Subgroups — #27303448

Order 13568.

462 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811003	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810426	C₂	1	2	yes	maximal
3	C₂×#16810426	C₂	1	2	yes	maximal
4	#20272793	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1942	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₁₂	—	2	1	no	—
7	C₂²×Dic₂₁₂	C₂²	1	2	yes	—
8	C₂×#13133217	—	2	1	no	—
9	C₂²×#11719908	C₂²	1	2	yes	—
10	#16811003	—	2	4	no	—
11	#16810426	—	2	4	no	—
12	#16810426	—	2	4	no	—
13	C₂×#107×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×#107×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#389	—	2	1	no	—
17	C₅₃×#340	—	2	1	no	—
18	C₂²×Dic₁₀₆	—	2	2	no	—
19	C₂×Dic₂₁₂	—	2	4	no	—
20	C₂×Dic₂₁₂	—	2	2	no	—
21	C₂×Dic₂₁₂	—	2	2	no	—
22	C₂×Dic₂₁₂	—	2	2	no	—
23	C₂×Dic₂₁₂	—	2	2	no	—
24	C₂×Dic₂₁₂	—	2	2	no	—
25	C₂×Dic₂₁₂	—	4	2	no	—
26	C₂×Dic₂₁₂	—	4	2	no	—
27	C₂×Dic₂₁₂	D₄	1	4	yes	—
28	#13133217	—	2	4	no	—
29	#13133217	—	4	2	no	—
30	C₂×C₄⋊Dic₅₃	—	4	1	no	—
31	C₂×C₄⋊Dic₅₃	—	4	1	no	—
32	C₂×#11719908	—	2	2	no	—
33	C₂×#11719908	—	2	2	no	—
34	C₂×#11719908	—	2	2	no	—
35	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
37	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
38	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
39	C₂×C₄⋊C₄×C₅₃	—	4	1	no	—
40	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
41	#107×C₅₃	—	2	1	no	—
42	#107×C₅₃	—	2	2	no	—
43	#107×C₅₃	D₄	1	2	yes	—
44	C₄×C₈×C₅₃	—	2	2	no	—
45	#95×C₅₃	—	4	1	no	—
46	C₂²×C₄×C₅₃	—	4	1	no	—
47	C₂²×C₄×C₅₃	—	4	1	no	—
48	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
51	C₂²×Dic₅₃	—	4	2	no	—
52	C₂×Dic₁₀₆	—	2	4	no	—
53	C₂×Dic₁₀₆	—	4	2	no	—
54	C₂×Dic₁₀₆	—	4	2	no	—
55	C₂×Dic₁₀₆	—	4	2	no	—
56	C₂×Dic₁₀₆	—	4	2	no	—
57	C₂×Dic₁₀₆	—	4	2	no	—
58	C₂×C₈×C₅₃	—	2	1	no	—
59	C₂×C₈×C₅₃	—	2	1	no	—
60	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
61	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
63	C₂×C₈×C₅₃	#41	1	2	yes	—
64	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
65	C₄⋊C₄×C₅₃	—	2	2	no	—
66	C₄⋊C₄×C₅₃	—	4	1	no	—
67	C₄⋊C₄×C₅₃	—	4	1	no	—
68	C₄⋊C₄×C₅₃	—	4	1	no	—
69	C₄⋊C₄×C₅₃	—	8	2	no	—
70	C₄²×C₅₃	—	2	1	no	—
71	C₄²×C₅₃	—	2	1	no	—
72	C₄⋊Dic₅₃	—	4	4	no	—
73	C₄⋊Dic₅₃	—	4	4	no	—
74	Dic₂₁₂	—	2	4	no	—
75	Dic₂₁₂	—	2	4	no	—
76	Dic₂₁₂	—	4	8	no	—
77	Dic₂₁₂	—	4	2	no	—
78	Dic₂₁₂	—	4	2	no	—
79	#11719908	—	4	2	no	—
80	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
81	C₂×C₄×C₅₃	—	2	1	no	—
82	C₂×C₄×C₅₃	—	2	1	no	—
83	C₂×C₄×C₅₃	—	2	1	no	—
84	C₂×C₄×C₅₃	—	2	1	no	—
85	C₂×C₄×C₅₃	—	4	2	no	—
86	C₂×C₄×C₅₃	—	8	1	no	—
87	C₂×C₄×C₅₃	—	8	1	no	—
88	C₂×C₄×C₅₃	—	8	1	no	—
89	C₂×C₄×C₅₃	—	8	1	no	—
90	C₂×C₄×C₅₃	—	8	1	no	—
91	C₂×C₄×C₅₃	SD₃₂	1	2	yes	—
92	C₂×C₄×C₅₃	Q₃₂	1	2	yes	—
93	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
94	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
95	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
96	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
97	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
98	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
99	C₂×Dic₅₃	—	4	4	no	—
100	C₂×Dic₅₃	—	8	2	no	—
101	C₂×Dic₅₃	—	8	2	no	—
102	Dic₁₀₆	—	4	4	no	—
103	Dic₁₀₆	—	4	4	no	—
104	Dic₁₀₆	—	8	2	no	—
105	Dic₁₀₆	—	8	2	no	—
106	C₈×C₅₃	—	2	2	no	—
107	C₈×C₅₃	#121	1	2	yes	—
108	C₈×C₅₃	#123	1	2	yes	—
109	#13409	—	53	1	no	maximal, Sylow
110	C₂²×C₅₃	#447	1	1	yes	characteristic
111	C₂²×C₅₃	#459	1	1	yes	characteristic
112	C₂²×C₅₃	#486	1	1	yes	characteristic
113	C₂²×C₅₃	C₂×SD₃₂	1	1	yes	characteristic
114	C₂²×C₅₃	C₂×Q₃₂	1	1	yes	characteristic
115	C₂²×C₅₃	#510	1	1	yes	characteristic
116	C₂²×C₅₃	#510	1	1	yes	characteristic
117	C₄×C₅₃	—	2	2	no	—
118	C₄×C₅₃	—	2	2	no	—
119	C₄×C₅₃	—	8	2	no	—
120	C₄×C₅₃	—	16	1	no	—
121	C₄×C₅₃	#394	1	1	yes	characteristic
122	C₄×C₅₃	#450	1	1	yes	characteristic
123	C₄×C₅₃	#460	1	1	yes	characteristic
124	C₄×C₅₃	#490	1	1	yes	characteristic
125	Dic₅₃	—	8	4	no	—
126	C₂×#494	—	53	1	no	—
127	C₂×#358	—	53	2	no	—
128	C₂×#358	—	53	2	no	—
129	#1942	D₅₃	1	1	yes	characteristic
130	#1374	—	53	1	no	—
131	C₂×C₅₃	#2041	1	1	yes	characteristic
132	C₂×C₅₃	#2185	1	1	yes	characteristic
133	C₂×C₅₃	#2188	1	1	yes	characteristic
134	C₂×C₅₃	#2208	1	1	yes	characteristic
135	C₂×C₅₃	#2209	1	1	yes	characteristic
136	C₂×C₅₃	#2241	1	1	yes	characteristic
137	C₂×C₅₃	#2245	1	1	yes	characteristic
138	C₂²×Q₁₆	—	53	2	no	—
139	C₂²×Q₁₆	—	106	1	no	—
140	C₂×C₄⋊Q₈	—	106	1	no	—
141	C₂²×C₁₆	—	53	2	no	—
142	#494	—	106	4	no	—
143	C₂×#107	C₂×D₅₃	1	1	yes	characteristic
144	C₂×#107	C₂×D₅₃	1	1	yes	characteristic
145	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
146	#389	—	2	1	no	—
147	#358	—	106	4	no	—
148	#358	—	106	4	no	—
149	#340	—	2	1	no	—
150	C₅₃	#13409	1	1	yes	characteristic, complemented, Sylow
151	C₂²×Q₈	—	106	2	no	—
152	C₂×Q₁₆	—	53	4	no	—
153	C₂×Q₁₆	—	106	4	no	—
154	C₂×Q₁₆	—	106	2	no	—
155	C₂×Q₁₆	—	106	2	no	—
156	C₂×Q₁₆	—	106	2	no	—
157	C₂×Q₁₆	—	106	2	no	—
158	C₂×Q₁₆	—	106	2	no	—
159	C₂×Q₁₆	—	212	2	no	—
160	C₂×Q₁₆	—	212	2	no	—
161	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
162	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
163	C₄⋊Q₈	—	106	4	no	—
164	C₄⋊Q₈	—	212	2	no	—
165	C₂×C₄⋊C₄	—	2	1	no	—
166	C₂×C₄⋊C₄	—	2	1	no	—
167	C₂×C₄⋊C₄	—	4	1	no	—
168	C₂×C₄⋊C₄	—	212	1	no	—
169	C₂×C₄⋊C₄	—	212	1	no	—
170	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
171	C₂×C₁₆	—	106	2	no	—
172	C₂×C₁₆	—	106	2	no	—
173	C₂×C₁₆	—	106	2	no	—
174	#107	—	2	1	no	—
175	#107	—	2	2	no	—
176	#107	D₂₁₂	1	2	yes	—
177	C₄×C₈	—	2	2	no	—
178	#95	—	4	1	no	—
179	C₂×Q₈	—	106	4	no	—
180	C₂×Q₈	—	212	2	no	—
181	C₂×Q₈	—	212	2	no	—
182	C₂×Q₈	—	212	2	no	—
183	C₂×Q₈	—	212	2	no	—
184	C₂×Q₈	—	212	2	no	—
185	C₂²×C₄	—	4	1	no	—
186	C₂²×C₄	—	4	1	no	—
187	C₂²×C₄	—	212	2	no	—
188	C₂²×C₄	#11719922	1	1	yes	characteristic
189	C₂²×C₄	#11719922	1	1	yes	characteristic
190	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
191	Q₁₆	—	106	4	no	—
192	Q₁₆	—	106	4	no	—
193	Q₁₆	—	212	8	no	—
194	Q₁₆	—	212	2	no	—
195	Q₁₆	—	212	2	no	—
196	C₂×C₈	—	2	1	no	—
197	C₂×C₈	—	2	1	no	—
198	C₂×C₈	#11719924	1	2	yes	—
199	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
200	C₂×C₈	#11719945	1	1	yes	characteristic
201	C₂×C₈	D₄×D₅₃	1	2	yes	—
202	C₂×C₈	#11719949	1	2	yes	—
203	C₄⋊C₄	—	2	2	no	—
204	C₄⋊C₄	—	4	1	no	—
205	C₄⋊C₄	—	4	1	no	—
206	C₄⋊C₄	—	4	1	no	—
207	C₄⋊C₄	—	8	2	no	—
208	C₄⋊C₄	—	212	4	no	—
209	C₄⋊C₄	—	212	4	no	—
210	C₄²	—	2	1	no	—
211	C₄²	—	2	1	no	—
212	C₁₆	—	212	2	no	—
213	C₂³	C₂×#11719922	1	1	yes	characteristic
214	Q₈	—	424	2	no	—
215	Q₈	—	424	2	no	—
216	Q₈	—	212	4	no	—
217	Q₈	—	212	4	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	8	1	no	—
224	C₂×C₄	—	8	1	no	—
225	C₂×C₄	—	8	1	no	—
226	C₂×C₄	—	8	1	no	—
227	C₂×C₄	—	8	1	no	—
228	C₂×C₄	—	424	2	no	—
229	C₂×C₄	—	424	2	no	—
230	C₂×C₄	—	212	4	no	—
231	C₂×C₄	#13133161	1	2	yes	—
232	C₂×C₄	#13133163	1	2	yes	—
233	C₂×C₄	#13133185	1	1	yes	characteristic
234	C₂×C₄	#13133191	1	1	yes	characteristic
235	C₂×C₄	#13133206	1	1	yes	characteristic
236	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
237	C₂×C₄	#13133242	1	1	yes	characteristic
238	C₂×C₄	#13133242	1	1	yes	characteristic
239	C₈	—	2	2	no	—
240	C₈	#13133192	1	2	yes	—
241	C₈	#13133202	1	2	yes	—
242	C₂²	#16810756	1	1	yes	characteristic
243	C₂²	#16810818	1	1	yes	characteristic
244	C₂²	#16810971	1	1	yes	characteristic
245	C₂²	C₂×#13133161	1	1	yes	characteristic
246	C₂²	C₂×#13133163	1	1	yes	characteristic
247	C₂²	#16811063	1	1	yes	characteristic
248	C₂²	#16811063	1	1	yes	characteristic
249	C₄	—	2	2	no	—
250	C₄	—	2	2	no	—
251	C₄	—	8	2	no	—
252	C₄	—	16	1	no	—
253	C₄	—	424	4	no	—
254	C₄	#16810540	1	1	yes	characteristic
255	C₄	#16810773	1	1	yes	characteristic
256	C₄	#16810823	1	1	yes	characteristic
257	C₄	#16810989	1	1	yes	characteristic
258	C₂	#20275949	1	1	yes	characteristic
259	C₂	#20276631	1	1	yes	characteristic
260	C₂	#20276648	1	1	yes	characteristic
261	C₂	#20276760	1	1	yes	characteristic
262	C₂	#20276765	1	1	yes	characteristic
263	C₂	#20276944	1	1	yes	characteristic
264	C₂	#20276962	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11719922
commutator, cosocle	C₂²×C₈×C₅₃	C₂³
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1942	C₂
socle, prime radical	C₂³×C₅₃	C₂×D₈
hypercenter	#1942	D₅₃
hypocenter	C₅₃	#13409
LCS	G ▷ C₂²×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1942

Complements

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

N	H ≅ G/N	type
150. C₅₃	#13409	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811003	2
C₂×#16810426	2
#20272793	2
C₅₃×#1942	2
#13409	53