Subgroups — #27271281

Order 13568.

246 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274879	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810374	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133151	C₄	1	2	yes	—
5	C₂×#13133149	—	2	1	no	—
6	C₂×C₄⋊#88322	—	2	1	no	—
7	#16810489	—	2	1	no	—
8	#16810374	—	2	1	no	—
9	#16810374	C₄	1	2	yes	complemented
10	C₂²×C₁₆×C₅₃	—	2	1	no	—
11	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₅₃×C₄⋊C₁₆	—	2	1	no	—
13	C₅₃×C₄⋊C₁₆	C₄	1	2	yes	complemented
14	C₂²×#88322	—	4	1	no	—
15	C₂×C₄⋊Dic₅₃	—	2	2	no	—
16	#13133151	—	4	2	no	—
17	#13133149	—	4	1	no	—
18	#13133149	—	4	1	no	—
19	C₄⋊#88322	—	2	2	no	—
20	C₄⋊#88322	—	4	1	no	—
21	#13133124	—	4	1	no	—
22	#13133124	—	4	1	no	—
23	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
25	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
26	C₂×C₁₆×C₅₃	—	2	2	no	—
27	C₂×C₁₆×C₅₃	—	4	1	no	—
28	C₂×C₁₆×C₅₃	—	4	1	no	—
29	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
32	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
33	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
34	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
35	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
36	C₂²×Dic₅₃	—	4	2	no	—
37	C₂×C₈×C₅₃	—	2	1	no	—
38	C₂×C₈×C₅₃	—	2	1	no	—
39	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
40	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₅₃	D₈	1	1	yes	characteristic
44	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
45	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
46	C₂×C₈×C₅₃	Q₁₆	1	1	yes	characteristic
47	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
48	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
49	C₄²×C₅₃	SD₁₆	1	1	yes	characteristic
50	C₄²×C₅₃	Q₁₆	1	1	yes	characteristic
51	C₄⋊Dic₅₃	—	4	2	no	—
52	C₄⋊Dic₅₃	—	4	2	no	—
53	C₂×#88322	—	4	2	no	—
54	C₂×#88322	—	8	2	no	—
55	C₁₆×C₅₃	—	4	2	no	—
56	C₂³×C₅₃	#102	1	1	yes	characteristic
57	C₂×C₄×C₅₃	—	2	2	no	—
58	C₂×C₄×C₅₃	—	2	2	no	—
59	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
60	C₂×C₄×C₅₃	#100	1	1	yes	characteristic
61	C₂×C₄×C₅₃	#101	1	1	yes	characteristic
62	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
63	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
64	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
65	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
66	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
67	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
68	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
69	C₂×Dic₅₃	—	8	2	no	—
70	C₂×Dic₅₃	—	8	2	no	—
71	C₂×Dic₅₃	—	8	2	no	—
72	C₈×C₅₃	—	2	1	no	—
73	C₈×C₅₃	—	2	1	no	—
74	C₈×C₅₃	#102	1	1	yes	characteristic
75	C₈×C₅₃	#103	1	1	yes	characteristic
76	C₈×C₅₃	#106	1	1	yes	characteristic
77	C₈×C₅₃	#107	1	1	yes	characteristic
78	#88322	—	8	2	no	—
79	#7275	—	53	1	no	maximal, Sylow
80	C₂²×C₅₃	#331	1	1	yes	characteristic
81	C₂²×C₅₃	#332	1	1	yes	characteristic
82	C₂²×C₅₃	#340	1	1	yes	characteristic
83	C₂²×C₅₃	#359	1	2	yes	—
84	C₂²×C₅₃	#360	1	2	yes	—
85	C₄×C₅₃	—	2	1	no	—
86	C₄×C₅₃	—	2	1	no	—
87	C₄×C₅₃	—	2	1	no	—
88	C₄×C₅₃	—	2	1	no	—
89	C₄×C₅₃	#332	1	1	yes	characteristic
90	C₄×C₅₃	#333	1	1	yes	characteristic
91	C₄×C₅₃	#340	1	1	yes	characteristic
92	C₄×C₅₃	#343	1	1	yes	characteristic
93	Dic₅₃	—	16	2	no	—
94	C₂×C₄⋊C₁₆	D₅₃	1	1	yes	characteristic
95	#1846	—	53	1	no	—
96	C₂×#334	—	53	1	no	—
97	C₂×C₅₃	#1296	1	1	yes	characteristic
98	C₂×C₅₃	#1360	1	2	yes	—
99	C₂×C₅₃	#1363	1	2	yes	—
100	C₂×C₅₃	#1382	1	1	yes	characteristic
101	C₂×C₅₃	#1386	1	1	yes	characteristic
102	C₂²×C₁₆	—	2	1	no	—
103	C₂×#107	—	53	2	no	—
104	C₂×#106	—	106	1	no	—
105	C₂×C₄⋊C₈	—	106	1	no	—
106	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
107	#382	—	106	1	no	—
108	C₄⋊C₁₆	—	2	1	no	—
109	C₄⋊C₁₆	Dic₅₃	1	2	yes	complemented
110	#334	—	53	2	no	—
111	#334	—	106	1	no	—
112	C₅₃	#7275	1	1	yes	characteristic, complemented, Sylow
113	C₂²×C₈	—	212	1	no	—
114	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
115	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
116	C₂×C₄⋊C₄	—	106	2	no	—
117	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
118	C₂×C₁₆	—	2	2	no	—
119	C₂×C₁₆	—	4	1	no	—
120	C₂×C₁₆	—	4	1	no	—
121	#107	—	212	2	no	—
122	#106	—	212	1	no	—
123	#106	—	212	1	no	—
124	C₄⋊C₈	—	106	2	no	—
125	C₄⋊C₈	—	212	1	no	—
126	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
127	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
128	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
129	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
130	#95	—	212	1	no	—
131	#95	—	212	1	no	—
132	C₂²×C₄	—	212	2	no	—
133	C₂²×C₄	#11719913	1	1	yes	characteristic
134	C₂²×C₄	D₄₂₄	1	1	yes	characteristic
135	C₂²×C₄	#11719921	1	1	yes	characteristic
136	C₂×C₈	—	2	1	no	—
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	—	424	2	no	—
139	C₂×C₈	—	212	2	no	—
140	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
141	C₂×C₈	#11719919	1	1	yes	characteristic
142	C₂×C₈	#11719919	1	1	yes	characteristic
143	C₂×C₈	#11719922	1	1	yes	characteristic
144	C₂×C₈	#11719923	1	1	yes	characteristic
145	C₂×C₈	#11719924	1	1	yes	characteristic
146	C₂×C₈	#11719925	1	1	yes	characteristic
147	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
148	C₄⋊C₄	—	212	2	no	—
149	C₄⋊C₄	—	212	2	no	—
150	C₄²	#11719913	1	1	yes	characteristic
151	C₄²	Dic₂₁₂	1	1	yes	characteristic
152	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
153	C₄²	#11719921	1	1	yes	characteristic
154	C₁₆	—	4	2	no	—
155	C₂³	#13133139	1	1	yes	characteristic
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	424	2	no	—
159	C₂×C₄	—	424	2	no	—
160	C₂×C₄	—	424	2	no	—
161	C₂×C₄	#13133124	1	1	yes	characteristic
162	C₂×C₄	#13133133	1	1	yes	characteristic
163	C₂×C₄	#13133135	1	1	yes	characteristic
164	C₂×C₄	#13133137	1	1	yes	characteristic
165	C₂×C₄	#13133139	1	1	yes	characteristic
166	C₂×C₄	#13133140	1	1	yes	characteristic
167	C₂×C₄	#13133142	1	1	yes	characteristic
168	C₂×C₄	#13133142	1	1	yes	characteristic
169	C₂×C₄	#13133149	1	1	yes	characteristic
170	C₂×C₄	#13133151	1	1	yes	characteristic
171	C₈	—	2	1	no	—
172	C₈	—	2	1	no	—
173	C₈	—	424	2	no	—
174	C₈	#13133138	1	1	yes	characteristic
175	C₈	#13133141	1	1	yes	characteristic
176	C₈	#13133148	1	1	yes	characteristic
177	C₈	#13133150	1	1	yes	characteristic
178	C₂²	#16810366	1	1	yes	characteristic
179	C₂²	#16810368	1	1	yes	characteristic
180	C₂²	#16810386	1	1	yes	characteristic
181	C₂²	#16810431	1	2	yes	—
182	C₂²	#16810434	1	2	yes	—
183	C₄	—	2	1	no	—
184	C₄	—	2	1	no	—
185	C₄	—	2	1	no	—
186	C₄	—	2	1	no	—
187	C₄	—	848	2	no	—
188	C₄	#16810370	1	1	yes	characteristic
189	C₄	#16810371	1	1	yes	characteristic
190	C₄	#16810385	1	1	yes	characteristic
191	C₄	#16810391	1	1	yes	characteristic
192	C₂	#20272599	1	1	yes	characteristic
193	C₂	#20272759	1	2	yes	—
194	C₂	#20272766	1	2	yes	—
195	C₂	#20272811	1	1	yes	characteristic
196	C₂	#20272821	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13133139
commutator	C₂×C₈×C₅₃	C₄²
Frattini	C₂×C₄×C₈	C₂×D₅₃
Fitting	C₂×C₅₃×C₄⋊C₁₆	C₂
socle, prime radical	C₂³×C₅₃	#102
cosocle	C₂×C₄×C₈×C₅₃	C₂²
hypercenter	C₂×C₄⋊C₁₆	D₅₃
hypocenter	C₅₃	#7275
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
9. #16810374	C₄	semidirect
13. C₅₃×C₄⋊C₁₆	C₄	semidirect
109. C₄⋊C₁₆	Dic₅₃	semidirect
112. C₅₃	#7275	semidirect

Maximal subgroups

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

subgroup	index
#20274879	2
C₂×#16810374	2
C₂×C₅₃×C₄⋊C₁₆	2
#7275	53