Subgroups — #28311139

Order 13568.

450 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20291047	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄⋊#11719910	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16811522	C₂	1	2	yes	maximal
4	#20289770	C₂	1	2	yes	maximal, complemented
5	C₄×#13133225	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133291	C₂²	1	2	yes	—
7	C₂²×#11719910	C₂²	1	2	yes	—
8	C₄⋊#11719910	—	2	1	no	—
9	C₄⋊#11719910	—	2	1	no	—
10	C₄⋊#11719910	—	2	1	no	—
11	C₄⋊#11719910	—	2	1	no	—
12	#16811522	—	2	4	no	—
13	C₂×#13133225	C₄	1	1	yes	characteristic
14	C₂×#13133225	C₂²	1	1	yes	characteristic
15	C₄×#11719945	C₄	1	1	yes	characteristic
16	#16810581	C₄	1	1	yes	characteristic
17	#16810581	C₂²	1	1	yes	characteristic
18	#16810580	—	2	1	no	—
19	#16810580	—	2	1	no	—
20	C₄×#11719912	—	2	1	no	—
21	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
22	C₄×C₈×D₅₃	—	2	1	no	—
23	C₄×M₁₆×C₅₃	C₄	1	1	yes	characteristic
24	#13133291	—	2	2	no	—
25	#13133291	—	2	2	no	—
26	C₂×#11719910	—	2	2	no	—
27	C₂×#11719910	—	2	2	no	—
28	C₂×#11719910	—	2	2	no	—
29	C₂×#11719910	—	2	2	no	—
30	C₂×#11719945	C₂×C₄	1	1	yes	characteristic
31	#13133225	—	2	2	no	—
32	#13133225	—	2	1	no	—
33	#13133225	—	2	1	no	—
34	#13133225	—	2	2	no	—
35	#13133225	D₄	1	2	yes	—
36	#13133225	Q₈	1	2	yes	—
37	C₂×#11719912	—	2	1	no	—
38	C₂×#11719912	—	2	1	no	—
39	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
40	C₂²×#88322	C₂³	1	1	yes	characteristic
41	C₂×C₈×D₅₃	—	2	1	no	—
42	C₂×C₈×D₅₃	—	2	1	no	—
43	C₄×Dic₁₀₆	C₈	1	1	yes	characteristic
44	C₄×Dic₁₀₆	C₈	1	1	yes	characteristic
45	C₄×D₂₁₂	C₈	1	1	yes	characteristic
46	C₄×D₂₁₂	C₈	1	1	yes	characteristic
47	C₄×C₂²⋊D₅₃	—	2	1	no	—
48	C₄×C₂²⋊D₅₃	—	2	1	no	—
49	#13133176	C₂×C₄	1	1	yes	characteristic
50	#13133174	—	2	1	no	—
51	C₄²×D₅₃	—	2	1	no	—
52	#13133127	—	2	1	no	—
53	#13133127	—	2	1	no	—
54	C₄×#88322	D₄	1	1	yes	characteristic
55	C₄×#88322	D₄	1	1	yes	characteristic
56	C₄×#88322	D₄	1	1	yes	characteristic
57	C₄×#88322	D₄	1	1	yes	characteristic
58	C₈×Dic₅₃	—	2	1	no	—
59	C₈×Dic₅₃	—	2	1	no	—
60	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
61	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
63	#97×C₅₃	—	2	1	no	—
64	C₄×C₈×C₅₃	—	2	1	no	—
65	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₂²⋊D₅₃	—	2	1	no	—
69	#11719945	—	2	1	no	—
70	#11719945	—	2	1	no	—
71	#11719945	C₄⋊C₄	1	2	yes	—
72	#11719945	M₁₆	1	2	yes	—
73	C₂×D₂₁₂	C₂×C₈	1	1	yes	characteristic
74	C₂×C₄×D₅₃	—	2	1	no	—
75	C₂×C₄×D₅₃	—	2	1	no	—
76	C₂×C₄×D₅₃	—	2	1	no	—
77	C₂×Dic₁₀₆	C₂×C₈	1	1	yes	characteristic
78	M₁₆×C₅₃	—	2	1	no	—
79	M₁₆×C₅₃	—	2	1	no	—
80	M₁₆×C₅₃	C₄⋊C₄	1	2	yes	—
81	M₁₆×C₅₃	M₁₆	1	2	yes	—
82	C₂×C₈×C₅₃	—	2	1	no	—
83	C₂×C₈×C₅₃	—	2	1	no	—
84	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
85	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
86	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
87	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
88	C₂²⋊Dic₅₃	—	2	1	no	—
89	#11719921	—	2	1	no	—
90	#11719921	—	2	1	no	—
91	C₄⋊Dic₅₃	C₂×C₈	1	1	yes	characteristic
92	C₄⋊Dic₅₃	C₂×C₈	1	1	yes	characteristic
93	#11719919	—	2	1	no	—
94	#11719919	—	2	1	no	—
95	C₄×Dic₅₃	—	2	1	no	—
96	C₄×Dic₅₃	—	2	1	no	—
97	C₄×Dic₅₃	—	2	1	no	—
98	C₂×#88322	—	2	1	no	—
99	C₂×#88322	—	2	1	no	—
100	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
101	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
102	C₂×#88322	C₂×D₄	1	1	yes	characteristic
103	C₂×#88322	C₂×D₄	1	1	yes	characteristic
104	C₂×#88322	C₂×D₄	1	1	yes	characteristic
105	C₂×#88322	C₂×Q₈	1	1	yes	characteristic
106	C₂×#88322	#41	1	1	yes	characteristic
107	C₂×#88322	#41	1	1	yes	characteristic
108	#11719912	—	2	4	no	—
109	#11719912	—	2	2	no	—
110	#11719912	—	2	2	no	—
111	C₈×D₅₃	—	2	4	no	—
112	C₈×D₅₃	—	2	2	no	—
113	C₈×D₅₃	—	2	2	no	—
114	#11719910	—	4	2	no	—
115	#11719910	—	4	2	no	—
116	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
117	C₂²×D₅₃	—	2	1	no	—
118	C₂×C₄×C₅₃	—	2	2	no	—
119	C₂×C₄×C₅₃	—	4	1	no	—
120	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
121	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
123	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
124	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
127	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
128	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
129	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
130	C₂²⋊D₅₃	—	2	2	no	—
131	C₂²⋊D₅₃	—	2	2	no	—
132	C₂×Dic₅₃	—	2	1	no	—
133	C₂×Dic₅₃	—	2	1	no	—
134	C₂×Dic₅₃	—	2	1	no	—
135	D₂₁₂	—	2	1	no	—
136	D₂₁₂	C₄⋊C₈	1	2	yes	—
137	C₄×D₅₃	—	2	2	no	—
138	C₄×D₅₃	—	2	2	no	—
139	C₄×D₅₃	—	4	2	no	—
140	C₄×D₅₃	—	4	2	no	—
141	Dic₁₀₆	—	2	1	no	—
142	Dic₁₀₆	C₄⋊C₈	1	2	yes	—
143	C₈×C₅₃	—	2	2	no	—
144	C₈×C₅₃	—	2	2	no	—
145	#88322	—	2	1	no	—
146	#88322	—	2	1	no	—
147	#88322	#121	1	1	yes	characteristic
148	#88322	#121	1	1	yes	characteristic
149	#88322	C₂²⋊Q₈	1	1	yes	characteristic
150	#88322	C₂²⋊Q₈	1	1	yes	characteristic
151	#12787	—	53	1	no	maximal, Sylow
152	C₂²×C₅₃	—	2	1	no	—
153	C₂²×C₅₃	—	2	1	no	—
154	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
155	C₂²×C₅₃	#423	1	1	yes	characteristic
156	C₂²×C₅₃	#432	1	1	yes	characteristic
157	C₂×D₅₃	—	2	2	no	—
158	C₂×D₅₃	—	4	1	no	—
159	C₂×D₅₃	—	4	1	no	—
160	C₄×C₅₃	—	2	1	no	—
161	C₄×C₅₃	—	2	1	no	—
162	C₄×C₅₃	—	2	1	no	—
163	C₄×C₅₃	#350	1	2	yes	complemented
164	C₄×C₅₃	#380	1	1	yes	characteristic
165	C₄×C₅₃	#407	1	1	yes	characteristic
166	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
167	C₄×C₅₃	#431	1	1	yes	characteristic
168	Dic₅₃	—	2	2	no	—
169	Dic₅₃	—	4	1	no	—
170	Dic₅₃	—	4	1	no	—
171	C₄×#131	—	53	1	no	—
172	#2150	—	53	1	no	—
173	C₂×C₄⋊C₁₆	—	53	1	no	—
174	C₂×#350	—	53	2	no	—
175	#1379	—	53	2	no	—
176	C₂×C₅₃	—	2	1	no	—
177	C₂×C₅₃	—	2	1	no	—
178	C₂×C₅₃	#1797	1	1	yes	characteristic
179	C₂×C₅₃	C₂×#350	1	1	yes	characteristic
180	C₂×C₅₃	#2117	1	1	yes	characteristic
181	D₅₃	—	4	2	no	—
182	C₂×#131	—	53	1	no	—
183	C₂×#131	—	53	1	no	—
184	C₄×#41	—	53	1	no	—
185	C₂×M₃₂	—	53	2	no	—
186	C₂²×C₁₆	—	53	2	no	—
187	#405	—	53	1	no	—
188	#405	—	53	1	no	—
189	#405	—	106	1	no	—
190	#405	—	106	1	no	—
191	C₄×M₁₆	—	106	1	no	—
192	C₄×M₁₆	#6157	1	1	yes	characteristic
193	C₂×C₄×C₈	—	53	1	no	—
194	C₂×C₄×C₈	—	106	1	no	—
195	C₄⋊C₁₆	—	106	1	no	—
196	C₄⋊C₁₆	—	106	1	no	—
197	C₄⋊C₁₆	—	106	1	no	—
198	C₄⋊C₁₆	—	106	1	no	—
199	#350	—	106	4	no	—
200	C₅₃	#12787	1	1	yes	characteristic, complemented, Sylow
201	C₂×#41	—	53	1	no	—
202	#131	—	53	2	no	—
203	#131	—	53	2	no	—
204	#131	—	106	1	no	—
205	#131	—	106	1	no	—
206	#131	—	106	2	no	—
207	#131	—	106	2	no	—
208	C₂×M₁₆	—	106	1	no	—
209	C₂×M₁₆	—	106	1	no	—
210	C₂×M₁₆	C₂×#6157	1	1	yes	characteristic
211	C₂×M₁₆	C₂×#6157	1	1	yes	characteristic
212	C₂²×C₈	—	53	1	no	—
213	C₂²×C₈	—	53	1	no	—
214	C₂²×C₈	—	106	1	no	—
215	C₂²×C₈	—	106	1	no	—
216	C₄×Q₈	—	53	1	no	—
217	C₄×Q₈	—	53	1	no	—
218	C₄×D₄	—	53	1	no	—
219	C₄×D₄	—	53	1	no	—
220	C₄×D₄	—	106	1	no	—
221	C₄×D₄	—	106	1	no	—
222	#117	—	53	1	no	—
223	#117	—	106	1	no	—
224	C₂×C₄²	—	106	1	no	—
225	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
226	M₃₂	—	106	2	no	—
227	M₃₂	—	106	2	no	—
228	C₂×C₁₆	—	106	2	no	—
229	C₂×C₁₆	—	106	2	no	—
230	C₂×C₁₆	—	106	2	no	—
231	C₂×C₁₆	—	106	2	no	—
232	#97	—	2	1	no	—
233	#97	—	106	1	no	—
234	#97	—	106	1	no	—
235	C₄×C₈	—	2	1	no	—
236	C₄×C₈	—	53	1	no	—
237	C₄×C₈	—	53	1	no	—
238	C₄×C₈	—	53	1	no	—
239	C₄×C₈	—	53	1	no	—
240	C₄×C₈	—	106	1	no	—
241	C₄×C₈	—	106	1	no	—
242	#41	—	53	2	no	—
243	#41	—	53	2	no	—
244	#41	—	106	1	no	—
245	#41	—	106	1	no	—
246	C₂×Q₈	—	53	1	no	—
247	C₂×D₄	—	53	1	no	—
248	C₂×D₄	—	106	1	no	—
249	C₂²×C₄	—	106	1	no	—
250	C₂²×C₄	—	106	1	no	—
251	C₂²×C₄	—	106	1	no	—
252	C₂²×C₄	#11719936	1	1	yes	characteristic
253	C₂²×C₄	#11719936	1	1	yes	characteristic
254	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
255	M₁₆	—	2	1	no	—
256	M₁₆	—	2	1	no	—
257	M₁₆	—	106	2	no	—
258	M₁₆	—	106	2	no	—
259	M₁₆	—	106	4	no	—
260	M₁₆	#11719936	1	2	yes	—
261	M₁₆	C₄⋊#6157	1	2	yes	—
262	C₂×C₈	—	2	1	no	—
263	C₂×C₈	—	2	1	no	—
264	C₂×C₈	—	53	1	no	—
265	C₂×C₈	—	53	1	no	—
266	C₂×C₈	—	53	1	no	—
267	C₂×C₈	—	53	1	no	—
268	C₂×C₈	—	53	1	no	—
269	C₂×C₈	—	53	1	no	—
270	C₂×C₈	—	53	1	no	—
271	C₂×C₈	—	53	1	no	—
272	C₂×C₈	—	106	2	no	—
273	C₂×C₈	—	106	2	no	—
274	C₂×C₈	—	106	4	no	—
275	C₂×C₈	—	106	1	no	—
276	C₂×C₈	—	106	1	no	—
277	C₄⋊C₄	—	53	1	no	—
278	C₄⋊C₄	—	53	1	no	—
279	C₄⋊C₄	—	106	1	no	—
280	C₄⋊C₄	—	106	1	no	—
281	C₂²⋊C₄	—	106	1	no	—
282	C₂²⋊C₄	—	106	1	no	—
283	C₂²⋊C₄	—	106	1	no	—
284	C₄²	—	106	1	no	—
285	C₄²	—	106	1	no	—
286	C₄²	—	106	1	no	—
287	C₄²	D₅₃⋊C₈	1	1	yes	characteristic
288	C₄²	D₅₃⋊C₈	1	1	yes	characteristic
289	C₄²	C₂²⋊#6157	1	1	yes	characteristic
290	C₄²	C₂²⋊#6157	1	1	yes	characteristic
291	C₁₆	—	212	2	no	—
292	C₁₆	—	212	2	no	—
293	C₂³	—	106	1	no	—
294	C₂³	C₂×#11719936	1	1	yes	characteristic
295	Q₈	—	53	2	no	—
296	Q₈	—	106	1	no	—
297	D₄	—	53	2	no	—
298	D₄	—	106	2	no	—
299	D₄	—	106	2	no	—
300	D₄	—	106	1	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	4	1	no	—
303	C₂×C₄	—	106	1	no	—
304	C₂×C₄	—	106	2	no	—
305	C₂×C₄	—	106	2	no	—
306	C₂×C₄	—	106	1	no	—
307	C₂×C₄	—	106	1	no	—
308	C₂×C₄	—	212	2	no	—
309	C₂×C₄	—	212	2	no	—
310	C₂×C₄	#13133274	1	1	yes	characteristic
311	C₂×C₄	#13133274	1	1	yes	characteristic
312	C₂×C₄	#13133274	1	1	yes	characteristic
313	C₂×C₄	#13133274	1	1	yes	characteristic
314	C₂×C₄	C₂×C₂²⋊#6157	1	1	yes	characteristic
315	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
316	C₂×C₄	#13133296	1	1	yes	characteristic
317	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
318	C₂×C₄	D₅₃⋊M₁₆	1	1	yes	characteristic
319	C₂×C₄	C₂×#11719936	1	1	yes	characteristic
320	C₈	—	2	2	no	—
321	C₈	—	2	2	no	—
322	C₈	—	53	1	no	—
323	C₈	—	53	1	no	—
324	C₈	—	53	1	no	—
325	C₈	—	53	1	no	—
326	C₈	—	106	1	no	—
327	C₈	—	106	1	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	106	2	no	—
331	C₂²	—	212	1	no	—
332	C₂²	—	212	1	no	—
333	C₂²	C₂×#13133274	1	1	yes	characteristic
334	C₂²	#16811627	1	1	yes	characteristic
335	C₂²	#16811645	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	106	2	no	—
340	C₄	—	212	1	no	—
341	C₄	—	212	1	no	—
342	C₄	#16811522	1	2	yes	complemented
343	C₄	#16811565	1	1	yes	characteristic
344	C₄	#16811594	1	1	yes	characteristic
345	C₄	D₅₃⋊(C₄⋊C₈)	1	1	yes	characteristic
346	C₄	#16811642	1	1	yes	characteristic
347	C₂	—	2	1	no	—
348	C₂	—	2	1	no	—
349	C₂	—	212	2	no	—
350	C₂	#20290570	1	1	yes	characteristic
351	C₂	C₂×#16811522	1	1	yes	characteristic
352	C₂	#20290959	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊#6157
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#6157
Fitting	C₄×M₁₆×C₅₃	C₄
socle	C₂²×C₅₃	C₂×C₂²⋊C₈
prime radical	C₂×#11719945	C₂×C₄
cosocle	C₂²×#88322	C₂³
hypercenter	C₄×M₁₆	#6157
hypocenter	C₅₃	#12787
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
1. #20291047	C₂	semidirect
2. C₂×C₄⋊#11719910	C₂	semidirect
4. #20289770	C₂	semidirect
163. C₄×C₅₃	#350	semidirect
200. C₅₃	#12787	semidirect
342. C₄	#16811522	semidirect

Maximal subgroups

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

subgroup	index
#20291047	2
C₂×C₄⋊#11719910	2
C₂×#16811522	2
#20289770	2
C₄×#13133225	2
#12787	53