Finite Group Databasebeta

← #28485551

Subgroups — #28485551

Order 15104.

580 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21433207C211yesmaximal, characteristic, complemented
2#21425024C211yesmaximal, characteristic, complemented
3#21424116C211yesmaximal, characteristic, complemented
4#21422764C211yesmaximal, characteristic, complemented
5#21421997C211yesmaximal, characteristic, complemented
6#21421970C211yesmaximal, characteristic, complemented
7C59×#3038C211yesmaximal, characteristic
8Q8×C59⋊C8—21no—
9D4×C59⋊C8—22no—
10D4×C59⋊C8—21no—
11#16837455—21no—
12#16837455C2211yescharacteristic
13#16837426C2211yescharacteristic
14C2×C4⋊(C59⋊C8)—21no—
15C2×C4⋊(C59⋊C8)C2211yescharacteristic
16C2×#14273654—21no—
17C2×#14273648—22no—
18C4×C4⋊Dic59C2211yescharacteristic
19#16837161—21no—
20#16837146—21no—
21#16837146—21no—
22C2×AGL(1,ℤ/8)×C59—21no—
23C2×#127×C59C2211yescharacteristic
24C4×#41×C59C2211yescharacteristic
25C59×#460—21no—
26C59×#460C412yescomplemented
27C59×#459—21no—
28C59×#459C412yescomplemented
29C59×#423C2211yescharacteristic
30C59×#418—21no—
31C2×#11745664—21no—
32C22×C59⋊C8—22no—
33C22×C59⋊C8D412yes—
34C2×C4⋊Dic59—21no—
35C2×C4×Dic59—21no—
36C4⋊(C59⋊C8)—22no—
37C4⋊(C59⋊C8)—21no—
38C4⋊(C59⋊C8)—21no—
39C4⋊(C59⋊C8)—21no—
40C4⋊(C59⋊C8)—21no—
41C4⋊(C59⋊C8)D411yescharacteristic
42C4⋊(C59⋊C8)D411yescharacteristic
43#14273654—42no—
44#14273648—44no—
45C22⋊(C59⋊C8)—22no—
46C22⋊(C59⋊C8)—21no—
47C4×C59⋊C8—22no—
48C4×C59⋊C8—21no—
49C59⋊#95—21no—
50C2×#41×C59D411yescharacteristic
51C22×D4×C59—21no—
52C22×D4×C59—21no—
53C22×D4×C59—21no—
54AGL(1,ℤ/8)×C59—42no—
55AGL(1,ℤ/8)×C59—42no—
56C2×SD16×C59—22no—
57C2×D8×C59—22no—
58C2×M16×C59—21no—
59#127×C59—21no—
60#127×C59—21no—
61#127×C59—21no—
62#127×C59C2×C412yes—
63C4×Q8×C59C2×C411yescharacteristic
64C4×Q8×C59D411yescharacteristic
65C4×D4×C59—21no—
66C4×D4×C59—21no—
67C4×D4×C59C2×C411yescharacteristic
68C4×D4×C59D411yescharacteristic
69#117×C59—21no—
70#117×C59D411yescharacteristic
71C2×C42×C59—21no—
72C2×C42×C59C2311yescharacteristic
73C4⋊C8×C59—21no—
74C4⋊C8×C59C2×C412yes—
75#102×C59—22no—
76#102×C59—22no—
77C24×C59—41no—
78#41×C59—21no—
79#41×C59—21no—
80#41×C59D812yes—
81#41×C59SD1612yes—
82C2×Q8×C59C22⋊C411yescharacteristic
83C2×D4×C59—21no—
84C2×D4×C59—22no—
85C2×D4×C59—22no—
86C2×D4×C59—22no—
87C2×D4×C59—42no—
88C2×D4×C59—42no—
89C2×D4×C59—41no—
90C2×D4×C59—41no—
91C2×D4×C59—41no—
92C2×D4×C59—41no—
93C2×D4×C59—41no—
94C2×D4×C59—41no—
95C2×D4×C59—41no—
96C2×D4×C59—82no—
97C2×D4×C59C22⋊C411yescharacteristic
98C22×C4×C59—21no—
99C22×C4×C59—21no—
100C22×C4×C59—21no—
101C22×C4×C59C2×D411yescharacteristic
102C22×C4×C59C2×D411yescharacteristic
103C22×C4×C59#4111yescharacteristic
104C22×Dic59—41no—
105SD16×C59—44no—
106D8×C59—44no—
107M16×C59—41no—
108M16×C59—41no—
109C2×C8×C59—22no—
110C4⋊C4×C59—21no—
111C4⋊C4×C59—21no—
112C4⋊C4×C59C22⋊C411yescharacteristic
113C4⋊C4×C59C22⋊C411yescharacteristic
114C22⋊C4×C59—21no—
115C22⋊C4×C59—21no—
116C22⋊C4×C59—21no—
117C42×C59—21no—
118C42×C59—21no—
119C42×C59—21no—
120C42×C59C22×C411yescharacteristic
121C42×C59C2×D411yescharacteristic
122C42×C59C2×D411yescharacteristic
123C42×C59#4111yescharacteristic
124C4⋊Dic59—22no—
125C4⋊Dic59—41no—
126C4×Dic59—41no—
127C4×Dic59—41no—
128#11745664—41no—
129#11745664—41no—
130C2×C59⋊C8—21no—
131C2×C59⋊C8—22no—
132C2×C59⋊C8—21no—
133C2×C59⋊C8—42no—
134C2×C59⋊C8—41no—
135C2×C59⋊C8—82no—
136C23×C59—21no—
137C23×C59—42no—
138C23×C59—82no—
139C23×C59—82no—
140C23×C59—81no—
141C23×C59—81no—
142C23×C59#12111yescharacteristic
143Q8×C59—21no—
144Q8×C59#10212yescomplemented
145D4×C59—22no—
146D4×C59—22no—
147D4×C59—21no—
148D4×C59—44no—
149D4×C59—42no—
150D4×C59—42no—
151D4×C59—82no—
152D4×C59—82no—
153D4×C59—81no—
154D4×C59—81no—
155D4×C59—81no—
156D4×C59—81no—
157D4×C59#10212yescomplemented
158C2×C4×C59—21no—
159C2×C4×C59—21no—
160C2×C4×C59—21no—
161C2×C4×C59—22no—
162C2×C4×C59—22no—
163C2×C4×C59—21no—
164C2×C4×C59—21no—
165C2×C4×C59—41no—
166C2×C4×C59—41no—
167C2×C4×C59—41no—
168C2×C4×C59—81no—
169C2×C4×C59C2×C22⋊C411yescharacteristic
170C2×C4×C59C4×D411yescharacteristic
171C2×C4×C59C4×D411yescharacteristic
172C2×C4×C59#12011yescharacteristic
173C2×C4×C59#12111yescharacteristic
174C2×C4×C59#12311yescharacteristic
175C2×C4×C59C2×D811yescharacteristic
176C2×C4×C59C2×SD1611yescharacteristic
177C2×C4×C59#13511yescharacteristic
178C2×C4×C59#13511yescharacteristic
179C2×Dic59—42no—
180C2×Dic59—81no—
181C2×Dic59—81no—
182C8×C59—42no—
183C59⋊C8—42no—
184C59⋊C8—41no—
185C59⋊C8—41no—
186C59⋊C8—81no—
187#9318—591nomaximal, Sylow
188C22×C59—21no—
189C22×C59—21no—
190C22×C59—22no—
191C22×C59—41no—
192C22×C59—41no—
193C22×C59—82no—
194C22×C59—82no—
195C22×C59—82no—
196C22×C59—162no—
197C22×C59—162no—
198C22×C59#38611yescharacteristic
199C22×C59#46511yescharacteristic
200C22×C59#46611yescharacteristic
201C4×C59—21no—
202C4×C59—22no—
203C4×C59—21no—
204C4×C59—21no—
205C4×C59—41no—
206C4×C59—41no—
207C4×C59C2×#10211yescharacteristic
208C4×C59#41611yescharacteristic
209C4×C59#45311yescharacteristic
210C4×C59#45411yescharacteristic
211C4×C59#46511yescharacteristic
212C4×C59#46611yescharacteristic
213Dic59—82no—
214#3038D5911yescharacteristic
215#2988—591no—
216#1982—591no—
217#1806—591no—
218#1545—591no—
219#1303—591no—
220#1292—591no—
221C2×C59—21no—
222C2×C59—21no—
223C2×C59—42no—
224C2×C59—162no—
225C2×C59#188311yescharacteristic
226C2×C59#189311yescharacteristic
227C2×C59#190311yescharacteristic
228C2×AGL(1,ℤ/8)—21no—
229C2×#127C2×D5911yescharacteristic
230C4×#41C2×D5911yescharacteristic
231#460—21no—
232#460Dic5912yescomplemented
233#459—21no—
234#459Dic5912yescomplemented
235C8×Q8—1181no—
236C8×D4—1182no—
237C8×D4—1181no—
238#431—591no—
239#431—1181no—
240#423—591no—
241#423C2×D5911yescharacteristic
242C2×C4⋊C8—591no—
243C2×C4⋊C8—1181no—
244C2×#104—1181no—
245#418—21no—
246C2×#102—1182no—
247C4×C4⋊C4—591no—
248#339—1181no—
249#331—1181no—
250#331—1181no—
251C59#931811yescharacteristic, complemented, Sylow
252C2×#41C22⋊D5911yescharacteristic
253C22×D4—21no—
254C22×D4—21no—
255C22×D4—21no—
256AGL(1,ℤ/8)—42no—
257AGL(1,ℤ/8)—42no—
258C2×SD16—22no—
259C2×D8—22no—
260C2×M16—21no—
261C2×M16—1181no—
262C22×C8—592no—
263C22×C8—1182no—
264#127—21no—
265#127—21no—
266#127—21no—
267#127C2×Dic5912yes—
268C4×Q8C2×Dic5911yescharacteristic
269C4×Q8C22⋊D5911yescharacteristic
270C4×D4—21no—
271C4×D4—21no—
272C4×D4C2×Dic5911yescharacteristic
273C4×D4C22⋊D5911yescharacteristic
274#117—21no—
275#117C22⋊D5911yescharacteristic
276C2×C4⋊C4—1181no—
277C2×C42—21no—
278C2×C42—1181no—
279C2×C42C22×D5911yescharacteristic
280C4⋊C8—21no—
281C4⋊C8—591no—
282C4⋊C8—591no—
283C4⋊C8—1182no—
284C4⋊C8—1181no—
285C4⋊C8—1181no—
286C4⋊C8—1181no—
287C4⋊C8—1181no—
288C4⋊C8C2×Dic5912yes—
289#104—2362no—
290#102—22no—
291#102—22no—
292#102—2364no—
293C22⋊C8—1182no—
294C22⋊C8—1181no—
295C4×C8—1182no—
296C4×C8—1181no—
297#95—1181no—
298C24—41no—
299#41—21no—
300#41—21no—
301#41#1174566912yes—
302#41#1174567012yes—
303C2×Q8C22⋊Dic5911yescharacteristic
304C2×D4—21no—
305C2×D4—22no—
306C2×D4—22no—
307C2×D4—22no—
308C2×D4—42no—
309C2×D4—42no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4—41no—
313C2×D4—41no—
314C2×D4—41no—
315C2×D4—41no—
316C2×D4—41no—
317C2×D4—82no—
318C2×D4C22⋊Dic5911yescharacteristic
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—2361no—
323C22×C4D4×D5911yescharacteristic
324C22×C4#1174568711yescharacteristic
325C22×C4C2×C22⋊D5911yescharacteristic
326SD16—44no—
327D8—44no—
328M16—41no—
329M16—41no—
330M16—2361no—
331M16—2361no—
332C2×C8—22no—
333C2×C8—1181no—
334C2×C8—1181no—
335C2×C8—1182no—
336C2×C8—4722no—
337C2×C8—2361no—
338C2×C8—2362no—
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—1182no—
342C4⋊C4—2361no—
343C4⋊C4C22⋊Dic5911yescharacteristic
344C4⋊C4C22⋊Dic5911yescharacteristic
345C22⋊C4—21no—
346C22⋊C4—21no—
347C22⋊C4—21no—
348C42—21no—
349C42—21no—
350C42—21no—
351C42—2361no—
352C42—2361no—
353C42D4×D5911yescharacteristic
354C42#1174568711yescharacteristic
355C42C22×Dic5911yescharacteristic
356C42C2×C22⋊D5911yescharacteristic
357C23—21no—
358C23—42no—
359C23—82no—
360C23—82no—
361C23—81no—
362C23—81no—
363C23#1427369911yescharacteristic
364Q8—21no—
365Q8#1427364812yescomplemented
366D4—22no—
367D4—22no—
368D4—21no—
369D4—44no—
370D4—42no—
371D4—42no—
372D4—82no—
373D4—82no—
374D4—81no—
375D4—81no—
376D4—81no—
377D4—81no—
378D4#1427364812yescomplemented
379C2×C4—21no—
380C2×C4—22no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—22no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—81no—
390C2×C4—4721no—
391C2×C4—4721no—
392C2×C4—2362no—
393C2×C4C2×C22⋊Dic5911yescharacteristic
394C2×C4D4×Dic5911yescharacteristic
395C2×C4D4×Dic5911yescharacteristic
396C2×C4#1427369511yescharacteristic
397C2×C4#1427369911yescharacteristic
398C2×C4#1427370811yescharacteristic
399C2×C4C2×#1174566911yescharacteristic
400C2×C4C2×#1174567011yescharacteristic
401C2×C4#1427374711yescharacteristic
402C2×C4#1427374711yescharacteristic
403C8—42no—
404C8—4721no—
405C8—2361no—
406C8—2362no—
407C8—2361no—
408C22—21no—
409C22—21no—
410C22—22no—
411C22—41no—
412C22—41no—
413C22—82no—
414C22—82no—
415C22—82no—
416C22—162no—
417C22—162no—
418C22#1683728211yescharacteristic
419C22#1683762911yescharacteristic
420C22#1683763611yescharacteristic
421C4—21no—
422C4—22no—
423C4—21no—
424C4—21no—
425C4—41no—
426C4—41no—
427C4—4722no—
428C4C2×#1427364811yescharacteristic
429C4#1683739811yescharacteristic
430C4#1683756811yescharacteristic
431C4#1683757311yescharacteristic
432C4#1683762911yescharacteristic
433C4#1683763611yescharacteristic
434C2—21no—
435C2—21no—
436C2—42no—
437C2—162no—
438C2#2142447411yescharacteristic
439C2#2142452511yescharacteristic
440C2#2142457811yescharacteristic
subgroupG/H
centerC22#16837282
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
Fitting, prime radicalC59×#3038C2
socleC22×C59#386
cosocleC2×C42×C59C23
hypercenter#3038D59
hypocenterC59#9318
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3038

Complements

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

NH ≅ G/Ntype
1. #21433207C2semidirect
2. #21425024C2semidirect
3. #21424116C2semidirect
4. #21422764C2semidirect
5. #21421997C2semidirect
6. #21421970C2semidirect
26. C59×#460C4semidirect
28. C59×#459C4semidirect
144. Q8×C59#102semidirect
157. D4×C59#102semidirect
232. #460Dic59semidirect
234. #459Dic59semidirect
251. C59#9318semidirect
365. Q8#14273648semidirect
378. D4#14273648semidirect

Maximal subgroups

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

subgroupindex
#214332072
#214250242
#214241162
#214227642
#214219972
#214219702
C59×#30382
#931859