Subgroups — #27536732

Order 13568.

826 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133157	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810578	C₂	1	1	yes	maximal, characteristic
3	#20276314	C₂	1	4	yes	maximal, complemented
4	#20276311	C₂	1	4	yes	maximal, complemented
5	M₃₂×Dic₅₃	C₂	1	4	yes	maximal, complemented
6	C₂²×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11719917	C₄	1	1	yes	characteristic
8	C₂×#13133176	C₄	1	1	yes	characteristic
9	C₂×#13133157	C₄	1	8	yes	complemented
10	C₂×#13133157	C₂²	1	2	yes	—
11	C₂×#13133157	C₂²	1	2	yes	—
12	C₂²×#11719908	C₂²	1	2	yes	—
13	#16810578	C₄	1	2	yes	—
14	#16810578	C₄	1	2	yes	—
15	#16810578	C₂²	1	2	yes	—
16	#16810578	C₂²	1	2	yes	—
17	C₂×#13133127	C₄	1	2	yes	—
18	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
19	#16810398	C₂²	1	8	yes	complemented
20	C₁₆×Dic₅₃	C₂²	1	8	yes	complemented
21	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₂×M₃₂×C₅₃	C₄	1	8	yes	complemented
23	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
24	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
25	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
26	C₂×#11719917	C₈	1	8	yes	—
27	C₂×#11719917	C₂×C₄	1	2	yes	—
28	C₂×#11719917	C₂×C₄	1	2	yes	—
29	C₂²×#88322	C₂×C₄	1	2	yes	—
30	#13133176	C₈	1	2	yes	—
31	#13133176	C₈	1	2	yes	—
32	#13133176	C₂×C₄	1	2	yes	—
33	#13133176	C₂×C₄	1	2	yes	—
34	C₂×C₄⋊Dic₅₃	C₈	1	2	yes	—
35	C₂×C₂²⋊Dic₅₃	C₈	1	2	yes	—
36	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
37	#13133157	C₂×C₄	1	8	yes	—
38	#13133157	C₂×C₄	1	8	yes	complemented
39	C₂×#11719908	C₂×C₄	1	8	yes	complemented
40	C₂×#11719908	C₂³	1	4	yes	—
41	#13133127	C₂×C₄	1	4	yes	—
42	#13133127	C₂×C₄	1	4	yes	—
43	C₈×Dic₅₃	C₂×C₄	1	4	yes	—
44	C₈×Dic₅₃	C₂³	1	4	yes	—
45	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
47	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
48	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
50	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
51	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
53	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
54	M₃₂×C₅₃	C₂×C₄	1	8	yes	—
55	M₃₂×C₅₃	C₂×C₄	1	8	yes	complemented
56	C₂×C₁₆×C₅₃	C₂×C₄	1	8	yes	complemented
57	C₂×C₁₆×C₅₃	C₂³	1	4	yes	—
58	C₂⁴×C₅₃	C₂×C₈	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	C₄²	1	4	yes	—
60	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	C₂×C₈	1	4	yes	—
64	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×Dic₅₃	C₂×C₈	1	4	yes	—
68	C₂×C₈×C₅₃	C₄²	1	4	yes	—
69	C₂×C₈×C₅₃	C₄²	1	4	yes	—
70	C₂×C₈×C₅₃	C₄²	1	4	yes	—
71	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
75	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
77	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
81	C₂²⋊Dic₅₃	C₂×C₈	1	4	yes	—
82	C₂²⋊Dic₅₃	C₂×C₈	1	4	yes	—
83	C₄⋊Dic₅₃	C₂×C₈	1	4	yes	—
84	C₄⋊Dic₅₃	C₂×C₈	1	4	yes	—
85	C₄×Dic₅₃	C₂×C₈	1	4	yes	—
86	C₄×Dic₅₃	C₂²×C₄	1	4	yes	—
87	#11719917	C₂×C₈	1	8	yes	—
88	#11719917	C₂×C₈	1	8	yes	—
89	C₂×#88322	C₂×C₈	1	8	yes	—
90	C₂×#88322	C₂²×C₄	1	4	yes	—
91	C₁₆×C₅₃	C₂²×C₄	1	8	yes	—
92	#11719908	C₂²×C₄	1	8	yes	—
93	C₂³×C₅₃	—	4	2	no	—
94	C₂³×C₅₃	C₄×C₈	1	4	yes	—
95	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
96	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
97	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
98	C₂×C₄×C₅₃	C₄×C₈	1	4	yes	—
99	C₂×C₄×C₅₃	C₄×C₈	1	4	yes	—
100	C₂×C₄×C₅₃	C₄×C₈	1	4	yes	—
101	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
102	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
103	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
104	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
105	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
106	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
107	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
108	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
109	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
110	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
111	C₂×Dic₅₃	—	4	4	no	—
112	C₂×Dic₅₃	C₂²×C₈	1	8	yes	—
113	C₈×C₅₃	C₂×C₄²	1	2	yes	—
114	C₈×C₅₃	C₂×C₄²	1	2	yes	—
115	C₈×C₅₃	C₂×C₄²	1	2	yes	—
116	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
117	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
118	#88322	C₂²×C₈	1	8	yes	—
119	#33324	—	53	1	no	maximal, Sylow
120	C₂²×C₅₃	—	2	2	no	—
121	C₂²×C₅₃	—	2	2	no	—
122	C₂²×C₅₃	—	2	2	no	—
123	C₂²×C₅₃	—	4	4	no	—
124	C₂²×C₅₃	C₂×C₄×C₈	1	2	yes	—
125	C₂²×C₅₃	C₂×C₄×C₈	1	2	yes	—
126	C₂²×C₅₃	C₂×C₄×C₈	1	2	yes	—
127	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
128	C₄×C₅₃	C₂×C₄×C₈	1	2	yes	—
129	C₄×C₅₃	C₂×C₄×C₈	1	2	yes	—
130	C₄×C₅₃	C₂×C₄×C₈	1	2	yes	—
131	C₄×C₅₃	C₂²×C₄²	1	1	yes	characteristic
132	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
133	Dic₅₃	—	2	8	no	—
134	C₂²×M₃₂	—	53	1	no	—
135	C₂²×M₃₂	D₅₃	1	1	yes	characteristic
136	C₂×#405	—	53	1	no	—
137	#2108	—	53	4	no	—
138	#2108	—	53	4	no	—
139	C₄×M₃₂	—	53	4	no	—
140	C₂×C₅₃	—	2	2	no	—
141	C₂×C₅₃	—	2	2	no	—
142	C₂×C₅₃	—	2	2	no	—
143	C₂×C₅₃	C₂²×C₄×C₈	1	1	yes	characteristic
144	C₂×C₅₃	#3407	1	2	yes	—
145	C₂²×M₁₆	—	53	1	no	—
146	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
147	C₂×#117	—	53	1	no	—
148	C₂×M₃₂	—	53	8	no	—
149	C₂×M₃₂	—	53	2	no	—
150	C₂×M₃₂	—	53	2	no	—
151	C₂×M₃₂	Dic₅₃	1	8	yes	complemented
152	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
153	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
154	C₂²×C₁₆	—	53	2	no	—
155	C₂²×C₁₆	C₂×D₅₃	1	2	yes	—
156	#405	—	53	2	no	—
157	#405	—	53	2	no	—
158	#405	—	53	2	no	—
159	#405	—	53	2	no	—
160	C₂×#97	—	53	2	no	—
161	C₂×C₄×C₈	—	53	2	no	—
162	#346	—	53	8	no	—
163	C₄×C₁₆	—	53	8	no	—
164	C₅₃	#33324	1	1	yes	characteristic, complemented, Sylow
165	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
166	C₂×M₁₆	—	53	2	no	—
167	C₂×M₁₆	—	53	2	no	—
168	C₂×M₁₆	—	53	8	no	—
169	C₂²×C₈	—	53	2	no	—
170	C₂²×C₈	C₄×D₅₃	1	4	yes	—
171	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
172	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
173	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
174	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
175	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
176	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
177	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
178	#117	—	53	2	no	—
179	#117	—	53	2	no	—
180	#117	—	53	2	no	—
181	#117	—	53	2	no	—
182	C₂×C₄⋊C₄	—	53	2	no	—
183	C₂×C₂²⋊C₄	—	53	2	no	—
184	C₂×C₄²	—	53	2	no	—
185	M₃₂	—	53	8	no	—
186	M₃₂	—	53	8	no	—
187	M₃₂	C₂×Dic₅₃	1	8	yes	—
188	M₃₂	C₂×Dic₅₃	1	8	yes	complemented
189	C₂×C₁₆	—	53	8	no	—
190	C₂×C₁₆	—	53	4	no	—
191	C₂×C₁₆	C₂×Dic₅₃	1	8	yes	complemented
192	C₂×C₁₆	C₂²×D₅₃	1	4	yes	—
193	#97	—	53	4	no	—
194	#97	—	53	4	no	—
195	C₄×C₈	—	53	4	no	—
196	C₄×C₈	—	53	4	no	—
197	C₂⁴	C₈×D₅₃	1	1	yes	characteristic
198	C₂²×C₄	—	53	4	no	—
199	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
200	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
201	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
202	C₂²×C₄	C₈×D₅₃	1	4	yes	—
203	C₂²×C₄	C₄×Dic₅₃	1	4	yes	—
204	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
205	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
206	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
207	M₁₆	—	53	8	no	—
208	M₁₆	—	53	8	no	—
209	C₂×C₈	—	53	8	no	—
210	C₂×C₈	—	53	4	no	—
211	C₂×C₈	C₄×Dic₅₃	1	4	yes	—
212	C₂×C₈	C₄×Dic₅₃	1	4	yes	—
213	C₂×C₈	C₄×Dic₅₃	1	4	yes	—
214	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
215	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
216	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
217	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
218	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
219	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
220	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
221	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
222	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
223	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
224	C₄⋊C₄	—	53	4	no	—
225	C₄⋊C₄	—	53	4	no	—
226	C₂²⋊C₄	—	53	4	no	—
227	C₂²⋊C₄	—	53	4	no	—
228	C₄²	—	53	4	no	—
229	C₄²	—	53	4	no	—
230	C₁₆	—	53	8	no	—
231	C₁₆	C₂²×Dic₅₃	1	8	yes	—
232	C₂³	—	4	2	no	—
233	C₂³	C₈×Dic₅₃	1	4	yes	—
234	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
235	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
236	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
237	C₂×C₄	—	53	8	no	—
238	C₂×C₄	—	212	4	no	—
239	C₂×C₄	C₈×Dic₅₃	1	4	yes	—
240	C₂×C₄	C₈×Dic₅₃	1	4	yes	—
241	C₂×C₄	C₈×Dic₅₃	1	4	yes	—
242	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
243	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
244	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
245	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
246	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
247	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
248	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
249	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
250	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
251	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
252	C₈	—	53	8	no	—
253	C₈	C₂×C₄×Dic₅₃	1	2	yes	—
254	C₈	C₂×C₄×Dic₅₃	1	2	yes	—
255	C₈	C₂×C₄×Dic₅₃	1	2	yes	—
256	C₈	C₂²×C₄×D₅₃	1	1	yes	characteristic
257	C₈	C₂³×Dic₅₃	1	1	yes	characteristic
258	C₂²	—	2	2	no	—
259	C₂²	—	2	2	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	4	4	no	—
262	C₂²	C₂×C₈×Dic₅₃	1	2	yes	—
263	C₂²	C₂×C₈×Dic₅₃	1	2	yes	—
264	C₂²	C₂×C₈×Dic₅₃	1	2	yes	—
265	C₂²	C₂²×C₈×D₅₃	1	1	yes	characteristic
266	C₄	—	106	8	no	—
267	C₄	C₂×C₈×Dic₅₃	1	2	yes	—
268	C₄	C₂×C₈×Dic₅₃	1	2	yes	—
269	C₄	C₂×C₈×Dic₅₃	1	2	yes	—
270	C₄	C₂²×C₄×Dic₅₃	1	1	yes	characteristic
271	C₄	C₂²×C₈×D₅₃	1	1	yes	characteristic
272	C₂	—	2	2	no	—
273	C₂	—	2	2	no	—
274	C₂	—	2	2	no	—
275	C₂	C₂²×C₈×Dic₅₃	1	1	yes	characteristic
276	C₂	#20287817	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅₃
commutator	C₂×C₅₃	C₂²×C₄×C₈
Fitting	C₂²×M₃₂×C₅₃	C₂
socle	C₂²×C₅₃	C₂³×C₈
prime radical	C₂⁴×C₅₃	C₂×C₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂²×M₃₂	D₅₃
hypocenter	C₅₃	#33324
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
3. #20276314	C₂	semidirect
4. #20276311	C₂	semidirect
5. M₃₂×Dic₅₃	C₂	semidirect
9. C₂×#13133157	C₄	semidirect
19. #16810398	C₂²	semidirect
20. C₁₆×Dic₅₃	C₂²	semidirect
22. C₂×M₃₂×C₅₃	C₄	semidirect
38. #13133157	C₂×C₄	semidirect
39. C₂×#11719908	C₂×C₄	semidirect
55. M₃₂×C₅₃	C₂×C₄	semidirect
56. C₂×C₁₆×C₅₃	C₂×C₄	semidirect
151. C₂×M₃₂	Dic₅₃	semidirect
164. C₅₃	#33324	semidirect
188. M₃₂	C₂×Dic₅₃	semidirect
191. C₂×C₁₆	C₂×Dic₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13133157	2
C₂×#16810578	2
#20276314	2
#20276311	2
M₃₂×Dic₅₃	2
C₂²×M₃₂×C₅₃	2
#33324	53