Subgroups — #27539580

Order 13568.

490 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20285229	C₂	1	1	yes	maximal, characteristic
2	#20276995	C₂	1	2	yes	maximal, complemented
3	#20276991	C₂	1	4	yes	maximal, complemented
4	#20276988	C₂	1	2	yes	maximal, complemented
5	#20276497	C₂	1	4	yes	maximal
6	C₂×C₄×#11719908	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#3160	C₂	1	1	yes	maximal, characteristic
8	#16811180	—	2	1	no	—
9	C₂²×#11719908	C₂²	1	2	yes	—
10	#16811028	C₂²	1	2	yes	—
11	#16811025	—	2	1	no	—
12	#16811025	C₂²	1	2	yes	—
13	#16811021	—	2	1	no	—
14	#16810661	—	2	1	no	—
15	#16810661	C₂²	1	2	yes	—
16	#16810450	C₂²	1	8	yes	—
17	#16810448	C₂²	1	8	yes	—
18	C₄×#11719908	C₂²	1	1	yes	characteristic
19	C₄×#11719908	C₂²	1	1	yes	characteristic
20	C₄×#11719908	C₂²	1	2	yes	—
21	C₅₃×#533	—	2	1	no	—
22	C₅₃×#500	C₂²	1	2	yes	—
23	C₅₃×#499	—	2	1	no	—
24	C₅₃×#499	C₂²	1	2	yes	—
25	C₅₃×#498	—	2	1	no	—
26	C₅₃×#427	—	2	1	no	—
27	C₅₃×#427	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
29	#13133217	—	2	2	no	—
30	#13133209	—	2	2	no	—
31	#13133196	—	4	2	no	—
32	C₄×Dic₁₀₆	—	4	2	no	—
33	#13133176	—	2	2	no	—
34	C₂×#11719908	Q₈	1	8	yes	complemented
35	C₂×#11719908	C₂³	1	2	yes	—
36	C₂×#11719908	C₂³	1	2	yes	—
37	#13133151	—	2	2	no	—
38	#13133151	C₂³	1	4	yes	—
39	#13133149	—	2	2	no	—
40	#13133149	—	2	2	no	—
41	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
43	C₄⋊Q₈×C₅₃	—	2	2	no	—
44	#125×C₅₃	—	2	2	no	—
45	C₂²⋊Q₈×C₅₃	—	4	2	no	—
46	C₄×Q₈×C₅₃	—	4	2	no	—
47	#117×C₅₃	—	2	2	no	—
48	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
49	#107×C₅₃	—	2	2	no	—
50	#107×C₅₃	C₂³	1	4	yes	—
51	#106×C₅₃	—	2	2	no	—
52	#106×C₅₃	—	2	2	no	—
53	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
54	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
55	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
56	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
57	C₂×Q₈×C₅₃	—	4	2	no	—
58	C₂²×C₄×C₅₃	D₈	1	2	yes	—
59	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
60	C₂×Dic₁₀₆	—	4	2	no	—
61	C₂×C₈×C₅₃	Q₁₆	1	4	yes	—
62	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
64	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
65	C₂×C₈×C₅₃	C₂×Q₈	1	4	yes	—
66	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
67	C₄⋊C₄×C₅₃	—	2	4	no	—
68	C₄⋊C₄×C₅₃	—	4	4	no	—
69	C₄⋊C₄×C₅₃	—	4	2	no	—
70	C₂²⋊C₄×C₅₃	—	4	2	no	—
71	C₄²×C₅₃	—	4	2	no	—
72	C₄²×C₅₃	D₈	1	2	yes	—
73	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
75	C₂²⋊Dic₅₃	—	4	2	no	—
76	C₄⋊Dic₅₃	—	2	4	no	—
77	C₄⋊Dic₅₃	—	4	2	no	—
78	#11719919	—	4	4	no	—
79	C₄×Dic₅₃	—	4	2	no	—
80	#11719908	C₂×Q₈	1	4	yes	—
81	#11719908	C₂×Q₈	1	4	yes	—
82	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
83	Q₈×C₅₃	—	8	4	no	—
84	C₂×C₄×C₅₃	—	2	4	no	—
85	C₂×C₄×C₅₃	—	4	4	no	—
86	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
87	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
88	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
89	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
90	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
91	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
92	C₂×Dic₅₃	—	4	4	no	—
93	Dic₁₀₆	—	8	4	no	—
94	C₈×C₅₃	C₄⋊Q₈	1	2	yes	—
95	C₈×C₅₃	C₂×Q₁₆	1	2	yes	—
96	C₈×C₅₃	C₂×Q₁₆	1	2	yes	—
97	C₈×C₅₃	C₂²×Q₈	1	2	yes	—
98	#33726	—	53	1	no	maximal, Sylow
99	C₂²×C₅₃	—	2	2	no	—
100	C₂²×C₅₃	#500	1	2	yes	—
101	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
102	C₄×C₅₃	—	2	2	no	—
103	C₄×C₅₃	—	8	4	no	—
104	C₄×C₅₃	#500	1	2	yes	—
105	C₄×C₅₃	#508	1	4	yes	—
106	C₄×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
107	C₄×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
108	Dic₅₃	—	8	4	no	—
109	#3160	—	53	1	no	—
110	#3160	D₅₃	1	1	yes	characteristic
111	#2254	—	53	2	no	—
112	#2253	—	53	4	no	—
113	#2252	—	53	2	no	—
114	#2158	—	53	4	no	—
115	C₂×C₄×C₁₆	—	53	1	no	—
116	C₂×C₅₃	—	2	2	no	—
117	C₂×C₅₃	C₂×#500	1	1	yes	characteristic
118	C₂×C₅₃	C₂×#508	1	2	yes	—
119	#533	—	2	1	no	—
120	#533	—	106	1	no	—
121	C₂²×C₁₆	—	53	2	no	—
122	#500	—	53	2	no	—
123	#500	C₂×D₅₃	1	2	yes	—
124	#499	—	2	1	no	—
125	#499	—	53	2	no	—
126	#499	—	106	1	no	—
127	#499	C₂×D₅₃	1	2	yes	—
128	#498	—	2	1	no	—
129	#498	—	106	1	no	—
130	#427	—	2	1	no	—
131	#427	—	53	2	no	—
132	#427	—	106	1	no	—
133	#427	C₂×D₅₃	1	2	yes	—
134	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
135	#367	—	53	8	no	—
136	#366	—	53	8	no	—
137	C₄×C₁₆	—	53	1	no	—
138	C₄×C₁₆	—	53	2	no	—
139	C₄×C₁₆	—	53	1	no	—
140	C₅₃	#33726	1	1	yes	characteristic, complemented, Sylow
141	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
142	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
143	C₄⋊Q₈	—	2	2	no	—
144	C₄⋊Q₈	—	106	2	no	—
145	#125	—	2	2	no	—
146	#125	—	106	2	no	—
147	C₂²⋊Q₈	—	4	2	no	—
148	C₂²⋊Q₈	—	212	2	no	—
149	C₄×Q₈	—	4	2	no	—
150	C₄×Q₈	—	212	2	no	—
151	#117	—	2	2	no	—
152	#117	—	106	2	no	—
153	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
154	C₂×C₁₆	—	53	8	no	—
155	C₂×C₁₆	—	53	2	no	—
156	C₂×C₁₆	—	53	2	no	—
157	#107	—	2	2	no	—
158	#107	—	53	4	no	—
159	#107	—	106	2	no	—
160	#107	C₂²×D₅₃	1	4	yes	—
161	#106	—	2	2	no	—
162	#106	—	2	2	no	—
163	#106	—	106	2	no	—
164	#106	—	106	2	no	—
165	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
166	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
167	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
168	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
169	C₂×Q₈	—	4	2	no	—
170	C₂×Q₈	—	212	2	no	—
171	C₂²×C₄	#11719922	1	2	yes	—
172	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
173	C₂×C₈	#11719925	1	4	yes	—
174	C₂×C₈	Q₈×D₅₃	1	4	yes	—
175	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
176	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
177	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
178	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
179	C₄⋊C₄	—	2	4	no	—
180	C₄⋊C₄	—	4	4	no	—
181	C₄⋊C₄	—	4	2	no	—
182	C₄⋊C₄	—	106	4	no	—
183	C₄⋊C₄	—	212	4	no	—
184	C₄⋊C₄	—	212	2	no	—
185	C₂²⋊C₄	—	4	2	no	—
186	C₂²⋊C₄	—	212	2	no	—
187	C₄²	—	4	2	no	—
188	C₄²	—	212	2	no	—
189	C₄²	#11719922	1	2	yes	—
190	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
191	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
192	C₁₆	—	53	4	no	—
193	C₁₆	—	53	4	no	—
194	C₂³	C₂×#11719922	1	1	yes	characteristic
195	Q₈	—	8	4	no	—
196	Q₈	—	424	4	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	4	4	no	—
199	C₂×C₄	—	212	4	no	—
200	C₂×C₄	#13133215	1	2	yes	—
201	C₂×C₄	C₂×#11719922	1	2	yes	—
202	C₂×C₄	C₂×#11719922	1	2	yes	—
203	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
204	C₂×C₄	C₂×#11719925	1	2	yes	—
205	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
206	C₈	#13133215	1	2	yes	—
207	C₈	C₂×#11719925	1	2	yes	—
208	C₈	C₂×#11719925	1	2	yes	—
209	C₈	C₂×Q₈×D₅₃	1	2	yes	—
210	C₂²	—	2	2	no	—
211	C₂²	#16811026	1	2	yes	—
212	C₂²	C₂²×#11719922	1	1	yes	characteristic
213	C₄	—	2	2	no	—
214	C₄	—	8	4	no	—
215	C₄	—	424	4	no	—
216	C₄	#16811026	1	2	yes	—
217	C₄	#16811053	1	4	yes	—
218	C₄	C₂×#13133215	1	1	yes	characteristic
219	C₄	C₂²×#11719925	1	1	yes	characteristic
220	C₂	—	2	2	no	—
221	C₂	C₂×#16811026	1	1	yes	characteristic
222	C₂	C₂×#16811053	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#11719922
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#3160	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂³×C₅₃	C₂×D₈
hypercenter	#3160	D₅₃
hypocenter	C₅₃	#33726
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3160

Complements

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

N	H ≅ G/N	type
2. #20276995	C₂	semidirect
3. #20276991	C₂	semidirect
4. #20276988	C₂	semidirect
34. C₂×#11719908	Q₈	semidirect
140. C₅₃	#33726	semidirect

Maximal subgroups

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

subgroup	index
#20285229	2
#20276995	2
#20276991	2
#20276988	2
#20276497	2
C₂×C₄×#11719908	2
C₅₃×#3160	2
#33726	53