Subgroups — #28303537

Order 13568.

362 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	#20290648	C₂	1	1	yes	maximal, characteristic
2	#20290647	C₂	1	1	yes	maximal, characteristic
3	#20289545	C₂	1	4	yes	maximal
4	#20274459	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133284	—	2	1	no	—
6	C₂×#13133284	—	2	1	no	—
7	C₂×C₄⋊#88324	—	2	1	no	—
8	C₂×C₄⋊#88324	—	2	1	no	—
9	C₂×C₄⋊#88324	C₂²	1	2	yes	—
10	C₂×C₄⋊#88324	C₂²	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	—	2	2	no	—
17	C₂²×#88324	—	2	2	no	—
18	#13133284	—	4	2	no	—
19	#13133284	—	4	2	no	—
20	C₄⋊#88324	—	2	4	no	—
21	C₄⋊#88324	—	2	4	no	—
22	C₄⋊#88324	—	4	2	no	—
23	C₄⋊#88324	—	4	2	no	—
24	C₂²×#88322	C₂×C₄	1	2	yes	—
25	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
27	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
28	C₂×C₄×Dic₅₃	Q₈	1	1	yes	characteristic
29	C₂×C₄×Dic₅₃	Q₈	1	1	yes	characteristic
30	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
31	C₈×Dic₅₃	—	2	2	no	—
32	C₈×Dic₅₃	—	2	2	no	—
33	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
36	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
37	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
38	C₂²×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
39	C₂²×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
40	C₂²×Dic₅₃	C₂×D₄	1	1	yes	characteristic
41	C₂²×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
42	C₂×#88324	—	4	2	no	—
43	C₂×#88324	—	4	2	no	—
44	C₂×#88324	—	4	2	no	—
45	C₂×#88324	—	4	2	no	—
46	C₂×#88324	—	4	2	no	—
47	C₂×#88324	—	4	2	no	—
48	C₂×C₈×C₅₃	—	2	2	no	—
49	C₂×C₈×C₅₃	—	2	2	no	—
50	C₂×C₈×C₅₃	—	2	2	no	—
51	C₄²×C₅₃	—	2	2	no	—
52	C₄×Dic₅₃	—	2	1	no	—
53	C₄×Dic₅₃	—	2	1	no	—
54	C₄×Dic₅₃	—	4	1	no	—
55	C₄×Dic₅₃	—	4	1	no	—
56	C₄×Dic₅₃	D₈	1	2	yes	—
57	C₄×Dic₅₃	SD₁₆	1	2	yes	—
58	C₄×Dic₅₃	SD₁₆	1	2	yes	—
59	C₄×Dic₅₃	Q₁₆	1	2	yes	—
60	C₄×Dic₅₃	#41	1	2	yes	—
61	C₄×Dic₅₃	#41	1	2	yes	—
62	C₂×#88322	—	2	2	no	—
63	C₂×#88322	—	2	2	no	—
64	C₂×#88322	—	2	2	no	—
65	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₅₃	—	2	1	no	—
67	C₂×C₄×C₅₃	—	2	1	no	—
68	C₂×C₄×C₅₃	—	2	1	no	—
69	C₂×C₄×C₅₃	—	2	1	no	—
70	C₂×C₄×C₅₃	#106	1	2	yes	—
71	C₂×C₄×C₅₃	#107	1	2	yes	—
72	C₂×C₄×C₅₃	#117	1	2	yes	—
73	C₂×C₄×C₅₃	#131	1	2	yes	—
74	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
75	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
76	C₂×Dic₅₃	—	2	1	no	—
77	C₂×Dic₅₃	—	2	1	no	—
78	C₂×Dic₅₃	—	2	1	no	—
79	C₂×Dic₅₃	—	2	1	no	—
80	C₂×Dic₅₃	—	2	1	no	—
81	C₂×Dic₅₃	—	2	1	no	—
82	C₂×Dic₅₃	#106	1	2	yes	—
83	C₂×Dic₅₃	#107	1	2	yes	—
84	C₂×Dic₅₃	#124	1	1	yes	characteristic
85	C₂×Dic₅₃	#124	1	1	yes	characteristic
86	C₂×Dic₅₃	#125	1	2	yes	—
87	C₂×Dic₅₃	C₂×D₈	1	1	yes	characteristic
88	C₂×Dic₅₃	C₂×SD₁₆	1	1	yes	characteristic
89	C₂×Dic₅₃	C₂×SD₁₆	1	1	yes	characteristic
90	C₂×Dic₅₃	C₂×Q₁₆	1	1	yes	characteristic
91	#88324	—	8	2	no	—
92	#88324	—	8	2	no	—
93	C₈×C₅₃	—	4	2	no	—
94	#88322	—	4	2	no	—
95	#8787	—	53	1	no	maximal, Sylow
96	C₂²×C₅₃	#382	1	1	yes	characteristic
97	C₂²×C₅₃	#424	1	1	yes	characteristic
98	C₂²×C₅₃	#424	1	1	yes	characteristic
99	C₂²×C₅₃	C₂×#106	1	1	yes	characteristic
100	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
101	C₂²×C₅₃	#430	1	1	yes	characteristic
102	C₂²×C₅₃	#430	1	1	yes	characteristic
103	C₄×C₅₃	—	2	2	no	—
104	C₄×C₅₃	—	2	2	no	—
105	C₄×C₅₃	#433	1	2	yes	—
106	C₄×C₅₃	#433	1	2	yes	—
107	Dic₅₃	—	2	2	no	—
108	Dic₅₃	—	2	2	no	—
109	Dic₅₃	—	2	2	no	—
110	Dic₅₃	#486	1	1	yes	characteristic
111	Dic₅₃	#486	1	1	yes	characteristic
112	Dic₅₃	#487	1	1	yes	characteristic
113	Dic₅₃	#487	1	1	yes	characteristic
114	#1842	—	53	1	no	—
115	#1842	—	53	1	no	—
116	#1744	—	53	1	no	—
117	#1277	—	53	4	no	—
118	C₂×C₅₃	#1570	1	1	yes	characteristic
119	C₂×C₅₃	#1570	1	1	yes	characteristic
120	C₂×C₅₃	#1571	1	1	yes	characteristic
121	C₂×C₅₃	#1571	1	1	yes	characteristic
122	C₂×C₅₃	#1832	1	1	yes	characteristic
123	C₂×C₅₃	#1834	1	1	yes	characteristic
124	C₂×C₅₃	#1838	1	1	yes	characteristic
125	C₂×C₄⋊C₈	—	53	2	no	—
126	C₂×C₄⋊C₈	—	53	2	no	—
127	C₂×C₄⋊C₈	—	106	1	no	—
128	C₂×C₄⋊C₈	—	106	1	no	—
129	C₂×C₄⋊C₈	—	106	1	no	—
130	C₂×C₄⋊C₈	—	106	1	no	—
131	C₂×C₄×C₈	—	53	2	no	—
132	C₄³	—	53	1	no	—
133	#336	—	53	2	no	—
134	#336	—	53	1	no	—
135	#336	#6157	1	1	yes	characteristic
136	C₅₃	#8787	1	1	yes	characteristic, complemented, Sylow
137	C₂²×C₈	—	53	2	no	—
138	C₂²×C₈	—	106	2	no	—
139	C₂²×C₈	—	106	2	no	—
140	C₂²×C₈	C₂×#6157	1	2	yes	—
141	C₂×C₄²	—	53	1	no	—
142	C₂×C₄²	—	53	1	no	—
143	C₂×C₄²	—	53	1	no	—
144	C₂×C₄²	—	53	1	no	—
145	C₂×C₄²	—	53	1	no	—
146	C₂×C₄²	—	53	1	no	—
147	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
148	C₄⋊C₈	—	106	4	no	—
149	C₄⋊C₈	—	106	4	no	—
150	C₄⋊C₈	—	212	2	no	—
151	C₄⋊C₈	—	212	2	no	—
152	C₄⋊C₈	—	212	2	no	—
153	C₄⋊C₈	—	212	2	no	—
154	C₄×C₈	—	106	2	no	—
155	C₄×C₈	—	106	2	no	—
156	C₂²×C₄	—	53	1	no	—
157	C₂²×C₄	—	53	1	no	—
158	C₂²×C₄	—	53	1	no	—
159	C₂²×C₄	—	53	1	no	—
160	C₂²×C₄	C₄⋊#6157	1	1	yes	characteristic
161	C₂²×C₄	C₄⋊#6157	1	1	yes	characteristic
162	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	2	2	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	106	2	no	—
167	C₂×C₈	—	106	2	no	—
168	C₂×C₈	—	106	2	no	—
169	C₂×C₈	—	212	2	no	—
170	C₂×C₈	—	212	2	no	—
171	C₂×C₈	—	212	2	no	—
172	C₂×C₈	—	212	2	no	—
173	C₂×C₈	—	212	2	no	—
174	C₂×C₈	—	212	2	no	—
175	C₄²	—	2	2	no	—
176	C₄²	—	53	2	no	—
177	C₄²	—	53	2	no	—
178	C₄²	—	53	2	no	—
179	C₄²	—	53	2	no	—
180	C₄²	—	53	2	no	—
181	C₄²	—	53	2	no	—
182	C₄²	—	106	1	no	—
183	C₄²	—	106	1	no	—
184	C₄²	—	212	1	no	—
185	C₄²	—	212	1	no	—
186	C₂³	C₂×C₄⋊#6157	1	1	yes	characteristic
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	53	1	no	—
192	C₂×C₄	—	53	1	no	—
193	C₂×C₄	—	53	1	no	—
194	C₂×C₄	—	53	1	no	—
195	C₂×C₄	—	53	1	no	—
196	C₂×C₄	—	53	1	no	—
197	C₂×C₄	—	53	2	no	—
198	C₂×C₄	—	53	2	no	—
199	C₂×C₄	—	53	2	no	—
200	C₂×C₄	—	106	1	no	—
201	C₂×C₄	—	106	1	no	—
202	C₂×C₄	—	106	1	no	—
203	C₂×C₄	—	106	1	no	—
204	C₂×C₄	—	106	1	no	—
205	C₂×C₄	—	106	1	no	—
206	C₂×C₄	#13133285	1	2	yes	—
207	C₂×C₄	#13133286	1	2	yes	—
208	C₂×C₄	#13133296	1	2	yes	—
209	C₂×C₄	#13133304	1	1	yes	characteristic
210	C₂×C₄	#13133304	1	2	yes	—
211	C₂×C₄	#13133305	1	1	yes	characteristic
212	C₈	—	4	2	no	—
213	C₈	—	424	2	no	—
214	C₈	—	424	2	no	—
215	C₈	—	212	2	no	—
216	C₂²	#16811567	1	1	yes	characteristic
217	C₂²	#16811629	1	1	yes	characteristic
218	C₂²	#16811630	1	1	yes	characteristic
219	C₂²	C₂×#13133285	1	1	yes	characteristic
220	C₂²	C₂×#13133286	1	1	yes	characteristic
221	C₂²	#16811638	1	1	yes	characteristic
222	C₂²	#16811638	1	1	yes	characteristic
223	C₄	—	2	2	no	—
224	C₄	—	2	2	no	—
225	C₄	—	53	1	no	—
226	C₄	—	53	1	no	—
227	C₄	—	53	1	no	—
228	C₄	—	53	1	no	—
229	C₄	—	106	2	no	—
230	C₄	—	106	2	no	—
231	C₄	—	106	2	no	—
232	C₄	#16811647	1	2	yes	—
233	C₄	#16811648	1	2	yes	—
234	C₂	#20290282	1	1	yes	characteristic
235	C₂	#20290283	1	1	yes	characteristic
236	C₂	#20290284	1	1	yes	characteristic
237	C₂	#20290285	1	1	yes	characteristic
238	C₂	#20290633	1	1	yes	characteristic
239	C₂	#20290636	1	1	yes	characteristic
240	C₂	#20290641	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₄⋊#6157
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×#6157
Fitting	C₅₃×#336	C₄
socle, prime radical	C₂³×C₅₃	C₂×C₄⋊C₄
cosocle	C₂×C₄×Dic₅₃	C₂³
hypercenter	#336	#6157
hypocenter	C₅₃	#8787
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

Complements

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

N	H ≅ G/N	type
136. C₅₃	#8787	semidirect

Maximal subgroups

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

subgroup	index
#20290648	2
#20290647	2
#20289545	2
#20274459	2
#8787	53