Finite Group Databasebeta

← #28507331

Subgroups — #28507331

Order 15104.

484 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 93 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#21428410C211yesmaximal, characteristic
2#21424082C211yesmaximal, characteristic
3#100×Dic59C211yesmaximal, characteristic
4#21422215C211yesmaximal, characteristic
5#21422214C211yesmaximal, characteristic
6#21422213C211yesmaximal, characteristic
7C59×#1390C211yesmaximal, characteristic
8C2×D4×Dic59C2211yescharacteristic
9C22×Dic59⋊C4C411yescharacteristic, complemented
10C2×#14273644C2211yescharacteristic
11C2×#14273642C2211yescharacteristic
12M16×Dic59—21no—
13#16837344—21no—
14#16837338—21no—
15#16837299—21no—
16#16837282C2211yescharacteristic
17#16837277C411yescharacteristic, complemented
18#16837275C411yescharacteristic, complemented
19#16837260—21no—
20#16837260—21no—
21#16837260C411yescharacteristic, complemented
22#16837170—21no—
23#16837168—21no—
24C2×#100×C59C2211yescharacteristic
25C2×#99×C59C2211yescharacteristic
26C59×#386C2211yescharacteristic
27C23×Dic59C2×C411yescharacteristic
28C23×Dic59C2×C411yescharacteristic
29C23×Dic59D411yescharacteristic
30C2×#11745664—21no—
31D4×Dic59—41no—
32D4×Dic59—41no—
33C2×C4⋊Dic59—21no—
34C2×C4⋊Dic59D411yescharacteristic
35C2×Dic59⋊C4—21no—
36C2×Dic59⋊C4—21no—
37C2×Dic59⋊C4—21no—
38C2×Dic59⋊C4—21no—
39C2×Dic59⋊C4—41no—
40C2×Dic59⋊C4—41no—
41C2×Dic59⋊C4—41no—
42C2×C22⋊Dic59—21no—
43C2×C22⋊Dic59—21no—
44C2×C22⋊Dic59C2×C411yescharacteristic
45C2×C22⋊Dic59C2×C411yescharacteristic
46C2×C22⋊Dic59D411yescharacteristic
47C2×C4×Dic59D411yescharacteristic
48#14273644—21no—
49#14273644Q812yes—
50#14273642—21no—
51#14273642—21no—
52#14273637—41no—
53C8×Dic59—41no—
54C59⋊#95—21no—
55C59⋊#95—21no—
56C59⋊#95—21no—
57C59⋊#95—21no—
58C59⋊#95—41no—
59C22×D4×C59C2311yescharacteristic
60C23×C4×C59C2×C411yescharacteristic
61C2×M16×C59—21no—
62C2×C22⋊C4×C59—21no—
63C2×C22⋊C4×C59—21no—
64C2×C22⋊C4×C59C2×C411yescharacteristic
65#100×C59—21no—
66#100×C59Q812yes—
67#99×C59—21no—
68#99×C59—21no—
69#95×C59—21no—
70C24×C59C22×C411yescharacteristic
71C24×C59C2×D411yescharacteristic
72C2×D4×C59—41no—
73C2×D4×C59—41no—
74C2×D4×C59C2×Q811yescharacteristic
75C2×D4×C59C2×Q811yescharacteristic
76C2×D4×C59#4111yescharacteristic
77C2×D4×C59#4111yescharacteristic
78C22×C4×C59—21no—
79C22×C4×C59—21no—
80C22×C4×C59—41no—
81C22×C4×C59—41no—
82C22×C4×C59—81no—
83C22×C4×C59C2×D411yescharacteristic
84C22×Dic59—21no—
85C22×Dic59—21no—
86C22×Dic59—21no—
87C22×Dic59—21no—
88C22×Dic59—21no—
89C22×Dic59—21no—
90C22×Dic59—42no—
91C22×Dic59—42no—
92C22×Dic59—41no—
93C22×Dic59—41no—
94C22×Dic59—81no—
95C22×Dic59C22⋊C411yescharacteristic
96C22×Dic59C22⋊C411yescharacteristic
97C22×Dic59C22⋊C411yescharacteristic
98C22×Dic59C22⋊C411yescharacteristic
99M16×C59—22no—
100M16×C59—41no—
101C2×C8×C59—41no—
102C22⋊C4×C59—41no—
103C22⋊C4×C59—41no—
104C22⋊C4×C59—41no—
105C22⋊C4×C59—81no—
106C22⋊Dic59—41no—
107C22⋊Dic59—41no—
108C22⋊Dic59—41no—
109C22⋊Dic59—41no—
110C22⋊Dic59—41no—
111C22⋊Dic59—81no—
112C4⋊Dic59—41no—
113C4⋊Dic59—81no—
114Dic59⋊C4—42no—
115Dic59⋊C4—42no—
116Dic59⋊C4—41no—
117Dic59⋊C4—41no—
118Dic59⋊C4—81no—
119Dic59⋊C4—82no—
120C4×Dic59—21no—
121C4×Dic59—21no—
122#11745664—22no—
123#11745664—41no—
124C2×C59⋊C8—41no—
125C23×C59—21no—
126C23×C59—21no—
127C23×C59—21no—
128C23×C59—21no—
129C23×C59—41no—
130C23×C59—41no—
131C23×C59—41no—
132C23×C59—41no—
133C23×C59C2×C22⋊C411yescharacteristic
134C23×C59C4×Q811yescharacteristic
135C23×C59C4×Q811yescharacteristic
136C23×C59#12411yescharacteristic
137C23×C59C4⋊Q811yescharacteristic
138D4×C59—41no—
139D4×C59—41no—
140D4×C59—41no—
141D4×C59—41no—
142C2×C4×C59—21no—
143C2×C4×C59—21no—
144C2×C4×C59—41no—
145C2×C4×C59—41no—
146C2×C4×C59—41no—
147C2×C4×C59—41no—
148C2×C4×C59—41no—
149C2×C4×C59—41no—
150C2×C4×C59—82no—
151C2×C4×C59—81no—
152C2×C4×C59—81no—
153C2×C4×C59—81no—
154C2×C4×C59C22⋊Q811yescharacteristic
155C2×C4×C59C22⋊Q811yescharacteristic
156C2×Dic59—22no—
157C2×Dic59—21no—
158C2×Dic59—21no—
159C2×Dic59—21no—
160C2×Dic59—42no—
161C2×Dic59—42no—
162C2×Dic59—42no—
163C2×Dic59—41no—
164C2×Dic59—41no—
165C2×Dic59—41no—
166C2×Dic59—41no—
167C2×Dic59—41no—
168C2×Dic59—41no—
169C2×Dic59—41no—
170C2×Dic59—41no—
171C2×Dic59—41no—
172C2×Dic59—81no—
173C2×Dic59—82no—
174C2×Dic59—81no—
175C2×Dic59—81no—
176C2×Dic59—81no—
177C2×Dic59#9911yescharacteristic
178C2×Dic59#9911yescharacteristic
179C8×C59—42no—
180C59⋊C8—42no—
181#13292—591nomaximal, Sylow
182C22×C59—21no—
183C22×C59—21no—
184C22×C59—21no—
185C22×C59—21no—
186C22×C59—21no—
187C22×C59—21no—
188C22×C59—41no—
189C22×C59—41no—
190C22×C59—41no—
191C22×C59—41no—
192C22×C59—42no—
193C22×C59—41no—
194C22×C59—41no—
195C22×C59—41no—
196C22×C59—41no—
197C22×C59—81no—
198C22×C59#39111yescharacteristic
199C22×C59C2×#9911yescharacteristic
200C22×C59#41011yescharacteristic
201C4×C59—21no—
202C4×C59—21no—
203C4×C59—81no—
204C4×C59—81no—
205C4×C59—161no—
206Dic59—21no—
207Dic59—41no—
208Dic59—41no—
209Dic59—41no—
210Dic59—81no—
211Dic59—81no—
212Dic59—161no—
213Dic59#35212yes—
214#2487—591no—
215#1799—591no—
216C4×#100—591no—
217#1390—591no—
218#1390—591no—
219#1390—591no—
220#1390D5911yescharacteristic
221C2×C59—21no—
222C2×C59—21no—
223C2×C59—41no—
224C2×C59—41no—
225C2×C59—41no—
226C2×C59—41no—
227C2×C59#199611yescharacteristic
228C2×C59C2×#35211yescharacteristic
229C2×C59#212211yescharacteristic
230C2×C4×D4—591no—
231C22×C4⋊C4—591no—
232C2×#100—591no—
233C2×#100C2×D5911yescharacteristic
234C2×#99—591no—
235C2×#99C2×D5911yescharacteristic
236C4×M16—1181no—
237#400—1181no—
238#399—1181no—
239#389—1181no—
240#386—591no—
241#386C2×D5911yescharacteristic
242#385—591no—
243#385—591no—
244#380—591no—
245#380—1181no—
246#380—1181no—
247#343—1181no—
248#342—1181no—
249C59#1329211yescharacteristic, complemented, Sylow
250C22×D4C22×D5911yescharacteristic
251C23×C4—591no—
252C23×C4—591no—
253C23×C4—591no—
254C23×C4C4×D5911yescharacteristic
255C2×M16—21no—
256C2×M16—1181no—
257C4×D4—2361no—
258C4×D4—2361no—
259C2×C4⋊C4—591no—
260C2×C4⋊C4—1181no—
261C2×C4⋊C4—1181no—
262C2×C4⋊C4—1181no—
263C2×C4⋊C4—1181no—
264C2×C4⋊C4—1181no—
265C2×C4⋊C4—2361no—
266C2×C4⋊C4—2361no—
267C2×C4⋊C4—2361no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—591no—
271C2×C22⋊C4—591no—
272C2×C22⋊C4—591no—
273C2×C22⋊C4—1181no—
274C2×C22⋊C4—1181no—
275C2×C22⋊C4C4×D5911yescharacteristic
276C2×C42—591no—
277#100—21no—
278#100—592no—
279#100—1181no—
280#100Dic11812yes—
281#99—21no—
282#99—21no—
283#99—1181no—
284#99—1181no—
285#97—2361no—
286C4×C8—2361no—
287#95—21no—
288#95—1181no—
289#95—1181no—
290#95—1181no—
291#95—1181no—
292#95—2361no—
293C24C2×C4×D5911yescharacteristic
294C24D4×D5911yescharacteristic
295C2×D4—41no—
296C2×D4—41no—
297C2×D4C2×Dic11811yescharacteristic
298C2×D4#1174568511yescharacteristic
299C2×D4#1174568711yescharacteristic
300C2×D4Q8×D5911yescharacteristic
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—81no—
306C22×C4—591no—
307C22×C4—591no—
308C22×C4—591no—
309C22×C4—591no—
310C22×C4—1181no—
311C22×C4—1181no—
312C22×C4—1181no—
313C22×C4—1181no—
314C22×C4—1181no—
315C22×C4—1181no—
316C22×C4—4721no—
317C22×C4—2362no—
318C22×C4—2362no—
319C22×C4—2361no—
320C22×C4—2361no—
321C22×C4D4×D5911yescharacteristic
322M16—22no—
323M16—41no—
324M16—1182no—
325M16—2361no—
326C2×C8—41no—
327C2×C8—2361no—
328C4⋊C4—4721no—
329C4⋊C4—4721no—
330C4⋊C4—4722no—
331C4⋊C4—2361no—
332C4⋊C4—2361no—
333C4⋊C4—2361no—
334C4⋊C4—2362no—
335C4⋊C4—2362no—
336C22⋊C4—41no—
337C22⋊C4—41no—
338C22⋊C4—41no—
339C22⋊C4—81no—
340C22⋊C4—4721no—
341C22⋊C4—2361no—
342C22⋊C4—2361no—
343C22⋊C4—2361no—
344C22⋊C4—2361no—
345C22⋊C4—2361no—
346C42—1181no—
347C42—1181no—
348C23—21no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23C22⋊C4×D5911yescharacteristic
357C23#1427369211yescharacteristic
358C23C4×Dic11811yescharacteristic
359C23#1427371311yescharacteristic
360C23#1427372611yescharacteristic
361D4—41no—
362D4—41no—
363D4—41no—
364D4—41no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—82no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—591no—
378C2×C4—591no—
379C2×C4—1182no—
380C2×C4—1181no—
381C2×C4—1181no—
382C2×C4—1181no—
383C2×C4—4721no—
384C2×C4—4721no—
385C2×C4—4721no—
386C2×C4—4721no—
387C2×C4—4722no—
388C2×C4—2362no—
389C2×C4—2362no—
390C2×C4—2362no—
391C2×C4—2361no—
392C2×C4—2361no—
393C2×C4—2361no—
394C2×C4—2361no—
395C2×C4—2361no—
396C2×C4—2361no—
397C2×C4—2361no—
398C2×C4—2361no—
399C2×C4—2361no—
400C2×C4#1427370411yescharacteristic
401C2×C4#1427370511yescharacteristic
402C8—42no—
403C8—2362no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—42no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22—81no—
420C22#1683730611yescharacteristic
421C22#99×D5911yescharacteristic
422C22#1683737411yescharacteristic
423C4—21no—
424C4—21no—
425C4—81no—
426C4—81no—
427C4—161no—
428C4—592no—
429C4—1181no—
430C4—9441no—
431C4—4721no—
432C4—4721no—
433C4—2361no—
434C4—2361no—
435C4—2361no—
436C2—21no—
437C2—21no—
438C2—41no—
439C2—41no—
440C2—41no—
441C2—41no—
442C2#2142507911yescharacteristic
443C2#352×D5911yescharacteristic
444C2#2142574411yescharacteristic
subgroupG/H
centerC22#99×D59
commutatorC24×C59C22×C4
FrattiniC22×D4C22×D59
FittingC59×#1390C2
socleC22×C59C2×#99
prime radical, cosocleC22×D4×C59C23
hypercenter#1390D59
hypocenterC59#13292
LCSG ▷ C24×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ #1390

Complements

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

NH ≅ G/Ntype
9. C22×Dic59⋊C4C4semidirect
17. #16837277C4semidirect
18. #16837275C4semidirect
21. #16837260C4semidirect
249. C59#13292semidirect

Maximal subgroups

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

subgroupindex
#214284102
#214240822
#100×Dic592
#214222152
#214222142
#214222132
C59×#13902
#1329259