Subgroups — #28305605

Order 13568.

404 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20290647	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#88324	C₂	1	1	yes	maximal, characteristic
3	C₂×#16811487	C₂	1	2	yes	maximal
4	#20289545	C₂	1	2	yes	maximal
5	C₄⋊C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#88324	—	2	1	no	—
7	C₂×C₄⋊#88324	—	2	1	no	—
8	C₂×C₄⋊#88324	C₂²	1	2	yes	—
9	C₂×C₄⋊#88324	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊#88324	C₂²	1	1	yes	characteristic
11	C₂×C₄×#88324	—	2	1	no	—
12	#16811489	—	2	1	no	—
13	#16811487	—	2	4	no	—
14	C₂×C₄⋊#88322	C₄	1	1	yes	characteristic
15	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
16	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
17	#16810377	C₄	1	2	yes	—
18	C₂×C₄⋊C₈×C₅₃	C₄	1	1	yes	characteristic
19	C₂²×#88324	—	2	1	no	—
20	C₂²×#88324	—	2	1	no	—
21	C₂²×#88324	—	2	2	no	—
22	C₄⋊#88324	—	2	1	no	—
23	C₄⋊#88324	—	2	1	no	—
24	C₄⋊#88324	—	2	2	no	—
25	C₄⋊#88324	—	2	2	no	—
26	C₄⋊#88324	—	4	2	no	—
27	C₄⋊#88324	—	4	2	no	—
28	C₄⋊#88324	D₄	1	2	yes	—
29	C₄⋊#88324	Q₈	1	2	yes	—
30	C₄×#88324	—	4	2	no	—
31	C₂²×#88322	C₂×C₄	1	2	yes	—
32	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
34	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
35	C₂×C₄×Dic₅₃	Q₈	1	1	yes	characteristic
36	C₂×C₄×Dic₅₃	Q₈	1	1	yes	characteristic
37	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
38	C₄⋊#88322	C₈	1	4	yes	complemented
39	C₈×Dic₅₃	—	2	2	no	—
40	C₈×Dic₅₃	—	2	2	no	—
41	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₄⋊C₈×C₅₃	—	2	2	no	—
44	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
47	C₂²×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
48	C₂²×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
49	C₂²×Dic₅₃	C₂×D₄	1	1	yes	characteristic
50	C₂²×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
51	C₂×#88324	—	2	2	no	—
52	C₂×#88324	—	2	2	no	—
53	C₂×#88324	—	4	1	no	—
54	C₂×#88324	—	4	1	no	—
55	C₂×#88324	—	4	1	no	—
56	C₂×#88324	—	4	1	no	—
57	C₂×#88324	—	4	2	no	—
58	C₂×#88324	—	4	2	no	—
59	C₂×#88324	—	4	2	no	—
60	C₂×C₈×C₅₃	—	2	2	no	—
61	C₂×C₈×C₅₃	—	2	2	no	—
62	C₂×C₈×C₅₃	—	2	2	no	—
63	C₄²×C₅₃	C₂×C₈	1	2	yes	—
64	C₄²×C₅₃	M₁₆	1	2	yes	—
65	C₄×Dic₅₃	—	2	2	no	—
66	C₄×Dic₅₃	—	2	1	no	—
67	C₄×Dic₅₃	—	2	1	no	—
68	C₄×Dic₅₃	—	2	1	no	—
69	C₄×Dic₅₃	—	2	1	no	—
70	C₄×Dic₅₃	—	4	1	no	—
71	C₄×Dic₅₃	D₈	1	2	yes	—
72	C₄×Dic₅₃	Q₁₆	1	2	yes	—
73	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
74	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
75	C₄×Dic₅₃	#41	1	1	yes	characteristic
76	C₄×Dic₅₃	#41	1	1	yes	characteristic
77	C₂×#88322	—	2	2	no	—
78	C₂×#88322	—	2	2	no	—
79	C₂×#88322	C₂×C₈	1	4	yes	—
80	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₅₃	—	2	1	no	—
82	C₂×C₄×C₅₃	—	2	1	no	—
83	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
84	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
85	C₂×C₄×C₅₃	#107	1	2	yes	—
86	C₂×C₄×C₅₃	#108	1	2	yes	—
87	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
88	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
89	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
90	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
91	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
92	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
93	C₂×Dic₅₃	—	2	1	no	—
94	C₂×Dic₅₃	—	2	1	no	—
95	C₂×Dic₅₃	—	2	1	no	—
96	C₂×Dic₅₃	—	2	1	no	—
97	C₂×Dic₅₃	—	2	1	no	—
98	C₂×Dic₅₃	—	2	1	no	—
99	C₂×Dic₅₃	—	2	1	no	—
100	C₂×Dic₅₃	#107	1	2	yes	—
101	C₂×Dic₅₃	#121	1	1	yes	characteristic
102	C₂×Dic₅₃	C₂²⋊Q₈	1	1	yes	characteristic
103	C₂×Dic₅₃	#125	1	1	yes	characteristic
104	C₂×Dic₅₃	C₄⋊Q₈	1	1	yes	characteristic
105	C₂×Dic₅₃	C₂×D₈	1	1	yes	characteristic
106	C₂×Dic₅₃	C₂×Q₁₆	1	1	yes	characteristic
107	C₂×Dic₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
108	C₂×Dic₅₃	#137	1	1	yes	characteristic
109	#88324	—	4	2	no	—
110	#88324	—	4	2	no	—
111	#88324	—	8	2	no	—
112	C₈×C₅₃	—	4	2	no	—
113	#88322	—	2	4	no	—
114	#9601	—	53	1	no	maximal, Sylow
115	C₂²×C₅₃	#389	1	1	yes	characteristic
116	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
117	C₂²×C₅₃	#424	1	1	yes	characteristic
118	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
119	C₂²×C₅₃	#428	1	1	yes	characteristic
120	C₂²×C₅₃	C₂×#108	1	1	yes	characteristic
121	C₂²×C₅₃	#430	1	1	yes	characteristic
122	C₄×C₅₃	—	2	2	no	—
123	C₄×C₅₃	#335	1	4	yes	complemented
124	C₄×C₅₃	C₈×D₄	1	1	yes	characteristic
125	C₄×C₅₃	#436	1	1	yes	characteristic
126	C₄×C₅₃	C₈×Q₈	1	1	yes	characteristic
127	C₄×C₅₃	#446	1	1	yes	characteristic
128	Dic₅₃	—	2	2	no	—
129	Dic₅₃	—	2	1	no	—
130	Dic₅₃	—	2	1	no	—
131	Dic₅₃	—	4	1	no	—
132	Dic₅₃	#459	1	1	yes	characteristic
133	Dic₅₃	#462	1	1	yes	characteristic
134	Dic₅₃	#474	1	1	yes	characteristic
135	Dic₅₃	#477	1	1	yes	characteristic
136	#1842	—	53	1	no	—
137	C₄×C₄⋊C₈	—	53	1	no	—
138	C₄×C₄⋊C₈	—	53	1	no	—
139	C₂×#335	—	53	2	no	—
140	#1277	—	53	2	no	—
141	C₂×C₅₃	C₂×#335	1	1	yes	characteristic
142	C₂×C₅₃	#1565	1	1	yes	characteristic
143	C₂×C₅₃	#1571	1	1	yes	characteristic
144	C₂×C₅₃	#1574	1	1	yes	characteristic
145	C₂×C₅₃	#1939	1	1	yes	characteristic
146	C₂×C₅₃	#1947	1	1	yes	characteristic
147	C₂×C₅₃	#1950	1	1	yes	characteristic
148	C₂×C₄⋊C₈	—	53	1	no	—
149	C₂×C₄⋊C₈	—	53	1	no	—
150	C₂×C₄⋊C₈	—	53	1	no	—
151	C₂×C₄⋊C₈	—	53	2	no	—
152	C₂×C₄⋊C₈	—	106	1	no	—
153	C₂×C₄⋊C₈	—	106	1	no	—
154	C₂×C₄⋊C₈	#6157	1	1	yes	characteristic
155	C₂×C₄×C₈	—	53	2	no	—
156	C₂×C₄×C₈	—	106	1	no	—
157	C₄³	—	53	1	no	—
158	#336	—	53	2	no	—
159	#336	—	106	1	no	—
160	#335	—	106	4	no	—
161	C₅₃	#9601	1	1	yes	characteristic, complemented, Sylow
162	C₂²×C₈	—	53	2	no	—
163	C₂²×C₈	—	106	1	no	—
164	C₂²×C₈	—	106	1	no	—
165	C₂²×C₈	—	106	2	no	—
166	C₂²×C₈	C₂×#6157	1	2	yes	—
167	C₂×C₄²	—	53	1	no	—
168	C₂×C₄²	—	53	1	no	—
169	C₂×C₄²	—	53	1	no	—
170	C₂×C₄²	—	53	1	no	—
171	C₂×C₄²	—	53	1	no	—
172	C₂×C₄²	—	53	1	no	—
173	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
174	C₄⋊C₈	—	2	2	no	—
175	C₄⋊C₈	—	53	2	no	—
176	C₄⋊C₈	—	53	2	no	—
177	C₄⋊C₈	—	53	4	no	—
178	C₄⋊C₈	—	106	1	no	—
179	C₄⋊C₈	—	106	1	no	—
180	C₄⋊C₈	—	106	2	no	—
181	C₄⋊C₈	—	106	2	no	—
182	C₄⋊C₈	—	212	2	no	—
183	C₄⋊C₈	—	212	2	no	—
184	C₄×C₈	—	106	2	no	—
185	C₄×C₈	—	106	2	no	—
186	C₄×C₈	—	212	2	no	—
187	C₂²×C₄	—	53	1	no	—
188	C₂²×C₄	—	53	1	no	—
189	C₂²×C₄	—	53	1	no	—
190	C₂²×C₄	—	53	1	no	—
191	C₂²×C₄	C₄⋊#6157	1	1	yes	characteristic
192	C₂²×C₄	C₄⋊#6157	1	1	yes	characteristic
193	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	53	4	no	—
198	C₂×C₈	—	106	2	no	—
199	C₂×C₈	—	106	2	no	—
200	C₂×C₈	—	106	2	no	—
201	C₂×C₈	—	106	2	no	—
202	C₂×C₈	—	212	2	no	—
203	C₂×C₈	—	212	2	no	—
204	C₂×C₈	—	212	2	no	—
205	C₂×C₈	—	212	1	no	—
206	C₂×C₈	—	212	1	no	—
207	C₂×C₈	—	212	1	no	—
208	C₂×C₈	—	212	1	no	—
209	C₄²	—	53	2	no	—
210	C₄²	—	53	2	no	—
211	C₄²	—	53	1	no	—
212	C₄²	—	53	1	no	—
213	C₄²	—	53	1	no	—
214	C₄²	—	53	1	no	—
215	C₄²	—	106	1	no	—
216	C₄²	—	106	1	no	—
217	C₄²	—	106	2	no	—
218	C₄²	—	106	1	no	—
219	C₄²	—	106	1	no	—
220	C₄²	—	212	1	no	—
221	C₄²	D₅₃⋊C₈	1	2	yes	—
222	C₄²	#11719940	1	2	yes	—
223	C₂³	C₂×C₄⋊#6157	1	1	yes	characteristic
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	53	1	no	—
227	C₂×C₄	—	53	1	no	—
228	C₂×C₄	—	53	1	no	—
229	C₂×C₄	—	53	1	no	—
230	C₂×C₄	—	53	1	no	—
231	C₂×C₄	—	53	1	no	—
232	C₂×C₄	—	53	1	no	—
233	C₂×C₄	—	53	1	no	—
234	C₂×C₄	—	53	2	no	—
235	C₂×C₄	—	106	1	no	—
236	C₂×C₄	—	106	1	no	—
237	C₂×C₄	—	106	1	no	—
238	C₂×C₄	—	106	1	no	—
239	C₂×C₄	—	106	1	no	—
240	C₂×C₄	—	106	1	no	—
241	C₂×C₄	—	106	1	no	—
242	C₂×C₄	#13133284	1	2	yes	—
243	C₂×C₄	#13133284	1	2	yes	—
244	C₂×C₄	#13133286	1	2	yes	—
245	C₂×C₄	#13133287	1	2	yes	—
246	C₂×C₄	D₄×#6157	1	1	yes	characteristic
247	C₂×C₄	Q₈×#6157	1	1	yes	characteristic
248	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
249	C₂×C₄	C₂×#11719940	1	1	yes	characteristic
250	C₂×C₄	#13133304	1	1	yes	characteristic
251	C₂×C₄	#13133305	1	1	yes	characteristic
252	C₈	—	4	2	no	—
253	C₈	—	106	4	no	—
254	C₈	—	424	2	no	—
255	C₈	—	212	2	no	—
256	C₈	—	212	2	no	—
257	C₂²	#16811574	1	1	yes	characteristic
258	C₂²	C₂×#13133284	1	1	yes	characteristic
259	C₂²	#16811630	1	1	yes	characteristic
260	C₂²	C₂×#13133286	1	1	yes	characteristic
261	C₂²	#16811634	1	1	yes	characteristic
262	C₂²	C₂×#13133287	1	1	yes	characteristic
263	C₂²	#16811638	1	1	yes	characteristic
264	C₄	—	2	2	no	—
265	C₄	—	53	1	no	—
266	C₄	—	53	1	no	—
267	C₄	—	53	1	no	—
268	C₄	—	53	1	no	—
269	C₄	—	106	1	no	—
270	C₄	—	106	1	no	—
271	C₄	—	106	2	no	—
272	C₄	—	212	1	no	—
273	C₄	#16811487	1	4	yes	complemented
274	C₄	#16811651	1	1	yes	characteristic
275	C₄	#16811658	1	1	yes	characteristic
276	C₄	#16811672	1	1	yes	characteristic
277	C₄	#16811674	1	1	yes	characteristic
278	C₂	C₂×#16811487	1	1	yes	characteristic
279	C₂	#20290268	1	1	yes	characteristic
280	C₂	#20290285	1	1	yes	characteristic
281	C₂	#20290292	1	1	yes	characteristic
282	C₂	#20290813	1	1	yes	characteristic
283	C₂	#20290826	1	1	yes	characteristic
284	C₂	#20290830	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₄⋊#6157
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#6157
Fitting	C₂×C₄⋊C₈×C₅₃	C₄
socle, prime radical	C₂³×C₅₃	C₂×C₄⋊C₄
cosocle	C₂×C₄×Dic₅₃	C₂³
hypercenter	C₂×C₄⋊C₈	#6157
hypocenter	C₅₃	#9601
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
38. C₄⋊#88322	C₈	semidirect
123. C₄×C₅₃	#335	semidirect
161. C₅₃	#9601	semidirect
273. C₄	#16811487	semidirect

Maximal subgroups

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

subgroup	index
#20290647	2
C₄×C₄⋊#88324	2
C₂×#16811487	2
#20289545	2
C₄⋊C₈×Dic₅₃	2
#9601	53