Subgroups — #27277162

Order 13568.

438 conjugacy classes of subgroups.

The subgroup lattice has 258 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	#20274843	C₂	1	1	yes	maximal, characteristic
3	#20274721	C₂	1	2	yes	maximal
4	#20272599	C₂	1	2	yes	maximal
5	C₂×C₅₃×#334	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
7	C₈×Dic₁₀₆	—	2	2	no	—
8	C₈×Dic₁₀₆	C₄	1	4	yes	complemented
9	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
10	C₂×#13133146	—	2	1	no	—
11	C₂×#13133142	—	2	2	no	—
12	C₂×C₈×Dic₅₃	—	2	1	no	—
13	#16810493	C₂²	1	2	yes	—
14	#16810386	—	2	2	no	—
15	#16810386	—	2	1	no	—
16	#16810386	—	2	1	no	—
17	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₅₃×#334	—	2	2	no	—
20	C₂²×Dic₁₀₆	D₄	1	1	yes	characteristic
21	C₂²×#88322	—	2	1	no	—
22	C₄×Dic₁₀₆	—	2	1	no	—
23	C₄×Dic₁₀₆	—	2	1	no	—
24	C₄×Dic₁₀₆	C₂×C₄	1	2	yes	—
25	C₄×Dic₁₀₆	Q₈	1	2	yes	—
26	C₂×C₄⋊Dic₅₃	—	2	2	no	—
27	C₂×C₄⋊Dic₅₃	—	2	2	no	—
28	C₂×C₄⋊Dic₅₃	—	2	2	no	—
29	C₂×C₄⋊Dic₅₃	D₄	1	1	yes	characteristic
30	C₂×#11719919	—	2	1	no	—
31	C₂×C₄×Dic₅₃	—	2	1	no	—
32	C₄⋊#88322	—	2	1	no	—
33	C₄⋊#88322	C₂×C₄	1	2	yes	—
34	#13133146	—	2	4	no	—
35	#13133142	—	4	4	no	—
36	C₈×Dic₅₃	—	2	4	no	—
37	C₂²×C₈×C₅₃	—	2	2	no	—
38	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
41	C₄⋊C₈×C₅₃	—	2	2	no	—
42	C₄⋊C₈×C₅₃	—	2	2	no	—
43	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
44	C₄×C₈×C₅₃	D₄	1	2	yes	—
45	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
48	C₂²×Dic₅₃	—	2	1	no	—
49	C₂²×Dic₅₃	—	2	1	no	—
50	C₂²×Dic₅₃	—	4	2	no	—
51	C₂×Dic₁₀₆	—	2	1	no	—
52	C₂×Dic₁₀₆	—	2	2	no	—
53	C₂×Dic₁₀₆	C₄⋊C₄	1	2	yes	—
54	C₂×Dic₁₀₆	SD₁₆	1	4	yes	—
55	C₂×C₈×C₅₃	—	2	2	no	—
56	C₂×C₈×C₅₃	—	2	2	no	—
57	C₂×C₈×C₅₃	—	4	2	no	—
58	C₂×C₈×C₅₃	—	4	2	no	—
59	C₂×C₈×C₅₃	—	4	2	no	—
60	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
61	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
62	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
63	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
64	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
65	C₄²×C₅₃	#41	1	1	yes	characteristic
66	C₄⋊Dic₅₃	—	2	1	no	—
67	C₄⋊Dic₅₃	—	4	2	no	—
68	C₄⋊Dic₅₃	—	4	2	no	—
69	C₄⋊Dic₅₃	—	4	2	no	—
70	C₄⋊Dic₅₃	—	4	2	no	—
71	C₄⋊Dic₅₃	—	8	2	no	—
72	C₄⋊Dic₅₃	C₄⋊C₄	1	2	yes	—
73	#11719919	—	2	2	no	—
74	#11719919	—	2	2	no	—
75	C₄×Dic₅₃	—	2	2	no	—
76	C₄×Dic₅₃	—	2	2	no	—
77	C₂×#88322	—	2	2	no	—
78	C₂×#88322	—	4	2	no	—
79	C₂³×C₅₃	#121	1	1	yes	characteristic
80	C₂×C₄×C₅₃	—	2	1	no	—
81	C₂×C₄×C₅₃	—	2	1	no	—
82	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
85	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
86	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
87	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
88	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
89	C₂×C₄×C₅₃	#135	1	2	yes	—
90	C₂×C₄×C₅₃	#135	1	2	yes	—
91	C₂×C₄×C₅₃	#137	1	2	yes	—
92	C₂×Dic₅₃	—	2	2	no	—
93	C₂×Dic₅₃	—	2	4	no	—
94	C₂×Dic₅₃	—	2	2	no	—
95	C₂×Dic₅₃	—	4	1	no	—
96	C₂×Dic₅₃	—	4	1	no	—
97	C₂×Dic₅₃	—	8	2	no	—
98	C₂×Dic₅₃	—	8	2	no	—
99	C₂×Dic₅₃	—	8	2	no	—
100	Dic₁₀₆	—	2	2	no	—
101	Dic₁₀₆	—	2	4	no	—
102	Dic₁₀₆	#106	1	4	yes	—
103	C₈×C₅₃	—	2	2	no	—
104	C₈×C₅₃	—	2	2	no	—
105	C₈×C₅₃	—	8	2	no	—
106	#88322	—	4	2	no	—
107	#8647	—	53	1	no	maximal, Sylow
108	C₂²×C₅₃	#380	1	1	yes	characteristic
109	C₂²×C₅₃	#464	1	2	yes	—
110	C₂²×C₅₃	#465	1	2	yes	—
111	C₂²×C₅₃	#472	1	1	yes	characteristic
112	C₂²×C₅₃	#473	1	1	yes	characteristic
113	C₄×C₅₃	—	2	1	no	—
114	C₄×C₅₃	—	2	1	no	—
115	C₄×C₅₃	#416	1	1	yes	characteristic
116	C₄×C₅₃	#419	1	1	yes	characteristic
117	C₄×C₅₃	C₂×#106	1	1	yes	characteristic
118	C₄×C₅₃	#427	1	1	yes	characteristic
119	C₄×C₅₃	#465	1	2	yes	—
120	C₄×C₅₃	#479	1	2	yes	—
121	Dic₅₃	—	2	4	no	—
122	Dic₅₃	—	4	2	no	—
123	Dic₅₃	—	16	2	no	—
124	C₂×C₈×Q₈	—	53	1	no	—
125	#1834	—	53	1	no	—
126	#1803	—	53	2	no	—
127	C₂×#334	D₅₃	1	1	yes	characteristic
128	#1296	—	53	2	no	—
129	C₂×C₅₃	#1678	1	2	yes	—
130	C₂×C₅₃	#1680	1	2	yes	—
131	C₂×C₅₃	#1805	1	1	yes	characteristic
132	C₂×C₅₃	#1807	1	1	yes	characteristic
133	C₂×C₅₃	#1816	1	1	yes	characteristic
134	C₂×C₄×Q₈	—	53	1	no	—
135	C₈×Q₈	—	53	4	no	—
136	C₈×Q₈	—	106	2	no	—
137	C₂×C₄⋊C₈	—	53	1	no	—
138	C₂×C₄⋊C₈	—	106	1	no	—
139	C₂×C₄⋊C₈	C₂×D₅₃	1	2	yes	—
140	C₂×#103	—	106	2	no	—
141	C₂×C₄×C₈	—	106	1	no	—
142	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
143	#384	—	53	2	no	—
144	#340	—	106	2	no	—
145	#340	—	106	1	no	—
146	#340	—	106	1	no	—
147	#334	—	2	2	no	—
148	C₅₃	#8647	1	1	yes	characteristic, complemented, Sylow
149	C₂²×Q₈	—	53	1	no	—
150	C₂²×C₈	—	2	2	no	—
151	C₂²×C₈	—	106	1	no	—
152	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
153	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
154	C₄×Q₈	—	53	2	no	—
155	C₄×Q₈	—	53	2	no	—
156	C₄×Q₈	—	106	1	no	—
157	C₄×Q₈	—	106	1	no	—
158	C₂×C₄⋊C₄	—	53	1	no	—
159	C₂×C₄⋊C₄	—	106	1	no	—
160	C₂×C₄⋊C₄	—	106	2	no	—
161	C₂×C₄⋊C₄	—	106	2	no	—
162	C₂×C₄⋊C₄	—	106	2	no	—
163	C₂×C₄²	—	106	1	no	—
164	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
165	C₄⋊C₈	—	2	2	no	—
166	C₄⋊C₈	—	2	2	no	—
167	C₄⋊C₈	—	53	2	no	—
168	C₄⋊C₈	—	106	4	no	—
169	C₄⋊C₈	—	106	1	no	—
170	#103	—	212	4	no	—
171	C₄×C₈	—	106	4	no	—
172	C₄×C₈	C₄×D₅₃	1	2	yes	—
173	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
174	C₂×Q₈	—	53	2	no	—
175	C₂×Q₈	—	53	4	no	—
176	C₂×Q₈	—	106	1	no	—
177	C₂×Q₈	—	106	2	no	—
178	C₂²×C₄	—	106	1	no	—
179	C₂²×C₄	—	106	1	no	—
180	C₂²×C₄	—	212	2	no	—
181	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
182	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
183	C₂²×C₄	#11719949	1	1	yes	characteristic
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	4	2	no	—
187	C₂×C₈	—	4	2	no	—
188	C₂×C₈	—	4	2	no	—
189	C₂×C₈	—	106	2	no	—
190	C₂×C₈	—	212	2	no	—
191	C₂×C₈	#11719921	1	2	yes	—
192	C₂×C₈	#11719921	1	2	yes	—
193	C₄⋊C₄	—	53	2	no	—
194	C₄⋊C₄	—	106	2	no	—
195	C₄⋊C₄	—	106	1	no	—
196	C₄⋊C₄	—	106	2	no	—
197	C₄⋊C₄	—	424	2	no	—
198	C₄⋊C₄	—	212	2	no	—
199	C₄⋊C₄	—	212	2	no	—
200	C₄⋊C₄	—	212	2	no	—
201	C₄⋊C₄	—	212	2	no	—
202	C₄²	—	106	2	no	—
203	C₄²	—	106	2	no	—
204	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
205	C₄²	#11719947	1	1	yes	characteristic
206	C₄²	Q₈×D₅₃	1	1	yes	characteristic
207	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
208	C₂³	#13133192	1	1	yes	characteristic
209	Q₈	—	53	4	no	—
210	Q₈	—	106	4	no	—
211	Q₈	—	106	2	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	106	4	no	—
215	C₂×C₄	—	106	2	no	—
216	C₂×C₄	—	106	2	no	—
217	C₂×C₄	—	424	2	no	—
218	C₂×C₄	—	424	2	no	—
219	C₂×C₄	—	424	2	no	—
220	C₂×C₄	—	212	1	no	—
221	C₂×C₄	—	212	1	no	—
222	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
223	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
224	C₂×C₄	#13133175	1	1	yes	characteristic
225	C₂×C₄	#13133188	1	1	yes	characteristic
226	C₂×C₄	#13133193	1	1	yes	characteristic
227	C₂×C₄	#13133197	1	1	yes	characteristic
228	C₂×C₄	SD₁₆×D₅₃	1	2	yes	—
229	C₂×C₄	#13133238	1	2	yes	—
230	C₂×C₄	#13133239	1	2	yes	—
231	C₂×C₄	#13133250	1	2	yes	—
232	C₈	—	2	2	no	—
233	C₈	—	2	2	no	—
234	C₈	—	8	2	no	—
235	C₈	—	212	2	no	—
236	C₂²	#16810480	1	1	yes	characteristic
237	C₂²	#16810846	1	2	yes	—
238	C₂²	#16810853	1	2	yes	—
239	C₂²	#16810894	1	1	yes	characteristic
240	C₂²	#16810899	1	1	yes	characteristic
241	C₄	—	2	1	no	—
242	C₄	—	2	1	no	—
243	C₄	—	848	2	no	—
244	C₄	—	106	4	no	—
245	C₄	—	212	2	no	—
246	C₄	#16810620	1	1	yes	characteristic
247	C₄	#16810633	1	1	yes	characteristic
248	C₄	#106×D₅₃	1	1	yes	characteristic
249	C₄	#16810658	1	1	yes	characteristic
250	C₄	#16810849	1	2	yes	—
251	C₄	#16810931	1	2	yes	—
252	C₂	#20274132	1	2	yes	—
253	C₂	#20274146	1	2	yes	—
254	C₂	#20274731	1	1	yes	characteristic
255	C₂	#20274737	1	1	yes	characteristic
256	C₂	#20274776	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13133192
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₅₃×#334	C₂
socle, prime radical	C₂³×C₅₃	#121
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×#334	D₅₃
hypocenter	C₅₃	#8647
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
8. C₈×Dic₁₀₆	C₄	semidirect
148. C₅₃	#8647	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
#20274843	2
#20274721	2
#20272599	2
C₂×C₅₃×#334	2
#8647	53