Finite Group Databasebeta

← #20279871

Subgroups — #20279871

Order 6784.

672 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133195C211yesmaximal, characteristic, complemented
2C2×C22⋊C4×D53C211yesmaximal, characteristic
3#16810552C211yesmaximal, characteristic, complemented
4#16810551C211yesmaximal, characteristic, complemented
5#16810536C211yesmaximal, characteristic, complemented
6#16810535C211yesmaximal, characteristic, complemented
7#16810517C211yesmaximal, characteristic, complemented
8#16810514C211yesmaximal, characteristic, complemented
9#16810513C211yesmaximal, characteristic, complemented
10#16810485C211yesmaximal, characteristic, complemented
11#16810485C211yesmaximal, characteristic, complemented
12#16810478C211yesmaximal, characteristic
13#16810478C211yesmaximal, characteristic
14C4×#11719921C211yesmaximal, characteristic, complemented
15C53×#388C211yesmaximal, characteristic, complemented
16C24×D53C2211yescharacteristic
17C22×Dic106C2211yescharacteristic, complemented
18C22×C4×D53C2211yescharacteristic
19C22×C4×D53C2211yescharacteristic
20#13133195—24no—
21C2×#11719919C2211yescharacteristic, complemented
22C2×#11719919C2211yescharacteristic, complemented
23C2×C22⋊Dic53C2211yescharacteristic
24C2×C22⋊Dic53C2211yescharacteristic
25C2×C22⋊Dic53C2211yescharacteristic
26C2×#11719921C2211yescharacteristic
27C2×#11719921C2211yescharacteristic
28C2×#11719921C2211yescharacteristic
29C2×#11719921C2211yescharacteristic
30C2×#11719921C2211yescharacteristic
31C2×#11719921C2211yescharacteristic
32C2×#11719921C2211yescharacteristic
33C2×#11719921C2211yescharacteristic
34C2×#11719921C2211yescharacteristic
35C2×#11719921C2211yescharacteristic
36C22⋊C4×D53—24no—
37C2×C4×Dic53C2211yescharacteristic, complemented
38#13133124C2211yescharacteristic, complemented
39#13133124C2211yescharacteristic, complemented
40#13133124C2211yescharacteristic, complemented
41#13133124C2211yescharacteristic, complemented
42#13133124C2211yescharacteristic, complemented
43#13133124C2211yescharacteristic, complemented
44#13133124C2211yescharacteristic, complemented
45#13133124C2211yescharacteristic, complemented
46C2×C4⋊C4×C53C2211yescharacteristic, complemented
47C2×C22⋊C4×C53C2211yescharacteristic
48C2×C22⋊C4×C53C2211yescharacteristic
49C2×C22⋊C4×C53C2211yescharacteristic
50C2×C42×C53C2211yescharacteristic, complemented
51#95×C53C2211yescharacteristic, complemented
52#95×C53C2211yescharacteristic, complemented
53C24×C53C2311yescharacteristic
54C23×D53—22no—
55C23×D53—22no—
56C23×D53—22no—
57C23×D53—22no—
58C23×D53—22no—
59C23×D53—41no—
60C23×D53—41no—
61C23×D53C2311yescharacteristic
62C23×D53C2311yescharacteristic
63C22×C4×C53C2311yescharacteristic
64C22×C4×C53C2311yescharacteristic
65C22×C4×C53C2311yescharacteristic
66C22×C4×C53C2311yescharacteristic
67C22×C4×C53C2311yescharacteristic
68C22×C4×C53C2311yescharacteristic
69C22×Dic53C2311yescharacteristic
70C22×Dic53C2311yescharacteristic
71C22×Dic53C2311yescharacteristic
72C22×Dic53C2311yescharacteristic
73C22×Dic53C2311yescharacteristic
74C22×Dic53C2311yescharacteristic
75C2×C4×D53—22no—
76C2×C4×D53—22no—
77C2×C4×D53—22no—
78C2×C4×D53—41no—
79C2×C4×D53—41no—
80C2×C4×D53D414yes—
81C2×Dic106—22no—
82C2×Dic106—41no—
83C2×Dic106—41no—
84C4⋊C4×C53—22no—
85C22⋊C4×C53—22no—
86C22⋊C4×C53—22no—
87C22⋊C4×C53—41no—
88C42×C53—41no—
89C22⋊Dic53—22no—
90C22⋊Dic53—22no—
91C22⋊Dic53—41no—
92#11719921—22no—
93#11719921—22no—
94#11719921—22no—
95#11719921—22no—
96#11719921—22no—
97#11719921—22no—
98#11719921—22no—
99#11719921—22no—
100#11719921—41no—
101#11719921—41no—
102#11719919—22no—
103#11719919—22no—
104C4×Dic53—41no—
105C23×C53—21no—
106C23×C53—21no—
107C23×C53—21no—
108C23×C53—21no—
109C23×C53—21no—
110C23×C53—21no—
111C23×C53#4112yes—
112C23×C53C2411yescharacteristic
113C22×D53—21no—
114C22×D53—21no—
115C22×D53—21no—
116C22×D53—21no—
117C22×D53—28no—
118C22×D53—21no—
119C22×D53—21no—
120C22×D53—21no—
121C22×D53—21no—
122C22×D53—21no—
123C22×D53—21no—
124C22×D53—44no—
125C22×D53—44no—
126C22×D53—44no—
127C22×D53—44no—
128C22×D53—44no—
129C22×D53—44no—
130C22×D53C2×D412yes—
131C22×D53#4112yes—
132C22×D53#4112yes—
133C22×D53#4112yes—
134C2×C4×C53—21no—
135C2×C4×C53—21no—
136C2×C4×C53—21no—
137C2×C4×C53—21no—
138C2×C4×C53—21no—
139C2×C4×C53—21no—
140C2×C4×C53—21no—
141C2×C4×C53—21no—
142C2×C4×C53—21no—
143C2×C4×C53—21no—
144C2×C4×C53—21no—
145C2×C4×C53—21no—
146C2×C4×C53—21no—
147C2×C4×C53—21no—
148C2×C4×C53—21no—
149C2×C4×C53—21no—
150C2×C4×C53—21no—
151C2×C4×C53C2×D412yes—
152C2×Dic53—21no—
153C2×Dic53—21no—
154C2×Dic53—21no—
155C2×Dic53—21no—
156C2×Dic53—21no—
157C2×Dic53—21no—
158C2×Dic53—21no—
159C2×Dic53—21no—
160C2×Dic53—21no—
161C2×Dic53—21no—
162C2×Dic53—21no—
163C2×Dic53—21no—
164C2×Dic53—21no—
165C2×Dic53—21no—
166C2×Dic53—21no—
167C2×Dic53—21no—
168C2×Dic53—21no—
169C2×Dic53C2×D412yes—
170C4×D53—44no—
171C4×D53—44no—
172Dic106—44no—
173C22×C53—22no—
174C22×C53—22no—
175C22×C53—22no—
176C22×C53—41no—
177C22×C53—41no—
178C22×C53—41no—
179C22×C53—41no—
180C22×C53C22×D411yescharacteristic
181C22×C53C2×#4111yescharacteristic
182C22×C53C2×#4111yescharacteristic
183C22×C53C2×#4111yescharacteristic
184C22×C53C2×#4111yescharacteristic
185C22×C53H(5,𝔽2)11yescharacteristic
186C22×C53H(5,𝔽2)11yescharacteristic
187C2×D53—22no—
188C2×D53—22no—
189C2×D53—22no—
190C2×D53—22no—
191C2×D53—22no—
192C2×D53—22no—
193C2×D53—22no—
194C2×D53—22no—
195C2×D53—22no—
196C2×D53—416no—
197C2×D53—41no—
198C2×D53—41no—
199C2×D53—41no—
200C2×D53—41no—
201C2×D53#12414yes—
202C4×C53—22no—
203C4×C53—41no—
204C4×C53—41no—
205C4×C53—41no—
206C4×C53—41no—
207C4×C53—41no—
208Dic53—22no—
209Dic53—41no—
210Dic53—41no—
211Dic53—41no—
212Dic53—41no—
213Dic53—41no—
214#2561—531nomaximal, Sylow
215C2×C53—42no—
216C2×C53#52511yescharacteristic
217C2×C53C2×#12411yescharacteristic
218C2×C53#53411yescharacteristic
219C2×C53#55111yescharacteristic
220C2×C53#55111yescharacteristic
221C2×C53#55111yescharacteristic
222C2×C53#55111yescharacteristic
223D53—24no—
224D53—42no—
225C2×C22⋊Q8—531no—
226C22×C22⋊C4—531no—
227#397—531no—
228#397—531no—
229#393—531no—
230#393—531no—
231#388—531no—
232#388—531no—
233#388—531no—
234#388D5311yescharacteristic, complemented
235#381—531no—
236#381—531no—
237#379—531no—
238#379—531no—
239C4×C22⋊C4—531no—
240C53#256111yescharacteristic, complemented, Sylow
241C25—531no—
242C22×Q8—531no—
243C23×C4—531no—
244C23×C4—531no—
245C22⋊Q8—1064no—
246C2×C4⋊C4—531no—
247C2×C4⋊C4—531no—
248C2×C4⋊C4C2×D5311yescharacteristic, complemented
249C2×C22⋊C4—531no—
250C2×C22⋊C4—531no—
251C2×C22⋊C4—531no—
252C2×C22⋊C4—531no—
253C2×C22⋊C4—531no—
254C2×C22⋊C4—531no—
255C2×C22⋊C4—531no—
256C2×C22⋊C4—531no—
257C2×C22⋊C4—531no—
258C2×C22⋊C4—531no—
259C2×C22⋊C4—531no—
260C2×C22⋊C4—531no—
261C2×C22⋊C4—531no—
262C2×C22⋊C4—1064no—
263C2×C22⋊C4C2×D5311yescharacteristic
264C2×C22⋊C4C2×D5311yescharacteristic
265C2×C22⋊C4C2×D5311yescharacteristic
266C2×C42—531no—
267C2×C42C2×D5311yescharacteristic, complemented
268#95—531no—
269#95—531no—
270#95—531no—
271#95—531no—
272#95—531no—
273#95—531no—
274#95—531no—
275#95—531no—
276#95C2×D5311yescharacteristic, complemented
277#95C2×D5311yescharacteristic, complemented
278C24—531no—
279C24—531no—
280C24—1062no—
281C24—1062no—
282C24—1062no—
283C24—1062no—
284C24—1062no—
285C24—2121no—
286C24—2121no—
287C24C22×D5311yescharacteristic
288C2×Q8—1062no—
289C2×Q8—2121no—
290C2×Q8—2121no—
291C22×C4—531no—
292C22×C4—534no—
293C22×C4—531no—
294C22×C4—531no—
295C22×C4—531no—
296C22×C4—531no—
297C22×C4—531no—
298C22×C4—1062no—
299C22×C4—1062no—
300C22×C4—1062no—
301C22×C4—2121no—
302C22×C4—2121no—
303C22×C4C22×D5311yescharacteristic
304C22×C4C22×D5311yescharacteristic
305C22×C4C22×D5311yescharacteristic
306C22×C4C22×D5311yescharacteristic
307C22×C4C22×D5311yescharacteristic
308C22×C4C22×D5311yescharacteristic
309C4⋊C4—22no—
310C4⋊C4—1062no—
311C4⋊C4—1062no—
312C22⋊C4—22no—
313C22⋊C4—22no—
314C22⋊C4—41no—
315C22⋊C4—1062no—
316C22⋊C4—1062no—
317C22⋊C4—1062no—
318C22⋊C4—1062no—
319C22⋊C4—1062no—
320C22⋊C4—1062no—
321C22⋊C4—1062no—
322C22⋊C4—1062no—
323C22⋊C4—1062no—
324C22⋊C4—1062no—
325C22⋊C4—2121no—
326C22⋊C4—2121no—
327C22⋊C4—2121no—
328C42—41no—
329C42—2121no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—532no—
337C23—532no—
338C23—532no—
339C23—532no—
340C23—1068no—
341C23—1061no—
342C23—1061no—
343C23—1061no—
344C23—1061no—
345C23—1061no—
346C23—1061no—
347C23—1061no—
348C23—1061no—
349C23—1061no—
350C23—1061no—
351C23—2124no—
352C23—2124no—
353C23—2124no—
354C23—2124no—
355C23—2124no—
356C23—2124no—
357C23#1171994512yes—
358C23C23×D5311yescharacteristic
359Q8—2124no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—532no—
378C2×C4—1061no—
379C2×C4—1061no—
380C2×C4—1061no—
381C2×C4—1061no—
382C2×C4—1061no—
383C2×C4—1061no—
384C2×C4—1061no—
385C2×C4—1061no—
386C2×C4—1061no—
387C2×C4—1061no—
388C2×C4—1061no—
389C2×C4—1061no—
390C2×C4—1061no—
391C2×C4—1061no—
392C2×C4—1061no—
393C2×C4—1061no—
394C2×C4—1061no—
395C2×C4—2124no—
396C2×C4—2124no—
397C2×C4D4×D5312yes—
398C22—22no—
399C22—22no—
400C22—22no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—534no—
406C22—1062no—
407C22—1062no—
408C22—1062no—
409C22—1062no—
410C22—1062no—
411C22—1062no—
412C22—1062no—
413C22—1062no—
414C22—1062no—
415C22—21216no—
416C22—2121no—
417C22—2121no—
418C22—2121no—
419C22—2121no—
420C22C2×D4×D5311yescharacteristic
421C22#41×D5311yescharacteristic
422C22#41×D5311yescharacteristic
423C22#41×D5311yescharacteristic
424C22C2×#1171994511yescharacteristic
425C22#1313326211yescharacteristic
426C22#1313326311yescharacteristic
427C4—22no—
428C4—41no—
429C4—41no—
430C4—41no—
431C4—41no—
432C4—41no—
433C4—1062no—
434C4—2121no—
435C4—2121no—
436C4—2121no—
437C4—2121no—
438C4—2121no—
439C2—42no—
440C2—1064no—
441C2—2122no—
442C2#1681113211yescharacteristic
443C2#124×D5311yescharacteristic
444C2#1681118111yescharacteristic
445C2#1681129711yescharacteristic
446C2#1681129811yescharacteristic
447C2#1681130111yescharacteristic
448C2#1681130211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#388C2
hypercenter#388D53
hypocenterC53#2561
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
1. C2×#13133195C2semidirect
3. #16810552C2semidirect
4. #16810551C2semidirect
5. #16810536C2semidirect
6. #16810535C2semidirect
7. #16810517C2semidirect
8. #16810514C2semidirect
9. #16810513C2semidirect
10. #16810485C2semidirect
11. #16810485C2semidirect
14. C4×#11719921C2semidirect
15. C53×#388C2semidirect
17. C22×Dic106C22semidirect
21. C2×#11719919C22semidirect
22. C2×#11719919C22semidirect
37. C2×C4×Dic53C22semidirect
38. #13133124C22semidirect
39. #13133124C22semidirect
40. #13133124C22semidirect
41. #13133124C22semidirect
42. #13133124C22semidirect
43. #13133124C22semidirect
44. #13133124C22semidirect
45. #13133124C22semidirect
46. C2×C4⋊C4×C53C22semidirect
50. C2×C42×C53C22semidirect
51. #95×C53C22semidirect
52. #95×C53C22semidirect
234. #388D53semidirect
240. C53#2561semidirect
248. C2×C4⋊C4C2×D53semidirect
267. C2×C42C2×D53semidirect
276. #95C2×D53semidirect
277. #95C2×D53semidirect