Subgroups — #28310266

Order 13568.

586 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133291	C₂	1	1	yes	maximal, characteristic
2	C₂²×#13133291	C₂	1	1	yes	maximal, characteristic
3	#20289770	C₂	1	4	yes	maximal, complemented
4	C₂×#16810581	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133291	—	2	1	no	—
6	C₂×#13133291	—	2	1	no	—
7	C₂×#13133291	—	2	1	no	—
8	C₂×#13133291	—	2	2	no	—
9	C₂×#13133291	—	2	2	no	—
10	C₂×#13133291	C₄	1	4	yes	complemented
11	C₂×#13133291	C₄	1	4	yes	complemented
12	C₂×#13133291	C₂²	1	2	yes	—
13	C₂²×#11719910	—	2	1	no	—
14	C₂²×#11719910	C₂²	1	2	yes	—
15	C₂³×#88322	C₂²	1	1	yes	characteristic
16	C₂×#13133176	C₄	1	1	yes	characteristic
17	#16810581	—	2	1	no	—
18	#16810581	—	2	1	no	—
19	#16810581	C₄	1	2	yes	—
20	#16810581	C₂²	1	2	yes	—
21	C₂×#13133127	—	2	1	no	—
22	C₂×C₈×Dic₅₃	—	2	1	no	—
23	C₂²×M₁₆×C₅₃	C₄	1	1	yes	characteristic
24	#13133291	—	2	4	no	—
25	#13133291	—	2	4	no	—
26	#13133291	—	2	4	no	—
27	#13133291	—	2	2	no	—
28	#13133291	C₂×C₄	1	4	yes	complemented
29	C₂×#11719910	—	2	4	no	—
30	C₂×#11719910	—	2	2	no	—
31	C₂×#11719910	—	2	2	no	—
32	C₂×#11719910	—	2	2	no	—
33	C₂×#11719910	C₂×C₄	1	4	yes	complemented
34	C₂²×#88322	C₂×C₄	1	2	yes	—
35	C₂²×#88322	C₂×C₄	1	2	yes	—
36	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
37	C₂²×#88322	D₄	1	1	yes	characteristic
38	C₂²×#88322	D₄	1	1	yes	characteristic
39	C₂²×#88322	D₄	1	1	yes	characteristic
40	C₂²×#88322	D₄	1	1	yes	characteristic
41	C₂²×#88322	D₄	1	2	yes	—
42	C₂²×#88322	Q₈	1	2	yes	—
43	C₂²×#88322	C₂³	1	1	yes	characteristic
44	#13133176	—	2	1	no	—
45	#13133176	—	2	1	no	—
46	#13133176	C₈	1	2	yes	—
47	#13133176	C₂×C₄	1	2	yes	—
48	C₂×C₄⋊Dic₅₃	C₈	1	2	yes	—
49	C₂×C₂²⋊Dic₅₃	—	2	1	no	—
50	C₂×C₄×Dic₅₃	—	2	1	no	—
51	#13133127	—	2	2	no	—
52	#13133127	—	2	2	no	—
53	C₈×Dic₅₃	—	2	2	no	—
54	C₈×Dic₅₃	—	2	2	no	—
55	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
56	C₂×M₁₆×C₅₃	—	2	1	no	—
57	C₂×M₁₆×C₅₃	—	2	2	no	—
58	C₂×M₁₆×C₅₃	C₈	1	4	yes	—
59	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
60	C₂²×C₈×C₅₃	—	2	1	no	—
61	C₂⁴×C₅₃	M₁₆	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
63	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
66	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
67	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
69	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
70	C₂²×C₄×C₅₃	M₁₆	1	2	yes	—
71	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
72	C₂²×Dic₅₃	—	2	2	no	—
73	M₁₆×C₅₃	—	2	4	no	—
74	M₁₆×C₅₃	—	2	2	no	—
75	M₁₆×C₅₃	C₂×C₈	1	4	yes	—
76	C₂×C₈×C₅₃	—	2	4	no	—
77	C₂×C₈×C₅₃	—	2	2	no	—
78	C₂²⋊Dic₅₃	—	2	2	no	—
79	C₂²⋊Dic₅₃	—	2	2	no	—
80	C₄⋊Dic₅₃	—	2	2	no	—
81	C₄⋊Dic₅₃	C₂×C₈	1	4	yes	—
82	C₄×Dic₅₃	—	2	2	no	—
83	C₄×Dic₅₃	—	2	2	no	—
84	C₂×#88322	C₄²	1	2	yes	—
85	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×#88322	C₂²⋊C₄	1	2	yes	—
88	C₂×#88322	C₂²⋊C₄	1	2	yes	—
89	C₂×#88322	C₂²⋊C₄	1	2	yes	—
90	C₂×#88322	C₂²⋊C₄	1	2	yes	—
91	C₂×#88322	C₄⋊C₄	1	2	yes	—
92	C₂×#88322	C₄⋊C₄	1	2	yes	—
93	C₂×#88322	C₄⋊C₄	1	2	yes	—
94	C₂×#88322	C₄⋊C₄	1	2	yes	—
95	C₂×#88322	C₄⋊C₄	1	2	yes	—
96	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
97	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
98	C₂×#88322	C₂×D₄	1	1	yes	characteristic
99	C₂×#88322	C₂×D₄	1	1	yes	characteristic
100	C₂×#88322	C₂×D₄	1	1	yes	characteristic
101	C₂×#88322	C₂×Q₈	1	1	yes	characteristic
102	#11719910	—	2	4	no	—
103	#11719910	—	2	4	no	—
104	C₂³×C₅₃	—	4	2	no	—
105	C₂³×C₅₃	#97	1	2	yes	—
106	C₂³×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
107	C₂³×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
108	C₂³×C₅₃	C₄⋊C₈	1	2	yes	—
109	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
110	C₂×C₄×C₅₃	#95	1	2	yes	—
111	C₂×C₄×C₅₃	#95	1	2	yes	—
112	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
113	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
114	C₂×C₄×C₅₃	#97	1	2	yes	—
115	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
116	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
117	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
118	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
119	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
120	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
121	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
122	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
123	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
126	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
127	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
128	C₂×Dic₅₃	—	2	4	no	—
129	C₂×Dic₅₃	—	4	4	no	—
130	C₈×C₅₃	—	2	4	no	—
131	#88322	#95	1	2	yes	—
132	#88322	#95	1	2	yes	—
133	#88322	C₂×C₂²⋊C₄	1	1	yes	characteristic
134	#88322	C₂×C₂²⋊C₄	1	1	yes	characteristic
135	#88322	C₂×C₄⋊C₄	1	1	yes	characteristic
136	#88322	C₂×C₄⋊C₄	1	1	yes	characteristic
137	#12554	—	53	1	no	maximal, Sylow
138	C₂²×C₅₃	—	2	2	no	—
139	C₂²×C₅₃	—	2	2	no	—
140	C₂²×C₅₃	—	2	2	no	—
141	C₂²×C₅₃	—	4	2	no	—
142	C₂²×C₅₃	—	4	2	no	—
143	C₂²×C₅₃	#336	1	2	yes	—
144	C₂²×C₅₃	#336	1	2	yes	—
145	C₂²×C₅₃	C₂×#97	1	1	yes	characteristic
146	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
147	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
148	C₄×C₅₃	#336	1	2	yes	—
149	C₄×C₅₃	#336	1	2	yes	—
150	C₄×C₅₃	C₂×#95	1	1	yes	characteristic
151	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
152	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
153	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
154	Dic₅₃	—	4	4	no	—
155	C₂²×M₃₂	—	53	1	no	—
156	C₂²×M₃₂	—	53	1	no	—
157	C₂×#405	—	53	1	no	—
158	#1379	—	53	4	no	—
159	C₂×C₅₃	—	2	2	no	—
160	C₂×C₅₃	—	2	2	no	—
161	C₂×C₅₃	—	2	1	no	—
162	C₂×C₅₃	—	2	1	no	—
163	C₂×C₅₃	C₂×#336	1	1	yes	characteristic
164	C₂×C₅₃	#2117	1	2	yes	—
165	C₂²×M₁₆	#6157	1	1	yes	characteristic
166	C₂³×C₈	—	53	1	no	—
167	C₂×#117	—	53	1	no	—
168	C₂×M₃₂	—	53	4	no	—
169	C₂×M₃₂	—	53	2	no	—
170	C₂×M₃₂	—	53	4	no	—
171	C₂×M₃₂	—	106	2	no	—
172	C₂×M₃₂	—	106	1	no	—
173	C₂×M₃₂	—	106	1	no	—
174	C₂×M₃₂	—	106	2	no	—
175	C₂×M₃₂	—	106	1	no	—
176	C₂²×C₁₆	—	53	2	no	—
177	C₂²×C₁₆	—	106	1	no	—
178	#405	—	53	2	no	—
179	#405	—	53	2	no	—
180	#405	—	106	1	no	—
181	#405	—	106	1	no	—
182	C₂×#97	—	106	1	no	—
183	C₂×C₄×C₈	—	106	1	no	—
184	C₅₃	#12554	1	1	yes	characteristic, complemented, Sylow
185	C₂³×C₄	C₂×#6157	1	1	yes	characteristic
186	C₂×M₁₆	—	2	2	no	—
187	C₂×M₁₆	—	2	1	no	—
188	C₂×M₁₆	#88324	1	4	yes	—
189	C₂×M₁₆	C₂×#6157	1	2	yes	—
190	C₂²×C₈	—	2	1	no	—
191	C₂²×C₈	—	53	2	no	—
192	C₂²×C₈	—	53	2	no	—
193	C₂²×C₈	—	53	2	no	—
194	C₂²×C₈	—	53	2	no	—
195	C₂²×C₈	—	53	1	no	—
196	C₂²×C₈	—	53	1	no	—
197	C₂²×C₈	—	53	1	no	—
198	C₂²×C₈	—	53	1	no	—
199	C₂²×C₈	—	53	1	no	—
200	C₂²×C₈	—	53	1	no	—
201	#117	—	53	2	no	—
202	#117	—	53	2	no	—
203	#117	—	106	1	no	—
204	#117	—	106	1	no	—
205	C₂×C₄⋊C₄	—	53	2	no	—
206	C₂×C₂²⋊C₄	—	106	1	no	—
207	C₂×C₄²	—	106	1	no	—
208	M₃₂	—	53	4	no	—
209	M₃₂	—	106	4	no	—
210	M₃₂	—	106	4	no	—
211	M₃₂	—	106	4	no	—
212	M₃₂	—	106	2	no	—
213	C₂×C₁₆	—	53	4	no	—
214	C₂×C₁₆	—	106	2	no	—
215	C₂×C₁₆	—	106	2	no	—
216	C₂×C₁₆	—	106	4	no	—
217	C₂×C₁₆	—	106	2	no	—
218	#97	—	106	2	no	—
219	#97	—	106	2	no	—
220	C₄×C₈	—	106	2	no	—
221	C₄×C₈	—	106	2	no	—
222	C₂⁴	#11719936	1	1	yes	characteristic
223	C₂²×C₄	—	106	2	no	—
224	C₂²×C₄	D₅₃⋊C₈	1	1	yes	characteristic
225	C₂²×C₄	D₅₃⋊C₈	1	1	yes	characteristic
226	C₂²×C₄	#11719936	1	1	yes	characteristic
227	C₂²×C₄	C₄×#6157	1	2	yes	—
228	C₂²×C₄	C₄⋊#6157	1	2	yes	—
229	C₂²×C₄	C₂×#88324	1	2	yes	—
230	C₂²×C₄	#11719940	1	2	yes	—
231	C₂²×C₄	C₂²⋊#6157	1	1	yes	characteristic
232	C₂²×C₄	C₂²⋊#6157	1	1	yes	characteristic
233	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
234	M₁₆	—	2	4	no	—
235	M₁₆	—	2	2	no	—
236	M₁₆	C₂×#88324	1	4	yes	—
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	53	2	no	—
240	C₂×C₈	—	53	2	no	—
241	C₂×C₈	—	53	2	no	—
242	C₂×C₈	—	53	2	no	—
243	C₂×C₈	—	53	2	no	—
244	C₂×C₈	—	53	2	no	—
245	C₂×C₈	—	53	2	no	—
246	C₂×C₈	—	53	2	no	—
247	C₂×C₈	—	53	1	no	—
248	C₂×C₈	—	53	1	no	—
249	C₂×C₈	—	53	2	no	—
250	C₂×C₈	—	53	1	no	—
251	C₂×C₈	—	53	1	no	—
252	C₂×C₈	—	53	1	no	—
253	C₂×C₈	—	53	1	no	—
254	C₂×C₈	—	53	1	no	—
255	C₂×C₈	—	53	2	no	—
256	C₂×C₈	—	53	1	no	—
257	C₄⋊C₄	—	53	4	no	—
258	C₄⋊C₄	—	106	2	no	—
259	C₂²⋊C₄	—	106	2	no	—
260	C₂²⋊C₄	—	106	2	no	—
261	C₄²	—	106	2	no	—
262	C₄²	—	106	2	no	—
263	C₁₆	—	106	4	no	—
264	C₁₆	—	106	4	no	—
265	C₂³	—	4	2	no	—
266	C₂³	#13133272	1	2	yes	—
267	C₂³	#13133274	1	1	yes	characteristic
268	C₂³	#13133274	1	1	yes	characteristic
269	C₂³	C₄⋊#88324	1	2	yes	—
270	C₂³	C₂×#11719936	1	1	yes	characteristic
271	C₂×C₄	—	106	4	no	—
272	C₂×C₄	—	212	4	no	—
273	C₂×C₄	(C₂×#6157)⋊C₄	1	2	yes	—
274	C₂×C₄	(C₂×#6157)⋊C₄	1	2	yes	—
275	C₂×C₄	C₄×#88324	1	2	yes	—
276	C₂×C₄	C₄×#88324	1	2	yes	—
277	C₂×C₄	#13133272	1	2	yes	—
278	C₂×C₄	C₂²⋊#88324	1	2	yes	—
279	C₂×C₄	C₂²⋊#88324	1	2	yes	—
280	C₂×C₄	#13133274	1	1	yes	characteristic
281	C₂×C₄	#13133274	1	1	yes	characteristic
282	C₂×C₄	C₄⋊#88324	1	2	yes	—
283	C₂×C₄	#13133284	1	2	yes	—
284	C₂×C₄	#13133284	1	2	yes	—
285	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
286	C₂×C₄	C₂×C₂²⋊#6157	1	1	yes	characteristic
287	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
288	C₂×C₄	C₂²×#88324	1	1	yes	characteristic
289	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
290	C₂×C₄	C₂×#11719940	1	1	yes	characteristic
291	C₈	—	2	4	no	—
292	C₈	—	53	2	no	—
293	C₈	—	53	2	no	—
294	C₈	—	53	1	no	—
295	C₈	—	53	1	no	—
296	C₈	—	53	1	no	—
297	C₈	—	53	1	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	4	2	no	—
302	C₂²	—	4	2	no	—
303	C₂²	#16811489	1	2	yes	—
304	C₂²	#16811489	1	2	yes	—
305	C₂²	C₂×#13133272	1	1	yes	characteristic
306	C₂²	C₂×#13133274	1	1	yes	characteristic
307	C₂²	C₂×C₄⋊#88324	1	1	yes	characteristic
308	C₄	—	212	4	no	—
309	C₄	#16811489	1	2	yes	—
310	C₄	#16811489	1	2	yes	—
311	C₄	C₂×(C₂×#6157)⋊C₄	1	1	yes	characteristic
312	C₄	C₂×C₄×#88324	1	1	yes	characteristic
313	C₄	C₂×C₂²⋊#88324	1	1	yes	characteristic
314	C₄	C₂×#13133284	1	1	yes	characteristic
315	C₂	—	2	1	no	—
316	C₂	—	2	1	no	—
317	C₂	—	2	2	no	—
318	C₂	—	2	2	no	—
319	C₂	C₂×#16811489	1	1	yes	characteristic
320	C₂	#20290959	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊#6157
commutator	C₄×C₅₃	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×#6157
Fitting	C₂²×M₁₆×C₅₃	C₄
socle	C₂²×C₅₃	C₂×C₂²⋊C₈
prime radical	C₂⁴×C₅₃	M₁₆
cosocle	C₂²×#88322	C₂³
hypercenter	C₂²×M₁₆	#6157
hypocenter	C₅₃	#12554
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
3. #20289770	C₂	semidirect
10. C₂×#13133291	C₄	semidirect
11. C₂×#13133291	C₄	semidirect
28. #13133291	C₂×C₄	semidirect
33. C₂×#11719910	C₂×C₄	semidirect
184. C₅₃	#12554	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13133291	2
#20289770	2
C₂×#16810581	2
#12554	53