Subgroups — #28328348

Order 13568.

646 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811682	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16811681	C₂	1	1	yes	maximal, characteristic, complemented
3	#20291106	C₂	1	4	yes	maximal, complemented
4	#20291103	C₂	1	4	yes	maximal, complemented
5	#20291058	C₂	1	1	yes	maximal, characteristic, complemented
6	#20290959	C₂	1	1	yes	maximal, characteristic, complemented
7	#20290957	C₂	1	1	yes	maximal, characteristic, complemented
8	#20290924	C₂	1	1	yes	maximal, characteristic, complemented
9	#20283614	C₂	1	1	yes	maximal, characteristic
10	#16811682	—	2	1	no	—
11	#16811682	—	2	1	no	—
12	#16811682	C₂²	1	2	yes	complemented
13	#16811682	C₂²	1	2	yes	complemented
14	#16811681	—	2	1	no	—
15	#16811681	—	2	1	no	—
16	#16811681	C₂²	1	2	yes	complemented
17	#16811681	C₂²	1	2	yes	complemented
18	C₂×#13133292	—	2	1	no	—
19	C₂×#13133291	—	2	1	no	—
20	C₂×#13133291	C₂²	1	1	yes	characteristic
21	C₂×#13133291	C₂²	1	1	yes	characteristic
22	C₂×#13133291	C₂²	1	1	yes	characteristic
23	C₂×D₅₃⋊C₁₆	—	2	1	no	—
24	C₂²×#11719910	—	2	1	no	—
25	C₂²×#11719910	C₂²	1	1	yes	characteristic
26	#16811546	C₂²	1	4	yes	complemented
27	#16811522	C₂²	1	4	yes	complemented
28	#16811521	C₂²	1	4	yes	complemented
29	#16811511	C₂²	1	2	yes	complemented
30	C₄×#11719910	C₂²	1	2	yes	complemented
31	#16811421	C₄	1	1	yes	characteristic
32	C₂×#13133225	C₂²	1	1	yes	characteristic
33	C₂×#13133224	C₂²	1	1	yes	characteristic
34	#16810742	C₄	1	4	yes	—
35	#16810737	C₂²	1	4	yes	—
36	#16810671	C₄	1	1	yes	characteristic
37	#16810641	C₄	1	1	yes	characteristic
38	#16810579	C₂²	1	1	yes	characteristic
39	C₅₃×#421	C₄	1	1	yes	characteristic
40	#13133292	—	2	2	no	—
41	#13133292	—	2	2	no	—
42	#13133291	—	2	1	no	—
43	#13133291	—	2	1	no	—
44	#13133291	—	2	1	no	—
45	#13133291	—	2	1	no	—
46	#13133291	—	2	2	no	—
47	#13133291	—	2	2	no	—
48	#13133291	C₂³	1	2	yes	—
49	#13133291	C₂³	1	2	yes	—
50	D₅₃⋊C₁₆	—	2	2	no	—
51	D₅₃⋊C₁₆	—	2	2	no	—
52	C₂×#11719910	—	2	1	no	—
53	C₂×#11719910	—	2	1	no	—
54	C₂×#11719910	—	2	2	no	—
55	C₂×#11719910	—	2	2	no	—
56	C₂×#11719910	D₄	1	1	yes	characteristic
57	C₂×#11719910	D₄	1	1	yes	characteristic
58	C₂×#11719910	D₄	1	1	yes	characteristic
59	C₂×#11719910	D₄	1	1	yes	characteristic
60	C₂×#11719910	C₂³	1	2	yes	—
61	C₂×#11719910	C₂³	1	1	yes	characteristic
62	C₂×#11719910	C₂³	1	1	yes	characteristic
63	C₂×#11719945	C₂×C₄	1	1	yes	characteristic
64	#13133248	C₈	1	2	yes	—
65	#13133247	C₈	1	2	yes	—
66	#13133243	C₈	1	2	yes	—
67	#13133242	C₈	1	2	yes	—
68	#13133237	C₂×C₄	1	4	yes	—
69	#13133225	—	2	1	no	—
70	#13133225	—	2	1	no	—
71	#13133225	C₂×C₄	1	2	yes	—
72	#13133225	C₂³	1	2	yes	—
73	#13133224	—	2	1	no	—
74	#13133224	—	2	1	no	—
75	#13133224	C₂×C₄	1	2	yes	—
76	#13133224	C₂³	1	2	yes	—
77	C₂×#11719917	—	2	1	no	—
78	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
79	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
80	C₂×#11719912	—	2	1	no	—
81	C₂²×#88322	—	2	1	no	—
82	C₂²×#88322	C₂³	1	1	yes	characteristic
83	C₂×C₈×D₅₃	—	2	1	no	—
84	#13133152	C₂×C₄	1	4	yes	—
85	#13133145	C₂×C₄	1	4	yes	—
86	#13133128	C₂×C₄	1	2	yes	—
87	C₄×#88322	C₂³	1	2	yes	—
88	C₂×#41×C₅₃	C₂×C₄	1	1	yes	characteristic
89	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
90	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
91	#104×C₅₃	C₂×C₄	1	4	yes	—
92	#41×C₅₃	—	2	1	no	—
93	#41×C₅₃	—	2	1	no	—
94	#41×C₅₃	C₂×C₈	1	2	yes	—
95	#41×C₅₃	C₂²×C₄	1	2	yes	—
96	C₂×Q₈×C₅₃	C₂×C₈	1	1	yes	characteristic
97	C₂×D₄×C₅₃	—	2	1	no	—
98	C₂×D₄×C₅₃	C₂×C₈	1	1	yes	characteristic
99	C₂²×C₄×C₅₃	—	2	1	no	—
100	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₂²⋊D₅₃	—	2	1	no	—
102	#11719945	—	2	1	no	—
103	#11719945	—	2	1	no	—
104	#11719945	C₂×C₈	1	2	yes	—
105	#11719945	C₂²×C₄	1	2	yes	—
106	C₂×D₂₁₂	C₂×C₈	1	1	yes	characteristic
107	C₂×C₄×D₅₃	—	2	1	no	—
108	C₂×Dic₁₀₆	C₂×C₈	1	1	yes	characteristic
109	M₁₆×C₅₃	—	2	1	no	—
110	M₁₆×C₅₃	C₂²×C₄	1	2	yes	—
111	C₂×C₈×C₅₃	—	2	1	no	—
112	C₄⋊C₄×C₅₃	—	2	1	no	—
113	C₂²⋊C₄×C₅₃	—	2	1	no	—
114	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
115	#11719925	C₂×C₈	1	4	yes	—
116	#11719924	C₂×C₈	1	4	yes	—
117	#11719923	C₂×C₈	1	4	yes	—
118	#11719922	C₂×C₈	1	4	yes	—
119	#11719917	—	2	1	no	—
120	#11719917	—	2	2	no	—
121	#11719917	—	2	2	no	—
122	#11719917	C₂×C₈	1	2	yes	—
123	#11719917	C₂×C₈	1	2	yes	—
124	#11719917	C₂²×C₄	1	2	yes	—
125	C₂×#88322	—	2	1	no	—
126	C₂×#88322	—	2	2	no	—
127	C₂×#88322	—	2	2	no	—
128	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
129	C₂×#88322	C₂²×C₄	1	2	yes	—
130	C₂×#88322	C₂×D₄	1	1	yes	characteristic
131	C₂×#88322	C₂×D₄	1	1	yes	characteristic
132	C₂×#88322	#41	1	1	yes	characteristic
133	C₂×#88322	#41	1	1	yes	characteristic
134	C₂×#88322	C₂⁴	1	1	yes	characteristic
135	#11719912	—	2	2	no	—
136	#11719912	—	2	2	no	—
137	C₈×D₅₃	—	2	2	no	—
138	C₈×D₅₃	—	2	2	no	—
139	#11719910	—	2	2	no	—
140	#11719910	—	2	2	no	—
141	#11719910	—	2	2	no	—
142	#11719910	C₂×D₄	1	1	yes	characteristic
143	#11719910	C₂×D₄	1	1	yes	characteristic
144	#11719910	C₂×D₄	1	1	yes	characteristic
145	#11719910	C₂×D₄	1	1	yes	characteristic
146	C₂³×C₅₃	—	2	1	no	—
147	C₂³×C₅₃	#131	1	1	yes	characteristic
148	C₂²×D₅₃	—	2	1	no	—
149	Q₈×C₅₃	—	2	1	no	—
150	Q₈×C₅₃	C₂²×C₈	1	2	yes	—
151	D₄×C₅₃	—	2	2	no	—
152	D₄×C₅₃	—	2	2	no	—
153	D₄×C₅₃	—	2	1	no	—
154	D₄×C₅₃	C₂²×C₈	1	2	yes	—
155	C₂×C₄×C₅₃	—	2	1	no	—
156	C₂×C₄×C₅₃	—	2	2	no	—
157	C₂×C₄×C₅₃	—	2	2	no	—
158	C₂×C₄×C₅₃	—	2	2	no	—
159	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
160	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
161	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
162	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
163	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
164	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
165	C₂²⋊D₅₃	—	2	2	no	—
166	C₂²⋊D₅₃	—	2	2	no	—
167	C₂×Dic₅₃	—	2	1	no	—
168	D₂₁₂	—	2	1	no	—
169	D₂₁₂	C₂²×C₈	1	2	yes	—
170	C₄×D₅₃	—	2	2	no	—
171	C₄×D₅₃	—	2	2	no	—
172	Dic₁₀₆	—	2	1	no	—
173	Dic₁₀₆	C₂²×C₈	1	2	yes	—
174	C₈×C₅₃	—	2	2	no	—
175	#88322	—	2	2	no	—
176	#88322	C₄×D₄	1	1	yes	characteristic
177	#88322	C₄×D₄	1	1	yes	characteristic
178	#88322	C₂²×C₈	1	4	yes	—
179	#88322	C₂²×D₄	1	1	yes	characteristic
180	#88322	C₂×#41	1	1	yes	characteristic
181	#33441	—	53	1	no	maximal, Sylow
182	C₂²×C₅₃	—	2	2	no	—
183	C₂²×C₅₃	—	4	1	no	—
184	C₂²×C₅₃	—	4	2	no	—
185	C₂²×C₅₃	C₈×D₄	1	1	yes	characteristic
186	C₂²×C₅₃	C₈×D₄	1	1	yes	characteristic
187	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
188	C₂×D₅₃	—	2	2	no	—
189	C₂×D₅₃	—	4	2	no	—
190	C₄×C₅₃	—	2	2	no	—
191	C₄×C₅₃	—	4	2	no	—
192	C₄×C₅₃	C₈×D₄	1	1	yes	characteristic
193	C₄×C₅₃	C₈×D₄	1	1	yes	characteristic
194	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
195	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
196	Dic₅₃	—	2	2	no	—
197	C₂×#504	—	53	1	no	—
198	C₂×#504	—	53	1	no	—
199	#2955	—	53	1	no	—
200	#2171	—	53	4	no	—
201	#2170	—	53	4	no	—
202	#2153	—	53	1	no	—
203	#2117	—	53	1	no	—
204	#2117	—	53	1	no	—
205	#2108	—	53	1	no	—
206	C₂×C₅₃	—	2	1	no	—
207	C₂×C₅₃	—	2	1	no	—
208	C₂×C₅₃	—	2	1	no	—
209	C₂×C₅₃	—	4	2	no	—
210	C₂×C₅₃	C₂×C₈×D₄	1	1	yes	characteristic
211	D₅₃	—	4	2	no	—
212	#575	—	53	1	no	—
213	C₂×#131	—	53	1	no	—
214	C₂×#131	—	53	1	no	—
215	#504	—	53	2	no	—
216	#504	—	53	2	no	—
217	#504	—	53	2	no	—
218	#504	—	53	2	no	—
219	#504	—	106	1	no	—
220	#504	—	106	1	no	—
221	#504	—	106	1	no	—
222	#504	—	106	1	no	—
223	C₂×M₃₂	—	53	1	no	—
224	C₂×M₃₂	—	53	1	no	—
225	C₂×M₃₂	—	53	1	no	—
226	C₂×M₃₂	—	106	1	no	—
227	C₂×M₃₂	—	106	1	no	—
228	C₂²×C₁₆	—	53	1	no	—
229	C₂²×C₁₆	—	106	1	no	—
230	C₂²×C₁₆	—	106	1	no	—
231	#444	—	53	4	no	—
232	#443	—	53	4	no	—
233	#430	—	53	1	no	—
234	#421	—	53	1	no	—
235	#421	#6157	1	1	yes	characteristic
236	#405	—	53	1	no	—
237	#364	—	53	4	no	—
238	#350	—	53	4	no	—
239	#350	—	53	4	no	—
240	#346	—	53	2	no	—
241	C₄×C₁₆	—	53	2	no	—
242	C₅₃	#33441	1	1	yes	characteristic, complemented, Sylow
243	C₂×#41	—	53	1	no	—
244	C₂×#41	C₂×#6157	1	1	yes	characteristic
245	#137	—	53	2	no	—
246	#137	—	53	2	no	—
247	AGL(1,ℤ/8)	—	53	2	no	—
248	AGL(1,ℤ/8)	—	53	2	no	—
249	#135	—	53	4	no	—
250	#131	—	53	2	no	—
251	#131	—	53	2	no	—
252	#131	—	53	2	no	—
253	#131	—	53	2	no	—
254	#131	—	106	1	no	—
255	#131	—	106	1	no	—
256	#131	—	106	1	no	—
257	#131	—	106	1	no	—
258	C₂×M₁₆	—	53	1	no	—
259	C₂×M₁₆	—	53	1	no	—
260	C₂×M₁₆	—	106	1	no	—
261	C₂×M₁₆	—	106	1	no	—
262	C₂×M₁₆	C₂×#6157	1	1	yes	characteristic
263	C₂²×C₈	—	53	1	no	—
264	C₂²×C₈	—	106	1	no	—
265	C₂²×C₈	—	106	1	no	—
266	#117	C₂×#6157	1	1	yes	characteristic
267	M₃₂	—	53	2	no	—
268	M₃₂	—	53	2	no	—
269	M₃₂	—	106	2	no	—
270	M₃₂	—	106	2	no	—
271	M₃₂	—	106	2	no	—
272	M₃₂	—	106	2	no	—
273	M₃₂	—	106	1	no	—
274	M₃₂	—	106	1	no	—
275	M₃₂	—	106	1	no	—
276	M₃₂	—	106	1	no	—
277	C₂×C₁₆	—	53	1	no	—
278	C₂×C₁₆	—	53	1	no	—
279	C₂×C₁₆	—	53	1	no	—
280	C₂×C₁₆	—	53	1	no	—
281	C₂×C₁₆	—	53	1	no	—
282	C₂×C₁₆	—	53	1	no	—
283	C₂×C₁₆	—	53	2	no	—
284	C₂×C₁₆	—	106	2	no	—
285	C₂×C₁₆	—	106	2	no	—
286	C₂×C₁₆	—	106	2	no	—
287	C₂×C₁₆	—	106	2	no	—
288	C₂×C₁₆	—	106	1	no	—
289	C₂×C₁₆	—	106	1	no	—
290	#108	—	53	4	no	—
291	#104	—	53	4	no	—
292	#104	C₂×#6157	1	4	yes	—
293	#97	—	53	2	no	—
294	C₄×C₈	—	53	2	no	—
295	#41	—	2	1	no	—
296	#41	—	2	1	no	—
297	#41	—	53	2	no	—
298	#41	—	53	2	no	—
299	#41	—	106	1	no	—
300	#41	—	106	1	no	—
301	#41	D₅₃⋊C₈	1	2	yes	—
302	#41	C₂²×#6157	1	2	yes	—
303	C₂×Q₈	—	53	1	no	—
304	C₂×Q₈	D₅₃⋊C₈	1	1	yes	characteristic
305	C₂×D₄	—	2	1	no	—
306	C₂×D₄	—	53	1	no	—
307	C₂×D₄	—	106	1	no	—
308	C₂×D₄	D₅₃⋊C₈	1	1	yes	characteristic
309	C₂²×C₄	—	2	1	no	—
310	C₂²×C₄	—	106	1	no	—
311	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
312	Q₁₆	—	53	4	no	—
313	SD₁₆	—	53	4	no	—
314	SD₁₆	—	53	4	no	—
315	D₈	—	53	4	no	—
316	M₁₆	—	2	1	no	—
317	M₁₆	—	53	2	no	—
318	M₁₆	—	53	2	no	—
319	M₁₆	—	53	2	no	—
320	M₁₆	—	106	2	no	—
321	M₁₆	—	106	2	no	—
322	M₁₆	—	106	2	no	—
323	M₁₆	—	106	2	no	—
324	M₁₆	—	106	1	no	—
325	M₁₆	C₂²×#6157	1	2	yes	—
326	C₂×C₈	—	2	1	no	—
327	C₂×C₈	—	53	1	no	—
328	C₂×C₈	—	53	1	no	—
329	C₂×C₈	—	53	1	no	—
330	C₂×C₈	—	53	1	no	—
331	C₂×C₈	—	53	1	no	—
332	C₂×C₈	—	53	2	no	—
333	C₂×C₈	—	53	1	no	—
334	C₂×C₈	—	106	2	no	—
335	C₂×C₈	—	106	2	no	—
336	C₂×C₈	—	106	2	no	—
337	C₂×C₈	—	106	2	no	—
338	C₂×C₈	—	106	1	no	—
339	C₄⋊C₄	—	2	1	no	—
340	C₂²⋊C₄	—	2	1	no	—
341	C₄²	C₂²×#6157	1	2	yes	—
342	C₁₆	—	53	1	no	—
343	C₁₆	—	53	1	no	—
344	C₁₆	—	53	1	no	—
345	C₁₆	—	53	1	no	—
346	C₁₆	—	106	2	no	—
347	C₁₆	—	106	2	no	—
348	C₁₆	—	106	2	no	—
349	C₂³	—	2	1	no	—
350	C₂³	—	106	1	no	—
351	C₂³	#13133304	1	1	yes	characteristic
352	Q₈	—	2	1	no	—
353	Q₈	—	53	2	no	—
354	Q₈	—	106	1	no	—
355	Q₈	C₂×D₅₃⋊C₈	1	2	yes	—
356	D₄	—	2	2	no	—
357	D₄	—	2	2	no	—
358	D₄	—	2	1	no	—
359	D₄	—	53	2	no	—
360	D₄	—	106	2	no	—
361	D₄	—	106	2	no	—
362	D₄	—	106	1	no	—
363	D₄	C₂×D₅₃⋊C₈	1	2	yes	—
364	C₂×C₄	—	2	1	no	—
365	C₂×C₄	—	2	2	no	—
366	C₂×C₄	—	2	2	no	—
367	C₂×C₄	—	2	2	no	—
368	C₂×C₄	—	106	1	no	—
369	C₂×C₄	—	106	2	no	—
370	C₂×C₄	—	106	2	no	—
371	C₂×C₄	D₄×#6157	1	1	yes	characteristic
372	C₂×C₄	D₄×#6157	1	1	yes	characteristic
373	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
374	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
375	C₂×C₄	#13133304	1	1	yes	characteristic
376	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
377	C₈	—	2	2	no	—
378	C₈	—	53	1	no	—
379	C₈	—	53	1	no	—
380	C₈	—	53	1	no	—
381	C₈	—	53	1	no	—
382	C₈	—	53	4	no	—
383	C₈	—	106	2	no	—
384	C₂²	—	2	2	no	—
385	C₂²	—	4	1	no	—
386	C₂²	—	4	2	no	—
387	C₂²	—	106	2	no	—
388	C₂²	—	212	2	no	—
389	C₂²	#16811650	1	1	yes	characteristic
390	C₂²	#16811650	1	1	yes	characteristic
391	C₂²	C₂×#13133304	1	1	yes	characteristic
392	C₄	—	2	2	no	—
393	C₄	—	4	2	no	—
394	C₄	—	106	2	no	—
395	C₄	#16811650	1	1	yes	characteristic
396	C₄	#16811650	1	1	yes	characteristic
397	C₄	C₂×D₄×#6157	1	1	yes	characteristic
398	C₄	C₂²×D₅₃⋊C₈	1	1	yes	characteristic
399	C₂	—	2	1	no	—
400	C₂	—	2	1	no	—
401	C₂	—	2	1	no	—
402	C₂	—	4	2	no	—
403	C₂	—	212	2	no	—
404	C₂	C₂×#16811650	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₄×#6157
commutator	C₄×C₅₃	C₂³×C₈
Frattini	C₂×C₄	C₂³×#6157
Fitting	C₅₃×#421	C₄
socle	C₂×C₅₃	C₂×C₈×D₄
prime radical	#16811421	C₄
cosocle	C₂×#88322	C₂⁴
hypercenter	#421	#6157
hypocenter	C₅₃	#33441
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #421

Complements

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

N	H ≅ G/N	type
1. C₂×#16811682	C₂	semidirect
2. C₂×#16811681	C₂	semidirect
3. #20291106	C₂	semidirect
4. #20291103	C₂	semidirect
5. #20291058	C₂	semidirect
6. #20290959	C₂	semidirect
7. #20290957	C₂	semidirect
8. #20290924	C₂	semidirect
12. #16811682	C₂²	semidirect
13. #16811682	C₂²	semidirect
16. #16811681	C₂²	semidirect
17. #16811681	C₂²	semidirect
26. #16811546	C₂²	semidirect
27. #16811522	C₂²	semidirect
28. #16811521	C₂²	semidirect
29. #16811511	C₂²	semidirect
30. C₄×#11719910	C₂²	semidirect
242. C₅₃	#33441	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811682	2
C₂×#16811681	2
#20291106	2
#20291103	2
#20291058	2
#20290959	2
#20290957	2
#20290924	2
#20283614	2
#33441	53