Subgroups — #28301674

Order 13568.

514 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20290006	C₂	1	2	yes	maximal
2	#20289918	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16811464	C₂	1	1	yes	maximal, characteristic, complemented
4	#20289548	C₂	1	2	yes	maximal, complemented
5	#131×Dic₅₃	C₂	1	1	yes	maximal, characteristic
6	#16811639	C₂²	1	2	yes	—
7	C₂×C₄×#88324	C₂²	1	2	yes	—
8	Q₈⋊#88324	C₄	1	4	yes	—
9	D₄⋊#88324	C₄	1	4	yes	—
10	#16811464	—	2	1	no	—
11	#16811464	—	2	1	no	—
12	#16811464	—	2	1	no	—
13	C₈×#88324	—	2	1	no	—
14	C₂×#13133224	C₄	1	1	yes	characteristic
15	#41×Dic₅₃	C₂²	1	1	yes	characteristic
16	#16810581	—	2	1	no	—
17	#16810581	C₄	1	1	yes	characteristic
18	#16810578	—	2	1	no	—
19	#16810578	C₄	1	1	yes	characteristic
20	M₁₆×Dic₅₃	—	2	1	no	—
21	M₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
22	C₂×C₈×Dic₅₃	—	2	1	no	—
23	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
24	C₂×#131×C₅₃	C₄	1	1	yes	characteristic
25	C₂²×#88324	—	2	2	no	—
26	C₄⋊#88324	C₂×C₄	1	4	yes	—
27	C₂²⋊#88324	—	2	2	no	—
28	C₄×#88324	—	2	2	no	—
29	C₄×#88324	—	2	2	no	—
30	C₄×#88324	C₂×C₄	1	4	yes	—
31	#13133224	—	2	2	no	—
32	#13133224	C₈	1	4	yes	complemented
33	C₂×#11719917	—	2	1	no	—
34	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
35	C₂²×#88322	—	2	1	no	—
36	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
37	Q₈×Dic₅₃	C₂×C₄	1	2	yes	—
38	D₄×Dic₅₃	—	2	2	no	—
39	D₄×Dic₅₃	C₂×C₄	1	2	yes	—
40	#13133176	—	2	1	no	—
41	#13133176	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄×Dic₅₃	—	2	1	no	—
43	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
44	#13133127	—	2	1	no	—
45	#13133127	—	2	1	no	—
46	#13133127	—	2	1	no	—
47	C₈×Dic₅₃	—	2	1	no	—
48	C₈×Dic₅₃	—	2	1	no	—
49	C₈×Dic₅₃	—	2	1	no	—
50	C₈×Dic₅₃	D₄	1	1	yes	characteristic
51	C₈×Dic₅₃	D₄	1	1	yes	characteristic
52	C₈×Dic₅₃	D₄	1	1	yes	characteristic
53	C₈×Dic₅₃	D₄	1	1	yes	characteristic
54	C₂×#41×C₅₃	C₂×C₄	1	1	yes	characteristic
55	#131×C₅₃	—	2	2	no	—
56	#131×C₅₃	C₈	1	4	yes	complemented
57	C₂×M₁₆×C₅₃	—	2	1	no	—
58	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
59	C₂²×C₈×C₅₃	—	2	1	no	—
60	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
61	#41×C₅₃	—	2	1	no	—
62	#41×C₅₃	—	2	1	no	—
63	#41×C₅₃	C₂×C₈	1	2	yes	—
64	#41×C₅₃	C₂×C₈	1	2	yes	—
65	C₂×Q₈×C₅₃	C₄²	1	1	yes	characteristic
66	C₂×D₄×C₅₃	—	2	1	no	—
67	C₂×D₄×C₅₃	C₄²	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	—	2	1	no	—
69	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×Dic₅₃	—	2	2	no	—
71	C₂²×Dic₅₃	#41	1	2	yes	—
72	C₂×#88324	—	2	4	no	—
73	C₂×#88324	—	2	4	no	—
74	C₂×#88324	—	4	4	no	—
75	M₁₆×C₅₃	—	2	1	no	—
76	M₁₆×C₅₃	—	2	4	no	—
77	M₁₆×C₅₃	C₂×C₈	1	2	yes	—
78	C₂×C₈×C₅₃	—	2	1	no	—
79	C₂×C₈×C₅₃	—	2	4	no	—
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×C₈	1	2	yes	—
83	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
84	C₂²⋊Dic₅₃	—	2	1	no	—
85	C₂²⋊Dic₅₃	—	2	2	no	—
86	C₄⋊Dic₅₃	—	2	2	no	—
87	C₄⋊Dic₅₃	C₄²	1	2	yes	—
88	C₄×Dic₅₃	—	2	1	no	—
89	C₄×Dic₅₃	—	2	2	no	—
90	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
91	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
92	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
93	#11719917	—	2	1	no	—
94	#11719917	—	2	4	no	—
95	#11719917	C₂×C₈	1	2	yes	—
96	C₂×#88322	—	2	1	no	—
97	C₂×#88322	—	2	4	no	—
98	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
99	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
100	C₂×#88322	C₂×C₈	1	2	yes	—
101	C₂×#88322	M₁₆	1	2	yes	—
102	C₂³×C₅₃	—	2	1	no	—
103	C₂³×C₅₃	#117	1	1	yes	characteristic
104	Q₈×C₅₃	—	2	1	no	—
105	Q₈×C₅₃	C₄×C₈	1	2	yes	—
106	D₄×C₅₃	—	2	2	no	—
107	D₄×C₅₃	—	2	2	no	—
108	D₄×C₅₃	—	2	1	no	—
109	D₄×C₅₃	C₄×C₈	1	2	yes	—
110	C₂×C₄×C₅₃	—	2	1	no	—
111	C₂×C₄×C₅₃	—	2	2	no	—
112	C₂×C₄×C₅₃	—	2	2	no	—
113	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
114	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
116	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
117	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
118	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
119	C₂×Dic₅₃	—	2	4	no	—
120	C₂×Dic₅₃	—	2	2	no	—
121	C₂×Dic₅₃	—	4	4	no	—
122	C₂×Dic₅₃	C₄×D₄	1	2	yes	—
123	C₂×Dic₅₃	C₄×D₄	1	2	yes	—
124	#88324	—	4	4	no	—
125	#88324	—	4	4	no	—
126	C₈×C₅₃	—	2	2	no	—
127	C₈×C₅₃	C₂²⋊C₈	1	2	yes	—
128	C₈×C₅₃	C₂²⋊C₈	1	2	yes	—
129	#88322	—	2	2	no	—
130	#88322	C₂²⋊C₈	1	2	yes	—
131	#88322	C₂²⋊C₈	1	2	yes	—
132	#8185	—	53	1	no	maximal, Sylow
133	C₂²×C₅₃	—	2	2	no	—
134	C₂²×C₅₃	—	2	2	no	—
135	C₂²×C₅₃	—	4	2	no	—
136	C₂²×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
137	C₂²×C₅₃	#431	1	1	yes	characteristic
138	C₂²×C₅₃	#431	1	1	yes	characteristic
139	C₄×C₅₃	—	2	2	no	—
140	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
141	C₄×C₅₃	C₄×M₁₆	1	1	yes	characteristic
142	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
143	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
144	Dic₅₃	—	2	2	no	—
145	Dic₅₃	—	4	2	no	—
146	Dic₅₃	#444	1	4	yes	—
147	C₄×#131	—	53	1	no	—
148	#1476	—	53	2	no	—
149	#1450	—	53	1	no	—
150	C₂×#322	—	53	1	no	—
151	#1278	—	53	2	no	—
152	C₂×C₅₃	—	2	1	no	—
153	C₂×C₅₃	—	2	1	no	—
154	C₂×C₅₃	—	4	2	no	—
155	C₂×C₅₃	C₄×C₂²⋊C₈	1	1	yes	characteristic
156	C₂×C₅₃	#1765	1	2	yes	—
157	C₂×#131	—	53	1	no	—
158	C₂×#131	#6157	1	1	yes	characteristic
159	C₄×#41	—	53	1	no	—
160	#431	—	53	2	no	—
161	#405	—	53	1	no	—
162	#405	—	53	1	no	—
163	#405	—	106	1	no	—
164	#405	—	106	1	no	—
165	C₄×M₁₆	—	53	1	no	—
166	C₄×M₁₆	—	106	1	no	—
167	C₂×C₄×C₈	—	53	1	no	—
168	C₂×C₄×C₈	—	53	2	no	—
169	C₂×C₄×C₈	—	106	1	no	—
170	Q₈⋊C₈	—	53	4	no	—
171	D₄⋊C₈	—	53	4	no	—
172	#322	—	106	1	no	—
173	#322	—	106	1	no	—
174	#322	—	106	1	no	—
175	C₈²	—	106	1	no	—
176	C₅₃	#8185	1	1	yes	characteristic, complemented, Sylow
177	C₂×#41	C₂×#6157	1	1	yes	characteristic
178	#131	—	2	2	no	—
179	#131	—	53	4	no	—
180	#131	—	106	2	no	—
181	#131	#88324	1	4	yes	complemented
182	C₂×M₁₆	—	2	1	no	—
183	C₂×M₁₆	—	53	1	no	—
184	C₂×M₁₆	—	106	1	no	—
185	C₂×M₁₆	C₂×#6157	1	1	yes	characteristic
186	C₂²×C₈	—	2	1	no	—
187	C₂²×C₈	—	53	1	no	—
188	C₂²×C₈	—	106	1	no	—
189	C₂²×C₈	—	106	2	no	—
190	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
191	C₄×Q₈	—	53	2	no	—
192	C₄×D₄	—	53	2	no	—
193	C₄×D₄	—	106	2	no	—
194	#117	—	53	1	no	—
195	#117	—	106	1	no	—
196	C₂×C₄²	—	53	1	no	—
197	C₂×C₄²	—	106	1	no	—
198	C₄⋊C₈	—	53	4	no	—
199	C₂²⋊C₈	—	106	2	no	—
200	#97	—	106	1	no	—
201	#97	—	106	1	no	—
202	#97	—	106	1	no	—
203	C₄×C₈	—	53	1	no	—
204	C₄×C₈	—	53	1	no	—
205	C₄×C₈	—	53	1	no	—
206	C₄×C₈	—	53	1	no	—
207	C₄×C₈	—	53	4	no	—
208	C₄×C₈	—	106	1	no	—
209	C₄×C₈	—	106	2	no	—
210	C₄×C₈	—	106	2	no	—
211	C₄×C₈	—	106	1	no	—
212	C₄×C₈	—	106	1	no	—
213	#41	—	2	1	no	—
214	#41	—	2	1	no	—
215	#41	D₅₃⋊C₈	1	2	yes	—
216	#41	C₂×#88324	1	2	yes	—
217	C₂×Q₈	C₄×#6157	1	1	yes	characteristic
218	C₂×D₄	—	2	1	no	—
219	C₂×D₄	C₄×#6157	1	1	yes	characteristic
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	—	53	2	no	—
222	C₂²×C₄	—	106	2	no	—
223	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
224	M₁₆	—	2	1	no	—
225	M₁₆	—	2	4	no	—
226	M₁₆	—	53	2	no	—
227	M₁₆	—	106	4	no	—
228	M₁₆	—	106	1	no	—
229	M₁₆	C₂×#88324	1	2	yes	—
230	C₂×C₈	—	2	1	no	—
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	—	53	1	no	—
233	C₂×C₈	—	53	1	no	—
234	C₂×C₈	—	53	2	no	—
235	C₂×C₈	—	53	2	no	—
236	C₂×C₈	—	106	4	no	—
237	C₂×C₈	—	106	1	no	—
238	C₂×C₈	—	106	4	no	—
239	C₂×C₈	—	106	4	no	—
240	C₂×C₈	—	212	4	no	—
241	C₂×C₈	C₂×#88324	1	2	yes	—
242	C₂×C₈	#11719940	1	2	yes	—
243	C₂×C₈	C₂²⋊#6157	1	1	yes	characteristic
244	C₂×C₈	C₂²⋊#6157	1	1	yes	characteristic
245	C₄⋊C₄	—	53	2	no	—
246	C₄⋊C₄	—	106	2	no	—
247	C₂²⋊C₄	—	106	2	no	—
248	C₂²⋊C₄	—	106	1	no	—
249	C₄²	—	53	2	no	—
250	C₄²	—	53	1	no	—
251	C₄²	—	53	1	no	—
252	C₄²	—	106	2	no	—
253	C₄²	—	106	1	no	—
254	C₂³	—	2	1	no	—
255	C₂³	#13133296	1	1	yes	characteristic
256	Q₈	—	2	1	no	—
257	Q₈	C₄×#88324	1	2	yes	—
258	D₄	—	2	2	no	—
259	D₄	—	2	2	no	—
260	D₄	—	2	1	no	—
261	D₄	C₄×#88324	1	2	yes	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	53	2	no	—
266	C₂×C₄	—	53	2	no	—
267	C₂×C₄	—	106	4	no	—
268	C₂×C₄	—	106	2	no	—
269	C₂×C₄	—	212	4	no	—
270	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
271	C₂×C₄	C₂×C₂²⋊#6157	1	1	yes	characteristic
272	C₂×C₄	C₂²×#88324	1	1	yes	characteristic
273	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
274	C₂×C₄	C₂×#11719940	1	1	yes	characteristic
275	C₂×C₄	D₅₃⋊M₁₆	1	1	yes	characteristic
276	C₈	—	2	2	no	—
277	C₈	—	53	2	no	—
278	C₈	—	53	2	no	—
279	C₈	—	106	2	no	—
280	C₈	—	212	4	no	—
281	C₈	—	212	4	no	—
282	C₈	C₂²⋊#88324	1	2	yes	—
283	C₈	C₂²⋊#88324	1	2	yes	—
284	C₂²	—	2	2	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	4	2	no	—
287	C₂²	C₄×C₂²⋊#6157	1	1	yes	characteristic
288	C₂²	#16811639	1	1	yes	characteristic
289	C₂²	#16811640	1	1	yes	characteristic
290	C₄	—	2	2	no	—
291	C₄	—	53	4	no	—
292	C₄	—	106	2	no	—
293	C₄	—	212	2	no	—
294	C₄	C₂×C₄×#88324	1	1	yes	characteristic
295	C₄	C₄×#11719940	1	1	yes	characteristic
296	C₄	C₂×C₂²⋊#88324	1	1	yes	characteristic
297	C₄	D₅₃⋊(C₂²⋊C₈)	1	1	yes	characteristic
298	C₂	—	2	1	no	—
299	C₂	—	2	1	no	—
300	C₂	—	4	2	no	—
301	C₂	C₄×C₂²⋊#88324	1	1	yes	characteristic
302	C₂	#20290493	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊#6157
commutator	C₄×C₅₃	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×#6157
Fitting	C₂×#131×C₅₃	C₄
socle	C₂²×C₅₃	C₄×C₂²⋊C₄
prime radical	C₂×#41×C₅₃	C₂×C₄
cosocle	C₂×C₄×Dic₅₃	C₂³
hypercenter	C₂×#131	#6157
hypocenter	C₅₃	#8185
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
2. #20289918	C₂	semidirect
3. C₂×#16811464	C₂	semidirect
4. #20289548	C₂	semidirect
32. #13133224	C₈	semidirect
56. #131×C₅₃	C₈	semidirect
176. C₅₃	#8185	semidirect
181. #131	#88324	semidirect

Maximal subgroups

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

subgroup	index
#20290006	2
#20289918	2
C₂×#16811464	2
#20289548	2
#131×Dic₅₃	2
#8185	53