Subgroups — #27284107

Order 13568.

458 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	C₄×#13133209	C₂	1	1	yes	maximal, characteristic
2	#20274867	C₂	1	1	yes	maximal, characteristic
3	#20272555	C₂	1	4	yes	maximal
4	C₅₃×#1744	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133209	C₄	1	1	yes	characteristic
6	C₂×C₄⋊#88322	—	2	2	no	—
7	C₂×C₄⋊#88322	C₂²	1	2	yes	—
8	#16810521	C₄	1	2	yes	—
9	#16810507	—	2	2	no	—
10	#16810489	C₄	1	1	yes	characteristic
11	C₄×#11719919	—	2	2	no	—
12	C₄×C₄⋊Dic₅₃	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
14	C₄³×C₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×#336	—	2	2	no	—
16	C₂²×#88322	—	2	2	no	—
17	#13133209	—	2	2	no	—
18	#13133209	C₈	1	4	yes	complemented
19	C₂×C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
20	C₂×C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
21	C₂×#11719919	—	2	2	no	—
22	C₂×#11719919	—	2	2	no	—
23	C₂×C₄×Dic₅₃	—	2	2	no	—
24	C₄⋊#88322	—	2	4	no	—
25	C₄⋊#88322	—	4	4	no	—
26	#13133124	—	2	2	no	—
27	#13133124	—	2	2	no	—
28	#13133124	—	2	2	no	—
29	C₂²×C₈×C₅₃	—	2	2	no	—
30	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₅₃	D₄	1	2	yes	—
33	C₂×C₄²×C₅₃	D₄	1	2	yes	—
34	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
35	C₄×C₈×C₅₃	—	2	2	no	—
36	C₄×C₈×C₅₃	Q₈	1	4	yes	—
37	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
38	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
39	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
42	C₂²×Dic₅₃	—	2	2	no	—
43	C₂²×Dic₅₃	—	2	2	no	—
44	C₂×C₈×C₅₃	—	2	4	no	—
45	C₂×C₈×C₅₃	—	4	4	no	—
46	C₄²×C₅₃	—	2	2	no	—
47	C₄²×C₅₃	—	2	2	no	—
48	C₄²×C₅₃	—	2	2	no	—
49	C₄²×C₅₃	—	4	2	no	—
50	C₄²×C₅₃	C₂×C₈	1	2	yes	—
51	C₄²×C₅₃	M₁₆	1	2	yes	—
52	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
53	C₄²×C₅₃	#41	1	2	yes	—
54	C₄⋊Dic₅₃	—	2	2	no	—
55	C₄⋊Dic₅₃	—	2	4	no	—
56	C₄⋊Dic₅₃	C₂×C₈	1	4	yes	—
57	#11719919	—	2	8	no	—
58	#11719919	—	4	4	no	—
59	C₄×Dic₅₃	—	4	2	no	—
60	C₄×Dic₅₃	—	4	2	no	—
61	C₂×#88322	—	4	2	no	—
62	C₂×#88322	—	4	4	no	—
63	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₅₃	—	2	2	no	—
65	C₂×C₄×C₅₃	—	2	2	no	—
66	C₂×C₄×C₅₃	—	2	2	no	—
67	C₂×C₄×C₅₃	—	2	2	no	—
68	C₂×C₄×C₅₃	—	2	2	no	—
69	C₂×C₄×C₅₃	—	2	2	no	—
70	C₂×C₄×C₅₃	C₂²⋊C₈	1	4	yes	—
71	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
72	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
73	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
77	C₂×C₄×C₅₃	#131	1	2	yes	—
78	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
79	C₂×C₄×C₅₃	#137	1	2	yes	—
80	C₂×Dic₅₃	—	2	4	no	—
81	C₂×Dic₅₃	—	4	2	no	—
82	C₂×Dic₅₃	—	4	2	no	—
83	C₂×Dic₅₃	—	4	2	no	—
84	C₂×Dic₅₃	—	4	2	no	—
85	C₂×Dic₅₃	—	4	2	no	—
86	C₈×C₅₃	—	4	4	no	—
87	#88322	—	8	2	no	—
88	#9872	—	53	1	no	maximal, Sylow
89	C₂²×C₅₃	#391	1	1	yes	characteristic
90	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
91	C₂²×C₅₃	#408	1	1	yes	characteristic
92	C₂²×C₅₃	#417	1	2	yes	—
93	C₂²×C₅₃	#421	1	2	yes	—
94	C₄×C₅₃	—	2	4	no	—
95	C₄×C₅₃	—	2	2	no	—
96	C₄×C₅₃	—	4	2	no	—
97	C₄×C₅₃	C₈×Q₈	1	2	yes	—
98	C₄×C₅₃	#446	1	2	yes	—
99	C₄×C₅₃	#490	1	1	yes	characteristic
100	C₄×C₅₃	#491	1	1	yes	characteristic
101	C₄×C₅₃	#501	1	2	yes	—
102	Dic₅₃	—	4	4	no	—
103	Dic₅₃	—	8	2	no	—
104	C₄×#125	—	53	1	no	—
105	#1842	—	53	1	no	—
106	#1744	D₅₃	1	1	yes	characteristic
107	#1279	—	53	4	no	—
108	C₂×C₅₃	#1477	1	2	yes	—
109	C₂×C₅₃	#1495	1	2	yes	—
110	C₂×C₅₃	#1984	1	1	yes	characteristic
111	C₂×C₅₃	#1992	1	1	yes	characteristic
112	C₂×C₅₃	#1995	1	1	yes	characteristic
113	C₂×#125	—	53	1	no	—
114	C₂×C₄⋊C₈	—	53	2	no	—
115	C₂×C₄⋊C₈	—	106	2	no	—
116	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
117	#389	—	53	2	no	—
118	#387	—	106	2	no	—
119	#382	—	53	1	no	—
120	C₄×C₄⋊C₄	—	53	2	no	—
121	C₄×C₄⋊C₄	—	106	2	no	—
122	C₄³	C₂×D₅₃	1	1	yes	characteristic
123	#336	—	2	2	no	—
124	C₅₃	#9872	1	1	yes	characteristic, complemented, Sylow
125	C₂²×C₈	—	2	2	no	—
126	C₂²×C₈	—	106	2	no	—
127	#125	—	53	4	no	—
128	#125	—	106	2	no	—
129	C₂×C₄⋊C₄	—	53	2	no	—
130	C₂×C₄⋊C₄	—	53	2	no	—
131	C₂×C₄⋊C₄	—	106	2	no	—
132	C₂×C₄⋊C₄	—	106	2	no	—
133	C₂×C₄²	—	106	2	no	—
134	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
135	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
136	C₂×C₄²	D₂₁₂	1	2	yes	—
137	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
138	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
139	C₄⋊C₈	—	106	4	no	—
140	C₄⋊C₈	—	212	4	no	—
141	C₄×C₈	—	2	2	no	—
142	C₄×C₈	Dic₁₀₆	1	4	yes	—
143	#95	—	106	2	no	—
144	#95	—	106	2	no	—
145	#95	—	106	2	no	—
146	C₂²×C₄	—	106	2	no	—
147	C₂²×C₄	—	106	2	no	—
148	C₂²×C₄	#11719921	1	2	yes	—
149	C₂²×C₄	#11719921	1	2	yes	—
150	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
151	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
152	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
153	C₂×C₈	—	2	4	no	—
154	C₂×C₈	—	4	4	no	—
155	C₂×C₈	—	212	2	no	—
156	C₂×C₈	—	212	4	no	—
157	C₄⋊C₄	—	53	4	no	—
158	C₄⋊C₄	—	106	8	no	—
159	C₄⋊C₄	—	106	2	no	—
160	C₄⋊C₄	—	106	4	no	—
161	C₄⋊C₄	—	212	4	no	—
162	C₄²	—	2	2	no	—
163	C₄²	—	2	2	no	—
164	C₄²	—	2	2	no	—
165	C₄²	—	4	2	no	—
166	C₄²	—	212	2	no	—
167	C₄²	—	212	2	no	—
168	C₄²	C₈×D₅₃	1	2	yes	—
169	C₄²	#11719912	1	2	yes	—
170	C₄²	C₂×Dic₁₀₆	1	2	yes	—
171	C₄²	#11719945	1	2	yes	—
172	C₂³	C₂×#11719921	1	1	yes	characteristic
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	106	4	no	—
180	C₂×C₄	—	212	2	no	—
181	C₂×C₄	—	212	2	no	—
182	C₂×C₄	—	212	2	no	—
183	C₂×C₄	—	212	2	no	—
184	C₂×C₄	—	212	2	no	—
185	C₂×C₄	#13133129	1	4	yes	—
186	C₂×C₄	C₄×Dic₁₀₆	1	2	yes	—
187	C₂×C₄	#13133196	1	2	yes	—
188	C₂×C₄	#13133205	1	1	yes	characteristic
189	C₂×C₄	#13133217	1	1	yes	characteristic
190	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
191	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
192	C₂×C₄	#13133223	1	2	yes	—
193	C₂×C₄	#13133245	1	2	yes	—
194	C₂×C₄	#13133250	1	2	yes	—
195	C₈	—	4	4	no	—
196	C₈	—	424	2	no	—
197	C₂²	#16810529	1	1	yes	characteristic
198	C₂²	C₂×#13133129	1	1	yes	characteristic
199	C₂²	#16810589	1	1	yes	characteristic
200	C₂²	#16810625	1	2	yes	—
201	C₂²	#16810641	1	2	yes	—
202	C₄	—	2	4	no	—
203	C₄	—	2	2	no	—
204	C₄	—	4	2	no	—
205	C₄	—	424	2	no	—
206	C₄	—	212	4	no	—
207	C₄	C₈×Dic₁₀₆	1	2	yes	—
208	C₄	#16810753	1	2	yes	—
209	C₄	#16810988	1	1	yes	characteristic
210	C₄	#16810992	1	1	yes	characteristic
211	C₄	#16811032	1	2	yes	—
212	C₂	#20273103	1	2	yes	—
213	C₂	#20273181	1	2	yes	—
214	C₂	#20275651	1	1	yes	characteristic
215	C₂	#20275684	1	1	yes	characteristic
216	C₂	#20275696	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1744	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1744	D₅₃
hypocenter	C₅₃	#9872
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1744

Complements

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

N	H ≅ G/N	type
18. #13133209	C₈	semidirect
124. C₅₃	#9872	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13133209	2
#20274867	2
#20272555	2
C₅₃×#1744	2
#9872	53