Subgroups — #27282649

Order 13568.

378 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20275433	C₂	1	2	yes	maximal
2	C₄×#13133149	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810374	C₂	1	1	yes	maximal, characteristic
4	#20272555	C₂	1	2	yes	maximal
5	C₂×C₅₃×#322	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133149	C₄	1	2	yes	—
7	C₂×C₄⋊#88322	C₂²	1	2	yes	—
8	C₂×#13133146	—	2	2	no	—
9	C₄×C₄⋊Dic₅₃	C₂²	1	2	yes	—
10	#16810386	C₄	1	1	yes	characteristic
11	#16810386	C₄	1	1	yes	characteristic
12	#16810377	—	2	2	no	—
13	#16810374	—	2	2	no	—
14	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#322	—	2	2	no	—
18	C₂²×#88322	—	2	2	no	—
19	C₂×C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
20	C₂×C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
21	C₂×C₄×Dic₅₃	—	2	2	no	—
22	#13133149	—	2	4	no	—
23	C₄⋊#88322	—	2	2	no	—
24	C₄⋊#88322	—	2	2	no	—
25	#13133146	—	4	4	no	—
26	#13133124	—	2	2	no	—
27	C₂²×C₈×C₅₃	—	2	1	no	—
28	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₅₃	D₄	1	2	yes	—
31	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
32	C₄×C₈×C₅₃	—	2	1	no	—
33	C₄×C₈×C₅₃	—	2	1	no	—
34	C₄×C₈×C₅₃	—	2	1	no	—
35	C₄×C₈×C₅₃	D₄	1	2	yes	—
36	C₄×C₈×C₅₃	Q₈	1	2	yes	—
37	C₄×C₈×C₅₃	Q₈	1	2	yes	—
38	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
41	C₂²×Dic₅₃	—	2	2	no	—
42	C₂²×Dic₅₃	—	2	2	no	—
43	C₂×C₈×C₅₃	—	2	2	no	—
44	C₂×C₈×C₅₃	—	2	2	no	—
45	C₂×C₈×C₅₃	—	2	2	no	—
46	C₂×C₈×C₅₃	—	2	2	no	—
47	C₂×C₈×C₅₃	—	2	2	no	—
48	C₂×C₈×C₅₃	—	4	1	no	—
49	C₂×C₈×C₅₃	—	4	1	no	—
50	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
51	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
52	C₂×C₈×C₅₃	SD₁₆	1	4	yes	—
53	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
54	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
55	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
56	C₄²×C₅₃	#41	1	1	yes	characteristic
57	C₄⋊Dic₅₃	—	2	2	no	—
58	C₄⋊Dic₅₃	—	2	2	no	—
59	C₄⋊Dic₅₃	—	2	2	no	—
60	C₄⋊Dic₅₃	M₁₆	1	4	yes	—
61	C₄×Dic₅₃	—	4	2	no	—
62	C₄×Dic₅₃	—	4	2	no	—
63	C₂×#88322	—	4	2	no	—
64	C₂×#88322	—	4	2	no	—
65	C₂×#88322	—	4	2	no	—
66	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
67	C₂×C₄×C₅₃	—	2	1	no	—
68	C₂×C₄×C₅₃	—	2	1	no	—
69	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
71	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
72	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
73	C₂×C₄×C₅₃	#125	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
76	C₂×C₄×C₅₃	#131	1	2	yes	—
77	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
78	C₂×C₄×C₅₃	#135	1	2	yes	—
79	C₂×Dic₅₃	—	2	4	no	—
80	C₂×Dic₅₃	—	4	2	no	—
81	C₂×Dic₅₃	—	4	2	no	—
82	C₂×Dic₅₃	—	4	2	no	—
83	C₂×Dic₅₃	—	4	2	no	—
84	C₂×Dic₅₃	—	4	2	no	—
85	C₈×C₅₃	—	2	2	no	—
86	C₈×C₅₃	—	2	2	no	—
87	C₈×C₅₃	—	2	4	no	—
88	C₈×C₅₃	—	4	2	no	—
89	#88322	—	8	2	no	—
90	#9597	—	53	1	no	maximal, Sylow
91	C₂²×C₅₃	#389	1	1	yes	characteristic
92	C₂²×C₅₃	#435	1	2	yes	—
93	C₂²×C₅₃	C₄×SD₁₆	1	2	yes	—
94	C₂²×C₅₃	#444	1	1	yes	characteristic
95	C₂²×C₅₃	#444	1	1	yes	characteristic
96	C₄×C₅₃	—	2	1	no	—
97	C₄×C₅₃	—	2	1	no	—
98	C₄×C₅₃	#423	1	1	yes	characteristic
99	C₄×C₅₃	#424	1	1	yes	characteristic
100	C₄×C₅₃	#446	1	2	yes	—
101	C₄×C₅₃	#465	1	2	yes	—
102	C₄×C₅₃	#498	1	1	yes	characteristic
103	C₄×C₅₃	#499	1	1	yes	characteristic
104	Dic₅₃	—	4	4	no	—
105	Dic₅₃	—	8	2	no	—
106	#1939	—	53	2	no	—
107	C₄×#106	—	53	1	no	—
108	C₂×#334	—	53	1	no	—
109	C₂×#322	D₅₃	1	1	yes	characteristic
110	#1279	—	53	2	no	—
111	C₂×C₅₃	#1582	1	2	yes	—
112	C₂×C₅₃	#1583	1	2	yes	—
113	C₂×C₅₃	#1940	1	1	yes	characteristic
114	C₂×C₅₃	#1941	1	1	yes	characteristic
115	C₂×C₅₃	#1946	1	1	yes	characteristic
116	C₂×#106	—	53	2	no	—
117	C₂×C₄⋊C₈	—	53	2	no	—
118	C₂×C₄⋊C₈	—	106	2	no	—
119	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
120	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
121	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
122	C₄×C₄⋊C₄	—	53	2	no	—
123	#340	—	53	1	no	—
124	#340	—	53	1	no	—
125	#336	—	106	2	no	—
126	#334	—	106	2	no	—
127	#322	—	2	2	no	—
128	C₅₃	#9597	1	1	yes	characteristic, complemented, Sylow
129	C₂²×C₈	—	2	1	no	—
130	C₂²×C₈	—	106	2	no	—
131	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
132	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
133	C₂²×C₈	D₂₁₂	1	2	yes	—
134	C₂×C₄⋊C₄	—	53	2	no	—
135	C₂×C₄⋊C₄	—	53	2	no	—
136	C₂×C₄²	—	106	2	no	—
137	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
138	#106	—	106	4	no	—
139	C₄⋊C₈	—	106	2	no	—
140	C₄⋊C₈	—	106	2	no	—
141	C₄⋊C₈	—	212	4	no	—
142	C₄×C₈	—	2	1	no	—
143	C₄×C₈	—	2	1	no	—
144	C₄×C₈	—	2	1	no	—
145	C₄×C₈	Dic₁₀₆	1	2	yes	—
146	C₄×C₈	Dic₁₀₆	1	2	yes	—
147	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
148	#95	—	106	2	no	—
149	C₂²×C₄	—	106	2	no	—
150	C₂²×C₄	—	106	2	no	—
151	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
152	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
153	C₂²×C₄	#11719945	1	1	yes	characteristic
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	2	2	no	—
159	C₂×C₈	—	4	1	no	—
160	C₂×C₈	—	4	1	no	—
161	C₂×C₈	—	212	2	no	—
162	C₂×C₈	—	212	2	no	—
163	C₂×C₈	—	212	2	no	—
164	C₂×C₈	#11719913	1	4	yes	—
165	C₂×C₈	#11719919	1	2	yes	—
166	C₂×C₈	#11719919	1	2	yes	—
167	C₄⋊C₄	—	53	4	no	—
168	C₄⋊C₄	—	106	2	no	—
169	C₄⋊C₄	—	106	2	no	—
170	C₄⋊C₄	—	106	2	no	—
171	C₄²	—	212	2	no	—
172	C₄²	—	212	2	no	—
173	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
174	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
175	C₄²	#11719945	1	1	yes	characteristic
176	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
177	C₂³	C₄×D₂₁₂	1	1	yes	characteristic
178	C₂×C₄	—	2	1	no	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	106	4	no	—
181	C₂×C₄	—	212	2	no	—
182	C₂×C₄	—	212	2	no	—
183	C₂×C₄	—	212	2	no	—
184	C₂×C₄	—	212	2	no	—
185	C₂×C₄	—	212	2	no	—
186	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
187	C₂×C₄	C₄×Dic₁₀₆	1	1	yes	characteristic
188	C₂×C₄	#13133191	1	1	yes	characteristic
189	C₂×C₄	#13133196	1	1	yes	characteristic
190	C₂×C₄	#13133209	1	1	yes	characteristic
191	C₂×C₄	#13133217	1	1	yes	characteristic
192	C₂×C₄	M₁₆×D₅₃	1	2	yes	—
193	C₂×C₄	#13133225	1	2	yes	—
194	C₂×C₄	C₂×#11719913	1	2	yes	—
195	C₂×C₄	#13133239	1	2	yes	—
196	C₈	—	2	4	no	—
197	C₈	—	2	2	no	—
198	C₈	—	2	2	no	—
199	C₈	—	4	2	no	—
200	C₈	—	424	2	no	—
201	C₂²	#16810521	1	1	yes	characteristic
202	C₂²	#16810696	1	2	yes	—
203	C₂²	C₄×#11719913	1	2	yes	—
204	C₂²	#16810746	1	1	yes	characteristic
205	C₂²	#16810746	1	1	yes	characteristic
206	C₄	—	2	1	no	—
207	C₄	—	2	1	no	—
208	C₄	—	424	2	no	—
209	C₄	—	212	4	no	—
210	C₄	#16810646	1	1	yes	characteristic
211	C₄	#16810651	1	1	yes	characteristic
212	C₄	#16810751	1	2	yes	—
213	C₄	#16810852	1	2	yes	—
214	C₄	#16811021	1	1	yes	characteristic
215	C₄	#16811025	1	1	yes	characteristic
216	C₂	#20273553	1	2	yes	—
217	C₂	#20273560	1	2	yes	—
218	C₂	#20275438	1	1	yes	characteristic
219	C₂	#20275443	1	1	yes	characteristic
220	C₂	#20275468	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₁₂
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₅₃×#322	C₂
socle, prime radical	C₂³×C₅₃	C₄×D₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×#322	D₅₃
hypocenter	C₅₃	#9597
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
128. C₅₃	#9597	semidirect

Maximal subgroups

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

subgroup	index
#20275433	2
C₄×#13133149	2
C₂×#16810374	2
#20272555	2
C₂×C₅₃×#322	2
#9597	53