Subgroups — #27354953

Order 13568.

544 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283810	C₂	1	2	yes	maximal
2	#20273587	C₂	1	4	yes	maximal, complemented
3	#20273566	C₂	1	4	yes	maximal, complemented
4	#20273478	C₂	1	1	yes	maximal, characteristic
5	#20273188	C₂	1	2	yes	maximal
6	#20273007	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#3170	C₂	1	1	yes	maximal, characteristic
8	#16810754	—	2	4	no	—
9	#16810754	C₂²	1	4	yes	—
10	Q₈×#88322	C₂²	1	4	yes	—
11	#16810684	—	2	4	no	—
12	#16810680	C₂²	1	2	yes	—
13	#16810676	C₂²	1	1	yes	characteristic
14	#16810676	C₂²	1	1	yes	characteristic
15	#16810376	C₂²	1	2	yes	—
16	#16810374	C₂²	1	2	yes	—
17	Q₈⋊#88322	C₂²	1	8	yes	—
18	#16810345	C₂²	1	2	yes	—
19	#16810345	C₂²	1	2	yes	—
20	C₅₃×#533	C₂²	1	2	yes	—
21	C₅₃×#500	C₄	1	1	yes	characteristic
22	C₅₃×#494	C₄	1	1	yes	characteristic
23	C₅₃×#487	C₄	1	2	yes	—
24	C₅₃×#467	C₄	1	2	yes	—
25	C₅₃×#467	C₄	1	2	yes	—
26	C₄×Q₁₆×C₅₃	C₂²	1	2	yes	—
27	C₄×Q₁₆×C₅₃	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
29	C₄⋊#88322	—	2	1	no	—
30	C₄⋊#88322	—	2	4	no	—
31	C₄⋊#88322	—	2	4	no	—
32	C₄⋊#88322	D₄	1	2	yes	—
33	C₄⋊#88322	C₂³	1	4	yes	—
34	C₂²⋊#88322	—	2	2	no	—
35	C₂²⋊#88322	—	2	1	no	—
36	C₂²⋊#88322	D₄	1	2	yes	—
37	#13133128	—	2	2	no	—
38	#13133128	—	2	4	no	—
39	C₄×#88322	—	2	4	no	—
40	C₄×#88322	C₂³	1	2	yes	—
41	C₄×#88322	C₂³	1	2	yes	—
42	C₂×Q₁₆×C₅₃	C₂×C₄	1	2	yes	—
43	C₂×Q₁₆×C₅₃	C₂×C₄	1	2	yes	—
44	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
46	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
47	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
48	#125×C₅₃	—	2	2	no	—
49	C₂²⋊Q₈×C₅₃	—	2	2	no	—
50	C₂²⋊Q₈×C₅₃	C₂×C₄	1	4	yes	—
51	C₄×Q₈×C₅₃	—	2	2	no	—
52	C₄×Q₈×C₅₃	C₂³	1	4	yes	—
53	#117×C₅₃	—	2	2	no	—
54	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
55	#107×C₅₃	C₂×C₄	1	2	yes	—
56	#107×C₅₃	C₂×C₄	1	2	yes	—
57	#103×C₅₃	C₂×C₄	1	4	yes	—
58	#103×C₅₃	C₂×C₄	1	4	yes	—
59	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
60	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
61	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
62	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
63	C₂×Q₈×C₅₃	—	2	2	no	—
64	C₂×Q₈×C₅₃	C₂²×C₄	1	4	yes	—
65	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
68	Q₁₆×C₅₃	—	4	2	no	—
69	Q₁₆×C₅₃	—	4	2	no	—
70	C₂×C₈×C₅₃	—	4	1	no	—
71	C₂×C₈×C₅₃	—	4	1	no	—
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	1	yes	characteristic
75	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
76	C₄⋊C₄×C₅₃	—	2	2	no	—
77	C₄⋊C₄×C₅₃	—	2	2	no	—
78	C₄⋊C₄×C₅₃	—	2	2	no	—
79	C₄⋊C₄×C₅₃	—	2	4	no	—
80	C₄⋊C₄×C₅₃	—	2	4	no	—
81	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
82	C₂²⋊C₄×C₅₃	—	2	4	no	—
83	C₄²×C₅₃	—	2	4	no	—
84	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
85	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₅₃	#41	1	1	yes	characteristic
87	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
88	C₂×#88322	—	2	4	no	—
89	C₂×#88322	—	2	2	no	—
90	C₂×#88322	C₂×D₄	1	4	yes	—
91	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
92	Q₈×C₅₃	—	2	4	no	—
93	Q₈×C₅₃	—	2	4	no	—
94	Q₈×C₅₃	—	8	2	no	—
95	C₂×C₄×C₅₃	—	2	1	no	—
96	C₂×C₄×C₅₃	—	2	1	no	—
97	C₂×C₄×C₅₃	—	2	2	no	—
98	C₂×C₄×C₅₃	—	2	4	no	—
99	C₂×C₄×C₅₃	—	2	4	no	—
100	C₂×C₄×C₅₃	—	4	1	no	—
101	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
102	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
103	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
104	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
106	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
107	C₈×C₅₃	—	2	1	no	—
108	C₈×C₅₃	—	2	1	no	—
109	C₈×C₅₃	—	2	1	no	—
110	C₈×C₅₃	—	2	1	no	—
111	#88322	—	2	4	no	—
112	#88322	—	4	2	no	—
113	#88322	—	4	4	no	—
114	#17991	—	53	1	no	maximal, Sylow
115	C₂²×C₅₃	—	2	1	no	—
116	C₂²×C₅₃	—	2	1	no	—
117	C₂²×C₅₃	—	2	1	no	—
118	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
119	C₄×C₅₃	—	2	1	no	—
120	C₄×C₅₃	—	2	1	no	—
121	C₄×C₅₃	—	2	1	no	—
122	C₄×C₅₃	—	2	1	no	—
123	C₄×C₅₃	—	4	4	no	—
124	C₄×C₅₃	—	4	4	no	—
125	C₄×C₅₃	#568	1	1	yes	characteristic
126	C₄×C₅₃	#568	1	1	yes	characteristic
127	C₄×C₅₃	#578	1	2	yes	—
128	#3170	D₅₃	1	1	yes	characteristic
129	#2985	—	53	2	no	—
130	#1587	—	53	4	no	—
131	#1584	—	53	4	no	—
132	#1568	—	53	1	no	—
133	#1497	—	53	2	no	—
134	#1452	—	53	1	no	—
135	C₂×C₅₃	—	4	1	no	—
136	C₂×C₅₃	#2932	1	1	yes	characteristic
137	C₂×C₅₃	#2950	1	1	yes	characteristic
138	C₂×C₅₃	#2957	1	1	yes	characteristic
139	#533	C₂×D₅₃	1	2	yes	—
140	#500	Dic₅₃	1	1	yes	characteristic
141	#494	Dic₅₃	1	1	yes	characteristic
142	#487	Dic₅₃	1	2	yes	—
143	#467	Dic₅₃	1	2	yes	—
144	#467	Dic₅₃	1	2	yes	—
145	#446	—	53	4	no	—
146	#446	—	106	4	no	—
147	C₈×Q₈	—	53	4	no	—
148	C₄×Q₁₆	C₂×D₅₃	1	2	yes	—
149	C₄×Q₁₆	C₂×D₅₃	1	2	yes	—
150	#433	—	106	4	no	—
151	#432	—	53	2	no	—
152	#431	—	53	1	no	—
153	#431	—	53	1	no	—
154	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
155	#335	—	53	2	no	—
156	#334	—	53	2	no	—
157	Q₈⋊C₈	—	53	8	no	—
158	#322	—	53	2	no	—
159	#322	—	53	2	no	—
160	C₅₃	#17991	1	1	yes	characteristic, complemented, Sylow
161	C₂×Q₁₆	C₂×Dic₅₃	1	2	yes	—
162	C₂×Q₁₆	C₂×Dic₅₃	1	2	yes	—
163	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
164	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
165	C₄⋊Q₈	C₂×Dic₅₃	1	2	yes	—
166	C₄⋊Q₈	C₂×Dic₅₃	1	2	yes	—
167	#125	—	2	2	no	—
168	C₂²⋊Q₈	—	2	2	no	—
169	C₂²⋊Q₈	C₂×Dic₅₃	1	4	yes	—
170	C₄×Q₈	—	2	2	no	—
171	C₄×Q₈	C₂²×D₅₃	1	4	yes	—
172	#117	—	2	2	no	—
173	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
174	#107	C₂×Dic₅₃	1	2	yes	—
175	#107	C₂×Dic₅₃	1	2	yes	—
176	C₄⋊C₈	—	53	2	no	—
177	C₄⋊C₈	—	53	4	no	—
178	C₄⋊C₈	—	106	4	no	—
179	C₄⋊C₈	—	106	4	no	—
180	C₄⋊C₈	—	106	1	no	—
181	#103	C₂×Dic₅₃	1	4	yes	—
182	#103	C₂×Dic₅₃	1	4	yes	—
183	C₂²⋊C₈	—	53	2	no	—
184	C₂²⋊C₈	—	106	2	no	—
185	C₂²⋊C₈	—	106	1	no	—
186	#97	—	106	4	no	—
187	#97	—	106	2	no	—
188	C₄×C₈	—	53	2	no	—
189	C₄×C₈	—	53	2	no	—
190	C₄×C₈	—	106	4	no	—
191	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
192	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
193	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
194	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
195	C₂×Q₈	—	2	2	no	—
196	C₂×Q₈	C₂²×Dic₅₃	1	4	yes	—
197	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
198	C₂²×C₄	#11719947	1	1	yes	characteristic
199	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
200	Q₁₆	—	4	2	no	—
201	Q₁₆	—	4	2	no	—
202	C₂×C₈	—	4	1	no	—
203	C₂×C₈	—	4	1	no	—
204	C₂×C₈	—	53	4	no	—
205	C₂×C₈	—	106	4	no	—
206	C₂×C₈	—	106	2	no	—
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	C₂²×Dic₅₃	1	1	yes	characteristic
211	C₄⋊C₄	—	2	2	no	—
212	C₄⋊C₄	—	2	2	no	—
213	C₄⋊C₄	—	2	2	no	—
214	C₄⋊C₄	—	2	4	no	—
215	C₄⋊C₄	—	2	4	no	—
216	C₄⋊C₄	C₂²×Dic₅₃	1	4	yes	—
217	C₂²⋊C₄	—	2	4	no	—
218	C₄²	—	2	4	no	—
219	C₄²	D₄×D₅₃	1	1	yes	characteristic
220	C₄²	#11719947	1	1	yes	characteristic
221	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
222	C₄²	C₂³×D₅₃	1	1	yes	characteristic
223	C₂³	D₄×Dic₅₃	1	1	yes	characteristic
224	Q₈	—	2	4	no	—
225	Q₈	—	2	4	no	—
226	Q₈	—	8	2	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	4	1	no	—
233	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
234	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
235	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
236	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
237	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
238	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
239	C₈	—	2	1	no	—
240	C₈	—	2	1	no	—
241	C₈	—	2	1	no	—
242	C₈	—	2	1	no	—
243	C₈	—	106	4	no	—
244	C₈	—	212	2	no	—
245	C₈	—	212	4	no	—
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	—	2	1	no	—
249	C₂²	C₂×D₄×Dic₅₃	1	1	yes	characteristic
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	2	1	no	—
254	C₄	—	4	4	no	—
255	C₄	—	4	4	no	—
256	C₄	#16811390	1	1	yes	characteristic
257	C₄	#16811390	1	1	yes	characteristic
258	C₄	#16811436	1	2	yes	—
259	C₂	—	4	1	no	—
260	C₂	#20283433	1	1	yes	characteristic
261	C₂	#20283583	1	1	yes	characteristic
262	C₂	#20283632	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#3170	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂³×C₅₃	C₄×D₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#3170	D₅₃
hypocenter	C₅₃	#17991
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3170

Complements

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

N	H ≅ G/N	type
2. #20273587	C₂	semidirect
3. #20273566	C₂	semidirect
160. C₅₃	#17991	semidirect

Maximal subgroups

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

subgroup	index
#20283810	2
#20273587	2
#20273566	2
#20273478	2
#20273188	2
#20273007	2
C₅₃×#3170	2
#17991	53