Finite Group Databasebeta

← #28484665

Subgroups — #28484665

Order 15104.

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

normal subgroups only

This group has 580 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
1#21433137C211yesmaximal, characteristic, complemented
2#21429980C211yesmaximal, characteristic
3#21422611C212yesmaximal, complemented
4#21421949C212yesmaximal, complemented
5C4×#99×C59C211yesmaximal, characteristic, complemented
6C2×#14273701C2211yescharacteristic
7#16837480—24no—
8#16837455—21no—
9#16837366—21no—
10#16837366C2212yes—
11#16837341C411yescharacteristic, complemented
12#16837338C411yescharacteristic, complemented
13#16837319C411yescharacteristic, complemented
14#16837317—21no—
15#16837311C411yescharacteristic, complemented
16#16837310—21no—
17#16837293—21no—
18#16837281—21no—
19#16837265C2211yescharacteristic
20#16837135—24no—
21C2×C4×D4×C59C2211yescharacteristic
22C2×#99×C59—21no—
23C4×C22⋊C4×C59C2212yes—
24C59×#342—21no—
25C22×D236C2×C411yescharacteristic
26C22×C22⋊D59—21no—
27C22×C22⋊D59—21no—
28C2×#11745664—22no—
29C2×#11745664—22no—
30#14273701—22no—
31#14273701—22no—
32C2×C4⋊Dic59C2×C411yescharacteristic
33C2×Dic59⋊C4—21no—
34C2×C22⋊Dic59—21no—
35C2×C22⋊Dic59—21no—
36C2×(C2×D59)⋊C4—21no—
37C2×(C2×D59)⋊C4—21no—
38C2×(C2×D59)⋊C4—21no—
39C2×(C2×D59)⋊C4—21no—
40C4⋊(C59⋊C8)—22no—
41C22⋊(C59⋊C8)—22no—
42C22⋊(C59⋊C8)—22no—
43C22⋊(C59⋊C8)D414yescomplemented
44C4×C59⋊C8—22no—
45C59⋊#95—21no—
46C59⋊#95—21no—
47C59⋊#95—21no—
48C59⋊#95C2×C411yescharacteristic
49C59⋊#95C2×C411yescharacteristic
50C22×D4×C59D411yescharacteristic
51C23×C4×C59D411yescharacteristic
52C23×C4×C59C2311yescharacteristic
53C4×D4×C59—22no—
54C4×D4×C59—22no—
55C2×C4⋊C4×C59C2×C411yescharacteristic
56C2×C22⋊C4×C59—22no—
57C2×C22⋊C4×C59C2×C411yescharacteristic
58C2×C22⋊C4×C59D411yescharacteristic
59C2×C42×C59—22no—
60C2×C42×C59D411yescharacteristic
61#99×C59—24no—
62#95×C59—22no—
63C24×C59C22⋊C411yescharacteristic
64C24×C59C2×D411yescharacteristic
65C23×D59—21no—
66C2×D4×C59—22no—
67C2×D4×C59—22no—
68C2×D4×C59—41no—
69C22×C4×C59—21no—
70C22×C4×C59—21no—
71C22×C4×C59—42no—
72C22×C4×C59—81no—
73C22×C4×C59—81no—
74C22×C4×C59C22⋊C411yescharacteristic
75C22×C4×C59C22⋊C411yescharacteristic
76C22×C4×C59C22⋊C411yescharacteristic
77C22×C4×C59C22×C411yescharacteristic
78C22×C4×C59C2×D412yes—
79C22×C4×C59C2×D411yescharacteristic
80C22×C4×C59#4112yes—
81C2×C22⋊D59—42no—
82C2×C22⋊D59—42no—
83C2×C22⋊D59—81no—
84C2×C22⋊D59—81no—
85C2×C22⋊D59—81no—
86C2×C22⋊D59—81no—
87C22×Dic59—21no—
88C22×Dic59—21no—
89C22×Dic59—21no—
90C2×D236—22no—
91C2×D236—81no—
92C4⋊C4×C59—22no—
93C22⋊C4×C59—22no—
94C22⋊C4×C59—22no—
95C22⋊C4×C59—24no—
96C22⋊C4×C59—42no—
97C42×C59—21no—
98C42×C59—21no—
99C42×C59—82no—
100C22⋊Dic59—81no—
101C22⋊Dic59—81no—
102(C2×D59)⋊C4—42no—
103(C2×D59)⋊C4—41no—
104(C2×D59)⋊C4—41no—
105(C2×D59)⋊C4—81no—
106(C2×D59)⋊C4—81no—
107C4⋊Dic59—22no—
108Dic59⋊C4—81no—
109#11745664—82no—
110#11745664—82no—
111C2×C59⋊C8—24no—
112C23×C59—21no—
113C23×C59—21no—
114C23×C59—21no—
115C23×C59—21no—
116C23×C59—21no—
117C23×C59—41no—
118C23×C59—41no—
119C23×C59—41no—
120C23×C59—41no—
121C23×C59C2×C22⋊C411yescharacteristic
122C23×C59#12112yes—
123C22×D59—41no—
124C22×D59—41no—
125C22×D59—41no—
126C22×D59—41no—
127C22×D59—41no—
128C22×D59—41no—
129C22×D59—41no—
130D4×C59—44no—
131C2×C4×C59—21no—
132C2×C4×C59—21no—
133C2×C4×C59—21no—
134C2×C4×C59—21no—
135C2×C4×C59—42no—
136C2×C4×C59—42no—
137C2×C4×C59—41no—
138C2×C4×C59—41no—
139C2×C4×C59—42no—
140C2×C4×C59—41no—
141C2×C4×C59—41no—
142C2×C4×C59—44no—
143C2×C4×C59—81no—
144C2×C4×C59—82no—
145C2×C4×C59—82no—
146C2×C4×C59#10412yes—
147C2×C4×C59C4×D412yes—
148C2×C4×C59#12011yescharacteristic
149C2×C4×C59#12311yescharacteristic
150C22⋊D59—84no—
151C22⋊D59—84no—
152C2×Dic59—41no—
153C2×Dic59—41no—
154C2×Dic59—41no—
155C2×Dic59—41no—
156C2×Dic59—41no—
157C2×Dic59—41no—
158C2×Dic59—41no—
159C2×Dic59—41no—
160C2×Dic59—41no—
161D236—82no—
162C59⋊C8—44no—
163#9186—591nomaximal, Sylow
164C22×C59—21no—
165C22×C59—21no—
166C22×C59—22no—
167C22×C59—22no—
168C22×C59—22no—
169C22×C59—42no—
170C22×C59—41no—
171C22×C59—41no—
172C22×C59—41no—
173C22×C59—81no—
174C22×C59—81no—
175C22×C59—81no—
176C22×C59#38611yescharacteristic
177C22×C59C2×#10411yescharacteristic
178C22×C59#42111yescharacteristic
179C2×D59—81no—
180C2×D59—81no—
181C2×D59—81no—
182C2×D59—81no—
183C2×D59—81no—
184C2×D59—81no—
185C2×D59—81no—
186C4×C59—22no—
187C4×C59—21no—
188C4×C59—22no—
189C4×C59—21no—
190C4×C59—41no—
191C4×C59—41no—
192C4×C59—84no—
193Dic59—81no—
194Dic59—81no—
195Dic59—81no—
196#2975—591no—
197#2635—591no—
198C4×#99D5911yescharacteristic, complemented
199#1507—592no—
200#1284—592no—
201C2×C59—21no—
202C2×C59—21no—
203C2×C59—42no—
204C2×C59—81no—
205C2×C59#187111yescharacteristic
206C2×C59#191011yescharacteristic
207C2×C59#191311yescharacteristic
208D59—161no—
209C2×#121—591no—
210C2×C4×D4C2×D5911yescharacteristic
211#436—1184no—
212#431—1181no—
213C2×#99—21no—
214#407—592no—
215#407—1181no—
216#400—591no—
217#399—591no—
218#394—591no—
219#394—1181no—
220#392—591no—
221#392—1181no—
222#388—1181no—
223#386—1181no—
224#381—591no—
225C4×C22⋊C4C2×D5912yes—
226#342—21no—
227#327—1184no—
228C59#918611yescharacteristic, complemented, Sylow
229C22×D4—591no—
230C22×D4—1181no—
231C22×D4—1181no—
232C22×D4C22⋊D5911yescharacteristic
233C23×C4D23611yescharacteristic
234C23×C4C22×D5911yescharacteristic
235C2×M16—1182no—
236C2×M16—1182no—
237#121—1182no—
238#121—1182no—
239C4×D4—22no—
240C4×D4—22no—
241C2×C4⋊C4—591no—
242C2×C4⋊C4—1181no—
243C2×C4⋊C4C4×D5911yescharacteristic
244C2×C22⋊C4—22no—
245C2×C22⋊C4—1181no—
246C2×C22⋊C4—1181no—
247C2×C22⋊C4—1181no—
248C2×C22⋊C4—1181no—
249C2×C22⋊C4—1181no—
250C2×C22⋊C4—1181no—
251C2×C22⋊C4C4×D5911yescharacteristic
252C2×C22⋊C4C22⋊D5911yescharacteristic
253C2×C42—22no—
254C2×C42D23611yescharacteristic
255C4⋊C8—1182no—
256#99—24no—
257C22⋊C8—594no—
258C22⋊C8—1182no—
259C22⋊C8—1182no—
260C4×C8—1182no—
261#95—22no—
262#95—591no—
263#95—591no—
264#95—1181no—
265#95—1181no—
266#95—1181no—
267C24—1181no—
268C24(C2×D59)⋊C411yescharacteristic
269C24C2×C22⋊D5911yescharacteristic
270C2×D4—22no—
271C2×D4—22no—
272C2×D4—41no—
273C2×D4—1182no—
274C2×D4—4721no—
275C2×D4—4721no—
276C2×D4—4721no—
277C2×D4—4721no—
278C2×D4—4721no—
279C2×D4—2362no—
280C2×D4—2362no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—42no—
284C22×C4—81no—
285C22×C4—81no—
286C22×C4—1181no—
287C22×C4—1181no—
288C22×C4—1181no—
289C22×C4(C2×D59)⋊C411yescharacteristic
290C22×C4(C2×D59)⋊C411yescharacteristic
291C22×C4(C2×D59)⋊C411yescharacteristic
292C22×C4C2×C4×D5911yescharacteristic
293C22×C4C2×D23611yescharacteristic
294C22×C4D4×D5912yes—
295C22×C4#1174568712yes—
296M16—4722no—
297M16—4722no—
298C2×C8—1184no—
299C4⋊C4—22no—
300C4⋊C4—1182no—
301C4⋊C4—4721no—
302C22⋊C4—22no—
303C22⋊C4—22no—
304C22⋊C4—24no—
305C22⋊C4—42no—
306C22⋊C4—4721no—
307C22⋊C4—4721no—
308C22⋊C4—4721no—
309C22⋊C4—4721no—
310C22⋊C4—2362no—
311C22⋊C4—2361no—
312C22⋊C4—2361no—
313C42—21no—
314C42—21no—
315C42—82no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—41no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—2361no—
326C23—2361no—
327C23—2361no—
328C23—2361no—
329C23—2361no—
330C23—2361no—
331C23—2361no—
332C23C2×(C2×D59)⋊C411yescharacteristic
333C23#1427369912yes—
334D4—44no—
335D4—4722no—
336D4—4724no—
337D4—4724no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—44no—
350C2×C4—81no—
351C2×C4—82no—
352C2×C4—82no—
353C2×C4—2361no—
354C2×C4—2361no—
355C2×C4—2361no—
356C2×C4—2361no—
357C2×C4—2361no—
358C2×C4—2361no—
359C2×C4—2361no—
360C2×C4—2361no—
361C2×C4—2361no—
362C2×C4#1427365512yes—
363C2×C4(C22⋊D59)⋊C412yes—
364C2×C4#1427369711yescharacteristic
365C2×C4#1427371111yescharacteristic
366C8—2364no—
367C22—21no—
368C22—21no—
369C22—22no—
370C22—22no—
371C22—22no—
372C22—42no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—81no—
377C22—81no—
378C22—81no—
379C22—4721no—
380C22—4721no—
381C22—4721no—
382C22—4721no—
383C22—4721no—
384C22—4721no—
385C22—4721no—
386C22#1683727911yescharacteristic
387C22C2×#1427365511yescharacteristic
388C22#1683742011yescharacteristic
389C4—22no—
390C4—21no—
391C4—21no—
392C4—22no—
393C4—41no—
394C4—41no—
395C4—84no—
396C4—4721no—
397C4—4721no—
398C4—4721no—
399C2—21no—
400C2—21no—
401C2—42no—
402C2—81no—
403C2—9441no—
404C2#2142439611yescharacteristic
405C2#2142462111yescharacteristic
406C2#2142463611yescharacteristic
subgroupG/H
centerC22#16837279
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC4×#99×C59C2
socleC22×C59#386
prime radical#21429980C2
cosocleC23×C4×C59C23
hypercenterC4×#99D59
hypocenterC59#9186
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ C4×#99

Complements

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

NH ≅ G/Ntype
1. #21433137C2semidirect
3. #21422611C2semidirect
4. #21421949C2semidirect
5. C4×#99×C59C2semidirect
11. #16837341C4semidirect
12. #16837338C4semidirect
13. #16837319C4semidirect
15. #16837311C4semidirect
43. C22⋊(C59⋊C8)D4semidirect
198. C4×#99D59semidirect
228. C59#9186semidirect

Maximal subgroups

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

subgroupindex
#214331372
#214299802
#214226112
#214219492
C4×#99×C592
#918659