Subgroups — #27539592

Order 13568.

482 conjugacy classes of subgroups.

The subgroup lattice has 216 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	#20277002	C₂	1	4	yes	maximal, complemented
3	#20276998	C₂	1	4	yes	maximal, complemented
4	#20276497	C₂	1	4	yes	maximal
5	C₂×#16810399	C₂	1	1	yes	maximal, characteristic
6	C₅₃×#3160	C₂	1	1	yes	maximal, characteristic
7	#16811180	—	2	1	no	—
8	C₂²×#11719908	C₂²	1	2	yes	—
9	#16811028	C₂²	1	2	yes	—
10	#16811025	—	2	1	no	—
11	#16811025	C₂²	1	2	yes	—
12	#16811021	—	2	1	no	—
13	#16810661	—	2	1	no	—
14	#16810661	C₂²	1	2	yes	—
15	#16810450	C₂²	1	8	yes	—
16	#16810448	C₂²	1	8	yes	—
17	#16810399	C₂²	1	2	yes	—
18	#16810399	C₂²	1	2	yes	—
19	C₅₃×#533	—	2	1	no	—
20	C₅₃×#500	C₂²	1	2	yes	—
21	C₅₃×#499	—	2	1	no	—
22	C₅₃×#499	C₂²	1	2	yes	—
23	C₅₃×#498	—	2	1	no	—
24	C₅₃×#427	—	2	1	no	—
25	C₅₃×#427	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
27	#13133217	—	2	2	no	—
28	#13133209	—	2	2	no	—
29	#13133196	—	4	2	no	—
30	C₄×Dic₁₀₆	—	4	2	no	—
31	#13133176	—	2	2	no	—
32	C₂×#11719908	Q₈	1	8	yes	complemented
33	C₂×#11719908	C₂³	1	4	yes	—
34	#13133151	—	2	2	no	—
35	#13133151	C₂³	1	4	yes	—
36	#13133149	—	2	2	no	—
37	#13133149	—	2	2	no	—
38	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
40	C₄⋊Q₈×C₅₃	—	2	2	no	—
41	#125×C₅₃	—	2	2	no	—
42	C₂²⋊Q₈×C₅₃	—	4	2	no	—
43	C₄×Q₈×C₅₃	—	4	2	no	—
44	#117×C₅₃	—	2	2	no	—
45	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
46	#107×C₅₃	—	2	2	no	—
47	#107×C₅₃	C₂³	1	4	yes	—
48	#106×C₅₃	—	2	2	no	—
49	#106×C₅₃	—	2	2	no	—
50	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
51	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
52	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
53	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
54	C₂×Q₈×C₅₃	—	4	2	no	—
55	C₂²×C₄×C₅₃	D₈	1	2	yes	—
56	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
57	C₂×Dic₁₀₆	—	4	2	no	—
58	C₂×C₈×C₅₃	Q₁₆	1	4	yes	—
59	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₅₃	C₂×Q₈	1	4	yes	—
63	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
64	C₄⋊C₄×C₅₃	—	2	4	no	—
65	C₄⋊C₄×C₅₃	—	4	4	no	—
66	C₄⋊C₄×C₅₃	—	4	2	no	—
67	C₂²⋊C₄×C₅₃	—	4	2	no	—
68	C₄²×C₅₃	—	4	2	no	—
69	C₄²×C₅₃	D₈	1	2	yes	—
70	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
71	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
72	C₂²⋊Dic₅₃	—	4	2	no	—
73	C₄⋊Dic₅₃	—	2	4	no	—
74	C₄⋊Dic₅₃	—	4	2	no	—
75	#11719919	—	4	4	no	—
76	C₄×Dic₅₃	—	4	2	no	—
77	#11719908	C₂×Q₈	1	8	yes	—
78	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
79	Q₈×C₅₃	—	8	4	no	—
80	C₂×C₄×C₅₃	—	4	2	no	—
81	C₂×C₄×C₅₃	—	4	4	no	—
82	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
83	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
84	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
85	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
86	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
87	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
88	C₂×Dic₅₃	—	4	4	no	—
89	Dic₁₀₆	—	8	4	no	—
90	C₈×C₅₃	C₄⋊Q₈	1	2	yes	—
91	C₈×C₅₃	C₂×Q₁₆	1	2	yes	—
92	C₈×C₅₃	C₂×Q₁₆	1	2	yes	—
93	C₈×C₅₃	C₂²×Q₈	1	2	yes	—
94	#33728	—	53	1	no	maximal, Sylow
95	C₂²×C₅₃	—	2	2	no	—
96	C₂²×C₅₃	#500	1	2	yes	—
97	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
98	C₄×C₅₃	—	2	2	no	—
99	C₄×C₅₃	—	2	2	no	—
100	C₄×C₅₃	—	8	4	no	—
101	C₄×C₅₃	#500	1	2	yes	—
102	C₄×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
103	C₄×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
104	Dic₅₃	—	8	4	no	—
105	#3160	—	53	1	no	—
106	#3160	D₅₃	1	1	yes	characteristic
107	#2255	—	53	4	no	—
108	#2255	—	53	4	no	—
109	#2158	—	53	4	no	—
110	C₂×#346	—	53	1	no	—
111	C₂×C₅₃	—	2	2	no	—
112	C₂×C₅₃	C₂×#500	1	1	yes	characteristic
113	C₂×C₅₃	#3414	1	2	yes	—
114	#533	—	2	1	no	—
115	#533	—	106	1	no	—
116	C₂²×C₁₆	—	53	2	no	—
117	#500	—	53	2	no	—
118	#500	C₂×D₅₃	1	2	yes	—
119	#499	—	2	1	no	—
120	#499	—	53	2	no	—
121	#499	—	106	1	no	—
122	#499	C₂×D₅₃	1	2	yes	—
123	#498	—	2	1	no	—
124	#498	—	106	1	no	—
125	#427	—	2	1	no	—
126	#427	—	53	2	no	—
127	#427	—	106	1	no	—
128	#427	C₂×D₅₃	1	2	yes	—
129	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
130	#367	—	53	8	no	—
131	#366	—	53	8	no	—
132	#346	—	53	2	no	—
133	#346	—	53	2	no	—
134	C₅₃	#33728	1	1	yes	characteristic, complemented, Sylow
135	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
136	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
137	C₄⋊Q₈	—	2	2	no	—
138	C₄⋊Q₈	—	106	2	no	—
139	#125	—	2	2	no	—
140	#125	—	106	2	no	—
141	C₂²⋊Q₈	—	4	2	no	—
142	C₂²⋊Q₈	—	212	2	no	—
143	C₄×Q₈	—	4	2	no	—
144	C₄×Q₈	—	212	2	no	—
145	#117	—	2	2	no	—
146	#117	—	106	2	no	—
147	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
148	C₂×C₁₆	—	53	8	no	—
149	C₂×C₁₆	—	53	4	no	—
150	#107	—	2	2	no	—
151	#107	—	53	4	no	—
152	#107	—	106	2	no	—
153	#107	C₂²×D₅₃	1	4	yes	—
154	#106	—	2	2	no	—
155	#106	—	2	2	no	—
156	#106	—	106	2	no	—
157	#106	—	106	2	no	—
158	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
159	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
160	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
161	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
162	C₂×Q₈	—	4	2	no	—
163	C₂×Q₈	—	212	2	no	—
164	C₂²×C₄	#11719922	1	2	yes	—
165	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
166	C₂×C₈	#11719925	1	4	yes	—
167	C₂×C₈	Q₈×D₅₃	1	4	yes	—
168	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
169	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
170	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
171	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
172	C₄⋊C₄	—	2	4	no	—
173	C₄⋊C₄	—	4	4	no	—
174	C₄⋊C₄	—	4	2	no	—
175	C₄⋊C₄	—	106	4	no	—
176	C₄⋊C₄	—	212	4	no	—
177	C₄⋊C₄	—	212	2	no	—
178	C₂²⋊C₄	—	4	2	no	—
179	C₂²⋊C₄	—	212	2	no	—
180	C₄²	—	4	2	no	—
181	C₄²	—	212	2	no	—
182	C₄²	#11719922	1	2	yes	—
183	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
184	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
185	C₁₆	—	53	8	no	—
186	C₂³	C₂×#11719922	1	1	yes	characteristic
187	Q₈	—	8	4	no	—
188	Q₈	—	424	4	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	—	4	4	no	—
191	C₂×C₄	—	212	4	no	—
192	C₂×C₄	#13133215	1	2	yes	—
193	C₂×C₄	C₂×#11719922	1	2	yes	—
194	C₂×C₄	C₂×#11719922	1	2	yes	—
195	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
196	C₂×C₄	C₂×#11719925	1	2	yes	—
197	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
198	C₈	#13133215	1	2	yes	—
199	C₈	C₂×#11719925	1	2	yes	—
200	C₈	C₂×#11719925	1	2	yes	—
201	C₈	C₂×Q₈×D₅₃	1	2	yes	—
202	C₂²	—	2	2	no	—
203	C₂²	#16811026	1	2	yes	—
204	C₂²	C₂²×#11719922	1	1	yes	characteristic
205	C₄	—	2	2	no	—
206	C₄	—	2	2	no	—
207	C₄	—	8	4	no	—
208	C₄	—	424	4	no	—
209	C₄	#16811026	1	2	yes	—
210	C₄	C₂×#13133215	1	1	yes	characteristic
211	C₄	C₂²×#11719925	1	1	yes	characteristic
212	C₂	—	2	2	no	—
213	C₂	C₂×#16811026	1	1	yes	characteristic
214	C₂	#20287849	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	#33728
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. #20277002	C₂	semidirect
3. #20276998	C₂	semidirect
32. C₂×#11719908	Q₈	semidirect
134. C₅₃	#33728	semidirect

Maximal subgroups

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

subgroup	index
#20285229	2
#20277002	2
#20276998	2
#20276497	2
C₂×#16810399	2
C₅₃×#3160	2
#33728	53