Finite Group Databasebeta

← #28485151

Subgroups — #28485151

Order 15104.

566 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 111 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×#16837721C211yesmaximal, characteristic
2#21426975C211yesmaximal, characteristic, complemented
3#21424931C211yesmaximal, characteristic
4#21424184C211yesmaximal, characteristic, complemented
5#21422663C211yesmaximal, characteristic
6#21421943C211yesmaximal, characteristic, complemented
7C59×#1289C211yesmaximal, characteristic, complemented
8C2×#14273701C2211yescharacteristic
9C22×C4⋊Dic59C2211yescharacteristic
10#16837721—22no—
11#16837721—21no—
12#16837721—21no—
13C2×#14273659C412yescomplemented
14C2×#14273649C412yescomplemented
15C2×#14273641—21no—
16C2×#14273641—21no—
17C2×C22⋊(C59⋊C8)C2211yescharacteristic
18#16837298—21no—
19#16837287—22no—
20#16837260—21no—
21C4⋊C4×Dic59—21no—
22C22⋊C4×Dic59—21no—
23C4×C22⋊Dic59C2211yescharacteristic
24#16837164—22no—
25#16837135—22no—
26C2×C22⋊C8×C59C2211yescharacteristic
27C59×#380C2211yescharacteristic
28C59×#380C2211yescharacteristic
29C22×D236C2×C411yescharacteristic
30C22×C22⋊D59—21no—
31C23×Dic59—21no—
32C22×C59⋊C8—21no—
33#14273701—21no—
34#14273701—21no—
35#14273701—21no—
36#14273701—21no—
37C2×C4⋊Dic59—21no—
38C2×C4⋊Dic59—21no—
39C2×C4⋊Dic59—21no—
40C2×C4⋊Dic59C2×C411yescharacteristic
41C2×C4⋊Dic59C2×C412yes—
42C2×C4⋊Dic59D412yes—
43C2×C4⋊Dic59D411yescharacteristic
44C2×C4⋊Dic59D411yescharacteristic
45C2×Dic59⋊C4—22no—
46C2×C22⋊Dic59D411yescharacteristic
47C2×C22⋊Dic59D411yescharacteristic
48C2×(C2×D59)⋊C4—21no—
49C2×C4×Dic59—21no—
50C2×C4×Dic59—22no—
51#14273659—22no—
52#14273659—22no—
53#14273649—22no—
54#14273649—22no—
55#14273641—42no—
56#14273641—42no—
57C22⋊(C59⋊C8)—21no—
58C22⋊(C59⋊C8)—21no—
59C59⋊#95—22no—
60C59⋊#95—22no—
61C59⋊#95—21no—
62C23×C4×C59C2311yescharacteristic
63C22×C8×C59C2×C412yes—
64C2×C4⋊C4×C59—21no—
65C2×C4⋊C4×C59—21no—
66C2×C22⋊C4×C59—21no—
67C2×C22⋊C4×C59—21no—
68C22⋊C8×C59—21no—
69C22⋊C8×C59—21no—
70#95×C59—21no—
71#95×C59D412yes—
72C24×C59C2×D411yescharacteristic
73C23×D59—21no—
74C22×C4×C59—22no—
75C22×C4×C59—42no—
76C22×C4×C59—41no—
77C22×C4×C59C22×C411yescharacteristic
78C22×C4×C59C2×D411yescharacteristic
79C22×C4×C59C2×D411yescharacteristic
80C22×C4×C59C2×D411yescharacteristic
81C22×C4×C59#4111yescharacteristic
82C22×C4×C59#4111yescharacteristic
83C2×C22⋊D59—41no—
84C2×C22⋊D59—41no—
85C2×C22⋊D59—41no—
86C2×C22⋊D59—41no—
87C2×C22⋊D59—41no—
88C2×C22⋊D59—41no—
89C22×Dic59—21no—
90C22×Dic59—21no—
91C22×Dic59—21no—
92C22×Dic59—22no—
93C22×Dic59—22no—
94C22×Dic59—21no—
95C22×Dic59—21no—
96C22×Dic59—82no—
97C2×D236—21no—
98C2×D236—21no—
99C2×D236—42no—
100C2×C8×C59—22no—
101C2×C8×C59—42no—
102C4⋊C4×C59—42no—
103C4⋊C4×C59—42no—
104C22⋊C4×C59—41no—
105C22⋊C4×C59—41no—
106C22⋊C4×C59—41no—
107C22⋊C4×C59—41no—
108C22⋊Dic59—21no—
109C22⋊Dic59—21no—
110C22⋊Dic59—21no—
111C22⋊Dic59SD1612yes—
112(C2×D59)⋊C4—41no—
113(C2×D59)⋊C4—41no—
114C4⋊Dic59—22no—
115C4⋊Dic59—22no—
116C4⋊Dic59—21no—
117C4⋊Dic59—21no—
118C4⋊Dic59—21no—
119C4⋊Dic59—21no—
120C4⋊Dic59C22⋊C412yes—
121C4⋊Dic59C22⋊C412yes—
122Dic59⋊C4—82no—
123C4×Dic59—42no—
124C4×Dic59—82no—
125C2×C59⋊C8—42no—
126C2×C59⋊C8—41no—
127C23×C59—41no—
128C23×C59—41no—
129C23×C59C4×D411yescharacteristic
130C23×C59#12011yescharacteristic
131C23×C59#12111yescharacteristic
132C23×C59C2×SD1611yescharacteristic
133C23×C59#13511yescharacteristic
134C23×C59AGL(1,ℤ/8)11yescharacteristic
135C23×C59#13711yescharacteristic
136C22×D59—41no—
137C22×D59—41no—
138C22×D59—41no—
139C22×D59—81no—
140C22×D59—81no—
141C2×C4×C59—22no—
142C2×C4×C59—22no—
143C2×C4×C59—22no—
144C2×C4×C59—41no—
145C2×C4×C59—41no—
146C2×C4×C59—41no—
147C2×C4×C59—42no—
148C2×C4×C59—42no—
149C2×C4×C59—42no—
150C2×C4×C59—81no—
151C2×C4×C59—81no—
152C2×C4×C59—81no—
153C2×C4×C59C2×C22⋊C411yescharacteristic
154C2×C4×C59C4×D411yescharacteristic
155C2×C4×C59#12111yescharacteristic
156C2×C4×C59#12311yescharacteristic
157C22⋊D59—81no—
158C22⋊D59—81no—
159C22⋊D59—81no—
160C22⋊D59—81no—
161C2×Dic59—22no—
162C2×Dic59—22no—
163C2×Dic59—22no—
164C2×Dic59—41no—
165C2×Dic59—41no—
166C2×Dic59—42no—
167C2×Dic59—42no—
168C2×Dic59—41no—
169C2×Dic59—41no—
170C2×Dic59—41no—
171C2×Dic59—41no—
172C2×Dic59—41no—
173C2×Dic59—41no—
174C2×Dic59—82no—
175C2×Dic59—82no—
176D236—42no—
177D236—81no—
178C8×C59—42no—
179C59⋊C8—81no—
180#9261—591nomaximal, Sylow
181C22×C59—21no—
182C22×C59—21no—
183C22×C59—21no—
184C22×C59—21no—
185C22×C59—21no—
186C22×C59—21no—
187C22×C59—41no—
188C22×C59—41no—
189C22×C59—41no—
190C22×C59—41no—
191C22×C59#38611yescharacteristic
192C22×C59C4×SD1611yescharacteristic
193C22×C59#44011yescharacteristic
194C22×C59#44811yescharacteristic
195C22×C59#44911yescharacteristic
196C22×C59#46011yescharacteristic
197C22×C59#46411yescharacteristic
198C2×D59—81no—
199C2×D59—81no—
200C2×D59—81no—
201C2×D59—161no—
202C2×D59—161no—
203C4×C59—21no—
204C4×C59—21no—
205C4×C59—21no—
206C4×C59—82no—
207C4×C59—161no—
208C4×C59#41012yes—
209Dic59—42no—
210Dic59—42no—
211Dic59—42no—
212Dic59—81no—
213Dic59—81no—
214C2×#481—591no—
215#2345—591no—
216#1964—591no—
217#1824—591no—
218#1519—591no—
219#1289D5911yescharacteristic, complemented
220#1282—591no—
221C2×C59—21no—
222C2×C59—21no—
223C2×C59—21no—
224C2×C59—21no—
225C2×C59—21no—
226C2×C59—21no—
227C2×C59#188011yescharacteristic
228C2×C59#189911yescharacteristic
229C2×C59#190411yescharacteristic
230D59—161no—
231C2×#121—591no—
232C22×C4⋊C4—591no—
233#481—1182no—
234#481—1181no—
235#481—1181no—
236C2×#106—592no—
237C2×#102—592no—
238C2×#99—1181no—
239C2×#99—1181no—
240C2×C22⋊C8—591no—
241C2×C22⋊C8C2×D5911yescharacteristic
242#389—1181no—
243#387—1182no—
244#380—1181no—
245#380C2×D5911yescharacteristic
246#380C2×D5911yescharacteristic
247C4×C4⋊C4—1181no—
248C4×C22⋊C4—591no—
249C4×C22⋊C4—1181no—
250#340—1182no—
251#327—1182no—
252C59#926111yescharacteristic, complemented, Sylow
253C22×D4—591no—
254C22×D4—1181no—
255C23×C4—1181no—
256C23×C4C22×D5911yescharacteristic
257C22×C8—1181no—
258C22×C8C4×D5912yes—
259#121—1181no—
260#121—1181no—
261#121—1181no—
262#121—1181no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—592no—
266C2×C4⋊C4—592no—
267C2×C4⋊C4—591no—
268C2×C4⋊C4—591no—
269C2×C4⋊C4—591no—
270C2×C4⋊C4—1181no—
271C2×C4⋊C4—1181no—
272C2×C4⋊C4—1181no—
273C2×C4⋊C4—1182no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—591no—
277C2×C22⋊C4—591no—
278C2×C22⋊C4—1181no—
279C2×C42—1182no—
280C2×C42—1181no—
281#106—1182no—
282#106—1182no—
283#102—1182no—
284#102—1182no—
285#99—2362no—
286#99—2362no—
287C22⋊C8—21no—
288C22⋊C8—21no—
289C22⋊C8—1181no—
290C22⋊C8—1181no—
291#95—21no—
292#95—1182no—
293#95—1182no—
294#95—1181no—
295#95D23612yes—
296C24—1181no—
297C24D4×D5911yescharacteristic
298C2×D4—1181no—
299C2×D4—1181no—
300C2×D4—2362no—
301C2×D4—2361no—
302C2×D4—2361no—
303C2×D4—2361no—
304C2×D4—2361no—
305C2×D4—2361no—
306C2×D4—2361no—
307C22×C4—22no—
308C22×C4—42no—
309C22×C4—41no—
310C22×C4—1182no—
311C22×C4—1182no—
312C22×C4—1181no—
313C22×C4—1181no—
314C22×C4—1181no—
315C22×C4—1181no—
316C22×C4—1181no—
317C22×C4—4722no—
318C22×C4C2×C4×D5911yescharacteristic
319C22×C4C2×D23611yescharacteristic
320C22×C4#1174568511yescharacteristic
321C22×C4D4×D5911yescharacteristic
322C22×C4D4×D5911yescharacteristic
323C22×C4#1174568911yescharacteristic
324C2×C8—22no—
325C2×C8—42no—
326C2×C8—2362no—
327C2×C8—2361no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—592no—
331C4⋊C4—592no—
332C4⋊C4—1182no—
333C4⋊C4—1182no—
334C4⋊C4—1181no—
335C4⋊C4—1181no—
336C4⋊C4—1181no—
337C4⋊C4—1181no—
338C4⋊C4—4722no—
339C22⋊C4—41no—
340C22⋊C4—41no—
341C22⋊C4—41no—
342C22⋊C4—41no—
343C22⋊C4—592no—
344C22⋊C4—1181no—
345C22⋊C4—1181no—
346C22⋊C4—1181no—
347C22⋊C4—2361no—
348C22⋊C4—2361no—
349C42—4722no—
350C42—2362no—
351C23—41no—
352C23—41no—
353C23—4721no—
354C23—4721no—
355C23—2361no—
356C23—2361no—
357C23—2361no—
358C23#1427368911yescharacteristic
359C23#1427369711yescharacteristic
360C23#1427370011yescharacteristic
361C23SD16×D5911yescharacteristic
362C23#1427374811yescharacteristic
363C23#1427375111yescharacteristic
364C23#1427375911yescharacteristic
365D4—4721no—
366D4—4721no—
367D4—4721no—
368D4—4721no—
369D4—4721no—
370D4—2362no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—1182no—
384C2×C4—1182no—
385C2×C4—1182no—
386C2×C4—4722no—
387C2×C4—4722no—
388C2×C4—2361no—
389C2×C4—2361no—
390C2×C4—2361no—
391C2×C4—2361no—
392C2×C4—2361no—
393C2×C4—2361no—
394C2×C4—2361no—
395C2×C4—2362no—
396C2×C4—2361no—
397C2×C4—2362no—
398C2×C4C22⋊C4×D5911yescharacteristic
399C2×C4C4×D23611yescharacteristic
400C2×C4#1427370211yescharacteristic
401C2×C4#1427371211yescharacteristic
402C8—42no—
403C8—4721no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—9441no—
415C22—9441no—
416C22—4721no—
417C22—4721no—
418C22—4721no—
419C22#1683728011yescharacteristic
420C22#1683748911yescharacteristic
421C22#1683750111yescharacteristic
422C22#1683754211yescharacteristic
423C22#1683754811yescharacteristic
424C22#1683760111yescharacteristic
425C22#1683762711yescharacteristic
426C4—21no—
427C4—21no—
428C4—21no—
429C4—82no—
430C4—161no—
431C4—4721no—
432C4—4721no—
433C4—2362no—
434C4—2362no—
435C4—2362no—
436C4#1683737412yes—
437C2—21no—
438C2—21no—
439C2—21no—
440C2—21no—
441C2—21no—
442C2—21no—
443C2—9441no—
444C2#2142445211yescharacteristic
445C2#2142455311yescharacteristic
446C2#2142458411yescharacteristic
subgroupG/H
centerC22#16837280
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×#1289C2
socleC22×C59#386
prime radicalC2×#14273701C22
cosocleC23×C4×C59C23
hypercenter#1289D59
hypocenterC59#9261
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1289

Complements

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

NH ≅ G/Ntype
2. #21426975C2semidirect
4. #21424184C2semidirect
6. #21421943C2semidirect
7. C59×#1289C2semidirect
13. C2×#14273659C4semidirect
14. C2×#14273649C4semidirect
219. #1289D59semidirect
252. C59#9261semidirect

Maximal subgroups

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

subgroupindex
C2×#168377212
#214269752
#214249312
#214241842
#214226632
#214219432
C59×#12892
#926159