Subgroups — #27282812

Order 13568.

322 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274887	C₂	1	1	yes	maximal, characteristic
2	C₈×C₄⋊Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	#20273490	C₂	1	2	yes	maximal
4	#20272617	C₂	1	2	yes	maximal
5	C₅₃×#1564	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133153	—	2	1	no	—
7	C₂×#13133153	—	2	1	no	—
8	#16810647	C₂²	1	2	yes	—
9	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
10	C₂×C₈×Dic₅₃	—	2	1	no	—
11	C₄×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
12	#16810377	—	2	1	no	—
13	#16810376	—	2	4	no	—
14	C₅₃×#431	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#335	C₄	1	2	yes	complemented
17	C₅₃×#334	C₄	1	2	yes	complemented
18	C₂×#11719917	—	2	2	no	—
19	C₂²×#88322	—	2	1	no	—
20	C₂×C₄⋊Dic₅₃	—	2	1	no	—
21	C₂×C₄×Dic₅₃	—	2	1	no	—
22	#13133153	—	4	2	no	—
23	#13133153	—	4	2	no	—
24	C₄⋊#88322	—	2	1	no	—
25	C₄⋊#88322	—	2	2	no	—
26	C₄⋊#88322	—	2	2	no	—
27	C₄⋊#88322	D₄	1	2	yes	—
28	C₈×Dic₅₃	—	4	2	no	—
29	#13133124	Q₈	1	2	yes	—
30	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
33	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
34	C₂²⋊C₈×C₅₃	—	2	2	no	—
35	C₄×C₈×C₅₃	—	2	2	no	—
36	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
39	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
43	C₂²×Dic₅₃	—	2	2	no	—
44	C₂×C₈×C₅₃	—	2	1	no	—
45	C₂×C₈×C₅₃	—	2	1	no	—
46	C₂×C₈×C₅₃	—	2	4	no	—
47	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₈×C₅₃	D₈	1	2	yes	—
52	C₂×C₈×C₅₃	Q₁₆	1	2	yes	—
53	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
54	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
55	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
56	C₄²×C₅₃	#41	1	1	yes	characteristic
57	C₄⋊Dic₅₃	—	2	4	no	—
58	C₄×Dic₅₃	—	4	1	no	—
59	C₄×Dic₅₃	—	4	1	no	—
60	#11719917	—	4	2	no	—
61	#11719917	—	4	2	no	—
62	C₂×#88322	—	2	2	no	—
63	C₂×#88322	—	4	1	no	—
64	C₂×#88322	—	4	1	no	—
65	C₂×#88322	—	4	2	no	—
66	C₂³×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₅₃	—	2	2	no	—
68	C₂×C₄×C₅₃	—	4	1	no	—
69	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
71	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
72	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
73	C₂×C₄×C₅₃	#125	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
77	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
78	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
79	C₂×Dic₅₃	—	2	4	no	—
80	C₂×Dic₅₃	—	4	2	no	—
81	C₂×Dic₅₃	—	4	2	no	—
82	C₈×C₅₃	—	2	1	no	—
83	C₈×C₅₃	—	2	1	no	—
84	C₈×C₅₃	—	4	4	no	—
85	C₈×C₅₃	#107	1	2	yes	—
86	C₈×C₅₃	#107	1	2	yes	—
87	#88322	—	4	2	no	—
88	#88322	—	8	2	no	—
89	#9627	—	53	1	no	maximal, Sylow
90	C₂²×C₅₃	—	2	1	no	—
91	C₂²×C₅₃	—	2	1	no	—
92	C₂²×C₅₃	#389	1	1	yes	characteristic
93	C₂²×C₅₃	#474	1	1	yes	characteristic
94	C₂²×C₅₃	#477	1	1	yes	characteristic
95	C₄×C₅₃	—	2	1	no	—
96	C₄×C₅₃	—	2	1	no	—
97	C₄×C₅₃	—	2	1	no	—
98	C₄×C₅₃	C₂×#107	1	1	yes	characteristic
99	C₄×C₅₃	#428	1	1	yes	characteristic
100	C₄×C₅₃	#443	1	2	yes	—
101	C₄×C₅₃	#459	1	1	yes	characteristic
102	C₄×C₅₃	#462	1	1	yes	characteristic
103	Dic₅₃	—	4	4	no	—
104	#1848	—	53	1	no	—
105	C₈×C₄⋊C₄	—	53	1	no	—
106	#1571	—	53	2	no	—
107	#1564	D₅₃	1	1	yes	characteristic
108	#1303	—	53	2	no	—
109	C₂×C₅₃	—	2	1	no	—
110	C₂×C₅₃	—	2	1	no	—
111	C₂×C₅₃	#1945	1	1	yes	characteristic
112	C₂×C₅₃	#1947	1	1	yes	characteristic
113	C₂×C₅₃	#1953	1	1	yes	characteristic
114	#431	C₂×D₅₃	1	2	yes	—
115	C₂×#108	—	106	1	no	—
116	C₂×#108	—	106	1	no	—
117	#423	—	53	2	no	—
118	C₂×C₄⋊C₈	—	53	1	no	—
119	C₂×C₄×C₈	—	106	1	no	—
120	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
121	C₄×C₄⋊C₄	—	53	1	no	—
122	#336	—	106	1	no	—
123	#335	—	106	4	no	—
124	#335	Dic₅₃	1	2	yes	complemented
125	#334	Dic₅₃	1	2	yes	complemented
126	C₅₃	#9627	1	1	yes	characteristic, complemented, Sylow
127	C₂×M₁₆	—	106	2	no	—
128	C₂²×C₈	—	106	1	no	—
129	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
130	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
131	C₂×C₄⋊C₄	—	106	1	no	—
132	C₂×C₄²	—	106	1	no	—
133	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
134	#108	—	212	2	no	—
135	#108	—	212	2	no	—
136	C₄⋊C₈	—	53	2	no	—
137	C₄⋊C₈	—	106	2	no	—
138	C₄⋊C₈	—	106	1	no	—
139	C₄⋊C₈	—	106	2	no	—
140	C₄⋊C₈	C₂×Dic₅₃	1	4	yes	—
141	C₂²⋊C₈	—	2	2	no	—
142	C₄×C₈	—	2	2	no	—
143	C₄×C₈	—	212	2	no	—
144	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
145	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
146	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
147	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
148	#95	—	53	2	no	—
149	C₂²×C₄	—	106	2	no	—
150	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
151	C₂²×C₄	#11719947	1	1	yes	characteristic
152	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
153	M₁₆	—	212	2	no	—
154	M₁₆	—	212	2	no	—
155	C₂×C₈	—	2	1	no	—
156	C₂×C₈	—	2	1	no	—
157	C₂×C₈	—	2	4	no	—
158	C₂×C₈	—	106	2	no	—
159	C₂×C₈	—	212	2	no	—
160	C₂×C₈	—	212	1	no	—
161	C₂×C₈	—	212	1	no	—
162	C₂×C₈	D₄₂₄	1	2	yes	—
163	C₂×C₈	Dic₂₁₂	1	2	yes	—
164	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
165	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
166	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
167	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
168	C₄⋊C₄	—	106	4	no	—
169	C₄²	—	212	1	no	—
170	C₄²	—	212	1	no	—
171	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
172	C₄²	D₄×D₅₃	1	1	yes	characteristic
173	C₄²	#11719949	1	1	yes	characteristic
174	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
175	C₂³	#13133197	1	1	yes	characteristic
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	4	1	no	—
178	C₂×C₄	—	106	4	no	—
179	C₂×C₄	—	212	2	no	—
180	C₂×C₄	—	212	2	no	—
181	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
182	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
183	C₂×C₄	Q₈×Dic₅₃	1	1	yes	characteristic
184	C₂×C₄	#13133192	1	1	yes	characteristic
185	C₂×C₄	#13133208	1	1	yes	characteristic
186	C₂×C₄	#13133216	1	1	yes	characteristic
187	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
188	C₂×C₄	C₂×Dic₂₁₂	1	1	yes	characteristic
189	C₂×C₄	#13133245	1	1	yes	characteristic
190	C₂×C₄	#13133250	1	1	yes	characteristic
191	C₈	—	2	1	no	—
192	C₈	—	2	1	no	—
193	C₈	—	4	4	no	—
194	C₈	—	424	2	no	—
195	C₈	—	212	2	no	—
196	C₈	#13133151	1	2	yes	—
197	C₈	#13133151	1	2	yes	—
198	C₂²	—	2	1	no	—
199	C₂²	—	2	1	no	—
200	C₂²	#16810522	1	1	yes	characteristic
201	C₂²	#16810904	1	1	yes	characteristic
202	C₂²	#16810919	1	1	yes	characteristic
203	C₄	—	2	1	no	—
204	C₄	—	2	1	no	—
205	C₄	—	2	1	no	—
206	C₄	—	212	4	no	—
207	C₄	C₂×#13133151	1	1	yes	characteristic
208	C₄	#16810665	1	1	yes	characteristic
209	C₄	#16810740	1	2	yes	—
210	C₄	#16810819	1	1	yes	characteristic
211	C₄	#16810834	1	1	yes	characteristic
212	C₂	—	2	1	no	—
213	C₂	—	2	1	no	—
214	C₂	#20275464	1	1	yes	characteristic
215	C₂	#20275474	1	1	yes	characteristic
216	C₂	#20275506	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13133192
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1564	C₂
socle	C₂²×C₅₃	#389
prime radical	C₂³×C₅₃	C₂²⋊Q₈
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1564	D₅₃
hypocenter	C₅₃	#9627
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
16. C₅₃×#335	C₄	semidirect
17. C₅₃×#334	C₄	semidirect
124. #335	Dic₅₃	semidirect
125. #334	Dic₅₃	semidirect
126. C₅₃	#9627	semidirect

Maximal subgroups

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

subgroup	index
#20274887	2
C₈×C₄⋊Dic₅₃	2
#20273490	2
#20272617	2
C₅₃×#1564	2
#9627	53