Subgroups — #27302694

Order 13568.

414 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#11719908	C₂	1	1	yes	maximal, characteristic
2	#20274875	C₂	1	1	yes	maximal, characteristic
3	#20272806	C₂	1	4	yes	maximal
4	C₂×C₅₃×#500	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719908	C₂²	1	2	yes	—
6	C₂×#13133151	—	2	1	no	—
7	C₂×#13133151	C₂²	1	2	yes	—
8	#16810493	—	2	1	no	—
9	C₄×#11719908	—	2	2	no	—
10	C₂×C₄⋊Q₈×C₅₃	—	2	1	no	—
11	C₅₃×#500	—	2	2	no	—
12	C₅₃×#500	C₄	1	4	yes	complemented
13	C₂×#107×C₅₃	—	2	1	no	—
14	C₂×#107×C₅₃	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₅₃	—	2	2	no	—
17	C₂×C₄⋊Dic₅₃	—	4	1	no	—
18	C₂×C₄⋊Dic₅₃	—	4	1	no	—
19	C₂×#11719908	—	2	2	no	—
20	C₂×#11719908	—	2	2	no	—
21	C₂×#11719908	Q₈	1	4	yes	complemented
22	#13133151	—	2	4	no	—
23	#13133151	—	4	2	no	—
24	C₂²×Q₈×C₅₃	—	4	1	no	—
25	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
27	C₄⋊Q₈×C₅₃	—	2	4	no	—
28	C₄⋊Q₈×C₅₃	—	4	2	no	—
29	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
30	C₂×C₄⋊C₄×C₅₃	—	4	1	no	—
31	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
32	#107×C₅₃	—	2	4	no	—
33	#107×C₅₃	—	2	2	no	—
34	#107×C₅₃	C₂×C₄	1	4	yes	—
35	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
36	C₄×C₈×C₅₃	D₄	1	2	yes	—
37	C₂×Q₈×C₅₃	—	4	4	no	—
38	C₂×Q₈×C₅₃	—	8	2	no	—
39	C₂×Q₈×C₅₃	—	8	2	no	—
40	C₂²×C₄×C₅₃	—	4	2	no	—
41	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
43	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
44	C₂²×Dic₅₃	—	4	2	no	—
45	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
46	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
47	C₂×C₈×C₅₃	Q₁₆	1	2	yes	—
48	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
51	C₂×C₈×C₅₃	#41	1	2	yes	—
52	C₄⋊C₄×C₅₃	—	2	4	no	—
53	C₄⋊C₄×C₅₃	—	4	4	no	—
54	C₄⋊C₄×C₅₃	—	4	2	no	—
55	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
56	C₄²×C₅₃	SD₁₆	1	2	yes	—
57	C₄⋊Dic₅₃	—	4	2	no	—
58	C₄⋊Dic₅₃	—	4	2	no	—
59	C₄⋊Dic₅₃	—	8	2	no	—
60	C₄⋊Dic₅₃	—	8	2	no	—
61	#11719908	—	2	4	no	—
62	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
63	Q₈×C₅₃	—	8	8	no	—
64	C₂×C₄×C₅₃	—	4	4	no	—
65	C₂×C₄×C₅₃	—	8	2	no	—
66	C₂×C₄×C₅₃	—	8	2	no	—
67	C₂×C₄×C₅₃	#102	1	2	yes	—
68	C₂×C₄×C₅₃	#102	1	2	yes	—
69	C₂×C₄×C₅₃	D₁₆	1	2	yes	—
70	C₂×C₄×C₅₃	SD₃₂	1	2	yes	—
71	C₂×C₄×C₅₃	SD₃₂	1	2	yes	—
72	C₂×C₄×C₅₃	Q₃₂	1	2	yes	—
73	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
77	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
78	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
79	C₂×Dic₅₃	—	8	2	no	—
80	C₂×Dic₅₃	—	8	2	no	—
81	C₂×Dic₅₃	—	8	2	no	—
82	C₈×C₅₃	#103	1	4	yes	—
83	C₈×C₅₃	C₄×Q₈	1	2	yes	—
84	C₈×C₅₃	C₂²⋊Q₈	1	2	yes	—
85	#13282	—	53	1	no	maximal, Sylow
86	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
87	C₂²×C₅₃	#486	1	1	yes	characteristic
88	C₂²×C₅₃	#500	1	1	yes	characteristic
89	C₂²×C₅₃	C₂×D₁₆	1	1	yes	characteristic
90	C₂²×C₅₃	C₂×SD₃₂	1	1	yes	characteristic
91	C₂²×C₅₃	C₂×SD₃₂	1	1	yes	characteristic
92	C₂²×C₅₃	C₂×Q₃₂	1	1	yes	characteristic
93	C₄×C₅₃	—	8	4	no	—
94	C₄×C₅₃	#357	1	4	yes	—
95	C₄×C₅₃	#358	1	4	yes	—
96	C₄×C₅₃	#391	1	1	yes	characteristic
97	C₄×C₅₃	C₂×#103	1	1	yes	characteristic
98	C₄×C₅₃	#487	1	1	yes	characteristic
99	C₄×C₅₃	#498	1	1	yes	characteristic
100	Dic₅₃	—	16	2	no	—
101	C₂×#500	D₅₃	1	1	yes	characteristic
102	C₂×C₄×C₁₆	—	53	1	no	—
103	#1845	—	53	1	no	—
104	#1380	—	53	4	no	—
105	C₂×C₅₃	#1990	1	1	yes	characteristic
106	C₂×C₅₃	C₂×#357	1	1	yes	characteristic
107	C₂×C₅₃	C₂×#358	1	1	yes	characteristic
108	C₂×C₅₃	#2240	1	1	yes	characteristic
109	C₂×C₅₃	#2241	1	1	yes	characteristic
110	C₂×C₅₃	#2252	1	1	yes	characteristic
111	C₂×C₅₃	#2254	1	1	yes	characteristic
112	C₂×C₄⋊Q₈	—	2	1	no	—
113	C₂²×C₁₆	—	53	2	no	—
114	#500	—	2	2	no	—
115	#500	Dic₅₃	1	4	yes	complemented
116	C₂×#107	—	2	1	no	—
117	C₂×#107	—	53	2	no	—
118	C₂×#107	—	106	1	no	—
119	C₂×#107	C₂×D₅₃	1	2	yes	—
120	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
121	#384	—	106	1	no	—
122	C₄×C₁₆	—	106	2	no	—
123	C₅₃	#13282	1	1	yes	characteristic, complemented, Sylow
124	C₂²×Q₈	—	4	1	no	—
125	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
126	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
127	C₄⋊Q₈	—	2	4	no	—
128	C₄⋊Q₈	—	4	2	no	—
129	C₂×C₄⋊C₄	—	2	2	no	—
130	C₂×C₄⋊C₄	—	4	1	no	—
131	C₂×C₄⋊C₄	—	106	2	no	—
132	C₂×C₄⋊C₄	—	212	1	no	—
133	C₂×C₄⋊C₄	—	212	1	no	—
134	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
135	C₂×C₁₆	—	53	4	no	—
136	C₂×C₁₆	—	106	2	no	—
137	C₂×C₁₆	—	106	2	no	—
138	#107	—	2	2	no	—
139	#107	—	2	4	no	—
140	#107	—	106	4	no	—
141	#107	—	212	2	no	—
142	#107	C₂×Dic₅₃	1	4	yes	—
143	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
144	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
145	C₂×Q₈	—	4	4	no	—
146	C₂×Q₈	—	8	2	no	—
147	C₂×Q₈	—	8	2	no	—
148	C₂²×C₄	—	4	2	no	—
149	C₂²×C₄	—	212	2	no	—
150	C₂²×C₄	#11719922	1	1	yes	characteristic
151	C₂²×C₄	#11719922	1	1	yes	characteristic
152	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
153	C₂×C₈	#11719924	1	2	yes	—
154	C₂×C₈	#11719925	1	2	yes	—
155	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
156	C₂×C₈	Q₈×D₅₃	1	2	yes	—
157	C₂×C₈	#11719949	1	2	yes	—
158	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
159	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
160	C₄⋊C₄	—	2	4	no	—
161	C₄⋊C₄	—	4	4	no	—
162	C₄⋊C₄	—	4	2	no	—
163	C₄⋊C₄	—	424	2	no	—
164	C₄⋊C₄	—	424	2	no	—
165	C₄⋊C₄	—	212	2	no	—
166	C₄⋊C₄	—	212	2	no	—
167	C₄²	#11719923	1	2	yes	—
168	C₄²	C₂²⋊Dic₅₃	1	2	yes	—
169	C₁₆	—	106	4	no	—
170	C₂³	C₂×#11719922	1	1	yes	characteristic
171	Q₈	—	8	8	no	—
172	C₂×C₄	—	4	4	no	—
173	C₂×C₄	—	8	2	no	—
174	C₂×C₄	—	8	2	no	—
175	C₂×C₄	—	424	2	no	—
176	C₂×C₄	—	424	2	no	—
177	C₂×C₄	—	424	2	no	—
178	C₂×C₄	#13133138	1	2	yes	—
179	C₂×C₄	#13133138	1	2	yes	—
180	C₂×C₄	#13133158	1	2	yes	—
181	C₂×C₄	#13133160	1	2	yes	—
182	C₂×C₄	#13133161	1	2	yes	—
183	C₂×C₄	#13133163	1	2	yes	—
184	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
185	C₂×C₄	#13133206	1	1	yes	characteristic
186	C₂×C₄	#13133215	1	1	yes	characteristic
187	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
188	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
189	C₂×C₄	C₂×#11719925	1	1	yes	characteristic
190	C₈	#13133141	1	4	yes	—
191	C₈	Q₈×Dic₅₃	1	2	yes	—
192	C₈	#13133193	1	2	yes	—
193	C₂²	C₂×#13133138	1	1	yes	characteristic
194	C₂²	#16810971	1	1	yes	characteristic
195	C₂²	#16811026	1	1	yes	characteristic
196	C₂²	C₂×#13133158	1	1	yes	characteristic
197	C₂²	C₂×#13133160	1	1	yes	characteristic
198	C₂²	C₂×#13133161	1	1	yes	characteristic
199	C₂²	C₂×#13133163	1	1	yes	characteristic
200	C₄	—	8	4	no	—
201	C₄	—	848	2	no	—
202	C₄	#16810424	1	4	yes	—
203	C₄	#16810427	1	4	yes	—
204	C₄	#16810530	1	1	yes	characteristic
205	C₄	C₂×#13133141	1	1	yes	characteristic
206	C₄	#16810975	1	1	yes	characteristic
207	C₄	#16811019	1	1	yes	characteristic
208	C₂	#20275677	1	1	yes	characteristic
209	C₂	C₂×#16810424	1	1	yes	characteristic
210	C₂	C₂×#16810427	1	1	yes	characteristic
211	C₂	#20276940	1	1	yes	characteristic
212	C₂	#20276944	1	1	yes	characteristic
213	C₂	#20276988	1	1	yes	characteristic
214	C₂	#20276995	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11719922
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#500	C₂
socle, prime radical	C₂³×C₅₃	C₂×D₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#500	D₅₃
hypocenter	C₅₃	#13282
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
12. C₅₃×#500	C₄	semidirect
21. C₂×#11719908	Q₈	semidirect
115. #500	Dic₅₃	semidirect
123. C₅₃	#13282	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#11719908	2
#20274875	2
#20272806	2
C₂×C₅₃×#500	2
#13282	53