Subgroups — #27277013

Order 13568.

398 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic
2	#20274708	C₂	1	2	yes	maximal
3	#20273471	C₂	1	1	yes	maximal, characteristic
4	#20272589	C₂	1	2	yes	maximal
5	C₅₃×#1566	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
7	C₈×Dic₁₀₆	—	2	1	no	—
8	C₈×Dic₁₀₆	—	2	1	no	—
9	C₈×Dic₁₀₆	—	2	1	no	—
10	C₈×Dic₁₀₆	—	2	1	no	—
11	#16810647	C₂²	1	2	yes	—
12	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
13	C₂×#13133146	C₄	1	2	yes	—
14	C₂×#13133135	—	2	2	no	—
15	C₂×C₈×Dic₅₃	C₄	1	2	yes	—
16	#16810391	—	2	2	no	—
17	#16810374	—	2	1	no	—
18	#16810374	—	2	1	no	—
19	C₅₃×#423	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₅₃×#334	—	2	1	no	—
22	C₅₃×#334	—	2	1	no	—
23	C₂²×Dic₁₀₆	D₄	1	1	yes	characteristic
24	C₂×#11719917	—	2	2	no	—
25	C₂²×#88322	C₂×C₄	1	2	yes	—
26	C₄×Dic₁₀₆	—	2	1	no	—
27	C₄×Dic₁₀₆	—	2	1	no	—
28	C₄×Dic₁₀₆	—	2	1	no	—
29	C₄×Dic₁₀₆	—	2	1	no	—
30	C₂×C₄⋊Dic₅₃	D₄	1	1	yes	characteristic
31	C₂×#11719919	Q₈	1	2	yes	—
32	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
33	C₄⋊#88322	—	2	1	no	—
34	C₄⋊#88322	—	2	1	no	—
35	C₄⋊#88322	—	2	2	no	—
36	C₄⋊#88322	—	2	2	no	—
37	#13133146	—	2	2	no	—
38	#13133146	—	2	2	no	—
39	#13133135	—	4	4	no	—
40	C₈×Dic₅₃	—	2	2	no	—
41	C₈×Dic₅₃	—	2	2	no	—
42	C₂×M₁₆×C₅₃	—	2	2	no	—
43	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
46	C₄⋊C₈×C₅₃	—	2	2	no	—
47	C₄⋊C₈×C₅₃	—	2	2	no	—
48	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
49	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
50	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
51	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
55	C₂²×Dic₅₃	C₄⋊C₄	1	2	yes	—
56	C₂²×Dic₅₃	C₄⋊C₄	1	2	yes	—
57	C₂×Dic₁₀₆	—	2	1	no	—
58	C₂×Dic₁₀₆	—	2	1	no	—
59	C₂×Dic₁₀₆	—	2	2	no	—
60	C₂×Dic₁₀₆	SD₁₆	1	4	yes	—
61	M₁₆×C₅₃	—	4	2	no	—
62	M₁₆×C₅₃	—	4	2	no	—
63	C₂×C₈×C₅₃	—	2	2	no	—
64	C₂×C₈×C₅₃	—	4	1	no	—
65	C₂×C₈×C₅₃	—	4	2	no	—
66	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
70	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
71	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
72	C₄²×C₅₃	#41	1	1	yes	characteristic
73	C₄²×C₅₃	#41	1	1	yes	characteristic
74	C₄⋊Dic₅₃	—	2	1	no	—
75	C₄⋊Dic₅₃	—	2	1	no	—
76	#11719919	—	2	2	no	—
77	#11719919	—	2	2	no	—
78	C₄×Dic₅₃	—	2	2	no	—
79	C₄×Dic₅₃	—	2	2	no	—
80	#11719917	—	4	2	no	—
81	#11719917	—	4	2	no	—
82	C₂×#88322	—	2	2	no	—
83	C₂×#88322	—	4	2	no	—
84	C₂×#88322	—	4	2	no	—
85	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₅₃	—	2	1	no	—
87	C₂×C₄×C₅₃	—	2	1	no	—
88	C₂×C₄×C₅₃	—	4	1	no	—
89	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
91	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
92	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
93	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
94	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
95	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
96	C₂×C₄×C₅₃	#135	1	2	yes	—
97	C₂×Dic₅₃	—	2	2	no	—
98	C₂×Dic₅₃	—	2	2	no	—
99	C₂×Dic₅₃	—	2	2	no	—
100	C₂×Dic₅₃	—	4	2	no	—
101	C₂×Dic₅₃	#106	1	4	yes	—
102	Dic₁₀₆	—	2	4	no	—
103	Dic₁₀₆	—	2	4	no	—
104	C₈×C₅₃	—	2	1	no	—
105	C₈×C₅₃	—	2	1	no	—
106	C₈×C₅₃	—	2	1	no	—
107	C₈×C₅₃	—	8	2	no	—
108	C₈×C₅₃	#101	1	2	yes	complemented
109	#88322	—	4	2	no	—
110	#88322	—	8	2	no	—
111	#8615	—	53	1	no	maximal, Sylow
112	C₂²×C₅₃	—	2	1	no	—
113	C₂²×C₅₃	—	2	1	no	—
114	C₂²×C₅₃	#380	1	1	yes	characteristic
115	C₂²×C₅₃	C₂×#106	1	1	yes	characteristic
116	C₂²×C₅₃	#427	1	1	yes	characteristic
117	C₄×C₅₃	—	2	1	no	—
118	C₄×C₅₃	—	2	1	no	—
119	C₄×C₅₃	—	2	1	no	—
120	C₄×C₅₃	C₂×#101	1	1	yes	characteristic
121	C₄×C₅₃	#413	1	1	yes	characteristic
122	C₄×C₅₃	#465	1	2	yes	—
123	C₄×C₅₃	#465	1	2	yes	—
124	Dic₅₃	—	2	4	no	—
125	Dic₅₃	—	4	2	no	—
126	C₂×C₈×Q₈	—	53	1	no	—
127	#1801	—	53	2	no	—
128	#1566	—	53	1	no	—
129	#1566	D₅₃	1	1	yes	characteristic
130	#1292	—	53	2	no	—
131	C₂×C₅₃	—	2	1	no	—
132	C₂×C₅₃	—	2	1	no	—
133	C₂×C₅₃	#1801	1	1	yes	characteristic
134	C₂×C₅₃	#1807	1	1	yes	characteristic
135	C₂×C₅₃	#1813	1	1	yes	characteristic
136	C₂×C₄×Q₈	—	53	1	no	—
137	C₈×Q₈	—	106	1	no	—
138	C₈×Q₈	—	106	1	no	—
139	C₈×Q₈	—	106	1	no	—
140	C₈×Q₈	—	106	1	no	—
141	#423	—	53	2	no	—
142	#423	C₂×D₅₃	1	2	yes	—
143	C₂×C₄⋊C₈	—	53	2	no	—
144	C₂×C₄⋊C₈	—	53	1	no	—
145	C₂×#101	—	106	2	no	—
146	C₂×C₄×C₈	—	53	2	no	—
147	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
148	#343	—	106	2	no	—
149	#334	—	2	1	no	—
150	#334	—	2	1	no	—
151	#334	—	106	1	no	—
152	#334	—	106	1	no	—
153	C₅₃	#8615	1	1	yes	characteristic, complemented, Sylow
154	C₂²×Q₈	—	53	1	no	—
155	C₂×M₁₆	—	2	2	no	—
156	C₂×M₁₆	—	106	2	no	—
157	C₂²×C₈	—	53	2	no	—
158	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
159	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
160	C₄×Q₈	—	106	1	no	—
161	C₄×Q₈	—	106	1	no	—
162	C₄×Q₈	—	106	1	no	—
163	C₄×Q₈	—	106	1	no	—
164	C₂×C₄⋊C₄	—	53	1	no	—
165	C₂×C₄⋊C₄	—	53	2	no	—
166	C₂×C₄²	—	53	2	no	—
167	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
168	C₄⋊C₈	—	2	2	no	—
169	C₄⋊C₈	—	2	2	no	—
170	C₄⋊C₈	—	106	2	no	—
171	C₄⋊C₈	—	106	2	no	—
172	C₄⋊C₈	—	106	1	no	—
173	C₄⋊C₈	—	106	2	no	—
174	C₄⋊C₈	—	106	2	no	—
175	C₄⋊C₈	—	106	1	no	—
176	#101	—	212	4	no	—
177	C₄×C₈	—	106	2	no	—
178	C₄×C₈	—	106	2	no	—
179	C₄×C₈	D₂₁₂	1	1	yes	characteristic
180	C₄×C₈	D₂₁₂	1	1	yes	characteristic
181	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
182	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
183	C₂×Q₈	—	53	4	no	—
184	C₂×Q₈	—	106	1	no	—
185	C₂×Q₈	—	106	1	no	—
186	C₂×Q₈	—	106	2	no	—
187	C₂²×C₄	—	53	2	no	—
188	C₂²×C₄	—	53	2	no	—
189	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
190	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
191	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
192	M₁₆	—	4	2	no	—
193	M₁₆	—	4	2	no	—
194	M₁₆	—	212	2	no	—
195	M₁₆	—	212	2	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	4	1	no	—
198	C₂×C₈	—	4	2	no	—
199	C₂×C₈	—	106	2	no	—
200	C₂×C₈	—	212	2	no	—
201	C₂×C₈	—	212	2	no	—
202	C₂×C₈	#11719921	1	1	yes	characteristic
203	C₂×C₈	#11719921	1	1	yes	characteristic
204	C₂×C₈	#11719921	1	1	yes	characteristic
205	C₂×C₈	#11719921	1	1	yes	characteristic
206	C₄⋊C₄	—	106	1	no	—
207	C₄⋊C₄	—	106	1	no	—
208	C₄⋊C₄	—	106	2	no	—
209	C₄⋊C₄	—	106	2	no	—
210	C₄²	—	106	2	no	—
211	C₄²	—	106	2	no	—
212	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
213	C₄²	#11719947	1	1	yes	characteristic
214	C₄²	#11719949	1	1	yes	characteristic
215	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
216	C₂³	C₄⋊C₄×D₅₃	1	1	yes	characteristic
217	Q₈	—	106	4	no	—
218	Q₈	—	106	4	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	4	1	no	—
222	C₂×C₄	—	53	4	no	—
223	C₂×C₄	—	106	2	no	—
224	C₂×C₄	—	106	2	no	—
225	C₂×C₄	—	106	2	no	—
226	C₂×C₄	—	212	2	no	—
227	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
228	C₂×C₄	#13133175	1	1	yes	characteristic
229	C₂×C₄	#13133188	1	1	yes	characteristic
230	C₂×C₄	#13133192	1	1	yes	characteristic
231	C₂×C₄	#13133193	1	1	yes	characteristic
232	C₂×C₄	#13133197	1	1	yes	characteristic
233	C₂×C₄	SD₁₆×D₅₃	1	2	yes	—
234	C₂×C₄	#13133238	1	2	yes	—
235	C₈	—	2	1	no	—
236	C₈	—	2	1	no	—
237	C₈	—	2	1	no	—
238	C₈	—	8	2	no	—
239	C₈	—	424	2	no	—
240	C₈	—	212	2	no	—
241	C₈	#13133135	1	2	yes	complemented
242	C₂²	—	2	1	no	—
243	C₂²	—	2	1	no	—
244	C₂²	#16810480	1	1	yes	characteristic
245	C₂²	#106×D₅₃	1	1	yes	characteristic
246	C₂²	#16810658	1	1	yes	characteristic
247	C₄	—	2	1	no	—
248	C₄	—	2	1	no	—
249	C₄	—	2	1	no	—
250	C₄	—	106	4	no	—
251	C₄	—	212	2	no	—
252	C₄	C₂×#13133135	1	1	yes	characteristic
253	C₄	#16810605	1	1	yes	characteristic
254	C₄	#16810849	1	2	yes	—
255	C₄	#16810853	1	2	yes	—
256	C₂	—	2	1	no	—
257	C₂	—	2	1	no	—
258	C₂	#20274708	1	1	yes	characteristic
259	C₂	#20274737	1	1	yes	characteristic
260	C₂	#20274762	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16810480
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1566	C₂
socle	C₂²×C₅₃	#380
prime radical	C₂³×C₅₃	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1566	D₅₃
hypocenter	C₅₃	#8615
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1566

Complements

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

N	H ≅ G/N	type
108. C₈×C₅₃	#101	semidirect
153. C₅₃	#8615	semidirect
241. C₈	#13133135	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×Dic₁₀₆	2
#20274708	2
#20273471	2
#20272589	2
C₅₃×#1566	2
#8615	53