Subgroups — #27302316

Order 13568.

514 conjugacy classes of subgroups.

The subgroup lattice has 268 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×C₄×#11719908	C₂	1	1	yes	maximal, characteristic, complemented
4	#20274879	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	C₂²	1	2	yes	—
8	C₂×#13133149	—	2	1	no	—
9	#16810489	—	2	1	no	—
10	#16810427	—	2	4	no	—
11	#16810424	—	2	4	no	—
12	C₄×#11719908	—	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×#11719908	—	2	2	no	—
22	C₂×#11719908	—	2	2	no	—
23	C₂×#11719908	D₄	1	4	yes	—
24	#13133151	—	2	2	no	—
25	#13133151	—	2	2	no	—
26	#13133149	—	4	2	no	—
27	#13133124	—	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	#11719908	—	2	4	no	—
84	C₂³×C₅₃	—	4	2	no	—
85	C₂³×C₅₃	—	8	1	no	—
86	C₂³×C₅₃	—	8	1	no	—
87	C₂³×C₅₃	—	8	4	no	—
88	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
89	Q₈×C₅₃	—	4	2	no	—
90	Q₈×C₅₃	—	4	2	no	—
91	Q₈×C₅₃	—	8	1	no	—
92	Q₈×C₅₃	—	8	1	no	—
93	D₄×C₅₃	—	4	2	no	—
94	D₄×C₅₃	—	4	2	no	—
95	D₄×C₅₃	—	8	1	no	—
96	D₄×C₅₃	—	8	1	no	—
97	C₂×C₄×C₅₃	—	2	4	no	—
98	C₂×C₄×C₅₃	—	4	2	no	—
99	C₂×C₄×C₅₃	—	8	1	no	—
100	C₂×C₄×C₅₃	—	8	1	no	—
101	C₂×C₄×C₅₃	#102	1	4	yes	—
102	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
104	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
106	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
107	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
108	C₂×Dic₅₃	—	8	2	no	—
109	C₂×Dic₅₃	—	8	4	no	—
110	C₈×C₅₃	#102	1	4	yes	—
111	C₈×C₅₃	C₄×D₄	1	2	yes	—
112	C₈×C₅₃	#121	1	2	yes	—
113	#13204	—	53	1	no	maximal, Sylow
114	C₂²×C₅₃	—	8	4	no	—
115	C₂²×C₅₃	—	8	2	no	—
116	C₂²×C₅₃	—	16	2	no	—
117	C₂²×C₅₃	—	16	2	no	—
118	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
119	C₂²×C₅₃	#486	1	1	yes	characteristic
120	C₂²×C₅₃	#493	1	1	yes	characteristic
121	C₂²×C₅₃	#508	1	4	yes	—
122	C₄×C₅₃	—	2	4	no	—
123	C₄×C₅₃	—	8	2	no	—
124	C₄×C₅₃	#390	1	1	yes	characteristic
125	C₄×C₅₃	C₂×#102	1	1	yes	characteristic
126	C₄×C₅₃	#486	1	1	yes	characteristic
127	C₄×C₅₃	#492	1	1	yes	characteristic
128	Dic₅₃	—	16	2	no	—
129	C₂×#495	D₅₃	1	1	yes	characteristic
130	C₂×#358	—	53	2	no	—
131	C₂×#357	—	53	2	no	—
132	C₂×C₄×C₁₆	—	53	1	no	—
133	#1846	—	53	1	no	—
134	C₂×C₅₃	—	16	2	no	—
135	C₂×C₅₃	#1957	1	1	yes	characteristic
136	C₂×C₅₃	#2138	1	2	yes	—
137	C₂×C₅₃	#2242	1	2	yes	—
138	C₂×C₅₃	#2249	1	2	yes	—
139	C₂²×Q₁₆	C₂×D₅₃	1	1	yes	characteristic
140	C₂²×SD₁₆	—	2	1	no	—
141	C₂²×D₈	C₂×D₅₃	1	1	yes	characteristic
142	C₂×#124	—	2	1	no	—
143	C₂²×C₁₆	—	53	2	no	—
144	#495	—	2	2	no	—
145	#495	Dic₅₃	1	4	yes	complemented
146	C₂×#107	—	53	2	no	—
147	C₂×#106	—	106	1	no	—
148	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
149	#382	—	106	1	no	—
150	#358	—	106	4	no	—
151	#357	—	106	4	no	—
152	C₄×C₁₆	—	106	2	no	—
153	C₅₃	#13204	1	1	yes	characteristic, complemented, Sylow
154	C₂²×Q₈	—	2	1	no	—
155	C₂²×D₄	—	2	1	no	—
156	C₂×Q₁₆	—	2	2	no	—
157	C₂×Q₁₆	—	2	2	no	—
158	C₂×Q₁₆	—	2	1	no	—
159	C₂×Q₁₆	C₂×Dic₅₃	1	2	yes	—
160	C₂×SD₁₆	—	2	4	no	—
161	C₂×SD₁₆	—	4	2	no	—
162	C₂×SD₁₆	—	4	2	no	—
163	C₂×D₈	—	2	2	no	—
164	C₂×D₈	—	2	2	no	—
165	C₂×D₈	—	2	1	no	—
166	C₂×D₈	C₂×Dic₅₃	1	2	yes	—
167	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
168	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
169	#124	—	2	4	no	—
170	#124	—	4	2	no	—
171	C₂×C₄⋊C₄	—	106	2	no	—
172	C₂×C₂²⋊C₄	—	4	2	no	—
173	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
174	C₂×C₁₆	—	53	4	no	—
175	C₂×C₁₆	—	106	2	no	—
176	C₂×C₁₆	—	106	2	no	—
177	#107	—	106	2	no	—
178	#107	—	106	2	no	—
179	#106	—	212	2	no	—
180	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
181	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
182	#95	—	212	2	no	—
183	C₂⁴	—	4	1	no	—
184	C₂×Q₈	—	2	2	no	—
185	C₂×Q₈	—	4	1	no	—
186	C₂×Q₈	—	4	1	no	—
187	C₂×Q₈	—	4	1	no	—
188	C₂×Q₈	—	4	1	no	—
189	C₂×Q₈	—	4	1	no	—
190	C₂×D₄	—	2	2	no	—
191	C₂×D₄	—	4	1	no	—
192	C₂×D₄	—	4	1	no	—
193	C₂×D₄	—	4	1	no	—
194	C₂×D₄	—	4	1	no	—
195	C₂×D₄	—	4	1	no	—
196	C₂²×C₄	—	4	1	no	—
197	C₂²×C₄	—	212	2	no	—
198	C₂²×C₄	#11719922	1	2	yes	—
199	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
200	Q₁₆	—	2	4	no	—
201	Q₁₆	—	4	2	no	—
202	SD₁₆	—	4	8	no	—
203	D₈	—	2	4	no	—
204	D₈	—	4	2	no	—
205	C₂×C₈	#11719922	1	2	yes	—
206	C₂×C₈	#11719923	1	2	yes	—
207	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
208	C₂×C₈	D₄×D₅₃	1	2	yes	—
209	C₂×C₈	#11719947	1	2	yes	—
210	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
211	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
212	C₄⋊C₄	—	212	2	no	—
213	C₄⋊C₄	—	212	2	no	—
214	C₂²⋊C₄	—	4	8	no	—
215	C₄²	#11719923	1	2	yes	—
216	C₄²	C₂²⋊Dic₅₃	1	2	yes	—
217	C₁₆	—	106	4	no	—
218	C₂³	—	4	2	no	—
219	C₂³	—	8	4	no	—
220	C₂³	—	8	1	no	—
221	C₂³	—	8	1	no	—
222	C₂³	C₂×#11719922	1	1	yes	characteristic
223	Q₈	—	4	2	no	—
224	Q₈	—	4	2	no	—
225	Q₈	—	8	1	no	—
226	Q₈	—	8	1	no	—
227	D₄	—	4	2	no	—
228	D₄	—	4	2	no	—
229	D₄	—	8	1	no	—
230	D₄	—	8	1	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	8	1	no	—
234	C₂×C₄	—	8	1	no	—
235	C₂×C₄	—	424	4	no	—
236	C₂×C₄	—	424	2	no	—
237	C₂×C₄	#13133138	1	4	yes	—
238	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
239	C₂×C₄	#13133204	1	1	yes	characteristic
240	C₂×C₄	#13133213	1	1	yes	characteristic
241	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
242	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
243	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
244	C₈	#13133138	1	4	yes	—
245	C₈	D₄×Dic₅₃	1	2	yes	—
246	C₈	#13133188	1	2	yes	—
247	C₂²	—	8	2	no	—
248	C₂²	—	8	4	no	—
249	C₂²	—	16	2	no	—
250	C₂²	—	16	2	no	—
251	C₂²	C₂×#13133138	1	1	yes	characteristic
252	C₂²	#16810969	1	1	yes	characteristic
253	C₂²	#16810999	1	1	yes	characteristic
254	C₂²	#16811053	1	4	yes	—
255	C₄	—	2	4	no	—
256	C₄	—	8	2	no	—
257	C₄	—	848	2	no	—
258	C₄	#16810526	1	1	yes	characteristic
259	C₄	C₂×#13133138	1	1	yes	characteristic
260	C₄	#16810969	1	1	yes	characteristic
261	C₄	#16810995	1	1	yes	characteristic
262	C₂	—	16	2	no	—
263	C₂	#20275522	1	1	yes	characteristic
264	C₂	#20276422	1	2	yes	—
265	C₂	#20276946	1	2	yes	—
266	C₂	#20276974	1	2	yes	—

	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	#13204
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×C₄×#11719908	C₂	semidirect
4. #20274879	C₂	semidirect
18. C₅₃×#495	C₄	semidirect
145. #495	Dic₅₃	semidirect
153. C₅₃	#13204	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×C₄×#11719908	2
#20274879	2
C₂×C₅₃×#495	2
#13204	53