Subgroups — #27372054

Order 13568.

710 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20285663	C₂	1	2	yes	maximal
2	#20285301	C₂	1	1	yes	maximal, characteristic
3	#20285257	C₂	1	2	yes	maximal
4	#20274391	C₂	1	4	yes	maximal, complemented
5	#20274332	C₂	1	2	yes	maximal, complemented
6	#20274313	C₂	1	2	yes	maximal, complemented
7	C₈×#11719917	C₂	1	1	yes	maximal, characteristic
8	C₂×C₅₃×#500	C₂	1	1	yes	maximal, characteristic
9	#16811180	C₂²	1	2	yes	—
10	#16811029	—	2	4	no	—
11	#16811028	C₂²	1	1	yes	characteristic
12	#16811026	C₂²	1	4	yes	—
13	#16811019	C₂²	1	4	yes	—
14	#16811003	C₂²	1	1	yes	characteristic
15	#16811002	C₂²	1	4	yes	—
16	#16810974	C₂²	1	2	yes	—
17	#16810973	C₂²	1	4	yes	—
18	#16810864	—	2	2	no	—
19	#16810830	—	2	4	no	—
20	#16810779	—	2	4	no	—
21	C₄×Dic₂₁₂	—	2	2	no	—
22	#16810676	C₂²	1	1	yes	characteristic
23	#16810663	—	2	4	no	—
24	C₄×#11719917	C₂²	1	1	yes	characteristic
25	#16810345	C₂²	1	2	yes	—
26	C₈×#88322	C₂²	1	2	yes	—
27	C₂×C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
28	C₅₃×#500	—	2	2	no	—
29	C₅₃×#500	C₂²	1	2	yes	—
30	C₅₃×#500	C₂²	1	2	yes	—
31	C₂×#107×C₅₃	—	2	2	no	—
32	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
33	C₂×Dic₂₁₂	—	2	2	no	—
34	C₂×#11719925	—	2	8	no	—
35	C₂×#11719917	D₄	1	2	yes	—
36	#13133217	C₂³	1	2	yes	—
37	#13133217	C₂³	1	2	yes	—
38	#13133209	—	2	2	no	—
39	#13133196	—	2	2	no	—
40	#13133196	—	2	2	no	—
41	C₄×Dic₁₀₆	—	2	2	no	—
42	C₄×Dic₁₀₆	—	2	2	no	—
43	#13133176	—	2	2	no	—
44	#13133151	—	2	2	no	—
45	#13133150	—	2	8	no	—
46	#13133148	—	2	8	no	—
47	C₄⋊#88322	D₄	1	2	yes	—
48	#13133142	—	2	4	no	—
49	#13133141	—	2	8	no	—
50	C₂²⋊#88322	D₄	1	2	yes	—
51	#13133128	D₄	1	2	yes	—
52	C₄×#88322	C₂³	1	2	yes	—
53	C₄×#88322	C₂³	1	2	yes	—
54	C₂²×Q₈×C₅₃	—	2	2	no	—
55	C₂²×C₈×C₅₃	D₄	1	2	yes	—
56	C₄⋊Q₈×C₅₃	—	2	2	no	—
57	C₄⋊Q₈×C₅₃	—	2	2	no	—
58	C₄⋊Q₈×C₅₃	—	2	2	no	—
59	C₄⋊Q₈×C₅₃	C₂³	1	4	yes	—
60	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
61	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
62	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
63	#107×C₅₃	—	2	4	no	—
64	#107×C₅₃	—	2	4	no	—
65	C₄×C₈×C₅₃	D₄	1	2	yes	—
66	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
68	C₂×Q₈×C₅₃	—	2	4	no	—
69	C₂×Q₈×C₅₃	—	4	2	no	—
70	C₂×Q₈×C₅₃	—	8	4	no	—
71	C₂²×C₄×C₅₃	—	4	2	no	—
72	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
75	C₂×Dic₁₀₆	—	2	2	no	—
76	C₂×Dic₁₀₆	—	2	2	no	—
77	C₂×C₈×C₅₃	—	2	4	no	—
78	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
79	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
80	C₄⋊C₄×C₅₃	—	2	4	no	—
81	C₄⋊C₄×C₅₃	—	2	4	no	—
82	C₄⋊C₄×C₅₃	—	4	2	no	—
83	C₄⋊C₄×C₅₃	—	4	2	no	—
84	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
85	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
88	C₂²⋊Dic₅₃	—	4	2	no	—
89	#11719925	—	8	8	no	—
90	C₄⋊Dic₅₃	—	2	2	no	—
91	C₄⋊Dic₅₃	—	2	2	no	—
92	C₄⋊Dic₅₃	—	2	2	no	—
93	C₄⋊Dic₅₃	—	2	2	no	—
94	#11719919	—	4	2	no	—
95	#11719919	—	4	2	no	—
96	C₄×Dic₅₃	—	4	2	no	—
97	#11719917	Q₁₆	1	8	yes	complemented
98	C₂×#88322	C₂×D₄	1	4	yes	—
99	C₂×#88322	C₂×D₄	1	4	yes	—
100	Dic₂₁₂	—	4	4	no	—
101	C₂³×C₅₃	#127	1	1	yes	characteristic
102	Q₈×C₅₃	—	4	8	no	—
103	Q₈×C₅₃	—	8	4	no	—
104	C₂×C₄×C₅₃	—	2	2	no	—
105	C₂×C₄×C₅₃	—	2	2	no	—
106	C₂×C₄×C₅₃	—	4	4	no	—
107	C₂×C₄×C₅₃	—	8	4	no	—
108	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
109	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
110	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
111	C₂×C₄×C₅₃	C₂×Q₁₆	1	4	yes	—
112	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
115	C₂×Dic₅₃	—	4	2	no	—
116	C₂×Dic₅₃	—	4	2	no	—
117	Dic₁₀₆	—	4	4	no	—
118	Dic₁₀₆	—	4	4	no	—
119	C₈×C₅₃	—	2	2	no	—
120	C₈×C₅₃	#137	1	4	yes	—
121	#88322	—	2	4	no	—
122	#88322	C₂×Q₁₆	1	8	yes	—
123	#19633	—	53	1	no	maximal, Sylow
124	C₂²×C₅₃	—	2	1	no	—
125	C₂²×C₅₃	—	2	1	no	—
126	C₂²×C₅₃	#494	1	2	yes	—
127	C₂²×C₅₃	C₂×#127	1	1	yes	characteristic
128	C₄×C₅₃	—	2	1	no	—
129	C₄×C₅₃	—	2	1	no	—
130	C₄×C₅₃	—	8	4	no	—
131	C₄×C₅₃	#494	1	2	yes	—
132	C₄×C₅₃	C₂²×Q₁₆	1	2	yes	—
133	C₄×C₅₃	C₂×#137	1	2	yes	—
134	C₄×C₅₃	#576	1	2	yes	—
135	Dic₅₃	—	8	2	no	—
136	Dic₅₃	—	8	2	no	—
137	#3209	—	53	2	no	—
138	#3170	—	53	1	no	—
139	#3165	—	53	2	no	—
140	C₂×#500	D₅₃	1	1	yes	characteristic
141	#1723	—	53	4	no	—
142	#1713	—	53	2	no	—
143	#1707	—	53	2	no	—
144	C₈×M₁₆	—	53	1	no	—
145	C₂×C₅₃	—	2	2	no	—
146	C₂×C₅₃	C₂×#494	1	1	yes	characteristic
147	C₂×C₅₃	#3152	1	1	yes	characteristic
148	C₂×C₅₃	#3155	1	1	yes	characteristic
149	#533	—	53	2	no	—
150	C₂×C₄⋊Q₈	C₂×D₅₃	1	2	yes	—
151	#501	—	106	4	no	—
152	#500	—	2	2	no	—
153	#500	—	53	4	no	—
154	#500	—	53	1	no	—
155	#500	C₂×D₅₃	1	2	yes	—
156	#500	C₂×D₅₃	1	2	yes	—
157	#498	—	53	4	no	—
158	#494	—	53	4	no	—
159	#494	—	53	1	no	—
160	#487	—	53	4	no	—
161	#487	—	53	2	no	—
162	#467	—	106	2	no	—
163	#462	—	106	4	no	—
164	#451	—	106	4	no	—
165	C₄×Q₁₆	—	106	2	no	—
166	#431	—	53	1	no	—
167	#428	—	106	4	no	—
168	C₂×#107	—	2	2	no	—
169	C₄×M₁₆	—	53	1	no	—
170	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
171	#322	—	53	2	no	—
172	C₈²	—	53	2	no	—
173	C₅₃	#19633	1	1	yes	characteristic, complemented, Sylow
174	C₂²×Q₈	—	2	2	no	—
175	C₂×Q₁₆	—	106	8	no	—
176	C₂×Q₁₆	—	106	2	no	—
177	C₂×M₁₆	—	53	2	no	—
178	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
179	C₄⋊Q₈	—	2	2	no	—
180	C₄⋊Q₈	—	2	2	no	—
181	C₄⋊Q₈	—	2	2	no	—
182	C₄⋊Q₈	—	53	2	no	—
183	C₄⋊Q₈	—	53	2	no	—
184	C₄⋊Q₈	C₂²×D₅₃	1	4	yes	—
185	#125	—	106	2	no	—
186	C₂²⋊Q₈	—	106	2	no	—
187	C₂²⋊Q₈	—	106	2	no	—
188	C₄×Q₈	—	106	2	no	—
189	C₄×Q₈	—	106	2	no	—
190	#117	—	106	2	no	—
191	C₂×C₄⋊C₄	—	2	2	no	—
192	C₂×C₄⋊C₄	—	2	2	no	—
193	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
194	#107	—	2	4	no	—
195	#107	—	2	4	no	—
196	#107	—	106	8	no	—
197	#107	—	106	2	no	—
198	#106	—	106	8	no	—
199	C₄⋊C₈	—	53	2	no	—
200	#103	—	106	8	no	—
201	#103	—	106	4	no	—
202	C₂²⋊C₈	—	53	2	no	—
203	#97	—	53	2	no	—
204	C₄×C₈	—	53	2	no	—
205	C₄×C₈	—	53	2	no	—
206	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
207	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
208	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
209	C₂×Q₈	—	2	4	no	—
210	C₂×Q₈	—	4	2	no	—
211	C₂×Q₈	—	8	4	no	—
212	C₂×Q₈	—	106	2	no	—
213	C₂×Q₈	—	106	2	no	—
214	C₂²×C₄	—	4	2	no	—
215	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
216	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
217	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
218	Q₁₆	—	424	8	no	—
219	Q₁₆	—	212	4	no	—
220	M₁₆	—	53	8	no	—
221	C₂×C₈	—	2	4	no	—
222	C₂×C₈	—	53	4	no	—
223	C₂×C₈	—	53	4	no	—
224	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
225	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
226	C₄⋊C₄	—	2	4	no	—
227	C₄⋊C₄	—	2	4	no	—
228	C₄⋊C₄	—	4	2	no	—
229	C₄⋊C₄	—	4	2	no	—
230	C₄⋊C₄	—	106	2	no	—
231	C₄⋊C₄	—	106	2	no	—
232	C₄⋊C₄	—	106	2	no	—
233	C₄⋊C₄	—	106	2	no	—
234	C₄⋊C₄	—	212	2	no	—
235	C₄⋊C₄	—	212	2	no	—
236	C₂²⋊C₄	—	212	2	no	—
237	C₄²	—	212	2	no	—
238	C₄²	D₄×D₅₃	1	1	yes	characteristic
239	C₄²	D₄×D₅₃	1	1	yes	characteristic
240	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
241	C₄²	C₂³×D₅₃	1	1	yes	characteristic
242	C₂³	#13133213	1	1	yes	characteristic
243	Q₈	—	4	8	no	—
244	Q₈	—	8	4	no	—
245	Q₈	—	212	4	no	—
246	Q₈	—	212	4	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	—	8	4	no	—
251	C₂×C₄	—	212	2	no	—
252	C₂×C₄	—	212	2	no	—
253	C₂×C₄	#13133213	1	1	yes	characteristic
254	C₂×C₄	#13133213	1	1	yes	characteristic
255	C₂×C₄	#13133213	1	1	yes	characteristic
256	C₂×C₄	Q₁₆×D₅₃	1	4	yes	—
257	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
258	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
259	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
260	C₈	—	2	2	no	—
261	C₈	—	53	8	no	—
262	C₈	—	106	4	no	—
263	C₈	#13133248	1	4	yes	—
264	C₂²	—	2	1	no	—
265	C₂²	—	2	1	no	—
266	C₂²	#16811001	1	2	yes	—
267	C₂²	C₂×#13133213	1	1	yes	characteristic
268	C₄	—	2	1	no	—
269	C₄	—	2	1	no	—
270	C₄	—	8	4	no	—
271	C₄	—	424	2	no	—
272	C₄	—	424	2	no	—
273	C₄	#16811001	1	2	yes	—
274	C₄	C₂×Q₁₆×D₅₃	1	2	yes	—
275	C₄	C₂×#13133248	1	2	yes	—
276	C₄	#16811424	1	2	yes	—
277	C₂	—	2	2	no	—
278	C₂	C₂×#16811001	1	1	yes	characteristic
279	C₂	#20285172	1	1	yes	characteristic
280	C₂	#20285196	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13133213
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₂×C₅₃×#500	C₂
socle	C₂²×C₅₃	C₂×#127
prime radical	C₂³×C₅₃	#127
hypercenter	C₂×#500	D₅₃
hypocenter	C₅₃	#19633
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
4. #20274391	C₂	semidirect
5. #20274332	C₂	semidirect
6. #20274313	C₂	semidirect
97. #11719917	Q₁₆	semidirect
173. C₅₃	#19633	semidirect

Maximal subgroups

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

subgroup	index
#20285663	2
#20285301	2
#20285257	2
#20274391	2
#20274332	2
#20274313	2
C₈×#11719917	2
C₂×C₅₃×#500	2
#19633	53