Subgroups — #27382207

Order 13568.

994 conjugacy classes of subgroups.

The subgroup lattice has 272 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	#20283133	C₂	1	2	yes	maximal
3	C₂×#16810581	C₂	1	1	yes	maximal, characteristic
4	#20274623	C₂	1	4	yes	maximal, complemented
5	#108×Dic₅₃	C₂	1	4	yes	maximal, complemented
6	C₂×#16810387	C₂	1	2	yes	maximal
7	C₂×C₅₃×#430	C₂	1	1	yes	maximal, characteristic
8	C₂²×#11719917	C₂²	1	2	yes	—
9	C₂³×#88322	C₂²	1	1	yes	characteristic
10	C₂×#13133176	C₂²	1	1	yes	characteristic
11	#16810672	C₄	1	8	yes	complemented
12	C₂×#13133153	C₂²	1	4	yes	—
13	#16810581	—	2	4	no	—
14	#16810581	C₄	1	8	yes	—
15	#16810578	—	2	4	no	—
16	M₁₆×Dic₅₃	C₂²	1	8	yes	—
17	C₂×#13133127	C₂²	1	2	yes	—
18	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
19	#16810387	C₂²	1	4	yes	—
20	#16810387	C₂²	1	4	yes	—
21	C₂²×M₁₆×C₅₃	C₂²	1	2	yes	—
22	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
23	C₅₃×#430	C₄	1	8	yes	complemented
24	C₂×#108×C₅₃	C₂²	1	4	yes	—
25	C₂×#11719917	—	2	2	no	—
26	C₂×#11719917	—	2	4	no	—
27	C₂×#11719917	C₂×C₄	1	8	yes	—
28	C₂×#11719917	C₂³	1	4	yes	—
29	C₂²×#88322	—	2	2	no	—
30	C₂²×#88322	—	2	2	no	—
31	C₂²×#88322	—	2	2	no	—
32	C₂²×#88322	C₂×C₄	1	4	yes	—
33	C₂²×#88322	C₂³	1	1	yes	characteristic
34	C₂²×#88322	C₂³	1	1	yes	characteristic
35	#13133176	C₂×C₄	1	4	yes	—
36	#13133176	D₄	1	4	yes	—
37	C₂×C₄⋊Dic₅₃	Q₈	1	2	yes	—
38	C₂×C₂²⋊Dic₅₃	Q₈	1	2	yes	—
39	C₂×C₄×Dic₅₃	C₂³	1	2	yes	—
40	#13133153	C₂×C₄	1	16	yes	—
41	#13133127	—	2	8	no	—
42	#13133127	C₂×C₄	1	8	yes	—
43	C₈×Dic₅₃	—	2	8	no	—
44	C₈×Dic₅₃	C₂×C₄	1	8	yes	—
45	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
46	C₂×M₁₆×C₅₃	—	2	2	no	—
47	C₂×M₁₆×C₅₃	—	2	4	no	—
48	C₂×M₁₆×C₅₃	C₂×C₄	1	4	yes	—
49	C₂×M₁₆×C₅₃	C₂×C₄	1	4	yes	—
50	C₂×M₁₆×C₅₃	C₂×C₄	1	8	yes	—
51	C₂×M₁₆×C₅₃	C₂×C₄	1	4	yes	—
52	C₂×M₁₆×C₅₃	C₂³	1	4	yes	—
53	C₂²×C₈×C₅₃	—	2	2	no	—
54	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
55	#108×C₅₃	C₂×C₄	1	16	yes	—
56	C₂⁴×C₅₃	C₂×Q₈	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
58	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	#41	1	2	yes	—
63	C₂²×C₄×C₅₃	#41	1	2	yes	—
64	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
65	C₂²×Dic₅₃	C₂×Q₈	1	4	yes	—
66	M₁₆×C₅₃	—	2	8	no	—
67	M₁₆×C₅₃	—	2	4	no	—
68	M₁₆×C₅₃	C₄²	1	16	yes	—
69	M₁₆×C₅₃	C₂²×C₄	1	8	yes	—
70	C₂×C₈×C₅₃	—	2	8	no	—
71	C₂×C₈×C₅₃	—	2	4	no	—
72	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
73	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
74	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
75	C₂²⋊Dic₅₃	C₄⋊C₄	1	8	yes	—
76	C₄⋊Dic₅₃	C₄⋊C₄	1	8	yes	—
77	C₄×Dic₅₃	C₂²×C₄	1	4	yes	—
78	C₄×Dic₅₃	C₂×D₄	1	4	yes	—
79	#11719917	—	2	8	no	—
80	#11719917	—	2	4	no	—
81	#11719917	C₂²×C₄	1	8	yes	—
82	C₂×#88322	—	2	8	no	—
83	C₂×#88322	—	2	8	no	—
84	C₂×#88322	—	2	4	no	—
85	C₂×#88322	—	2	1	no	—
86	C₂×#88322	—	2	1	no	—
87	C₂×#88322	—	2	1	no	—
88	C₂×#88322	—	2	1	no	—
89	C₂×#88322	C₂²×C₄	1	2	yes	—
90	C₂×#88322	C₂²×C₄	1	2	yes	—
91	C₂³×C₅₃	—	4	2	no	—
92	C₂³×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
93	C₂³×C₅₃	C₄×Q₈	1	2	yes	—
94	C₂³×C₅₃	C₄×Q₈	1	2	yes	—
95	C₂³×C₅₃	C₂²×Q₈	1	1	yes	characteristic
96	C₂×C₄×C₅₃	C₂×C₄²	1	4	yes	—
97	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
98	C₂×C₄×C₅₃	#117	1	4	yes	—
99	C₂×C₄×C₅₃	C₄×D₄	1	4	yes	—
100	C₂×C₄×C₅₃	C₄×D₄	1	4	yes	—
101	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
102	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
103	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
104	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
106	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
107	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
108	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
109	C₂×Dic₅₃	—	4	4	no	—
110	C₂×Dic₅₃	C₂×C₄⋊C₄	1	8	yes	—
111	C₈×C₅₃	—	2	8	no	—
112	C₈×C₅₃	C₂×C₄²	1	8	yes	—
113	#88322	—	2	2	no	—
114	#88322	—	2	2	no	—
115	#88322	—	2	8	no	—
116	#20459	—	53	1	no	maximal, Sylow
117	C₂²×C₅₃	—	2	2	no	—
118	C₂²×C₅₃	—	2	4	no	—
119	C₂²×C₅₃	—	4	2	no	—
120	C₂²×C₅₃	—	4	2	no	—
121	C₂²×C₅₃	C₄×C₄⋊C₄	1	4	yes	—
122	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
123	C₂²×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
124	C₂²×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
125	C₄×C₅₃	C₄×C₄⋊C₄	1	4	yes	—
126	C₄×C₅₃	C₂²×C₄²	1	1	yes	characteristic
127	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
128	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
129	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
130	Dic₅₃	—	2	8	no	—
131	C₂×#430	—	53	1	no	—
132	C₂×#430	D₅₃	1	1	yes	characteristic
133	#2873	—	53	2	no	—
134	C₂×#405	—	53	1	no	—
135	#1778	—	53	4	no	—
136	C₄×#108	—	53	4	no	—
137	C₂×#341	—	53	2	no	—
138	C₂×C₅₃	—	2	1	no	—
139	C₂×C₅₃	—	2	1	no	—
140	C₂×C₅₃	—	2	4	no	—
141	C₂×C₅₃	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
142	C₂×C₅₃	#2916	1	2	yes	—
143	C₂²×M₁₆	—	53	2	no	—
144	C₂²×M₁₆	C₂×D₅₃	1	2	yes	—
145	C₂²×M₁₆	C₂×D₅₃	1	1	yes	characteristic
146	C₂³×C₈	—	53	1	no	—
147	C₂×#117	—	53	1	no	—
148	#430	—	53	8	no	—
149	#430	Dic₅₃	1	8	yes	complemented
150	C₂×#108	—	53	4	no	—
151	C₂×#108	C₂×D₅₃	1	4	yes	—
152	#405	—	53	8	no	—
153	#405	—	106	4	no	—
154	#405	—	106	4	no	—
155	C₄×M₁₆	—	53	8	no	—
156	C₂×#97	—	53	2	no	—
157	C₂×C₄×C₈	—	53	2	no	—
158	#341	—	53	4	no	—
159	#341	—	53	4	no	—
160	C₅₃	#20459	1	1	yes	characteristic, complemented, Sylow
161	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
162	C₂×M₁₆	—	2	2	no	—
163	C₂×M₁₆	—	2	4	no	—
164	C₂×M₁₆	—	53	8	no	—
165	C₂×M₁₆	—	53	4	no	—
166	C₂×M₁₆	—	106	2	no	—
167	C₂×M₁₆	—	106	4	no	—
168	C₂×M₁₆	C₄×D₅₃	1	4	yes	—
169	C₂×M₁₆	C₄×D₅₃	1	4	yes	—
170	C₂×M₁₆	C₂×Dic₅₃	1	8	yes	—
171	C₂×M₁₆	C₂×Dic₅₃	1	4	yes	—
172	C₂×M₁₆	C₂²×D₅₃	1	4	yes	—
173	C₂²×C₈	—	2	2	no	—
174	C₂²×C₈	—	53	4	no	—
175	C₂²×C₈	—	53	1	no	—
176	C₂²×C₈	—	53	1	no	—
177	C₂²×C₈	—	106	2	no	—
178	C₂²×C₈	—	106	2	no	—
179	C₂²×C₈	—	106	2	no	—
180	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
181	#117	—	53	4	no	—
182	#117	—	53	4	no	—
183	C₂×C₄⋊C₄	—	53	2	no	—
184	C₂×C₂²⋊C₄	—	53	2	no	—
185	C₂×C₄²	—	53	2	no	—
186	#108	—	53	16	no	—
187	#108	C₂×Dic₅₃	1	16	yes	—
188	#97	—	53	8	no	—
189	#97	—	106	8	no	—
190	C₄×C₈	—	53	8	no	—
191	C₄×C₈	—	106	8	no	—
192	C₂⁴	Q₈×D₅₃	1	1	yes	characteristic
193	C₂²×C₄	—	53	4	no	—
194	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
195	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
196	C₂²×C₄	D₄×D₅₃	1	2	yes	—
197	C₂²×C₄	#11719947	1	2	yes	—
198	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
199	C₂²×C₄	#11719949	1	2	yes	—
200	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
201	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
202	M₁₆	—	2	8	no	—
203	M₁₆	—	2	4	no	—
204	M₁₆	—	53	8	no	—
205	M₁₆	—	106	8	no	—
206	M₁₆	—	106	4	no	—
207	M₁₆	C₄×Dic₅₃	1	16	yes	—
208	M₁₆	C₂²×Dic₅₃	1	8	yes	—
209	C₂×C₈	—	2	8	no	—
210	C₂×C₈	—	2	4	no	—
211	C₂×C₈	—	53	2	no	—
212	C₂×C₈	—	53	2	no	—
213	C₂×C₈	—	106	8	no	—
214	C₂×C₈	—	106	8	no	—
215	C₂×C₈	—	106	4	no	—
216	C₂×C₈	—	106	1	no	—
217	C₂×C₈	—	106	1	no	—
218	C₂×C₈	—	106	1	no	—
219	C₂×C₈	—	106	1	no	—
220	C₂×C₈	C₂×C₄×D₅₃	1	4	yes	—
221	C₂×C₈	C₂×C₄×D₅₃	1	4	yes	—
222	C₂×C₈	C₂²×Dic₅₃	1	4	yes	—
223	C₄⋊C₄	—	53	8	no	—
224	C₂²⋊C₄	—	53	8	no	—
225	C₄²	—	53	4	no	—
226	C₄²	—	53	4	no	—
227	C₂³	—	4	2	no	—
228	C₂³	C₄⋊C₄×D₅₃	1	2	yes	—
229	C₂³	#13133182	1	2	yes	—
230	C₂³	Q₈×Dic₅₃	1	2	yes	—
231	C₂³	C₂×Q₈×D₅₃	1	1	yes	characteristic
232	C₂×C₄	—	53	8	no	—
233	C₂×C₄	—	212	4	no	—
234	C₂×C₄	C₂×C₄×Dic₅₃	1	4	yes	—
235	C₂×C₄	C₄⋊C₄×D₅₃	1	2	yes	—
236	C₂×C₄	#13133175	1	4	yes	—
237	C₂×C₄	#13133179	1	4	yes	—
238	C₂×C₄	D₄×Dic₅₃	1	4	yes	—
239	C₂×C₄	#13133182	1	2	yes	—
240	C₂×C₄	Q₈×Dic₅₃	1	2	yes	—
241	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
242	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
243	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
244	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
245	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
246	C₂×C₄	C₂×#11719949	1	1	yes	characteristic
247	C₈	—	2	8	no	—
248	C₈	—	106	2	no	—
249	C₈	—	106	2	no	—
250	C₈	—	106	8	no	—
251	C₈	C₂×C₄×Dic₅₃	1	8	yes	—
252	C₂²	—	2	2	no	—
253	C₂²	—	2	4	no	—
254	C₂²	—	4	2	no	—
255	C₂²	—	4	2	no	—
256	C₂²	C₄⋊C₄×Dic₅₃	1	4	yes	—
257	C₂²	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
258	C₂²	C₂×#13133182	1	1	yes	characteristic
259	C₂²	C₂×Q₈×Dic₅₃	1	1	yes	characteristic
260	C₄	—	106	8	no	—
261	C₄	C₄⋊C₄×Dic₅₃	1	4	yes	—
262	C₄	C₂²×C₄×Dic₅₃	1	1	yes	characteristic
263	C₄	C₂×#13133175	1	1	yes	characteristic
264	C₄	C₂×#13133179	1	1	yes	characteristic
265	C₄	C₂×D₄×Dic₅₃	1	1	yes	characteristic
266	C₂	—	2	1	no	—
267	C₂	—	2	1	no	—
268	C₂	—	2	4	no	—
269	C₂	C₂×C₄⋊C₄×Dic₅₃	1	1	yes	characteristic
270	C₂	#20283325	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₄×C₅₃	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₅₃×#430	C₂
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₅₃	C₂×Q₈
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂×#430	D₅₃
hypocenter	C₅₃	#20459
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#430

Complements

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

N	H ≅ G/N	type
4. #20274623	C₂	semidirect
5. #108×Dic₅₃	C₂	semidirect
11. #16810672	C₄	semidirect
23. C₅₃×#430	C₄	semidirect
149. #430	Dic₅₃	semidirect
160. C₅₃	#20459	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
#20283133	2
C₂×#16810581	2
#20274623	2
#108×Dic₅₃	2
C₂×#16810387	2
C₂×C₅₃×#430	2
#20459	53