Subgroups — #27539527

Order 13568.

814 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811064	C₂	1	4	yes	maximal, complemented
2	#20285143	C₂	1	1	yes	maximal, characteristic
3	#20276984	C₂	1	4	yes	maximal, complemented
4	#20276972	C₂	1	2	yes	maximal, complemented
5	#20276967	C₂	1	2	yes	maximal, complemented
6	C₄×#13133157	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₅₃×#494	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13133239	—	2	1	no	—
9	C₂×#13133239	C₂²	1	2	yes	—
10	#16811173	—	2	1	no	—
11	#16811064	—	2	16	no	—
12	C₂×#13133163	C₂²	1	8	yes	—
13	C₂×#13133160	C₂²	1	8	yes	—
14	C₂×#13133157	C₂²	1	2	yes	—
15	#16811007	—	2	1	no	—
16	#16811007	C₂²	1	2	yes	—
17	#16811003	C₂²	1	1	yes	characteristic
18	#16811000	C₂²	1	1	yes	characteristic
19	#16810996	—	2	1	no	—
20	#16810399	C₂²	1	2	yes	—
21	C₄×#11719908	C₂²	1	2	yes	—
22	C₂²×Q₁₆×C₅₃	—	2	1	no	—
23	C₂²×Q₁₆×C₅₃	C₂²	1	2	yes	—
24	C₂×C₄⋊Q₈×C₅₃	—	2	1	no	—
25	C₅₃×#494	—	2	1	no	—
26	C₅₃×#494	—	2	1	no	—
27	C₅₃×#494	C₂²	1	2	yes	—
28	C₅₃×#494	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
30	C₂×#11719945	—	2	2	no	—
31	#13133239	—	2	8	no	—
32	#13133239	—	4	4	no	—
33	C₂×Dic₂₁₂	—	2	1	no	—
34	C₂×Dic₂₁₂	C₂³	1	2	yes	—
35	C₂×#11719913	—	2	2	no	—
36	C₂×#11719913	—	2	2	no	—
37	C₂×D₄₂₄	—	2	1	no	—
38	C₂×D₄₂₄	C₂³	1	2	yes	—
39	#13133217	—	2	1	no	—
40	#13133214	—	2	1	no	—
41	#13133205	—	2	2	no	—
42	#13133191	—	4	2	no	—
43	C₄×D₂₁₂	—	4	2	no	—
44	#13133163	—	2	16	no	—
45	#13133160	—	2	16	no	—
46	#13133157	D₄	1	8	yes	complemented
47	C₂×#11719908	C₂³	1	4	yes	—
48	C₂²×Q₈×C₅₃	—	2	2	no	—
49	C₂×Q₁₆×C₅₃	—	2	2	no	—
50	C₂×Q₁₆×C₅₃	—	2	2	no	—
51	C₂×Q₁₆×C₅₃	—	2	2	no	—
52	C₂×Q₁₆×C₅₃	—	2	8	no	—
53	C₂×Q₁₆×C₅₃	—	4	4	no	—
54	C₂×Q₁₆×C₅₃	C₂³	1	4	yes	—
55	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
56	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
57	C₄⋊Q₈×C₅₃	—	2	2	no	—
58	C₄⋊Q₈×C₅₃	—	2	2	no	—
59	C₄⋊Q₈×C₅₃	—	4	2	no	—
60	C₂×C₄⋊C₄×C₅₃	—	4	2	no	—
61	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
62	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
63	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
64	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
65	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
66	C₂×Q₈×C₅₃	—	2	4	no	—
67	C₂×Q₈×C₅₃	—	4	2	no	—
68	C₂×Q₈×C₅₃	—	4	4	no	—
69	C₂×Q₈×C₅₃	—	4	4	no	—
70	C₂²×C₄×C₅₃	—	4	2	no	—
71	C₂²×C₄×C₅₃	D₈	1	2	yes	—
72	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
73	C₂×C₂²⋊D₅₃	—	4	2	no	—
74	#11719945	—	4	4	no	—
75	#11719945	—	4	4	no	—
76	C₂×D₂₁₂	—	2	2	no	—
77	C₂×D₂₁₂	—	4	2	no	—
78	C₂×C₄×D₅₃	—	4	2	no	—
79	C₂×Dic₁₀₆	—	2	2	no	—
80	Q₁₆×C₅₃	—	2	8	no	—
81	Q₁₆×C₅₃	—	4	4	no	—
82	Q₁₆×C₅₃	—	4	4	no	—
83	Q₁₆×C₅₃	—	4	4	no	—
84	C₂×C₈×C₅₃	D₈	1	4	yes	—
85	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
87	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
90	C₄⋊C₄×C₅₃	—	4	4	no	—
91	C₄⋊C₄×C₅₃	—	4	4	no	—
92	C₄²×C₅₃	D₈	1	2	yes	—
93	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
95	#11719921	—	4	4	no	—
96	C₄⋊Dic₅₃	—	4	2	no	—
97	Dic₂₁₂	—	2	4	no	—
98	Dic₂₁₂	—	4	2	no	—
99	D₄₂₄	—	2	4	no	—
100	D₄₂₄	—	4	2	no	—
101	#11719913	—	4	4	no	—
102	#11719913	—	4	4	no	—
103	#11719908	C₂×D₄	1	8	yes	—
104	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
105	C₂²×D₅₃	—	4	2	no	—
106	Q₈×C₅₃	—	4	4	no	—
107	Q₈×C₅₃	—	4	4	no	—
108	Q₈×C₅₃	—	8	4	no	—
109	C₂×C₄×C₅₃	—	2	4	no	—
110	C₂×C₄×C₅₃	—	4	4	no	—
111	C₂×C₄×C₅₃	—	8	4	no	—
112	C₂×C₄×C₅₃	#127	1	2	yes	—
113	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
114	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
115	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
116	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
117	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
118	C₂²⋊D₅₃	—	8	4	no	—
119	C₂×Dic₅₃	—	4	2	no	—
120	D₂₁₂	—	4	2	no	—
121	D₂₁₂	—	4	2	no	—
122	D₂₁₂	—	8	4	no	—
123	C₄×D₅₃	—	8	4	no	—
124	Dic₁₀₆	—	4	2	no	—
125	Dic₁₀₆	—	4	2	no	—
126	C₈×C₅₃	#127	1	2	yes	—
127	C₈×C₅₃	C₂×D₈	1	2	yes	—
128	C₈×C₅₃	C₂×D₈	1	2	yes	—
129	C₈×C₅₃	C₂²×D₄	1	2	yes	—
130	#33718	—	53	1	no	maximal, Sylow
131	C₂²×C₅₃	—	2	2	no	—
132	C₂²×C₅₃	#493	1	2	yes	—
133	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
134	C₂×D₅₃	—	8	2	no	—
135	C₂×D₅₃	—	8	4	no	—
136	C₄×C₅₃	—	2	2	no	—
137	C₄×C₅₃	—	8	4	no	—
138	C₄×C₅₃	#493	1	2	yes	—
139	C₄×C₅₃	#510	1	4	yes	complemented
140	C₄×C₅₃	C₂×#127	1	1	yes	characteristic
141	C₄×C₅₃	C₂²×D₈	1	1	yes	characteristic
142	Dic₅₃	—	8	2	no	—
143	C₂×#510	—	53	4	no	—
144	#3147	—	53	1	no	—
145	C₂×#494	D₅₃	1	1	yes	characteristic, complemented
146	#2251	—	53	4	no	—
147	#2248	—	53	2	no	—
148	#2247	—	53	2	no	—
149	C₄×M₃₂	—	53	1	no	—
150	C₂×C₅₃	—	2	2	no	—
151	C₂×C₅₃	C₂×#493	1	1	yes	characteristic
152	C₂×C₅₃	C₂×#510	1	2	yes	—
153	D₅₃	—	16	2	no	—
154	C₂×#135	—	53	2	no	—
155	C₂×#135	—	106	1	no	—
156	C₂²×Q₁₆	—	2	1	no	—
157	C₂²×Q₁₆	C₂×D₅₃	1	2	yes	—
158	#532	—	106	1	no	—
159	C₂×C₄⋊Q₈	—	2	1	no	—
160	#510	—	106	16	no	—
161	C₂×Q₃₂	—	53	8	no	—
162	C₂×SD₃₂	—	53	8	no	—
163	C₂×M₃₂	—	53	2	no	—
164	#495	—	53	2	no	—
165	#495	—	106	1	no	—
166	#494	—	2	1	no	—
167	#494	—	2	1	no	—
168	#494	—	53	1	no	—
169	#494	C₂×D₅₃	1	2	yes	—
170	#494	C₂×D₅₃	1	2	yes	—
171	#493	—	53	1	no	—
172	#492	—	106	1	no	—
173	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
174	#346	—	53	2	no	—
175	C₄×C₁₆	—	53	2	no	—
176	C₅₃	#33718	1	1	yes	characteristic, complemented, Sylow
177	C₂×#41	—	106	2	no	—
178	C₂²×Q₈	—	2	2	no	—
179	#135	—	106	8	no	—
180	#135	—	212	4	no	—
181	C₂×Q₁₆	—	2	8	no	—
182	C₂×Q₁₆	—	2	2	no	—
183	C₂×Q₁₆	—	2	2	no	—
184	C₂×Q₁₆	—	2	2	no	—
185	C₂×Q₁₆	—	4	4	no	—
186	C₂×Q₁₆	—	53	2	no	—
187	C₂×Q₁₆	—	106	1	no	—
188	C₂×Q₁₆	C₂²×D₅₃	1	4	yes	—
189	C₂×SD₁₆	—	106	2	no	—
190	C₂×SD₁₆	—	106	2	no	—
191	C₂×D₈	—	53	2	no	—
192	C₂×D₈	—	106	1	no	—
193	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
194	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
195	C₄⋊Q₈	—	2	2	no	—
196	C₄⋊Q₈	—	2	2	no	—
197	C₄⋊Q₈	—	4	2	no	—
198	C₄⋊Q₈	—	106	1	no	—
199	#127	—	106	1	no	—
200	#124	—	106	2	no	—
201	#121	—	212	2	no	—
202	C₄×D₄	—	212	2	no	—
203	C₂×C₄⋊C₄	—	4	2	no	—
204	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
205	Q₃₂	—	106	16	no	—
206	SD₃₂	—	106	16	no	—
207	M₃₂	—	53	8	no	—
208	C₂×C₁₆	—	53	4	no	—
209	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
210	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
211	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
212	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
213	#41	—	212	4	no	—
214	#41	—	212	4	no	—
215	C₂×Q₈	—	2	4	no	—
216	C₂×Q₈	—	4	2	no	—
217	C₂×Q₈	—	4	4	no	—
218	C₂×Q₈	—	4	4	no	—
219	C₂×Q₈	—	106	2	no	—
220	C₂×D₄	—	106	2	no	—
221	C₂×D₄	—	212	2	no	—
222	C₂×D₄	—	212	2	no	—
223	C₂²×C₄	—	4	2	no	—
224	C₂²×C₄	—	212	2	no	—
225	C₂²×C₄	#11719922	1	2	yes	—
226	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
227	Q₁₆	—	2	8	no	—
228	Q₁₆	—	4	4	no	—
229	Q₁₆	—	4	4	no	—
230	Q₁₆	—	4	4	no	—
231	Q₁₆	—	106	4	no	—
232	Q₁₆	—	212	2	no	—
233	SD₁₆	—	212	4	no	—
234	SD₁₆	—	212	4	no	—
235	D₈	—	106	4	no	—
236	D₈	—	212	2	no	—
237	C₂×C₈	#11719922	1	4	yes	—
238	C₂×C₈	D₄×D₅₃	1	4	yes	—
239	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
240	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
241	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
242	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
243	C₄⋊C₄	—	4	4	no	—
244	C₄⋊C₄	—	4	4	no	—
245	C₄⋊C₄	—	212	2	no	—
246	C₂²⋊C₄	—	212	4	no	—
247	C₄²	#11719922	1	2	yes	—
248	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
249	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
250	C₁₆	—	53	8	no	—
251	C₂³	—	212	2	no	—
252	C₂³	C₂×#11719922	1	1	yes	characteristic
253	Q₈	—	4	4	no	—
254	Q₈	—	4	4	no	—
255	Q₈	—	8	4	no	—
256	Q₈	—	212	2	no	—
257	Q₈	—	212	2	no	—
258	D₄	—	424	4	no	—
259	D₄	—	424	4	no	—
260	D₄	—	212	2	no	—
261	D₄	—	212	2	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	4	4	no	—
264	C₂×C₄	—	8	4	no	—
265	C₂×C₄	—	424	4	no	—
266	C₂×C₄	—	212	2	no	—
267	C₂×C₄	#13133213	1	2	yes	—
268	C₂×C₄	C₂×#11719922	1	2	yes	—
269	C₂×C₄	C₂×#11719922	1	2	yes	—
270	C₂×C₄	C₂×#11719922	1	2	yes	—
271	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
272	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
273	C₈	#13133213	1	2	yes	—
274	C₈	C₂×#11719922	1	2	yes	—
275	C₈	C₂×#11719922	1	2	yes	—
276	C₈	C₂×D₄×D₅₃	1	2	yes	—
277	C₂²	—	2	2	no	—
278	C₂²	—	424	4	no	—
279	C₂²	—	424	2	no	—
280	C₂²	#16810999	1	2	yes	—
281	C₂²	C₂²×#11719922	1	1	yes	characteristic
282	C₄	—	2	2	no	—
283	C₄	—	8	4	no	—
284	C₄	—	424	2	no	—
285	C₄	#16810999	1	2	yes	—
286	C₄	#16811064	1	4	yes	complemented
287	C₄	C₂×#13133213	1	1	yes	characteristic
288	C₄	C₂²×#11719922	1	1	yes	characteristic
289	C₂	—	2	2	no	—
290	C₂	—	848	2	no	—
291	C₂	C₂×#16810999	1	1	yes	characteristic
292	C₂	C₂×#16811064	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×#11719922
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂×C₅₃×#494	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	#20285143	C₂
hypercenter	C₂×#494	D₅₃
hypocenter	C₅₃	#33718
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#494

Complements

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

N	H ≅ G/N	type
1. C₂×#16811064	C₂	semidirect
3. #20276984	C₂	semidirect
4. #20276972	C₂	semidirect
5. #20276967	C₂	semidirect
6. C₄×#13133157	C₂	semidirect
7. C₂×C₅₃×#494	C₂	semidirect
46. #13133157	D₄	semidirect
139. C₄×C₅₃	#510	semidirect
145. C₂×#494	D₅₃	semidirect
176. C₅₃	#33718	semidirect
286. C₄	#16811064	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811064	2
#20285143	2
#20276984	2
#20276972	2
#20276967	2
C₄×#13133157	2
C₂×C₅₃×#494	2
#33718	53