Subgroups — #27299394

Order 13568.

412 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810746	C₂	1	1	yes	maximal, characteristic
2	#20276462	C₂	1	1	yes	maximal, characteristic
3	#20276442	C₂	1	1	yes	maximal, characteristic
4	C₂×#16810401	C₂	1	1	yes	maximal, characteristic, complemented
5	#20272826	C₂	1	1	yes	maximal, characteristic, complemented
6	#20272818	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#1778	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13133239	C₂²	1	1	yes	characteristic
9	C₂×#13133223	—	2	1	no	—
10	C₂×#13133157	C₂²	1	1	yes	characteristic
11	C₂×#13133157	C₂²	1	1	yes	characteristic
12	#16810746	—	2	2	no	—
13	#16810746	C₄	1	4	yes	—
14	C₂×#13133153	C₂²	1	1	yes	characteristic
15	C₂×#13133145	—	2	1	no	—
16	#16810438	C₄	1	2	yes	complemented
17	#16810435	C₄	1	2	yes	complemented
18	#16810429	C₄	1	4	yes	complemented
19	#16810401	—	2	2	no	—
20	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×#108×C₅₃	C₂²	1	1	yes	characteristic
22	C₄×M₁₆×C₅₃	—	2	1	no	—
23	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
24	C₅₃×#341	—	2	1	no	—
25	C₂×#11719945	—	2	1	no	—
26	#13133239	—	2	1	no	—
27	#13133239	—	2	1	no	—
28	#13133239	C₂×C₄	1	2	yes	—
29	#13133239	C₂×C₄	1	2	yes	—
30	C₂×Dic₂₁₂	C₂×C₄	1	1	yes	characteristic
31	C₂×#11719913	—	2	1	no	—
32	C₂×D₄₂₄	C₂×C₄	1	1	yes	characteristic
33	#13133223	—	2	4	no	—
34	#13133223	—	4	2	no	—
35	C₂×#11719917	—	2	1	no	—
36	C₂×#11719912	—	4	1	no	—
37	C₂×C₈×D₅₃	—	4	1	no	—
38	#13133157	—	2	1	no	—
39	#13133157	—	2	1	no	—
40	#13133157	—	2	1	no	—
41	#13133157	C₂×C₄	1	2	yes	—
42	C₂×#11719908	—	2	1	no	—
43	C₂×#11719908	C₂×C₄	1	2	yes	—
44	#13133153	—	2	1	no	—
45	#13133153	C₂×C₄	1	2	yes	—
46	#13133145	—	2	4	no	—
47	C₂×M₁₆×C₅₃	—	2	1	no	—
48	C₂×M₁₆×C₅₃	—	2	1	no	—
49	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
50	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
52	#108×C₅₃	—	2	1	no	—
53	#108×C₅₃	—	2	1	no	—
54	#108×C₅₃	C₂×C₄	1	2	yes	—
55	#108×C₅₃	C₂×C₄	1	2	yes	—
56	#97×C₅₃	—	4	1	no	—
57	#97×C₅₃	C₂×C₄	1	2	yes	—
58	#97×C₅₃	D₄	1	2	yes	—
59	C₄×C₈×C₅₃	—	4	1	no	—
60	C₂²×C₄×C₅₃	—	2	1	no	—
61	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
62	C₂×C₂²⋊D₅₃	—	4	1	no	—
63	#11719945	—	2	2	no	—
64	#11719945	—	2	2	no	—
65	#11719945	—	4	1	no	—
66	#11719945	—	4	1	no	—
67	C₂×D₂₁₂	—	2	1	no	—
68	C₂×C₄×D₅₃	—	4	1	no	—
69	C₂×Dic₁₀₆	—	2	1	no	—
70	M₁₆×C₅₃	—	2	2	no	—
71	M₁₆×C₅₃	—	2	2	no	—
72	M₁₆×C₅₃	—	4	1	no	—
73	M₁₆×C₅₃	—	4	1	no	—
74	C₂×C₈×C₅₃	—	2	1	no	—
75	C₂×C₈×C₅₃	—	2	1	no	—
76	C₂×C₈×C₅₃	—	4	1	no	—
77	C₂×C₈×C₅₃	—	4	1	no	—
78	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
79	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
84	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
85	C₄²×C₅₃	—	2	1	no	—
86	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
87	#11719917	—	2	2	no	—
88	#11719917	—	4	1	no	—
89	C₂×#88322	—	4	1	no	—
90	Dic₂₁₂	—	2	1	no	—
91	Dic₂₁₂	C₄²	1	2	yes	—
92	D₄₂₄	—	2	1	no	—
93	D₄₂₄	C₄²	1	2	yes	—
94	#11719913	—	2	2	no	—
95	#11719913	—	2	2	no	—
96	#11719912	—	4	4	no	—
97	C₈×D₅₃	—	4	4	no	—
98	#11719908	—	2	2	no	—
99	#11719908	—	2	2	no	—
100	C₂³×C₅₃	#137	1	1	yes	characteristic
101	C₂²×D₅₃	—	4	1	no	—
102	C₂×C₄×C₅₃	—	2	2	no	—
103	C₂×C₄×C₅₃	—	8	1	no	—
104	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
108	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
109	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
110	C₂²⋊D₅₃	—	4	2	no	—
111	C₂²⋊D₅₃	—	4	2	no	—
112	C₂×Dic₅₃	—	4	1	no	—
113	D₂₁₂	—	2	2	no	—
114	D₂₁₂	—	4	1	no	—
115	C₄×D₅₃	—	4	2	no	—
116	C₄×D₅₃	—	4	2	no	—
117	Dic₁₀₆	—	2	2	no	—
118	Dic₁₀₆	—	4	1	no	—
119	C₈×C₅₃	—	2	2	no	—
120	C₈×C₅₃	—	4	2	no	—
121	C₈×C₅₃	—	4	2	no	—
122	C₈×C₅₃	C₂×C₄²	1	1	yes	characteristic
123	C₈×C₅₃	#117	1	1	yes	characteristic
124	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
125	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
126	#88322	—	4	2	no	—
127	#12651	—	53	1	no	maximal, Sylow
128	C₂²×C₅₃	—	4	1	no	—
129	C₂²×C₅₃	#417	1	1	yes	characteristic
130	C₂²×C₅₃	#440	1	1	yes	characteristic
131	C₂²×C₅₃	#440	1	1	yes	characteristic
132	C₂×D₅₃	—	4	2	no	—
133	C₂×D₅₃	—	8	2	no	—
134	C₄×C₅₃	—	4	2	no	—
135	C₄×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
136	C₄×C₅₃	#418	1	1	yes	characteristic
137	C₄×C₅₃	#442	1	1	yes	characteristic
138	C₄×C₅₃	#442	1	1	yes	characteristic
139	Dic₅₃	—	4	2	no	—
140	C₂×#444	—	53	1	no	—
141	#2148	—	53	1	no	—
142	#2143	—	53	1	no	—
143	C₂×#347	—	53	1	no	—
144	#1778	D₅₃	1	1	yes	characteristic, complemented
145	#1388	—	53	1	no	—
146	#1385	—	53	1	no	—
147	C₂×C₅₃	—	2	1	no	—
148	C₂×C₅₃	—	2	1	no	—
149	C₂×C₅₃	—	2	1	no	—
150	C₂×C₅₃	#1762	1	1	yes	characteristic
151	D₅₃	—	8	2	no	—
152	C₂×#135	—	53	1	no	—
153	C₂×#131	—	106	1	no	—
154	C₂×M₃₂	—	53	1	no	—
155	C₂×M₃₂	—	53	1	no	—
156	#444	—	53	4	no	—
157	#444	—	106	2	no	—
158	C₂×#108	—	53	1	no	—
159	C₂×#108	C₂×D₅₃	1	1	yes	characteristic
160	C₂×#108	C₂×D₅₃	1	1	yes	characteristic
161	C₂×#104	—	106	1	no	—
162	C₄×M₁₆	—	2	1	no	—
163	C₂×#97	C₂×D₅₃	1	1	yes	characteristic
164	#362	—	53	2	no	—
165	#361	—	53	2	no	—
166	#359	—	53	4	no	—
167	#347	—	106	2	no	—
168	#341	—	2	1	no	—
169	C₅₃	#12651	1	1	yes	characteristic, complemented, Sylow
170	C₂×#41	—	106	1	no	—
171	#135	—	53	2	no	—
172	#135	—	53	2	no	—
173	#135	—	106	1	no	—
174	#135	—	106	1	no	—
175	C₂×Q₁₆	—	53	1	no	—
176	C₂×SD₁₆	—	106	1	no	—
177	C₂×D₈	—	53	1	no	—
178	#131	—	106	4	no	—
179	#131	—	212	2	no	—
180	C₂×M₁₆	—	2	1	no	—
181	C₂×M₁₆	—	2	1	no	—
182	C₂×M₁₆	—	106	1	no	—
183	C₂×M₁₆	—	212	1	no	—
184	C₂²×C₈	—	212	1	no	—
185	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
186	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
187	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
188	M₃₂	—	53	2	no	—
189	M₃₂	—	106	1	no	—
190	M₃₂	—	106	1	no	—
191	M₃₂	—	106	1	no	—
192	C₂×C₁₆	—	53	2	no	—
193	C₂×C₁₆	—	106	1	no	—
194	#108	—	2	1	no	—
195	#108	—	2	1	no	—
196	#108	—	53	2	no	—
197	#108	—	106	1	no	—
198	#108	C₄×D₅₃	1	2	yes	—
199	#108	C₄×D₅₃	1	2	yes	—
200	#104	—	106	4	no	—
201	#97	—	4	1	no	—
202	#97	C₄×D₅₃	1	2	yes	—
203	#97	D₂₁₂	1	2	yes	—
204	C₄×C₈	—	4	1	no	—
205	#41	—	106	2	no	—
206	#41	—	106	2	no	—
207	#41	—	212	1	no	—
208	#41	—	212	1	no	—
209	C₂×Q₈	—	106	1	no	—
210	C₂×D₄	—	106	1	no	—
211	C₂×D₄	—	212	1	no	—
212	C₂²×C₄	—	2	1	no	—
213	C₂²×C₄	—	212	1	no	—
214	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
215	Q₁₆	—	53	2	no	—
216	Q₁₆	—	106	1	no	—
217	SD₁₆	—	106	2	no	—
218	SD₁₆	—	106	2	no	—
219	D₈	—	53	2	no	—
220	D₈	—	106	1	no	—
221	M₁₆	—	2	2	no	—
222	M₁₆	—	2	2	no	—
223	M₁₆	—	4	1	no	—
224	M₁₆	—	4	1	no	—
225	M₁₆	—	106	2	no	—
226	M₁₆	—	212	4	no	—
227	M₁₆	—	212	1	no	—
228	C₂×C₈	—	2	1	no	—
229	C₂×C₈	—	2	1	no	—
230	C₂×C₈	—	4	1	no	—
231	C₂×C₈	—	4	1	no	—
232	C₂×C₈	—	212	4	no	—
233	C₂×C₈	—	212	1	no	—
234	C₂×C₈	#11719921	1	2	yes	—
235	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
236	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
237	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
238	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
239	C₂×C₈	#11719945	1	1	yes	characteristic
240	C₂×C₈	#11719945	1	1	yes	characteristic
241	C₄²	—	2	1	no	—
242	C₄²	#11719921	1	2	yes	—
243	C₁₆	—	106	2	no	—
244	C₁₆	—	106	2	no	—
245	C₂³	—	212	1	no	—
246	C₂³	#13133248	1	1	yes	characteristic
247	Q₈	—	106	2	no	—
248	Q₈	—	212	1	no	—
249	D₄	—	106	2	no	—
250	D₄	—	212	2	no	—
251	D₄	—	212	2	no	—
252	D₄	—	212	1	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	8	1	no	—
255	C₂×C₄	—	212	1	no	—
256	C₂×C₄	—	212	2	no	—
257	C₂×C₄	—	212	2	no	—
258	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
259	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
260	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
261	C₂×C₄	#13133242	1	1	yes	characteristic
262	C₂×C₄	#13133242	1	1	yes	characteristic
263	C₂×C₄	#13133243	1	1	yes	characteristic
264	C₈	—	2	2	no	—
265	C₈	—	4	2	no	—
266	C₈	—	4	2	no	—
267	C₈	—	212	2	no	—
268	C₈	C₄²×D₅₃	1	1	yes	characteristic
269	C₈	#13133174	1	1	yes	characteristic
270	C₈	C₄×D₂₁₂	1	1	yes	characteristic
271	C₈	C₄×D₂₁₂	1	1	yes	characteristic
272	C₂²	—	4	1	no	—
273	C₂²	—	424	2	no	—
274	C₂²	—	212	2	no	—
275	C₂²	#16810622	1	1	yes	characteristic
276	C₂²	#16810721	1	1	yes	characteristic
277	C₂²	#16810721	1	1	yes	characteristic
278	C₄	—	4	2	no	—
279	C₄	—	212	2	no	—
280	C₄	C₄×#11719921	1	1	yes	characteristic
281	C₄	#16810627	1	1	yes	characteristic
282	C₄	#16810733	1	1	yes	characteristic
283	C₄	#16810733	1	1	yes	characteristic
284	C₂	—	2	1	no	—
285	C₂	—	2	1	no	—
286	C₂	—	2	1	no	—
287	C₂	—	424	2	no	—
288	C₂	#20274543	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16810733
commutator	C₈×C₅₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1778	C₂
socle	C₂×C₅₃	#1762
prime radical	C₂×#13133239	C₂²
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1778	D₅₃
hypocenter	C₅₃	#12651
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1778

Complements

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

N	H ≅ G/N	type
4. C₂×#16810401	C₂	semidirect
5. #20272826	C₂	semidirect
6. #20272818	C₂	semidirect
7. C₅₃×#1778	C₂	semidirect
16. #16810438	C₄	semidirect
17. #16810435	C₄	semidirect
18. #16810429	C₄	semidirect
144. #1778	D₅₃	semidirect
169. C₅₃	#12651	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810746	2
#20276462	2
#20276442	2
C₂×#16810401	2
#20272826	2
#20272818	2
C₅₃×#1778	2
#12651	53