Finite Group Databasebeta

← #28484626

Subgroups — #28484626

Order 15104.

620 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21427006C211yesmaximal, characteristic, complemented
2#21422656C211yesmaximal, characteristic, complemented
3#21422591C211yesmaximal, characteristic, complemented
4#21422487C211yesmaximal, characteristic, complemented
5#21421997C211yesmaximal, characteristic, complemented
6#21421956C211yesmaximal, characteristic, complemented
7C59×#3036C211yesmaximal, characteristic
8C2×D4×Dic59—21no—
9#16837455C2211yescharacteristic
10#16837455C2211yescharacteristic
11#16837301—21no—
12#16837277—22no—
13#16837268C2211yescharacteristic
14#16837262—21no—
15C4×C4⋊Dic59C2211yescharacteristic
16C2×C59⋊#95—21no—
17#16837136—22no—
18#16837136—21no—
19#16837136—21no—
20D4⋊(C59⋊C8)—22no—
21C2×AGL(1,ℤ/8)×C59—21no—
22C59×#532C2211yescharacteristic
23C2×C4×D4×C59C2211yescharacteristic
24C59×#463—22no—
25C59×#461C412yescomplemented
26C59×#459C412yescomplemented
27C59×#423C2211yescharacteristic
28C59×#417—21no—
29C23×Dic59—22no—
30D4×Dic59—42no—
31D4×Dic59—42no—
32C2×C4⋊Dic59—21no—
33C2×C4⋊Dic59—21no—
34C2×C4⋊Dic59—22no—
35C2×C22⋊Dic59—22no—
36C2×C4×Dic59—21no—
37C4⋊(C59⋊C8)—21no—
38C4⋊(C59⋊C8)—21no—
39C22⋊(C59⋊C8)—21no—
40C22⋊(C59⋊C8)—21no—
41C4×C59⋊C8—21no—
42C4×C59⋊C8—21no—
43C59⋊#95—22no—
44C59⋊#95—21no—
45C59⋊#95D411yescharacteristic
46C59⋊#95D411yescharacteristic
47C59⋊#95D412yes—
48C2×#41×C59—21no—
49C22×D4×C59D411yescharacteristic
50C23×C4×C59—21no—
51AGL(1,ℤ/8)×C59—42no—
52AGL(1,ℤ/8)×C59—42no—
53C2×SD16×C59—22no—
54C2×D8×C59—22no—
55C2×M16×C59—21no—
56C4⋊Q8×C59C2×C411yescharacteristic, complemented
57#127×C59C2×C411yescharacteristic, complemented
58#124×C59—21no—
59#121×C59—21no—
60#121×C59—21no—
61C4×D4×C59—21no—
62C4×D4×C59—21no—
63C4×D4×C59—21no—
64C4×D4×C59—21no—
65C4×D4×C59C2×C412yes—
66C4×D4×C59D412yes—
67C2×C4⋊C4×C59D411yescharacteristic
68C2×C22⋊C4×C59—21no—
69C2×C42×C59C2311yescharacteristic
70C4⋊C8×C59—21no—
71C4⋊C8×C59C2×C412yescomplemented
72#103×C59—22no—
73#102×C59—22no—
74C24×C59—21no—
75#41×C59—41no—
76#41×C59—41no—
77#41×C59—41no—
78#41×C59—41no—
79C2×Q8×C59—21no—
80C2×D4×C59—21no—
81C2×D4×C59—21no—
82C2×D4×C59—21no—
83C2×D4×C59—21no—
84C2×D4×C59—21no—
85C2×D4×C59—21no—
86C2×D4×C59—41no—
87C2×D4×C59C22⋊C412yes—
88C2×D4×C59D812yes—
89C2×D4×C59SD1612yes—
90C22×C4×C59—21no—
91C22×C4×C59—22no—
92C22×C4×C59—22no—
93C22×C4×C59—41no—
94C22×C4×C59—82no—
95C22×C4×C59C2×D411yescharacteristic
96C22×C4×C59C2×D411yescharacteristic
97C22×C4×C59C2×D411yescharacteristic
98C22×Dic59—21no—
99C22×Dic59—22no—
100C22×Dic59—21no—
101C22×Dic59—44no—
102C22×Dic59—44no—
103C22×Dic59—82no—
104SD16×C59—44no—
105D8×C59—44no—
106M16×C59—41no—
107M16×C59—41no—
108C2×C8×C59—22no—
109C4⋊C4×C59—21no—
110C4⋊C4×C59—21no—
111C4⋊C4×C59—21no—
112C4⋊C4×C59C22⋊C412yes—
113C22⋊C4×C59—22no—
114C22⋊C4×C59—22no—
115C22⋊C4×C59—41no—
116C22⋊C4×C59—41no—
117C42×C59C22×C411yescharacteristic
118C42×C59C2×D411yescharacteristic
119C42×C59#4111yescharacteristic
120C42×C59#4111yescharacteristic
121C22⋊Dic59—44no—
122C4⋊Dic59—42no—
123C4⋊Dic59—41no—
124C4⋊Dic59—41no—
125C4⋊Dic59—82no—
126C4×Dic59—41no—
127C4×Dic59—41no—
128C2×C59⋊C8—41no—
129C2×C59⋊C8—41no—
130C23×C59—22no—
131C23×C59—22no—
132C23×C59—22no—
133C23×C59—42no—
134C23×C59—42no—
135C23×C59—41no—
136C23×C59#12011yescharacteristic
137Q8×C59—41no—
138Q8×C59—41no—
139D4×C59—22no—
140D4×C59—22no—
141D4×C59—22no—
142D4×C59—41no—
143D4×C59—41no—
144D4×C59—81no—
145D4×C59—81no—
146D4×C59—81no—
147D4×C59—81no—
148D4×C59#10214yes—
149C2×C4×C59—22no—
150C2×C4×C59—41no—
151C2×C4×C59—41no—
152C2×C4×C59—42no—
153C2×C4×C59—41no—
154C2×C4×C59—82no—
155C2×C4×C59—82no—
156C2×C4×C59—81no—
157C2×C4×C59—81no—
158C2×C4×C59C2×C22⋊C411yescharacteristic
159C2×C4×C59C4×D411yescharacteristic
160C2×C4×C59C4×D411yescharacteristic
161C2×C4×C59#12111yescharacteristic
162C2×C4×C59#12111yescharacteristic
163C2×C4×C59#12311yescharacteristic
164C2×C4×C59C2×D811yescharacteristic
165C2×C4×C59C2×SD1611yescharacteristic
166C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
167C2×C4×C59#13711yescharacteristic
168C2×Dic59—22no—
169C2×Dic59—41no—
170C2×Dic59—41no—
171C2×Dic59—41no—
172C2×Dic59—41no—
173C2×Dic59—42no—
174C2×Dic59—42no—
175C2×Dic59—41no—
176C2×Dic59—42no—
177C2×Dic59—88no—
178C8×C59—42no—
179C59⋊C8—81no—
180C59⋊C8—81no—
181#9179—591nomaximal, Sylow
182C22×C59—21no—
183C22×C59—21no—
184C22×C59—24no—
185C22×C59—44no—
186C22×C59—44no—
187C22×C59—42no—
188C22×C59—42no—
189C22×C59—81no—
190C22×C59—81no—
191C22×C59—81no—
192C22×C59#38611yescharacteristic
193C22×C59#44711yescharacteristic
194C22×C59#44811yescharacteristic
195C4×C59—21no—
196C4×C59—21no—
197C4×C59—21no—
198C4×C59—21no—
199C4×C59—42no—
200C4×C59—81no—
201C4×C59C2×#10211yescharacteristic
202C4×C59#41711yescharacteristic
203C4×C59#48011yescharacteristic
204C4×C59#48111yescharacteristic
205Dic59—42no—
206Dic59—81no—
207Dic59—82no—
208#3036D5911yescharacteristic
209#2349—591no—
210#1517—591no—
211#1502—591no—
212#1478—591no—
213#1303—591no—
214#1286—591no—
215C2×C59—21no—
216C2×C59—21no—
217C2×C59—44no—
218C2×C59—161no—
219C2×C59#187411yescharacteristic
220C2×C59#190411yescharacteristic
221C2×C59#190611yescharacteristic
222C2×AGL(1,ℤ/8)—21no—
223#532C2×D5911yescharacteristic
224C2×C4×D4—1181no—
225C2×C4×D4C2×D5911yescharacteristic
226#463—22no—
227#461Dic5912yescomplemented
228#459Dic5912yescomplemented
229#431—591no—
230#431—591no—
231#423C2×D5911yescharacteristic
232#417—21no—
233#389—1181no—
234#385—1182no—
235#382—591no—
236#380—1181no—
237C4×C4⋊C4—591no—
238C2×#95—1181no—
239#327—1181no—
240#327—1181no—
241#327—1182no—
242D4⋊C8—1182no—
243C59#917911yescharacteristic, complemented, Sylow
244C2×#41—21no—
245C22×D4C22⋊D5911yescharacteristic
246C23×C4—21no—
247C23×C4—1182no—
248AGL(1,ℤ/8)—42no—
249AGL(1,ℤ/8)—42no—
250C2×SD16—22no—
251C2×D8—22no—
252C2×M16—21no—
253C4⋊Q8C2×Dic5911yescharacteristic, complemented
254#127C2×Dic5911yescharacteristic, complemented
255#124—21no—
256#121—21no—
257#121—21no—
258C4×D4—21no—
259C4×D4—21no—
260C4×D4—21no—
261C4×D4—21no—
262C4×D4—2362no—
263C4×D4—2362no—
264C4×D4C2×Dic5912yes—
265C4×D4C22⋊D5912yes—
266C2×C4⋊C4—1182no—
267C2×C4⋊C4—1181no—
268C2×C4⋊C4—1181no—
269C2×C4⋊C4C22⋊D5911yescharacteristic
270C2×C22⋊C4—21no—
271C2×C22⋊C4—1182no—
272C2×C42—1181no—
273C2×C42C22×D5911yescharacteristic
274C4⋊C8—21no—
275C4⋊C8—1181no—
276C4⋊C8—1181no—
277C4⋊C8C2×Dic5912yescomplemented
278#103—22no—
279#102—22no—
280C22⋊C8—1181no—
281C22⋊C8—1181no—
282C4×C8—1181no—
283C4×C8—1181no—
284#95—591no—
285#95—591no—
286#95—592no—
287#95—1182no—
288#95—1181no—
289C24—21no—
290#41—41no—
291#41—41no—
292#41—41no—
293#41—41no—
294C2×Q8—21no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—21no—
301C2×D4—41no—
302C2×D4#1174566912yes—
303C2×D4#1174567012yes—
304C2×D4C22⋊Dic5912yes—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—41no—
309C22×C4—82no—
310C22×C4—1181no—
311C22×C4—1182no—
312C22×C4—1181no—
313C22×C4—4722no—
314C22×C4—2364no—
315C22×C4—2364no—
316C22×C4D4×D5911yescharacteristic
317C22×C4D4×D5911yescharacteristic
318C22×C4C2×C22⋊D5911yescharacteristic
319SD16—44no—
320D8—44no—
321M16—41no—
322M16—41no—
323C2×C8—22no—
324C2×C8—2361no—
325C2×C8—2361no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—4722no—
330C4⋊C4—2362no—
331C4⋊C4—2361no—
332C4⋊C4—2361no—
333C4⋊C4C22⋊Dic5912yes—
334C22⋊C4—22no—
335C22⋊C4—22no—
336C22⋊C4—41no—
337C22⋊C4—41no—
338C22⋊C4—2364no—
339C42—2361no—
340C42—2361no—
341C42#1174568711yescharacteristic
342C42#1174568711yescharacteristic
343C42C22×Dic5911yescharacteristic
344C42C2×C22⋊D5911yescharacteristic
345C23—22no—
346C23—22no—
347C23—22no—
348C23—42no—
349C23—42no—
350C23—41no—
351C23#1427369511yescharacteristic
352Q8—41no—
353Q8—41no—
354D4—22no—
355D4—22no—
356D4—22no—
357D4—41no—
358D4—41no—
359D4—81no—
360D4—81no—
361D4—81no—
362D4—81no—
363D4#1427364814yes—
364C2×C4—22no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—41no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—1182no—
374C2×C4—4728no—
375C2×C4—2361no—
376C2×C4—2362no—
377C2×C4—2361no—
378C2×C4—2362no—
379C2×C4—2361no—
380C2×C4—2362no—
381C2×C4—2361no—
382C2×C4—2361no—
383C2×C4C2×C22⋊Dic5911yescharacteristic
384C2×C4D4×Dic5911yescharacteristic
385C2×C4D4×Dic5911yescharacteristic
386C2×C4#1427369911yescharacteristic
387C2×C4#1427369911yescharacteristic
388C2×C4#1427370811yescharacteristic
389C2×C4C2×#1174566911yescharacteristic
390C2×C4C2×#1174567011yescharacteristic
391C2×C4#1427375211yescharacteristic
392C2×C4#1427375711yescharacteristic
393C8—42no—
394C8—4721no—
395C8—4721no—
396C22—21no—
397C22—21no—
398C22—24no—
399C22—44no—
400C22—44no—
401C22—42no—
402C22—42no—
403C22—81no—
404C22—81no—
405C22—81no—
406C22#1683728211yescharacteristic
407C22#1683753511yescharacteristic
408C22#1683754011yescharacteristic
409C4—21no—
410C4—21no—
411C4—21no—
412C4—21no—
413C4—42no—
414C4—81no—
415C4—4722no—
416C4—4721no—
417C4—2362no—
418C4C2×#1427364811yescharacteristic
419C4#1683740311yescharacteristic
420C4#1683771311yescharacteristic
421C4#1683771811yescharacteristic
422C2—21no—
423C2—21no—
424C2—44no—
425C2—161no—
426C2#2142441411yescharacteristic
427C2#2142458511yescharacteristic
428C2#2142459911yescharacteristic
subgroupG/H
centerC22#16837282
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
Fitting, prime radicalC59×#3036C2
socleC22×C59#386
cosocleC2×C42×C59C23
hypercenter#3036D59
hypocenterC59#9179
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3036

Complements

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

NH ≅ G/Ntype
1. #21427006C2semidirect
2. #21422656C2semidirect
3. #21422591C2semidirect
4. #21422487C2semidirect
5. #21421997C2semidirect
6. #21421956C2semidirect
25. C59×#461C4semidirect
26. C59×#459C4semidirect
56. C4⋊Q8×C59C2×C4semidirect
57. #127×C59C2×C4semidirect
71. C4⋊C8×C59C2×C4semidirect
227. #461Dic59semidirect
228. #459Dic59semidirect
243. C59#9179semidirect
253. C4⋊Q8C2×Dic59semidirect
254. #127C2×Dic59semidirect
277. C4⋊C8C2×Dic59semidirect

Maximal subgroups

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

subgroupindex
#214270062
#214226562
#214225912
#214224872
#214219972
#214219562
C59×#30362
#917959