Subgroups — #28311140

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	#20291049	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16811544	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16811521	C₂	1	2	yes	maximal
4	#20289770	C₂	1	2	yes	maximal, complemented
5	#131×Dic₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133291	C₂²	1	2	yes	—
7	C₂²×#11719910	C₂²	1	2	yes	—
8	#16811544	—	2	1	no	—
9	#16811544	—	2	1	no	—
10	#16811544	—	2	1	no	—
11	#16811544	—	2	1	no	—
12	#16811521	—	2	4	no	—
13	C₂×#13133224	C₂²	1	1	yes	characteristic
14	#41×Dic₅₃	C₄	1	1	yes	characteristic
15	#16810581	—	2	1	no	—
16	#16810581	C₂²	1	1	yes	characteristic
17	#16810578	—	2	1	no	—
18	#16810578	C₄	1	1	yes	characteristic
19	M₁₆×Dic₅₃	—	2	1	no	—
20	M₁₆×Dic₅₃	C₄	1	1	yes	characteristic
21	C₂×C₈×Dic₅₃	—	2	1	no	—
22	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
23	C₂×#131×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	#13133224	—	2	1	no	—
31	#13133224	—	2	1	no	—
32	#13133224	D₄	1	2	yes	—
33	#13133224	Q₈	1	2	yes	—
34	C₂×#11719917	—	2	1	no	—
35	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
36	C₂²×#88322	—	2	1	no	—
37	C₂²×#88322	C₂³	1	1	yes	characteristic
38	Q₈×Dic₅₃	C₈	1	1	yes	characteristic
39	Q₈×Dic₅₃	C₈	1	1	yes	characteristic
40	D₄×Dic₅₃	—	2	1	no	—
41	D₄×Dic₅₃	—	2	1	no	—
42	D₄×Dic₅₃	C₈	1	1	yes	characteristic
43	D₄×Dic₅₃	C₈	1	1	yes	characteristic
44	#13133176	—	2	1	no	—
45	#13133176	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×Dic₅₃	—	2	1	no	—
47	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
48	#13133127	—	2	1	no	—
49	#13133127	—	2	1	no	—
50	#13133127	—	2	1	no	—
51	C₈×Dic₅₃	—	2	1	no	—
52	C₈×Dic₅₃	—	2	1	no	—
53	C₈×Dic₅₃	—	2	1	no	—
54	C₈×Dic₅₃	D₄	1	1	yes	characteristic
55	C₈×Dic₅₃	D₄	1	1	yes	characteristic
56	C₈×Dic₅₃	D₄	1	1	yes	characteristic
57	C₈×Dic₅₃	D₄	1	1	yes	characteristic
58	C₂×#41×C₅₃	C₂×C₄	1	1	yes	characteristic
59	#131×C₅₃	—	2	2	no	—
60	#131×C₅₃	—	2	2	no	—
61	C₂×M₁₆×C₅₃	—	2	1	no	—
62	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
63	C₂²×C₈×C₅₃	—	2	1	no	—
64	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
65	#41×C₅₃	—	2	1	no	—
66	#41×C₅₃	—	2	1	no	—
67	#41×C₅₃	C₄⋊C₄	1	2	yes	—
68	#41×C₅₃	M₁₆	1	2	yes	—
69	C₂×Q₈×C₅₃	C₂×C₈	1	1	yes	characteristic
70	C₂×D₄×C₅₃	—	2	1	no	—
71	C₂×D₄×C₅₃	C₂×C₈	1	1	yes	characteristic
72	C₂²×C₄×C₅₃	—	2	1	no	—
73	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
74	C₂²×Dic₅₃	—	2	1	no	—
75	C₂²×Dic₅₃	—	2	1	no	—
76	C₂²×Dic₅₃	M₁₆	1	1	yes	characteristic
77	C₂²×Dic₅₃	M₁₆	1	1	yes	characteristic
78	M₁₆×C₅₃	—	2	4	no	—
79	M₁₆×C₅₃	—	2	1	no	—
80	M₁₆×C₅₃	M₁₆	1	2	yes	—
81	C₂×C₈×C₅₃	—	2	4	no	—
82	C₂×C₈×C₅₃	—	2	1	no	—
83	C₂×C₈×C₅₃	—	2	1	no	—
84	C₂×C₈×C₅₃	—	2	1	no	—
85	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
87	C₂²⋊Dic₅₃	—	2	1	no	—
88	C₂²⋊Dic₅₃	—	2	1	no	—
89	C₂²⋊Dic₅₃	—	2	1	no	—
90	C₄⋊Dic₅₃	—	2	1	no	—
91	C₄⋊Dic₅₃	—	2	1	no	—
92	C₄⋊Dic₅₃	C₂×C₈	1	1	yes	characteristic
93	C₄⋊Dic₅₃	C₂×C₈	1	1	yes	characteristic
94	C₄×Dic₅₃	—	2	1	no	—
95	C₄×Dic₅₃	—	2	1	no	—
96	C₄×Dic₅₃	—	2	1	no	—
97	C₄×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
98	C₄×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
99	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
100	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
101	#11719917	—	2	1	no	—
102	#11719917	—	2	2	no	—
103	#11719917	—	2	2	no	—
104	#11719917	C₄⋊C₄	1	2	yes	—
105	C₂×#88322	—	2	1	no	—
106	C₂×#88322	—	2	2	no	—
107	C₂×#88322	—	2	2	no	—
108	C₂×#88322	C₂×D₄	1	1	yes	characteristic
109	C₂×#88322	C₂×D₄	1	1	yes	characteristic
110	C₂×#88322	C₂×D₄	1	1	yes	characteristic
111	C₂×#88322	C₂×Q₈	1	1	yes	characteristic
112	C₂×#88322	#41	1	1	yes	characteristic
113	C₂×#88322	#41	1	1	yes	characteristic
114	#11719910	—	4	2	no	—
115	#11719910	—	4	2	no	—
116	C₂³×C₅₃	—	2	1	no	—
117	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
118	Q₈×C₅₃	—	2	1	no	—
119	Q₈×C₅₃	C₄⋊C₈	1	2	yes	—
120	D₄×C₅₃	—	2	2	no	—
121	D₄×C₅₃	—	2	2	no	—
122	D₄×C₅₃	—	2	1	no	—
123	D₄×C₅₃	C₄⋊C₈	1	2	yes	—
124	C₂×C₄×C₅₃	—	2	1	no	—
125	C₂×C₄×C₅₃	—	2	2	no	—
126	C₂×C₄×C₅₃	—	2	2	no	—
127	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
130	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
131	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
132	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
133	C₂×Dic₅₃	—	2	2	no	—
134	C₂×Dic₅₃	—	2	1	no	—
135	C₂×Dic₅₃	—	2	1	no	—
136	C₂×Dic₅₃	—	4	1	no	—
137	C₂×Dic₅₃	—	4	2	no	—
138	C₂×Dic₅₃	—	4	2	no	—
139	C₂×Dic₅₃	C₂²⋊C₈	1	1	yes	characteristic
140	C₂×Dic₅₃	C₂²⋊C₈	1	1	yes	characteristic
141	C₂×Dic₅₃	C₂²⋊C₈	1	1	yes	characteristic
142	C₂×Dic₅₃	C₂²⋊C₈	1	1	yes	characteristic
143	C₈×C₅₃	—	2	1	no	—
144	C₈×C₅₃	—	2	2	no	—
145	C₈×C₅₃	—	2	1	no	—
146	#88322	—	2	2	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	—	2	2	no	—
155	C₂²×C₅₃	—	4	1	no	—
156	C₂²×C₅₃	—	4	1	no	—
157	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
158	C₂²×C₅₃	#423	1	1	yes	characteristic
159	C₂²×C₅₃	#432	1	1	yes	characteristic
160	C₄×C₅₃	—	2	2	no	—
161	C₄×C₅₃	#380	1	1	yes	characteristic
162	C₄×C₅₃	#407	1	1	yes	characteristic
163	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
164	C₄×C₅₃	#431	1	1	yes	characteristic
165	Dic₅₃	—	2	1	no	—
166	Dic₅₃	—	2	1	no	—
167	Dic₅₃	—	2	1	no	—
168	Dic₅₃	—	4	1	no	—
169	Dic₅₃	—	4	1	no	—
170	Dic₅₃	#350	1	2	yes	complemented
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	—	4	2	no	—
179	C₂×C₅₃	#1797	1	1	yes	characteristic
180	C₂×C₅₃	C₂×#350	1	1	yes	characteristic
181	C₂×C₅₃	#2117	1	1	yes	characteristic
182	C₂×#131	—	53	1	no	—
183	C₂×#131	#6157	1	1	yes	characteristic
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	—	53	1	no	—
192	C₄×M₁₆	—	106	1	no	—
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	C₂×#6157	1	1	yes	characteristic
202	#131	—	2	2	no	—
203	#131	—	2	2	no	—
204	#131	—	53	2	no	—
205	#131	—	53	2	no	—
206	#131	—	106	1	no	—
207	#131	—	106	1	no	—
208	C₂×M₁₆	—	2	1	no	—
209	C₂×M₁₆	—	53	1	no	—
210	C₂×M₁₆	—	106	1	no	—
211	C₂×M₁₆	C₂×#6157	1	1	yes	characteristic
212	C₂²×C₈	—	2	1	no	—
213	C₂²×C₈	—	53	1	no	—
214	C₂²×C₈	—	106	1	no	—
215	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
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	—	53	1	no	—
225	C₂×C₄²	—	106	1	no	—
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	—	106	1	no	—
233	#97	—	106	1	no	—
234	#97	—	106	1	no	—
235	C₄×C₈	—	53	1	no	—
236	C₄×C₈	—	53	1	no	—
237	C₄×C₈	—	53	1	no	—
238	C₄×C₈	—	53	1	no	—
239	C₄×C₈	—	106	1	no	—
240	C₄×C₈	—	106	1	no	—
241	C₄×C₈	—	106	1	no	—
242	#41	—	2	1	no	—
243	#41	—	2	1	no	—
244	#41	C₄⋊#6157	1	2	yes	—
245	#41	#11719940	1	2	yes	—
246	C₂×Q₈	D₅₃⋊C₈	1	1	yes	characteristic
247	C₂×D₄	—	2	1	no	—
248	C₂×D₄	D₅₃⋊C₈	1	1	yes	characteristic
249	C₂²×C₄	—	2	1	no	—
250	C₂²×C₄	—	53	1	no	—
251	C₂²×C₄	—	53	1	no	—
252	C₂²×C₄	—	106	1	no	—
253	C₂²×C₄	—	106	1	no	—
254	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
255	M₁₆	—	2	4	no	—
256	M₁₆	—	2	1	no	—
257	M₁₆	—	53	2	no	—
258	M₁₆	—	106	2	no	—
259	M₁₆	—	106	2	no	—
260	M₁₆	—	106	1	no	—
261	M₁₆	#11719940	1	2	yes	—
262	C₂×C₈	—	2	4	no	—
263	C₂×C₈	—	2	1	no	—
264	C₂×C₈	—	2	1	no	—
265	C₂×C₈	—	2	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	1	no	—
275	C₂×C₈	C₂²⋊#6157	1	1	yes	characteristic
276	C₂×C₈	C₂²⋊#6157	1	1	yes	characteristic
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	—	53	1	no	—
285	C₄²	—	53	1	no	—
286	C₄²	—	53	1	no	—
287	C₄²	—	53	1	no	—
288	C₄²	—	106	1	no	—
289	C₄²	—	106	1	no	—
290	C₄²	—	106	1	no	—
291	C₁₆	—	212	2	no	—
292	C₁₆	—	212	2	no	—
293	C₂³	—	2	1	no	—
294	C₂³	D₅₃⋊M₁₆	1	1	yes	characteristic
295	Q₈	—	2	1	no	—
296	Q₈	#13133284	1	2	yes	—
297	D₄	—	2	2	no	—
298	D₄	—	2	2	no	—
299	D₄	—	2	1	no	—
300	D₄	#13133284	1	2	yes	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	53	1	no	—
305	C₂×C₄	—	53	1	no	—
306	C₂×C₄	—	53	1	no	—
307	C₂×C₄	—	53	1	no	—
308	C₂×C₄	—	106	2	no	—
309	C₂×C₄	—	106	1	no	—
310	C₂×C₄	—	106	1	no	—
311	C₂×C₄	—	212	2	no	—
312	C₂×C₄	—	212	1	no	—
313	C₂×C₄	—	212	2	no	—
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	C₂×#11719940	1	1	yes	characteristic
319	C₂×C₄	C₂×#11719940	1	1	yes	characteristic
320	C₈	—	2	1	no	—
321	C₈	—	2	2	no	—
322	C₈	—	2	1	no	—
323	C₈	—	53	1	no	—
324	C₈	—	53	1	no	—
325	C₈	—	53	1	no	—
326	C₈	—	53	1	no	—
327	C₈	—	106	2	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	2	no	—
331	C₂²	—	4	1	no	—
332	C₂²	—	4	1	no	—
333	C₂²	D₅₃⋊(C₂²⋊C₈)	1	1	yes	characteristic
334	C₂²	#16811626	1	1	yes	characteristic
335	C₂²	#16811643	1	1	yes	characteristic
336	C₄	—	2	2	no	—
337	C₄	—	53	2	no	—
338	C₄	—	106	1	no	—
339	C₄	—	106	1	no	—
340	C₄	—	106	1	no	—
341	C₄	—	212	1	no	—
342	C₄	—	212	1	no	—
343	C₄	#16811565	1	1	yes	characteristic
344	C₄	#16811593	1	1	yes	characteristic
345	C₄	C₂×#13133284	1	1	yes	characteristic
346	C₄	#16811640	1	1	yes	characteristic
347	C₂	—	2	1	no	—
348	C₂	—	2	1	no	—
349	C₂	—	4	2	no	—
350	C₂	#20290571	1	1	yes	characteristic
351	C₂	D₅₃⋊#350	1	1	yes	characteristic
352	C₂	#20290958	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	C₂×#131×C₅₃	C₄
socle	C₂²×C₅₃	C₂×C₂²⋊C₈
prime radical	C₂×#41×C₅₃	C₂×C₄
cosocle	C₂²×#88322	C₂³
hypercenter	C₂×#131	#6157
hypocenter	C₅₃	#12787
LCS	G ▷ C₂×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
1. #20291049	C₂	semidirect
2. C₂×#16811544	C₂	semidirect
4. #20289770	C₂	semidirect
170. Dic₅₃	#350	semidirect
200. C₅₃	#12787	semidirect

Maximal subgroups

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

subgroup	index
#20291049	2
C₂×#16811544	2
C₂×#16811521	2
#20289770	2
#131×Dic₅₃	2
#12787	53