Subgroups — #27302698

Order 13568.

390 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810399	C₂	1	1	yes	maximal, characteristic
2	#20274879	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	C₂²	1	2	yes	—
7	C₂×#13133149	—	2	1	no	—
8	#16810489	—	2	1	no	—
9	#16810399	—	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×#11719908	—	2	2	no	—
18	C₂×#11719908	—	2	2	no	—
19	C₂×#11719908	Q₈	1	4	yes	complemented
20	#13133151	—	2	4	no	—
21	#13133149	—	4	2	no	—
22	#13133124	—	4	1	no	—
23	#13133124	—	4	1	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	4	no	—
39	C₂²×C₄×C₅₃	—	4	2	no	—
40	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
43	C₂²×Dic₅₃	—	4	2	no	—
44	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
45	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
46	C₂×C₈×C₅₃	Q₁₆	1	2	yes	—
47	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
50	C₂×C₈×C₅₃	#41	1	2	yes	—
51	C₄⋊C₄×C₅₃	—	2	4	no	—
52	C₄⋊C₄×C₅₃	—	4	4	no	—
53	C₄⋊C₄×C₅₃	—	4	2	no	—
54	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
55	C₄²×C₅₃	SD₁₆	1	2	yes	—
56	C₄⋊Dic₅₃	—	4	2	no	—
57	C₄⋊Dic₅₃	—	4	2	no	—
58	#11719908	—	2	4	no	—
59	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
60	Q₈×C₅₃	—	8	8	no	—
61	C₂×C₄×C₅₃	—	2	2	no	—
62	C₂×C₄×C₅₃	—	2	2	no	—
63	C₂×C₄×C₅₃	—	4	4	no	—
64	C₂×C₄×C₅₃	—	8	4	no	—
65	C₂×C₄×C₅₃	#102	1	2	yes	—
66	C₂×C₄×C₅₃	#102	1	2	yes	—
67	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
69	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
70	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
71	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
72	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
73	C₂×Dic₅₃	—	8	2	no	—
74	C₂×Dic₅₃	—	8	2	no	—
75	C₂×Dic₅₃	—	8	2	no	—
76	C₈×C₅₃	#103	1	4	yes	—
77	C₈×C₅₃	C₄×Q₈	1	2	yes	—
78	C₈×C₅₃	C₂²⋊Q₈	1	2	yes	—
79	#13283	—	53	1	no	maximal, Sylow
80	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
81	C₂²×C₅₃	#486	1	1	yes	characteristic
82	C₂²×C₅₃	#500	1	1	yes	characteristic
83	C₂²×C₅₃	#509	1	2	yes	—
84	C₂²×C₅₃	#510	1	2	yes	—
85	C₄×C₅₃	—	2	2	no	—
86	C₄×C₅₃	—	2	2	no	—
87	C₄×C₅₃	—	8	4	no	—
88	C₄×C₅₃	#391	1	1	yes	characteristic
89	C₄×C₅₃	C₂×#103	1	1	yes	characteristic
90	C₄×C₅₃	#487	1	1	yes	characteristic
91	C₄×C₅₃	#498	1	1	yes	characteristic
92	Dic₅₃	—	16	2	no	—
93	C₂×#500	D₅₃	1	1	yes	characteristic
94	C₂×#346	—	53	1	no	—
95	#1846	—	53	1	no	—
96	#1380	—	53	4	no	—
97	C₂×C₅₃	#1990	1	1	yes	characteristic
98	C₂×C₅₃	#2140	1	1	yes	characteristic
99	C₂×C₅₃	#2141	1	1	yes	characteristic
100	C₂×C₅₃	#2243	1	2	yes	—
101	C₂×C₅₃	#2255	1	2	yes	—
102	C₂×C₄⋊Q₈	—	2	1	no	—
103	C₂²×C₁₆	—	53	2	no	—
104	#500	—	2	2	no	—
105	#500	Dic₅₃	1	4	yes	complemented
106	C₂×#107	—	2	1	no	—
107	C₂×#107	—	53	2	no	—
108	C₂×#107	C₂×D₅₃	1	2	yes	—
109	C₂×#106	—	106	1	no	—
110	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
111	#382	—	106	1	no	—
112	#346	—	106	2	no	—
113	C₅₃	#13283	1	1	yes	characteristic, complemented, Sylow
114	C₂²×Q₈	—	4	1	no	—
115	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
116	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
117	C₄⋊Q₈	—	2	4	no	—
118	C₄⋊Q₈	—	4	2	no	—
119	C₂×C₄⋊C₄	—	2	2	no	—
120	C₂×C₄⋊C₄	—	4	1	no	—
121	C₂×C₄⋊C₄	—	106	2	no	—
122	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
123	C₂×C₁₆	—	53	4	no	—
124	C₂×C₁₆	—	106	2	no	—
125	C₂×C₁₆	—	106	2	no	—
126	#107	—	2	2	no	—
127	#107	—	2	4	no	—
128	#107	—	106	4	no	—
129	#107	C₂×Dic₅₃	1	4	yes	—
130	#106	—	212	2	no	—
131	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
132	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
133	#95	—	212	1	no	—
134	#95	—	212	1	no	—
135	C₂×Q₈	—	4	4	no	—
136	C₂×Q₈	—	8	4	no	—
137	C₂²×C₄	—	4	2	no	—
138	C₂²×C₄	—	212	2	no	—
139	C₂²×C₄	#11719922	1	1	yes	characteristic
140	C₂²×C₄	#11719922	1	1	yes	characteristic
141	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
142	C₂×C₈	#11719924	1	2	yes	—
143	C₂×C₈	#11719925	1	2	yes	—
144	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
145	C₂×C₈	Q₈×D₅₃	1	2	yes	—
146	C₂×C₈	#11719949	1	2	yes	—
147	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
148	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
149	C₄⋊C₄	—	2	4	no	—
150	C₄⋊C₄	—	4	4	no	—
151	C₄⋊C₄	—	4	2	no	—
152	C₄⋊C₄	—	212	2	no	—
153	C₄⋊C₄	—	212	2	no	—
154	C₄²	#11719923	1	2	yes	—
155	C₄²	C₂²⋊Dic₅₃	1	2	yes	—
156	C₁₆	—	106	4	no	—
157	C₂³	C₂×#11719922	1	1	yes	characteristic
158	Q₈	—	8	8	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	4	4	no	—
162	C₂×C₄	—	8	4	no	—
163	C₂×C₄	—	424	2	no	—
164	C₂×C₄	—	424	2	no	—
165	C₂×C₄	—	424	2	no	—
166	C₂×C₄	#13133138	1	2	yes	—
167	C₂×C₄	#13133138	1	2	yes	—
168	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
169	C₂×C₄	#13133206	1	1	yes	characteristic
170	C₂×C₄	#13133215	1	1	yes	characteristic
171	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
172	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
173	C₂×C₄	C₂×#11719925	1	1	yes	characteristic
174	C₈	#13133141	1	4	yes	—
175	C₈	Q₈×Dic₅₃	1	2	yes	—
176	C₈	#13133193	1	2	yes	—
177	C₂²	C₂×#13133138	1	1	yes	characteristic
178	C₂²	#16810971	1	1	yes	characteristic
179	C₂²	#16811026	1	1	yes	characteristic
180	C₂²	#16811059	1	2	yes	—
181	C₂²	#16811064	1	2	yes	—
182	C₄	—	2	2	no	—
183	C₄	—	2	2	no	—
184	C₄	—	8	4	no	—
185	C₄	—	848	2	no	—
186	C₄	#16810530	1	1	yes	characteristic
187	C₄	C₂×#13133141	1	1	yes	characteristic
188	C₄	#16810975	1	1	yes	characteristic
189	C₄	#16811019	1	1	yes	characteristic
190	C₂	#20275677	1	1	yes	characteristic
191	C₂	#20276431	1	1	yes	characteristic
192	C₂	#20276435	1	1	yes	characteristic
193	C₂	#20276954	1	2	yes	—
194	C₂	#20276998	1	2	yes	—

	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	#13283
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
19. C₂×#11719908	Q₈	semidirect
105. #500	Dic₅₃	semidirect
113. C₅₃	#13283	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810399	2
#20274879	2
#20272806	2
C₂×C₅₃×#500	2
#13283	53