Subgroups — #27403093

Order 13568.

642 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283666	C₂	1	2	yes	maximal
2	C₂×C₈×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic
3	#20275489	C₂	1	4	yes	maximal
4	#20275488	C₂	1	2	yes	maximal
5	#20274851	C₂	1	2	yes	maximal
6	C₄×#13133153	C₂	1	1	yes	maximal, characteristic
7	#20274616	C₂	1	2	yes	maximal
8	C₅₃×#1837	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
10	#16810752	—	2	4	no	—
11	#16810750	—	2	2	no	—
12	C₈×Dic₁₀₆	C₄	1	4	yes	—
13	C₈×Dic₁₀₆	C₄	1	2	yes	—
14	C₈×Dic₁₀₆	C₄	1	2	yes	—
15	#16810747	—	2	2	no	—
16	C₂×#13133153	C₂²	1	2	yes	—
17	C₂×#13133152	C₂²	1	4	yes	—
18	#16810646	C₂²	1	4	yes	—
19	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
20	C₂×#13133146	C₂²	1	2	yes	—
21	C₄×#11719917	C₂²	1	2	yes	—
22	M₁₆×Dic₅₃	C₂²	1	4	yes	—
23	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
24	#16810388	C₂²	1	1	yes	characteristic
25	#16810388	C₂²	1	1	yes	characteristic
26	#16810387	C₂²	1	4	yes	—
27	C₅₃×#423	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
29	C₅₃×#341	C₂²	1	2	yes	—
30	C₅₃×#341	C₂²	1	2	yes	—
31	C₂²×Dic₁₀₆	Q₈	1	1	yes	characteristic
32	C₂×#11719917	C₂³	1	2	yes	—
33	C₂×#11719917	C₂³	1	2	yes	—
34	C₂²×#88322	C₂³	1	2	yes	—
35	C₄×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
36	C₄×Dic₁₀₆	C₂×C₄	1	2	yes	—
37	C₄×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
38	C₄×Dic₁₀₆	D₄	1	2	yes	—
39	C₄×Dic₁₀₆	D₄	1	1	yes	characteristic
40	C₄×Dic₁₀₆	D₄	1	1	yes	characteristic
41	C₂×C₄⋊Dic₅₃	Q₈	1	1	yes	characteristic
42	C₂×#11719919	Q₈	1	2	yes	—
43	C₂×C₄×Dic₅₃	C₂³	1	2	yes	—
44	#13133153	—	2	4	no	—
45	#13133152	—	2	8	no	—
46	C₄⋊#88322	C₂×C₄	1	2	yes	—
47	C₄⋊#88322	C₂×C₄	1	2	yes	—
48	#13133146	—	2	4	no	—
49	#13133146	—	2	4	no	—
50	#13133146	C₂×C₄	1	4	yes	—
51	#13133146	C₂×C₄	1	4	yes	—
52	#13133128	—	2	2	no	—
53	#13133127	—	2	4	no	—
54	C₄×#88322	—	2	2	no	—
55	C₈×Dic₅₃	—	2	4	no	—
56	C₈×Dic₅₃	C₂×C₄	1	4	yes	—
57	C₈×Dic₅₃	C₂×C₄	1	4	yes	—
58	C₂×M₁₆×C₅₃	C₂³	1	4	yes	—
59	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
60	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
61	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
62	C₄⋊C₈×C₅₃	—	2	2	no	—
63	C₄⋊C₈×C₅₃	—	2	2	no	—
64	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
65	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
66	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
69	C₂²×Dic₅₃	C₂×Q₈	1	2	yes	—
70	C₂²×Dic₅₃	C₂×Q₈	1	2	yes	—
71	C₂×Dic₁₀₆	—	2	4	no	—
72	C₂×Dic₁₀₆	C₄⋊C₄	1	1	yes	characteristic
73	C₂×Dic₁₀₆	C₄⋊C₄	1	2	yes	—
74	C₂×Dic₁₀₆	C₄⋊C₄	1	1	yes	characteristic
75	M₁₆×C₅₃	—	2	4	no	—
76	M₁₆×C₅₃	—	2	4	no	—
77	C₂×C₈×C₅₃	—	2	2	no	—
78	C₂×C₈×C₅₃	—	2	2	no	—
79	C₂×C₈×C₅₃	—	2	4	no	—
80	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
82	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
84	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
85	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
86	C₄⋊Dic₅₃	C₄⋊C₄	1	2	yes	—
87	C₄⋊Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
88	C₄⋊Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
89	#11719919	C₄⋊C₄	1	2	yes	—
90	#11719919	C₄⋊C₄	1	2	yes	—
91	#11719919	C₄⋊C₄	1	2	yes	—
92	#11719919	C₄⋊C₄	1	2	yes	—
93	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
94	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
95	C₄×Dic₅₃	C₂×D₄	1	2	yes	—
96	C₄×Dic₅₃	C₂×D₄	1	2	yes	—
97	#11719917	—	2	2	no	—
98	#11719917	—	2	2	no	—
99	#11719917	—	2	2	no	—
100	#11719917	#41	1	4	yes	—
101	C₂×#88322	—	2	2	no	—
102	C₂×#88322	—	2	4	no	—
103	C₂×#88322	—	2	2	no	—
104	C₂×#88322	C₂²×C₄	1	4	yes	—
105	C₂³×C₅₃	C₂²×Q₈	1	1	yes	characteristic
106	C₂×C₄×C₅₃	—	2	2	no	—
107	C₂×C₄×C₅₃	—	4	1	no	—
108	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
113	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
115	C₂×C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
116	C₂×Dic₅₃	—	2	4	no	—
117	C₂×Dic₅₃	—	4	2	no	—
118	C₂×Dic₅₃	C₂×C₄⋊C₄	1	2	yes	—
119	C₂×Dic₅₃	C₂×C₄⋊C₄	1	2	yes	—
120	C₂×Dic₅₃	C₂×C₄⋊C₄	1	2	yes	—
121	C₂×Dic₅₃	C₂×C₄⋊C₄	1	2	yes	—
122	Dic₁₀₆	—	2	2	no	—
123	Dic₁₀₆	—	2	4	no	—
124	Dic₁₀₆	#108	1	4	yes	complemented
125	C₈×C₅₃	—	2	2	no	—
126	C₈×C₅₃	—	2	2	no	—
127	C₈×C₅₃	—	4	4	no	—
128	#88322	—	2	4	no	—
129	#88322	—	2	4	no	—
130	#88322	—	4	2	no	—
131	#22746	—	53	1	no	maximal, Sylow
132	C₂²×C₅₃	—	2	1	no	—
133	C₂²×C₅₃	—	2	1	no	—
134	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
135	C₂²×C₅₃	#557	1	2	yes	—
136	C₄×C₅₃	—	2	1	no	—
137	C₄×C₅₃	—	2	1	no	—
138	C₄×C₅₃	—	2	1	no	—
139	C₄×C₅₃	C₂×#108	1	2	yes	—
140	C₄×C₅₃	C₄×#41	1	1	yes	characteristic
141	C₄×C₅₃	#519	1	1	yes	characteristic
142	C₄×C₅₃	#548	1	2	yes	—
143	Dic₅₃	—	2	2	no	—
144	Dic₅₃	—	2	2	no	—
145	Dic₅₃	—	2	2	no	—
146	Dic₅₃	C₂×#108	1	4	yes	—
147	Q₈×M₁₆	—	53	2	no	—
148	C₂×C₈×Q₈	—	53	1	no	—
149	#1950	—	53	2	no	—
150	#1950	—	53	4	no	—
151	#1837	—	53	2	no	—
152	#1837	D₅₃	1	1	yes	characteristic
153	C₄×#108	—	53	1	no	—
154	C₄×#108	—	53	2	no	—
155	C₂×C₅₃	—	2	2	no	—
156	C₂×C₅₃	#2351	1	1	yes	characteristic
157	C₂×C₅₃	C₂²×#108	1	1	yes	characteristic
158	C₂×C₅₃	#2916	1	1	yes	characteristic
159	C₂×C₄×Q₈	—	53	1	no	—
160	#446	—	106	4	no	—
161	#446	—	106	2	no	—
162	C₈×Q₈	—	53	2	no	—
163	C₈×Q₈	—	53	4	no	—
164	C₈×Q₈	—	53	2	no	—
165	C₈×Q₈	—	106	2	no	—
166	C₂×#108	—	53	4	no	—
167	C₂×#108	—	53	2	no	—
168	#423	—	53	4	no	—
169	#423	C₂×D₅₃	1	2	yes	—
170	C₂×C₄⋊C₈	—	53	2	no	—
171	C₂×C₄⋊C₈	—	53	1	no	—
172	C₄×M₁₆	—	53	4	no	—
173	C₄×M₁₆	—	53	2	no	—
174	C₂×C₄×C₈	—	53	2	no	—
175	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
176	#341	—	53	4	no	—
177	#341	—	53	1	no	—
178	#341	—	53	1	no	—
179	#341	C₂×D₅₃	1	2	yes	—
180	#341	C₂×D₅₃	1	2	yes	—
181	C₅₃	#22746	1	1	yes	characteristic, complemented, Sylow
182	C₂²×Q₈	—	53	1	no	—
183	C₂×M₁₆	—	53	2	no	—
184	C₂×M₁₆	—	53	2	no	—
185	C₂×M₁₆	C₂²×D₅₃	1	4	yes	—
186	C₂²×C₈	—	53	2	no	—
187	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
188	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
189	C₄×Q₈	—	53	1	no	—
190	C₄×Q₈	—	53	2	no	—
191	C₄×Q₈	—	53	1	no	—
192	C₄×Q₈	—	53	1	no	—
193	C₄×Q₈	—	53	2	no	—
194	C₄×Q₈	—	53	1	no	—
195	C₂×C₄⋊C₄	—	53	2	no	—
196	C₂×C₄⋊C₄	—	53	1	no	—
197	C₂×C₄²	—	53	2	no	—
198	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
199	#108	—	106	4	no	—
200	#108	—	106	8	no	—
201	C₄⋊C₈	—	2	2	no	—
202	C₄⋊C₈	—	2	2	no	—
203	C₄⋊C₈	—	53	4	no	—
204	C₄⋊C₈	—	53	4	no	—
205	C₄⋊C₈	—	53	2	no	—
206	C₄⋊C₈	—	53	2	no	—
207	C₄⋊C₈	—	106	4	no	—
208	C₄⋊C₈	—	106	4	no	—
209	#97	—	106	4	no	—
210	#97	—	106	2	no	—
211	C₄×C₈	—	53	4	no	—
212	C₄×C₈	—	53	4	no	—
213	C₄×C₈	—	106	4	no	—
214	C₄×C₈	—	106	2	no	—
215	C₄×C₈	C₄×D₅₃	1	2	yes	—
216	C₄×C₈	C₄×D₅₃	1	2	yes	—
217	C₂×Q₈	—	53	1	no	—
218	C₂×Q₈	—	53	2	no	—
219	C₂×Q₈	—	53	1	no	—
220	C₂×Q₈	—	106	4	no	—
221	C₂²×C₄	—	53	2	no	—
222	C₂²×C₄	—	53	2	no	—
223	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
224	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
225	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
226	M₁₆	—	2	4	no	—
227	M₁₆	—	2	4	no	—
228	M₁₆	—	53	4	no	—
229	M₁₆	—	106	2	no	—
230	M₁₆	—	106	2	no	—
231	M₁₆	—	106	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	—	53	4	no	—
236	C₂×C₈	—	106	2	no	—
237	C₂×C₈	—	106	4	no	—
238	C₂×C₈	—	106	2	no	—
239	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
240	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
241	C₄⋊C₄	—	53	2	no	—
242	C₄⋊C₄	—	53	2	no	—
243	C₄⋊C₄	—	53	2	no	—
244	C₄⋊C₄	—	53	2	no	—
245	C₄⋊C₄	—	53	2	no	—
246	C₄⋊C₄	—	53	1	no	—
247	C₄⋊C₄	—	53	1	no	—
248	C₄²	—	53	2	no	—
249	C₄²	—	53	2	no	—
250	C₄²	—	53	2	no	—
251	C₄²	—	53	2	no	—
252	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
253	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
254	C₄²	D₄×D₅₃	1	1	yes	characteristic
255	C₄²	D₄×D₅₃	1	1	yes	characteristic
256	C₂³	C₂×Q₈×D₅₃	1	1	yes	characteristic
257	Q₈	—	53	4	no	—
258	Q₈	—	106	4	no	—
259	Q₈	—	106	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	4	1	no	—
262	C₂×C₄	—	53	2	no	—
263	C₂×C₄	—	53	2	no	—
264	C₂×C₄	—	53	2	no	—
265	C₂×C₄	—	53	2	no	—
266	C₂×C₄	—	106	4	no	—
267	C₂×C₄	—	212	2	no	—
268	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
269	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
270	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
271	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
272	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
273	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
274	C₂×C₄	#41×D₅₃	1	2	yes	—
275	C₂×C₄	#13133264	1	2	yes	—
276	C₈	—	2	2	no	—
277	C₈	—	2	2	no	—
278	C₈	—	4	4	no	—
279	C₈	—	106	4	no	—
280	C₈	—	106	4	no	—
281	C₈	—	212	2	no	—
282	C₂²	—	2	1	no	—
283	C₂²	—	2	1	no	—
284	C₂²	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
285	C₂²	#16811350	1	2	yes	—
286	C₄	—	2	1	no	—
287	C₄	—	2	1	no	—
288	C₄	—	2	1	no	—
289	C₄	—	53	4	no	—
290	C₄	—	106	2	no	—
291	C₄	—	106	2	no	—
292	C₄	—	106	2	no	—
293	C₄	#108×D₅₃	1	2	yes	—
294	C₄	#16811091	1	1	yes	characteristic
295	C₄	#16811100	1	1	yes	characteristic
296	C₄	#16811281	1	2	yes	—
297	C₂	—	2	2	no	—
298	C₂	#20277632	1	1	yes	characteristic
299	C₂	C₂×#108×D₅₃	1	1	yes	characteristic
300	C₂	#20283323	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1837	C₂
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂³×C₅₃	C₂²×Q₈
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1837	D₅₃
hypocenter	C₅₃	#22746
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1837

Complements

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

N	H ≅ G/N	type
124. Dic₁₀₆	#108	semidirect
181. C₅₃	#22746	semidirect

Maximal subgroups

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

subgroup	index
#20283666	2
C₂×C₈×Dic₁₀₆	2
#20275489	2
#20275488	2
#20274851	2
C₄×#13133153	2
#20274616	2
C₅₃×#1837	2
#22746	53