Subgroups — #27283314

Order 13568.

454 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811028	C₂	1	1	yes	maximal, characteristic
2	#20275662	C₂	1	2	yes	maximal
3	C₈×C₄⋊Dic₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#16810371	C₂	1	2	yes	maximal
5	C₂×C₅₃×#335	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133217	C₂²	1	2	yes	—
7	#16811028	—	2	2	no	—
8	#16811028	C₄	1	4	yes	complemented
9	C₂×#13133151	—	2	1	no	—
10	C₂×#13133151	—	2	1	no	—
11	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
12	C₂×C₈×Dic₅₃	—	2	1	no	—
13	C₄×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
14	#16810386	—	2	4	no	—
15	#16810377	—	2	1	no	—
16	#16810371	—	2	4	no	—
17	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₅₃×#335	—	2	2	no	—
20	C₂²×Dic₁₀₆	—	2	2	no	—
21	C₂²×#88322	—	2	1	no	—
22	#13133217	—	2	2	no	—
23	#13133217	—	2	2	no	—
24	#13133217	—	2	2	no	—
25	#13133217	C₂×C₄	1	4	yes	—
26	C₂×C₄⋊Dic₅₃	—	2	2	no	—
27	C₂×C₄⋊Dic₅₃	—	2	2	no	—
28	C₂×C₄⋊Dic₅₃	D₄	1	2	yes	—
29	C₂×C₄×Dic₅₃	—	2	1	no	—
30	#13133151	—	2	4	no	—
31	#13133151	—	2	4	no	—
32	C₄⋊#88322	—	2	1	no	—
33	C₄⋊#88322	D₄	1	2	yes	—
34	C₈×Dic₅₃	—	4	2	no	—
35	#13133124	—	2	1	no	—
36	C₂²×C₈×C₅₃	—	2	2	no	—
37	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
40	C₄⋊C₈×C₅₃	—	2	2	no	—
41	C₄⋊C₈×C₅₃	—	2	2	no	—
42	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
43	C₄×C₈×C₅₃	D₄	1	2	yes	—
44	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
47	C₂²×Dic₅₃	—	2	2	no	—
48	C₂²×Dic₅₃	—	4	2	no	—
49	C₂×Dic₁₀₆	—	2	4	no	—
50	C₂×Dic₁₀₆	—	4	2	no	—
51	C₂×Dic₁₀₆	—	8	4	no	—
52	C₂×C₈×C₅₃	—	2	1	no	—
53	C₂×C₈×C₅₃	—	2	1	no	—
54	C₂×C₈×C₅₃	—	4	2	no	—
55	C₂×C₈×C₅₃	—	4	4	no	—
56	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
57	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
58	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
59	C₂×C₈×C₅₃	Q₁₆	1	2	yes	—
60	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
61	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
62	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
63	C₄²×C₅₃	#41	1	1	yes	characteristic
64	C₄⋊Dic₅₃	—	2	2	no	—
65	C₄⋊Dic₅₃	—	2	4	no	—
66	C₄⋊Dic₅₃	—	2	4	no	—
67	C₄⋊Dic₅₃	—	4	2	no	—
68	C₄⋊Dic₅₃	—	4	2	no	—
69	C₄⋊Dic₅₃	Q₁₆	1	4	yes	—
70	C₄×Dic₅₃	—	4	1	no	—
71	C₄×Dic₅₃	—	4	1	no	—
72	C₂×#88322	—	2	2	no	—
73	C₂×#88322	—	4	1	no	—
74	C₂×#88322	—	4	1	no	—
75	C₂³×C₅₃	#121	1	1	yes	characteristic
76	C₂×C₄×C₅₃	—	2	2	no	—
77	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
79	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
80	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
81	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
82	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
83	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
84	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
85	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
86	C₂×C₄×C₅₃	#135	1	2	yes	—
87	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
88	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
89	C₂×Dic₅₃	—	2	4	no	—
90	C₂×Dic₅₃	—	4	2	no	—
91	C₂×Dic₅₃	—	4	2	no	—
92	C₂×Dic₅₃	—	4	4	no	—
93	C₂×Dic₅₃	—	8	4	no	—
94	Dic₁₀₆	—	8	4	no	—
95	Dic₁₀₆	—	8	4	no	—
96	C₈×C₅₃	—	2	2	no	—
97	C₈×C₅₃	—	8	2	no	—
98	C₈×C₅₃	#103	1	4	yes	—
99	#88322	—	4	2	no	—
100	#9730	—	53	1	no	maximal, Sylow
101	C₂²×C₅₃	#390	1	1	yes	characteristic
102	C₂²×C₅₃	#461	1	1	yes	characteristic
103	C₂²×C₅₃	#462	1	1	yes	characteristic
104	C₂²×C₅₃	#467	1	2	yes	—
105	C₂²×C₅₃	#472	1	2	yes	—
106	C₄×C₅₃	—	2	1	no	—
107	C₄×C₅₃	—	2	1	no	—
108	C₄×C₅₃	C₂×#103	1	1	yes	characteristic
109	C₄×C₅₃	#419	1	1	yes	characteristic
110	C₄×C₅₃	C₄×Q₁₆	1	2	yes	—
111	C₄×C₅₃	#461	1	1	yes	characteristic
112	C₄×C₅₃	#462	1	1	yes	characteristic
113	C₄×C₅₃	#494	1	1	yes	characteristic
114	C₄×C₅₃	#495	1	1	yes	characteristic
115	Dic₅₃	—	4	4	no	—
116	Dic₅₃	—	8	4	no	—
117	C₂×#500	—	53	1	no	—
118	#1987	—	53	2	no	—
119	C₈×C₄⋊C₄	—	53	1	no	—
120	C₂×#335	D₅₃	1	1	yes	characteristic
121	C₂×#333	—	53	2	no	—
122	C₂×C₅₃	#1675	1	2	yes	—
123	C₂×C₅₃	#1676	1	2	yes	—
124	C₂×C₅₃	#1969	1	1	yes	characteristic
125	C₂×C₅₃	#1978	1	1	yes	characteristic
126	C₂×C₅₃	#1981	1	1	yes	characteristic
127	C₂×C₄⋊Q₈	—	53	2	no	—
128	#500	—	53	4	no	—
129	#500	—	106	2	no	—
130	C₂×#107	—	106	1	no	—
131	C₂×#107	—	106	1	no	—
132	C₂×C₄⋊C₈	—	53	1	no	—
133	C₂×C₄⋊C₈	C₂×D₅₃	1	2	yes	—
134	C₂×C₄×C₈	—	106	1	no	—
135	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
136	C₄×C₄⋊C₄	—	53	1	no	—
137	#340	—	106	4	no	—
138	#336	—	106	1	no	—
139	#335	—	2	2	no	—
140	#333	—	106	4	no	—
141	C₅₃	#9730	1	1	yes	characteristic, complemented, Sylow
142	C₂²×Q₈	—	106	2	no	—
143	C₂²×C₈	—	2	2	no	—
144	C₂²×C₈	—	106	1	no	—
145	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
146	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
147	C₄⋊Q₈	—	53	4	no	—
148	C₄⋊Q₈	—	106	2	no	—
149	C₄⋊Q₈	—	106	2	no	—
150	C₄⋊Q₈	—	106	2	no	—
151	C₂×C₄⋊C₄	—	53	2	no	—
152	C₂×C₄⋊C₄	—	106	2	no	—
153	C₂×C₄⋊C₄	—	106	2	no	—
154	C₂×C₄²	—	106	1	no	—
155	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
156	#107	—	106	4	no	—
157	#107	—	106	4	no	—
158	C₄⋊C₈	—	2	2	no	—
159	C₄⋊C₈	—	2	2	no	—
160	C₄⋊C₈	—	53	2	no	—
161	C₄⋊C₈	—	106	1	no	—
162	C₄×C₈	—	212	2	no	—
163	C₄×C₈	C₄×D₅₃	1	2	yes	—
164	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
165	#95	—	106	1	no	—
166	C₂×Q₈	—	106	4	no	—
167	C₂×Q₈	—	424	4	no	—
168	C₂×Q₈	—	212	2	no	—
169	C₂²×C₄	—	106	2	no	—
170	C₂²×C₄	—	212	2	no	—
171	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
172	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
173	C₂²×C₄	#11719949	1	1	yes	characteristic
174	C₂×C₈	—	2	1	no	—
175	C₂×C₈	—	2	1	no	—
176	C₂×C₈	—	4	2	no	—
177	C₂×C₈	—	4	4	no	—
178	C₂×C₈	—	106	2	no	—
179	C₂×C₈	—	212	1	no	—
180	C₂×C₈	—	212	1	no	—
181	C₂×C₈	#11719913	1	2	yes	—
182	C₂×C₈	Dic₂₁₂	1	2	yes	—
183	C₂×C₈	#11719921	1	2	yes	—
184	C₂×C₈	#11719921	1	2	yes	—
185	C₄⋊C₄	—	53	4	no	—
186	C₄⋊C₄	—	106	2	no	—
187	C₄⋊C₄	—	106	4	no	—
188	C₄⋊C₄	—	106	4	no	—
189	C₄⋊C₄	—	212	2	no	—
190	C₄⋊C₄	—	212	2	no	—
191	C₄²	—	212	1	no	—
192	C₄²	—	212	1	no	—
193	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
194	C₄²	D₄×D₅₃	1	1	yes	characteristic
195	C₄²	#11719949	1	1	yes	characteristic
196	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
197	C₂³	#13133192	1	1	yes	characteristic
198	Q₈	—	424	4	no	—
199	Q₈	—	424	4	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	106	4	no	—
202	C₂×C₄	—	424	4	no	—
203	C₂×C₄	—	212	4	no	—
204	C₂×C₄	—	212	2	no	—
205	C₂×C₄	—	212	2	no	—
206	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
207	C₂×C₄	#13133179	1	1	yes	characteristic
208	C₂×C₄	#13133179	1	1	yes	characteristic
209	C₂×C₄	#13133192	1	1	yes	characteristic
210	C₂×C₄	#13133206	1	1	yes	characteristic
211	C₂×C₄	#13133213	1	1	yes	characteristic
212	C₂×C₄	C₂×#11719913	1	1	yes	characteristic
213	C₂×C₄	Q₁₆×D₅₃	1	2	yes	—
214	C₂×C₄	C₂×Dic₂₁₂	1	1	yes	characteristic
215	C₂×C₄	#13133236	1	2	yes	—
216	C₂×C₄	#13133250	1	1	yes	characteristic
217	C₂×C₄	#13133250	1	1	yes	characteristic
218	C₈	—	2	2	no	—
219	C₈	—	8	2	no	—
220	C₈	—	212	2	no	—
221	C₈	#13133142	1	4	yes	—
222	C₂²	#16810524	1	1	yes	characteristic
223	C₂²	#16810829	1	1	yes	characteristic
224	C₂²	#16810834	1	1	yes	characteristic
225	C₂²	#16810865	1	2	yes	—
226	C₂²	#16810894	1	2	yes	—
227	C₄	—	2	1	no	—
228	C₄	—	2	1	no	—
229	C₄	—	424	4	no	—
230	C₄	—	212	4	no	—
231	C₄	C₂×#13133142	1	1	yes	characteristic
232	C₄	#16810633	1	1	yes	characteristic
233	C₄	#16810717	1	2	yes	—
234	C₄	#16810829	1	1	yes	characteristic
235	C₄	#16810834	1	1	yes	characteristic
236	C₄	#16811001	1	1	yes	characteristic
237	C₄	#16811006	1	1	yes	characteristic
238	C₂	#20274115	1	2	yes	—
239	C₂	#20274120	1	2	yes	—
240	C₂	#20275574	1	1	yes	characteristic
241	C₂	#20275623	1	1	yes	characteristic
242	C₂	#20275637	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13133192
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₅₃×#335	C₂
socle, prime radical	C₂³×C₅₃	#121
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×#335	D₅₃
hypocenter	C₅₃	#9730
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
8. #16811028	C₄	semidirect
141. C₅₃	#9730	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811028	2
#20275662	2
C₈×C₄⋊Dic₅₃	2
C₂×#16810371	2
C₂×C₅₃×#335	2
#9730	53