Subgroups — #27273571

Order 13568.

574 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810680	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810676	C₂	1	1	yes	maximal, characteristic
3	#20272549	C₂	1	4	yes	maximal, complemented
4	C₂×C₅₃×#431	C₂	1	1	yes	maximal, characteristic
5	#16810680	—	2	2	no	—
6	#16810680	C₄	1	4	yes	—
7	#16810676	—	2	2	no	—
8	#16810676	—	2	2	no	—
9	C₂×C₄⋊#88322	C₄	1	2	yes	—
10	C₂×C₄⋊#88322	C₂²	1	2	yes	—
11	C₂×C₂²⋊#88322	—	2	1	no	—
12	C₂×C₂²⋊#88322	C₄	1	2	yes	—
13	C₂×#13133128	—	2	1	no	—
14	C₂×C₄×#88322	C₂²	1	2	yes	—
15	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#431	—	2	2	no	—
17	C₅₃×#431	C₄	1	4	yes	—
18	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
19	C₂×C₂²⋊C₈×C₅₃	—	2	1	no	—
20	C₂×C₄×C₈×C₅₃	—	2	1	no	—
21	C₂²×#88322	—	2	2	no	—
22	C₂²×#88322	C₂×C₄	1	4	yes	—
23	C₄⋊#88322	—	2	4	no	—
24	C₄⋊#88322	—	2	2	no	—
25	C₄⋊#88322	C₂×C₄	1	4	yes	—
26	C₂²⋊#88322	—	2	4	no	—
27	C₂²⋊#88322	—	2	4	no	—
28	#13133128	—	2	4	no	—
29	C₄×#88322	—	2	2	no	—
30	C₄×#88322	—	2	2	no	—
31	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
33	C₂³×C₄×C₅₃	Q₈	1	1	yes	characteristic
34	C₂²×C₈×C₅₃	—	2	2	no	—
35	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₅₃	D₄	1	2	yes	—
39	C₂×C₄²×C₅₃	D₄	1	2	yes	—
40	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
42	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
43	C₄⋊C₈×C₅₃	—	2	2	no	—
44	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
45	C₂²⋊C₈×C₅₃	—	2	4	no	—
46	C₄×C₈×C₅₃	—	2	4	no	—
47	C₂⁴×C₅₃	C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	—	2	4	no	—
49	C₂²×C₄×C₅₃	—	2	4	no	—
50	C₂²×C₄×C₅₃	—	4	2	no	—
51	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
52	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
53	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
54	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
55	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
59	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
60	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
63	C₂×C₈×C₅₃	—	2	4	no	—
64	C₂×C₈×C₅₃	—	4	4	no	—
65	C₄²×C₅₃	C₄²	1	2	yes	—
66	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
67	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
68	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
69	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
70	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
71	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
72	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
73	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
75	C₂×#88322	—	2	4	no	—
76	C₂×#88322	—	2	4	no	—
77	C₂×#88322	—	2	8	no	—
78	C₂×#88322	—	4	4	no	—
79	C₂³×C₅₃	—	2	2	no	—
80	C₂³×C₅₃	—	2	2	no	—
81	C₂³×C₅₃	—	2	2	no	—
82	C₂³×C₅₃	#95	1	2	yes	—
83	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₅₃	—	2	2	no	—
85	C₂×C₄×C₅₃	—	2	2	no	—
86	C₂×C₄×C₅₃	—	2	2	no	—
87	C₂×C₄×C₅₃	—	2	2	no	—
88	C₂×C₄×C₅₃	—	2	4	no	—
89	C₂×C₄×C₅₃	—	2	4	no	—
90	C₂×C₄×C₅₃	—	2	4	no	—
91	C₂×C₄×C₅₃	—	2	4	no	—
92	C₂×C₄×C₅₃	—	4	2	no	—
93	C₂×C₄×C₅₃	—	4	2	no	—
94	C₂×C₄×C₅₃	#95	1	2	yes	—
95	C₂×C₄×C₅₃	#95	1	2	yes	—
96	C₂×C₄×C₅₃	#95	1	2	yes	—
97	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
98	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₅₃	#131	1	4	yes	—
104	C₂×C₄×C₅₃	#135	1	4	yes	—
105	C₈×C₅₃	—	4	4	no	—
106	#88322	—	4	4	no	—
107	#88322	—	4	4	no	—
108	#7878	—	53	1	no	maximal, Sylow
109	C₂²×C₅₃	—	2	2	no	—
110	C₂²×C₅₃	—	2	2	no	—
111	C₂²×C₅₃	—	2	2	no	—
112	C₂²×C₅₃	—	4	4	no	—
113	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
114	C₂²×C₅₃	#424	1	2	yes	—
115	C₂²×C₅₃	#427	1	2	yes	—
116	C₂²×C₅₃	#430	1	2	yes	—
117	C₄×C₅₃	—	2	2	no	—
118	C₄×C₅₃	—	2	2	no	—
119	C₄×C₅₃	—	2	2	no	—
120	C₄×C₅₃	—	2	1	no	—
121	C₄×C₅₃	—	2	1	no	—
122	C₄×C₅₃	#405	1	2	yes	—
123	C₄×C₅₃	#408	1	2	yes	—
124	C₄×C₅₃	#416	1	2	yes	—
125	C₄×C₅₃	#416	1	2	yes	—
126	C₂×#432	—	53	1	no	—
127	C₂×#431	—	53	1	no	—
128	C₂×#431	D₅₃	1	1	yes	characteristic
129	#1277	—	53	4	no	—
130	C₂×C₅₃	—	4	2	no	—
131	C₂×C₅₃	#1572	1	4	yes	—
132	C₂×C₅₃	#1729	1	1	yes	characteristic
133	C₂×C₅₃	#1735	1	1	yes	characteristic
134	C₂×C₅₃	#1738	1	1	yes	characteristic
135	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
136	#432	—	53	4	no	—
137	#432	—	106	2	no	—
138	#431	—	2	2	no	—
139	#431	—	106	2	no	—
140	#431	—	106	2	no	—
141	#431	Dic₅₃	1	4	yes	—
142	C₂×C₄⋊C₈	—	53	2	no	—
143	C₂×C₄⋊C₈	—	53	2	no	—
144	C₂×C₄⋊C₈	C₂×D₅₃	1	2	yes	—
145	C₂×C₂²⋊C₈	—	2	1	no	—
146	C₂×C₂²⋊C₈	—	53	2	no	—
147	C₂×C₂²⋊C₈	—	106	1	no	—
148	C₂×#97	—	106	1	no	—
149	C₂×C₄×C₈	—	2	1	no	—
150	C₂×C₄×C₈	—	53	2	no	—
151	C₅₃	#7878	1	1	yes	characteristic, complemented, Sylow
152	C₂³×C₄	Dic₁₀₆	1	1	yes	characteristic
153	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
154	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
155	C₂²×C₈	—	2	2	no	—
156	C₂²×C₈	—	53	4	no	—
157	C₂²×C₈	—	106	2	no	—
158	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
159	C₂×C₄²	C₄×D₅₃	1	2	yes	—
160	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
161	C₂×C₄²	D₂₁₂	1	2	yes	—
162	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
163	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
164	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
165	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
166	C₄⋊C₈	—	2	2	no	—
167	C₄⋊C₈	—	53	4	no	—
168	C₄⋊C₈	—	106	2	no	—
169	C₄⋊C₈	—	106	4	no	—
170	C₄⋊C₈	C₂×Dic₅₃	1	4	yes	—
171	C₂²⋊C₈	—	2	4	no	—
172	C₂²⋊C₈	—	106	4	no	—
173	C₂²⋊C₈	—	106	4	no	—
174	#97	—	106	4	no	—
175	C₄×C₈	—	2	4	no	—
176	C₄×C₈	—	106	2	no	—
177	C₄×C₈	—	106	2	no	—
178	C₂⁴	#11719919	1	1	yes	characteristic
179	C₂²×C₄	—	2	4	no	—
180	C₂²×C₄	—	2	4	no	—
181	C₂²×C₄	—	4	2	no	—
182	C₂²×C₄	C₄×Dic₅₃	1	2	yes	—
183	C₂²×C₄	#11719919	1	1	yes	characteristic
184	C₂²×C₄	#11719919	1	1	yes	characteristic
185	C₂²×C₄	#11719919	1	1	yes	characteristic
186	C₂²×C₄	#11719919	1	2	yes	—
187	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
188	C₂²×C₄	#11719921	1	2	yes	—
189	C₂²×C₄	#11719921	1	2	yes	—
190	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
191	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
192	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
193	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
194	C₂×C₈	—	2	4	no	—
195	C₂×C₈	—	4	4	no	—
196	C₂×C₈	—	106	4	no	—
197	C₂×C₈	—	106	4	no	—
198	C₂×C₈	—	106	8	no	—
199	C₂×C₈	—	212	4	no	—
200	C₄²	C₄×Dic₅₃	1	2	yes	—
201	C₄²	#11719919	1	2	yes	—
202	C₄²	C₄⋊Dic₅₃	1	2	yes	—
203	C₄²	#11719921	1	2	yes	—
204	C₄²	#11719921	1	2	yes	—
205	C₄²	C₂²⋊Dic₅₃	1	2	yes	—
206	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
207	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
208	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
209	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
210	C₂³	—	2	2	no	—
211	C₂³	—	2	2	no	—
212	C₂³	—	2	2	no	—
213	C₂³	#13133124	1	2	yes	—
214	C₂³	C₂×#11719919	1	1	yes	characteristic
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	4	no	—
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	#13133124	1	2	yes	—
226	C₂×C₄	#13133124	1	2	yes	—
227	C₂×C₄	#13133124	1	2	yes	—
228	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
229	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
230	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
231	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
232	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
233	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
234	C₂×C₄	#13133223	1	4	yes	—
235	C₂×C₄	#13133237	1	4	yes	—
236	C₈	—	4	4	no	—
237	C₈	—	212	4	no	—
238	C₈	—	212	4	no	—
239	C₂²	—	2	2	no	—
240	C₂²	—	2	2	no	—
241	C₂²	—	2	2	no	—
242	C₂²	—	4	4	no	—
243	C₂²	C₂×#13133124	1	1	yes	characteristic
244	C₂²	#16810649	1	2	yes	—
245	C₂²	#16810659	1	2	yes	—
246	C₂²	#16810671	1	2	yes	—
247	C₄	—	2	2	no	—
248	C₄	—	2	1	no	—
249	C₄	—	2	1	no	—
250	C₄	—	2	2	no	—
251	C₄	—	2	2	no	—
252	C₄	#16810578	1	2	yes	—
253	C₄	#16810589	1	2	yes	—
254	C₄	#16810617	1	2	yes	—
255	C₄	#16810619	1	2	yes	—
256	C₂	—	4	2	no	—
257	C₂	#20273493	1	4	yes	—
258	C₂	#20274411	1	1	yes	characteristic
259	C₂	#20274431	1	1	yes	characteristic
260	C₂	#20274441	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₅₃×#431	C₂
socle	C₂³×C₅₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₅₃	C₄⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×#431	D₅₃
hypocenter	C₅₃	#7878
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
3. #20272549	C₂	semidirect
151. C₅₃	#7878	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810680	2
C₂×#16810676	2
#20272549	2
C₂×C₅₃×#431	2
#7878	53