Subgroups — #27302358

Order 13568.

490 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810439	C₂	1	2	yes	maximal, complemented
2	C₂×#16810436	C₂	1	2	yes	maximal
3	C₄×#13133157	C₂	1	1	yes	maximal, characteristic, complemented
4	#20274887	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₅₃×#495	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133157	C₂²	1	2	yes	—
7	C₂×#13133153	—	2	1	no	—
8	C₂×#13133153	C₂²	1	2	yes	—
9	#16810647	—	2	1	no	—
10	#16810439	—	2	4	no	—
11	#16810436	—	2	4	no	—
12	#16810399	—	2	1	no	—
13	C₄×#11719908	—	2	1	no	—
14	C₂²×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
15	C₂²×SD₁₆×C₅₃	—	2	1	no	—
16	C₂²×D₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×#124×C₅₃	—	2	1	no	—
18	C₅₃×#495	—	2	1	no	—
19	C₅₃×#495	—	2	1	no	—
20	C₅₃×#495	C₄	1	2	yes	—
21	C₅₃×#495	C₄	1	2	yes	—
22	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×#11719917	—	2	2	no	—
24	#13133157	—	2	2	no	—
25	#13133157	D₄	1	4	yes	—
26	C₂×#11719908	—	2	2	no	—
27	#13133153	—	2	2	no	—
28	#13133153	—	2	2	no	—
29	#13133153	—	4	2	no	—
30	C₄⋊#88322	—	4	2	no	—
31	C₂²×Q₈×C₅₃	—	2	1	no	—
32	C₂²×D₄×C₅₃	—	2	1	no	—
33	C₂×Q₁₆×C₅₃	—	2	1	no	—
34	C₂×Q₁₆×C₅₃	—	2	2	no	—
35	C₂×Q₁₆×C₅₃	—	2	2	no	—
36	C₂×Q₁₆×C₅₃	C₂×C₄	1	2	yes	—
37	C₂×SD₁₆×C₅₃	—	2	2	no	—
38	C₂×SD₁₆×C₅₃	—	2	2	no	—
39	C₂×SD₁₆×C₅₃	—	4	2	no	—
40	C₂×SD₁₆×C₅₃	—	4	2	no	—
41	C₂×D₈×C₅₃	—	2	1	no	—
42	C₂×D₈×C₅₃	—	2	2	no	—
43	C₂×D₈×C₅₃	—	2	2	no	—
44	C₂×D₈×C₅₃	C₂×C₄	1	2	yes	—
45	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
47	#124×C₅₃	—	2	2	no	—
48	#124×C₅₃	—	2	2	no	—
49	#124×C₅₃	—	4	2	no	—
50	C₂×C₂²⋊C₄×C₅₃	—	4	2	no	—
51	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
52	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
55	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
56	C₂⁴×C₅₃	—	4	1	no	—
57	C₂×Q₈×C₅₃	—	2	2	no	—
58	C₂×Q₈×C₅₃	—	4	1	no	—
59	C₂×Q₈×C₅₃	—	4	1	no	—
60	C₂×Q₈×C₅₃	—	4	1	no	—
61	C₂×Q₈×C₅₃	—	4	1	no	—
62	C₂×Q₈×C₅₃	—	4	1	no	—
63	C₂×D₄×C₅₃	—	2	2	no	—
64	C₂×D₄×C₅₃	—	4	1	no	—
65	C₂×D₄×C₅₃	—	4	1	no	—
66	C₂×D₄×C₅₃	—	4	1	no	—
67	C₂×D₄×C₅₃	—	4	1	no	—
68	C₂×D₄×C₅₃	—	4	1	no	—
69	C₂²×C₄×C₅₃	—	4	1	no	—
70	C₂²×C₄×C₅₃	SD₁₆	1	2	yes	—
71	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
72	Q₁₆×C₅₃	—	2	4	no	—
73	Q₁₆×C₅₃	—	4	2	no	—
74	SD₁₆×C₅₃	—	4	4	no	—
75	SD₁₆×C₅₃	—	4	4	no	—
76	D₈×C₅₃	—	2	4	no	—
77	D₈×C₅₃	—	4	2	no	—
78	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₈×C₅₃	D₈	1	2	yes	—
81	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
82	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
84	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₅₃	#41	1	2	yes	—
86	C₂²⋊C₄×C₅₃	—	4	4	no	—
87	C₂²⋊C₄×C₅₃	—	4	4	no	—
88	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
89	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
90	C₄²×C₅₃	D₈	1	2	yes	—
91	#11719917	—	4	2	no	—
92	#11719917	—	4	2	no	—
93	C₂×#88322	—	4	2	no	—
94	#11719908	—	2	4	no	—
95	C₂³×C₅₃	—	4	2	no	—
96	C₂³×C₅₃	—	8	4	no	—
97	C₂³×C₅₃	—	8	1	no	—
98	C₂³×C₅₃	—	8	1	no	—
99	C₂³×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
100	Q₈×C₅₃	—	4	2	no	—
101	Q₈×C₅₃	—	4	2	no	—
102	Q₈×C₅₃	—	8	1	no	—
103	Q₈×C₅₃	—	8	1	no	—
104	D₄×C₅₃	—	4	2	no	—
105	D₄×C₅₃	—	4	2	no	—
106	D₄×C₅₃	—	8	1	no	—
107	D₄×C₅₃	—	8	1	no	—
108	C₂×C₄×C₅₃	—	4	2	no	—
109	C₂×C₄×C₅₃	—	4	2	no	—
110	C₂×C₄×C₅₃	—	8	1	no	—
111	C₂×C₄×C₅₃	—	8	1	no	—
112	C₂×C₄×C₅₃	#102	1	2	yes	—
113	C₂×C₄×C₅₃	#102	1	2	yes	—
114	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
116	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
117	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
118	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
119	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
120	C₈×C₅₃	#102	1	2	yes	—
121	C₈×C₅₃	#102	1	2	yes	—
122	C₈×C₅₃	C₄×D₄	1	2	yes	—
123	C₈×C₅₃	#121	1	2	yes	—
124	#88322	—	8	2	no	—
125	#13214	—	53	1	no	maximal, Sylow
126	C₂²×C₅₃	—	2	2	no	—
127	C₂²×C₅₃	—	8	4	no	—
128	C₂²×C₅₃	—	8	2	no	—
129	C₂²×C₅₃	—	16	2	no	—
130	C₂²×C₅₃	—	16	2	no	—
131	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
132	C₂²×C₅₃	#486	1	1	yes	characteristic
133	C₂²×C₅₃	#492	1	1	yes	characteristic
134	C₄×C₅₃	—	2	2	no	—
135	C₄×C₅₃	—	2	2	no	—
136	C₄×C₅₃	—	8	2	no	—
137	C₄×C₅₃	#390	1	1	yes	characteristic
138	C₄×C₅₃	C₂×#102	1	1	yes	characteristic
139	C₄×C₅₃	#486	1	1	yes	characteristic
140	C₄×C₅₃	#493	1	1	yes	characteristic
141	C₂×#495	D₅₃	1	1	yes	characteristic
142	C₂×#362	—	53	2	no	—
143	C₂×#361	—	53	2	no	—
144	C₄×M₃₂	—	53	1	no	—
145	#1848	—	53	1	no	—
146	C₂×C₅₃	—	2	1	no	—
147	C₂×C₅₃	—	2	1	no	—
148	C₂×C₅₃	—	16	2	no	—
149	C₂×C₅₃	#1957	1	1	yes	characteristic
150	C₂×C₅₃	#2148	1	2	yes	—
151	C₂²×Q₁₆	C₂×D₅₃	1	1	yes	characteristic
152	C₂²×SD₁₆	—	2	1	no	—
153	C₂²×D₈	C₂×D₅₃	1	1	yes	characteristic
154	C₂×#124	—	2	1	no	—
155	C₂×M₃₂	—	53	2	no	—
156	#495	—	2	1	no	—
157	#495	—	2	1	no	—
158	#495	Dic₅₃	1	2	yes	—
159	#495	Dic₅₃	1	2	yes	—
160	C₂×#108	—	53	2	no	—
161	C₂×#108	—	106	1	no	—
162	#423	—	106	1	no	—
163	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
164	#362	—	106	4	no	—
165	#361	—	106	4	no	—
166	#346	—	106	1	no	—
167	C₄×C₁₆	—	106	1	no	—
168	C₅₃	#13214	1	1	yes	characteristic, complemented, Sylow
169	C₂²×Q₈	—	2	1	no	—
170	C₂²×D₄	—	2	1	no	—
171	C₂×Q₁₆	—	2	2	no	—
172	C₂×Q₁₆	—	2	2	no	—
173	C₂×Q₁₆	—	2	1	no	—
174	C₂×Q₁₆	C₂×Dic₅₃	1	2	yes	—
175	C₂×SD₁₆	—	2	2	no	—
176	C₂×SD₁₆	—	2	2	no	—
177	C₂×SD₁₆	—	4	2	no	—
178	C₂×SD₁₆	—	4	2	no	—
179	C₂×D₈	—	2	2	no	—
180	C₂×D₈	—	2	2	no	—
181	C₂×D₈	—	2	1	no	—
182	C₂×D₈	C₂×Dic₅₃	1	2	yes	—
183	C₂×M₁₆	—	106	2	no	—
184	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
185	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
186	#124	—	2	2	no	—
187	#124	—	2	2	no	—
188	#124	—	4	2	no	—
189	C₂×C₂²⋊C₄	—	4	2	no	—
190	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
191	M₃₂	—	53	4	no	—
192	M₃₂	—	106	2	no	—
193	C₂×C₁₆	—	106	2	no	—
194	#108	—	106	2	no	—
195	#108	—	106	2	no	—
196	#108	—	212	2	no	—
197	C₄⋊C₈	—	212	2	no	—
198	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
199	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
200	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
201	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
202	C₂⁴	—	4	1	no	—
203	C₂×Q₈	—	2	2	no	—
204	C₂×Q₈	—	4	1	no	—
205	C₂×Q₈	—	4	1	no	—
206	C₂×Q₈	—	4	1	no	—
207	C₂×Q₈	—	4	1	no	—
208	C₂×Q₈	—	4	1	no	—
209	C₂×D₄	—	2	2	no	—
210	C₂×D₄	—	4	1	no	—
211	C₂×D₄	—	4	1	no	—
212	C₂×D₄	—	4	1	no	—
213	C₂×D₄	—	4	1	no	—
214	C₂×D₄	—	4	1	no	—
215	C₂²×C₄	—	4	1	no	—
216	C₂²×C₄	#11719923	1	2	yes	—
217	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
218	Q₁₆	—	2	4	no	—
219	Q₁₆	—	4	2	no	—
220	SD₁₆	—	4	4	no	—
221	SD₁₆	—	4	4	no	—
222	D₈	—	2	4	no	—
223	D₈	—	4	2	no	—
224	M₁₆	—	212	2	no	—
225	M₁₆	—	212	2	no	—
226	C₂×C₈	—	212	2	no	—
227	C₂×C₈	#11719922	1	2	yes	—
228	C₂×C₈	#11719923	1	2	yes	—
229	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
230	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
231	C₂×C₈	D₄×D₅₃	1	2	yes	—
232	C₂×C₈	#11719947	1	2	yes	—
233	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
234	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
235	C₂²⋊C₄	—	4	4	no	—
236	C₂²⋊C₄	—	4	4	no	—
237	C₄²	#11719922	1	2	yes	—
238	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
239	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
240	C₁₆	—	106	4	no	—
241	C₂³	—	4	2	no	—
242	C₂³	—	8	4	no	—
243	C₂³	—	8	1	no	—
244	C₂³	—	8	1	no	—
245	C₂³	C₂×#11719923	1	1	yes	characteristic
246	Q₈	—	4	2	no	—
247	Q₈	—	4	2	no	—
248	Q₈	—	8	1	no	—
249	Q₈	—	8	1	no	—
250	D₄	—	4	2	no	—
251	D₄	—	4	2	no	—
252	D₄	—	8	1	no	—
253	D₄	—	8	1	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	8	1	no	—
257	C₂×C₄	—	8	1	no	—
258	C₂×C₄	#13133138	1	2	yes	—
259	C₂×C₄	#13133138	1	2	yes	—
260	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
261	C₂×C₄	#13133204	1	1	yes	characteristic
262	C₂×C₄	#13133213	1	1	yes	characteristic
263	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
264	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
265	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
266	C₈	—	424	2	no	—
267	C₈	#13133138	1	2	yes	—
268	C₈	#13133138	1	2	yes	—
269	C₈	D₄×Dic₅₃	1	2	yes	—
270	C₈	#13133188	1	2	yes	—
271	C₂²	—	2	2	no	—
272	C₂²	—	8	2	no	—
273	C₂²	—	8	4	no	—
274	C₂²	—	16	2	no	—
275	C₂²	—	16	2	no	—
276	C₂²	C₂×#13133138	1	1	yes	characteristic
277	C₂²	#16810969	1	1	yes	characteristic
278	C₂²	#16810995	1	1	yes	characteristic
279	C₄	—	2	2	no	—
280	C₄	—	2	2	no	—
281	C₄	—	8	2	no	—
282	C₄	#16810526	1	1	yes	characteristic
283	C₄	C₂×#13133138	1	1	yes	characteristic
284	C₄	#16810969	1	1	yes	characteristic
285	C₄	#16810999	1	1	yes	characteristic
286	C₂	—	2	1	no	—
287	C₂	—	2	1	no	—
288	C₂	—	16	2	no	—
289	C₂	#20275522	1	1	yes	characteristic
290	C₂	#20276464	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#13133138
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#495	C₂
socle	C₂²×C₅₃	C₂×#102
prime radical	C₂²×D₈×C₅₃	C₂²
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#495	D₅₃
hypocenter	C₅₃	#13214
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#495

Complements

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

N	H ≅ G/N	type
1. C₂×#16810439	C₂	semidirect
3. C₄×#13133157	C₂	semidirect
4. #20274887	C₂	semidirect
168. C₅₃	#13214	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810439	2
C₂×#16810436	2
C₄×#13133157	2
#20274887	2
C₂×C₅₃×#495	2
#13214	53