Subgroups — #27302651

Order 13568.

366 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811021	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810444	C₂	1	1	yes	maximal, characteristic
3	#20272820	C₂	1	2	yes	maximal
4	#20272815	C₂	1	2	yes	maximal
5	C₅₃×#1772	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133217	—	2	1	no	—
7	C₂×#13133157	C₂²	1	2	yes	—
8	#16811021	—	2	2	no	—
9	#16811021	C₄	1	4	yes	complemented
10	C₂×#13133149	—	2	1	no	—
11	C₂×#13133149	C₂²	1	2	yes	—
12	#16810444	—	2	2	no	—
13	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×#427	C₂²	1	1	yes	characteristic
15	C₅₃×#405	—	2	1	no	—
16	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#339	—	2	1	no	—
18	C₂²×Dic₁₀₆	—	4	1	no	—
19	#13133217	—	2	4	no	—
20	#13133217	—	4	2	no	—
21	C₂×C₄⋊Dic₅₃	—	2	2	no	—
22	C₂×C₄⋊Dic₅₃	—	4	1	no	—
23	#13133157	—	2	2	no	—
24	#13133157	—	2	2	no	—
25	C₂×#11719908	—	2	2	no	—
26	#13133149	—	2	4	no	—
27	#13133149	—	2	2	no	—
28	#13133149	C₂×C₄	1	4	yes	—
29	C₂×M₁₆×C₅₃	—	2	1	no	—
30	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
32	#117×C₅₃	—	2	1	no	—
33	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
34	#108×C₅₃	—	2	1	no	—
35	#108×C₅₃	Q₈	1	2	yes	complemented
36	#107×C₅₃	Q₈	1	2	yes	complemented
37	#106×C₅₃	—	2	1	no	—
38	#97×C₅₃	—	4	1	no	—
39	C₄×C₈×C₅₃	—	4	1	no	—
40	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
41	C₄×C₈×C₅₃	D₄	1	2	yes	—
42	C₂²×C₄×C₅₃	—	2	1	no	—
43	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
44	C₂²×Dic₅₃	—	4	2	no	—
45	C₂×Dic₁₀₆	—	4	4	no	—
46	C₂×Dic₁₀₆	—	8	4	no	—
47	M₁₆×C₅₃	—	2	2	no	—
48	M₁₆×C₅₃	—	4	1	no	—
49	C₂×C₈×C₅₃	—	4	1	no	—
50	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
51	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
52	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
53	C₂×C₈×C₅₃	Q₁₆	1	2	yes	—
54	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
59	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
60	C₄⋊C₄×C₅₃	—	2	2	no	—
61	C₂²⋊C₄×C₅₃	—	4	1	no	—
62	C₄²×C₅₃	—	2	1	no	—
63	C₄²×C₅₃	—	4	1	no	—
64	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
65	C₄⋊Dic₅₃	—	2	4	no	—
66	C₄⋊Dic₅₃	—	4	4	no	—
67	C₄⋊Dic₅₃	—	4	2	no	—
68	#11719908	—	4	2	no	—
69	C₂³×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
70	C₂×C₄×C₅₃	—	2	4	no	—
71	C₂×C₄×C₅₃	—	2	2	no	—
72	C₂×C₄×C₅₃	—	4	2	no	—
73	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
77	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
78	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
79	C₂×Dic₅₃	—	4	4	no	—
80	C₂×Dic₅₃	—	8	4	no	—
81	Dic₁₀₆	—	8	8	no	—
82	C₈×C₅₃	—	4	2	no	—
83	C₈×C₅₃	#103	1	2	yes	—
84	C₈×C₅₃	#103	1	2	yes	—
85	C₈×C₅₃	C₄×Q₈	1	1	yes	characteristic
86	C₈×C₅₃	C₄×Q₈	1	1	yes	characteristic
87	C₈×C₅₃	#124	1	1	yes	characteristic
88	C₈×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
89	#13272	—	53	1	no	maximal, Sylow
90	C₂²×C₅₃	—	2	2	no	—
91	C₂²×C₅₃	#418	1	1	yes	characteristic
92	C₂²×C₅₃	#476	1	1	yes	characteristic
93	C₂²×C₅₃	#477	1	1	yes	characteristic
94	C₄×C₅₃	—	2	4	no	—
95	C₄×C₅₃	—	8	2	no	—
96	C₄×C₅₃	#391	1	1	yes	characteristic
97	C₄×C₅₃	C₂×#103	1	1	yes	characteristic
98	C₄×C₅₃	#476	1	1	yes	characteristic
99	C₄×C₅₃	#477	1	1	yes	characteristic
100	Dic₅₃	—	8	4	no	—
101	C₂×#498	—	53	1	no	—
102	C₂×#364	—	53	1	no	—
103	#1772	D₅₃	1	1	yes	characteristic
104	#1386	—	53	2	no	—
105	#1383	—	53	2	no	—
106	C₂×C₅₃	—	2	1	no	—
107	C₂×C₅₃	—	2	1	no	—
108	C₂×C₅₃	#1987	1	1	yes	characteristic
109	C₂×C₅₃	#2230	1	2	yes	—
110	C₂×C₄⋊Q₈	—	106	1	no	—
111	C₂×M₃₂	—	53	2	no	—
112	#498	—	53	4	no	—
113	#498	—	106	2	no	—
114	C₂×#108	C₂×D₅₃	1	1	yes	characteristic
115	#427	C₂×D₅₃	1	1	yes	characteristic
116	C₂×#106	—	53	2	no	—
117	C₂×#106	—	106	1	no	—
118	#405	—	2	1	no	—
119	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
120	#364	—	106	2	no	—
121	#339	—	2	1	no	—
122	C₅₃	#13272	1	1	yes	characteristic, complemented, Sylow
123	C₂²×Q₈	—	212	1	no	—
124	C₂×M₁₆	—	2	1	no	—
125	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
126	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
127	C₄⋊Q₈	—	106	4	no	—
128	C₄⋊Q₈	—	212	2	no	—
129	#117	—	2	1	no	—
130	C₂×C₄⋊C₄	—	106	2	no	—
131	C₂×C₄⋊C₄	—	212	1	no	—
132	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
133	M₃₂	—	106	2	no	—
134	M₃₂	—	106	2	no	—
135	C₂×C₁₆	—	106	2	no	—
136	#108	—	2	1	no	—
137	#108	Dic₁₀₆	1	2	yes	complemented
138	#107	Dic₁₀₆	1	2	yes	complemented
139	#106	—	2	1	no	—
140	#106	—	53	4	no	—
141	#106	—	106	2	no	—
142	#106	—	106	4	no	—
143	#97	—	4	1	no	—
144	C₄×C₈	—	4	1	no	—
145	C₄×C₈	C₄×D₅₃	1	2	yes	—
146	C₄×C₈	D₂₁₂	1	2	yes	—
147	C₂×Q₈	—	424	4	no	—
148	C₂×Q₈	—	212	4	no	—
149	C₂²×C₄	—	2	1	no	—
150	C₂²×C₄	—	212	2	no	—
151	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
152	M₁₆	—	2	2	no	—
153	M₁₆	—	4	1	no	—
154	C₂×C₈	—	4	1	no	—
155	C₂×C₈	#11719921	1	2	yes	—
156	C₂×C₈	#11719923	1	1	yes	characteristic
157	C₂×C₈	#11719923	1	1	yes	characteristic
158	C₂×C₈	#11719925	1	2	yes	—
159	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
160	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
161	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
162	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
163	C₂×C₈	#11719945	1	1	yes	characteristic
164	C₂×C₈	#11719945	1	1	yes	characteristic
165	C₄⋊C₄	—	2	2	no	—
166	C₄⋊C₄	—	106	4	no	—
167	C₄⋊C₄	—	212	4	no	—
168	C₄⋊C₄	—	212	2	no	—
169	C₂²⋊C₄	—	4	1	no	—
170	C₄²	—	2	1	no	—
171	C₄²	—	4	1	no	—
172	C₄²	#11719921	1	2	yes	—
173	C₁₆	—	212	2	no	—
174	C₂³	#13133242	1	1	yes	characteristic
175	Q₈	—	424	8	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	—	424	4	no	—
180	C₂×C₄	—	212	4	no	—
181	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
182	C₂×C₄	#13133196	1	1	yes	characteristic
183	C₂×C₄	#13133196	1	1	yes	characteristic
184	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
185	C₂×C₄	C₂×#11719925	1	1	yes	characteristic
186	C₂×C₄	#13133242	1	1	yes	characteristic
187	C₈	—	4	2	no	—
188	C₈	#13133140	1	2	yes	—
189	C₈	#13133140	1	2	yes	—
190	C₈	C₄×Dic₁₀₆	1	1	yes	characteristic
191	C₈	C₄×Dic₁₀₆	1	1	yes	characteristic
192	C₈	#13133205	1	1	yes	characteristic
193	C₈	#13133217	1	1	yes	characteristic
194	C₂²	—	2	2	no	—
195	C₂²	#16810627	1	1	yes	characteristic
196	C₂²	#16810913	1	1	yes	characteristic
197	C₂²	#16810918	1	1	yes	characteristic
198	C₄	—	2	4	no	—
199	C₄	—	8	2	no	—
200	C₄	—	424	4	no	—
201	C₄	#16810529	1	1	yes	characteristic
202	C₄	C₂×#13133140	1	1	yes	characteristic
203	C₄	#16810913	1	1	yes	characteristic
204	C₄	#16810918	1	1	yes	characteristic
205	C₂	—	2	1	no	—
206	C₂	—	2	1	no	—
207	C₂	#20275663	1	1	yes	characteristic
208	C₂	#20276886	1	2	yes	—

	subgroup	G/H
center	C₂²	#16810913
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1772	C₂
socle	C₂²×C₅₃	#476
prime radical	C₂³×C₅₃	AGL(1,ℤ/8)
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1772	D₅₃
hypocenter	C₅₃	#13272
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1772

Complements

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

N	H ≅ G/N	type
9. #16811021	C₄	semidirect
35. #108×C₅₃	Q₈	semidirect
36. #107×C₅₃	Q₈	semidirect
122. C₅₃	#13272	semidirect
137. #108	Dic₁₀₆	semidirect
138. #107	Dic₁₀₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811021	2
C₂×#16810444	2
#20272820	2
#20272815	2
C₅₃×#1772	2
#13272	53