Subgroups — #27274620

Order 13568.

506 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20273029	C₂	1	2	yes	maximal
2	#20272998	C₂	1	2	yes	maximal, complemented
3	#20272560	C₂	1	2	yes	maximal, complemented
4	C₂×C₅₃×#405	C₂	1	1	yes	maximal, characteristic
5	#16810676	C₂²	1	4	yes	—
6	#16810349	C₄	1	4	yes	—
7	#16810347	C₄	1	4	yes	—
8	#16810344	—	2	2	no	—
9	C₈×#88322	—	2	2	no	—
10	C₂²×M₁₆×C₅₃	C₄	1	1	yes	characteristic
11	C₂³×C₈×C₅₃	C₄	1	1	yes	characteristic
12	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
13	C₅₃×#405	—	2	2	no	—
14	C₅₃×#405	—	2	2	no	—
15	C₂×#97×C₅₃	C₄	1	1	yes	characteristic
16	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
18	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₄⋊#88322	—	2	4	no	—
20	C₂²⋊#88322	C₂×C₄	1	8	yes	—
21	C₄×#88322	—	2	4	no	—
22	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂×M₁₆×C₅₃	—	2	2	no	—
24	C₂×M₁₆×C₅₃	—	2	2	no	—
25	C₂×M₁₆×C₅₃	C₈	1	4	yes	complemented
26	C₂²×C₈×C₅₃	—	2	1	no	—
27	C₂²×C₈×C₅₃	—	2	1	no	—
28	C₂²×C₈×C₅₃	—	2	2	no	—
29	C₂²×C₈×C₅₃	C₈	1	4	yes	complemented
30	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
34	#117×C₅₃	—	2	2	no	—
35	#117×C₅₃	—	2	2	no	—
36	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
37	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
40	#97×C₅₃	—	2	2	no	—
41	#97×C₅₃	—	2	2	no	—
42	C₄×C₈×C₅₃	—	2	1	no	—
43	C₄×C₈×C₅₃	—	2	1	no	—
44	C₄×C₈×C₅₃	D₄	1	2	yes	—
45	C₄×C₈×C₅₃	D₄	1	2	yes	—
46	C₂⁴×C₅₃	C₄²	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	—	2	1	no	—
48	C₂²×C₄×C₅₃	—	2	1	no	—
49	C₂²×C₄×C₅₃	—	2	1	no	—
50	C₂²×C₄×C₅₃	—	2	1	no	—
51	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
53	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
54	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
55	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
56	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
57	M₁₆×C₅₃	—	2	8	no	—
58	C₂×C₈×C₅₃	—	2	1	no	—
59	C₂×C₈×C₅₃	—	2	1	no	—
60	C₂×C₈×C₅₃	—	2	1	no	—
61	C₂×C₈×C₅₃	—	2	1	no	—
62	C₂×C₈×C₅₃	—	2	4	no	—
63	C₂×C₈×C₅₃	—	2	4	no	—
64	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
66	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
68	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
69	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
70	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
71	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
72	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
73	C₄⋊C₄×C₅₃	—	2	4	no	—
74	C₂²⋊C₄×C₅₃	—	2	4	no	—
75	C₄²×C₅₃	—	2	1	no	—
76	C₄²×C₅₃	—	2	1	no	—
77	C₄²×C₅₃	C₂×D₄	1	2	yes	—
78	C₄²×C₅₃	#41	1	2	yes	—
79	C₂×#88322	—	2	8	no	—
80	C₂³×C₅₃	—	2	1	no	—
81	C₂³×C₅₃	—	2	1	no	—
82	C₂³×C₅₃	—	4	2	no	—
83	C₂³×C₅₃	C₄×C₈	1	2	yes	—
84	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
85	C₂×C₄×C₅₃	—	2	1	no	—
86	C₂×C₄×C₅₃	—	2	1	no	—
87	C₂×C₄×C₅₃	—	2	2	no	—
88	C₂×C₄×C₅₃	—	2	2	no	—
89	C₂×C₄×C₅₃	—	2	2	no	—
90	C₂×C₄×C₅₃	—	2	2	no	—
91	C₂×C₄×C₅₃	—	2	2	no	—
92	C₂×C₄×C₅₃	—	4	2	no	—
93	C₂×C₄×C₅₃	—	4	2	no	—
94	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
95	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
97	C₂×C₄×C₅₃	C₄×D₄	1	4	yes	—
98	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
99	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
100	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
101	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
102	C₈×C₅₃	—	2	2	no	—
103	C₈×C₅₃	—	2	2	no	—
104	C₈×C₅₃	C₂²⋊C₈	1	2	yes	—
105	C₈×C₅₃	C₂²⋊C₈	1	2	yes	—
106	C₈×C₅₃	C₂²⋊C₈	1	4	yes	—
107	#88322	—	4	8	no	—
108	#8156	—	53	1	no	maximal, Sylow
109	C₂²×C₅₃	—	2	2	no	—
110	C₂²×C₅₃	—	2	2	no	—
111	C₂²×C₅₃	—	4	2	no	—
112	C₂²×C₅₃	—	4	2	no	—
113	C₂²×C₅₃	—	4	2	no	—
114	C₂²×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
115	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
116	C₂²×C₅₃	C₄×M₁₆	1	1	yes	characteristic
117	C₄×C₅₃	—	2	2	no	—
118	C₄×C₅₃	—	2	4	no	—
119	C₄×C₅₃	—	4	2	no	—
120	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
121	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
122	C₄×C₅₃	#431	1	1	yes	characteristic
123	C₄×C₅₃	#431	1	1	yes	characteristic
124	C₂×#405	D₅₃	1	1	yes	characteristic
125	#1458	—	53	2	no	—
126	#1450	—	53	2	no	—
127	#1281	—	53	2	no	—
128	C₂×C₅₃	—	2	1	no	—
129	C₂×C₅₃	—	2	1	no	—
130	C₂×C₅₃	—	4	2	no	—
131	C₂×C₅₃	C₄×C₂²⋊C₈	1	1	yes	characteristic
132	C₂×C₅₃	#1757	1	2	yes	—
133	C₂²×M₁₆	Dic₅₃	1	1	yes	characteristic
134	C₂³×C₈	Dic₅₃	1	1	yes	characteristic
135	C₂×#117	C₂×D₅₃	1	1	yes	characteristic
136	#431	—	53	4	no	—
137	#405	—	2	2	no	—
138	#405	—	2	2	no	—
139	C₂×#97	Dic₅₃	1	1	yes	characteristic
140	C₂×#97	C₂×D₅₃	1	1	yes	characteristic
141	C₂×C₄×C₈	Dic₅₃	1	1	yes	characteristic
142	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
143	#324	—	53	4	no	—
144	#323	—	53	4	no	—
145	#322	—	106	2	no	—
146	C₈²	—	106	2	no	—
147	C₅₃	#8156	1	1	yes	characteristic, complemented, Sylow
148	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
149	C₂×M₁₆	—	2	2	no	—
150	C₂×M₁₆	—	2	2	no	—
151	C₂×M₁₆	#88322	1	4	yes	complemented
152	C₂²×C₈	—	2	2	no	—
153	C₂²×C₈	—	2	1	no	—
154	C₂²×C₈	—	2	1	no	—
155	C₂²×C₈	#88322	1	4	yes	complemented
156	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
157	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
158	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
159	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
160	#117	—	2	2	no	—
161	#117	—	2	2	no	—
162	C₂×C₄⋊C₄	C₄×D₅₃	1	2	yes	—
163	C₂×C₂²⋊C₄	C₄×D₅₃	1	2	yes	—
164	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
165	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
166	C₄⋊C₈	—	106	4	no	—
167	C₂²⋊C₈	—	53	8	no	—
168	#97	—	2	2	no	—
169	#97	—	2	2	no	—
170	C₄×C₈	—	2	1	no	—
171	C₄×C₈	—	2	1	no	—
172	C₄×C₈	—	106	4	no	—
173	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
174	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
175	C₂⁴	C₄×Dic₅₃	1	1	yes	characteristic
176	C₂²×C₄	—	2	1	no	—
177	C₂²×C₄	—	2	1	no	—
178	C₂²×C₄	—	2	1	no	—
179	C₂²×C₄	—	2	1	no	—
180	C₂²×C₄	C₂×#88322	1	2	yes	—
181	C₂²×C₄	C₂×#88322	1	2	yes	—
182	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
183	C₂²×C₄	C₄×Dic₅₃	1	2	yes	—
184	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
185	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
186	M₁₆	—	2	8	no	—
187	C₂×C₈	—	2	1	no	—
188	C₂×C₈	—	2	1	no	—
189	C₂×C₈	—	2	1	no	—
190	C₂×C₈	—	2	1	no	—
191	C₂×C₈	—	2	4	no	—
192	C₂×C₈	—	2	4	no	—
193	C₂×C₈	—	106	8	no	—
194	C₂×C₈	C₂×#88322	1	2	yes	—
195	C₂×C₈	C₂×#88322	1	2	yes	—
196	C₂×C₈	C₂×#88322	1	2	yes	—
197	C₂×C₈	C₂×#88322	1	2	yes	—
198	C₂×C₈	#11719917	1	2	yes	—
199	C₂×C₈	#11719917	1	2	yes	—
200	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
201	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
202	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
203	C₄⋊C₄	—	2	4	no	—
204	C₂²⋊C₄	—	2	4	no	—
205	C₄²	—	2	1	no	—
206	C₄²	—	2	1	no	—
207	C₄²	#11719945	1	2	yes	—
208	C₄²	C₂×C₂²⋊D₅₃	1	2	yes	—
209	C₂³	—	2	1	no	—
210	C₂³	—	2	1	no	—
211	C₂³	—	4	2	no	—
212	C₂³	C₄×#88322	1	2	yes	—
213	C₂³	C₂×C₄×Dic₅₃	1	1	yes	characteristic
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	C₄×#88322	1	2	yes	—
224	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
225	C₂×C₄	#13133176	1	1	yes	characteristic
226	C₂×C₄	C₄×C₂²⋊D₅₃	1	4	yes	—
227	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
228	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
229	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
230	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
231	C₈	—	2	2	no	—
232	C₈	—	2	2	no	—
233	C₈	—	212	8	no	—
234	C₈	C₂²⋊#88322	1	2	yes	—
235	C₈	C₂²⋊#88322	1	2	yes	—
236	C₈	C₂²⋊#88322	1	4	yes	—
237	C₂²	—	2	2	no	—
238	C₂²	—	2	2	no	—
239	C₂²	—	4	2	no	—
240	C₂²	—	4	2	no	—
241	C₂²	—	4	2	no	—
242	C₂²	C₄×C₂²⋊Dic₅₃	1	1	yes	characteristic
243	C₂²	C₂×C₄×#88322	1	1	yes	characteristic
244	C₂²	C₄×#11719917	1	1	yes	characteristic
245	C₄	—	2	2	no	—
246	C₄	—	2	4	no	—
247	C₄	—	4	2	no	—
248	C₄	C₂×C₂²⋊#88322	1	1	yes	characteristic
249	C₄	C₂×C₂²⋊#88322	1	1	yes	characteristic
250	C₄	#16810676	1	1	yes	characteristic
251	C₄	#16810676	1	1	yes	characteristic
252	C₂	—	2	1	no	—
253	C₂	—	2	1	no	—
254	C₂	—	4	2	no	—
255	C₂	C₄×C₂²⋊#88322	1	1	yes	characteristic
256	C₂	#20274515	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂²⋊Dic₅₃
commutator	C₂²×C₅₃	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₅₃×#405	C₂
socle	C₂²×C₅₃	C₄×C₂²⋊C₄
prime radical	C₂⁴×C₅₃	C₄²
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×#405	D₅₃
hypocenter	C₅₃	#8156
LCS	G ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
2. #20272998	C₂	semidirect
3. #20272560	C₂	semidirect
25. C₂×M₁₆×C₅₃	C₈	semidirect
29. C₂²×C₈×C₅₃	C₈	semidirect
147. C₅₃	#8156	semidirect
151. C₂×M₁₆	#88322	semidirect
155. C₂²×C₈	#88322	semidirect

Maximal subgroups

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

subgroup	index
#20273029	2
#20272998	2
#20272560	2
C₂×C₅₃×#405	2
#8156	53