Subgroups — #27536671

Order 13568.

862 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×#16811028	C₂	1	1	yes	maximal, characteristic
2	C₂×#16811019	C₂	1	2	yes	maximal
3	#20283092	C₂	1	2	yes	maximal
4	#20274878	C₂	1	8	yes	maximal
5	C₄×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₅₃×#500	C₂	1	1	yes	maximal, characteristic
7	C₂×#13133217	C₂²	1	2	yes	—
8	#16811028	—	2	4	no	—
9	#16811019	—	2	8	no	—
10	C₂×#13133151	C₂²	1	4	yes	—
11	C₂×#13133148	C₂²	1	8	yes	—
12	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
13	C₂×C₈×Dic₅₃	C₂²	1	4	yes	—
14	#16810530	—	2	4	no	—
15	#16810528	—	2	4	no	—
16	#16810488	C₂²	1	8	yes	—
17	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
19	C₅₃×#500	—	2	4	no	—
20	C₂×#107×C₅₃	C₂²	1	4	yes	—
21	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₂²×Dic₁₀₆	—	2	2	no	—
23	C₂²×#88322	C₂³	1	2	yes	—
24	#13133217	—	2	4	no	—
25	#13133217	—	2	4	no	—
26	C₂×C₄⋊Dic₅₃	—	2	2	no	—
27	C₂×C₄⋊Dic₅₃	C₂³	1	4	yes	—
28	C₂×C₄×Dic₅₃	D₄	1	2	yes	—
29	C₂×C₄×Dic₅₃	D₄	1	2	yes	—
30	C₂×C₄×Dic₅₃	C₂³	1	2	yes	—
31	#13133151	—	2	8	no	—
32	#13133148	—	2	16	no	—
33	C₄×#88322	D₄	1	4	yes	—
34	C₈×Dic₅₃	Q₈	1	16	yes	complemented
35	#13133124	—	2	16	no	—
36	C₂²×Q₈×C₅₃	—	2	2	no	—
37	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
38	C₄⋊Q₈×C₅₃	—	2	4	no	—
39	C₄⋊Q₈×C₅₃	—	2	4	no	—
40	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
41	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
42	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
43	#107×C₅₃	—	2	8	no	—
44	C₄×C₈×C₅₃	D₄	1	4	yes	—
45	C₂×Q₈×C₅₃	—	4	4	no	—
46	C₂×Q₈×C₅₃	—	4	8	no	—
47	C₂²×C₄×C₅₃	—	2	4	no	—
48	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
51	C₂²×Dic₅₃	—	2	4	no	—
52	C₂²×Dic₅₃	C₂×D₄	1	4	yes	—
53	C₂×Dic₁₀₆	—	4	4	no	—
54	C₂×Dic₁₀₆	—	4	8	no	—
55	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
56	C₂×C₈×C₅₃	C₂×Q₈	1	8	yes	—
57	C₄⋊C₄×C₅₃	—	2	4	no	—
58	C₄⋊C₄×C₅₃	—	2	4	no	—
59	C₄⋊C₄×C₅₃	—	4	4	no	—
60	C₄²×C₅₃	C₂×D₄	1	2	yes	—
61	C₄²×C₅₃	C₂×D₄	1	2	yes	—
62	C₄⋊Dic₅₃	—	2	4	no	—
63	C₄⋊Dic₅₃	—	2	4	no	—
64	C₄⋊Dic₅₃	—	4	4	no	—
65	C₄×Dic₅₃	—	2	4	no	—
66	C₄×Dic₅₃	—	2	4	no	—
67	C₄×Dic₅₃	C₂×Q₈	1	8	yes	—
68	C₂×#88322	C₂×D₄	1	4	yes	—
69	C₂×#88322	C₂×Q₈	1	8	yes	—
70	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
71	Q₈×C₅₃	—	8	8	no	—
72	C₂×C₄×C₅₃	—	4	4	no	—
73	C₂×C₄×C₅₃	—	4	8	no	—
74	C₂×C₄×C₅₃	#127	1	2	yes	—
75	C₂×C₄×C₅₃	#127	1	2	yes	—
76	C₂×C₄×C₅₃	#135	1	4	yes	—
77	C₂×C₄×C₅₃	#135	1	4	yes	—
78	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
79	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
80	C₂×C₄×C₅₃	C₂²×Q₈	1	4	yes	—
81	C₂×Dic₅₃	—	2	4	no	—
82	C₂×Dic₅₃	—	2	4	no	—
83	C₂×Dic₅₃	—	4	4	no	—
84	C₂×Dic₅₃	—	4	8	no	—
85	C₂×Dic₅₃	C₄⋊Q₈	1	8	yes	—
86	Dic₁₀₆	—	8	8	no	—
87	C₈×C₅₃	C₄⋊Q₈	1	8	yes	—
88	#88322	C₄⋊Q₈	1	8	yes	—
89	#33313	—	53	1	no	maximal, Sylow
90	C₂²×C₅₃	C₂×#127	1	1	yes	characteristic
91	C₂²×C₅₃	C₂×C₄⋊Q₈	1	2	yes	—
92	C₂²×C₅₃	C₂×#135	1	2	yes	—
93	C₂²×C₅₃	C₂×#135	1	2	yes	—
94	C₄×C₅₃	—	8	4	no	—
95	C₄×C₅₃	#495	1	4	yes	—
96	C₄×C₅₃	#495	1	4	yes	—
97	C₄×C₅₃	C₂×C₄⋊Q₈	1	2	yes	—
98	C₄×C₅₃	C₂×C₄⋊Q₈	1	2	yes	—
99	Dic₅₃	—	2	8	no	—
100	Dic₅₃	—	8	4	no	—
101	C₂×#500	—	53	1	no	—
102	C₂×#500	D₅₃	1	1	yes	characteristic
103	C₂×#498	—	53	2	no	—
104	#2862	—	53	2	no	—
105	#1846	—	53	8	no	—
106	C₄²×C₈	—	53	1	no	—
107	C₂×C₅₃	#2868	1	1	yes	characteristic
108	C₂×C₅₃	C₂×#495	1	1	yes	characteristic
109	C₂×C₅₃	C₂×#495	1	1	yes	characteristic
110	C₂×C₅₃	#3160	1	4	yes	—
111	C₂×C₄⋊Q₈	—	53	2	no	—
112	C₂×C₄⋊Q₈	C₂×D₅₃	1	2	yes	—
113	#500	—	2	4	no	—
114	#500	—	106	4	no	—
115	#498	—	106	8	no	—
116	C₂×#107	—	53	4	no	—
117	C₂×#107	C₂×D₅₃	1	4	yes	—
118	C₂×#106	—	53	8	no	—
119	C₂×C₄×C₈	—	53	4	no	—
120	C₂×C₄×C₈	—	53	1	no	—
121	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
122	#391	—	106	4	no	—
123	#391	—	106	4	no	—
124	#382	—	53	8	no	—
125	C₄³	—	53	1	no	—
126	C₅₃	#33313	1	1	yes	characteristic, complemented, Sylow
127	C₂²×Q₈	—	2	2	no	—
128	C₂²×Q₈	—	106	2	no	—
129	C₂²×C₈	—	53	2	no	—
130	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
131	C₄⋊Q₈	—	2	4	no	—
132	C₄⋊Q₈	—	2	4	no	—
133	C₄⋊Q₈	—	106	4	no	—
134	C₄⋊Q₈	—	106	4	no	—
135	C₂×C₄⋊C₄	—	2	2	no	—
136	C₂×C₄⋊C₄	—	53	4	no	—
137	C₂×C₄⋊C₄	—	106	2	no	—
138	C₂×C₄⋊C₄	C₂²×D₅₃	1	4	yes	—
139	C₂×C₄²	—	53	2	no	—
140	C₂×C₄²	—	53	2	no	—
141	C₂×C₄²	—	53	2	no	—
142	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
143	#107	—	2	8	no	—
144	#107	—	106	8	no	—
145	#106	—	106	16	no	—
146	C₄×C₈	—	53	16	no	—
147	C₄×C₈	—	53	4	no	—
148	C₄×C₈	C₂²⋊D₅₃	1	4	yes	—
149	#95	—	106	16	no	—
150	C₂×Q₈	—	4	4	no	—
151	C₂×Q₈	—	4	8	no	—
152	C₂×Q₈	—	212	4	no	—
153	C₂×Q₈	—	212	8	no	—
154	C₂²×C₄	—	2	4	no	—
155	C₂²×C₄	—	53	4	no	—
156	C₂²×C₄	—	106	4	no	—
157	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
158	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
159	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
160	C₂×C₈	—	53	4	no	—
161	C₂×C₈	—	53	8	no	—
162	C₂×C₈	Q₈×D₅₃	1	8	yes	—
163	C₂×C₈	C₂×C₂²⋊D₅₃	1	4	yes	—
164	C₄⋊C₄	—	2	4	no	—
165	C₄⋊C₄	—	2	4	no	—
166	C₄⋊C₄	—	4	4	no	—
167	C₄⋊C₄	—	106	4	no	—
168	C₄⋊C₄	—	106	4	no	—
169	C₄⋊C₄	—	212	4	no	—
170	C₄²	—	53	8	no	—
171	C₄²	—	106	4	no	—
172	C₄²	—	106	4	no	—
173	C₄²	D₄×D₅₃	1	2	yes	—
174	C₄²	C₂×C₂²⋊D₅₃	1	2	yes	—
175	C₂³	C₂×D₄×D₅₃	1	1	yes	characteristic
176	Q₈	—	8	8	no	—
177	Q₈	—	424	8	no	—
178	C₂×C₄	—	4	8	no	—
179	C₂×C₄	—	4	4	no	—
180	C₂×C₄	—	53	8	no	—
181	C₂×C₄	—	106	4	no	—
182	C₂×C₄	—	106	4	no	—
183	C₂×C₄	—	212	8	no	—
184	C₂×C₄	—	212	4	no	—
185	C₂×C₄	#13133213	1	2	yes	—
186	C₂×C₄	#13133213	1	2	yes	—
187	C₂×C₄	#13133236	1	4	yes	—
188	C₂×C₄	#13133240	1	4	yes	—
189	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
190	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
191	C₂×C₄	C₂×Q₈×D₅₃	1	4	yes	—
192	C₈	—	53	8	no	—
193	C₈	#13133215	1	8	yes	—
194	C₂²	C₂×#13133213	1	1	yes	characteristic
195	C₂²	C₄⋊Q₈×D₅₃	1	2	yes	—
196	C₂²	C₂×#13133236	1	2	yes	—
197	C₂²	C₂×#13133240	1	2	yes	—
198	C₄	—	8	4	no	—
199	C₄	—	106	8	no	—
200	C₄	—	424	4	no	—
201	C₄	#16811006	1	4	yes	—
202	C₄	#16811008	1	4	yes	—
203	C₄	C₄⋊Q₈×D₅₃	1	2	yes	—
204	C₄	C₂×#13133215	1	2	yes	—
205	C₂	#20283111	1	1	yes	characteristic
206	C₂	C₂×#16811006	1	1	yes	characteristic
207	C₂	C₂×#16811008	1	1	yes	characteristic
208	C₂	#20285226	1	4	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₅₃
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₅₃×#500	C₂
socle, prime radical	C₂³×C₅₃	C₂²×D₄
hypercenter	C₂×#500	D₅₃
hypocenter	C₅₃	#33313
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
34. C₈×Dic₅₃	Q₈	semidirect
126. C₅₃	#33313	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811028	2
C₂×#16811019	2
#20283092	2
#20274878	2
C₄×C₈×Dic₅₃	2
C₂×C₅₃×#500	2
#33313	53