Subgroups — #27403101

Order 13568.

622 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	M₁₆×Dic₁₀₆	C₂	1	2	yes	maximal
2	C₂×#16810753	C₂	1	1	yes	maximal, characteristic
3	#20275490	C₂	1	2	yes	maximal
4	#20275487	C₂	1	4	yes	maximal
5	#20274854	C₂	1	2	yes	maximal
6	#20274853	C₂	1	1	yes	maximal, characteristic
7	#20274620	C₂	1	2	yes	maximal
8	C₄×#108×C₅₃	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
10	#16810753	—	2	2	no	—
11	#16810753	C₄	1	4	yes	—
12	#16810753	C₄	1	4	yes	—
13	#16810751	—	2	4	no	—
14	C₈×Dic₁₀₆	—	2	2	no	—
15	C₂×#13133152	C₂²	1	4	yes	—
16	#16810647	C₂²	1	2	yes	—
17	#16810646	C₂²	1	4	yes	—
18	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
19	C₂×#13133146	C₂²	1	2	yes	—
20	M₁₆×Dic₅₃	C₂²	1	4	yes	—
21	C₂×#13133127	C₂²	1	2	yes	—
22	#16810388	C₂²	1	2	yes	—
23	#16810388	C₂²	1	2	yes	—
24	#16810387	C₂²	1	4	yes	—
25	C₂×#108×C₅₃	C₂²	1	2	yes	—
26	C₄×M₁₆×C₅₃	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
28	C₅₃×#341	C₂²	1	1	yes	characteristic
29	C₅₃×#341	C₂²	1	1	yes	characteristic
30	C₂²×Dic₁₀₆	Q₈	1	1	yes	characteristic
31	C₂×#11719917	C₂³	1	4	yes	—
32	C₂²×#88322	C₂³	1	2	yes	—
33	C₄×Dic₁₀₆	C₂×C₄	1	2	yes	—
34	C₄×Dic₁₀₆	C₂×C₄	1	2	yes	—
35	C₄×Dic₁₀₆	D₄	1	2	yes	—
36	C₄×Dic₁₀₆	D₄	1	2	yes	—
37	C₂×C₄⋊Dic₅₃	Q₈	1	1	yes	characteristic
38	C₂×#11719919	Q₈	1	2	yes	—
39	C₂×C₄×Dic₅₃	C₂³	1	2	yes	—
40	#13133152	—	2	8	no	—
41	C₄⋊#88322	—	2	2	no	—
42	C₄⋊#88322	—	2	2	no	—
43	C₄⋊#88322	C₂×C₄	1	2	yes	—
44	C₄⋊#88322	C₂×C₄	1	2	yes	—
45	#13133146	—	2	4	no	—
46	#13133146	—	2	4	no	—
47	#13133146	C₂×C₄	1	8	yes	—
48	#13133127	—	2	4	no	—
49	#13133127	C₂×C₄	1	8	yes	—
50	C₈×Dic₅₃	—	2	4	no	—
51	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
52	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
53	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
56	#108×C₅₃	—	2	4	no	—
57	#97×C₅₃	—	2	2	no	—
58	C₄×C₈×C₅₃	—	2	2	no	—
59	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
60	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
61	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
64	C₂²×Dic₅₃	C₂×Q₈	1	2	yes	—
65	C₂²×Dic₅₃	C₂×Q₈	1	2	yes	—
66	C₂×Dic₁₀₆	—	4	2	no	—
67	C₂×Dic₁₀₆	C₄⋊C₄	1	2	yes	—
68	C₂×Dic₁₀₆	C₄⋊C₄	1	2	yes	—
69	M₁₆×C₅₃	—	2	2	no	—
70	M₁₆×C₅₃	—	2	2	no	—
71	M₁₆×C₅₃	—	2	2	no	—
72	M₁₆×C₅₃	#41	1	4	yes	—
73	C₂×C₈×C₅₃	—	2	2	no	—
74	C₂×C₈×C₅₃	—	2	2	no	—
75	C₂×C₈×C₅₃	—	2	2	no	—
76	C₂×C₈×C₅₃	—	2	2	no	—
77	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
79	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
80	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
81	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
83	C₄⋊Dic₅₃	C₄⋊C₄	1	2	yes	—
84	C₄⋊Dic₅₃	C₄⋊C₄	1	2	yes	—
85	#11719919	C₄⋊C₄	1	4	yes	—
86	#11719919	C₄⋊C₄	1	4	yes	—
87	C₄×Dic₅₃	C₂²×C₄	1	4	yes	—
88	C₄×Dic₅₃	C₂×D₄	1	4	yes	—
89	#11719917	—	2	4	no	—
90	#11719917	—	2	4	no	—
91	C₂×#88322	—	2	4	no	—
92	C₂×#88322	—	2	4	no	—
93	C₂×#88322	C₂²×C₄	1	4	yes	—
94	C₂³×C₅₃	C₂²×Q₈	1	1	yes	characteristic
95	C₂×C₄×C₅₃	—	2	2	no	—
96	C₂×C₄×C₅₃	—	4	1	no	—
97	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
102	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
103	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
104	C₂×C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
105	C₂×Dic₅₃	—	4	2	no	—
106	C₂×Dic₅₃	—	4	2	no	—
107	C₂×Dic₅₃	C₂×C₄⋊C₄	1	4	yes	—
108	C₂×Dic₅₃	C₂×C₄⋊C₄	1	4	yes	—
109	Dic₁₀₆	—	2	4	no	—
110	Dic₁₀₆	—	4	2	no	—
111	C₈×C₅₃	—	2	2	no	—
112	C₈×C₅₃	—	2	2	no	—
113	C₈×C₅₃	—	2	4	no	—
114	C₈×C₅₃	—	4	2	no	—
115	#88322	—	2	4	no	—
116	#88322	—	4	4	no	—
117	#22747	—	53	1	no	maximal, Sylow
118	C₂²×C₅₃	—	2	1	no	—
119	C₂²×C₅₃	—	2	1	no	—
120	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
121	C₂²×C₅₃	#557	1	2	yes	—
122	C₄×C₅₃	—	2	1	no	—
123	C₄×C₅₃	—	2	1	no	—
124	C₄×C₅₃	—	2	1	no	—
125	C₄×C₅₃	#430	1	2	yes	—
126	C₄×C₅₃	C₄×#41	1	1	yes	characteristic
127	C₄×C₅₃	#519	1	1	yes	characteristic
128	C₄×C₅₃	#548	1	2	yes	—
129	Dic₅₃	—	2	4	no	—
130	Dic₅₃	—	2	4	no	—
131	Q₈×M₁₆	—	53	2	no	—
132	C₂×#446	—	53	1	no	—
133	#1950	—	53	4	no	—
134	#1950	—	53	2	no	—
135	#1838	—	53	2	no	—
136	#1837	—	53	1	no	—
137	#1778	—	53	2	no	—
138	C₄×#108	D₅₃	1	1	yes	characteristic
139	C₂×C₅₃	—	2	2	no	—
140	C₂×C₅₃	#2351	1	1	yes	characteristic
141	C₂×C₅₃	C₂×#430	1	1	yes	characteristic
142	C₂×C₅₃	#2916	1	1	yes	characteristic
143	C₂×C₄×Q₈	—	53	1	no	—
144	#446	—	53	4	no	—
145	#446	—	53	4	no	—
146	#446	—	106	2	no	—
147	#446	—	106	4	no	—
148	C₈×Q₈	—	106	2	no	—
149	C₂×#108	—	53	4	no	—
150	C₂×#108	C₂×D₅₃	1	2	yes	—
151	#423	—	53	4	no	—
152	#423	—	53	2	no	—
153	C₂×C₄⋊C₈	—	53	2	no	—
154	C₂×C₄⋊C₈	—	53	1	no	—
155	C₄×M₁₆	—	53	4	no	—
156	C₄×M₁₆	C₂×D₅₃	1	2	yes	—
157	C₂×#97	—	53	2	no	—
158	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
159	#341	—	53	4	no	—
160	#341	—	53	2	no	—
161	#341	—	53	2	no	—
162	#341	C₂×D₅₃	1	1	yes	characteristic
163	#341	C₂×D₅₃	1	1	yes	characteristic
164	C₅₃	#22747	1	1	yes	characteristic, complemented, Sylow
165	C₂²×Q₈	—	53	1	no	—
166	C₂×M₁₆	—	53	4	no	—
167	C₂×M₁₆	C₂²×D₅₃	1	2	yes	—
168	C₂×M₁₆	C₂²×D₅₃	1	2	yes	—
169	C₂²×C₈	—	53	2	no	—
170	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
171	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
172	C₄×Q₈	—	53	2	no	—
173	C₄×Q₈	—	53	2	no	—
174	C₄×Q₈	—	53	2	no	—
175	C₄×Q₈	—	53	2	no	—
176	C₂×C₄⋊C₄	—	53	2	no	—
177	C₂×C₄⋊C₄	—	53	1	no	—
178	C₂×C₄²	—	53	2	no	—
179	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
180	#108	—	2	4	no	—
181	#108	—	106	8	no	—
182	C₄⋊C₈	—	53	8	no	—
183	C₄⋊C₈	—	53	2	no	—
184	C₄⋊C₈	—	53	2	no	—
185	C₄⋊C₈	—	106	4	no	—
186	C₄⋊C₈	—	106	4	no	—
187	C₄⋊C₈	—	106	2	no	—
188	C₄⋊C₈	—	106	2	no	—
189	#97	—	2	2	no	—
190	#97	—	53	8	no	—
191	#97	—	106	4	no	—
192	C₄×C₈	—	2	2	no	—
193	C₄×C₈	—	106	4	no	—
194	C₄×C₈	C₄×D₅₃	1	2	yes	—
195	C₄×C₈	C₄×D₅₃	1	2	yes	—
196	C₂×Q₈	—	53	2	no	—
197	C₂×Q₈	—	53	2	no	—
198	C₂×Q₈	—	212	2	no	—
199	C₂²×C₄	—	53	2	no	—
200	C₂²×C₄	—	53	2	no	—
201	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
202	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
203	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
204	M₁₆	—	2	2	no	—
205	M₁₆	—	2	2	no	—
206	M₁₆	—	2	2	no	—
207	M₁₆	—	106	4	no	—
208	M₁₆	—	106	4	no	—
209	M₁₆	#11719945	1	4	yes	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	53	4	no	—
215	C₂×C₈	—	106	4	no	—
216	C₂×C₈	—	106	4	no	—
217	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
218	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
219	C₄⋊C₄	—	53	4	no	—
220	C₄⋊C₄	—	53	4	no	—
221	C₄⋊C₄	—	53	2	no	—
222	C₄⋊C₄	—	53	2	no	—
223	C₄²	—	53	4	no	—
224	C₄²	—	53	4	no	—
225	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
226	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
227	C₄²	D₄×D₅₃	1	1	yes	characteristic
228	C₄²	D₄×D₅₃	1	1	yes	characteristic
229	C₂³	C₂×Q₈×D₅₃	1	1	yes	characteristic
230	Q₈	—	106	4	no	—
231	Q₈	—	212	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	4	1	no	—
234	C₂×C₄	—	53	4	no	—
235	C₂×C₄	—	53	4	no	—
236	C₂×C₄	—	212	2	no	—
237	C₂×C₄	—	212	2	no	—
238	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
239	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
240	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
241	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
242	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
243	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
244	C₂×C₄	C₂×#11719945	1	2	yes	—
245	C₂×C₄	#13133264	1	2	yes	—
246	C₈	—	2	4	no	—
247	C₈	—	2	2	no	—
248	C₈	—	2	2	no	—
249	C₈	—	4	2	no	—
250	C₈	—	106	4	no	—
251	C₈	—	212	4	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
255	C₂²	#16811351	1	2	yes	—
256	C₄	—	2	1	no	—
257	C₄	—	2	1	no	—
258	C₄	—	2	1	no	—
259	C₄	—	106	4	no	—
260	C₄	—	106	4	no	—
261	C₄	#16810670	1	2	yes	—
262	C₄	C₄×#11719945	1	1	yes	characteristic
263	C₄	#16811100	1	1	yes	characteristic
264	C₄	#16811282	1	2	yes	—
265	C₂	—	2	2	no	—
266	C₂	#20277635	1	1	yes	characteristic
267	C₂	C₂×#16810670	1	1	yes	characteristic
268	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	C₄×#108×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂³×C₅₃	C₂²×Q₈
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₄×#108	D₅₃
hypocenter	C₅₃	#22747
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#108

Complements

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

N	H ≅ G/N	type
164. C₅₃	#22747	semidirect

Maximal subgroups

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

subgroup	index
M₁₆×Dic₁₀₆	2
C₂×#16810753	2
#20275490	2
#20275487	2
#20274854	2
#20274853	2
#20274620	2
C₄×#108×C₅₃	2
#22747	53