Subgroups — #27302233

Order 13568.

310 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810447	C₂	1	2	yes	maximal
2	C₂×#16810444	C₂	1	1	yes	maximal, characteristic
3	#20274872	C₂	1	1	yes	maximal, characteristic
4	#20272820	C₂	1	2	yes	maximal
5	C₅₃×#1772	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133157	C₂²	1	2	yes	—
7	C₂×#13133149	—	2	1	no	—
8	C₂×#13133149	C₂²	1	2	yes	—
9	#16810493	—	2	1	no	—
10	#16810447	—	2	4	no	—
11	#16810444	C₄	1	4	yes	complemented
12	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
13	C₅₃×#427	C₂²	1	1	yes	characteristic
14	C₅₃×#405	—	2	1	no	—
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#339	—	2	1	no	—
17	C₂×C₄⋊Dic₅₃	—	2	2	no	—
18	C₂×C₄⋊Dic₅₃	—	4	2	no	—
19	#13133157	—	2	2	no	—
20	#13133157	C₂×C₄	1	4	yes	—
21	C₂×#11719908	—	2	2	no	—
22	#13133149	—	2	2	no	—
23	#13133149	—	2	2	no	—
24	#13133149	—	4	2	no	—
25	C₂×M₁₆×C₅₃	—	2	1	no	—
26	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
28	#117×C₅₃	—	2	1	no	—
29	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
30	#108×C₅₃	—	2	1	no	—
31	#108×C₅₃	D₄	1	2	yes	—
32	#107×C₅₃	Q₈	1	2	yes	—
33	#106×C₅₃	—	2	1	no	—
34	#97×C₅₃	—	4	1	no	—
35	C₄×C₈×C₅₃	—	4	1	no	—
36	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
37	C₄×C₈×C₅₃	Q₈	1	2	yes	—
38	C₂²×C₄×C₅₃	—	2	1	no	—
39	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
40	C₂²×Dic₅₃	—	4	2	no	—
41	M₁₆×C₅₃	—	2	2	no	—
42	M₁₆×C₅₃	—	4	1	no	—
43	C₂×C₈×C₅₃	—	4	1	no	—
44	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
45	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
46	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
47	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
48	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
53	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
54	C₄⋊C₄×C₅₃	—	2	2	no	—
55	C₂²⋊C₄×C₅₃	—	4	1	no	—
56	C₄²×C₅₃	—	2	1	no	—
57	C₄²×C₅₃	—	4	1	no	—
58	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
59	C₄⋊Dic₅₃	—	4	2	no	—
60	C₄⋊Dic₅₃	—	4	2	no	—
61	C₄⋊Dic₅₃	—	8	4	no	—
62	#11719908	—	2	4	no	—
63	C₂³×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
64	C₂×C₄×C₅₃	—	2	4	no	—
65	C₂×C₄×C₅₃	—	2	2	no	—
66	C₂×C₄×C₅₃	—	4	2	no	—
67	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
69	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
70	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
71	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
72	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
73	C₂×Dic₅₃	—	8	4	no	—
74	C₂×Dic₅₃	—	8	2	no	—
75	C₈×C₅₃	—	4	2	no	—
76	C₈×C₅₃	#106	1	2	yes	—
77	C₈×C₅₃	#106	1	2	yes	—
78	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
79	C₈×C₅₃	C₄×Q₈	1	1	yes	characteristic
80	C₈×C₅₃	#125	1	1	yes	characteristic
81	C₈×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
82	#13186	—	53	1	no	maximal, Sylow
83	C₂²×C₅₃	—	2	2	no	—
84	C₂²×C₅₃	#428	1	1	yes	characteristic
85	C₂²×C₅₃	#460	1	1	yes	characteristic
86	C₂²×C₅₃	#476	1	1	yes	characteristic
87	C₄×C₅₃	—	2	4	no	—
88	C₄×C₅₃	—	8	2	no	—
89	C₄×C₅₃	#389	1	1	yes	characteristic
90	C₄×C₅₃	C₂×#106	1	1	yes	characteristic
91	C₄×C₅₃	#461	1	1	yes	characteristic
92	C₄×C₅₃	#475	1	1	yes	characteristic
93	Dic₅₃	—	16	2	no	—
94	C₂×#365	—	53	2	no	—
95	C₂×#364	—	53	1	no	—
96	#1844	—	53	1	no	—
97	#1772	D₅₃	1	1	yes	characteristic
98	#1386	—	53	2	no	—
99	C₂×C₅₃	—	2	1	no	—
100	C₂×C₅₃	—	2	1	no	—
101	C₂×C₅₃	#1948	1	1	yes	characteristic
102	C₂×C₅₃	#2230	1	2	yes	—
103	C₂×M₃₂	—	53	2	no	—
104	C₂×#108	C₂×D₅₃	1	1	yes	characteristic
105	#427	C₂×D₅₃	1	1	yes	characteristic
106	C₂×#106	—	53	2	no	—
107	C₂×#106	—	106	1	no	—
108	#405	—	2	1	no	—
109	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
110	#384	—	106	1	no	—
111	#365	—	106	4	no	—
112	#364	—	53	4	no	—
113	#339	—	2	1	no	—
114	C₅₃	#13186	1	1	yes	characteristic, complemented, Sylow
115	C₂×M₁₆	—	2	1	no	—
116	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
117	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
118	#117	—	2	1	no	—
119	C₂×C₄⋊C₄	—	106	2	no	—
120	C₂×C₄⋊C₄	—	212	2	no	—
121	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
122	M₃₂	—	53	4	no	—
123	M₃₂	—	106	2	no	—
124	C₂×C₁₆	—	106	2	no	—
125	#108	—	2	1	no	—
126	#108	D₂₁₂	1	2	yes	—
127	#107	Dic₁₀₆	1	2	yes	—
128	#106	—	2	1	no	—
129	#106	—	106	2	no	—
130	#106	—	106	2	no	—
131	#106	—	212	2	no	—
132	#97	—	4	1	no	—
133	C₄×C₈	—	4	1	no	—
134	C₄×C₈	Dic₁₀₆	1	2	yes	—
135	C₄×C₈	C₄×D₅₃	1	2	yes	—
136	C₂²×C₄	—	2	1	no	—
137	C₂²×C₄	—	212	2	no	—
138	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
139	M₁₆	—	2	2	no	—
140	M₁₆	—	4	1	no	—
141	C₂×C₈	—	4	1	no	—
142	C₂×C₈	#11719919	1	2	yes	—
143	C₂×C₈	#11719923	1	1	yes	characteristic
144	C₂×C₈	#11719923	1	1	yes	characteristic
145	C₂×C₈	#11719924	1	2	yes	—
146	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
147	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
148	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
149	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
150	C₂×C₈	#11719945	1	1	yes	characteristic
151	C₂×C₈	#11719945	1	1	yes	characteristic
152	C₄⋊C₄	—	2	2	no	—
153	C₄⋊C₄	—	424	4	no	—
154	C₄⋊C₄	—	212	2	no	—
155	C₄⋊C₄	—	212	2	no	—
156	C₂²⋊C₄	—	4	1	no	—
157	C₄²	—	2	1	no	—
158	C₄²	—	4	1	no	—
159	C₄²	#11719919	1	2	yes	—
160	C₁₆	—	106	4	no	—
161	C₂³	#13133242	1	1	yes	characteristic
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	4	2	no	—
165	C₂×C₄	—	424	4	no	—
166	C₂×C₄	—	424	2	no	—
167	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
168	C₂×C₄	#13133191	1	1	yes	characteristic
169	C₂×C₄	#13133196	1	1	yes	characteristic
170	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
171	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
172	C₂×C₄	#13133247	1	1	yes	characteristic
173	C₈	—	4	2	no	—
174	C₈	#13133148	1	2	yes	—
175	C₈	#13133148	1	2	yes	—
176	C₈	C₄×D₂₁₂	1	1	yes	characteristic
177	C₈	C₄×Dic₁₀₆	1	1	yes	characteristic
178	C₈	#13133209	1	1	yes	characteristic
179	C₈	#13133217	1	1	yes	characteristic
180	C₂²	—	2	2	no	—
181	C₂²	#16810666	1	1	yes	characteristic
182	C₂²	#16810823	1	1	yes	characteristic
183	C₂²	#16810913	1	1	yes	characteristic
184	C₄	—	2	4	no	—
185	C₄	—	8	2	no	—
186	C₄	—	848	2	no	—
187	C₄	#16810521	1	1	yes	characteristic
188	C₄	C₂×#13133148	1	1	yes	characteristic
189	C₄	#16810828	1	1	yes	characteristic
190	C₄	#16810908	1	1	yes	characteristic
191	C₂	—	2	1	no	—
192	C₂	—	2	1	no	—
193	C₂	#20275478	1	1	yes	characteristic
194	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	#13186
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
11. #16810444	C₄	semidirect
114. C₅₃	#13186	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810447	2
C₂×#16810444	2
#20274872	2
#20272820	2
C₅₃×#1772	2
#13186	53