Subgroups — #27283126

Order 13568.

570 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	#20283113	C₂	1	1	yes	maximal, characteristic
2	#20274867	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810368	C₂	1	4	yes	maximal
4	C₄×C₄⋊C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133217	—	2	2	no	—
6	C₂×#13133217	C₂²	1	2	yes	—
7	C₂×C₄⋊#88322	—	2	2	no	—
8	C₂×C₄⋊#88322	C₂²	1	2	yes	—
9	#16810493	—	2	1	no	—
10	#16810493	—	2	1	no	—
11	#16810493	C₄	1	4	yes	—
12	#16810368	—	2	8	no	—
13	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₅₃	—	2	1	no	—
15	C₄³×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#336	—	2	1	no	—
17	C₂²×Dic₁₀₆	—	2	2	no	—
18	C₂²×#88322	—	2	2	no	—
19	#13133217	—	2	4	no	—
20	#13133217	—	2	4	no	—
21	#13133217	—	4	4	no	—
22	#13133217	—	4	4	no	—
23	C₂×C₄⋊Dic₅₃	—	2	4	no	—
24	C₂×C₄⋊Dic₅₃	—	2	4	no	—
25	C₂×C₄⋊Dic₅₃	—	2	2	no	—
26	C₂×C₄⋊Dic₅₃	C₂×C₄	1	4	yes	—
27	C₄⋊#88322	—	2	4	no	—
28	C₄⋊#88322	—	4	4	no	—
29	C₂²×C₈×C₅₃	—	2	2	no	—
30	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₅₃	D₄	1	2	yes	—
32	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
35	C₄⋊C₈×C₅₃	—	2	2	no	—
36	C₄⋊C₈×C₅₃	D₄	1	4	yes	—
37	C₄×C₈×C₅₃	—	4	2	no	—
38	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
39	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
40	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
43	C₂²×Dic₅₃	—	2	4	no	—
44	C₂×Dic₁₀₆	—	4	2	no	—
45	C₂×Dic₁₀₆	—	4	2	no	—
46	C₂×Dic₁₀₆	—	4	4	no	—
47	C₂×Dic₁₀₆	—	4	4	no	—
48	C₂×C₈×C₅₃	—	2	4	no	—
49	C₂×C₈×C₅₃	—	4	2	no	—
50	C₂×C₈×C₅₃	—	4	2	no	—
51	C₄²×C₅₃	—	2	4	no	—
52	C₄²×C₅₃	—	2	2	no	—
53	C₄²×C₅₃	D₈	1	4	yes	—
54	C₄²×C₅₃	SD₁₆	1	4	yes	—
55	C₄²×C₅₃	SD₁₆	1	2	yes	—
56	C₄²×C₅₃	Q₁₆	1	2	yes	—
57	C₄²×C₅₃	C₂×D₄	1	2	yes	—
58	C₄²×C₅₃	#41	1	2	yes	—
59	C₄⋊Dic₅₃	—	2	8	no	—
60	C₄⋊Dic₅₃	—	4	8	no	—
61	C₄⋊Dic₅₃	—	4	8	no	—
62	C₄⋊Dic₅₃	—	4	4	no	—
63	C₂×#88322	—	4	2	no	—
64	C₂×#88322	—	4	4	no	—
65	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	#101	1	4	yes	—
70	C₂×C₄×C₅₃	#102	1	4	yes	—
71	C₂×C₄×C₅₃	#102	1	4	yes	—
72	C₂×C₄×C₅₃	#103	1	4	yes	—
73	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
74	C₂×C₄×C₅₃	#121	1	2	yes	—
75	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
76	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
77	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
78	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
79	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
80	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
81	C₂×C₄×C₅₃	#137	1	2	yes	—
82	C₂×Dic₅₃	—	4	4	no	—
83	C₂×Dic₅₃	—	4	8	no	—
84	Dic₁₀₆	—	8	4	no	—
85	Dic₁₀₆	—	8	2	no	—
86	Dic₁₀₆	—	8	2	no	—
87	C₈×C₅₃	—	4	4	no	—
88	#88322	—	8	2	no	—
89	#9692	—	53	1	no	maximal, Sylow
90	C₂²×C₅₃	#390	1	1	yes	characteristic
91	C₂²×C₅₃	C₂×#101	1	2	yes	—
92	C₂²×C₅₃	C₂×#102	1	2	yes	—
93	C₂²×C₅₃	C₂×#103	1	1	yes	characteristic
94	C₂²×C₅₃	#419	1	1	yes	characteristic
95	C₄×C₅₃	—	2	4	no	—
96	C₄×C₅₃	—	2	2	no	—
97	C₄×C₅₃	#332	1	8	yes	complemented
98	C₄×C₅₃	#461	1	2	yes	—
99	C₄×C₅₃	#462	1	2	yes	—
100	C₄×C₅₃	#486	1	2	yes	—
101	C₄×C₅₃	#492	1	1	yes	characteristic
102	C₄×C₅₃	#493	1	1	yes	characteristic
103	Dic₅₃	—	8	4	no	—
104	#2868	—	53	1	no	—
105	#1842	—	53	1	no	—
106	C₄×C₄⋊C₈	D₅₃	1	1	yes	characteristic
107	C₂×#332	—	53	4	no	—
108	C₂×C₅₃	C₂×#332	1	2	yes	—
109	C₂×C₅₃	#1546	1	2	yes	—
110	C₂×C₅₃	#1957	1	1	yes	characteristic
111	C₂×C₅₃	#1966	1	1	yes	characteristic
112	C₂×C₅₃	#1981	1	1	yes	characteristic
113	C₂×C₄⋊Q₈	—	53	2	no	—
114	C₂×C₄⋊Q₈	—	106	2	no	—
115	C₂×C₄⋊C₈	—	53	2	no	—
116	C₂×C₄⋊C₈	—	106	2	no	—
117	C₂×C₄⋊C₈	C₂×D₅₃	1	2	yes	—
118	C₂×C₄×C₈	—	2	1	no	—
119	#384	—	53	4	no	—
120	#384	—	106	1	no	—
121	#384	—	106	1	no	—
122	C₄³	C₂×D₅₃	1	1	yes	characteristic
123	#336	—	2	1	no	—
124	#332	—	106	8	no	—
125	C₅₃	#9692	1	1	yes	characteristic, complemented, Sylow
126	C₂²×Q₈	—	106	2	no	—
127	C₂²×C₈	—	2	2	no	—
128	C₂²×C₈	—	106	2	no	—
129	C₄⋊Q₈	—	106	4	no	—
130	C₄⋊Q₈	—	106	4	no	—
131	C₄⋊Q₈	—	212	4	no	—
132	C₄⋊Q₈	—	212	4	no	—
133	C₂×C₄⋊C₄	—	53	4	no	—
134	C₂×C₄⋊C₄	—	106	2	no	—
135	C₂×C₄⋊C₄	—	106	4	no	—
136	C₂×C₄⋊C₄	—	106	4	no	—
137	C₂×C₄²	C₄×D₅₃	1	2	yes	—
138	C₂×C₄²	D₂₁₂	1	2	yes	—
139	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
140	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
141	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
142	C₄⋊C₈	—	2	2	no	—
143	C₄⋊C₈	—	106	4	no	—
144	C₄⋊C₈	—	212	4	no	—
145	C₄⋊C₈	D₂₁₂	1	4	yes	—
146	C₄×C₈	—	4	2	no	—
147	C₂×Q₈	—	212	2	no	—
148	C₂×Q₈	—	212	2	no	—
149	C₂×Q₈	—	212	4	no	—
150	C₂×Q₈	—	212	4	no	—
151	C₂²×C₄	—	106	4	no	—
152	C₂²×C₄	#11719921	1	2	yes	—
153	C₂²×C₄	#11719921	1	2	yes	—
154	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
155	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
156	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
157	C₂×C₈	—	2	4	no	—
158	C₂×C₈	—	4	2	no	—
159	C₂×C₈	—	4	2	no	—
160	C₂×C₈	—	212	2	no	—
161	C₂×C₈	—	212	4	no	—
162	C₄⋊C₄	—	106	8	no	—
163	C₄⋊C₄	—	212	8	no	—
164	C₄⋊C₄	—	212	8	no	—
165	C₄⋊C₄	—	212	4	no	—
166	C₄²	—	2	4	no	—
167	C₄²	—	2	2	no	—
168	C₄²	#11719913	1	4	yes	—
169	C₄²	D₄₂₄	1	4	yes	—
170	C₄²	#11719923	1	2	yes	—
171	C₄²	#11719925	1	2	yes	—
172	C₄²	C₂×D₂₁₂	1	2	yes	—
173	C₄²	#11719945	1	2	yes	—
174	C₂³	C₂×#11719921	1	1	yes	characteristic
175	Q₈	—	424	4	no	—
176	Q₈	—	424	2	no	—
177	Q₈	—	424	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	212	4	no	—
182	C₂×C₄	—	212	8	no	—
183	C₂×C₄	#13133135	1	4	yes	—
184	C₂×C₄	#13133139	1	4	yes	—
185	C₂×C₄	#13133139	1	4	yes	—
186	C₂×C₄	#13133140	1	4	yes	—
187	C₂×C₄	C₄×D₂₁₂	1	2	yes	—
188	C₂×C₄	#13133191	1	2	yes	—
189	C₂×C₄	#13133205	1	1	yes	characteristic
190	C₂×C₄	#13133214	1	1	yes	characteristic
191	C₂×C₄	C₂×D₄₂₄	1	2	yes	—
192	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
193	C₂×C₄	C₂×#11719913	1	2	yes	—
194	C₂×C₄	C₂×#11719925	1	1	yes	characteristic
195	C₂×C₄	#13133247	1	2	yes	—
196	C₈	—	4	4	no	—
197	C₈	—	424	2	no	—
198	C₂²	#16810525	1	1	yes	characteristic
199	C₂²	C₂×#13133135	1	2	yes	—
200	C₂²	C₂×#13133139	1	2	yes	—
201	C₂²	C₂×#13133140	1	1	yes	characteristic
202	C₂²	#16810631	1	1	yes	characteristic
203	C₄	—	2	2	no	—
204	C₄	—	2	4	no	—
205	C₄	—	424	4	no	—
206	C₄	#16810368	1	8	yes	complemented
207	C₄	#16810828	1	2	yes	—
208	C₄	#16810833	1	2	yes	—
209	C₄	#16810970	1	2	yes	—
210	C₄	#16810996	1	1	yes	characteristic
211	C₄	#16811000	1	1	yes	characteristic
212	C₂	C₂×#16810368	1	2	yes	—
213	C₂	#20273386	1	2	yes	—
214	C₂	#20275521	1	1	yes	characteristic
215	C₂	#20275559	1	1	yes	characteristic
216	C₂	#20275638	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11719921
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₄×C₄⋊C₈×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₄×C₄⋊C₈	D₅₃
hypocenter	C₅₃	#9692
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
97. C₄×C₅₃	#332	semidirect
125. C₅₃	#9692	semidirect
206. C₄	#16810368	semidirect

Maximal subgroups

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

subgroup	index
#20283113	2
#20274867	2
C₂×#16810368	2
C₄×C₄⋊C₈×C₅₃	2
#9692	53