Subgroups — #27537079

Order 13568.

706 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	C₂²×#13133157	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810578	C₂	1	1	yes	maximal, characteristic
3	#20276475	C₂	1	4	yes	maximal, complemented
4	#20276473	C₂	1	4	yes	maximal, complemented
5	#20276470	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	—	2	4	no	—
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	#16810441	C₂²	1	16	yes	complemented
20	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×M₃₂×C₅₃	—	2	4	no	—
22	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
23	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
24	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
25	C₂×#11719917	—	2	4	no	—
26	C₂×#11719917	C₂×C₄	1	2	yes	—
27	C₂×#11719917	C₂×C₄	1	2	yes	—
28	C₂²×#88322	C₂×C₄	1	2	yes	—
29	#13133176	C₈	1	2	yes	—
30	#13133176	C₈	1	2	yes	—
31	#13133176	C₂×C₄	1	2	yes	—
32	#13133176	C₂×C₄	1	2	yes	—
33	C₂×C₄⋊Dic₅₃	C₈	1	2	yes	—
34	C₂×C₂²⋊Dic₅₃	C₈	1	2	yes	—
35	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
36	#13133157	—	2	4	no	—
37	#13133157	—	2	4	no	—
38	C₂×#11719908	—	2	4	no	—
39	C₂×#11719908	C₂³	1	4	yes	—
40	#13133127	C₂×C₄	1	4	yes	—
41	#13133127	C₂×C₄	1	4	yes	—
42	C₈×Dic₅₃	C₂×C₄	1	4	yes	—
43	C₈×Dic₅₃	C₂³	1	4	yes	—
44	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₈×C₅₃	D₄	1	4	yes	—
49	C₂²×C₈×C₅₃	Q₈	1	4	yes	—
50	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
53	M₃₂×C₅₃	—	2	4	no	—
54	M₃₂×C₅₃	—	2	4	no	—
55	C₂×C₁₆×C₅₃	—	2	4	no	—
56	C₂×C₁₆×C₅₃	C₂³	1	4	yes	—
57	C₂⁴×C₅₃	C₂×C₈	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
59	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	M₁₆	1	4	yes	—
63	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
66	C₂²×Dic₅₃	C₂×C₈	1	4	yes	—
67	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
68	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
69	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
70	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
74	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
75	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
76	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
77	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
78	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
79	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
80	C₂²⋊Dic₅₃	C₂×C₈	1	4	yes	—
81	C₂²⋊Dic₅₃	C₂×C₈	1	4	yes	—
82	C₄⋊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	C₂²×C₄	1	4	yes	—
86	#11719917	—	2	4	no	—
87	#11719917	—	2	4	no	—
88	C₂×#88322	—	2	4	no	—
89	C₂×#88322	C₂²×C₄	1	4	yes	—
90	C₁₆×C₅₃	—	2	4	no	—
91	#11719908	—	2	4	no	—
92	C₂³×C₅₃	—	4	2	no	—
93	C₂³×C₅₃	C₄⋊C₈	1	4	yes	—
94	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
95	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
96	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
97	C₂×C₄×C₅₃	C₄⋊C₈	1	4	yes	—
98	C₂×C₄×C₅₃	C₄⋊C₈	1	4	yes	—
99	C₂×C₄×C₅₃	C₄⋊C₈	1	4	yes	—
100	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
101	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
102	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
103	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
104	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
106	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
107	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
108	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
109	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
110	C₂×Dic₅₃	—	4	4	no	—
111	C₂×Dic₅₃	C₂²×C₈	1	8	yes	—
112	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
113	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
114	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
115	C₈×C₅₃	C₂²×D₄	1	1	yes	characteristic
116	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
117	#88322	—	2	4	no	—
118	#33397	—	53	1	no	maximal, Sylow
119	C₂²×C₅₃	—	2	2	no	—
120	C₂²×C₅₃	—	2	2	no	—
121	C₂²×C₅₃	—	2	2	no	—
122	C₂²×C₅₃	—	4	4	no	—
123	C₂²×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
124	C₂²×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
125	C₂²×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
126	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
127	C₄×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
128	C₄×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
129	C₄×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
130	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
131	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
132	Dic₅₃	—	2	8	no	—
133	C₂²×M₃₂	—	53	1	no	—
134	C₂²×M₃₂	D₅₃	1	1	yes	characteristic
135	C₂×#405	—	53	1	no	—
136	#2151	—	53	4	no	—
137	#2151	—	53	4	no	—
138	#2150	—	53	4	no	—
139	C₂×C₅₃	—	2	2	no	—
140	C₂×C₅₃	—	2	2	no	—
141	C₂×C₅₃	—	2	2	no	—
142	C₂×C₅₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
143	C₂×C₅₃	#3407	1	2	yes	—
144	C₂²×M₁₆	—	53	1	no	—
145	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
146	C₂×#117	—	53	1	no	—
147	C₂×M₃₂	—	2	4	no	—
148	C₂×M₃₂	—	53	2	no	—
149	C₂×M₃₂	—	53	2	no	—
150	C₂×M₃₂	—	106	4	no	—
151	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
152	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
153	C₂²×C₁₆	—	53	2	no	—
154	C₂²×C₁₆	C₂×D₅₃	1	2	yes	—
155	#405	—	53	2	no	—
156	#405	—	53	2	no	—
157	#405	—	53	2	no	—
158	#405	—	53	2	no	—
159	C₂×#97	—	53	2	no	—
160	C₂×C₄×C₈	—	53	2	no	—
161	C₄⋊C₁₆	—	53	16	no	—
162	C₅₃	#33397	1	1	yes	characteristic, complemented, Sylow
163	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
164	C₂×M₁₆	—	53	2	no	—
165	C₂×M₁₆	—	53	2	no	—
166	C₂×M₁₆	—	106	4	no	—
167	C₂²×C₈	—	53	2	no	—
168	C₂²×C₈	Dic₁₀₆	1	4	yes	—
169	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
170	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
171	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
172	C₂²×C₈	C₂²⋊D₅₃	1	4	yes	—
173	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
174	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
175	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
176	#117	—	53	2	no	—
177	#117	—	53	2	no	—
178	#117	—	53	2	no	—
179	#117	—	53	2	no	—
180	C₂×C₄⋊C₄	—	53	2	no	—
181	C₂×C₂²⋊C₄	—	53	2	no	—
182	C₂×C₄²	—	53	2	no	—
183	M₃₂	—	2	4	no	—
184	M₃₂	—	2	4	no	—
185	M₃₂	—	106	4	no	—
186	M₃₂	—	106	4	no	—
187	C₂×C₁₆	—	2	4	no	—
188	C₂×C₁₆	—	53	4	no	—
189	C₂×C₁₆	—	106	4	no	—
190	C₂×C₁₆	C₂²×D₅₃	1	4	yes	—
191	#97	—	53	4	no	—
192	#97	—	53	4	no	—
193	C₄×C₈	—	53	4	no	—
194	C₄×C₈	—	53	4	no	—
195	C₂⁴	C₈×D₅₃	1	1	yes	characteristic
196	C₂²×C₄	—	53	4	no	—
197	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
198	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
199	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
200	C₂²×C₄	#11719912	1	4	yes	—
201	C₂²×C₄	#11719919	1	4	yes	—
202	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
203	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
204	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
205	M₁₆	—	106	4	no	—
206	M₁₆	—	106	4	no	—
207	C₂×C₈	—	53	4	no	—
208	C₂×C₈	—	106	4	no	—
209	C₂×C₈	#11719919	1	4	yes	—
210	C₂×C₈	#11719919	1	4	yes	—
211	C₂×C₈	#11719919	1	4	yes	—
212	C₂×C₈	C₂×Dic₁₀₆	1	2	yes	—
213	C₂×C₈	C₂×Dic₁₀₆	1	2	yes	—
214	C₂×C₈	C₂×Dic₁₀₆	1	2	yes	—
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	1	yes	characteristic
218	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
219	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
220	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
221	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
222	C₄⋊C₄	—	53	4	no	—
223	C₄⋊C₄	—	53	4	no	—
224	C₂²⋊C₄	—	53	4	no	—
225	C₂²⋊C₄	—	53	4	no	—
226	C₄²	—	53	4	no	—
227	C₄²	—	53	4	no	—
228	C₁₆	—	2	4	no	—
229	C₁₆	—	106	4	no	—
230	C₂³	—	4	2	no	—
231	C₂³	#13133146	1	4	yes	—
232	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
233	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
234	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
235	C₂×C₄	—	53	8	no	—
236	C₂×C₄	—	212	4	no	—
237	C₂×C₄	#13133146	1	4	yes	—
238	C₂×C₄	#13133146	1	4	yes	—
239	C₂×C₄	#13133146	1	4	yes	—
240	C₂×C₄	C₂×#11719919	1	2	yes	—
241	C₂×C₄	C₂×#11719919	1	2	yes	—
242	C₂×C₄	C₂×#11719919	1	2	yes	—
243	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
244	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
245	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
246	C₂×C₄	C₂×#11719912	1	2	yes	—
247	C₂×C₄	C₂×#11719912	1	2	yes	—
248	C₂×C₄	C₂×#11719912	1	2	yes	—
249	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
250	C₈	—	106	4	no	—
251	C₈	C₂×#11719919	1	2	yes	—
252	C₈	C₂×#11719919	1	2	yes	—
253	C₈	C₂×#11719919	1	2	yes	—
254	C₈	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
255	C₈	C₂²×Dic₁₀₆	1	1	yes	characteristic
256	C₂²	—	2	2	no	—
257	C₂²	—	2	2	no	—
258	C₂²	—	2	2	no	—
259	C₂²	—	4	4	no	—
260	C₂²	C₂×#13133146	1	2	yes	—
261	C₂²	C₂×#13133146	1	2	yes	—
262	C₂²	C₂×#13133146	1	2	yes	—
263	C₂²	C₂²×C₈×D₅₃	1	1	yes	characteristic
264	C₄	—	106	8	no	—
265	C₄	C₂×#13133146	1	2	yes	—
266	C₄	C₂×#13133146	1	2	yes	—
267	C₄	C₂×#13133146	1	2	yes	—
268	C₄	C₂²×#11719919	1	1	yes	characteristic
269	C₄	C₂²×#11719912	1	1	yes	characteristic
270	C₂	—	2	2	no	—
271	C₂	—	2	2	no	—
272	C₂	—	2	2	no	—
273	C₂	C₂²×#13133146	1	1	yes	characteristic
274	C₂	#20287817	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	C₂²×M₃₂×C₅₃	C₂
prime radical	C₂⁴×C₅₃	C₂×C₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂²×M₃₂	D₅₃
hypocenter	C₅₃	#33397
LCS	G ▷ C₂²×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. #20276475	C₂	semidirect
4. #20276473	C₂	semidirect
5. #20276470	C₂	semidirect
19. #16810441	C₂²	semidirect
162. C₅₃	#33397	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
#20276475	2
#20276473	2
#20276470	2
C₂²×M₃₂×C₅₃	2
#33397	53