Subgroups — #27274042

Order 13568.

634 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810647	C₂	1	2	yes	maximal
2	#20272590	C₂	1	4	yes	maximal, complemented
3	C₂×C₅₃×#431	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11719917	—	2	2	no	—
5	#16810647	—	2	2	no	—
6	#16810647	—	2	2	no	—
7	#16810647	C₄	1	4	yes	—
8	#16810647	C₂²	1	4	yes	—
9	C₂×C₄⋊#88322	—	2	2	no	—
10	C₂×C₄⋊#88322	C₄	1	4	yes	—
11	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
12	C₅₃×#431	—	2	2	no	—
13	C₅₃×#431	—	2	2	no	—
14	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
15	C₂×C₂²⋊C₈×C₅₃	C₄	1	2	yes	—
16	C₂×C₄×C₈×C₅₃	C₄	1	2	yes	—
17	C₂×#11719917	—	2	4	no	—
18	C₂×#11719917	—	2	4	no	—
19	C₂×#11719917	—	4	4	no	—
20	C₂×#11719917	—	4	4	no	—
21	C₂²×#88322	—	2	4	no	—
22	C₄⋊#88322	—	2	4	no	—
23	C₄⋊#88322	—	2	4	no	—
24	C₄⋊#88322	—	2	4	no	—
25	C₄⋊#88322	C₂×C₄	1	8	yes	—
26	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
28	C₂³×C₄×C₅₃	Q₈	1	1	yes	characteristic
29	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
30	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₅₃	D₄	1	2	yes	—
34	C₂×C₄²×C₅₃	D₄	1	2	yes	—
35	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
37	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
38	C₄⋊C₈×C₅₃	—	2	4	no	—
39	C₂²⋊C₈×C₅₃	—	2	4	no	—
40	C₄×C₈×C₅₃	—	2	2	no	—
41	C₄×C₈×C₅₃	—	2	2	no	—
42	C₂⁴×C₅₃	C₄⋊C₄	1	1	yes	characteristic
43	C₂²×C₄×C₅₃	—	4	2	no	—
44	C₂²×C₄×C₅₃	—	4	2	no	—
45	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
46	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
47	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
48	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
49	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
51	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
53	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	D₈	1	2	yes	—
55	C₂²×C₄×C₅₃	SD₁₆	1	4	yes	—
56	C₂²×C₄×C₅₃	Q₁₆	1	2	yes	—
57	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₈×C₅₃	—	2	4	no	—
61	C₂×C₈×C₅₃	—	4	4	no	—
62	C₄²×C₅₃	C₄²	1	1	yes	characteristic
63	C₄²×C₅₃	C₄²	1	1	yes	characteristic
64	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
65	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
66	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
67	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
68	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
69	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
70	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
71	C₄²×C₅₃	C₂×D₄	1	2	yes	—
72	#11719917	—	4	8	no	—
73	#11719917	—	4	4	no	—
74	#11719917	—	4	4	no	—
75	C₂×#88322	—	2	8	no	—
76	C₂×#88322	—	4	8	no	—
77	C₂³×C₅₃	—	4	1	no	—
78	C₂³×C₅₃	—	4	1	no	—
79	C₂³×C₅₃	#95	1	2	yes	—
80	C₂³×C₅₃	#106	1	2	yes	—
81	C₂³×C₅₃	#107	1	2	yes	—
82	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₅₃	—	2	4	no	—
84	C₂×C₄×C₅₃	—	2	2	no	—
85	C₂×C₄×C₅₃	—	2	4	no	—
86	C₂×C₄×C₅₃	—	2	2	no	—
87	C₂×C₄×C₅₃	—	4	2	no	—
88	C₂×C₄×C₅₃	—	4	2	no	—
89	C₂×C₄×C₅₃	—	4	1	no	—
90	C₂×C₄×C₅₃	—	4	1	no	—
91	C₂×C₄×C₅₃	#95	1	2	yes	—
92	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
93	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
94	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
95	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
96	C₂×C₄×C₅₃	#102	1	4	yes	—
97	C₂×C₄×C₅₃	#102	1	4	yes	—
98	C₂×C₄×C₅₃	#103	1	4	yes	—
99	C₂×C₄×C₅₃	#103	1	4	yes	—
100	C₂×C₄×C₅₃	#106	1	2	yes	—
101	C₂×C₄×C₅₃	#107	1	2	yes	—
102	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
103	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
104	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
106	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
107	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
108	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
109	C₈×C₅₃	—	4	4	no	—
110	#88322	—	4	8	no	—
111	#8001	—	53	1	no	maximal, Sylow
112	C₂²×C₅₃	—	2	2	no	—
113	C₂²×C₅₃	—	2	1	no	—
114	C₂²×C₅₃	—	2	2	no	—
115	C₂²×C₅₃	—	2	1	no	—
116	C₂²×C₅₃	—	4	2	no	—
117	C₂²×C₅₃	—	4	2	no	—
118	C₂²×C₅₃	#340	1	4	yes	—
119	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
120	C₂²×C₅₃	C₂×#106	1	1	yes	characteristic
121	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
122	C₄×C₅₃	—	2	1	no	—
123	C₄×C₅₃	—	2	2	no	—
124	C₄×C₅₃	—	2	2	no	—
125	C₄×C₅₃	—	2	1	no	—
126	C₄×C₅₃	#338	1	2	yes	—
127	C₄×C₅₃	#340	1	2	yes	—
128	C₄×C₅₃	#340	1	2	yes	—
129	C₄×C₅₃	#413	1	2	yes	—
130	C₄×C₅₃	C₂×#102	1	2	yes	—
131	C₄×C₅₃	C₂×#103	1	2	yes	—
132	C₂×#431	D₅₃	1	1	yes	characteristic
133	C₂×#423	—	53	2	no	—
134	#1292	—	53	4	no	—
135	C₂×C₅₃	—	2	4	no	—
136	C₂×C₅₃	—	2	1	no	—
137	C₂×C₅₃	—	2	1	no	—
138	C₂×C₅₃	C₂×#338	1	1	yes	characteristic
139	C₂×C₅₃	C₂×#340	1	1	yes	characteristic
140	C₂×C₅₃	#1742	1	1	yes	characteristic
141	C₂²×M₁₆	—	106	2	no	—
142	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
143	#431	—	2	2	no	—
144	#431	—	2	2	no	—
145	#423	—	53	4	no	—
146	#423	—	53	4	no	—
147	#423	—	106	2	no	—
148	#423	—	106	2	no	—
149	C₂×C₄⋊C₈	—	53	4	no	—
150	C₂×C₄⋊C₈	—	106	2	no	—
151	C₂×C₄⋊C₈	C₂×D₅₃	1	2	yes	—
152	C₂×C₂²⋊C₈	Dic₅₃	1	2	yes	—
153	C₂×C₄×C₈	Dic₅₃	1	2	yes	—
154	C₅₃	#8001	1	1	yes	characteristic, complemented, Sylow
155	C₂³×C₄	Dic₁₀₆	1	1	yes	characteristic
156	C₂³×C₄	D₂₁₂	1	1	yes	characteristic
157	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
158	C₂×M₁₆	—	106	4	no	—
159	C₂×M₁₆	—	106	4	no	—
160	C₂×M₁₆	—	212	4	no	—
161	C₂×M₁₆	—	212	4	no	—
162	C₂²×C₈	—	106	4	no	—
163	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
164	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
165	C₂×C₄²	C₄×D₅₃	1	2	yes	—
166	C₂×C₄²	C₄×D₅₃	1	2	yes	—
167	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
168	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
169	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
170	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
171	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
172	C₄⋊C₈	—	2	4	no	—
173	C₄⋊C₈	—	53	8	no	—
174	C₄⋊C₈	—	106	4	no	—
175	C₄⋊C₈	—	106	4	no	—
176	C₄⋊C₈	—	106	4	no	—
177	C₂²⋊C₈	—	2	4	no	—
178	C₄×C₈	—	2	2	no	—
179	C₄×C₈	—	2	2	no	—
180	C₂⁴	C₄⋊Dic₅₃	1	1	yes	characteristic
181	C₂²×C₄	—	4	2	no	—
182	C₂²×C₄	—	4	2	no	—
183	C₂²×C₄	#11719913	1	4	yes	—
184	C₂²×C₄	D₄₂₄	1	2	yes	—
185	C₂²×C₄	Dic₂₁₂	1	2	yes	—
186	C₂²×C₄	C₄×Dic₅₃	1	2	yes	—
187	C₂²×C₄	#11719919	1	2	yes	—
188	C₂²×C₄	#11719919	1	2	yes	—
189	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
190	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
191	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
192	C₂²×C₄	#11719921	1	2	yes	—
193	C₂²×C₄	#11719921	1	2	yes	—
194	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
195	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
196	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
197	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
198	M₁₆	—	212	8	no	—
199	M₁₆	—	212	4	no	—
200	M₁₆	—	212	4	no	—
201	C₂×C₈	—	2	4	no	—
202	C₂×C₈	—	4	4	no	—
203	C₂×C₈	—	106	8	no	—
204	C₂×C₈	—	212	8	no	—
205	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
206	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
207	C₄²	#11719919	1	2	yes	—
208	C₄²	#11719919	1	2	yes	—
209	C₄²	#11719921	1	2	yes	—
210	C₄²	#11719921	1	2	yes	—
211	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
212	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
213	C₄²	C₂×C₄×D₅₃	1	2	yes	—
214	C₄²	C₂×C₂²⋊D₅₃	1	2	yes	—
215	C₂³	—	4	1	no	—
216	C₂³	—	4	1	no	—
217	C₂³	#13133124	1	2	yes	—
218	C₂³	#13133149	1	2	yes	—
219	C₂³	#13133151	1	2	yes	—
220	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	4	1	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	4	1	no	—
229	C₂×C₄	#13133124	1	1	yes	characteristic
230	C₂×C₄	#13133124	1	2	yes	—
231	C₂×C₄	#13133124	1	1	yes	characteristic
232	C₂×C₄	#13133124	1	1	yes	characteristic
233	C₂×C₄	#13133124	1	1	yes	characteristic
234	C₂×C₄	#13133139	1	4	yes	—
235	C₂×C₄	#13133139	1	4	yes	—
236	C₂×C₄	#13133142	1	4	yes	—
237	C₂×C₄	#13133142	1	4	yes	—
238	C₂×C₄	#13133149	1	2	yes	—
239	C₂×C₄	#13133151	1	2	yes	—
240	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
241	C₂×C₄	C₂×#11719921	1	2	yes	—
242	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
243	C₂×C₄	C₂×#11719919	1	2	yes	—
244	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
245	C₂×C₄	C₂×#11719913	1	2	yes	—
246	C₂×C₄	C₂×Dic₂₁₂	1	1	yes	characteristic
247	C₈	—	4	4	no	—
248	C₈	—	212	8	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	2	1	no	—
251	C₂²	—	2	2	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	4	2	no	—
254	C₂²	—	4	2	no	—
255	C₂²	#16810386	1	4	yes	—
256	C₂²	C₂×#13133124	1	1	yes	characteristic
257	C₂²	C₂×#13133149	1	1	yes	characteristic
258	C₂²	C₂×#13133151	1	1	yes	characteristic
259	C₄	—	2	2	no	—
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	2	2	no	—
263	C₄	#16810381	1	2	yes	—
264	C₄	#16810386	1	2	yes	—
265	C₄	#16810386	1	2	yes	—
266	C₄	#16810607	1	2	yes	—
267	C₄	C₂×#13133139	1	2	yes	—
268	C₄	C₂×#13133142	1	2	yes	—
269	C₂	—	2	4	no	—
270	C₂	—	2	1	no	—
271	C₂	—	2	1	no	—
272	C₂	C₂×#16810381	1	1	yes	characteristic
273	C₂	C₂×#16810386	1	1	yes	characteristic
274	C₂	#20274454	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13133124
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₅₃×#431	C₂
socle	C₂²×C₅₃	C₂×#95
prime radical	C₂⁴×C₅₃	C₄⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×#431	D₅₃
hypocenter	C₅₃	#8001
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
2. #20272590	C₂	semidirect
154. C₅₃	#8001	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810647	2
#20272590	2
C₂×C₅₃×#431	2
#8001	53