Finite Group Databasebeta

← #28486326

Subgroups — #28486326

Order 15104.

712 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 712 conjugacy classes of subgroups — too many to draw. Only its 103 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21432541C211yesmaximal, characteristic
2#21427720C211yesmaximal, characteristic
3#21422628C212yesmaximal, complemented
4#21421948C212yesmaximal, complemented
5C59×#1782C211yesmaximal, characteristic
6C22×#11745664—21no—
7C2×#14273725C411yescharacteristic, complemented
8C2×#14273714C411yescharacteristic, complemented
9C2×#14273706C2211yescharacteristic
10#16837369—24no—
11#16837366—21no—
12#16837366C2212yes—
13#16837316—21no—
14#16837316—21no—
15#16837309C411yescharacteristic, complemented
16#16837305C411yescharacteristic, complemented
17#16837275—21no—
18#16837275—21no—
19C4×C22⋊Dic59C2211yescharacteristic
20#16837138—24no—
21C22×#41×C59C2211yescharacteristic
22C2×#99×C59—21no—
23C2×#99×C59—21no—
24C59×#381C2212yes—
25C22×Dic118C2×C411yescharacteristic
26C2×#11745664—22no—
27C2×#11745664—22no—
28C2×#11745664—42no—
29C2×#11745664—42no—
30C22×C59⋊C8—22no—
31#14273725—42no—
32#14273714—42no—
33#14273706—22no—
34#14273706—22no—
35C2×C4⋊Dic59C2×C411yescharacteristic
36C2×Dic59⋊C4—21no—
37C2×Dic59⋊C4—21no—
38C2×Dic59⋊C4—21no—
39C2×C22⋊Dic59—21no—
40C2×C22⋊Dic59—21no—
41C2×C22⋊Dic59—21no—
42C2×C22⋊Dic59—21no—
43C2×C22⋊Dic59—21no—
44C2×C22⋊Dic59—21no—
45C2×C4×Dic59C2×C411yescharacteristic
46C2×C4×Dic59C2×C411yescharacteristic
47C22⋊(C59⋊C8)—24no—
48C22⋊(C59⋊C8)—22no—
49C22⋊(C59⋊C8)D414yes—
50C59⋊#95—21no—
51C59⋊#95—21no—
52C59⋊#95—21no—
53C2×#41×C59—22no—
54C2×#41×C59—22no—
55C2×#41×C59—42no—
56C2×#41×C59—42no—
57C22×Q8×C59C2×C411yescharacteristic
58C22×D4×C59C2×C411yescharacteristic
59C22×D4×C59D411yescharacteristic
60C22×D4×C59D411yescharacteristic
61C23×C4×C59D411yescharacteristic
62C23×C4×C59D411yescharacteristic
63C23×C4×C59C2311yescharacteristic
64C2×C22⋊C4×C59—22no—
65#99×C59—42no—
66#99×C59—42no—
67#95×C59—22no—
68#95×C59—22no—
69C24×C59C22⋊C411yescharacteristic
70C24×C59C22⋊C411yescharacteristic
71C24×C59C2×D411yescharacteristic
72#41×C59—44no—
73#41×C59—44no—
74#41×C59—48no—
75C2×Q8×C59—22no—
76C2×Q8×C59—22no—
77C2×Q8×C59—41no—
78C2×D4×C59—22no—
79C2×D4×C59—22no—
80C2×D4×C59—22no—
81C2×D4×C59—22no—
82C2×D4×C59—22no—
83C2×D4×C59—22no—
84C2×D4×C59—41no—
85C2×D4×C59—41no—
86C2×D4×C59—41no—
87C22×C4×C59—22no—
88C22×C4×C59—22no—
89C22×C4×C59—21no—
90C22×C4×C59—21no—
91C22×C4×C59—21no—
92C22×C4×C59—21no—
93C22×C4×C59—41no—
94C22×C4×C59—41no—
95C22×C4×C59—41no—
96C22×C4×C59—41no—
97C22×C4×C59—42no—
98C22×C4×C59C22⋊C411yescharacteristic
99C22×C4×C59C22⋊C411yescharacteristic
100C22×C4×C59C22×C411yescharacteristic
101C22×C4×C59C2×D411yescharacteristic
102C22×C4×C59C2×D412yes—
103C22×C4×C59#4112yes—
104C22×Dic59—21no—
105C22×Dic59—21no—
106C22×Dic59—21no—
107C22×Dic59—21no—
108C2×Dic118—22no—
109C2×Dic118—41no—
110C2×Dic118—41no—
111C22⋊C4×C59—42no—
112C22⋊C4×C59—42no—
113C22⋊Dic59—42no—
114C22⋊Dic59—42no—
115C22⋊Dic59—42no—
116C22⋊Dic59—42no—
117C22⋊Dic59—42no—
118C22⋊Dic59—42no—
119C4⋊Dic59—22no—
120Dic59⋊C4—42no—
121Dic59⋊C4—42no—
122Dic59⋊C4—41no—
123Dic59⋊C4—41no—
124C4×Dic59—41no—
125C4×Dic59—41no—
126#11745664—44no—
127#11745664—42no—
128#11745664—42no—
129C2×C59⋊C8—24no—
130C2×C59⋊C8—44no—
131C23×C59—21no—
132C23×C59—21no—
133C23×C59—21no—
134C23×C59—21no—
135C23×C59—21no—
136C23×C59—21no—
137C23×C59—21no—
138C23×C59—41no—
139C23×C59—41no—
140C23×C59—41no—
141C23×C59—41no—
142C23×C59—41no—
143C23×C59—41no—
144C23×C59#9912yes—
145C23×C59C2×C22⋊C411yescharacteristic
146C23×C59#12112yes—
147Q8×C59—44no—
148D4×C59—44no—
149D4×C59—44no—
150D4×C59—44no—
151C2×C4×C59—22no—
152C2×C4×C59—22no—
153C2×C4×C59—24no—
154C2×C4×C59—21no—
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—21no—
162C2×C4×C59—21no—
163C2×C4×C59—42no—
164C2×C4×C59—42no—
165C2×C4×C59—42no—
166C2×C4×C59—42no—
167C2×C4×C59—82no—
168C2×C4×C59—82no—
169C2×C4×C59—82no—
170C2×C4×C59#9912yes—
171C2×C4×C59C4×D412yes—
172C2×C4×C59#12011yescharacteristic
173C2×C4×C59#12311yescharacteristic
174C2×Dic59—22no—
175C2×Dic59—41no—
176C2×Dic59—41no—
177C2×Dic59—41no—
178C2×Dic59—41no—
179C2×Dic59—41no—
180C2×Dic59—41no—
181C2×Dic59—41no—
182C2×Dic59—41no—
183C2×Dic59—41no—
184C2×Dic59—41no—
185C2×Dic59—41no—
186Dic118—82no—
187C59⋊C8—44no—
188#9443—591nomaximal, Sylow
189C22×C59—21no—
190C22×C59—21no—
191C22×C59—22no—
192C22×C59—22no—
193C22×C59—22no—
194C22×C59—22no—
195C22×C59—42no—
196C22×C59—42no—
197C22×C59—41no—
198C22×C59—41no—
199C22×C59—41no—
200C22×C59—41no—
201C22×C59—41no—
202C22×C59—81no—
203C22×C59—81no—
204C22×C59—81no—
205C22×C59—81no—
206C22×C59#38611yescharacteristic
207C22×C59C2×#9911yescharacteristic
208C22×C59C2×#9911yescharacteristic
209C4×C59—21no—
210C4×C59—22no—
211C4×C59—21no—
212C4×C59—22no—
213C4×C59—41no—
214C4×C59—162no—
215Dic59—42no—
216Dic59—81no—
217Dic59—81no—
218Dic59—81no—
219#2879—591no—
220#2432—591no—
221#1782D5911yescharacteristic
222#1511—592no—
223#1284—592no—
224C2×C59—21no—
225C2×C59—21no—
226C2×C59—42no—
227C2×C59—42no—
228C2×C59—81no—
229C2×C59#188411yescharacteristic
230C2×C59#189811yescharacteristic
231C2×C59#190911yescharacteristic
232C22×#41C2×D5911yescharacteristic
233C22×M16—1181no—
234C2×C4⋊Q8—591no—
235C2×#124—591no—
236C2×C22⋊Q8—591no—
237C2×#99—21no—
238C2×#99—21no—
239#408—1184no—
240#407—592no—
241#407—1181no—
242#394—1181no—
243#394—1181no—
244#391—591no—
245#390—591no—
246#385—1181no—
247#385—1181no—
248#381C2×D5912yes—
249C4×C22⋊C4—591no—
250#328—1184no—
251C59#944311yescharacteristic, complemented, Sylow
252C2×#41—22no—
253C2×#41—22no—
254C2×#41—42no—
255C2×#41—42no—
256C22×Q8—591no—
257C22×Q8C4×D5911yescharacteristic
258C22×D4C4×D5911yescharacteristic
259C22×D4C22⋊D5911yescharacteristic
260C22×D4C22⋊D5911yescharacteristic
261C23×C4D23611yescharacteristic
262C23×C4D23611yescharacteristic
263C23×C4C22×D5911yescharacteristic
264C2×M16—1182no—
265C2×M16—1182no—
266C2×M16—2362no—
267C2×M16—2362no—
268C22×C8—1182no—
269C4⋊Q8—2362no—
270#124—2362no—
271C22⋊Q8—1182no—
272C22⋊Q8—1182no—
273C2×C4⋊C4—591no—
274C2×C4⋊C4—1181no—
275C2×C4⋊C4—1181no—
276C2×C4⋊C4—1181no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—1181no—
279C2×C22⋊C4—1181no—
280C2×C22⋊C4—1181no—
281C2×C22⋊C4—1181no—
282C2×C22⋊C4—1181no—
283C2×C22⋊C4—1181no—
284C2×C42—591no—
285C2×C42—591no—
286#99—42no—
287#99—42no—
288C22⋊C8—594no—
289C22⋊C8—1184no—
290C22⋊C8—1182no—
291#95—22no—
292#95—22no—
293#95—1181no—
294#95—1181no—
295#95—1181no—
296C24(C2×D59)⋊C411yescharacteristic
297C24(C2×D59)⋊C411yescharacteristic
298C24C2×C22⋊D5911yescharacteristic
299#41—44no—
300#41—44no—
301#41—48no—
302C2×Q8—22no—
303C2×Q8—22no—
304C2×Q8—41no—
305C2×Q8—1182no—
306C2×Q8—2361no—
307C2×Q8—2361no—
308C2×D4—22no—
309C2×D4—22no—
310C2×D4—22no—
311C2×D4—22no—
312C2×D4—22no—
313C2×D4—22no—
314C2×D4—41no—
315C2×D4—41no—
316C2×D4—41no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—42no—
328C22×C4—1181no—
329C22×C4—1181no—
330C22×C4—1181no—
331C22×C4—1181no—
332C22×C4(C2×D59)⋊C411yescharacteristic
333C22×C4(C2×D59)⋊C411yescharacteristic
334C22×C4C2×C4×D5911yescharacteristic
335C22×C4C2×D23611yescharacteristic
336C22×C4D4×D5912yes—
337C22×C4#1174568712yes—
338M16—2364no—
339M16—2362no—
340M16—2362no—
341C2×C8—1184no—
342C2×C8—2364no—
343C4⋊C4—1182no—
344C4⋊C4—2362no—
345C4⋊C4—2362no—
346C4⋊C4—2361no—
347C4⋊C4—2361no—
348C22⋊C4—42no—
349C22⋊C4—42no—
350C22⋊C4—2362no—
351C22⋊C4—2362no—
352C22⋊C4—2362no—
353C22⋊C4—2362no—
354C22⋊C4—2362no—
355C22⋊C4—2362no—
356C42—2361no—
357C42—2361no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23—41no—
371C23#1427364112yes—
372C23C2×(C2×D59)⋊C411yescharacteristic
373C23#1427369912yes—
374Q8—44no—
375Q8—4722no—
376D4—44no—
377D4—44no—
378D4—44no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—24no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—1182no—
399C2×C4—2361no—
400C2×C4—2361no—
401C2×C4—2361no—
402C2×C4—2361no—
403C2×C4—2361no—
404C2×C4—2361no—
405C2×C4—2361no—
406C2×C4—2361no—
407C2×C4—2361no—
408C2×C4—2361no—
409C2×C4—2361no—
410C2×C4#1427364112yes—
411C2×C4(C22⋊D59)⋊C412yes—
412C2×C4#1427369711yescharacteristic
413C2×C4#1427371111yescharacteristic
414C8—2364no—
415C22—21no—
416C22—21no—
417C22—22no—
418C22—22no—
419C22—22no—
420C22—22no—
421C22—42no—
422C22—42no—
423C22—41no—
424C22—41no—
425C22—41no—
426C22—41no—
427C22—41no—
428C22—81no—
429C22—81no—
430C22—81no—
431C22—81no—
432C22#1683727911yescharacteristic
433C22C2×#1427364111yescharacteristic
434C22C2×#1427364111yescharacteristic
435C4—21no—
436C4—21no—
437C4—22no—
438C4—22no—
439C4—41no—
440C4—162no—
441C4—4721no—
442C4—4721no—
443C4—4721no—
444C4—2362no—
445C2—21no—
446C2—21no—
447C2—42no—
448C2—42no—
449C2—81no—
450C2#2142447711yescharacteristic
451C2#2142454711yescharacteristic
452C2#2142461611yescharacteristic
subgroupG/H
centerC22#16837279
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×#1782C2
socleC22×C59#386
prime radicalC22×#41×C59C22
cosocleC23×C4×C59C23
hypercenter#1782D59
hypocenterC59#9443
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1782

Complements

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

NH ≅ G/Ntype
3. #21422628C2semidirect
4. #21421948C2semidirect
7. C2×#14273725C4semidirect
8. C2×#14273714C4semidirect
15. #16837309C4semidirect
16. #16837305C4semidirect
251. C59#9443semidirect

Maximal subgroups

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

subgroupindex
#214325412
#214277202
#214226282
#214219482
C59×#17822
#944359