Finite Group Databasebeta

← #28489141

Subgroups — #28489141

Order 15104.

610 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 105 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#21434853C211yesmaximal, characteristic
2#21428570C211yesmaximal, characteristic, complemented
3#21424933C211yesmaximal, characteristic
4#21424186C211yesmaximal, characteristic, complemented
5#21422669C211yesmaximal, characteristic, complemented
6#21421942C211yesmaximal, characteristic, complemented
7C2×C59×#481C211yesmaximal, characteristic
8C2×#14273706C2211yescharacteristic
9#16837732—21no—
10#16837718—21no—
11#16837629—22no—
12#16837553—21no—
13#16837540—21no—
14C2×#14273642—22no—
15C2×C22⋊(C59⋊C8)C2211yescharacteristic
16#16837333—22no—
17#16837307—21no—
18#16837275—22no—
19#16837271—21no—
20#16837262C2211yescharacteristic
21C4×C22⋊Dic59C2211yescharacteristic
22#16837164—22no—
23#16837140—22no—
24C2×#121×C59C2211yescharacteristic
25C22×C4⋊C4×C59C2211yescharacteristic
26C59×#481—22no—
27C59×#481—22no—
28C2×#106×C59—21no—
29C2×#102×C59—21no—
30C2×C22⋊C8×C59C2211yescharacteristic
31C22×Dic118D411yescharacteristic
32C2×#11745670—22no—
33C22×C59⋊C8D412yes—
34#14273706—21no—
35#14273706—21no—
36#14273706—21no—
37#14273706—21no—
38C2×C4⋊Dic59—21no—
39C2×C4⋊Dic59D411yescharacteristic
40C2×Dic59⋊C4—21no—
41C2×Dic59⋊C4—24no—
42C2×C22⋊Dic59—21no—
43C2×C22⋊Dic59—21no—
44C2×C22⋊Dic59D412yes—
45C2×C4×Dic59—21no—
46#14273658—22no—
47#14273650—22no—
48#14273648—22no—
49#14273642—44no—
50C22⋊(C59⋊C8)—21no—
51C22⋊(C59⋊C8)—21no—
52C59⋊#95—24no—
53C59⋊#95—21no—
54C59⋊#95—21no—
55C22×D4×C59—21no—
56C22×D4×C59D411yescharacteristic
57C23×C4×C59—21no—
58C23×C4×C59C2311yescharacteristic
59C22×C8×C59—21no—
60#121×C59—21no—
61#121×C59—21no—
62#121×C59—21no—
63#121×C59—21no—
64C2×C4⋊C4×C59—21no—
65C2×C4⋊C4×C59—21no—
66C2×C4⋊C4×C59—21no—
67C2×C4⋊C4×C59—21no—
68C2×C4⋊C4×C59—21no—
69C2×C4⋊C4×C59D411yescharacteristic
70C2×C4⋊C4×C59D412yes—
71C2×C22⋊C4×C59—21no—
72#106×C59—24no—
73#102×C59—24no—
74C22⋊C8×C59—21no—
75C22⋊C8×C59D412yes—
76C24×C59—21no—
77C24×C59C2×D411yescharacteristic
78C2×D4×C59—21no—
79C2×D4×C59—21no—
80C2×D4×C59—42no—
81C2×D4×C59—41no—
82C2×D4×C59—41no—
83C2×D4×C59—42no—
84C2×D4×C59—42no—
85C22×C4×C59—22no—
86C22×C4×C59—21no—
87C22×C4×C59—42no—
88C22×C4×C59—41no—
89C22×C4×C59—41no—
90C22×C4×C59—82no—
91C22×C4×C59C2×D411yescharacteristic
92C22×C4×C59C2×D411yescharacteristic
93C22×C4×C59C2×D411yescharacteristic
94C22×C4×C59C2×D411yescharacteristic
95C22×C4×C59C2×D411yescharacteristic
96C22×C4×C59#4111yescharacteristic
97C22×Dic59—22no—
98C22×Dic59—21no—
99C22×Dic59—21no—
100C22×Dic59—41no—
101C2×Dic118—21no—
102C2×Dic118—21no—
103C2×Dic118—42no—
104C2×C8×C59—22no—
105C2×C8×C59—42no—
106C4⋊C4×C59—22no—
107C4⋊C4×C59—21no—
108C4⋊C4×C59—21no—
109C4⋊C4×C59—22no—
110C4⋊C4×C59—41no—
111C4⋊C4×C59—41no—
112C22⋊C4×C59—41no—
113C22⋊C4×C59—41no—
114C22⋊Dic59—22no—
115C22⋊Dic59—41no—
116C22⋊Dic59—41no—
117C22⋊Dic59—42no—
118C22⋊Dic59SD1614yescomplemented
119#11745670—82no—
120C4⋊Dic59—21no—
121C4⋊Dic59—21no—
122C4⋊Dic59—42no—
123Dic59⋊C4—41no—
124Dic59⋊C4—41no—
125Dic59⋊C4—84no—
126C4×Dic59—42no—
127C2×C59⋊C8—22no—
128C2×C59⋊C8—42no—
129C23×C59—41no—
130C23×C59—41no—
131C23×C59—41no—
132C23×C59—42no—
133C23×C59—81no—
134C23×C59—81no—
135C23×C59#12011yescharacteristic
136C23×C59#12111yescharacteristic
137C23×C59#12711yescharacteristic
138C23×C59C2×SD1612yes—
139C23×C59#13712yes—
140D4×C59—42no—
141D4×C59—42no—
142D4×C59—82no—
143D4×C59—81no—
144D4×C59—81no—
145C2×C4×C59—22no—
146C2×C4×C59—22no—
147C2×C4×C59—22no—
148C2×C4×C59—41no—
149C2×C4×C59—41no—
150C2×C4×C59—41no—
151C2×C4×C59—44no—
152C2×C4×C59—41no—
153C2×C4×C59—42no—
154C2×C4×C59—42no—
155C2×C4×C59—82no—
156C2×C4×C59—82no—
157C2×C4×C59#12011yescharacteristic
158C2×C4×C59#12011yescharacteristic
159C2×C4×C59#12111yescharacteristic
160C2×C4×C59#12111yescharacteristic
161C2×Dic59—24no—
162C2×Dic59—42no—
163C2×Dic59—42no—
164C2×Dic59—42no—
165C2×Dic59—42no—
166C2×Dic59—41no—
167C2×Dic59—41no—
168C2×Dic59—81no—
169C2×Dic59—81no—
170C2×Dic59—81no—
171Dic118—42no—
172Dic118—81no—
173C8×C59—42no—
174C59⋊C8—42no—
175#9951—591nomaximal, Sylow
176C22×C59—21no—
177C22×C59—21no—
178C22×C59—21no—
179C22×C59—21no—
180C22×C59—21no—
181C22×C59—21no—
182C22×C59—42no—
183C22×C59—42no—
184C22×C59—81no—
185C22×C59—81no—
186C22×C59—81no—
187C22×C59—162no—
188C22×C59#39211yescharacteristic
189C22×C59#44812yes—
190C22×C59#46112yes—
191C22×C59#49211yescharacteristic
192C22×C59#49711yescharacteristic
193C4×C59—21no—
194C4×C59—21no—
195C4×C59—21no—
196C4×C59—82no—
197C4×C59—81no—
198C4×C59#45812yes—
199Dic59—44no—
200Dic59—81no—
201Dic59—81no—
202Dic59—161no—
203#3187—591no—
204C2×#481D5911yescharacteristic
205#2502—591no—
206#1964—591no—
207#1824—591no—
208#1520—591no—
209#1282—591no—
210C2×C59—22no—
211C2×C59—21no—
212C2×C59—22no—
213C2×C59—21no—
214C2×C59—161no—
215C2×C59#200811yescharacteristic
216C2×C59#201311yescharacteristic
217C2×C59#202211yescharacteristic
218C2×C22⋊Q8—591no—
219C2×#121C2×D5911yescharacteristic
220C22×C4⋊C4C2×D5911yescharacteristic
221#483—1181no—
222#481—22no—
223#481—22no—
224#481—1181no—
225#465—1182no—
226#450—1181no—
227#448—1181no—
228C2×#106—21no—
229C2×#102—21no—
230C2×#99—1182no—
231C2×C22⋊C8—591no—
232C2×C22⋊C8C2×D5911yescharacteristic
233#398—1182no—
234#391—1181no—
235#385—1182no—
236#384—1181no—
237#380—591no—
238C4×C22⋊C4—591no—
239#340—1182no—
240#328—1182no—
241C59#995111yescharacteristic, complemented, Sylow
242C22×Q8—591no—
243C22×D4—21no—
244C22×D4C22⋊D5911yescharacteristic
245C23×C4—21no—
246C23×C4C22×D5911yescharacteristic
247C2×SD16—1182no—
248C22×C8—21no—
249C22×C8—592no—
250C22⋊Q8—1181no—
251C22⋊Q8—1181no—
252C22⋊Q8—1181no—
253C22⋊Q8—1181no—
254#121—21no—
255#121—21no—
256#121—21no—
257#121—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—591no—
264C2×C4⋊C4—1184no—
265C2×C4⋊C4—1181no—
266C2×C4⋊C4—1181no—
267C2×C4⋊C4C22⋊D5912yes—
268C2×C4⋊C4C22⋊D5911yescharacteristic
269C2×C22⋊C4—21no—
270C2×C22⋊C4—592no—
271C2×C22⋊C4—1181no—
272C2×C22⋊C4—1181no—
273C2×C42—1181no—
274#106—24no—
275#106—1182no—
276#103—1182no—
277#102—24no—
278#102—1182no—
279#99—2364no—
280C22⋊C8—21no—
281C22⋊C8—1181no—
282C22⋊C8—1181no—
283C22⋊C8C22⋊D5912yes—
284#95—1184no—
285#95—1181no—
286#95—1181no—
287C24—21no—
288C24D4×D5911yescharacteristic
289C2×Q8—1181no—
290C2×Q8—1181no—
291C2×Q8—2362no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—42no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—42no—
298C2×D4—42no—
299C22×C4—22no—
300C22×C4—21no—
301C22×C4—42no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—82no—
305C22×C4—1182no—
306C22×C4—1181no—
307C22×C4—1181no—
308C22×C4—2361no—
309C22×C4D4×D5911yescharacteristic
310C22×C4D4×D5911yescharacteristic
311C22×C4#1174568711yescharacteristic
312C22×C4C2×C22⋊D5911yescharacteristic
313C22×C4C2×C22⋊D5911yescharacteristic
314C22×C4C2×C22⋊D5911yescharacteristic
315SD16—4722no—
316C2×C8—22no—
317C2×C8—42no—
318C2×C8—1182no—
319C2×C8—2362no—
320C4⋊C4—21no—
321C4⋊C4—22no—
322C4⋊C4—21no—
323C4⋊C4—22no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—1181no—
327C4⋊C4—1181no—
328C4⋊C4—4724no—
329C4⋊C4—2362no—
330C4⋊C4—2361no—
331C4⋊C4—2361no—
332C22⋊C4—41no—
333C22⋊C4—41no—
334C22⋊C4—594no—
335C22⋊C4—1182no—
336C22⋊C4—2362no—
337C22⋊C4—2361no—
338C22⋊C4—2361no—
339C42—2362no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23—42no—
344C23—81no—
345C23—81no—
346C23#1427369511yescharacteristic
347C23#1427369911yescharacteristic
348C23#1427372311yescharacteristic
349C23SD16×D5912yes—
350C23#1427375612yes—
351Q8—4721no—
352Q8—2362no—
353D4—42no—
354D4—42no—
355D4—81no—
356D4—81no—
357D4—82no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—44no—
365C2×C4—41no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4—1184no—
371C2×C4—4721no—
372C2×C4—4721no—
373C2×C4—4721no—
374C2×C4—2362no—
375C2×C4—2362no—
376C2×C4—2361no—
377C2×C4—2361no—
378C2×C4—2362no—
379C2×C4—2362no—
380C2×C4#1427369511yescharacteristic
381C2×C4#1427369611yescharacteristic
382C2×C4#1427369811yescharacteristic
383C2×C4#1427369911yescharacteristic
384C8—42no—
385C8—2362no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—42no—
393C22—42no—
394C22—81no—
395C22—81no—
396C22—81no—
397C22—162no—
398C22#1683731011yescharacteristic
399C22#1683753912yes—
400C22#1683760512yes—
401C22#1683777311yescharacteristic
402C22#1683779311yescharacteristic
403C4—21no—
404C4—21no—
405C4—21no—
406C4—81no—
407C4—82no—
408C4—9441no—
409C4—4721no—
410C4—4721no—
411C4—2364no—
412C4#1683759112yes—
413C2—21no—
414C2—22no—
415C2—22no—
416C2—21no—
417C2—161no—
418C2#2142514711yescharacteristic
419C2#2142517711yescharacteristic
420C2#2142522411yescharacteristic
subgroupG/H
centerC22#16837310
commutator, cosocleC23×C4×C59C23
FrattiniC23×C4C22×D59
FittingC2×C59×#481C2
socleC22×C59#392
prime radicalC2×#121×C59C22
hypercenterC2×#481D59
hypocenterC59#9951
LCSG ▷ C23×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#481

Complements

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

NH ≅ G/Ntype
2. #21428570C2semidirect
4. #21424186C2semidirect
5. #21422669C2semidirect
6. #21421942C2semidirect
118. C22⋊Dic59SD16semidirect
241. C59#9951semidirect

Maximal subgroups

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

subgroupindex
#214348532
#214285702
#214249332
#214241862
#214226692
#214219422
C2×C59×#4812
#995159