Subgroups — #28328547

Order 13568.

452 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20291132	C₂	1	2	yes	maximal, complemented
2	#20291132	C₂	1	2	yes	maximal, complemented
3	#20291131	C₂	1	2	yes	maximal, complemented
4	#20291129	C₂	1	2	yes	maximal, complemented
5	#20291081	C₂	1	2	yes	maximal
6	#20291080	C₂	1	2	yes	maximal
7	#20291077	C₂	1	1	yes	maximal, characteristic
8	#20291073	C₂	1	1	yes	maximal, characteristic
9	#20283648	C₂	1	1	yes	maximal, characteristic
10	#16811544	C₂²	1	2	yes	—
11	#16811544	C₂²	1	2	yes	—
12	#16811544	C₂²	1	2	yes	—
13	#16811544	C₂²	1	2	yes	—
14	C₄⋊#11719910	C₂²	1	2	yes	—
15	C₄⋊#11719910	C₂²	1	2	yes	—
16	C₂²⋊#11719910	C₂²	1	2	yes	—
17	C₂²⋊#11719910	C₂²	1	2	yes	—
18	#16811512	C₂²	1	1	yes	characteristic
19	#16811512	C₂²	1	1	yes	characteristic
20	#16811512	C₂²	1	2	yes	—
21	#16811511	C₂²	1	2	yes	—
22	#16811509	C₂²	1	1	yes	characteristic
23	#16811509	C₂²	1	1	yes	characteristic
24	#16811509	C₂²	1	2	yes	—
25	C₄×#11719910	C₂²	1	2	yes	—
26	#16811176	C₄	1	1	yes	characteristic
27	#16810752	C₄	1	2	yes	—
28	#16810752	C₄	1	2	yes	—
29	#16810747	C₂²	1	2	yes	—
30	#16810747	C₂²	1	2	yes	—
31	#16810649	C₄	1	1	yes	characteristic
32	#16810646	C₄	1	1	yes	characteristic
33	#16810581	C₂²	1	1	yes	characteristic
34	#16810579	C₂²	1	1	yes	characteristic
35	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
36	C₅₃×#424	C₄	1	1	yes	characteristic
37	C₂×#11719910	C₂³	1	2	yes	—
38	C₂×#11719910	C₂³	1	2	yes	—
39	C₂×#11719910	C₂³	1	2	yes	—
40	C₂×#11719910	C₂³	1	2	yes	—
41	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
42	C₂²×#88322	C₂³	1	1	yes	characteristic
43	#13133216	C₈	1	1	yes	characteristic
44	#13133216	C₈	1	1	yes	characteristic
45	#13133208	C₈	1	1	yes	characteristic
46	#13133208	C₈	1	1	yes	characteristic
47	#13133194	C₈	1	2	yes	—
48	#13133194	C₈	1	2	yes	—
49	C₄×Dic₁₀₆	C₂×C₄	1	2	yes	—
50	C₄×Dic₁₀₆	C₂×C₄	1	2	yes	—
51	#13133176	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
53	#13133146	C₂×C₄	1	2	yes	—
54	#13133146	C₂×C₄	1	2	yes	—
55	#13133146	C₂×C₄	1	2	yes	—
56	#13133146	C₂×C₄	1	2	yes	—
57	#13133128	C₂×C₄	1	2	yes	—
58	#13133127	C₂×C₄	1	2	yes	—
59	C₄×#88322	C₂³	1	2	yes	—
60	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
61	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
62	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
63	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
64	C₈×Dic₅₃	C₂³	1	2	yes	—
65	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
66	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
67	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
68	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
69	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
70	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
71	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
72	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
73	C₂×Dic₁₀₆	C₂×C₈	1	2	yes	—
74	C₂×Dic₁₀₆	C₂×C₈	1	2	yes	—
75	M₁₆×C₅₃	—	4	1	no	—
76	C₂×C₈×C₅₃	—	4	1	no	—
77	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
80	C₄⋊C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
81	C₄⋊C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
82	C₂²⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
83	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
84	C₂²⋊Dic₅₃	C₂×C₈	1	2	yes	—
85	C₄⋊Dic₅₃	C₂×C₈	1	1	yes	characteristic
86	C₄⋊Dic₅₃	C₂×C₈	1	1	yes	characteristic
87	C₄⋊Dic₅₃	C₂×C₈	1	2	yes	—
88	C₄⋊Dic₅₃	C₂×C₈	1	2	yes	—
89	#11719919	C₂×C₈	1	2	yes	—
90	#11719919	C₂×C₈	1	2	yes	—
91	#11719919	C₂×C₈	1	2	yes	—
92	#11719919	C₂×C₈	1	2	yes	—
93	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
94	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
95	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
96	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
97	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
98	#11719917	—	4	1	no	—
99	C₂×#88322	—	2	2	no	—
100	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
101	C₂×#88322	C₂²×C₄	1	2	yes	—
102	C₂×#88322	C₂⁴	1	1	yes	characteristic
103	#11719910	—	2	2	no	—
104	#11719910	—	2	2	no	—
105	#11719910	—	2	2	no	—
106	#11719910	—	2	2	no	—
107	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
108	C₂×C₄×C₅₃	—	2	1	no	—
109	C₂×C₄×C₅₃	—	2	1	no	—
110	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
111	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
112	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
113	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
114	C₂×Dic₅₃	—	2	2	no	—
115	C₂×Dic₅₃	—	4	1	no	—
116	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
117	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
118	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
119	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
120	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
121	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
122	Dic₁₀₆	—	2	4	no	—
123	Dic₁₀₆	—	4	2	no	—
124	C₈×C₅₃	—	2	1	no	—
125	C₈×C₅₃	—	2	1	no	—
126	C₈×C₅₃	—	2	2	no	—
127	#88322	—	2	1	no	—
128	#88322	—	2	2	no	—
129	#88322	2₋¹⁺⁴	1	2	yes	—
130	#33499	—	53	1	no	maximal, Sylow
131	C₂²×C₅₃	—	2	1	no	—
132	C₂²×C₅₃	—	2	1	no	—
133	C₂²×C₅₃	—	2	1	no	—
134	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
135	C₄×C₅₃	—	2	2	no	—
136	C₄×C₅₃	—	2	2	no	—
137	C₄×C₅₃	—	2	1	no	—
138	C₄×C₅₃	#519	1	1	yes	characteristic
139	C₄×C₅₃	#568	1	1	yes	characteristic
140	Dic₅₃	—	2	2	no	—
141	Dic₅₃	—	2	1	no	—
142	Dic₅₃	—	2	1	no	—
143	Dic₅₃	—	2	2	no	—
144	Dic₅₃	—	2	1	no	—
145	Dic₅₃	#504	1	2	yes	—
146	#2960	—	53	1	no	—
147	#2183	—	53	2	no	—
148	#2183	—	53	2	no	—
149	#2183	—	53	2	no	—
150	Q₈×C₁₆	—	53	2	no	—
151	#2164	—	53	2	no	—
152	#2164	—	53	2	no	—
153	#2163	—	53	1	no	—
154	#2162	—	53	1	no	—
155	C₂×C₅₃	—	4	1	no	—
156	C₂×C₅₃	#2982	1	1	yes	characteristic
157	C₂×C₅₃	C₂×#504	1	1	yes	characteristic
158	C₂×C₅₃	#3407	1	1	yes	characteristic
159	#533	—	53	1	no	—
160	#446	—	53	2	no	—
161	#446	—	53	2	no	—
162	C₈×Q₈	—	53	2	no	—
163	C₈×Q₈	—	53	2	no	—
164	#424	—	53	1	no	—
165	#424	#6157	1	1	yes	characteristic
166	#423	—	53	1	no	—
167	#405	—	53	1	no	—
168	#405	—	53	1	no	—
169	C₂×C₄×C₈	—	53	1	no	—
170	C₄⋊C₁₆	—	53	2	no	—
171	C₄⋊C₁₆	—	53	2	no	—
172	C₄⋊C₁₆	—	53	2	no	—
173	C₄⋊C₁₆	—	53	2	no	—
174	C₄⋊C₁₆	—	53	2	no	—
175	C₄⋊C₁₆	—	53	2	no	—
176	C₂²⋊C₁₆	—	53	2	no	—
177	C₂²⋊C₁₆	—	53	2	no	—
178	#346	—	53	1	no	—
179	#346	—	53	1	no	—
180	#346	—	53	2	no	—
181	#346	—	53	2	no	—
182	C₄×C₁₆	—	53	1	no	—
183	C₄×C₁₆	—	53	1	no	—
184	C₄×C₁₆	—	53	2	no	—
185	C₄×C₁₆	—	53	2	no	—
186	C₅₃	#33499	1	1	yes	characteristic, complemented, Sylow
187	C₂×M₁₆	—	53	1	no	—
188	C₂×M₁₆	C₂×#6157	1	1	yes	characteristic
189	C₂²×C₈	—	53	1	no	—
190	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
191	C₄⋊Q₈	—	53	1	no	—
192	C₄⋊Q₈	—	53	1	no	—
193	#125	—	53	1	no	—
194	#125	—	53	1	no	—
195	C₂²⋊Q₈	—	53	2	no	—
196	C₂²⋊Q₈	—	53	2	no	—
197	C₄×Q₈	—	53	2	no	—
198	C₄×Q₈	—	53	2	no	—
199	#117	—	53	1	no	—
200	#117	C₂×#6157	1	1	yes	characteristic
201	C₂×C₄²	—	53	1	no	—
202	C₂×C₁₆	—	53	2	no	—
203	C₂×C₁₆	—	53	2	no	—
204	C₂×C₁₆	—	53	2	no	—
205	C₂×C₁₆	—	53	2	no	—
206	C₄⋊C₈	—	53	2	no	—
207	C₄⋊C₈	—	53	2	no	—
208	C₄⋊C₈	—	53	2	no	—
209	C₄⋊C₈	—	53	2	no	—
210	C₄⋊C₈	C₂×#6157	1	2	yes	—
211	C₄⋊C₈	C₂×#6157	1	2	yes	—
212	#97	—	53	2	no	—
213	#97	—	53	2	no	—
214	C₄×C₈	—	53	1	no	—
215	C₄×C₈	—	53	1	no	—
216	C₄×C₈	—	53	1	no	—
217	C₄×C₈	—	53	2	no	—
218	C₄×C₈	—	53	1	no	—
219	C₄×C₈	—	53	2	no	—
220	C₂×Q₈	—	53	2	no	—
221	C₂×Q₈	—	53	2	no	—
222	C₂²×C₄	—	53	1	no	—
223	C₂²×C₄	—	53	1	no	—
224	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
225	M₁₆	—	4	1	no	—
226	M₁₆	—	212	1	no	—
227	C₂×C₈	—	4	1	no	—
228	C₂×C₈	—	53	1	no	—
229	C₂×C₈	—	53	1	no	—
230	C₂×C₈	—	53	2	no	—
231	C₂×C₈	—	106	2	no	—
232	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
233	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
234	C₂×C₈	C₂²×#6157	1	2	yes	—
235	C₄⋊C₄	—	53	1	no	—
236	C₄⋊C₄	—	53	1	no	—
237	C₄⋊C₄	—	53	2	no	—
238	C₄⋊C₄	—	53	2	no	—
239	C₄⋊C₄	—	53	2	no	—
240	C₄⋊C₄	—	53	2	no	—
241	C₄⋊C₄	—	53	2	no	—
242	C₄⋊C₄	—	53	2	no	—
243	C₄⋊C₄	D₅₃⋊C₈	1	1	yes	characteristic
244	C₄⋊C₄	D₅₃⋊C₈	1	1	yes	characteristic
245	C₂²⋊C₄	—	53	2	no	—
246	C₂²⋊C₄	D₅₃⋊C₈	1	2	yes	—
247	C₄²	—	53	1	no	—
248	C₄²	—	53	1	no	—
249	C₄²	—	53	1	no	—
250	C₄²	—	53	1	no	—
251	C₄²	—	53	2	no	—
252	C₄²	C₂²×#6157	1	2	yes	—
253	C₁₆	—	106	2	no	—
254	C₁₆	—	106	2	no	—
255	C₁₆	—	106	2	no	—
256	C₁₆	—	106	2	no	—
257	C₂³	C₂×D₅₃⋊C₈	1	1	yes	characteristic
258	Q₈	—	106	4	no	—
259	Q₈	—	212	2	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	53	1	no	—
263	C₂×C₄	—	53	1	no	—
264	C₂×C₄	—	53	1	no	—
265	C₂×C₄	—	53	1	no	—
266	C₂×C₄	—	53	2	no	—
267	C₂×C₄	—	53	2	no	—
268	C₂×C₄	—	106	2	no	—
269	C₂×C₄	—	212	1	no	—
270	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
271	C₂×C₄	C₂×D₅₃⋊C₈	1	2	yes	—
272	C₂×C₄	C₂×D₅₃⋊C₈	1	2	yes	—
273	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
274	C₈	—	2	1	no	—
275	C₈	—	2	1	no	—
276	C₈	—	2	2	no	—
277	C₈	—	53	2	no	—
278	C₈	—	106	1	no	—
279	C₈	—	106	2	no	—
280	C₂²	—	2	1	no	—
281	C₂²	—	2	1	no	—
282	C₂²	—	2	1	no	—
283	C₂²	C₂²×D₅₃⋊C₈	1	1	yes	characteristic
284	C₄	—	2	1	no	—
285	C₄	—	2	2	no	—
286	C₄	—	2	2	no	—
287	C₄	—	53	2	no	—
288	C₄	—	106	1	no	—
289	C₄	—	106	2	no	—
290	C₄	—	106	1	no	—
291	C₄	—	106	1	no	—
292	C₄	—	106	2	no	—
293	C₄	#16811691	1	1	yes	characteristic
294	C₄	#16811701	1	1	yes	characteristic
295	C₂	—	4	1	no	—
296	C₂	#20291610	1	1	yes	characteristic
297	C₂	D₅₃⋊#504	1	1	yes	characteristic
298	C₂	#20291683	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#6157
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	C₅₃×#424	C₄
prime radical	C₂³×C₅₃	C₂²×C₈
cosocle	C₂×#88322	C₂⁴
hypercenter	#424	#6157
hypocenter	C₅₃	#33499
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #424

Complements

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

N	H ≅ G/N	type
1. #20291132	C₂	semidirect
2. #20291132	C₂	semidirect
3. #20291131	C₂	semidirect
4. #20291129	C₂	semidirect
186. C₅₃	#33499	semidirect

Maximal subgroups

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

subgroup	index
#20291132	2
#20291131	2
#20291129	2
#20291081	2
#20291080	2
#20291077	2
#20291073	2
#20283648	2
#33499	53