes	—
210	C₂	#20289920	1	1	yes	characteristic
211	C₂	#20289921	1	1	yes	characteristic
212	C₂	#20289930	1	1	yes	characteristic
213	C₂	#20289931	1	1	yes	characteristic
214	C₂	#20290428	1	1	yes	characteristic
215	C₂	#20290434	1	1	yes	characteristic
216	C₂	#20290435	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	#8133
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
21. C₄×#88324	C₈	semidirect
124. C₅₃	#8133	semidirect

Maximal subgroups

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

subgroup	index
#20290430	2
C₄²×#88324	2
#20289537	2
#20274459	2
#8133	53