Subgroups — #27302123

Order 13568.

326 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810449	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₆×Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	#20275449	C₂	1	2	yes	maximal
4	#20272806	C₂	1	2	yes	maximal
5	C₂×C₅₃×#366	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719908	C₂²	1	1	yes	characteristic
7	C₂×#13133151	C₂²	1	2	yes	—
8	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
9	#16810522	—	2	2	no	—
10	#16810449	—	2	2	no	—
11	C₁₆×Dic₅₃	—	2	2	no	—
12	#16810385	—	2	2	no	—
13	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×#107×C₅₃	C₂²	1	2	yes	—
15	C₅₃×#366	C₄	1	4	yes	complemented
16	C₂²×#88322	D₄	1	1	yes	characteristic
17	C₂×C₄⋊Dic₅₃	—	2	2	no	—
18	C₂×C₄⋊Dic₅₃	—	4	2	no	—
19	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
20	C₂×#11719908	—	2	1	no	—
21	C₂×#11719908	—	2	1	no	—
22	C₂×#11719908	Q₈	1	2	yes	—
23	#13133151	—	2	2	no	—
24	#13133151	—	2	2	no	—
25	C₈×Dic₅₃	—	2	1	no	—
26	C₈×Dic₅₃	—	2	1	no	—
27	#13133124	—	4	2	no	—
28	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
30	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
31	C₂×C₁₆×C₅₃	D₄	1	2	yes	—
32	C₂×C₁₆×C₅₃	Q₈	1	2	yes	—
33	#107×C₅₃	—	2	2	no	—
34	#107×C₅₃	C₂×C₄	1	4	yes	—
35	C₂²×C₄×C₅₃	—	4	2	no	—
36	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
37	C₂²×Dic₅₃	—	4	2	no	—
38	C₂²×Dic₅₃	D₈	1	1	yes	characteristic
39	C₂²×Dic₅₃	D₈	1	1	yes	characteristic
40	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
42	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
45	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
46	C₄⋊C₄×C₅₃	—	2	4	no	—
47	C₄⋊C₄×C₅₃	—	4	2	no	—
48	C₄⋊Dic₅₃	—	4	2	no	—
49	C₄⋊Dic₅₃	—	4	2	no	—
50	C₄⋊Dic₅₃	—	8	4	no	—
51	C₄×Dic₅₃	—	2	1	no	—
52	C₄×Dic₅₃	—	2	1	no	—
53	C₂×#88322	—	2	1	no	—
54	C₂×#88322	—	2	1	no	—
55	C₂×#88322	Q₁₆	1	2	yes	—
56	C₁₆×C₅₃	C₄⋊C₄	1	4	yes	complemented
57	#11719908	—	2	2	no	—
58	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
59	C₂×C₄×C₅₃	—	4	4	no	—
60	C₂×C₄×C₅₃	—	8	2	no	—
61	C₂×C₄×C₅₃	—	8	2	no	—
62	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
63	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
64	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
65	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
66	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
67	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
68	C₂×Dic₅₃	—	2	1	no	—
69	C₂×Dic₅₃	—	2	1	no	—
70	C₂×Dic₅₃	—	2	1	no	—
71	C₂×Dic₅₃	—	2	1	no	—
72	C₂×Dic₅₃	—	8	2	no	—
73	C₂×Dic₅₃	—	8	2	no	—
74	C₂×Dic₅₃	—	8	2	no	—
75	C₂×Dic₅₃	D₁₆	1	2	yes	—
76	C₂×Dic₅₃	Q₃₂	1	2	yes	—
77	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₈×C₅₃	C₄×Q₈	1	1	yes	characteristic
79	C₈×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
80	C₈×C₅₃	#125	1	1	yes	characteristic
81	#88322	—	2	2	no	—
82	#13165	—	53	1	no	maximal, Sylow
83	C₂²×C₅₃	C₄×D₈	1	1	yes	characteristic
84	C₂²×C₅₃	#466	1	1	yes	characteristic
85	C₂²×C₅₃	#500	1	1	yes	characteristic
86	C₂²×C₅₃	C₂×D₁₆	1	1	yes	characteristic
87	C₂²×C₅₃	C₂×Q₃₂	1	1	yes	characteristic
88	C₂²×C₅₃	#508	1	1	yes	characteristic
89	C₂²×C₅₃	#508	1	1	yes	characteristic
90	C₄×C₅₃	—	8	4	no	—
91	C₄×C₅₃	#389	1	1	yes	characteristic
92	C₄×C₅₃	C₄×Q₁₆	1	1	yes	characteristic
93	C₄×C₅₃	#467	1	1	yes	characteristic
94	C₄×C₅₃	#499	1	1	yes	characteristic
95	Dic₅₃	—	2	2	no	—
96	Dic₅₃	—	2	2	no	—
97	Dic₅₃	—	16	2	no	—
98	C₂×#366	—	53	1	no	—
99	C₂×#366	D₅₃	1	1	yes	characteristic
100	C₂×C₄×C₁₆	—	53	1	no	—
101	#1942	—	53	2	no	—
102	#1380	—	53	2	no	—
103	C₂×C₅₃	#1942	1	1	yes	characteristic
104	C₂×C₅₃	C₄×D₁₆	1	1	yes	characteristic
105	C₂×C₅₃	C₄×Q₃₂	1	1	yes	characteristic
106	C₂×C₅₃	#2215	1	1	yes	characteristic
107	C₂×C₅₃	#2216	1	1	yes	characteristic
108	C₂×C₅₃	#2252	1	1	yes	characteristic
109	C₂×C₅₃	#2253	1	1	yes	characteristic
110	C₂²×C₁₆	—	53	1	no	—
111	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
112	C₂×#107	—	53	2	no	—
113	C₂×#107	C₂×D₅₃	1	2	yes	—
114	C₂×C₄×C₈	—	53	1	no	—
115	#389	—	106	2	no	—
116	#366	—	106	2	no	—
117	#366	Dic₅₃	1	4	yes	complemented
118	C₄×C₁₆	—	106	2	no	—
119	#340	—	106	2	no	—
120	C₅₃	#13165	1	1	yes	characteristic, complemented, Sylow
121	C₂²×C₈	—	53	1	no	—
122	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
123	C₂×C₄⋊C₄	—	2	2	no	—
124	C₂×C₄⋊C₄	—	106	2	no	—
125	C₂×C₄⋊C₄	—	212	2	no	—
126	C₂×C₄²	—	53	1	no	—
127	C₂×C₁₆	—	53	2	no	—
128	C₂×C₁₆	—	106	1	no	—
129	C₂×C₁₆	—	106	1	no	—
130	C₂×C₁₆	Dic₁₀₆	1	2	yes	—
131	C₂×C₁₆	D₂₁₂	1	2	yes	—
132	C₂×C₁₆	C₂×Dic₅₃	1	2	yes	—
133	#107	—	2	2	no	—
134	#107	—	106	2	no	—
135	#107	—	106	2	no	—
136	#107	C₂×Dic₅₃	1	4	yes	—
137	C₄×C₈	—	106	1	no	—
138	C₄×C₈	—	106	1	no	—
139	#95	—	212	2	no	—
140	C₂²×C₄	—	4	2	no	—
141	C₂²×C₄	—	53	1	no	—
142	C₂²×C₄	—	53	1	no	—
143	C₂²×C₄	—	212	2	no	—
144	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
145	C₂×C₈	—	53	2	no	—
146	C₂×C₈	—	106	1	no	—
147	C₂×C₈	—	106	1	no	—
148	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
149	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
150	C₂×C₈	#11719947	1	1	yes	characteristic
151	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
152	C₂×C₈	#11719949	1	1	yes	characteristic
153	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
154	C₄⋊C₄	—	2	4	no	—
155	C₄⋊C₄	—	4	2	no	—
156	C₄⋊C₄	—	424	4	no	—
157	C₄⋊C₄	—	212	2	no	—
158	C₄⋊C₄	—	212	2	no	—
159	C₄²	—	106	1	no	—
160	C₄²	—	106	1	no	—
161	C₁₆	—	106	2	no	—
162	C₁₆	C₄⋊Dic₅₃	1	4	yes	complemented
163	C₂³	D₈×D₅₃	1	1	yes	characteristic
164	C₂×C₄	—	4	4	no	—
165	C₂×C₄	—	8	2	no	—
166	C₂×C₄	—	8	2	no	—
167	C₂×C₄	—	53	2	no	—
168	C₂×C₄	—	53	2	no	—
169	C₂×C₄	—	106	1	no	—
170	C₂×C₄	—	106	1	no	—
171	C₂×C₄	—	106	1	no	—
172	C₂×C₄	—	106	1	no	—
173	C₂×C₄	—	424	2	no	—
174	C₂×C₄	—	424	2	no	—
175	C₂×C₄	—	424	2	no	—
176	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
177	C₂×C₄	#13133192	1	1	yes	characteristic
178	C₂×C₄	#13133216	1	1	yes	characteristic
179	C₂×C₄	Q₁₆×D₅₃	1	1	yes	characteristic
180	C₂×C₄	#13133236	1	1	yes	characteristic
181	C₂×C₄	#13133240	1	1	yes	characteristic
182	C₈	—	106	2	no	—
183	C₈	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
184	C₈	Q₈×Dic₅₃	1	1	yes	characteristic
185	C₈	#13133197	1	1	yes	characteristic
186	C₈	#13133208	1	1	yes	characteristic
187	C₂²	D₈×Dic₅₃	1	1	yes	characteristic
188	C₂²	#16810860	1	1	yes	characteristic
189	C₂²	#16811027	1	1	yes	characteristic
190	C₂²	D₁₆×D₅₃	1	1	yes	characteristic
191	C₂²	Q₃₂×D₅₃	1	1	yes	characteristic
192	C₂²	#16811052	1	1	yes	characteristic
193	C₂²	#16811056	1	1	yes	characteristic
194	C₄	—	8	4	no	—
195	C₄	—	848	2	no	—
196	C₄	—	106	2	no	—
197	C₄	—	106	2	no	—
198	C₄	#16810522	1	1	yes	characteristic
199	C₄	Q₁₆×Dic₅₃	1	1	yes	characteristic
200	C₄	#16810865	1	1	yes	characteristic
201	C₄	#16811024	1	1	yes	characteristic
202	C₂	#20275449	1	1	yes	characteristic
203	C₂	D₁₆×Dic₅₃	1	1	yes	characteristic
204	C₂	Q₃₂×Dic₅₃	1	1	yes	characteristic
205	C₂	#20276802	1	1	yes	characteristic
206	C₂	#20276807	1	1	yes	characteristic
207	C₂	#20276989	1	1	yes	characteristic
208	C₂	#20276992	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₈×D₅₃
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#366	C₂
socle, prime radical	C₂³×C₅₃	C₂×D₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#366	D₅₃
hypocenter	C₅₃	#13165
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#366

Complements

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

N	H ≅ G/N	type
15. C₅₃×#366	C₄	semidirect
56. C₁₆×C₅₃	C₄⋊C₄	semidirect
117. #366	Dic₅₃	semidirect
120. C₅₃	#13165	semidirect
162. C₁₆	C₄⋊Dic₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810449	2
C₂×C₁₆×Dic₅₃	2
#20275449	2
#20272806	2
C₂×C₅₃×#366	2
#13165	53