Subgroups — #27537237

Order 13568.

528 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	#20276531	C₂	1	4	yes	maximal, complemented
2	#20276527	C₂	1	4	yes	maximal, complemented
3	C₂×#16810444	C₂	1	2	yes	maximal, complemented
4	#20276336	C₂	1	2	yes	maximal
5	#20276320	C₂	1	2	yes	maximal, complemented
6	C₄×#131×C₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂×#13133157	C₂²	1	4	yes	—
8	#16810444	C₂²	1	4	yes	—
9	#16810444	C₂²	1	4	yes	—
10	#16810405	C₂²	1	8	yes	—
11	#16810401	C₂²	1	8	yes	—
12	C₂×#131×C₅₃	C₄	1	1	yes	characteristic
13	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
14	C₄×#41×C₅₃	C₄	1	1	yes	characteristic
15	C₅₃×#405	C₄	1	1	yes	characteristic
16	C₅₃×#405	C₄	1	2	yes	—
17	C₅₃×#405	C₂²	1	1	yes	characteristic
18	C₅₃×#405	C₂²	1	2	yes	—
19	C₄×M₁₆×C₅₃	C₄	1	1	yes	characteristic
20	C₄×M₁₆×C₅₃	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₅₃	C₄	1	2	yes	—
22	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
23	#13133157	—	2	4	no	—
24	#13133157	C₂³	1	8	yes	—
25	C₂×#11719908	—	2	4	no	—
26	C₂×#41×C₅₃	C₂×C₄	1	1	yes	characteristic
27	#131×C₅₃	—	2	4	no	—
28	#131×C₅₃	—	2	2	no	—
29	#131×C₅₃	—	2	2	no	—
30	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
32	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
34	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
35	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
36	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
38	C₄×Q₈×C₅₃	—	2	1	no	—
39	C₄×D₄×C₅₃	—	2	1	no	—
40	C₄×D₄×C₅₃	—	2	2	no	—
41	#117×C₅₃	C₂×C₄	1	2	yes	—
42	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
45	#97×C₅₃	C₂×C₄	1	2	yes	—
46	#97×C₅₃	C₂×C₄	1	2	yes	—
47	#97×C₅₃	C₂³	1	2	yes	—
48	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
50	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
51	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
52	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
54	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
55	#41×C₅₃	—	2	2	no	—
56	#41×C₅₃	—	2	2	no	—
57	C₂×Q₈×C₅₃	M₁₆	1	1	yes	characteristic
58	C₂×D₄×C₅₃	M₁₆	1	2	yes	—
59	C₂×D₄×C₅₃	M₁₆	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	—	2	2	no	—
61	C₂²×C₄×C₅₃	—	2	1	no	—
62	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
64	M₁₆×C₅₃	—	2	2	no	—
65	M₁₆×C₅₃	—	2	2	no	—
66	M₁₆×C₅₃	—	2	4	no	—
67	M₁₆×C₅₃	—	2	4	no	—
68	C₂×C₈×C₅₃	—	2	2	no	—
69	C₂×C₈×C₅₃	—	2	2	no	—
70	C₂×C₈×C₅₃	—	2	4	no	—
71	C₂×C₈×C₅₃	M₁₆	1	4	yes	—
72	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
73	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
74	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₅₃	#41	1	2	yes	—
78	C₂×C₈×C₅₃	#41	1	2	yes	—
79	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
80	C₄⋊C₄×C₅₃	—	2	2	no	—
81	C₄⋊C₄×C₅₃	—	2	1	no	—
82	C₂²⋊C₄×C₅₃	—	2	2	no	—
83	C₂²⋊C₄×C₅₃	—	2	1	no	—
84	C₄²×C₅₃	—	2	2	no	—
85	C₄²×C₅₃	—	2	1	no	—
86	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
87	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
88	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
89	#11719908	—	2	8	no	—
90	C₂³×C₅₃	C₂×M₁₆	1	2	yes	—
91	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
92	Q₈×C₅₃	—	2	2	no	—
93	D₄×C₅₃	—	2	4	no	—
94	D₄×C₅₃	—	2	2	no	—
95	C₂×C₄×C₅₃	—	2	1	no	—
96	C₂×C₄×C₅₃	—	2	2	no	—
97	C₂×C₄×C₅₃	—	2	2	no	—
98	C₂×C₄×C₅₃	—	2	2	no	—
99	C₂×C₄×C₅₃	—	2	1	no	—
100	C₂×C₄×C₅₃	—	4	2	no	—
101	C₂×C₄×C₅₃	—	4	4	no	—
102	C₂×C₄×C₅₃	#117	1	2	yes	—
103	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
104	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
105	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
106	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
107	C₈×C₅₃	—	2	2	no	—
108	C₈×C₅₃	—	2	2	no	—
109	C₈×C₅₃	#117	1	1	yes	characteristic
110	C₈×C₅₃	#117	1	1	yes	characteristic
111	C₈×C₅₃	C₂×M₁₆	1	2	yes	—
112	C₈×C₅₃	C₂×M₁₆	1	2	yes	—
113	C₈×C₅₃	C₂×#41	1	2	yes	—
114	#33428	—	53	1	no	maximal, Sylow
115	C₂²×C₅₃	—	2	2	no	—
116	C₂²×C₅₃	—	4	1	no	—
117	C₂²×C₅₃	—	4	2	no	—
118	C₂²×C₅₃	#432	1	2	yes	—
119	C₂²×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
120	C₄×C₅₃	—	2	2	no	—
121	C₄×C₅₃	—	2	2	no	—
122	C₄×C₅₃	—	4	2	no	—
123	C₄×C₅₃	—	4	1	no	—
124	C₄×C₅₃	#432	1	2	yes	—
125	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
126	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
127	C₄×#131	D₅₃	1	1	yes	characteristic
128	#2166	—	53	4	no	—
129	#2165	—	53	4	no	—
130	C₂×#364	—	53	2	no	—
131	#2115	—	53	2	no	—
132	#2110	—	53	2	no	—
133	C₂×C₅₃	—	2	1	no	—
134	C₂×C₅₃	—	2	2	no	—
135	C₂×C₅₃	—	4	2	no	—
136	C₂×C₅₃	C₂×#432	1	1	yes	characteristic
137	C₂×#131	Dic₅₃	1	1	yes	characteristic
138	C₂×#131	C₂×D₅₃	1	1	yes	characteristic
139	C₄×#41	Dic₅₃	1	1	yes	characteristic
140	C₂×M₃₂	—	53	4	no	—
141	#405	Dic₅₃	1	1	yes	characteristic
142	#405	Dic₅₃	1	2	yes	—
143	#405	C₂×D₅₃	1	2	yes	—
144	#405	C₂×D₅₃	1	1	yes	characteristic
145	C₄×M₁₆	Dic₅₃	1	1	yes	characteristic
146	C₄×M₁₆	C₂×D₅₃	1	2	yes	—
147	C₂×C₄×C₈	Dic₅₃	1	2	yes	—
148	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
149	#364	—	53	4	no	—
150	#364	—	53	4	no	—
151	#349	—	53	8	no	—
152	#347	—	53	8	no	—
153	C₅₃	#33428	1	1	yes	characteristic, complemented, Sylow
154	C₂×#41	C₂×Dic₅₃	1	1	yes	characteristic
155	#131	—	2	2	no	—
156	#131	—	2	2	no	—
157	#131	—	2	4	no	—
158	C₂×M₁₆	C₂×Dic₅₃	1	1	yes	characteristic
159	C₂×M₁₆	C₂×Dic₅₃	1	2	yes	—
160	C₂×M₁₆	C₂×Dic₅₃	1	1	yes	characteristic
161	C₂×M₁₆	C₂²×D₅₃	1	2	yes	—
162	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
163	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
164	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
165	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
166	C₄×Q₈	—	2	1	no	—
167	C₄×D₄	—	2	2	no	—
168	C₄×D₄	—	2	1	no	—
169	#117	C₂×Dic₅₃	1	2	yes	—
170	#117	C₂×Dic₅₃	1	1	yes	characteristic
171	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
172	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
173	M₃₂	—	53	8	no	—
174	M₃₂	—	106	4	no	—
175	C₂×C₁₆	—	106	4	no	—
176	#97	C₂×Dic₅₃	1	2	yes	—
177	#97	C₂×Dic₅₃	1	2	yes	—
178	#97	C₂²×D₅₃	1	2	yes	—
179	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
180	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
181	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
182	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
183	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
184	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
185	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
186	#41	—	2	2	no	—
187	#41	—	2	2	no	—
188	C₂×Q₈	#11719917	1	1	yes	characteristic
189	C₂×D₄	#11719917	1	2	yes	—
190	C₂×D₄	#11719917	1	1	yes	characteristic
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	—	2	1	no	—
193	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
194	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
195	M₁₆	—	2	2	no	—
196	M₁₆	—	2	2	no	—
197	M₁₆	—	2	4	no	—
198	M₁₆	—	2	4	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	2	4	no	—
202	C₂×C₈	#11719917	1	4	yes	—
203	C₂×C₈	#11719945	1	2	yes	—
204	C₂×C₈	#11719945	1	2	yes	—
205	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
206	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
207	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
208	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
209	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
210	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
211	C₄⋊C₄	—	2	2	no	—
212	C₄⋊C₄	—	2	1	no	—
213	C₂²⋊C₄	—	2	2	no	—
214	C₂²⋊C₄	—	2	1	no	—
215	C₄²	—	2	2	no	—
216	C₄²	—	2	1	no	—
217	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
218	C₄²	C₂²×Dic₅₃	1	2	yes	—
219	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
220	C₁₆	—	106	8	no	—
221	C₂³	C₂×#11719917	1	2	yes	—
222	C₂³	C₂×#11719917	1	1	yes	characteristic
223	Q₈	—	2	2	no	—
224	D₄	—	2	4	no	—
225	D₄	—	2	2	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	4	4	no	—
233	C₂×C₄	#13133176	1	2	yes	—
234	C₂×C₄	C₂×#11719917	1	2	yes	—
235	C₂×C₄	C₂×#11719917	1	2	yes	—
236	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
237	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
238	C₈	—	2	2	no	—
239	C₈	—	2	2	no	—
240	C₈	#13133176	1	1	yes	characteristic
241	C₈	#13133176	1	1	yes	characteristic
242	C₈	C₂×#11719917	1	2	yes	—
243	C₈	C₂×#11719917	1	2	yes	—
244	C₈	C₂×#11719945	1	2	yes	—
245	C₂²	—	2	2	no	—
246	C₂²	—	4	1	no	—
247	C₂²	—	4	2	no	—
248	C₂²	#16810680	1	2	yes	—
249	C₂²	C₂²×#11719917	1	1	yes	characteristic
250	C₄	—	2	2	no	—
251	C₄	—	2	2	no	—
252	C₄	—	4	2	no	—
253	C₄	—	4	1	no	—
254	C₄	#16810680	1	2	yes	—
255	C₄	C₂×#13133176	1	1	yes	characteristic
256	C₄	C₂²×#11719917	1	1	yes	characteristic
257	C₂	—	2	1	no	—
258	C₂	—	2	2	no	—
259	C₂	—	4	2	no	—
260	C₂	C₂×#16810680	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#13133176
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₄×#131×C₅₃	C₂
socle	C₂×C₅₃	C₂×#432
prime radical	C₂×#41×C₅₃	C₂×C₄
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₄×#131	D₅₃
hypocenter	C₅₃	#33428
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈ ◁ #405 ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
1. #20276531	C₂	semidirect
2. #20276527	C₂	semidirect
3. C₂×#16810444	C₂	semidirect
5. #20276320	C₂	semidirect
153. C₅₃	#33428	semidirect

Maximal subgroups

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

subgroup	index
#20276531	2
#20276527	2
C₂×#16810444	2
#20276336	2
#20276320	2
C₄×#131×C₅₃	2
#33428	53