Subgroups — #27300243

Order 13568.

396 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810446	C₂	1	2	yes	maximal, complemented
2	#20274753	C₂	1	1	yes	maximal, characteristic
3	#20274740	C₂	1	1	yes	maximal, characteristic
4	#20272814	C₂	1	2	yes	maximal, complemented
5	C₂²×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133153	C₂²	1	2	yes	—
7	C₂×#13133149	C₂²	1	2	yes	—
8	#16810587	—	2	1	no	—
9	#16810483	—	2	1	no	—
10	#16810446	—	2	4	no	—
11	#16810388	—	2	1	no	—
12	#16810386	—	2	1	no	—
13	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×M₃₂×C₅₃	—	2	4	no	—
15	C₂×M₃₂×C₅₃	C₄	1	2	yes	complemented
16	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
18	C₂²×C₁₆×C₅₃	C₄	1	2	yes	complemented
19	C₂×#11719917	—	2	2	no	—
20	C₂×C₄⋊Dic₅₃	—	2	2	no	—
21	C₂×C₂²⋊Dic₅₃	—	4	1	no	—
22	#13133153	—	2	2	no	—
23	#13133153	—	2	2	no	—
24	#13133149	—	2	2	no	—
25	#13133149	—	2	2	no	—
26	C₂²⋊#88322	—	4	1	no	—
27	C₂²⋊#88322	—	4	1	no	—
28	#13133124	—	4	1	no	—
29	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₅₃	—	2	2	no	—
31	C₂²×C₈×C₅₃	—	2	2	no	—
32	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
36	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
37	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
38	M₃₂×C₅₃	—	2	2	no	—
39	M₃₂×C₅₃	—	2	4	no	—
40	M₃₂×C₅₃	D₄	1	4	yes	—
41	C₂×C₁₆×C₅₃	—	2	4	no	—
42	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
43	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
44	C₂⁴×C₅₃	SD₁₆	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	—	2	2	no	—
46	C₂²×C₄×C₅₃	—	2	2	no	—
47	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
53	C₂²×Dic₅₃	—	4	2	no	—
54	C₂×C₈×C₅₃	—	2	2	no	—
55	C₂×C₈×C₅₃	—	2	1	no	—
56	C₂×C₈×C₅₃	—	2	1	no	—
57	C₂×C₈×C₅₃	—	2	1	no	—
58	C₂×C₈×C₅₃	—	2	1	no	—
59	C₂×C₈×C₅₃	—	2	2	no	—
60	C₂×C₈×C₅₃	—	2	2	no	—
61	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
65	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
66	C₂×C₈×C₅₃	#41	1	2	yes	—
67	C₂²⋊Dic₅₃	—	8	2	no	—
68	C₄⋊Dic₅₃	—	4	2	no	—
69	C₄⋊Dic₅₃	—	4	2	no	—
70	#11719917	—	4	2	no	—
71	#11719917	—	4	2	no	—
72	C₂×#88322	—	4	2	no	—
73	C₁₆×C₅₃	—	2	2	no	—
74	C₁₆×C₅₃	C₂²⋊C₄	1	4	yes	complemented
75	C₂³×C₅₃	—	2	2	no	—
76	C₂³×C₅₃	—	2	2	no	—
77	C₂³×C₅₃	—	4	1	no	—
78	C₂³×C₅₃	#106	1	1	yes	characteristic
79	C₂³×C₅₃	#106	1	1	yes	characteristic
80	C₂³×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
81	C₂×C₄×C₅₃	—	2	2	no	—
82	C₂×C₄×C₅₃	—	2	1	no	—
83	C₂×C₄×C₅₃	—	2	1	no	—
84	C₂×C₄×C₅₃	—	2	1	no	—
85	C₂×C₄×C₅₃	—	2	1	no	—
86	C₂×C₄×C₅₃	—	2	2	no	—
87	C₂×C₄×C₅₃	—	2	2	no	—
88	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
89	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
90	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₅₃	#121	1	2	yes	—
92	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
93	C₂×C₄×C₅₃	#135	1	2	yes	—
94	C₂×Dic₅₃	—	8	2	no	—
95	C₂×Dic₅₃	—	8	2	no	—
96	C₂×Dic₅₃	—	8	2	no	—
97	C₈×C₅₃	—	2	1	no	—
98	C₈×C₅₃	—	2	1	no	—
99	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₈×C₅₃	#117	1	1	yes	characteristic
101	C₈×C₅₃	C₂²⋊Q₈	1	2	yes	—
102	#88322	—	8	2	no	—
103	#12831	—	53	1	no	maximal, Sylow
104	C₂²×C₅₃	—	2	1	no	—
105	C₂²×C₅₃	—	2	1	no	—
106	C₂²×C₅₃	—	2	1	no	—
107	C₂²×C₅₃	—	2	1	no	—
108	C₂²×C₅₃	—	2	1	no	—
109	C₂²×C₅₃	—	2	1	no	—
110	C₂²×C₅₃	—	2	1	no	—
111	C₂²×C₅₃	—	4	4	no	—
112	C₂²×C₅₃	#365	1	2	yes	complemented
113	C₂²×C₅₃	C₂×#106	1	1	yes	characteristic
114	C₂²×C₅₃	#465	1	2	yes	—
115	C₄×C₅₃	—	2	1	no	—
116	C₄×C₅₃	—	2	1	no	—
117	C₄×C₅₃	#380	1	1	yes	characteristic
118	C₄×C₅₃	#427	1	1	yes	characteristic
119	C₄×C₅₃	#465	1	2	yes	—
120	Dic₅₃	—	16	2	no	—
121	C₂²×M₃₂	D₅₃	1	1	yes	characteristic
122	C₂×#365	—	53	2	no	—
123	#1810	—	53	1	no	—
124	#1807	—	53	1	no	—
125	#1383	—	53	2	no	—
126	C₂×C₅₃	—	2	2	no	—
127	C₂×C₅₃	—	2	2	no	—
128	C₂×C₅₃	—	4	1	no	—
129	C₂×C₅₃	#1807	1	1	yes	characteristic
130	C₂×C₅₃	C₂×#365	1	1	yes	characteristic
131	C₂×C₅₃	#2155	1	1	yes	characteristic
132	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
133	C₂×M₃₂	—	2	4	no	—
134	C₂×M₃₂	Dic₅₃	1	2	yes	complemented
135	C₂×M₃₂	C₂×D₅₃	1	1	yes	characteristic
136	C₂×M₃₂	C₂×D₅₃	1	1	yes	characteristic
137	C₂²×C₁₆	Dic₅₃	1	2	yes	complemented
138	C₂×#108	—	53	2	no	—
139	C₂×#106	—	53	2	no	—
140	#407	—	106	1	no	—
141	#380	—	106	1	no	—
142	#365	—	106	4	no	—
143	#341	—	106	1	no	—
144	#340	—	106	1	no	—
145	C₅₃	#12831	1	1	yes	characteristic, complemented, Sylow
146	C₂³×C₄	D₂₁₂	1	1	yes	characteristic
147	C₂×M₁₆	—	106	2	no	—
148	C₂²×C₈	—	2	2	no	—
149	C₂²×C₈	—	2	2	no	—
150	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
151	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
152	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
153	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
154	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
155	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
156	C₂×C₄⋊C₄	—	106	2	no	—
157	C₂×C₂²⋊C₄	—	212	1	no	—
158	M₃₂	—	2	2	no	—
159	M₃₂	—	2	4	no	—
160	M₃₂	C₂²⋊D₅₃	1	4	yes	—
161	C₂×C₁₆	—	2	4	no	—
162	C₂×C₁₆	C₂×Dic₅₃	1	2	yes	—
163	C₂×C₁₆	C₂×Dic₅₃	1	2	yes	—
164	#108	—	106	2	no	—
165	#108	—	106	2	no	—
166	#106	—	106	2	no	—
167	#106	—	106	2	no	—
168	C₂²⋊C₈	—	212	1	no	—
169	C₂²⋊C₈	—	212	1	no	—
170	#95	—	212	1	no	—
171	C₂⁴	#11719913	1	1	yes	characteristic
172	C₂²×C₄	—	2	2	no	—
173	C₂²×C₄	—	2	2	no	—
174	C₂²×C₄	—	212	2	no	—
175	C₂²×C₄	#11719913	1	1	yes	characteristic
176	C₂²×C₄	#11719913	1	1	yes	characteristic
177	C₂²×C₄	#11719913	1	1	yes	characteristic
178	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
179	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
180	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
181	M₁₆	—	212	2	no	—
182	M₁₆	—	212	2	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	2	1	no	—
185	C₂×C₈	—	2	1	no	—
186	C₂×C₈	—	2	1	no	—
187	C₂×C₈	—	2	1	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	—	212	2	no	—
191	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
192	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
193	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
194	C₂×C₈	#11719945	1	2	yes	—
195	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
196	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
197	C₄⋊C₄	—	212	2	no	—
198	C₄⋊C₄	—	212	2	no	—
199	C₂²⋊C₄	—	424	2	no	—
200	C₁₆	—	2	2	no	—
201	C₁₆	C₂²⋊Dic₅₃	1	4	yes	complemented
202	C₂³	—	2	2	no	—
203	C₂³	—	2	2	no	—
204	C₂³	—	4	1	no	—
205	C₂³	#13133149	1	1	yes	characteristic
206	C₂³	#13133149	1	1	yes	characteristic
207	C₂³	C₂×#11719913	1	1	yes	characteristic
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	424	2	no	—
216	C₂×C₄	—	424	2	no	—
217	C₂×C₄	—	424	2	no	—
218	C₂×C₄	#13133149	1	1	yes	characteristic
219	C₂×C₄	#13133149	1	1	yes	characteristic
220	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
221	C₂×C₄	#13133191	1	2	yes	—
222	C₂×C₄	C₂×#11719913	1	1	yes	characteristic
223	C₂×C₄	#13133239	1	2	yes	—
224	C₈	—	2	1	no	—
225	C₈	—	2	1	no	—
226	C₈	—	424	2	no	—
227	C₈	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
228	C₈	#13133176	1	1	yes	characteristic
229	C₈	#13133196	1	2	yes	—
230	C₂²	—	2	1	no	—
231	C₂²	—	2	1	no	—
232	C₂²	—	2	1	no	—
233	C₂²	—	2	1	no	—
234	C₂²	—	2	1	no	—
235	C₂²	—	2	1	no	—
236	C₂²	—	2	1	no	—
237	C₂²	—	4	4	no	—
238	C₂²	#16810446	1	2	yes	complemented
239	C₂²	C₂×#13133149	1	1	yes	characteristic
240	C₂²	#16810852	1	2	yes	—
241	C₄	—	2	1	no	—
242	C₄	—	2	1	no	—
243	C₄	—	848	2	no	—
244	C₄	#16810483	1	1	yes	characteristic
245	C₄	#16810661	1	1	yes	characteristic
246	C₄	#16810852	1	2	yes	—
247	C₂	—	2	2	no	—
248	C₂	—	2	2	no	—
249	C₂	—	4	1	no	—
250	C₂	#20274740	1	1	yes	characteristic
251	C₂	C₂×#16810446	1	1	yes	characteristic
252	C₂	#20276488	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13133149
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂²×M₃₂×C₅₃	C₂
socle	C₂²×C₅₃	C₂×#106
prime radical	C₂⁴×C₅₃	SD₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂²×M₃₂	D₅₃
hypocenter	C₅₃	#12831
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
1. C₂×#16810446	C₂	semidirect
4. #20272814	C₂	semidirect
15. C₂×M₃₂×C₅₃	C₄	semidirect
18. C₂²×C₁₆×C₅₃	C₄	semidirect
74. C₁₆×C₅₃	C₂²⋊C₄	semidirect
112. C₂²×C₅₃	#365	semidirect
134. C₂×M₃₂	Dic₅₃	semidirect
137. C₂²×C₁₆	Dic₅₃	semidirect
145. C₅₃	#12831	semidirect
201. C₁₆	C₂²⋊Dic₅₃	semidirect
238. C₂²	#16810446	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810446	2
#20274753	2
#20274740	2
#20272814	2
C₂²×M₃₂×C₅₃	2
#12831	53