Subgroups — #27302262

Order 13568.

446 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283428	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810406	C₂	1	4	yes	maximal, complemented
3	C₄×#13133157	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#13133223	C₂²	1	2	yes	—
6	#16811173	C₄	1	1	yes	characteristic
7	C₂×#13133157	C₂²	1	2	yes	—
8	#16810702	—	2	1	no	—
9	#16810702	—	2	1	no	—
10	C₈×D₂₁₂	—	2	1	no	—
11	C₈×D₂₁₂	—	2	1	no	—
12	#16810647	C₄	1	1	yes	characteristic
13	#16810589	C₄	1	2	yes	—
14	#16810406	—	2	8	no	—
15	#16810399	—	2	1	no	—
16	C₄×#11719908	—	2	1	no	—
17	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₅₃×C₄⋊C₁₆	—	2	1	no	—
20	C₅₃×C₄⋊C₁₆	—	2	1	no	—
21	C₂×#11719945	C₂×C₄	1	2	yes	—
22	#13133223	—	2	4	no	—
23	#13133223	—	2	4	no	—
24	C₂×#11719917	C₂×C₄	1	2	yes	—
25	C₂×#11719912	—	2	2	no	—
26	C₂×C₈×D₅₃	—	2	2	no	—
27	#13133217	C₈	1	1	yes	characteristic
28	#13133214	C₈	1	1	yes	characteristic
29	#13133205	C₈	1	2	yes	—
30	#13133191	—	2	1	no	—
31	#13133191	—	2	1	no	—
32	C₄×D₂₁₂	—	2	1	no	—
33	C₄×D₂₁₂	—	2	1	no	—
34	#13133157	—	2	2	no	—
35	#13133157	D₄	1	4	yes	complemented
36	C₂×#11719908	—	2	2	no	—
37	C₄⋊#88322	—	2	1	no	—
38	C₄⋊#88322	—	2	1	no	—
39	#13133129	—	2	2	no	—
40	#13133129	—	2	2	no	—
41	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
43	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₁₆×C₅₃	—	2	2	no	—
45	C₂×C₁₆×C₅₃	—	2	4	no	—
46	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
47	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
48	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
49	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₂²⋊D₅₃	—	2	2	no	—
54	#11719945	—	2	4	no	—
55	#11719945	—	2	4	no	—
56	C₂×D₂₁₂	—	2	1	no	—
57	C₂×D₂₁₂	—	2	1	no	—
58	C₂×D₂₁₂	C₂×C₈	1	2	yes	—
59	C₂×C₄×D₅₃	—	2	2	no	—
60	C₂×Dic₁₀₆	C₂×C₈	1	2	yes	—
61	C₂×C₈×C₅₃	—	2	2	no	—
62	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
65	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
66	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
67	C₂×C₈×C₅₃	#41	1	2	yes	—
68	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
69	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
70	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
71	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
72	#11719921	—	2	2	no	—
73	#11719921	—	2	2	no	—
74	C₄⋊Dic₅₃	—	2	1	no	—
75	C₄⋊Dic₅₃	—	2	1	no	—
76	#11719917	—	2	4	no	—
77	C₂×#88322	—	2	2	no	—
78	#11719912	—	4	2	no	—
79	#11719912	—	4	2	no	—
80	C₈×D₅₃	—	4	4	no	—
81	C₁₆×C₅₃	—	4	2	no	—
82	#11719908	—	2	4	no	—
83	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
84	C₂²×D₅₃	—	2	2	no	—
85	C₂×C₄×C₅₃	—	2	2	no	—
86	C₂×C₄×C₅₃	—	4	1	no	—
87	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
88	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
89	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
90	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
91	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
93	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
94	C₂×C₄×C₅₃	#131	1	2	yes	—
95	C₂²⋊D₅₃	—	4	4	no	—
96	C₂×Dic₅₃	—	2	2	no	—
97	D₂₁₂	—	2	4	no	—
98	D₂₁₂	—	4	2	no	—
99	D₂₁₂	—	4	2	no	—
100	C₄×D₅₃	—	4	2	no	—
101	C₄×D₅₃	—	4	2	no	—
102	Dic₁₀₆	—	2	4	no	—
103	C₈×C₅₃	—	2	1	no	—
104	C₈×C₅₃	—	2	1	no	—
105	C₈×C₅₃	#121	1	2	yes	—
106	C₈×C₅₃	#124	1	1	yes	characteristic
107	C₈×C₅₃	#127	1	1	yes	characteristic
108	#88322	—	4	2	no	—
109	#13193	—	53	1	no	maximal, Sylow
110	C₂²×C₅₃	—	2	1	no	—
111	C₂²×C₅₃	—	2	1	no	—
112	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
113	C₂²×C₅₃	#436	1	2	yes	—
114	C₂×D₅₃	—	4	2	no	—
115	C₂×D₅₃	—	4	2	no	—
116	C₂×D₅₃	—	4	2	no	—
117	C₄×C₅₃	—	2	1	no	—
118	C₄×C₅₃	—	2	1	no	—
119	C₄×C₅₃	—	2	1	no	—
120	C₄×C₅₃	#350	1	2	yes	complemented
121	C₄×C₅₃	#390	1	1	yes	characteristic
122	C₄×C₅₃	#408	1	1	yes	characteristic
123	C₄×C₅₃	C₈×D₄	1	2	yes	—
124	Dic₅₃	—	4	2	no	—
125	#2931	—	53	1	no	—
126	C₂×C₄⋊C₁₆	D₅₃	1	1	yes	characteristic, complemented
127	C₂×#350	—	53	4	no	—
128	C₄×M₃₂	—	53	1	no	—
129	C₂×C₅₃	—	2	2	no	—
130	C₂×C₅₃	#1954	1	1	yes	characteristic
131	C₂×C₅₃	C₂×#350	1	1	yes	characteristic
132	C₂×C₅₃	#2117	1	1	yes	characteristic
133	D₅₃	—	8	2	no	—
134	C₂×#131	—	53	2	no	—
135	#532	—	53	1	no	—
136	C₂×M₃₂	—	53	2	no	—
137	C₂²×C₁₆	C₂×D₅₃	1	2	yes	—
138	#436	—	106	1	no	—
139	#436	—	106	1	no	—
140	C₈×D₄	—	106	1	no	—
141	C₈×D₄	—	106	1	no	—
142	#423	—	53	1	no	—
143	#408	—	53	2	no	—
144	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
145	C₄⋊C₁₆	—	2	1	no	—
146	C₄⋊C₁₆	—	2	1	no	—
147	#350	—	106	8	no	—
148	#346	—	106	1	no	—
149	C₄×C₁₆	—	106	1	no	—
150	C₅₃	#13193	1	1	yes	characteristic, complemented, Sylow
151	C₂×#41	—	53	2	no	—
152	#131	—	106	4	no	—
153	#131	—	106	4	no	—
154	C₂×M₁₆	—	53	2	no	—
155	C₂×M₁₆	—	106	2	no	—
156	C₂²×C₈	—	106	2	no	—
157	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
158	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
159	C₄⋊Q₈	—	53	1	no	—
160	#127	—	53	1	no	—
161	#124	—	53	2	no	—
162	#121	—	106	1	no	—
163	#121	—	106	1	no	—
164	C₄×D₄	—	106	1	no	—
165	C₄×D₄	—	106	1	no	—
166	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
167	M₃₂	—	53	4	no	—
168	M₃₂	—	106	2	no	—
169	C₂×C₁₆	—	2	4	no	—
170	C₂×C₁₆	—	2	2	no	—
171	C₂×C₁₆	—	106	2	no	—
172	C₄⋊C₈	—	106	1	no	—
173	C₄⋊C₈	—	106	1	no	—
174	C₂²⋊C₈	—	106	2	no	—
175	C₂²⋊C₈	—	106	2	no	—
176	C₄×C₈	D₂₁₂	1	1	yes	characteristic
177	C₄×C₈	D₂₁₂	1	1	yes	characteristic
178	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
179	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
180	#41	—	106	4	no	—
181	#41	—	106	4	no	—
182	C₂×Q₈	—	53	2	no	—
183	C₂×D₄	—	53	2	no	—
184	C₂×D₄	—	106	2	no	—
185	C₂×D₄	—	106	1	no	—
186	C₂×D₄	—	106	1	no	—
187	C₂²×C₄	—	106	2	no	—
188	C₂²×C₄	#11719912	1	1	yes	characteristic
189	C₂²×C₄	#11719912	1	1	yes	characteristic
190	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
191	M₁₆	—	106	4	no	—
192	M₁₆	—	212	2	no	—
193	M₁₆	—	212	2	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	106	2	no	—
196	C₂×C₈	—	212	4	no	—
197	C₂×C₈	#11719921	1	1	yes	characteristic
198	C₂×C₈	#11719921	1	1	yes	characteristic
199	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
200	C₂×C₈	D₄×D₅₃	1	2	yes	—
201	C₂×C₈	#11719949	1	2	yes	—
202	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
203	C₄⋊C₄	—	106	1	no	—
204	C₄⋊C₄	—	106	1	no	—
205	C₂²⋊C₄	—	106	2	no	—
206	C₂²⋊C₄	—	106	2	no	—
207	C₄²	C₈×D₅₃	1	1	yes	characteristic
208	C₄²	C₈×D₅₃	1	1	yes	characteristic
209	C₄²	#11719921	1	1	yes	characteristic
210	C₄²	#11719921	1	1	yes	characteristic
211	C₁₆	—	4	2	no	—
212	C₁₆	—	106	4	no	—
213	C₂³	—	106	2	no	—
214	C₂³	C₂×#11719912	1	1	yes	characteristic
215	Q₈	—	106	4	no	—
216	D₄	—	106	4	no	—
217	D₄	—	212	2	no	—
218	D₄	—	212	4	no	—
219	D₄	—	212	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	4	1	no	—
222	C₂×C₄	—	106	2	no	—
223	C₂×C₄	—	212	2	no	—
224	C₂×C₄	—	212	2	no	—
225	C₂×C₄	#13133129	1	1	yes	characteristic
226	C₂×C₄	#13133129	1	1	yes	characteristic
227	C₂×C₄	#13133129	1	1	yes	characteristic
228	C₂×C₄	#13133129	1	1	yes	characteristic
229	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
230	C₂×C₄	#13133179	1	2	yes	—
231	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
232	C₂×C₄	#13133225	1	2	yes	—
233	C₈	—	2	1	no	—
234	C₈	—	2	1	no	—
235	C₈	—	212	2	no	—
236	C₈	#13133192	1	2	yes	—
237	C₈	#13133206	1	1	yes	characteristic
238	C₈	#13133213	1	1	yes	characteristic
239	C₂²	—	2	1	no	—
240	C₂²	—	2	1	no	—
241	C₂²	—	212	2	no	—
242	C₂²	—	212	2	no	—
243	C₂²	—	212	2	no	—
244	C₂²	C₂×#13133129	1	1	yes	characteristic
245	C₂²	#16810700	1	2	yes	—
246	C₄	—	2	1	no	—
247	C₄	—	2	1	no	—
248	C₄	—	2	1	no	—
249	C₄	—	212	2	no	—
250	C₄	#16810406	1	2	yes	complemented
251	C₄	#16810524	1	1	yes	characteristic
252	C₄	#16810591	1	1	yes	characteristic
253	C₄	#16810688	1	2	yes	—
254	C₂	—	2	2	no	—
255	C₂	—	424	2	no	—
256	C₂	#20275508	1	1	yes	characteristic
257	C₂	C₂×#16810406	1	1	yes	characteristic
258	C₂	#20276342	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×C₄⋊C₁₆	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₈
prime radical	#16811173	C₄
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₅₃
hypocenter	C₅₃	#13193
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×#16810406	C₂	semidirect
3. C₄×#13133157	C₂	semidirect
4. C₂×C₅₃×C₄⋊C₁₆	C₂	semidirect
35. #13133157	D₄	semidirect
120. C₄×C₅₃	#350	semidirect
126. C₂×C₄⋊C₁₆	D₅₃	semidirect
150. C₅₃	#13193	semidirect
250. C₄	#16810406	semidirect

Maximal subgroups

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

subgroup	index
#20283428	2
C₂×#16810406	2
C₄×#13133157	2
C₂×C₅₃×C₄⋊C₁₆	2
#13193	53