Subgroups — #28328431

Order 13568.

490 conjugacy classes of subgroups.

The subgroup lattice has 310 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	#20291130	C₂	1	2	yes	maximal, complemented
5	#20291051	C₂	1	2	yes	maximal
6	#20291048	C₂	1	1	yes	maximal, characteristic
7	#20290925	C₂	1	2	yes	maximal
8	#20290916	C₂	1	1	yes	maximal, characteristic
9	#20283646	C₂	1	1	yes	maximal, characteristic
10	C₂×#13133292	C₂²	1	2	yes	—
11	C₂×D₅₃⋊C₁₆	C₂²	1	1	yes	characteristic
12	C₂×D₅₃⋊C₁₆	C₂²	1	1	yes	characteristic
13	#16811544	C₂²	1	1	yes	characteristic
14	#16811544	C₂²	1	2	yes	—
15	#16811544	C₂²	1	1	yes	characteristic
16	#16811544	C₂²	1	2	yes	—
17	C₄⋊#11719910	C₂²	1	2	yes	—
18	C₄⋊#11719910	C₂²	1	2	yes	—
19	C₄⋊#11719910	C₂²	1	2	yes	—
20	#16811512	C₂²	1	2	yes	—
21	#16811512	C₂²	1	2	yes	—
22	#16811511	C₂²	1	2	yes	—
23	#16811511	C₂²	1	2	yes	—
24	#16811509	C₂²	1	2	yes	—
25	C₄×#11719910	C₂²	1	2	yes	—
26	#16811177	C₄	1	1	yes	characteristic
27	#16810753	C₄	1	1	yes	characteristic
28	#16810751	C₄	1	2	yes	—
29	Q₈×#88322	C₂²	1	1	yes	characteristic
30	#16810747	C₂²	1	2	yes	—
31	#16810747	C₂²	1	1	yes	characteristic
32	#16810648	C₄	1	2	yes	—
33	#16810645	C₄	1	1	yes	characteristic
34	#16810580	C₂²	1	2	yes	—
35	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
36	C₅₃×#446	C₄	1	1	yes	characteristic
37	#13133292	—	2	4	no	—
38	D₅₃⋊C₁₆	—	2	2	no	—
39	D₅₃⋊C₁₆	Q₈	1	4	yes	complemented
40	C₂×#11719910	C₂³	1	2	yes	—
41	C₂×#11719910	C₂³	1	2	yes	—
42	C₂×#11719910	C₂³	1	2	yes	—
43	C₂×#11719910	C₂³	1	2	yes	—
44	C₂×#11719912	C₂×C₄	1	2	yes	—
45	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
47	#13133216	C₈	1	1	yes	characteristic
48	#13133215	C₈	1	1	yes	characteristic
49	#13133207	C₈	1	2	yes	—
50	#13133197	C₈	1	1	yes	characteristic
51	#13133195	C₈	1	2	yes	—
52	#13133193	C₈	1	1	yes	characteristic
53	C₄×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
54	C₄×Dic₁₀₆	C₂×C₄	1	2	yes	—
55	#13133174	C₂×C₄	1	2	yes	—
56	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
57	C₄⋊#88322	C₂×C₄	1	1	yes	characteristic
58	C₄⋊#88322	C₂×C₄	1	2	yes	—
59	#13133146	C₂×C₄	1	2	yes	—
60	#13133146	C₂×C₄	1	2	yes	—
61	#13133146	C₂×C₄	1	2	yes	—
62	#13133127	C₂×C₄	1	2	yes	—
63	C₄×#88322	C₂³	1	1	yes	characteristic
64	C₄×#88322	C₂³	1	2	yes	—
65	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
66	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
67	C₈×Dic₅₃	C₂³	1	2	yes	—
68	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
69	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
70	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
71	#97×C₅₃	C₂×C₄	1	2	yes	—
72	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
73	C₂×Q₈×C₅₃	C₂×C₈	1	1	yes	characteristic
74	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
75	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
76	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
77	C₂×Dic₁₀₆	C₂×C₈	1	2	yes	—
78	C₂×Dic₁₀₆	C₂×C₈	1	1	yes	characteristic
79	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
82	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
83	C₄⋊C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
84	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
85	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
86	#11719921	C₂×C₈	1	2	yes	—
87	#11719921	C₂×C₈	1	2	yes	—
88	C₄⋊Dic₅₃	C₂×C₈	1	1	yes	characteristic
89	C₄⋊Dic₅₃	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	#11719919	C₂×C₈	1	2	yes	—
94	#11719919	C₂×C₈	1	2	yes	—
95	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
96	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
97	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
98	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
99	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
100	C₂×#88322	C₂²×C₄	1	2	yes	—
101	C₂×#88322	C₂⁴	1	1	yes	characteristic
102	#11719912	—	2	4	no	—
103	C₈×D₅₃	—	2	2	no	—
104	C₈×D₅₃	C₂×Q₈	1	2	yes	—
105	C₈×D₅₃	#41	1	2	yes	—
106	#11719910	—	2	2	no	—
107	#11719910	—	2	2	no	—
108	#11719910	—	2	2	no	—
109	#11719910	C₂×Q₈	1	4	yes	—
110	C₂²×D₅₃	C₂²×C₈	1	1	yes	characteristic
111	Q₈×C₅₃	—	4	1	no	—
112	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
113	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
114	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
115	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
116	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
117	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
118	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
119	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
120	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
121	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
122	C₄×D₅₃	—	4	1	no	—
123	C₄×D₅₃	—	4	1	no	—
124	C₄×D₅₃	C₄×Q₈	1	2	yes	—
125	C₄×D₅₃	#131	1	2	yes	—
126	Dic₁₀₆	—	4	2	no	—
127	Dic₁₀₆	—	4	1	no	—
128	C₈×C₅₃	—	2	1	no	—
129	C₈×C₅₃	—	2	2	no	—
130	C₈×C₅₃	C₄×Q₈	1	2	yes	—
131	#88322	—	2	1	no	—
132	#88322	—	2	2	no	—
133	#88322	C₂²×Q₈	1	1	yes	characteristic
134	#88322	C₂×#41	1	1	yes	characteristic
135	#33475	—	53	1	no	maximal, Sylow
136	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
137	C₂×D₅₃	—	2	1	no	—
138	C₂×D₅₃	—	2	1	no	—
139	C₂×D₅₃	C₈×Q₈	1	2	yes	—
140	C₄×C₅₃	—	2	1	no	—
141	C₄×C₅₃	—	2	2	no	—
142	C₄×C₅₃	—	2	1	no	—
143	C₄×C₅₃	—	2	2	no	—
144	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
145	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
146	Dic₅₃	—	2	1	no	—
147	Dic₅₃	—	2	2	no	—
148	Dic₅₃	—	2	2	no	—
149	Dic₅₃	—	2	1	no	—
150	Dic₅₃	C₈×Q₈	1	2	yes	—
151	#2960	—	53	1	no	—
152	#2183	—	53	2	no	—
153	#2183	—	53	2	no	—
154	#2183	—	53	2	no	—
155	#2183	—	53	2	no	—
156	#2151	—	53	2	no	—
157	#2150	—	53	1	no	—
158	#2108	—	53	2	no	—
159	C₂×#346	—	53	1	no	—
160	C₂×C₅₃	C₂×C₈×Q₈	1	1	yes	characteristic
161	C₂×C₅₃	#3407	1	1	yes	characteristic
162	C₂×C₅₃	#3407	1	1	yes	characteristic
163	D₅₃	—	2	2	no	—
164	#533	—	53	1	no	—
165	C₂×M₃₂	—	53	2	no	—
166	C₂²×C₁₆	—	53	1	no	—
167	C₂²×C₁₆	—	53	1	no	—
168	#446	—	53	2	no	—
169	#446	—	53	1	no	—
170	#446	#6157	1	1	yes	characteristic
171	C₈×Q₈	—	53	1	no	—
172	C₈×Q₈	—	53	1	no	—
173	C₈×Q₈	—	53	2	no	—
174	#424	—	53	2	no	—
175	#423	—	53	1	no	—
176	#405	—	53	2	no	—
177	C₂×C₄×C₈	—	53	1	no	—
178	C₄⋊C₁₆	—	53	2	no	—
179	C₄⋊C₁₆	—	53	2	no	—
180	C₄⋊C₁₆	—	53	2	no	—
181	C₄⋊C₁₆	—	53	1	no	—
182	C₄⋊C₁₆	—	53	1	no	—
183	C₄⋊C₁₆	—	53	2	no	—
184	C₄⋊C₁₆	—	53	2	no	—
185	#346	—	53	2	no	—
186	#346	—	53	2	no	—
187	#346	—	53	2	no	—
188	#346	—	53	2	no	—
189	C₄×C₁₆	—	53	2	no	—
190	C₄×C₁₆	—	53	2	no	—
191	C₅₃	#33475	1	1	yes	characteristic, complemented, Sylow
192	C₂×M₁₆	—	53	2	no	—
193	C₂²×C₈	—	53	1	no	—
194	C₂²×C₈	—	53	1	no	—
195	C₄⋊Q₈	—	53	1	no	—
196	C₄⋊Q₈	—	53	1	no	—
197	#125	—	53	2	no	—
198	C₂²⋊Q₈	—	53	1	no	—
199	C₂²⋊Q₈	—	53	2	no	—
200	C₂²⋊Q₈	—	53	1	no	—
201	C₄×Q₈	—	53	2	no	—
202	C₄×Q₈	—	53	1	no	—
203	C₄×Q₈	C₂×#6157	1	1	yes	characteristic
204	#117	—	53	2	no	—
205	C₂×C₄²	—	53	1	no	—
206	M₃₂	—	106	4	no	—
207	C₂×C₁₆	—	53	2	no	—
208	C₂×C₁₆	—	53	4	no	—
209	C₂×C₁₆	—	53	2	no	—
210	C₂×C₁₆	—	53	2	no	—
211	C₂×C₁₆	—	53	2	no	—
212	C₂×C₁₆	—	106	2	no	—
213	C₄⋊C₈	—	53	2	no	—
214	C₄⋊C₈	—	53	2	no	—
215	C₄⋊C₈	—	53	2	no	—
216	C₄⋊C₈	—	53	1	no	—
217	C₄⋊C₈	—	53	2	no	—
218	C₄⋊C₈	C₂×#6157	1	1	yes	characteristic
219	C₄⋊C₈	C₂×#6157	1	2	yes	—
220	#97	—	53	2	no	—
221	#97	C₂×#6157	1	2	yes	—
222	C₄×C₈	—	53	1	no	—
223	C₄×C₈	—	53	2	no	—
224	C₄×C₈	—	53	1	no	—
225	C₄×C₈	—	53	2	no	—
226	C₄×C₈	—	53	1	no	—
227	C₄×C₈	C₂×#6157	1	1	yes	characteristic
228	C₂×Q₈	—	53	2	no	—
229	C₂×Q₈	—	53	1	no	—
230	C₂×Q₈	D₅₃⋊C₈	1	1	yes	characteristic
231	C₂²×C₄	—	53	1	no	—
232	C₂²×C₄	—	53	1	no	—
233	C₂²×C₄	—	53	1	no	—
234	M₁₆	—	106	4	no	—
235	C₂×C₈	—	53	1	no	—
236	C₂×C₈	—	53	2	no	—
237	C₂×C₈	—	53	2	no	—
238	C₂×C₈	—	53	1	no	—
239	C₂×C₈	—	53	2	no	—
240	C₂×C₈	—	106	2	no	—
241	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
242	C₂×C₈	C₂²×#6157	1	2	yes	—
243	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
244	C₄⋊C₄	—	53	2	no	—
245	C₄⋊C₄	—	53	2	no	—
246	C₄⋊C₄	—	53	2	no	—
247	C₄⋊C₄	—	53	2	no	—
248	C₄⋊C₄	—	53	2	no	—
249	C₄⋊C₄	—	53	2	no	—
250	C₄⋊C₄	—	53	1	no	—
251	C₄⋊C₄	D₅₃⋊C₈	1	2	yes	—
252	C₄⋊C₄	D₅₃⋊C₈	1	1	yes	characteristic
253	C₂²⋊C₄	—	53	2	no	—
254	C₂²⋊C₄	—	53	2	no	—
255	C₄²	—	53	1	no	—
256	C₄²	—	53	1	no	—
257	C₄²	—	53	1	no	—
258	C₄²	—	53	2	no	—
259	C₄²	C₂²×#6157	1	1	yes	characteristic
260	C₄²	C₂²×#6157	1	2	yes	—
261	C₁₆	—	53	4	no	—
262	C₁₆	—	106	2	no	—
263	C₁₆	—	106	2	no	—
264	C₁₆	—	106	2	no	—
265	C₂³	—	53	1	no	—
266	Q₈	—	4	1	no	—
267	Q₈	—	212	2	no	—
268	Q₈	—	212	1	no	—
269	C₂×C₄	—	53	1	no	—
270	C₂×C₄	—	53	2	no	—
271	C₂×C₄	—	53	2	no	—
272	C₂×C₄	—	53	1	no	—
273	C₂×C₄	—	53	1	no	—
274	C₂×C₄	—	53	2	no	—
275	C₂×C₄	—	53	2	no	—
276	C₂×C₄	—	212	1	no	—
277	C₂×C₄	—	212	1	no	—
278	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
279	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
280	C₂×C₄	C₂×D₅₃⋊C₈	1	2	yes	—
281	C₂×C₄	C₂×D₅₃⋊C₈	1	2	yes	—
282	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
283	C₈	—	2	1	no	—
284	C₈	—	2	2	no	—
285	C₈	—	53	1	no	—
286	C₈	—	53	1	no	—
287	C₈	—	106	1	no	—
288	C₈	—	106	2	no	—
289	C₈	Q₈×#6157	1	2	yes	—
290	C₂²	—	53	2	no	—
291	C₂²	—	106	1	no	—
292	C₂²	—	106	1	no	—
293	C₂²	C₂²×D₅₃⋊C₈	1	1	yes	characteristic
294	C₄	—	2	1	no	—
295	C₄	—	2	2	no	—
296	C₄	—	2	1	no	—
297	C₄	—	2	2	no	—
298	C₄	—	53	2	no	—
299	C₄	—	106	1	no	—
300	C₄	—	106	2	no	—
301	C₄	—	106	1	no	—
302	C₄	—	106	2	no	—
303	C₄	C₂×Q₈×#6157	1	1	yes	characteristic
304	C₄	D₅₃⋊#131	1	1	yes	characteristic
305	C₂	—	106	2	no	—
306	C₂	Q₈×D₅₃⋊C₈	1	1	yes	characteristic
307	C₂	#20291683	1	1	yes	characteristic
308	C₂	#20291684	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#6157
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	C₅₃×#446	C₄
prime radical	C₂²×D₅₃	C₂²×C₈
cosocle	C₂×#88322	C₂⁴
hypercenter	#446	#6157
hypocenter	C₅₃	#33475
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #446

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. #20291130	C₂	semidirect
39. D₅₃⋊C₁₆	Q₈	semidirect
191. C₅₃	#33475	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
#20291130	2
#20291051	2
#20291048	2
#20290925	2
#20290916	2
#20283646	2
#33475	53