Subgroups — #27292818

Order 13568.

698 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283279	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810650	C₂	1	2	yes	maximal
3	C₂×#16810393	C₂	1	2	yes	maximal
4	#20274412	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1740	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719917	C₂²	1	1	yes	characteristic
7	C₂²×#11719917	C₂²	1	2	yes	—
8	C₂³×#88322	C₂²	1	1	yes	characteristic
9	#16810650	—	2	4	no	—
10	#16810650	—	2	4	no	—
11	#16810650	—	2	4	no	—
12	#16810647	—	2	2	no	—
13	C₂×C₄⋊#88322	—	2	4	no	—
14	#16810393	—	2	4	no	—
15	#16810378	—	2	2	no	—
16	C₂×#117×C₅₃	C₂²	1	2	yes	—
17	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×#342	C₄	1	4	yes	—
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	—	2	2	no	—
24	C₂×#11719917	—	2	2	no	—
25	C₂×#11719917	—	2	2	no	—
26	C₂×#11719917	—	2	1	no	—
27	C₂×#11719917	—	2	2	no	—
28	C₂×#11719917	Q₈	1	2	yes	—
29	C₂²×#88322	—	2	1	no	—
30	C₂²×#88322	—	2	2	no	—
31	C₂²×#88322	—	2	2	no	—
32	C₂²×#88322	—	2	1	no	—
33	C₂²×#88322	—	2	1	no	—
34	C₂²×#88322	D₄	1	2	yes	—
35	C₂²×#88322	D₄	1	4	yes	—
36	C₄⋊#88322	—	2	16	no	—
37	C₄⋊#88322	—	2	8	no	—
38	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
39	#117×C₅₃	—	2	2	no	—
40	#117×C₅₃	—	2	2	no	—
41	#117×C₅₃	—	2	2	no	—
42	#117×C₅₃	—	2	2	no	—
43	#117×C₅₃	D₄	1	4	yes	—
44	#117×C₅₃	Q₈	1	4	yes	—
45	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
46	C₂×C₄⋊C₄×C₅₃	Q₈	1	2	yes	—
47	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
48	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
49	C₂×C₄²×C₅₃	—	2	2	no	—
50	C₂×C₄²×C₅₃	D₄	1	2	yes	—
51	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	—	2	4	no	—
53	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
54	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
55	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
58	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
61	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	#41	1	2	yes	—
63	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
64	C₄⋊C₄×C₅₃	—	2	4	no	—
65	C₄⋊C₄×C₅₃	—	2	4	no	—
66	C₄⋊C₄×C₅₃	—	2	4	no	—
67	C₂²⋊C₄×C₅₃	—	2	4	no	—
68	C₂²⋊C₄×C₅₃	—	2	4	no	—
69	C₂²⋊C₄×C₅₃	C₄⋊C₄	1	8	yes	—
70	C₄²×C₅₃	—	2	4	no	—
71	C₄²×C₅₃	—	2	2	no	—
72	C₄²×C₅₃	—	2	4	no	—
73	C₄²×C₅₃	—	2	2	no	—
74	#11719917	—	2	4	no	—
75	#11719917	—	4	2	no	—
76	#11719917	—	4	2	no	—
77	#11719917	—	4	2	no	—
78	#11719917	—	4	2	no	—
79	#11719917	—	4	2	no	—
80	C₂×#88322	—	2	4	no	—
81	C₂×#88322	—	2	4	no	—
82	C₂×#88322	—	2	4	no	—
83	C₂×#88322	—	2	2	no	—
84	C₂×#88322	—	2	4	no	—
85	C₂×#88322	—	2	2	no	—
86	C₂×#88322	—	4	1	no	—
87	C₂×#88322	—	4	2	no	—
88	C₂×#88322	—	4	1	no	—
89	C₂×#88322	—	4	1	no	—
90	C₂×#88322	—	4	1	no	—
91	C₂³×C₅₃	—	4	2	no	—
92	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂³×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
94	C₂³×C₅₃	C₄×D₄	1	2	yes	—
95	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
96	C₂³×C₅₃	C₄×Q₈	1	1	yes	characteristic
97	C₂×C₄×C₅₃	—	2	8	no	—
98	C₂×C₄×C₅₃	—	2	4	no	—
99	C₂×C₄×C₅₃	—	4	8	no	—
100	C₂×C₄×C₅₃	—	4	4	no	—
101	C₂×C₄×C₅₃	#120	1	2	yes	—
102	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
103	C₂×C₄×C₅₃	#121	1	2	yes	—
104	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
106	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
107	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
108	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
109	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
110	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
111	C₂×C₄×C₅₃	#123	1	2	yes	—
112	C₂×C₄×C₅₃	#125	1	1	yes	characteristic
113	C₂×C₄×C₅₃	#125	1	2	yes	—
114	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
115	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
116	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
117	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
118	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
119	#88322	—	4	2	no	—
120	#88322	—	4	4	no	—
121	#88322	—	4	2	no	—
122	#11345	—	53	1	no	maximal, Sylow
123	C₂²×C₅₃	—	2	2	no	—
124	C₂²×C₅₃	—	2	4	no	—
125	C₂²×C₅₃	—	4	2	no	—
126	C₂²×C₅₃	—	4	1	no	—
127	C₂²×C₅₃	—	4	1	no	—
128	C₂²×C₅₃	#384	1	1	yes	characteristic
129	C₂²×C₅₃	#385	1	2	yes	—
130	C₂²×C₅₃	#385	1	1	yes	characteristic
131	C₂²×C₅₃	#389	1	2	yes	—
132	C₂²×C₅₃	#389	1	1	yes	characteristic
133	C₄×C₅₃	—	4	8	no	—
134	C₄×C₅₃	—	4	4	no	—
135	C₄×C₅₃	#392	1	1	yes	characteristic
136	C₄×C₅₃	#395	1	2	yes	—
137	C₄×C₅₃	#395	1	1	yes	characteristic
138	C₄×C₅₃	#398	1	2	yes	—
139	C₄×C₅₃	#398	1	1	yes	characteristic
140	C₄×C₅₃	#399	1	1	yes	characteristic
141	#2905	—	53	1	no	—
142	C₂×#424	—	53	2	no	—
143	C₂×#344	—	53	2	no	—
144	#1740	D₅₃	1	1	yes	characteristic
145	#1729	—	53	1	no	—
146	C₂×C₅₃	—	2	2	no	—
147	C₂×C₅₃	—	2	1	no	—
148	C₂×C₅₃	—	2	1	no	—
149	C₂×C₅₃	—	2	2	no	—
150	C₂×C₅₃	#1444	1	1	yes	characteristic
151	C₂×C₅₃	#1868	1	2	yes	—
152	C₂²×M₁₆	—	53	2	no	—
153	C₂²×M₁₆	—	53	1	no	—
154	C₂³×C₈	—	53	1	no	—
155	C₂×#117	C₂×D₅₃	1	2	yes	—
156	C₂×#117	C₂×D₅₃	1	1	yes	characteristic
157	#424	—	106	4	no	—
158	#424	—	106	4	no	—
159	#424	—	106	4	no	—
160	#423	—	106	2	no	—
161	C₂×C₄⋊C₈	—	106	4	no	—
162	#344	—	106	4	no	—
163	#342	Dic₅₃	1	4	yes	—
164	#336	—	106	2	no	—
165	C₅₃	#11345	1	1	yes	characteristic, complemented, Sylow
166	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
167	C₂×M₁₆	—	53	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	—	106	2	no	—
173	C₂×M₁₆	—	106	1	no	—
174	C₂×M₁₆	—	106	2	no	—
175	C₂×M₁₆	—	106	2	no	—
176	C₂×M₁₆	—	106	2	no	—
177	C₂²×C₈	—	53	2	no	—
178	C₂²×C₈	—	53	4	no	—
179	C₂²×C₈	—	106	1	no	—
180	C₂²×C₈	—	106	2	no	—
181	C₂²×C₈	—	106	1	no	—
182	C₂²×C₈	—	106	2	no	—
183	C₂²×C₈	—	106	1	no	—
184	#117	—	2	2	no	—
185	#117	—	2	2	no	—
186	#117	—	2	2	no	—
187	#117	—	2	2	no	—
188	#117	Dic₁₀₆	1	4	yes	—
189	#117	D₂₁₂	1	4	yes	—
190	C₂×C₄⋊C₄	—	2	2	no	—
191	C₂×C₄⋊C₄	Dic₁₀₆	1	2	yes	—
192	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	2	yes	—
193	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	4	yes	—
194	C₂×C₄²	—	2	2	no	—
195	C₂×C₄²	D₂₁₂	1	2	yes	—
196	C₄⋊C₈	—	106	16	no	—
197	C₄⋊C₈	—	106	8	no	—
198	C₂⁴	C₂²×Dic₅₃	1	1	yes	characteristic
199	C₂²×C₄	—	2	4	no	—
200	C₂²×C₄	C₄⋊Dic₅₃	1	4	yes	—
201	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
202	C₂²×C₄	C₂×Dic₁₀₆	1	2	yes	—
203	C₂²×C₄	C₂×D₂₁₂	1	2	yes	—
204	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
205	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
206	C₂²×C₄	D₄×D₅₃	1	2	yes	—
207	C₂²×C₄	#11719947	1	2	yes	—
208	C₂²×C₄	#11719947	1	1	yes	characteristic
209	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
210	C₂²×C₄	#11719949	1	1	yes	characteristic
211	M₁₆	—	106	4	no	—
212	M₁₆	—	212	2	no	—
213	M₁₆	—	212	2	no	—
214	M₁₆	—	212	2	no	—
215	M₁₆	—	212	2	no	—
216	M₁₆	—	212	2	no	—
217	C₂×C₈	—	106	4	no	—
218	C₂×C₈	—	106	4	no	—
219	C₂×C₈	—	106	4	no	—
220	C₂×C₈	—	106	2	no	—
221	C₂×C₈	—	106	4	no	—
222	C₂×C₈	—	106	2	no	—
223	C₂×C₈	—	212	1	no	—
224	C₂×C₈	—	212	2	no	—
225	C₂×C₈	—	212	1	no	—
226	C₂×C₈	—	212	1	no	—
227	C₂×C₈	—	212	1	no	—
228	C₄⋊C₄	—	2	4	no	—
229	C₄⋊C₄	—	2	4	no	—
230	C₄⋊C₄	—	2	4	no	—
231	C₂²⋊C₄	—	2	4	no	—
232	C₂²⋊C₄	—	2	4	no	—
233	C₂²⋊C₄	C₄⋊Dic₅₃	1	8	yes	—
234	C₄²	—	2	4	no	—
235	C₄²	—	2	2	no	—
236	C₄²	—	2	4	no	—
237	C₄²	—	2	2	no	—
238	C₂³	—	4	2	no	—
239	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
240	C₂³	C₂×C₄⋊Dic₅₃	1	2	yes	—
241	C₂³	D₄×Dic₅₃	1	2	yes	—
242	C₂³	D₄×Dic₅₃	1	1	yes	characteristic
243	C₂³	Q₈×Dic₅₃	1	1	yes	characteristic
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	2	8	no	—
246	C₂×C₄	—	4	8	no	—
247	C₂×C₄	—	4	4	no	—
248	C₂×C₄	#13133187	1	2	yes	—
249	C₂×C₄	#13133187	1	1	yes	characteristic
250	C₂×C₄	#13133192	1	2	yes	—
251	C₂×C₄	#13133192	1	1	yes	characteristic
252	C₂×C₄	#13133194	1	2	yes	—
253	C₂×C₄	#13133194	1	2	yes	—
254	C₂×C₄	#13133194	1	2	yes	—
255	C₂×C₄	#13133197	1	1	yes	characteristic
256	C₂×C₄	#13133197	1	2	yes	—
257	C₂×C₄	#13133201	1	1	yes	characteristic
258	C₂×C₄	#13133201	1	2	yes	—
259	C₂×C₄	#13133208	1	1	yes	characteristic
260	C₂×C₄	#13133208	1	2	yes	—
261	C₂×C₄	#13133214	1	1	yes	characteristic
262	C₂×C₄	#13133216	1	1	yes	characteristic
263	C₂×C₄	#13133216	1	2	yes	—
264	C₂×C₄	#13133217	1	2	yes	—
265	C₂×C₄	#13133217	1	1	yes	characteristic
266	C₈	—	212	2	no	—
267	C₈	—	212	4	no	—
268	C₈	—	212	2	no	—
269	C₂²	—	2	2	no	—
270	C₂²	—	2	4	no	—
271	C₂²	—	4	2	no	—
272	C₂²	—	4	1	no	—
273	C₂²	—	4	1	no	—
274	C₂²	#16810493	1	1	yes	characteristic
275	C₂²	#16810498	1	2	yes	—
276	C₂²	#16810498	1	1	yes	characteristic
277	C₂²	#16810522	1	2	yes	—
278	C₂²	#16810522	1	1	yes	characteristic
279	C₄	—	4	8	no	—
280	C₄	—	4	4	no	—
281	C₄	#16810533	1	1	yes	characteristic
282	C₄	#16810544	1	2	yes	—
283	C₄	#16810544	1	1	yes	characteristic
284	C₄	#16810555	1	2	yes	—
285	C₄	#16810555	1	1	yes	characteristic
286	C₄	#16810560	1	1	yes	characteristic
287	C₂	—	2	2	no	—
288	C₂	—	2	1	no	—
289	C₂	—	2	2	no	—
290	C₂	—	2	1	no	—
291	C₂	#20272980	1	1	yes	characteristic
292	C₂	#20275003	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#13133187
commutator, prime radical	C₂⁴×C₅₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₅₃×#1740	C₂
socle	C₂²×C₅₃	#385
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	#1740	D₅₃
hypocenter	C₅₃	#11345
LCS	G ▷ C₂⁴×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #1740

Complements

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

N	H ≅ G/N	type
165. C₅₃	#11345	semidirect

Maximal subgroups

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

subgroup	index
#20283279	2
C₂×#16810650	2
C₂×#16810393	2
#20274412	2
C₅₃×#1740	2
#11345	53