Finite Group Databasebeta

← #28491847

Subgroups — #28491847

Order 15104.

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

normal subgroups only

This group has 466 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#21429107C211yesmaximal, characteristic
2#21425080C211yesmaximal, characteristic
3#21422669C211yesmaximal, characteristic
4C2×#16837138C211yesmaximal, characteristic
5#21421963C211yesmaximal, characteristic
6#21421943C211yesmaximal, characteristic
7C59×#1294C211yesmaximal, characteristic
8C2×#14273706—21no—
9C2×#14273706C2211yescharacteristic
10C2×C22⋊(C59⋊C8)C2211yescharacteristic
11#16837336—21no—
12#16837323—21no—
13#16837306—21no—
14#16837299—21no—
15#16837275—21no—
16#16837262C2211yescharacteristic
17C4×C22⋊Dic59C2211yescharacteristic
18#16837168—21no—
19#16837168—21no—
20#16837140—21no—
21#16837140—21no—
22#16837138—21no—
23#16837138—21no—
24C2×C22⋊C8×C59C2211yescharacteristic
25C59×#380C2211yescharacteristic
26C2×#95×C59C2211yescharacteristic
27C22×Dic118—21no—
28C22×Dic118Q811yescharacteristic, complemented
29C22×C59⋊C8—21no—
30#14273706—21no—
31#14273706—21no—
32#14273706—21no—
33#14273706—21no—
34#14273706—42no—
35#14273706—42no—
36C2×C4⋊Dic59—21no—
37C2×C4⋊Dic59—21no—
38C2×C4⋊Dic59Q811yescharacteristic, complemented
39C2×Dic59⋊C4—21no—
40C2×Dic59⋊C4—21no—
41C2×Dic59⋊C4—21no—
42C2×Dic59⋊C4—21no—
43C2×Dic59⋊C4—21no—
44C2×C22⋊Dic59—21no—
45C2×C22⋊Dic59—21no—
46C2×C22⋊Dic59D411yescharacteristic
47C2×C22⋊Dic59D411yescharacteristic
48C2×C4×Dic59—21no—
49C22⋊(C59⋊C8)—21no—
50C22⋊(C59⋊C8)—21no—
51C59⋊#95—21no—
52C59⋊#95—21no—
53C59⋊#95—21no—
54C59⋊#95D411yescharacteristic
55C59⋊#95D411yescharacteristic
56C23×C4×C59—21no—
57C23×C4×C59C2311yescharacteristic
58C22×C8×C59—21no—
59C2×C4⋊C4×C59—21no—
60C2×C22⋊C4×C59—21no—
61C22⋊C8×C59—21no—
62C22⋊C8×C59—21no—
63#95×C59—21no—
64#95×C59—21no—
65#95×C59D411yescharacteristic
66#95×C59D411yescharacteristic
67C24×C59C2×D411yescharacteristic
68C22×C4×C59—21no—
69C22×C4×C59—21no—
70C22×C4×C59—42no—
71C22×C4×C59—41no—
72C22×C4×C59—41no—
73C22×C4×C59—41no—
74C22×C4×C59—81no—
75C22×C4×C59—81no—
76C22×C4×C59C2×D411yescharacteristic
77C22×C4×C59C2×D411yescharacteristic
78C22×C4×C59C2×Q811yescharacteristic
79C22×C4×C59#4111yescharacteristic
80C22×C4×C59#4111yescharacteristic
81C22×C4×C59#4111yescharacteristic
82C22×Dic59—21no—
83C22×Dic59—21no—
84C22×Dic59—21no—
85C22×Dic59—21no—
86C22×Dic59—21no—
87C22×Dic59—21no—
88C2×Dic118—21no—
89C2×Dic118—21no—
90C2×Dic118—42no—
91C2×Dic118—42no—
92C2×Dic118—81no—
93C2×Dic118—81no—
94C2×C8×C59—41no—
95C2×C8×C59—81no—
96C4⋊C4×C59—81no—
97C22⋊C4×C59—41no—
98C22⋊C4×C59—41no—
99C22⋊Dic59—21no—
100C22⋊Dic59—21no—
101C22⋊Dic59—41no—
102C22⋊Dic59—41no—
103C22⋊Dic59—41no—
104C22⋊Dic59—41no—
105C22⋊Dic59SD1612yes—
106C22⋊Dic59Q1612yes—
107C4⋊Dic59—21no—
108C4⋊Dic59—21no—
109C4⋊Dic59—42no—
110C4⋊Dic59—81no—
111Dic59⋊C4—42no—
112Dic59⋊C4—42no—
113Dic59⋊C4—42no—
114Dic59⋊C4—41no—
115Dic59⋊C4—41no—
116Dic59⋊C4—81no—
117C4×Dic59—81no—
118C2×C59⋊C8—41no—
119C2×C59⋊C8—81no—
120C23×C59—41no—
121C23×C59—41no—
122C23×C59#12011yescharacteristic
123C23×C59C22⋊Q811yescharacteristic
124C23×C59#12411yescharacteristic
125C23×C59C2×SD1611yescharacteristic
126C23×C59C2×Q1611yescharacteristic
127C23×C59AGL(1,ℤ/8)11yescharacteristic
128C23×C59#13711yescharacteristic
129C2×C4×C59—22no—
130C2×C4×C59—41no—
131C2×C4×C59—41no—
132C2×C4×C59—41no—
133C2×C4×C59—41no—
134C2×C4×C59—41no—
135C2×C4×C59—41no—
136C2×C4×C59—41no—
137C2×C4×C59—41no—
138C2×C4×C59—41no—
139C2×C4×C59—41no—
140C2×C4×C59—41no—
141C2×C4×C59—82no—
142C2×C4×C59—82no—
143C2×C4×C59—81no—
144C2×C4×C59—81no—
145C2×C4×C59—81no—
146C2×C4×C59C22⋊Q811yescharacteristic
147C2×C4×C59C22⋊Q811yescharacteristic
148C2×C4×C59#12411yescharacteristic
149C2×C4×C59#12411yescharacteristic
150C2×Dic59—22no—
151C2×Dic59—22no—
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—
161C2×Dic59—41no—
162C2×Dic59—41no—
163C2×Dic59—41no—
164C2×Dic59—81no—
165C2×Dic59—81no—
166Dic118—84no—
167Dic118—81no—
168Dic118—81no—
169C8×C59—81no—
170C59⋊C8—81no—
171#10375—591nomaximal, Sylow
172C22×C59—21no—
173C22×C59—21no—
174C22×C59—21no—
175C22×C59—21no—
176C22×C59—21no—
177C22×C59—21no—
178C22×C59—41no—
179C22×C59—41no—
180C22×C59—41no—
181C22×C59—41no—
182C22×C59#39311yescharacteristic
183C22×C59#45011yescharacteristic
184C22×C59#45111yescharacteristic
185C22×C59#47511yescharacteristic
186C22×C59#47711yescharacteristic
187C22×C59#48711yescharacteristic
188C22×C59#48911yescharacteristic
189C4×C59—21no—
190C4×C59—21no—
191C4×C59—21no—
192C4×C59—21no—
193C4×C59—42no—
194C4×C59—81no—
195C4×C59—161no—
196Dic59—42no—
197Dic59—42no—
198Dic59—81no—
199Dic59—81no—
200Dic59—81no—
201Dic59—81no—
202#2549—591no—
203#1996—591no—
204#1520—591no—
205C2×#328—591no—
206#1294D5911yescharacteristic
207#1289—591no—
208#1282—591no—
209C2×C59—21no—
210C2×C59—21no—
211C2×C59—21no—
212C2×C59—21no—
213C2×C59—21no—
214C2×C59—21no—
215C2×C59#202411yescharacteristic
216C2×C59#203111yescharacteristic
217C2×C59#203211yescharacteristic
218C2×C22⋊Q8—591no—
219C2×C22⋊Q8—1181no—
220C2×C22⋊C8—591no—
221C2×C22⋊C8C2×D5911yescharacteristic
222#398—1181no—
223#395—1181no—
224#391—1181no—
225#389—1181no—
226#385—1181no—
227#380—591no—
228#380C2×D5911yescharacteristic
229C4×C22⋊C4—591no—
230C2×#95C2×D5911yescharacteristic
231#342—1181no—
232#342—1181no—
233#328—1181no—
234#328—1181no—
235#328—1181no—
236#328—1181no—
237C59#1037511yescharacteristic, complemented, Sylow
238C22×Q8—591no—
239C22×Q8—1181no—
240C23×C4—21no—
241C23×C4C22×D5911yescharacteristic
242C22×C8—21no—
243C22×C8—1181no—
244C22⋊Q8—1181no—
245C22⋊Q8—1181no—
246C22⋊Q8—1181no—
247C22⋊Q8—1181no—
248C22⋊Q8—2362no—
249C22⋊Q8—2362no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—591no—
252C2×C4⋊C4—1181no—
253C2×C4⋊C4—1181no—
254C2×C4⋊C4—1181no—
255C2×C4⋊C4—1181no—
256C2×C4⋊C4—1181no—
257C2×C4⋊C4—1181no—
258C2×C4⋊C4—1181no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—591no—
261C2×C22⋊C4—591no—
262C2×C22⋊C4—1181no—
263C2×C22⋊C4—1181no—
264C2×C42—1181no—
265C22⋊C8—21no—
266C22⋊C8—21no—
267C22⋊C8—1181no—
268C22⋊C8—1181no—
269#95—21no—
270#95—21no—
271#95—591no—
272#95—591no—
273#95—1181no—
274#95—1181no—
275#95—1181no—
276#95D23611yescharacteristic
277#95D23611yescharacteristic
278C24D4×D5911yescharacteristic
279C2×Q8—1181no—
280C2×Q8—1181no—
281C2×Q8—4721no—
282C2×Q8—4721no—
283C2×Q8—2362no—
284C2×Q8—2362no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—42no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—81no—
292C22×C4—81no—
293C22×C4—1181no—
294C22×C4—1181no—
295C22×C4—1181no—
296C22×C4—1181no—
297C22×C4—1181no—
298C22×C4—1181no—
299C22×C4C2×D23611yescharacteristic
300C22×C4#1174568511yescharacteristic
301C22×C4#1174568511yescharacteristic
302C22×C4D4×D5911yescharacteristic
303C22×C4#1174568711yescharacteristic
304C22×C4Q8×D5911yescharacteristic
305C2×C8—41no—
306C2×C8—81no—
307C2×C8—4721no—
308C2×C8—2361no—
309C4⋊C4—81no—
310C4⋊C4—1181no—
311C4⋊C4—1181no—
312C4⋊C4—4721no—
313C4⋊C4—4721no—
314C4⋊C4—2362no—
315C4⋊C4—2362no—
316C4⋊C4—2362no—
317C4⋊C4—2362no—
318C4⋊C4—2361no—
319C4⋊C4—2361no—
320C22⋊C4—41no—
321C22⋊C4—41no—
322C22⋊C4—592no—
323C22⋊C4—592no—
324C22⋊C4—1181no—
325C22⋊C4—1181no—
326C22⋊C4—2361no—
327C22⋊C4—2361no—
328C22⋊C4—2361no—
329C22⋊C4—2361no—
330C42—4721no—
331C23—41no—
332C23—41no—
333C23#1427369711yescharacteristic
334C23#1427370511yescharacteristic
335C23#1427371311yescharacteristic
336C23SD16×D5911yescharacteristic
337C23Q16×D5911yescharacteristic
338C23#1427375111yescharacteristic
339C23#1427375611yescharacteristic
340Q8—4721no—
341Q8—4721no—
342Q8—4724no—
343C2×C4—22no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—82no—
356C2×C4—82no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—1182no—
361C2×C4—1182no—
362C2×C4—4721no—
363C2×C4—4721no—
364C2×C4—2361no—
365C2×C4—2361no—
366C2×C4—2361no—
367C2×C4—2361no—
368C2×C4—2361no—
369C2×C4—2361no—
370C2×C4—2361no—
371C2×C4—2361no—
372C2×C4—2361no—
373C2×C4—2361no—
374C2×C4—2361no—
375C2×C4—2361no—
376C2×C4#1427370511yescharacteristic
377C2×C4#1427370711yescharacteristic
378C2×C4#1427371311yescharacteristic
379C2×C4#1427371511yescharacteristic
380C8—81no—
381C8—4721no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22#1683731511yescharacteristic
393C22#1683755411yescharacteristic
394C22#1683755911yescharacteristic
395C22#1683768611yescharacteristic
396C22#1683769611yescharacteristic
397C22#1683775111yescharacteristic
398C22#1683776011yescharacteristic
399C4—21no—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—42no—
404C4—81no—
405C4—161no—
406C4—4721no—
407C4—4721no—
408C4—4721no—
409C4—4721no—
410C4—2362no—
411C4—2362no—
412C2—21no—
413C2—21no—
414C2—21no—
415C2—21no—
416C2—21no—
417C2—21no—
418C2#2142523611yescharacteristic
419C2#2142526911yescharacteristic
420C2#2142527411yescharacteristic
subgroupG/H
centerC22#16837315
commutator, cosocleC23×C4×C59C23
FrattiniC23×C4C22×D59
FittingC59×#1294C2
socleC22×C59#393
prime radicalC24×C59C2×D4
hypercenter#1294D59
hypocenterC59#10375
LCSG ▷ C23×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1294

Complements

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

NH ≅ G/Ntype
28. C22×Dic118Q8semidirect
38. C2×C4⋊Dic59Q8semidirect
237. C59#10375semidirect

Maximal subgroups

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

subgroupindex
#214291072
#214250802
#214226692
C2×#168371382
#214219632
#214219432
C59×#12942
#1037559