Finite Group Databasebeta

← #28485748

Subgroups — #28485748

Order 15104.

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

normal subgroups only

This group has 444 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#21434040C211yesmaximal, characteristic
2#21432756C211yesmaximal, characteristic
3#21424192C211yesmaximal, characteristic
4#21422668C211yesmaximal, characteristic
5#21422003C211yesmaximal, characteristic
6#21421984C211yesmaximal, characteristic
7C59×#1472C211yesmaximal, characteristic
8C2×#14273686C2211yescharacteristic
9#16837740—21no—
10#16837740—21no—
11#16837735—21no—
12#16837730—21no—
13#16837460—22no—
14#16837456—21no—
15#16837452—21no—
16#16837442C411yescharacteristic, complemented
17#16837442C411yescharacteristic, complemented
18#16837430—21no—
19#16837411C411yescharacteristic, complemented
20#16837411C411yescharacteristic, complemented
21#16837366C2211yescharacteristic
22C2×C22⋊(C59⋊C8)C2211yescharacteristic
23#16837360—21no—
24#16837262C2211yescharacteristic
25#16837165—21no—
26#16837165—21no—
27#16837139—21no—
28#16837139—21no—
29C2×C22⋊Q8×C59C2211yescharacteristic
30C59×#407C2211yescharacteristic
31C2×C22⋊C8×C59C2211yescharacteristic
32C59×#324—22no—
33C2×#11745664C2×C411yescharacteristic
34C2×#11745664C2×C411yescharacteristic
35C22×C59⋊C8—21no—
36#14273686—21no—
37#14273686—21no—
38#14273686C2×C411yescharacteristic
39#14273686C2×C411yescharacteristic
40#14273686D411yescharacteristic
41#14273686D411yescharacteristic
42C2×C4⋊Dic59—21no—
43C2×C4⋊Dic59D411yescharacteristic
44C2×C4⋊Dic59D411yescharacteristic
45C2×C22⋊Dic59—21no—
46C2×C22⋊Dic59—21no—
47C2×C4×Dic59—21no—
48#14273660—21no—
49#14273660—21no—
50#14273658—21no—
51#14273658—21no—
52Dic59⋊C8—22no—
53Dic59⋊C8—22no—
54#14273651—21no—
55#14273651—21no—
56#14273651—21no—
57#14273651—21no—
58C22⋊(C59⋊C8)—21no—
59C22⋊(C59⋊C8)—21no—
60C22⋊(C59⋊C8)—21no—
61C22⋊(C59⋊C8)D411yescharacteristic
62C22⋊(C59⋊C8)D411yescharacteristic
63#14273637—22no—
64C8×Dic59—22no—
65C59⋊#95—21no—
66C22×Q8×C59C2×C411yescharacteristic
67C23×C4×C59C2311yescharacteristic
68C2×M16×C59—21no—
69C22×C8×C59—21no—
70C22⋊Q8×C59—21no—
71C22⋊Q8×C59—21no—
72C22⋊Q8×C59—21no—
73C22⋊Q8×C59—21no—
74C2×C4⋊C4×C59—21no—
75C2×C4⋊C4×C59C2×C411yescharacteristic
76C2×C22⋊C4×C59—21no—
77C22⋊C8×C59—21no—
78C22⋊C8×C59—21no—
79C22⋊C8×C59—21no—
80C22⋊C8×C59D412yes—
81C24×C59#4111yescharacteristic
82C2×Q8×C59—21no—
83C2×Q8×C59—21no—
84C2×Q8×C59—41no—
85C2×Q8×C59—41no—
86C22×C4×C59—21no—
87C22×C4×C59—21no—
88C22×C4×C59—41no—
89C22×C4×C59—41no—
90C22×C4×C59C22×C411yescharacteristic
91C22×C4×C59C2×D411yescharacteristic
92C22×C4×C59C2×D411yescharacteristic
93C22×C4×C59C2×D411yescharacteristic
94C22×C4×C59C2×D411yescharacteristic
95C22×C4×C59#4111yescharacteristic
96C22×Dic59—21no—
97C22×Dic59—21no—
98C22×Dic59—41no—
99M16×C59—41no—
100M16×C59—41no—
101C2×C8×C59—22no—
102C2×C8×C59—41no—
103C2×C8×C59—41no—
104C2×C8×C59—41no—
105C4⋊C4×C59—21no—
106C4⋊C4×C59—21no—
107C4⋊C4×C59—41no—
108C4⋊C4×C59—41no—
109C22⋊C4×C59—41no—
110C22⋊C4×C59—41no—
111C22⋊Dic59—21no—
112C22⋊Dic59—21no—
113C22⋊Dic59—21no—
114C22⋊Dic59—21no—
115C22⋊Dic59—81no—
116C4⋊Dic59—21no—
117C4⋊Dic59—21no—
118C4⋊Dic59—41no—
119C4⋊Dic59—41no—
120C4⋊Dic59C22⋊C411yescharacteristic
121C4⋊Dic59C22⋊C411yescharacteristic
122C4⋊Dic59C22⋊C411yescharacteristic
123C4⋊Dic59C22⋊C411yescharacteristic
124C4×Dic59—21no—
125C4×Dic59—21no—
126C4×Dic59—21no—
127C4×Dic59—21no—
128#11745664—41no—
129#11745664—41no—
130C2×C59⋊C8—21no—
131C2×C59⋊C8—21no—
132C2×C59⋊C8—21no—
133C2×C59⋊C8—21no—
134C2×C59⋊C8—41no—
135C2×C59⋊C8—41no—
136C23×C59—21no—
137C23×C59—21no—
138C23×C59—41no—
139C23×C59—41no—
140C23×C59C4×D411yescharacteristic
141C23×C59#12111yescharacteristic
142C23×C59#12311yescharacteristic
143Q8×C59—41no—
144Q8×C59—41no—
145Q8×C59—81no—
146C2×C4×C59—21no—
147C2×C4×C59—21no—
148C2×C4×C59—21no—
149C2×C4×C59—21no—
150C2×C4×C59—21no—
151C2×C4×C59—21no—
152C2×C4×C59—41no—
153C2×C4×C59—41no—
154C2×C4×C59—41no—
155C2×C4×C59—42no—
156C2×C4×C59—41no—
157C2×C4×C59—41no—
158C2×C4×C59—81no—
159C2×C4×C59C2×C22⋊C411yescharacteristic
160C2×C4×C59C4×D411yescharacteristic
161C2×C4×C59#12011yescharacteristic
162C2×C4×C59#12111yescharacteristic
163C2×Dic59—21no—
164C2×Dic59—21no—
165C2×Dic59—21no—
166C2×Dic59—21no—
167C2×Dic59—41no—
168C2×Dic59—41no—
169C2×Dic59—81no—
170C2×Dic59—81no—
171C2×Dic59—81no—
172C2×Dic59—81no—
173C8×C59—42no—
174C8×C59—81no—
175C59⋊C8—41no—
176C59⋊C8—41no—
177C59⋊C8—41no—
178C59⋊C8—41no—
179#9346—591nomaximal, Sylow
180C22×C59—21no—
181C22×C59—21no—
182C22×C59—21no—
183C22×C59—21no—
184C22×C59—21no—
185C22×C59—21no—
186C22×C59—21no—
187C22×C59—21no—
188C22×C59—81no—
189C22×C59—81no—
190C22×C59#38611yescharacteristic
191C22×C59#47111yescharacteristic
192C22×C59#47311yescharacteristic
193C4×C59—21no—
194C4×C59—21no—
195C4×C59—21no—
196C4×C59—81no—
197C4×C59—81no—
198C4×C59#41011yescharacteristic
199C4×C59#41011yescharacteristic
200Dic59—41no—
201Dic59—41no—
202Dic59—41no—
203Dic59—41no—
204Dic59—161no—
205#3095—591no—
206#2924—591no—
207#1825—591no—
208#1520—591no—
209#1472D5911yescharacteristic
210#1305—591no—
211#1298—591no—
212C2×C59—21no—
213C2×C59—21no—
214C2×C59—41no—
215C2×C59—41no—
216C2×C59#188111yescharacteristic
217C2×C59#189911yescharacteristic
218C2×C59#190811yescharacteristic
219C2×C22⋊Q8C2×D5911yescharacteristic
220C2×#117—591no—
221#485—1181no—
222#485—1181no—
223#484—1181no—
224#483—1181no—
225#433—1182no—
226#432—1181no—
227#431—1181no—
228#428—591no—
229#428—591no—
230#424—1181no—
231#419—591no—
232#419—591no—
233#407—591no—
234#407C2×D5911yescharacteristic
235C2×C22⋊C8—591no—
236C2×C22⋊C8C2×D5911yescharacteristic
237#405—1181no—
238#380—591no—
239#340—1181no—
240#340—1181no—
241#328—1181no—
242#328—1181no—
243#324—22no—
244C59#934611yescharacteristic, complemented, Sylow
245C22×Q8C4×D5911yescharacteristic
246C23×C4C22×D5911yescharacteristic
247C2×M16—21no—
248C2×M16—591no—
249C2×M16—591no—
250C22×C8—21no—
251C22×C8—1181no—
252C22⋊Q8—21no—
253C22⋊Q8—21no—
254C22⋊Q8—21no—
255C22⋊Q8—21no—
256#117—591no—
257#117—591no—
258#117—591no—
259#117—591no—
260#117—1181no—
261#117—1181no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—591no—
264C2×C4⋊C4—591no—
265C2×C4⋊C4—1181no—
266C2×C4⋊C4C4×D5911yescharacteristic
267C2×C22⋊C4—21no—
268C2×C22⋊C4—1181no—
269C2×C22⋊C4—1181no—
270C2×C42—1181no—
271#107—1181no—
272#107—1181no—
273#106—1181no—
274#106—1181no—
275C4⋊C8—1182no—
276C4⋊C8—1182no—
277#103—1181no—
278#103—1181no—
279#103—1181no—
280#103—1181no—
281C22⋊C8—21no—
282C22⋊C8—21no—
283C22⋊C8—21no—
284C22⋊C8—591no—
285C22⋊C8—591no—
286C22⋊C8—1181no—
287C22⋊C8—1181no—
288C22⋊C8—1181no—
289C22⋊C8C22⋊D5912yes—
290#97—1182no—
291C4×C8—1182no—
292#95—1181no—
293C24#1174568511yescharacteristic
294C2×Q8—21no—
295C2×Q8—21no—
296C2×Q8—41no—
297C2×Q8—41no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—1181no—
303C22×C4—1181no—
304C22×C4—2361no—
305C22×C4C2×C4×D5911yescharacteristic
306C22×C4D4×D5911yescharacteristic
307C22×C4D4×D5911yescharacteristic
308C22×C4D4×D5911yescharacteristic
309C22×C4#1174568711yescharacteristic
310C22×C4C2×C22⋊D5911yescharacteristic
311M16—41no—
312M16—41no—
313M16—2361no—
314M16—2361no—
315C2×C8—22no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—1181no—
320C2×C8—1181no—
321C2×C8—1181no—
322C2×C8—1181no—
323C2×C8—2361no—
324C2×C8—2361no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—591no—
330C4⋊C4—591no—
331C4⋊C4—591no—
332C4⋊C4—591no—
333C4⋊C4—1181no—
334C4⋊C4—1181no—
335C4⋊C4—2361no—
336C4⋊C4—2361no—
337C22⋊C4—41no—
338C22⋊C4—41no—
339C22⋊C4—1181no—
340C22⋊C4—1181no—
341C22⋊C4—1181no—
342C22⋊C4—1181no—
343C22⋊C4—4721no—
344C42—1181no—
345C42—1181no—
346C42—1181no—
347C42—1181no—
348C23—21no—
349C23—21no—
350C23—41no—
351C23—41no—
352C23C4×C22⋊D5911yescharacteristic
353C23#1427370011yescharacteristic
354C23#1427370911yescharacteristic
355Q8—41no—
356Q8—41no—
357Q8—81no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—81no—
371C2×C4—1181no—
372C2×C4—1181no—
373C2×C4—1181no—
374C2×C4—1181no—
375C2×C4—4721no—
376C2×C4—4721no—
377C2×C4—4721no—
378C2×C4—4721no—
379C2×C4—2361no—
380C2×C4—2361no—
381C2×C4C22⋊C4×D5911yescharacteristic
382C2×C4(C22⋊D59)⋊C411yescharacteristic
383C2×C4#1427369511yescharacteristic
384C2×C4#1427369811yescharacteristic
385C8—42no—
386C8—81no—
387C8—2361no—
388C8—2361no—
389C8—2361no—
390C8—2361no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—81no—
400C22—81no—
401C22#1683727811yescharacteristic
402C22#1683766811yescharacteristic
403C22#1683767611yescharacteristic
404C4—21no—
405C4—21no—
406C4—21no—
407C4—81no—
408C4—81no—
409C4—9441no—
410C4—2361no—
411C4—2361no—
412C4—2361no—
413C4—2361no—
414C4#1683737411yescharacteristic
415C4#1683737411yescharacteristic
416C2—21no—
417C2—21no—
418C2—41no—
419C2—41no—
420C2#2142446111yescharacteristic
421C2#2142455111yescharacteristic
422C2#2142460811yescharacteristic
subgroupG/H
centerC22#16837278
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×#1472C2
socleC22×C59#386
prime radicalC24×C59#41
cosocleC23×C4×C59C23
hypercenter#1472D59
hypocenterC59#9346
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
16. #16837442C4semidirect
17. #16837442C4semidirect
19. #16837411C4semidirect
20. #16837411C4semidirect
244. C59#9346semidirect

Maximal subgroups

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

subgroupindex
#214340402
#214327562
#214241922
#214226682
#214220032
#214219842
C59×#14722
#934659