Finite Group Databasebeta

← #28488392

Subgroups — #28488392

Order 15104.

522 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21424134C211yesmaximal, characteristic, complemented
2C8×C22⋊Dic59C211yesmaximal, characteristic, complemented
3#21422616C211yesmaximal, characteristic, complemented
4#21422612C211yesmaximal, characteristic
5#21422460C211yesmaximal, characteristic, complemented
6#21422456C211yesmaximal, characteristic
7C2×C59×#465C211yesmaximal, characteristic
8C2×#14273663—21no—
9#16837366C2211yescharacteristic
10#16837366C2211yescharacteristic
11C2×C22⋊(C59⋊C8)C2211yescharacteristic
12C2×C8×Dic59—21no—
13C4×C22⋊Dic59C2211yescharacteristic
14#16837167—21no—
15#16837156—21no—
16#16837139—22no—
17#16837136—22no—
18#16837128—22no—
19#16837126—22no—
20C22×SD16×C59C411yescharacteristic, complemented
21C23×C8×C59C2211yescharacteristic
22C2×C22⋊Q8×C59C2211yescharacteristic
23C2×#121×C59C2211yescharacteristic
24C59×#465—21no—
25C59×#465—21no—
26C59×#465—21no—
27C59×#465—21no—
28C2×#106×C59C411yescharacteristic, complemented
29C2×#103×C59C411yescharacteristic, complemented
30C2×#102×C59C411yescharacteristic, complemented
31C2×#11745664—21no—
32C2×#11745664—21no—
33C22×C59⋊C8—21no—
34C2×C22⋊Dic59—21no—
35C2×C4×Dic59—21no—
36#14273663—42no—
37C22⋊(C59⋊C8)—21no—
38C22⋊(C59⋊C8)—21no—
39C22⋊(C59⋊C8)—21no—
40C22⋊(C59⋊C8)—21no—
41C22⋊(C59⋊C8)—21no—
42C22⋊(C59⋊C8)D412yes—
43C8×Dic59—42no—
44C59⋊#95D412yes—
45C22×Q8×C59C2×C411yescharacteristic
46C22×D4×C59—21no—
47C22×D4×C59C2×C411yescharacteristic
48C23×C4×C59C2311yescharacteristic
49C2×SD16×C59—21no—
50C2×SD16×C59—21no—
51C2×SD16×C59—42no—
52C22×C8×C59—42no—
53C22×C8×C59C2×C411yescharacteristic
54C22×C8×C59C2×C411yescharacteristic
55C22×C8×C59D411yescharacteristic
56C22×C8×C59D411yescharacteristic
57C22×C8×C59D411yescharacteristic
58C22×C8×C59D411yescharacteristic
59C22⋊Q8×C59—21no—
60C22⋊Q8×C59—21no—
61C22⋊Q8×C59—21no—
62C22⋊Q8×C59—21no—
63#121×C59—21no—
64#121×C59—21no—
65#121×C59—21no—
66#121×C59—21no—
67C2×C4⋊C4×C59—21no—
68C2×C4⋊C4×C59C2×C411yescharacteristic
69C2×C4⋊C4×C59C2×C411yescharacteristic
70C2×C22⋊C4×C59—21no—
71C2×C22⋊C4×C59—21no—
72#106×C59—21no—
73#106×C59—21no—
74#103×C59—21no—
75#103×C59—21no—
76#102×C59—21no—
77#102×C59—21no—
78C24×C59—21no—
79C24×C59#4111yescharacteristic
80C2×Q8×C59—21no—
81C2×Q8×C59—21no—
82C2×Q8×C59—41no—
83C2×Q8×C59—41no—
84C2×D4×C59—21no—
85C2×D4×C59—21no—
86C2×D4×C59—41no—
87C2×D4×C59—41no—
88C2×D4×C59—41no—
89C2×D4×C59—41no—
90C2×D4×C59—82no—
91C22×C4×C59—21no—
92C22×C4×C59—21no—
93C22×C4×C59—21no—
94C22×C4×C59—41no—
95C22×C4×C59—41no—
96C22×C4×C59C22×C411yescharacteristic
97C22×C4×C59C2×D411yescharacteristic
98C22×C4×C59C2×D411yescharacteristic
99C22×C4×C59C2×D411yescharacteristic
100C22×C4×C59C2×D411yescharacteristic
101C22×C4×C59#4111yescharacteristic
102C22×Dic59—22no—
103SD16×C59—42no—
104SD16×C59—42no—
105C2×C8×C59—22no—
106C2×C8×C59—22no—
107C2×C8×C59—22no—
108C2×C8×C59—41no—
109C2×C8×C59—41no—
110C2×C8×C59—41no—
111C2×C8×C59C22⋊C411yescharacteristic
112C2×C8×C59C22⋊C411yescharacteristic
113C2×C8×C59C22⋊C411yescharacteristic
114C2×C8×C59C22⋊C411yescharacteristic
115C4⋊C4×C59—21no—
116C4⋊C4×C59—21no—
117C4⋊C4×C59—21no—
118C4⋊C4×C59—21no—
119C4⋊C4×C59—41no—
120C4⋊C4×C59—41no—
121C22⋊C4×C59—41no—
122C22⋊C4×C59—41no—
123C22⋊C4×C59—41no—
124C22⋊C4×C59—41no—
125C22⋊Dic59—24no—
126C4×Dic59—41no—
127C4×Dic59—41no—
128#11745664—41no—
129#11745664—41no—
130#11745664—41no—
131#11745664—41no—
132C2×C59⋊C8—22no—
133C2×C59⋊C8—41no—
134C2×C59⋊C8—41no—
135C2×C59⋊C8—41no—
136C2×C59⋊C8—41no—
137C23×C59—22no—
138C23×C59—22no—
139C23×C59—41no—
140C23×C59—41no—
141C23×C59—42no—
142C23×C59—81no—
143C23×C59—81no—
144C23×C59C4×D411yescharacteristic
145C23×C59#12111yescharacteristic
146C23×C59#12411yescharacteristic
147Q8×C59—41no—
148Q8×C59—41no—
149Q8×C59—81no—
150D4×C59—41no—
151D4×C59—41no—
152D4×C59—81no—
153D4×C59—82no—
154D4×C59—82no—
155C2×C4×C59—21no—
156C2×C4×C59—21no—
157C2×C4×C59—21no—
158C2×C4×C59—21no—
159C2×C4×C59—21no—
160C2×C4×C59—21no—
161C2×C4×C59—41no—
162C2×C4×C59—41no—
163C2×C4×C59—41no—
164C2×C4×C59—41no—
165C2×C4×C59—41no—
166C2×C4×C59—41no—
167C2×C4×C59—42no—
168C2×C4×C59—81no—
169C2×C4×C59—81no—
170C2×C4×C59C2×C22⋊C411yescharacteristic
171C2×C4×C59C4×D411yescharacteristic
172C2×C4×C59#12111yescharacteristic
173C2×C4×C59#12711yescharacteristic
174C2×Dic59—24no—
175C2×Dic59—42no—
176C2×Dic59—42no—
177C8×C59—21no—
178C8×C59—21no—
179C8×C59—21no—
180C8×C59#9912yes—
181C59⋊C8—42no—
182C59⋊C8—81no—
183C59⋊C8—81no—
184#9813—591nomaximal, Sylow
185C22×C59—21no—
186C22×C59—21no—
187C22×C59—21no—
188C22×C59—21no—
189C22×C59—21no—
190C22×C59—21no—
191C22×C59—21no—
192C22×C59—44no—
193C22×C59—81no—
194C22×C59—81no—
195C22×C59—81no—
196C22×C59—162no—
197C22×C59#39011yescharacteristic
198C22×C59#44312yes—
199C22×C59#47112yes—
200C4×C59—21no—
201C4×C59—21no—
202C4×C59—21no—
203C4×C59—81no—
204C4×C59—81no—
205C4×C59—81no—
206C4×C59C2×#9911yescharacteristic
207C4×C59#41011yescharacteristic
208Dic59—44no—
209C2×#465D5911yescharacteristic
210#1810—591no—
211C8×C22⋊C4—591no—
212#1508—591no—
213#1507—591no—
214#1472—591no—
215#1471—591no—
216C2×C59—22no—
217C2×C59—22no—
218C2×C59—41no—
219C2×C59—161no—
220C2×C59#196411yescharacteristic
221C2×C59#197411yescharacteristic
222C2×C59#197711yescharacteristic
223C22×SD16Dic5911yescharacteristic, complemented
224C23×C8C2×D5911yescharacteristic
225C2×C22⋊Q8C2×D5911yescharacteristic
226C2×#121C2×D5911yescharacteristic
227#465—21no—
228#465—21no—
229#465—21no—
230#465—21no—
231C2×#108—1181no—
232C2×#106Dic5911yescharacteristic, complemented
233C2×#103Dic5911yescharacteristic, complemented
234C2×#102Dic5911yescharacteristic, complemented
235#407—591no—
236#407—591no—
237C2×C22⋊C8—591no—
238C2×C4×C8—1181no—
239C4×C22⋊C4—591no—
240#341—1181no—
241#336—1181no—
242#328—1182no—
243#327—1182no—
244#324—1182no—
245#323—1182no—
246C59#981311yescharacteristic, complemented, Sylow
247C22×Q8C2×Dic5911yescharacteristic
248C22×D4—21no—
249C22×D4C2×Dic5911yescharacteristic
250C23×C4C22×D5911yescharacteristic
251C2×SD16—21no—
252C2×SD16—21no—
253C2×SD16—42no—
254C2×M16—1181no—
255C2×M16—1181no—
256C22×C8—42no—
257C22×C8—1181no—
258C22×C8C2×Dic5911yescharacteristic
259C22×C8C2×Dic5911yescharacteristic
260C22×C8C22⋊D5911yescharacteristic
261C22×C8C22⋊D5911yescharacteristic
262C22×C8C22⋊D5911yescharacteristic
263C22×C8C22⋊D5911yescharacteristic
264C22⋊Q8—21no—
265C22⋊Q8—21no—
266C22⋊Q8—21no—
267C22⋊Q8—21no—
268#121—21no—
269#121—21no—
270#121—21no—
271#121—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4C2×Dic5911yescharacteristic
274C2×C4⋊C4C2×Dic5911yescharacteristic
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—1181no—
278C2×C42—1181no—
279#108—2362no—
280#106—21no—
281#106—21no—
282#103—21no—
283#103—21no—
284#102—21no—
285#102—21no—
286C22⋊C8—592no—
287C22⋊C8—1181no—
288C22⋊C8—1181no—
289C22⋊C8—1181no—
290C22⋊C8—1181no—
291C22⋊C8—1181no—
292C4×C8—2362no—
293#95—592no—
294C24—21no—
295C24#1174568711yescharacteristic
296C2×Q8—21no—
297C2×Q8—21no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×D4—21no—
301C2×D4—21no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—82no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—1182no—
313C22×C4D4×D5911yescharacteristic
314C22×C4D4×D5911yescharacteristic
315C22×C4#1174568711yescharacteristic
316C22×C4C22×Dic5911yescharacteristic
317C22×C4C2×C22⋊D5911yescharacteristic
318C22×C4C2×C22⋊D5911yescharacteristic
319SD16—42no—
320SD16—42no—
321M16—2361no—
322M16—2361no—
323M16—2361no—
324M16—2361no—
325C2×C8—22no—
326C2×C8—22no—
327C2×C8—22no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—1182no—
332C2×C8—2361no—
333C2×C8—2361no—
334C2×C8—2361no—
335C2×C8—2361no—
336C2×C8C22⋊Dic5911yescharacteristic
337C2×C8C22⋊Dic5911yescharacteristic
338C2×C8C22⋊Dic5911yescharacteristic
339C2×C8C22⋊Dic5911yescharacteristic
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C22⋊C4—41no—
347C22⋊C4—41no—
348C22⋊C4—41no—
349C22⋊C4—41no—
350C22⋊C4—1184no—
351C42—2361no—
352C42—2361no—
353C23—22no—
354C23—22no—
355C23—41no—
356C23—41no—
357C23—42no—
358C23—81no—
359C23—81no—
360C23D4×Dic5911yescharacteristic
361C23#1427369811yescharacteristic
362C23#1427371411yescharacteristic
363Q8—41no—
364Q8—41no—
365Q8—81no—
366D4—41no—
367D4—41no—
368D4—81no—
369D4—82no—
370D4—82no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—42no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—1184no—
387C2×C4—2362no—
388C2×C4—2362no—
389C2×C4C2×C22⋊Dic5911yescharacteristic
390C2×C4D4×Dic5911yescharacteristic
391C2×C4#1427369811yescharacteristic
392C2×C4#1427372311yescharacteristic
393C8—21no—
394C8—21no—
395C8—21no—
396C8—4721no—
397C8—4721no—
398C8—2362no—
399C8#1427364212yes—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—44no—
408C22—81no—
409C22—81no—
410C22—81no—
411C22—162no—
412C22#1683730511yescharacteristic
413C22#1683751912yes—
414C22#1683766212yes—
415C4—21no—
416C4—21no—
417C4—21no—
418C4—81no—
419C4—81no—
420C4—81no—
421C4—2364no—
422C4C2×#1427364211yescharacteristic
423C4#1683737611yescharacteristic
424C2—22no—
425C2—22no—
426C2—41no—
427C2—161no—
428C2#2142493311yescharacteristic
429C2#2142498511yescharacteristic
430C2#2142500111yescharacteristic
subgroupG/H
centerC22#16837305
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC2×C59×#465C2
socleC22×C59#390
prime radicalC2×#121×C59C22
cosocleC23×C4×C59C23
hypercenterC2×#465D59
hypocenterC59#9813
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#465

Complements

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

NH ≅ G/Ntype
1. #21424134C2semidirect
2. C8×C22⋊Dic59C2semidirect
3. #21422616C2semidirect
5. #21422460C2semidirect
20. C22×SD16×C59C4semidirect
28. C2×#106×C59C4semidirect
29. C2×#103×C59C4semidirect
30. C2×#102×C59C4semidirect
223. C22×SD16Dic59semidirect
232. C2×#106Dic59semidirect
233. C2×#103Dic59semidirect
234. C2×#102Dic59semidirect
246. C59#9813semidirect

Maximal subgroups

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

subgroupindex
#214241342
C8×C22⋊Dic592
#214226162
#214226122
#214224602
#214224562
C2×C59×#4652
#981359