Subgroups — #27292970

Order 13568.

598 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810672	C₂	1	1	yes	maximal, characteristic
2	#20274424	C₂	1	2	yes	maximal
3	C₂×#16810380	C₂	1	1	yes	maximal, characteristic
4	#20274411	C₂	1	2	yes	maximal
5	C₅₃×#1729	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719917	C₂²	1	2	yes	—
7	C₂×#13133176	C₂²	1	2	yes	—
8	#16810672	—	2	2	no	—
9	#16810672	C₄	1	4	yes	complemented
10	C₂×#13133153	—	2	2	no	—
11	#16810382	—	2	4	no	—
12	#16810380	—	2	2	no	—
13	#16810377	—	2	4	no	—
14	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
15	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#336	—	2	2	no	—
18	C₂×#11719917	—	2	2	no	—
19	C₂×#11719917	—	2	2	no	—
20	C₂×#11719917	—	2	2	no	—
21	C₂×#11719917	—	2	2	no	—
22	C₂×#11719917	—	2	2	no	—
23	C₂×#11719917	C₂×C₄	1	4	yes	—
24	C₂²×#88322	—	2	2	no	—
25	#13133176	—	2	2	no	—
26	#13133176	—	2	2	no	—
27	#13133176	—	2	2	no	—
28	#13133176	—	2	2	no	—
29	C₂×C₄⋊Dic₅₃	—	2	2	no	—
30	C₂×C₂²⋊Dic₅₃	—	2	2	no	—
31	C₂×C₄×Dic₅₃	D₄	1	4	yes	—
32	#13133153	—	2	8	no	—
33	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
34	C₂×M₁₆×C₅₃	—	2	2	no	—
35	C₂×M₁₆×C₅₃	—	2	2	no	—
36	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
37	C₂×M₁₆×C₅₃	D₄	1	2	yes	—
38	C₂²×C₈×C₅₃	—	2	1	no	—
39	C₂²×C₈×C₅₃	—	2	2	no	—
40	C₂²×C₈×C₅₃	—	2	2	no	—
41	C₂²×C₈×C₅₃	—	2	1	no	—
42	C₂²×C₈×C₅₃	D₄	1	2	yes	—
43	C₂²×C₈×C₅₃	Q₈	1	2	yes	—
44	#117×C₅₃	—	2	2	no	—
45	#117×C₅₃	—	2	2	no	—
46	C₂×C₄⋊C₄×C₅₃	Q₈	1	2	yes	—
47	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
48	C₂×C₄²×C₅₃	—	2	1	no	—
49	C₂⁴×C₅₃	#41	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	—	2	2	no	—
51	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
55	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	#41	1	2	yes	—
58	C₂²×C₄×C₅₃	#41	1	2	yes	—
59	C₂²×C₄×C₅₃	#41	1	2	yes	—
60	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
61	C₂²×Dic₅₃	—	2	4	no	—
62	M₁₆×C₅₃	—	2	4	no	—
63	M₁₆×C₅₃	—	2	4	no	—
64	C₂×C₈×C₅₃	—	2	2	no	—
65	C₂×C₈×C₅₃	—	2	2	no	—
66	C₂×C₈×C₅₃	—	2	4	no	—
67	C₂×C₈×C₅₃	—	2	2	no	—
68	C₂×C₈×C₅₃	—	4	1	no	—
69	C₂×C₈×C₅₃	—	4	2	no	—
70	C₂×C₈×C₅₃	—	4	1	no	—
71	C₂×C₈×C₅₃	C₂²⋊C₄	1	4	yes	—
72	C₄⋊C₄×C₅₃	—	2	4	no	—
73	C₂²⋊C₄×C₅₃	—	4	2	no	—
74	C₄²×C₅₃	—	4	1	no	—
75	C₄²×C₅₃	—	4	1	no	—
76	C₂²⋊Dic₅₃	—	4	2	no	—
77	C₂²⋊Dic₅₃	—	4	2	no	—
78	C₄⋊Dic₅₃	—	4	2	no	—
79	C₄⋊Dic₅₃	—	4	2	no	—
80	C₄×Dic₅₃	—	2	4	no	—
81	C₄×Dic₅₃	—	2	4	no	—
82	#11719917	—	2	4	no	—
83	#11719917	—	2	4	no	—
84	#11719917	—	4	2	no	—
85	#11719917	—	4	2	no	—
86	C₂×#88322	—	2	4	no	—
87	C₂×#88322	—	4	2	no	—
88	C₂×#88322	—	4	2	no	—
89	C₂³×C₅₃	—	4	2	no	—
90	C₂³×C₅₃	C₄×Q₈	1	1	yes	characteristic
91	C₂³×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
92	C₂³×C₅₃	#123	1	2	yes	—
93	C₂³×C₅₃	#124	1	1	yes	characteristic
94	C₂³×C₅₃	#126	1	2	yes	—
95	C₂×C₄×C₅₃	—	4	4	no	—
96	C₂×C₄×C₅₃	—	4	2	no	—
97	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₅₃	#117	1	2	yes	—
99	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
100	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
101	C₂×C₄×C₅₃	#121	1	2	yes	—
102	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
103	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
104	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
105	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
106	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
107	C₂×C₄×C₅₃	#123	1	2	yes	—
108	C₂×C₄×C₅₃	#123	1	2	yes	—
109	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
110	C₂×C₄×C₅₃	#125	1	2	yes	—
111	C₂×C₄×C₅₃	#125	1	2	yes	—
112	C₂×C₄×C₅₃	#126	1	2	yes	—
113	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
114	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
115	C₂×Dic₅₃	—	4	8	no	—
116	C₂×Dic₅₃	—	4	4	no	—
117	C₈×C₅₃	—	2	4	no	—
118	C₈×C₅₃	—	4	2	no	—
119	#88322	—	4	4	no	—
120	#11374	—	53	1	no	maximal, Sylow
121	C₂²×C₅₃	—	2	4	no	—
122	C₂²×C₅₃	—	2	2	no	—
123	C₂²×C₅₃	—	4	1	no	—
124	C₂²×C₅₃	—	4	1	no	—
125	C₂²×C₅₃	—	4	2	no	—
126	C₂²×C₅₃	#387	1	2	yes	—
127	C₂²×C₅₃	#391	1	1	yes	characteristic
128	C₂²×C₅₃	#397	1	2	yes	—
129	C₂²×C₅₃	#399	1	1	yes	characteristic
130	C₂²×C₅₃	#400	1	1	yes	characteristic
131	C₄×C₅₃	—	8	2	no	—
132	C₄×C₅₃	#381	1	1	yes	characteristic
133	C₄×C₅₃	#387	1	2	yes	—
134	C₄×C₅₃	#390	1	1	yes	characteristic
135	C₄×C₅₃	#398	1	2	yes	—
136	C₄×C₅₃	#399	1	1	yes	characteristic
137	C₄×C₅₃	#400	1	1	yes	characteristic
138	Dic₅₃	—	8	4	no	—
139	C₂×#430	—	53	1	no	—
140	#1733	—	53	2	no	—
141	C₂×H(3,ℤ/4)	—	53	1	no	—
142	#1729	—	53	2	no	—
143	#1729	D₅₃	1	1	yes	characteristic
144	C₂×C₅₃	—	2	2	no	—
145	C₂×C₅₃	—	2	2	no	—
146	C₂×C₅₃	—	2	1	no	—
147	C₂×C₅₃	—	2	1	no	—
148	C₂×C₅₃	#1445	1	1	yes	characteristic
149	C₂×C₅₃	#2098	1	2	yes	—
150	C₂²×M₁₆	—	53	2	no	—
151	C₂²×M₁₆	C₂×D₅₃	1	1	yes	characteristic
152	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
153	C₂×#117	—	53	2	no	—
154	C₂×#117	C₂×D₅₃	1	1	yes	characteristic
155	#430	—	53	4	no	—
156	#430	—	106	2	no	—
157	C₂×#108	—	106	2	no	—
158	#339	—	106	4	no	—
159	H(3,ℤ/4)	—	106	2	no	—
160	#336	—	2	2	no	—
161	#336	—	106	4	no	—
162	C₅₃	#11374	1	1	yes	characteristic, complemented, Sylow
163	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
164	C₂×M₁₆	—	2	2	no	—
165	C₂×M₁₆	—	2	2	no	—
166	C₂×M₁₆	—	53	4	no	—
167	C₂×M₁₆	—	106	2	no	—
168	C₂×M₁₆	—	106	2	no	—
169	C₂×M₁₆	—	106	2	no	—
170	C₂×M₁₆	—	106	2	no	—
171	C₂×M₁₆	—	106	2	no	—
172	C₂×M₁₆	C₄×D₅₃	1	2	yes	—
173	C₂×M₁₆	D₂₁₂	1	2	yes	—
174	C₂²×C₈	—	2	1	no	—
175	C₂²×C₈	—	2	2	no	—
176	C₂²×C₈	—	2	2	no	—
177	C₂²×C₈	—	2	1	no	—
178	C₂²×C₈	—	106	2	no	—
179	C₂²×C₈	Dic₁₀₆	1	2	yes	—
180	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
181	#117	—	2	2	no	—
182	#117	—	2	2	no	—
183	#117	—	106	2	no	—
184	#117	—	106	2	no	—
185	#117	—	106	2	no	—
186	#117	—	106	2	no	—
187	C₂×C₄⋊C₄	—	106	2	no	—
188	C₂×C₄⋊C₄	Dic₁₀₆	1	2	yes	—
189	C₂×C₂²⋊C₄	—	2	1	no	—
190	C₂×C₂²⋊C₄	—	106	2	no	—
191	C₂×C₄²	—	2	1	no	—
192	C₂×C₄²	—	53	4	no	—
193	#108	—	106	8	no	—
194	C₂⁴	#11719945	1	1	yes	characteristic
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	—	106	4	no	—
197	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
198	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
199	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
200	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
201	C₂²×C₄	#11719945	1	1	yes	characteristic
202	C₂²×C₄	D₄×D₅₃	1	2	yes	—
203	C₂²×C₄	#11719947	1	2	yes	—
204	C₂²×C₄	#11719947	1	2	yes	—
205	C₂²×C₄	#11719949	1	2	yes	—
206	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
207	M₁₆	—	2	4	no	—
208	M₁₆	—	2	4	no	—
209	M₁₆	—	106	4	no	—
210	M₁₆	—	106	4	no	—
211	M₁₆	—	212	2	no	—
212	M₁₆	—	212	2	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	4	no	—
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	—	4	1	no	—
218	C₂×C₈	—	4	2	no	—
219	C₂×C₈	—	4	1	no	—
220	C₂×C₈	—	106	4	no	—
221	C₂×C₈	—	212	2	no	—
222	C₂×C₈	—	212	2	no	—
223	C₂×C₈	#11719921	1	4	yes	—
224	C₄⋊C₄	—	2	4	no	—
225	C₄⋊C₄	—	212	2	no	—
226	C₄⋊C₄	—	212	2	no	—
227	C₂²⋊C₄	—	4	2	no	—
228	C₂²⋊C₄	—	212	2	no	—
229	C₂²⋊C₄	—	212	2	no	—
230	C₄²	—	4	1	no	—
231	C₄²	—	4	1	no	—
232	C₄²	—	106	4	no	—
233	C₄²	—	106	4	no	—
234	C₂³	—	4	2	no	—
235	C₂³	C₄×Dic₁₀₆	1	1	yes	characteristic
236	C₂³	#13133196	1	1	yes	characteristic
237	C₂³	#13133199	1	2	yes	—
238	C₂³	#13133205	1	1	yes	characteristic
239	C₂³	#13133212	1	2	yes	—
240	C₂×C₄	—	4	4	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	—	212	8	no	—
243	C₂×C₄	—	212	4	no	—
244	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
245	C₂×C₄	#13133173	1	2	yes	—
246	C₂×C₄	#13133179	1	2	yes	—
247	C₂×C₄	C₄×Dic₁₀₆	1	1	yes	characteristic
248	C₂×C₄	#13133192	1	2	yes	—
249	C₂×C₄	#13133194	1	2	yes	—
250	C₂×C₄	#13133194	1	2	yes	—
251	C₂×C₄	#13133196	1	1	yes	characteristic
252	C₂×C₄	#13133198	1	1	yes	characteristic
253	C₂×C₄	#13133198	1	1	yes	characteristic
254	C₂×C₄	#13133199	1	2	yes	—
255	C₂×C₄	#13133201	1	2	yes	—
256	C₂×C₄	#13133206	1	1	yes	characteristic
257	C₂×C₄	#13133208	1	2	yes	—
258	C₂×C₄	#13133208	1	2	yes	—
259	C₂×C₄	#13133212	1	2	yes	—
260	C₂×C₄	#13133213	1	1	yes	characteristic
261	C₂×C₄	#13133217	1	1	yes	characteristic
262	C₈	—	2	4	no	—
263	C₈	—	4	2	no	—
264	C₈	—	212	4	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	2	4	no	—
267	C₂²	—	4	1	no	—
268	C₂²	—	4	1	no	—
269	C₂²	—	4	2	no	—
270	C₂²	#16810506	1	2	yes	—
271	C₂²	#16810529	1	1	yes	characteristic
272	C₂²	#16810550	1	2	yes	—
273	C₂²	#16810559	1	1	yes	characteristic
274	C₂²	#16810562	1	1	yes	characteristic
275	C₄	—	8	2	no	—
276	C₄	—	424	4	no	—
277	C₄	#16810484	1	1	yes	characteristic
278	C₄	#16810506	1	2	yes	—
279	C₄	#16810524	1	1	yes	characteristic
280	C₄	#16810555	1	2	yes	—
281	C₄	#16810559	1	1	yes	characteristic
282	C₄	#16810562	1	1	yes	characteristic
283	C₂	—	2	2	no	—
284	C₂	—	2	2	no	—
285	C₂	—	2	1	no	—
286	C₂	—	2	1	no	—
287	C₂	#20272984	1	1	yes	characteristic
288	C₂	#20276268	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#13133213
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₅₃×#1729	C₂
socle	C₂²×C₅₃	#399
prime radical	C₂⁴×C₅₃	#41
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	#1729	D₅₃
hypocenter	C₅₃	#11374
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #1729

Complements

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

N	H ≅ G/N	type
9. #16810672	C₄	semidirect
162. C₅₃	#11374	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810672	2
#20274424	2
C₂×#16810380	2
#20274411	2
C₅₃×#1729	2
#11374	53