Finite Group Databasebeta

← #28562863

Subgroups — #28562863

Order 15104.

1256 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 1256 conjugacy classes of subgroups — too many to draw. Only its 183 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#21434893C212yesmaximal, complemented
2#21429576C211yesmaximal, characteristic, complemented
3#21423133C214yesmaximal, complemented
4#21423031C214yesmaximal, complemented
5#21422612C212yesmaximal, complemented
6#21422426C211yesmaximal, characteristic, complemented
7C2×C59×#537C211yesmaximal, characteristic, complemented
8C2×#14273701C2211yescharacteristic
9#16837720—22no—
10#16837718C2214yescomplemented
11#16837653—24no—
12#16837536—22no—
13#16837535C2214yescomplemented
14#16837366C2212yes—
15#16837341C2212yescomplemented
16#16837319C2212yescomplemented
17#16837292—22no—
18#16837281—21no—
19#16837281—21no—
20C4×C22⋊Dic59C2211yescharacteristic
21#16837136C2218yescomplemented
22#16837128C2214yescomplemented
23C22×#41×C59—21no—
24C59×#537—24no—
25C59×#537—22no—
26C59×#537C2214yes—
27C2×C22⋊Q8×C59—22no—
28C2×#121×C59—22no—
29C2×#121×C59—21no—
30C2×#121×C59C2212yes—
31C22×C4⋊C4×C59C2211yescharacteristic
32C22×D236C2311yescharacteristic
33C22×C22⋊D59—21no—
34C2×#11745669—24no—
35C2×#11745664—22no—
36#14273701—22no—
37#14273701—22no—
38C2×C4⋊Dic59C2311yescharacteristic
39C2×C22⋊Dic59—21no—
40C2×(C2×D59)⋊C4—21no—
41C2×(C2×D59)⋊C4—22no—
42C2×(C2×D59)⋊C4—22no—
43C2×C4×Dic59—21no—
44#14273658—24no—
45#14273648—24no—
46#14273647—24no—
47C22⋊(C59⋊C8)—22no—
48C22⋊(C59⋊C8)C2314yes—
49C59⋊#95—22no—
50C59⋊#95—22no—
51C59⋊#95C2312yes—
52C2×#41×C59—24no—
53C2×#41×C59—24no—
54C2×#41×C59—48no—
55C22×Q8×C59—21no—
56C22×D4×C59—21no—
57C22×D4×C59—22no—
58C22×D4×C59—21no—
59C22×D4×C59D412yes—
60C23×C4×C59—22no—
61C23×C4×C59D411yescharacteristic
62C23×C4×C59D411yescharacteristic
63C23×C4×C59C2311yescharacteristic
64C22⋊Q8×C59—28no—
65C22⋊Q8×C59—28no—
66#121×C59—24no—
67#121×C59—28no—
68#121×C59—28no—
69#121×C59—24no—
70#121×C59—22no—
71#121×C59—22no—
72#121×C59D414yes—
73#121×C59C2314yes—
74C2×C4⋊C4×C59—21no—
75C2×C4⋊C4×C59—22no—
76C2×C4⋊C4×C59—22no—
77C2×C4⋊C4×C59D412yes—
78C2×C4⋊C4×C59D412yes—
79C2×C4⋊C4×C59C2312yes—
80C2×C22⋊C4×C59—22no—
81C2×C22⋊C4×C59—22no—
82C24×C59—22no—
83C24×C59C2×D411yescharacteristic
84C23×D59—21no—
85#41×C59—416no—
86#41×C59—48no—
87#41×C59—48no—
88C2×Q8×C59—24no—
89C2×Q8×C59—44no—
90C2×D4×C59—24no—
91C2×D4×C59—24no—
92C2×D4×C59—24no—
93C2×D4×C59—24no—
94C2×D4×C59—22no—
95C2×D4×C59—44no—
96C2×D4×C59—48no—
97C2×D4×C59—82no—
98C2×D4×C59—82no—
99C2×D4×C59C2×D414yes—
100C22×C4×C59—21no—
101C22×C4×C59—21no—
102C22×C4×C59—24no—
103C22×C4×C59—28no—
104C22×C4×C59—21no—
105C22×C4×C59—22no—
106C22×C4×C59—21no—
107C22×C4×C59—48no—
108C22×C4×C59—81no—
109C22×C4×C59—81no—
110C22×C4×C59C2×D412yes—
111C22×C4×C59C2×D412yes—
112C22×C4×C59C2×D411yescharacteristic
113C22×C4×C59C2×D412yes—
114C22×C4×C59C2×D412yes—
115C22×C4×C59C2411yescharacteristic
116C2×C22⋊D59—44no—
117C2×C22⋊D59—41no—
118C2×C22⋊D59—41no—
119C22×Dic59—21no—
120C22×Dic59—22no—
121C2×D236—22no—
122C2×D236—81no—
123C4⋊C4×C59—24no—
124C4⋊C4×C59—24no—
125C4⋊C4×C59—22no—
126C4⋊C4×C59—24no—
127C4⋊C4×C59C2×D414yes—
128C22⋊C4×C59—28no—
129C22⋊C4×C59—24no—
130C22⋊C4×C59—24no—
131C22⋊Dic59—42no—
132#11745669—84no—
133(C2×D59)⋊C4—42no—
134(C2×D59)⋊C4—44no—
135(C2×D59)⋊C4—82no—
136C4⋊Dic59—22no—
137C4×Dic59—81no—
138#11745664—82no—
139C2×C59⋊C8—24no—
140C23×C59—22no—
141C23×C59—22no—
142C23×C59—24no—
143C23×C59—41no—
144C23×C59—44no—
145C23×C59—44no—
146C23×C59—44no—
147C23×C59#12011yescharacteristic
148C23×C59#12011yescharacteristic
149C23×C59C22×D411yescharacteristic
150C22×D59—42no—
151C22×D59—42no—
152C22×D59—41no—
153C22×D59—42no—
154Q8×C59—48no—
155D4×C59—44no—
156D4×C59—48no—
157D4×C59—48no—
158D4×C59—48no—
159D4×C59—82no—
160D4×C59—84no—
161C2×C4×C59—24no—
162C2×C4×C59—24no—
163C2×C4×C59—22no—
164C2×C4×C59—22no—
165C2×C4×C59—22no—
166C2×C4×C59—24no—
167C2×C4×C59—42no—
168C2×C4×C59—41no—
169C2×C4×C59—41no—
170C2×C4×C59—42no—
171C2×C4×C59—416no—
172C2×C4×C59—44no—
173C2×C4×C59—42no—
174C2×C4×C59—42no—
175C2×C4×C59—81no—
176C2×C4×C59#12012yes—
177C2×C4×C59AGL(1,ℤ/8)14yes—
178C2×C4×C59C22×D412yes—
179C22⋊D59—82no—
180C22⋊D59—82no—
181C2×Dic59—42no—
182C2×Dic59—42no—
183C2×Dic59—42no—
184C2×Dic59—41no—
185C2×Dic59—42no—
186D236—44no—
187C59⋊C8—44no—
188#18324—591nomaximal, Sylow
189C22×C59—21no—
190C22×C59—21no—
191C22×C59—21no—
192C22×C59—21no—
193C22×C59—21no—
194C22×C59—21no—
195C22×C59—21no—
196C22×C59—44no—
197C22×C59—44no—
198C22×C59—44no—
199C22×C59—44no—
200C22×C59—88no—
201C22×C59#45412yes—
202C22×C59C2×#12011yescharacteristic
203C22×C59C2×AGL(1,ℤ/8)12yes—
204C2×D59—84no—
205C2×D59—81no—
206C2×D59—81no—
207C2×D59—81no—
208C4×C59—24no—
209C4×C59—22no—
210C4×C59—22no—
211C4×C59—44no—
212C4×C59—42no—
213Dic59—82no—
214Dic59—81no—
215C2×#537D5911yescharacteristic, complemented
216#3192—592no—
217#2591—591no—
218#1616—594no—
219#1598—594no—
220#1507—592no—
221#1463—591no—
222C2×C59—22no—
223C2×C59—22no—
224C2×C59—41no—
225C2×C59—84no—
226C2×C59#300311yescharacteristic
227C2×C59C2×#45411yescharacteristic
228C2×C59#302611yescharacteristic
229D59—161no—
230C22×#41—21no—
231#537—24no—
232#537—22no—
233#537C2×D5914yes—
234C2×C22⋊Q8—22no—
235C2×#121—22no—
236C2×#121—21no—
237C2×#121—591no—
238C2×#121C2×D5912yes—
239C22×C4⋊C4C2×D5911yescharacteristic
240#481—594no—
241#481—1182no—
242#469—1184no—
243#447—594no—
244#447—1182no—
245#407—592no—
246#400—592no—
247#394—592no—
248#388—1182no—
249#386—1181no—
250#386—1181no—
251C4×C22⋊C4—591no—
252#327—598no—
253#324—594no—
254C59#1832411yescharacteristic, complemented, Sylow
255C2×#41—24no—
256C2×#41—24no—
257C2×#41—48no—
258C22×Q8—21no—
259C22×D4—21no—
260C22×D4—22no—
261C22×D4—21no—
262C22×D4—591no—
263C22×D4—1181no—
264C22×D4C22⋊D5912yes—
265C23×C4—22no—
266C23×C4C22⋊D5911yescharacteristic
267C23×C4C22⋊D5911yescharacteristic
268C23×C4C22×D5911yescharacteristic
269C2×D8—1184no—
270C2×M16—1182no—
271C22⋊Q8—28no—
272C22⋊Q8—28no—
273#121—24no—
274#121—28no—
275#121—28no—
276#121—24no—
277#121—22no—
278#121—22no—
279#121—1182no—
280#121—1182no—
281#121C22⋊D5914yes—
282#121C22×D5914yes—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—591no—
287C2×C4⋊C4C22⋊D5912yes—
288C2×C4⋊C4C22⋊D5912yes—
289C2×C4⋊C4C22×D5912yes—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—1182no—
293C2×C22⋊C4—1181no—
294C2×C22⋊C4—1181no—
295C2×C22⋊C4—1182no—
296C2×C42—1181no—
297#106—1184no—
298#102—1184no—
299#102—1184no—
300C22⋊C8—594no—
301C22⋊C8—1182no—
302#95—592no—
303#95—1182no—
304#95—1182no—
305C24—22no—
306C24—1181no—
307C24C2×C22⋊D5911yescharacteristic
308#41—416no—
309#41—48no—
310#41—48no—
311C2×Q8—24no—
312C2×Q8—44no—
313C2×D4—24no—
314C2×D4—24no—
315C2×D4—24no—
316C2×D4—24no—
317C2×D4—22no—
318C2×D4—44no—
319C2×D4—48no—
320C2×D4—82no—
321C2×D4—82no—
322C2×D4—1182no—
323C2×D4—4721no—
324C2×D4—2364no—
325C2×D4—2361no—
326C2×D4—2361no—
327C2×D4C2×C22⋊D5914yes—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—24no—
331C22×C4—28no—
332C22×C4—21no—
333C22×C4—22no—
334C22×C4—21no—
335C22×C4—48no—
336C22×C4—81no—
337C22×C4—81no—
338C22×C4—1182no—
339C22×C4—1181no—
340C22×C4C2×C22⋊D5912yes—
341C22×C4C2×C22⋊D5912yes—
342C22×C4C2×C22⋊D5911yescharacteristic
343C22×C4C2×C22⋊D5912yes—
344C22×C4C2×C22⋊D5912yes—
345C22×C4C23×D5911yescharacteristic
346D8—4724no—
347M16—4722no—
348C2×C8—1184no—
349C4⋊C4—24no—
350C4⋊C4—24no—
351C4⋊C4—22no—
352C4⋊C4—24no—
353C4⋊C4—1182no—
354C4⋊C4C2×C22⋊D5914yes—
355C22⋊C4—28no—
356C22⋊C4—24no—
357C22⋊C4—24no—
358C22⋊C4—4722no—
359C22⋊C4—2362no—
360C22⋊C4—2362no—
361C22⋊C4—2364no—
362C42—4721no—
363C23—22no—
364C23—22no—
365C23—24no—
366C23—41no—
367C23—44no—
368C23—44no—
369C23—44no—
370C23—2362no—
371C23—2362no—
372C23—2361no—
373C23—2362no—
374C23#1427369611yescharacteristic
375C23#1427369611yescharacteristic
376C23C22×C22⋊D5911yescharacteristic
377Q8—48no—
378D4—44no—
379D4—48no—
380D4—48no—
381D4—48no—
382D4—82no—
383D4—84no—
384D4—4722no—
385D4—4722no—
386D4—2364no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—24no—
393C2×C4—41no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—41no—
397C2×C4—416no—
398C2×C4—44no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—81no—
402C2×C4—2362no—
403C2×C4—2362no—
404C2×C4—2362no—
405C2×C4—2361no—
406C2×C4—2362no—
407C2×C4#1427369612yes—
408C2×C4#1427375314yes—
409C2×C4C22×C22⋊D5912yes—
410C8—2364no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—44no—
419C22—44no—
420C22—44no—
421C22—44no—
422C22—88no—
423C22—4724no—
424C22—4721no—
425C22—4721no—
426C22—4721no—
427C22#1683757512yes—
428C22C2×#1427369611yescharacteristic
429C22C2×#1427375312yes—
430C4—22no—
431C4—22no—
432C4—24no—
433C4—42no—
434C4—44no—
435C4—4722no—
436C4—4721no—
437C2—22no—
438C2—22no—
439C2—41no—
440C2—84no—
441C2—9441no—
442C2#2143332311yescharacteristic
443C2C2×#1683757511yescharacteristic
444C2#2143349311yescharacteristic
subgroupG/H
centerC22C2×#14273696
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC2×C59×#537C2
socleC22×C59C2×#120
hypercenterC2×#537D59
hypocenterC59#18324
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#537

Complements

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

NH ≅ G/Ntype
1. #21434893C2semidirect
2. #21429576C2semidirect
3. #21423133C2semidirect
4. #21423031C2semidirect
5. #21422612C2semidirect
6. #21422426C2semidirect
7. C2×C59×#537C2semidirect
10. #16837718C22semidirect
13. #16837535C22semidirect
15. #16837341C22semidirect
16. #16837319C22semidirect
21. #16837136C22semidirect
22. #16837128C22semidirect
215. C2×#537D59semidirect
254. C59#18324semidirect

Maximal subgroups

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

subgroupindex
#214348932
#214295762
#214231332
#214230312
#214226122
#214224262
C2×C59×#5372
#1832459