Subgroups — #27354608

Order 13568.

750 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	#20283427	C₂	1	2	yes	maximal, complemented
2	#20273600	C₂	1	4	yes	maximal, complemented
3	D₈×#88322	C₂	1	4	yes	maximal, complemented
4	#20273474	C₂	1	1	yes	maximal, characteristic, complemented
5	#20273166	C₂	1	2	yes	maximal, complemented
6	C₂×C₈×#88322	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#3169	C₂	1	1	yes	maximal, characteristic
8	C₂×#13133224	—	2	2	no	—
9	#16810701	—	2	2	no	—
10	#16810701	C₂²	1	4	yes	—
11	D₄×#88322	—	2	2	no	—
12	D₄×#88322	C₂²	1	4	yes	—
13	#16810647	C₂²	1	2	yes	—
14	#16810590	—	2	2	no	—
15	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
16	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
17	#16810376	C₂²	1	4	yes	—
18	D₄⋊#88322	C₂²	1	8	yes	—
19	C₈×#88322	C₂²	1	2	yes	—
20	C₈×#88322	C₂²	1	2	yes	—
21	C₅₃×#532	C₂²	1	2	yes	—
22	C₅₃×#500	C₄	1	1	yes	characteristic
23	C₅₃×#493	C₄	1	1	yes	characteristic
24	C₅₃×#486	C₄	1	2	yes	—
25	C₅₃×#466	C₄	1	4	yes	—
26	C₄×D₈×C₅₃	C₂²	1	4	yes	—
27	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
28	#13133224	—	4	8	no	—
29	C₂×#11719917	—	2	4	no	—
30	C₂×#11719917	—	2	2	no	—
31	C₂²×#88322	—	2	4	no	—
32	C₂²×#88322	—	2	1	no	—
33	C₂²×#88322	D₄	1	1	yes	characteristic
34	C₂²×#88322	D₄	1	1	yes	characteristic
35	C₄⋊#88322	—	2	2	no	—
36	C₄⋊#88322	C₂³	1	4	yes	—
37	C₂²⋊#88322	—	2	4	no	—
38	C₂²⋊#88322	—	2	4	no	—
39	C₄×#88322	—	2	1	no	—
40	C₄×#88322	D₄	1	1	yes	characteristic
41	C₄×#88322	D₄	1	1	yes	characteristic
42	C₄×#88322	C₂³	1	2	yes	—
43	C₄×#88322	C₂³	1	1	yes	characteristic
44	C₄×#88322	C₂³	1	1	yes	characteristic
45	C₂×#41×C₅₃	—	2	2	no	—
46	C₂×D₈×C₅₃	C₂×C₄	1	4	yes	—
47	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
48	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
49	#127×C₅₃	C₂×C₄	1	2	yes	—
50	#124×C₅₃	—	2	2	no	—
51	#121×C₅₃	—	2	2	no	—
52	#121×C₅₃	C₂×C₄	1	4	yes	—
53	C₄×D₄×C₅₃	—	2	2	no	—
54	C₄×D₄×C₅₃	C₂³	1	4	yes	—
55	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
56	#107×C₅₃	C₂×C₄	1	4	yes	—
57	#102×C₅₃	C₂×C₄	1	8	yes	—
58	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
59	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
60	C₄×C₈×C₅₃	C₂³	1	2	yes	—
61	#41×C₅₃	—	4	4	no	—
62	#41×C₅₃	—	4	4	no	—
63	C₂×Q₈×C₅₃	—	2	2	no	—
64	C₂×D₄×C₅₃	—	2	4	no	—
65	C₂×D₄×C₅₃	—	2	2	no	—
66	C₂×D₄×C₅₃	—	2	2	no	—
67	C₂×D₄×C₅₃	C₂²×C₄	1	4	yes	—
68	C₂²×C₄×C₅₃	—	2	4	no	—
69	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
72	D₈×C₅₃	—	2	8	no	—
73	C₂×C₈×C₅₃	—	2	4	no	—
74	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
75	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
76	C₄⋊C₄×C₅₃	—	2	2	no	—
77	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
78	C₂²⋊C₄×C₅₃	—	2	4	no	—
79	C₂²⋊C₄×C₅₃	—	2	4	no	—
80	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
81	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₅₃	#41	1	1	yes	characteristic
83	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
84	#11719917	—	4	4	no	—
85	#11719917	—	4	8	no	—
86	C₂×#88322	—	2	4	no	—
87	C₂×#88322	—	2	2	no	—
88	C₂×#88322	—	2	2	no	—
89	C₂×#88322	—	2	4	no	—
90	C₂×#88322	—	4	8	no	—
91	C₂×#88322	D₈	1	4	yes	complemented
92	C₂×#88322	C₂×D₄	1	1	yes	characteristic
93	C₂×#88322	C₂×D₄	1	1	yes	characteristic
94	C₂×#88322	C₂×D₄	1	1	yes	characteristic
95	C₂×#88322	C₂×D₄	1	1	yes	characteristic
96	C₂³×C₅₃	—	2	4	no	—
97	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
98	Q₈×C₅₃	—	4	4	no	—
99	D₄×C₅₃	—	2	8	no	—
100	D₄×C₅₃	—	4	8	no	—
101	D₄×C₅₃	—	4	4	no	—
102	D₄×C₅₃	—	4	4	no	—
103	C₂×C₄×C₅₃	—	2	2	no	—
104	C₂×C₄×C₅₃	—	2	2	no	—
105	C₂×C₄×C₅₃	—	2	4	no	—
106	C₂×C₄×C₅₃	—	4	4	no	—
107	C₂×C₄×C₅₃	—	4	4	no	—
108	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
110	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
112	C₂×C₄×C₅₃	#135	1	2	yes	—
113	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
114	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
115	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
116	C₈×C₅₃	—	2	2	no	—
117	C₈×C₅₃	#131	1	4	yes	complemented
118	#88322	—	2	4	no	—
119	#88322	—	2	1	no	—
120	#88322	—	2	1	no	—
121	#88322	—	4	4	no	—
122	#88322	C₂×D₈	1	2	yes	—
123	#88322	C₂×D₈	1	2	yes	—
124	#17956	—	53	1	no	maximal, Sylow
125	C₂²×C₅₃	—	2	1	no	—
126	C₂²×C₅₃	—	2	1	no	—
127	C₂²×C₅₃	—	4	4	no	—
128	C₂²×C₅₃	—	4	4	no	—
129	C₂²×C₅₃	—	4	4	no	—
130	C₂²×C₅₃	C₄×D₈	1	2	yes	—
131	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
132	C₄×C₅₃	—	2	1	no	—
133	C₄×C₅₃	—	2	1	no	—
134	C₄×C₅₃	—	4	4	no	—
135	C₄×C₅₃	C₄×D₈	1	2	yes	—
136	C₄×C₅₃	#443	1	2	yes	—
137	C₄×C₅₃	C₂×#131	1	2	yes	—
138	C₄×C₅₃	C₂²×D₈	1	1	yes	characteristic
139	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
140	#3169	D₅₃	1	1	yes	characteristic
141	#2931	—	53	2	no	—
142	#1589	—	53	4	no	—
143	C₈×D₈	—	53	4	no	—
144	#1567	—	53	1	no	—
145	#1492	—	53	2	no	—
146	C₂×C₈²	—	53	1	no	—
147	C₂×C₅₃	—	2	2	no	—
148	C₂×C₅₃	—	8	4	no	—
149	C₂×C₅₃	#2931	1	1	yes	characteristic
150	C₂×C₅₃	C₂×C₄×D₈	1	1	yes	characteristic
151	C₂×C₅₃	C₂×#443	1	1	yes	characteristic
152	C₂×#131	—	106	2	no	—
153	#532	C₂×D₅₃	1	2	yes	—
154	#500	Dic₅₃	1	1	yes	characteristic
155	#493	Dic₅₃	1	1	yes	characteristic
156	#486	Dic₅₃	1	2	yes	—
157	#466	Dic₅₃	1	4	yes	—
158	C₄×D₈	C₂×D₅₃	1	4	yes	—
159	#436	—	53	4	no	—
160	#436	—	106	2	no	—
161	C₈×D₄	—	53	4	no	—
162	C₈×D₄	—	106	2	no	—
163	#423	—	53	2	no	—
164	#408	—	106	2	no	—
165	C₂×C₄×C₈	—	53	1	no	—
166	C₂×C₄×C₈	—	53	1	no	—
167	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
168	#335	—	53	4	no	—
169	D₄⋊C₈	—	53	8	no	—
170	C₈²	—	53	2	no	—
171	C₈²	—	53	2	no	—
172	C₅₃	#17956	1	1	yes	characteristic, complemented, Sylow
173	C₂×#41	—	2	2	no	—
174	C₂×D₈	C₂×Dic₅₃	1	4	yes	—
175	#131	—	212	8	no	—
176	C₂×M₁₆	—	106	4	no	—
177	C₂×M₁₆	—	106	2	no	—
178	C₂²×C₈	—	53	1	no	—
179	C₂²×C₈	—	53	1	no	—
180	C₂²×C₈	—	106	4	no	—
181	C₂²×C₈	—	106	1	no	—
182	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
183	C₄⋊Q₈	C₂×Dic₅₃	1	2	yes	—
184	#127	C₂×Dic₅₃	1	2	yes	—
185	#124	—	2	2	no	—
186	#121	—	2	2	no	—
187	#121	C₂×Dic₅₃	1	4	yes	—
188	C₄×D₄	—	2	2	no	—
189	C₄×D₄	C₂²×D₅₃	1	4	yes	—
190	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
191	#107	C₂×Dic₅₃	1	4	yes	—
192	C₄⋊C₈	—	53	4	no	—
193	C₄⋊C₈	—	106	2	no	—
194	#102	C₂×Dic₅₃	1	8	yes	—
195	C₂²⋊C₈	—	106	4	no	—
196	C₂²⋊C₈	—	106	4	no	—
197	C₄×C₈	—	53	1	no	—
198	C₄×C₈	—	53	1	no	—
199	C₄×C₈	—	53	1	no	—
200	C₄×C₈	—	53	1	no	—
201	C₄×C₈	—	53	2	no	—
202	C₄×C₈	—	106	1	no	—
203	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
204	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
205	C₄×C₈	C₂²×D₅₃	1	2	yes	—
206	#41	—	4	4	no	—
207	#41	—	4	4	no	—
208	C₂×Q₈	—	2	2	no	—
209	C₂×D₄	—	2	4	no	—
210	C₂×D₄	—	2	2	no	—
211	C₂×D₄	—	2	2	no	—
212	C₂×D₄	C₂²×Dic₅₃	1	4	yes	—
213	C₂²×C₄	—	2	4	no	—
214	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
215	C₂²×C₄	#11719947	1	1	yes	characteristic
216	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
217	D₈	—	2	8	no	—
218	M₁₆	—	212	8	no	—
219	M₁₆	—	212	4	no	—
220	C₂×C₈	—	2	4	no	—
221	C₂×C₈	—	53	4	no	—
222	C₂×C₈	—	53	1	no	—
223	C₂×C₈	—	53	1	no	—
224	C₂×C₈	—	53	1	no	—
225	C₂×C₈	—	53	1	no	—
226	C₂×C₈	—	106	4	no	—
227	C₂×C₈	—	106	2	no	—
228	C₂×C₈	—	106	2	no	—
229	C₂×C₈	—	106	4	no	—
230	C₂×C₈	—	212	8	no	—
231	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
232	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
233	C₄⋊C₄	—	2	2	no	—
234	C₄⋊C₄	C₂²×Dic₅₃	1	4	yes	—
235	C₂²⋊C₄	—	2	4	no	—
236	C₂²⋊C₄	—	2	4	no	—
237	C₄²	D₄×D₅₃	1	1	yes	characteristic
238	C₄²	#11719947	1	1	yes	characteristic
239	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
240	C₄²	C₂³×D₅₃	1	1	yes	characteristic
241	C₂³	—	2	4	no	—
242	C₂³	D₄×Dic₅₃	1	1	yes	characteristic
243	Q₈	—	4	4	no	—
244	D₄	—	2	8	no	—
245	D₄	—	4	8	no	—
246	D₄	—	4	4	no	—
247	D₄	—	4	4	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	4	4	no	—
252	C₂×C₄	—	4	4	no	—
253	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
254	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
255	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
256	C₂×C₄	D₈×D₅₃	1	2	yes	—
257	C₂×C₄	#13133236	1	2	yes	—
258	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
259	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
260	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
261	C₈	—	2	2	no	—
262	C₈	—	53	2	no	—
263	C₈	—	53	2	no	—
264	C₈	—	106	4	no	—
265	C₈	—	106	1	no	—
266	C₈	—	106	1	no	—
267	C₈	—	212	4	no	—
268	C₈	#13133224	1	4	yes	complemented
269	C₂²	—	2	1	no	—
270	C₂²	—	2	1	no	—
271	C₂²	—	4	4	no	—
272	C₂²	—	4	4	no	—
273	C₂²	—	4	4	no	—
274	C₂²	D₈×Dic₅₃	1	2	yes	—
275	C₂²	C₂×D₄×Dic₅₃	1	1	yes	characteristic
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	4	4	no	—
279	C₄	D₈×Dic₅₃	1	2	yes	—
280	C₄	#16810740	1	2	yes	—
281	C₄	C₂×#13133224	1	2	yes	—
282	C₄	C₂×D₈×D₅₃	1	1	yes	characteristic
283	C₄	C₂×#13133236	1	1	yes	characteristic
284	C₂	—	2	2	no	—
285	C₂	—	8	4	no	—
286	C₂	#20283427	1	1	yes	characteristic
287	C₂	C₂×D₈×Dic₅₃	1	1	yes	characteristic
288	C₂	C₂×#16810740	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting, prime radical	C₅₃×#3169	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#3169	D₅₃
hypocenter	C₅₃	#17956
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3169

Complements

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

N	H ≅ G/N	type
1. #20283427	C₂	semidirect
2. #20273600	C₂	semidirect
3. D₈×#88322	C₂	semidirect
4. #20273474	C₂	semidirect
5. #20273166	C₂	semidirect
6. C₂×C₈×#88322	C₂	semidirect
91. C₂×#88322	D₈	semidirect
117. C₈×C₅₃	#131	semidirect
172. C₅₃	#17956	semidirect
268. C₈	#13133224	semidirect

Maximal subgroups

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

subgroup	index
#20283427	2
#20273600	2
D₈×#88322	2
#20273474	2
#20273166	2
C₂×C₈×#88322	2
C₅₃×#3169	2
#17956	53