Subgroups — #27484886

Order 13568.

404 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20276293	C₂	1	4	yes	maximal
2	#20275505	C₂	1	2	yes	maximal
3	#20275503	C₂	1	2	yes	maximal
4	#20274890	C₂	1	1	yes	maximal, characteristic
5	#20274839	C₂	1	2	yes	maximal
6	#20274838	C₂	1	2	yes	maximal
7	#20274599	C₂	1	1	yes	maximal, characteristic
8	C₅₃×#1773	C₂	1	1	yes	maximal, characteristic
9	#16810671	C₂²	1	4	yes	—
10	C₂×#13133153	C₂²	1	2	yes	—
11	C₂×#13133153	C₂²	1	1	yes	characteristic
12	C₂×#13133153	C₂²	1	2	yes	—
13	#16810661	C₂²	1	1	yes	characteristic
14	#16810647	C₂²	1	2	yes	—
15	#16810578	C₂²	1	2	yes	—
16	#16810394	C₂²	1	4	yes	—
17	#16810393	C₂²	1	4	yes	—
18	#16810383	C₂²	1	2	yes	—
19	#16810379	C₂²	1	4	yes	—
20	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
21	C₅₃×#427	C₂²	1	1	yes	characteristic
22	C₅₃×#405	C₂²	1	2	yes	—
23	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
24	C₅₃×#339	C₂²	1	2	yes	—
25	C₂×#11719917	C₂³	1	2	yes	—
26	C₂×#11719917	C₂³	1	4	yes	—
27	#13133176	C₂³	1	2	yes	—
28	#13133153	—	2	2	no	—
29	#13133153	—	2	2	no	—
30	#13133153	—	2	2	no	—
31	#13133153	—	2	2	no	—
32	#13133153	—	2	2	no	—
33	#13133152	—	4	4	no	—
34	#13133151	—	2	1	no	—
35	#13133149	—	2	1	no	—
36	C₄⋊#88322	—	4	2	no	—
37	#13133127	—	2	2	no	—
38	C₈×Dic₅₃	—	2	2	no	—
39	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
40	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
41	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
42	#117×C₅₃	C₂³	1	2	yes	—
43	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
44	#108×C₅₃	—	2	2	no	—
45	#107×C₅₃	—	2	1	no	—
46	#106×C₅₃	—	2	1	no	—
47	#97×C₅₃	—	2	2	no	—
48	#97×C₅₃	Q₈	1	4	yes	—
49	C₄×C₈×C₅₃	—	2	2	no	—
50	C₂²×C₄×C₅₃	—	2	1	no	—
51	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
52	M₁₆×C₅₃	—	2	2	no	—
53	M₁₆×C₅₃	—	2	2	no	—
54	C₂×C₈×C₅₃	—	2	2	no	—
55	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
56	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
57	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
58	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
59	C₂×C₈×C₅₃	#41	1	2	yes	—
60	C₂×C₈×C₅₃	#41	1	2	yes	—
61	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
62	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
63	C₄⋊C₄×C₅₃	—	2	2	no	—
64	C₂²⋊C₄×C₅₃	—	2	2	no	—
65	C₄²×C₅₃	—	2	1	no	—
66	C₄²×C₅₃	—	2	2	no	—
67	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
68	C₂²⋊Dic₅₃	—	2	2	no	—
69	C₄⋊Dic₅₃	—	2	2	no	—
70	C₄×Dic₅₃	—	2	2	no	—
71	#11719917	—	2	2	no	—
72	#11719917	—	2	4	no	—
73	#11719917	—	2	4	no	—
74	#11719917	—	2	2	no	—
75	C₂×#88322	—	2	2	no	—
76	C₂×#88322	—	2	4	no	—
77	C₂³×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
78	C₂×C₄×C₅₃	—	2	2	no	—
79	C₂×C₄×C₅₃	—	4	2	no	—
80	C₂×C₄×C₅₃	—	8	1	no	—
81	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
82	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
83	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
84	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
85	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
86	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
87	C₂×Dic₅₃	—	4	2	no	—
88	C₈×C₅₃	—	4	2	no	—
89	C₈×C₅₃	#125	1	2	yes	—
90	C₈×C₅₃	#125	1	2	yes	—
91	C₈×C₅₃	#125	1	2	yes	—
92	C₈×C₅₃	#125	1	2	yes	—
93	#88322	—	4	2	no	—
94	#88322	—	4	4	no	—
95	#29231	—	53	1	no	maximal, Sylow
96	C₂²×C₅₃	—	4	1	no	—
97	C₂²×C₅₃	#544	1	1	yes	characteristic
98	C₂²×C₅₃	#556	1	1	yes	characteristic
99	C₂²×C₅₃	#556	1	1	yes	characteristic
100	C₄×C₅₃	—	4	2	no	—
101	C₄×C₅₃	—	8	2	no	—
102	C₄×C₅₃	C₂×#125	1	1	yes	characteristic
103	C₄×C₅₃	C₂×#125	1	1	yes	characteristic
104	C₄×C₅₃	#555	1	1	yes	characteristic
105	C₄×C₅₃	#559	1	1	yes	characteristic
106	Dic₅₃	—	8	2	no	—
107	#2102	—	53	4	no	—
108	#1953	—	53	2	no	—
109	#1953	—	53	2	no	—
110	#1849	—	53	1	no	—
111	#1833	—	53	2	no	—
112	#1833	—	53	2	no	—
113	#1773	—	53	1	no	—
114	#1773	D₅₃	1	1	yes	characteristic
115	C₂×C₅₃	—	2	1	no	—
116	C₂×C₅₃	—	2	1	no	—
117	C₂×C₅₃	—	2	1	no	—
118	C₂×C₅₃	#2571	1	1	yes	characteristic
119	#430	—	53	4	no	—
120	C₂×#108	—	53	2	no	—
121	C₂×#108	—	53	2	no	—
122	C₂×#108	—	53	1	no	—
123	C₂×#108	C₂×D₅₃	1	1	yes	characteristic
124	#427	—	53	1	no	—
125	#427	C₂×D₅₃	1	1	yes	characteristic
126	#423	—	53	2	no	—
127	#405	—	53	2	no	—
128	#405	C₂×D₅₃	1	2	yes	—
129	C₂×#97	C₂×D₅₃	1	1	yes	characteristic
130	#344	—	53	4	no	—
131	#344	—	53	4	no	—
132	#339	—	53	2	no	—
133	#339	C₂×D₅₃	1	2	yes	—
134	H(3,ℤ/4)	—	53	4	no	—
135	C₅₃	#29231	1	1	yes	characteristic, complemented, Sylow
136	C₂×M₁₆	—	53	4	no	—
137	C₂×M₁₆	—	53	2	no	—
138	C₂×M₁₆	C₂²×D₅₃	1	2	yes	—
139	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
140	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
141	#117	—	53	2	no	—
142	#117	C₂²×D₅₃	1	2	yes	—
143	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
144	#108	—	2	2	no	—
145	#108	—	106	2	no	—
146	#108	—	106	2	no	—
147	#108	—	106	2	no	—
148	#108	—	106	2	no	—
149	#108	—	106	2	no	—
150	#108	—	212	4	no	—
151	#107	—	2	1	no	—
152	#107	—	106	1	no	—
153	#106	—	2	1	no	—
154	#106	—	106	1	no	—
155	C₄⋊C₈	—	212	2	no	—
156	#97	—	2	2	no	—
157	#97	—	106	2	no	—
158	#97	Dic₁₀₆	1	4	yes	—
159	C₄×C₈	—	2	2	no	—
160	C₄×C₈	—	106	2	no	—
161	C₂²×C₄	—	2	1	no	—
162	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
163	M₁₆	—	2	2	no	—
164	M₁₆	—	2	2	no	—
165	M₁₆	—	106	2	no	—
166	M₁₆	—	106	4	no	—
167	M₁₆	—	106	4	no	—
168	M₁₆	—	106	2	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	106	2	no	—
171	C₂×C₈	—	106	4	no	—
172	C₂×C₈	C₂×Dic₁₀₆	1	2	yes	—
173	C₂×C₈	C₂×Dic₁₀₆	1	2	yes	—
174	C₂×C₈	#11719945	1	2	yes	—
175	C₂×C₈	#11719945	1	1	yes	characteristic
176	C₂×C₈	#11719945	1	1	yes	characteristic
177	C₂×C₈	#11719947	1	2	yes	—
178	C₂×C₈	#11719949	1	1	yes	characteristic
179	C₂×C₈	#11719949	1	1	yes	characteristic
180	C₄⋊C₄	—	2	2	no	—
181	C₄⋊C₄	—	106	2	no	—
182	C₂²⋊C₄	—	2	2	no	—
183	C₂²⋊C₄	—	106	2	no	—
184	C₄²	—	2	1	no	—
185	C₄²	—	2	2	no	—
186	C₄²	—	106	2	no	—
187	C₄²	C₂×Dic₁₀₆	1	2	yes	—
188	C₂³	#13133264	1	1	yes	characteristic
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	4	2	no	—
191	C₂×C₄	—	8	1	no	—
192	C₂×C₄	—	212	2	no	—
193	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
194	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
195	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
196	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
197	C₂×C₄	C₂×#11719949	1	1	yes	characteristic
198	C₂×C₄	#13133263	1	1	yes	characteristic
199	C₈	—	4	2	no	—
200	C₈	—	212	2	no	—
201	C₈	—	212	4	no	—
202	C₈	#13133208	1	2	yes	—
203	C₈	#13133208	1	2	yes	—
204	C₈	#13133209	1	2	yes	—
205	C₈	#13133209	1	2	yes	—
206	C₂²	—	4	1	no	—
207	C₂²	#16811251	1	1	yes	characteristic
208	C₂²	#16811343	1	1	yes	characteristic
209	C₂²	#16811346	1	1	yes	characteristic
210	C₄	—	4	2	no	—
211	C₄	—	8	2	no	—
212	C₄	—	424	2	no	—
213	C₄	C₂×#13133208	1	1	yes	characteristic
214	C₄	C₂×#13133209	1	1	yes	characteristic
215	C₄	#16811338	1	1	yes	characteristic
216	C₄	#16811358	1	1	yes	characteristic
217	C₂	—	2	1	no	—
218	C₂	—	2	1	no	—
219	C₂	—	2	1	no	—
220	C₂	#20279954	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16811338
commutator, cosocle	C₂²×C₄×C₅₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1773	C₂
socle	C₂×C₅₃	#2571
prime radical	C₂³×C₅₃	2₋¹⁺⁴
hypercenter	#1773	D₅₃
hypocenter	C₅₃	#29231
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1773

Complements

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

N	H ≅ G/N	type
135. C₅₃	#29231	semidirect

Maximal subgroups

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

subgroup	index
#20276293	2
#20275505	2
#20275503	2
#20274890	2
#20274839	2
#20274838	2
#20274599	2
C₅₃×#1773	2
#29231	53