Finite Group Databasebeta

← #28488393

Subgroups — #28488393

Order 15104.

522 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 522 conjugacy classes of subgroups — too many to draw. Only its 99 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
1C2×#16837633C211yesmaximal, characteristic
2#21424132C211yesmaximal, characteristic, complemented
3#21423870C211yesmaximal, characteristic, complemented
4#21422616C211yesmaximal, characteristic, complemented
5#21422613C211yesmaximal, characteristic
6#21422455C211yesmaximal, characteristic
7C59×#1472C211yesmaximal, characteristic, complemented
8C22×#11745671C411yescharacteristic, complemented
9C23×C59⋊C8C2211yescharacteristic
10C2×#14273701C2211yescharacteristic
11#16837633—21no—
12#16837633—21no—
13#16837633—21no—
14#16837633—21no—
15C2×#14273662—21no—
16C2×#14273658C411yescharacteristic, complemented
17C2×#14273651C411yescharacteristic, complemented
18C2×#14273647C411yescharacteristic, complemented
19#16837366C2211yescharacteristic
20C2×C8×Dic59—21no—
21C4×C22⋊Dic59C2211yescharacteristic
22#16837166—21no—
23#16837156—21no—
24#16837139—22no—
25#16837137—22no—
26#16837125—22no—
27C2×C22⋊Q8×C59C2211yescharacteristic
28C59×#407C2211yescharacteristic
29C2×C22⋊C8×C59C2211yescharacteristic
30C59×#324—22no—
31C22×D236C2×C411yescharacteristic
32C22×C22⋊D59—21no—
33C2×#11745671—21no—
34C2×#11745671—21no—
35C2×#11745671—42no—
36C2×#11745664—21no—
37C22×C59⋊C8—42no—
38C22×C59⋊C8C2×C411yescharacteristic
39C22×C59⋊C8C2×C411yescharacteristic
40C22×C59⋊C8D411yescharacteristic
41C22×C59⋊C8D411yescharacteristic
42C22×C59⋊C8D411yescharacteristic
43C22×C59⋊C8D411yescharacteristic
44#14273701—21no—
45#14273701—21no—
46#14273701—21no—
47#14273701—21no—
48C2×C4⋊Dic59C2×C411yescharacteristic
49C2×C22⋊Dic59—21no—
50C2×(C2×D59)⋊C4—21no—
51C2×C4×Dic59—21no—
52#14273662—42no—
53#14273658—21no—
54#14273658—21no—
55#14273651—21no—
56#14273651—21no—
57#14273647—21no—
58#14273647—21no—
59C22⋊(C59⋊C8)—21no—
60C22⋊(C59⋊C8)—21no—
61C8×Dic59—42no—
62C59⋊#95D412yes—
63C22×Q8×C59C2×C411yescharacteristic
64C23×C4×C59C2311yescharacteristic
65C2×M16×C59—21no—
66C22×C8×C59—21no—
67C22⋊Q8×C59—21no—
68C22⋊Q8×C59—21no—
69C22⋊Q8×C59—21no—
70C22⋊Q8×C59—21no—
71C2×C4⋊C4×C59—21no—
72C2×C4⋊C4×C59C2×C411yescharacteristic
73C2×C22⋊C4×C59—21no—
74C22⋊C8×C59—21no—
75C22⋊C8×C59—21no—
76C22⋊C8×C59—21no—
77C22⋊C8×C59D412yes—
78C24×C59#4111yescharacteristic
79C23×D59—21no—
80C2×Q8×C59—21no—
81C2×Q8×C59—21no—
82C2×Q8×C59—41no—
83C2×Q8×C59—41no—
84C22×C4×C59—21no—
85C22×C4×C59—21no—
86C22×C4×C59—41no—
87C22×C4×C59—41no—
88C22×C4×C59C22×C411yescharacteristic
89C22×C4×C59C2×D411yescharacteristic
90C22×C4×C59C2×D411yescharacteristic
91C22×C4×C59C2×D411yescharacteristic
92C22×C4×C59C2×D411yescharacteristic
93C22×C4×C59#4111yescharacteristic
94C2×C22⋊D59—41no—
95C2×C22⋊D59—41no—
96C2×C22⋊D59—82no—
97C22×Dic59—22no—
98C22×Dic59—21no—
99C2×D236—21no—
100C2×D236—21no—
101C2×D236—41no—
102C2×D236—41no—
103M16×C59—41no—
104M16×C59—41no—
105C2×C8×C59—22no—
106C2×C8×C59—41no—
107C2×C8×C59—41no—
108C2×C8×C59—41no—
109C4⋊C4×C59—21no—
110C4⋊C4×C59—21no—
111C4⋊C4×C59—41no—
112C4⋊C4×C59—41no—
113C22⋊C4×C59—41no—
114C22⋊C4×C59—41no—
115C22⋊Dic59—24no—
116#11745671—42no—
117#11745671—42no—
118(C2×D59)⋊C4—41no—
119(C2×D59)⋊C4—41no—
120C4⋊Dic59—21no—
121C4⋊Dic59—21no—
122C4×Dic59—41no—
123C4×Dic59—41no—
124#11745664—41no—
125#11745664—41no—
126C2×C59⋊C8—22no—
127C2×C59⋊C8—22no—
128C2×C59⋊C8—22no—
129C2×C59⋊C8—41no—
130C2×C59⋊C8—41no—
131C2×C59⋊C8—41no—
132C2×C59⋊C8—41no—
133C2×C59⋊C8C22⋊C411yescharacteristic
134C2×C59⋊C8C22⋊C411yescharacteristic
135C2×C59⋊C8C22⋊C411yescharacteristic
136C2×C59⋊C8C22⋊C411yescharacteristic
137C23×C59—22no—
138C23×C59—22no—
139C23×C59—41no—
140C23×C59C4×D411yescharacteristic
141C23×C59#12111yescharacteristic
142C23×C59#12411yescharacteristic
143C22×D59—41no—
144C22×D59—42no—
145C22×D59—81no—
146C22×D59—81no—
147Q8×C59—41no—
148Q8×C59—41no—
149Q8×C59—81no—
150C2×C4×C59—21no—
151C2×C4×C59—21no—
152C2×C4×C59—21no—
153C2×C4×C59—21no—
154C2×C4×C59—21no—
155C2×C4×C59—21no—
156C2×C4×C59—41no—
157C2×C4×C59—41no—
158C2×C4×C59—41no—
159C2×C4×C59—41no—
160C2×C4×C59—41no—
161C2×C4×C59—42no—
162C2×C4×C59—81no—
163C2×C4×C59C2×C22⋊C411yescharacteristic
164C2×C4×C59C4×D411yescharacteristic
165C2×C4×C59#12111yescharacteristic
166C2×C4×C59#12711yescharacteristic
167C22⋊D59—82no—
168C22⋊D59—82no—
169C2×Dic59—24no—
170C2×Dic59—42no—
171C2×Dic59—42no—
172C2×Dic59—41no—
173C2×Dic59—81no—
174D236—41no—
175D236—41no—
176D236—81no—
177C8×C59—42no—
178C8×C59—81no—
179C59⋊C8—21no—
180C59⋊C8—21no—
181C59⋊C8—21no—
182C59⋊C8—81no—
183C59⋊C8#9912yes—
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#39011yescharacteristic
194C22×C59#44312yes—
195C22×C59#47112yes—
196C2×D59—81no—
197C2×D59—81no—
198C2×D59—81no—
199C2×D59—162no—
200C4×C59—21no—
201C4×C59—21no—
202C4×C59—21no—
203C4×C59—81no—
204C4×C59—81no—
205C4×C59C2×#9911yescharacteristic
206C4×C59#41011yescharacteristic
207Dic59—44no—
208Dic59—81no—
209C2×#465—591no—
210#1810—591no—
211C8×C22⋊C4—591no—
212#1508—591no—
213#1507—591no—
214#1472D5911yescharacteristic, complemented
215#1471—591no—
216C2×C59—22no—
217C2×C59—22no—
218C2×C59—41no—
219C2×C59#196411yescharacteristic
220C2×C59#197411yescharacteristic
221C2×C59#197711yescharacteristic
222D59—161no—
223C22×SD16—591no—
224C23×C8—591no—
225C2×C22⋊Q8C2×D5911yescharacteristic
226C2×#121—591no—
227#465—1181no—
228#465—1181no—
229#465—1181no—
230#465—1181no—
231C2×#108—1181no—
232C2×#106—591no—
233C2×#103—591no—
234C2×#102—591no—
235#407—591no—
236#407C2×D5911yescharacteristic
237C2×C22⋊C8C2×D5911yescharacteristic
238C2×C4×C8—1181no—
239C4×C22⋊C4—591no—
240#341—1181no—
241#336—1181no—
242#328—1182no—
243#327—1182no—
244#324—22no—
245#323—1182no—
246C59#981311yescharacteristic, complemented, Sylow
247C22×Q8C4×D5911yescharacteristic
248C22×D4—591no—
249C22×D4—1181no—
250C23×C4C22×D5911yescharacteristic
251C2×SD16—1181no—
252C2×SD16—1181no—
253C2×SD16—2362no—
254C2×M16—21no—
255C2×M16—1181no—
256C22×C8—21no—
257C22×C8—591no—
258C22×C8—591no—
259C22×C8—591no—
260C22×C8—591no—
261C22×C8—591no—
262C22×C8—591no—
263C22×C8—2362no—
264C22⋊Q8—21no—
265C22⋊Q8—21no—
266C22⋊Q8—21no—
267C22⋊Q8—21no—
268#121—1181no—
269#121—1181no—
270#121—1181no—
271#121—1181no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—591no—
274C2×C4⋊C4C4×D5911yescharacteristic
275C2×C22⋊C4—21no—
276C2×C22⋊C4—1181no—
277C2×C22⋊C4—1181no—
278C2×C42—1181no—
279#108—2362no—
280#106—1181no—
281#106—1181no—
282#103—1181no—
283#103—1181no—
284#102—1181no—
285#102—1181no—
286C22⋊C8—21no—
287C22⋊C8—21no—
288C22⋊C8—21no—
289C22⋊C8—1181no—
290C22⋊C8—1181no—
291C22⋊C8C22⋊D5912yes—
292C4×C8—2362no—
293#95—592no—
294C24—1181no—
295C24#1174568511yescharacteristic
296C2×Q8—21no—
297C2×Q8—21no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×D4—1181no—
301C2×D4—1181no—
302C2×D4—4722no—
303C2×D4—2361no—
304C2×D4—2361no—
305C2×D4—2361no—
306C2×D4—2361no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—1182no—
312C22×C4—1181no—
313C22×C4C2×C4×D5911yescharacteristic
314C22×C4D4×D5911yescharacteristic
315C22×C4D4×D5911yescharacteristic
316C22×C4D4×D5911yescharacteristic
317C22×C4#1174568711yescharacteristic
318C22×C4C2×C22⋊D5911yescharacteristic
319SD16—2362no—
320SD16—2362no—
321M16—41no—
322M16—41no—
323M16—2361no—
324M16—2361no—
325C2×C8—22no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—591no—
330C2×C8—591no—
331C2×C8—591no—
332C2×C8—591no—
333C2×C8—1182no—
334C2×C8—1182no—
335C2×C8—1182no—
336C2×C8—2361no—
337C2×C8—2361no—
338C2×C8—2361no—
339C2×C8—2361no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—1181no—
345C4⋊C4—1181no—
346C22⋊C4—41no—
347C22⋊C4—41no—
348C22⋊C4—1184no—
349C22⋊C4—2361no—
350C22⋊C4—2361no—
351C42—2361no—
352C42—2361no—
353C23—22no—
354C23—22no—
355C23—41no—
356C23—4721no—
357C23—4721no—
358C23—2362no—
359C23—2361no—
360C23C4×C22⋊D5911yescharacteristic
361C23#1427370011yescharacteristic
362C23#1427371311yescharacteristic
363Q8—41no—
364Q8—41no—
365Q8—81no—
366D4—4721no—
367D4—4722no—
368D4—4722no—
369D4—2361no—
370D4—2361no—
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—81no—
384C2×C4—1184no—
385C2×C4—4721no—
386C2×C4—2361no—
387C2×C4—2362no—
388C2×C4—2362no—
389C2×C4C22⋊C4×D5911yescharacteristic
390C2×C4(C22⋊D59)⋊C411yescharacteristic
391C2×C4#1427369911yescharacteristic
392C2×C4#1427372311yescharacteristic
393C8—42no—
394C8—81no—
395C8—592no—
396C8—1181no—
397C8—1181no—
398C8—1181no—
399C8—4721no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—44no—
408C22—9442no—
409C22—4721no—
410C22—4721no—
411C22—4721no—
412C22#1683730211yescharacteristic
413C22#1683751712yes—
414C22#1683766812yes—
415C4—21no—
416C4—21no—
417C4—21no—
418C4—81no—
419C4—81no—
420C4—4721no—
421C4—2364no—
422C4#99×D5911yescharacteristic
423C4#1683737411yescharacteristic
424C2—22no—
425C2—22no—
426C2—41no—
427C2—9441no—
428C2#2142493011yescharacteristic
429C2#2142498311yescharacteristic
430C2#2142500011yescharacteristic
subgroupG/H
centerC22#16837302
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×#1472C2
socleC22×C59#390
prime radicalC2×#14273701C22
cosocleC23×C4×C59C23
hypercenter#1472D59
hypocenterC59#9813
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1472

Complements

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

NH ≅ G/Ntype
2. #21424132C2semidirect
3. #21423870C2semidirect
4. #21422616C2semidirect
7. C59×#1472C2semidirect
8. C22×#11745671C4semidirect
16. C2×#14273658C4semidirect
17. C2×#14273651C4semidirect
18. C2×#14273647C4semidirect
214. #1472D59semidirect
246. C59#9813semidirect

Maximal subgroups

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

subgroupindex
C2×#168376332
#214241322
#214238702
#214226162
#214226132
#214224552
C59×#14722
#981359