Subgroups — #27300537

Order 13568.

350 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810451	C₂	1	2	yes	maximal
2	C₂×#16810449	C₂	1	2	yes	maximal
3	#20274879	C₂	1	1	yes	maximal, characteristic
4	#20274875	C₂	1	1	yes	maximal, characteristic
5	C₂×C₅₃×#346	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133151	—	2	1	no	—
7	C₂×#13133151	C₂²	1	1	yes	characteristic
8	C₂×#13133151	C₂²	1	2	yes	—
9	C₂×#13133151	C₂²	1	1	yes	characteristic
10	C₂×#13133149	—	2	1	no	—
11	#16810493	—	2	1	no	—
12	#16810489	—	2	1	no	—
13	#16810451	—	2	4	no	—
14	#16810449	—	2	4	no	—
15	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#346	C₄	1	4	yes	complemented
18	C₂×C₄⋊Dic₅₃	—	2	1	no	—
19	C₂×C₄⋊Dic₅₃	—	2	1	no	—
20	C₂×C₄⋊Dic₅₃	—	2	2	no	—
21	C₂×C₄⋊Dic₅₃	—	4	2	no	—
22	#13133151	—	2	2	no	—
23	#13133151	—	2	2	no	—
24	#13133151	—	2	2	no	—
25	#13133151	—	2	2	no	—
26	#13133151	—	4	2	no	—
27	#13133149	—	4	2	no	—
28	#13133124	—	4	2	no	—
29	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
31	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
32	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	—
33	C₂×C₁₆×C₅₃	D₄	1	4	yes	—
34	C₂×C₁₆×C₅₃	Q₈	1	4	yes	—
35	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
36	C₄×C₈×C₅₃	Q₈	1	2	yes	—
37	C₂²×C₄×C₅₃	D₈	1	2	yes	—
38	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
39	C₂²×Dic₅₃	—	4	1	no	—
40	C₂²×Dic₅₃	—	4	1	no	—
41	C₂²×Dic₅₃	—	4	2	no	—
42	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
43	C₂×C₈×C₅₃	D₈	1	2	yes	—
44	C₂×C₈×C₅₃	Q₁₆	1	2	yes	—
45	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
47	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
48	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
49	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
50	C₄²×C₅₃	Q₁₆	1	2	yes	—
51	C₄⋊Dic₅₃	—	4	1	no	—
52	C₄⋊Dic₅₃	—	4	2	no	—
53	C₄⋊Dic₅₃	—	4	1	no	—
54	C₄⋊Dic₅₃	—	4	1	no	—
55	C₄⋊Dic₅₃	—	4	1	no	—
56	C₄⋊Dic₅₃	—	4	2	no	—
57	C₄⋊Dic₅₃	—	8	4	no	—
58	C₁₆×C₅₃	C₄⋊C₄	1	8	yes	complemented
59	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
60	C₂×C₄×C₅₃	—	2	2	no	—
61	C₂×C₄×C₅₃	—	2	2	no	—
62	C₂×C₄×C₅₃	#107	1	4	yes	—
63	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
65	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
66	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
67	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
68	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
69	C₂×Dic₅₃	—	8	1	no	—
70	C₂×Dic₅₃	—	8	2	no	—
71	C₂×Dic₅₃	—	8	1	no	—
72	C₂×Dic₅₃	—	8	2	no	—
73	C₂×Dic₅₃	—	8	4	no	—
74	C₂×Dic₅₃	—	8	2	no	—
75	C₈×C₅₃	#107	1	4	yes	—
76	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
77	C₈×C₅₃	C₄⋊Q₈	1	2	yes	—
78	#12908	—	53	1	no	maximal, Sylow
79	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
80	C₂²×C₅₃	#493	1	1	yes	characteristic
81	C₂²×C₅₃	#500	1	1	yes	characteristic
82	C₂²×C₅₃	#509	1	2	yes	—
83	C₂²×C₅₃	#510	1	2	yes	—
84	C₄×C₅₃	—	2	4	no	—
85	C₄×C₅₃	#384	1	1	yes	characteristic
86	C₄×C₅₃	C₂×#107	1	1	yes	characteristic
87	C₄×C₅₃	#494	1	1	yes	characteristic
88	C₄×C₅₃	#500	1	1	yes	characteristic
89	Dic₅₃	—	16	1	no	—
90	Dic₅₃	—	16	2	no	—
91	Dic₅₃	—	16	1	no	—
92	C₂×#367	—	53	2	no	—
93	C₂×#366	—	53	2	no	—
94	C₂×#346	D₅₃	1	1	yes	characteristic
95	#1846	—	53	1	no	—
96	#1845	—	53	1	no	—
97	C₂×C₅₃	#1845	1	1	yes	characteristic
98	C₂×C₅₃	#2159	1	2	yes	—
99	C₂×C₅₃	#2250	1	1	yes	characteristic
100	C₂×C₅₃	#2251	1	1	yes	characteristic
101	C₂×C₅₃	#2255	1	2	yes	—
102	C₂²×C₁₆	C₂×D₅₃	1	2	yes	—
103	C₂×#107	—	53	2	no	—
104	C₂×#107	—	53	1	no	—
105	C₂×#107	—	53	1	no	—
106	C₂×#107	—	106	1	no	—
107	C₂×#106	—	106	1	no	—
108	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
109	#384	—	106	1	no	—
110	#382	—	106	1	no	—
111	#367	—	106	4	no	—
112	#366	—	106	4	no	—
113	#346	Dic₅₃	1	4	yes	complemented
114	C₅₃	#12908	1	1	yes	characteristic, complemented, Sylow
115	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
116	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
117	C₂×C₄⋊C₄	—	106	1	no	—
118	C₂×C₄⋊C₄	—	106	1	no	—
119	C₂×C₄⋊C₄	—	106	2	no	—
120	C₂×C₄⋊C₄	—	212	2	no	—
121	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
122	C₂×C₁₆	Dic₁₀₆	1	4	yes	—
123	C₂×C₁₆	D₂₁₂	1	4	yes	—
124	C₂×C₁₆	C₂×Dic₅₃	1	4	yes	—
125	#107	—	106	2	no	—
126	#107	—	106	2	no	—
127	#107	—	106	2	no	—
128	#107	—	106	2	no	—
129	#107	—	212	2	no	—
130	#106	—	212	2	no	—
131	C₄×C₈	Dic₁₀₆	1	2	yes	—
132	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
133	#95	—	212	2	no	—
134	C₂²×C₄	—	212	1	no	—
135	C₂²×C₄	—	212	1	no	—
136	C₂²×C₄	—	212	2	no	—
137	C₂²×C₄	D₄₂₄	1	2	yes	—
138	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
139	C₂×C₈	D₄₂₄	1	2	yes	—
140	C₂×C₈	Dic₂₁₂	1	2	yes	—
141	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
142	C₂×C₈	C₂×Dic₁₀₆	1	2	yes	—
143	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
144	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
145	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
146	C₄⋊C₄	—	424	4	no	—
147	C₄⋊C₄	—	212	1	no	—
148	C₄⋊C₄	—	212	2	no	—
149	C₄⋊C₄	—	212	1	no	—
150	C₄⋊C₄	—	212	1	no	—
151	C₄⋊C₄	—	212	1	no	—
152	C₄⋊C₄	—	212	2	no	—
153	C₄²	Dic₂₁₂	1	2	yes	—
154	C₄²	C₄⋊Dic₅₃	1	2	yes	—
155	C₁₆	C₄⋊Dic₅₃	1	8	yes	complemented
156	C₂³	C₂×D₄₂₄	1	1	yes	characteristic
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	424	4	no	—
160	C₂×C₄	—	424	2	no	—
161	C₂×C₄	—	424	2	no	—
162	C₂×C₄	—	424	1	no	—
163	C₂×C₄	—	424	2	no	—
164	C₂×C₄	—	424	1	no	—
165	C₂×C₄	#13133151	1	4	yes	—
166	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
167	C₂×C₄	#13133214	1	1	yes	characteristic
168	C₂×C₄	#13133217	1	1	yes	characteristic
169	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
170	C₂×C₄	C₂×Dic₂₁₂	1	1	yes	characteristic
171	C₂×C₄	C₂×Dic₂₁₂	1	1	yes	characteristic
172	C₈	#13133151	1	4	yes	—
173	C₈	C₂×C₄⋊Dic₅₃	1	2	yes	—
174	C₈	#13133217	1	2	yes	—
175	C₂²	C₂×#13133151	1	1	yes	characteristic
176	C₂²	#16811000	1	1	yes	characteristic
177	C₂²	#16811028	1	1	yes	characteristic
178	C₂²	#16811061	1	2	yes	—
179	C₂²	#16811066	1	2	yes	—
180	C₄	—	2	4	no	—
181	C₄	—	848	2	no	—
182	C₄	—	848	1	no	—
183	C₄	—	848	1	no	—
184	C₄	#16810493	1	1	yes	characteristic
185	C₄	C₂×#13133151	1	1	yes	characteristic
186	C₄	#16811003	1	1	yes	characteristic
187	C₄	#16811028	1	1	yes	characteristic
188	C₂	#20274875	1	1	yes	characteristic
189	C₂	#20276503	1	2	yes	—
190	C₂	#20276981	1	1	yes	characteristic
191	C₂	#20276986	1	1	yes	characteristic
192	C₂	#20277001	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄₂₄
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#346	C₂
socle, prime radical	C₂³×C₅₃	C₂×D₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#346	D₅₃
hypocenter	C₅₃	#12908
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
17. C₅₃×#346	C₄	semidirect
58. C₁₆×C₅₃	C₄⋊C₄	semidirect
113. #346	Dic₅₃	semidirect
114. C₅₃	#12908	semidirect
155. 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×#16810451	2
C₂×#16810449	2
#20274879	2
#20274875	2
C₂×C₅₃×#346	2
#12908	53