Subgroups — #27302321

Order 13568.

522 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810427	C₂	1	2	yes	maximal, complemented
2	C₂×#16810424	C₂	1	2	yes	maximal
3	C₂×#16810399	C₂	1	1	yes	maximal, characteristic, complemented
4	#20274875	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₅₃×#495	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719908	C₂²	1	2	yes	—
7	C₂×#13133151	—	2	1	no	—
8	C₂×#13133151	C₂²	1	2	yes	—
9	#16810493	—	2	1	no	—
10	#16810427	—	2	4	no	—
11	#16810424	—	2	4	no	—
12	#16810399	—	2	2	no	—
13	C₂²×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
14	C₂²×SD₁₆×C₅₃	—	2	1	no	—
15	C₂²×D₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×#124×C₅₃	—	2	1	no	—
17	C₅₃×#495	—	2	2	no	—
18	C₅₃×#495	C₄	1	4	yes	complemented
19	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊Dic₅₃	—	2	2	no	—
21	C₂×C₄⋊Dic₅₃	—	4	2	no	—
22	C₂×#11719908	—	2	2	no	—
23	C₂×#11719908	—	2	2	no	—
24	C₂×#11719908	D₄	1	4	yes	—
25	#13133151	—	2	2	no	—
26	#13133151	—	2	2	no	—
27	#13133151	—	4	2	no	—
28	C₂²×Q₈×C₅₃	—	2	1	no	—
29	C₂²×D₄×C₅₃	—	2	1	no	—
30	C₂×Q₁₆×C₅₃	—	2	1	no	—
31	C₂×Q₁₆×C₅₃	—	2	2	no	—
32	C₂×Q₁₆×C₅₃	—	2	2	no	—
33	C₂×Q₁₆×C₅₃	C₂×C₄	1	2	yes	—
34	C₂×SD₁₆×C₅₃	—	2	4	no	—
35	C₂×SD₁₆×C₅₃	—	4	2	no	—
36	C₂×SD₁₆×C₅₃	—	4	2	no	—
37	C₂×D₈×C₅₃	—	2	1	no	—
38	C₂×D₈×C₅₃	—	2	2	no	—
39	C₂×D₈×C₅₃	—	2	2	no	—
40	C₂×D₈×C₅₃	C₂×C₄	1	2	yes	—
41	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
43	#124×C₅₃	—	2	4	no	—
44	#124×C₅₃	—	4	2	no	—
45	C₂×C₂²⋊C₄×C₅₃	—	4	2	no	—
46	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
47	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
48	C₄×C₈×C₅₃	D₄	1	2	yes	—
49	C₂⁴×C₅₃	—	4	1	no	—
50	C₂×Q₈×C₅₃	—	2	2	no	—
51	C₂×Q₈×C₅₃	—	4	1	no	—
52	C₂×Q₈×C₅₃	—	4	1	no	—
53	C₂×Q₈×C₅₃	—	4	1	no	—
54	C₂×Q₈×C₅₃	—	4	1	no	—
55	C₂×Q₈×C₅₃	—	4	1	no	—
56	C₂×D₄×C₅₃	—	2	2	no	—
57	C₂×D₄×C₅₃	—	4	1	no	—
58	C₂×D₄×C₅₃	—	4	1	no	—
59	C₂×D₄×C₅₃	—	4	1	no	—
60	C₂×D₄×C₅₃	—	4	1	no	—
61	C₂×D₄×C₅₃	—	4	1	no	—
62	C₂²×C₄×C₅₃	—	4	1	no	—
63	C₂²×C₄×C₅₃	D₈	1	2	yes	—
64	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×Dic₅₃	—	4	2	no	—
66	Q₁₆×C₅₃	—	2	4	no	—
67	Q₁₆×C₅₃	—	4	2	no	—
68	SD₁₆×C₅₃	—	4	8	no	—
69	D₈×C₅₃	—	2	4	no	—
70	D₈×C₅₃	—	4	2	no	—
71	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
72	C₂×C₈×C₅₃	D₈	1	2	yes	—
73	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
74	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
76	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₅₃	#41	1	2	yes	—
78	C₂²⋊C₄×C₅₃	—	4	8	no	—
79	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
80	C₄²×C₅₃	SD₁₆	1	2	yes	—
81	C₄⋊Dic₅₃	—	4	2	no	—
82	C₄⋊Dic₅₃	—	4	2	no	—
83	C₄⋊Dic₅₃	—	8	4	no	—
84	#11719908	—	2	4	no	—
85	C₂³×C₅₃	—	4	2	no	—
86	C₂³×C₅₃	—	8	1	no	—
87	C₂³×C₅₃	—	8	1	no	—
88	C₂³×C₅₃	—	8	2	no	—
89	C₂³×C₅₃	—	8	2	no	—
90	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
91	Q₈×C₅₃	—	4	2	no	—
92	Q₈×C₅₃	—	4	2	no	—
93	Q₈×C₅₃	—	8	1	no	—
94	Q₈×C₅₃	—	8	1	no	—
95	D₄×C₅₃	—	4	2	no	—
96	D₄×C₅₃	—	4	2	no	—
97	D₄×C₅₃	—	8	1	no	—
98	D₄×C₅₃	—	8	1	no	—
99	C₂×C₄×C₅₃	—	2	2	no	—
100	C₂×C₄×C₅₃	—	2	2	no	—
101	C₂×C₄×C₅₃	—	4	2	no	—
102	C₂×C₄×C₅₃	—	8	1	no	—
103	C₂×C₄×C₅₃	—	8	1	no	—
104	C₂×C₄×C₅₃	#102	1	4	yes	—
105	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
107	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
108	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
109	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
110	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
111	C₂×Dic₅₃	—	8	2	no	—
112	C₂×Dic₅₃	—	8	4	no	—
113	C₈×C₅₃	#102	1	4	yes	—
114	C₈×C₅₃	C₄×D₄	1	2	yes	—
115	C₈×C₅₃	#121	1	2	yes	—
116	#13205	—	53	1	no	maximal, Sylow
117	C₂²×C₅₃	—	8	4	no	—
118	C₂²×C₅₃	—	8	2	no	—
119	C₂²×C₅₃	—	16	2	no	—
120	C₂²×C₅₃	—	16	1	no	—
121	C₂²×C₅₃	—	16	1	no	—
122	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
123	C₂²×C₅₃	#486	1	1	yes	characteristic
124	C₂²×C₅₃	#493	1	1	yes	characteristic
125	C₂²×C₅₃	#509	1	2	yes	—
126	C₂²×C₅₃	#510	1	2	yes	—
127	C₄×C₅₃	—	2	4	no	—
128	C₄×C₅₃	—	8	2	no	—
129	C₄×C₅₃	#390	1	1	yes	characteristic
130	C₄×C₅₃	C₂×#102	1	1	yes	characteristic
131	C₄×C₅₃	#486	1	1	yes	characteristic
132	C₄×C₅₃	#492	1	1	yes	characteristic
133	Dic₅₃	—	16	2	no	—
134	C₂×#495	D₅₃	1	1	yes	characteristic
135	C₂×#358	—	53	2	no	—
136	C₂×#357	—	53	2	no	—
137	C₂×#346	—	53	1	no	—
138	#1845	—	53	1	no	—
139	C₂×C₅₃	—	16	2	no	—
140	C₂×C₅₃	#1957	1	1	yes	characteristic
141	C₂×C₅₃	#2139	1	2	yes	—
142	C₂×C₅₃	#2243	1	2	yes	—
143	C₂×C₅₃	#2250	1	1	yes	characteristic
144	C₂×C₅₃	#2251	1	1	yes	characteristic
145	C₂²×Q₁₆	C₂×D₅₃	1	1	yes	characteristic
146	C₂²×SD₁₆	—	2	1	no	—
147	C₂²×D₈	C₂×D₅₃	1	1	yes	characteristic
148	C₂×#124	—	2	1	no	—
149	C₂²×C₁₆	—	53	2	no	—
150	#495	—	2	2	no	—
151	#495	Dic₅₃	1	4	yes	complemented
152	C₂×#107	—	53	2	no	—
153	C₂×#107	—	106	1	no	—
154	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
155	#384	—	106	1	no	—
156	#358	—	106	4	no	—
157	#357	—	106	4	no	—
158	#346	—	106	2	no	—
159	C₅₃	#13205	1	1	yes	characteristic, complemented, Sylow
160	C₂²×Q₈	—	2	1	no	—
161	C₂²×D₄	—	2	1	no	—
162	C₂×Q₁₆	—	2	2	no	—
163	C₂×Q₁₆	—	2	2	no	—
164	C₂×Q₁₆	—	2	1	no	—
165	C₂×Q₁₆	C₂×Dic₅₃	1	2	yes	—
166	C₂×SD₁₆	—	2	4	no	—
167	C₂×SD₁₆	—	4	2	no	—
168	C₂×SD₁₆	—	4	2	no	—
169	C₂×D₈	—	2	2	no	—
170	C₂×D₈	—	2	2	no	—
171	C₂×D₈	—	2	1	no	—
172	C₂×D₈	C₂×Dic₅₃	1	2	yes	—
173	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
174	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
175	#124	—	2	4	no	—
176	#124	—	4	2	no	—
177	C₂×C₄⋊C₄	—	106	2	no	—
178	C₂×C₄⋊C₄	—	212	2	no	—
179	C₂×C₂²⋊C₄	—	4	2	no	—
180	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
181	C₂×C₁₆	—	53	4	no	—
182	C₂×C₁₆	—	106	2	no	—
183	C₂×C₁₆	—	106	2	no	—
184	#107	—	106	2	no	—
185	#107	—	106	2	no	—
186	#107	—	212	2	no	—
187	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
188	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
189	C₂⁴	—	4	1	no	—
190	C₂×Q₈	—	2	2	no	—
191	C₂×Q₈	—	4	1	no	—
192	C₂×Q₈	—	4	1	no	—
193	C₂×Q₈	—	4	1	no	—
194	C₂×Q₈	—	4	1	no	—
195	C₂×Q₈	—	4	1	no	—
196	C₂×D₄	—	2	2	no	—
197	C₂×D₄	—	4	1	no	—
198	C₂×D₄	—	4	1	no	—
199	C₂×D₄	—	4	1	no	—
200	C₂×D₄	—	4	1	no	—
201	C₂×D₄	—	4	1	no	—
202	C₂²×C₄	—	4	1	no	—
203	C₂²×C₄	—	212	2	no	—
204	C₂²×C₄	#11719922	1	2	yes	—
205	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
206	Q₁₆	—	2	4	no	—
207	Q₁₆	—	4	2	no	—
208	SD₁₆	—	4	8	no	—
209	D₈	—	2	4	no	—
210	D₈	—	4	2	no	—
211	C₂×C₈	#11719922	1	2	yes	—
212	C₂×C₈	#11719923	1	2	yes	—
213	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
214	C₂×C₈	D₄×D₅₃	1	2	yes	—
215	C₂×C₈	#11719947	1	2	yes	—
216	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
217	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
218	C₄⋊C₄	—	424	4	no	—
219	C₄⋊C₄	—	212	2	no	—
220	C₄⋊C₄	—	212	2	no	—
221	C₂²⋊C₄	—	4	8	no	—
222	C₄²	#11719923	1	2	yes	—
223	C₄²	C₂²⋊Dic₅₃	1	2	yes	—
224	C₁₆	—	106	4	no	—
225	C₂³	—	4	2	no	—
226	C₂³	—	8	2	no	—
227	C₂³	—	8	2	no	—
228	C₂³	—	8	1	no	—
229	C₂³	—	8	1	no	—
230	C₂³	C₂×#11719922	1	1	yes	characteristic
231	Q₈	—	4	2	no	—
232	Q₈	—	4	2	no	—
233	Q₈	—	8	1	no	—
234	Q₈	—	8	1	no	—
235	D₄	—	4	2	no	—
236	D₄	—	4	2	no	—
237	D₄	—	8	1	no	—
238	D₄	—	8	1	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	—	8	1	no	—
243	C₂×C₄	—	8	1	no	—
244	C₂×C₄	—	424	4	no	—
245	C₂×C₄	—	424	2	no	—
246	C₂×C₄	#13133138	1	4	yes	—
247	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
248	C₂×C₄	#13133204	1	1	yes	characteristic
249	C₂×C₄	#13133213	1	1	yes	characteristic
250	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
251	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
252	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
253	C₈	#13133138	1	4	yes	—
254	C₈	D₄×Dic₅₃	1	2	yes	—
255	C₈	#13133188	1	2	yes	—
256	C₂²	—	8	2	no	—
257	C₂²	—	8	4	no	—
258	C₂²	—	16	2	no	—
259	C₂²	—	16	1	no	—
260	C₂²	—	16	1	no	—
261	C₂²	C₂×#13133138	1	1	yes	characteristic
262	C₂²	#16810969	1	1	yes	characteristic
263	C₂²	#16810999	1	1	yes	characteristic
264	C₂²	#16811058	1	2	yes	—
265	C₂²	#16811063	1	2	yes	—
266	C₄	—	2	4	no	—
267	C₄	—	8	2	no	—
268	C₄	—	848	2	no	—
269	C₄	#16810526	1	1	yes	characteristic
270	C₄	C₂×#13133138	1	1	yes	characteristic
271	C₄	#16810969	1	1	yes	characteristic
272	C₄	#16810995	1	1	yes	characteristic
273	C₂	—	16	2	no	—
274	C₂	#20275522	1	1	yes	characteristic
275	C₂	#20276427	1	2	yes	—
276	C₂	#20276951	1	2	yes	—
277	C₂	#20276982	1	1	yes	characteristic
278	C₂	#20276987	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11719922
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#495	C₂
socle	C₂³×C₅₃	C₂×D₈
prime radical	C₂²×D₈×C₅₃	C₂²
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#495	D₅₃
hypocenter	C₅₃	#13205
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#495

Complements

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

N	H ≅ G/N	type
1. C₂×#16810427	C₂	semidirect
3. C₂×#16810399	C₂	semidirect
4. #20274875	C₂	semidirect
18. C₅₃×#495	C₄	semidirect
151. #495	Dic₅₃	semidirect
159. C₅₃	#13205	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810427	2
C₂×#16810424	2
C₂×#16810399	2
#20274875	2
C₂×C₅₃×#495	2
#13205	53