Finite Group Databasebeta

← #28481675

Subgroups — #28481675

Order 15104.

456 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 97 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#21432786C211yesmaximal, characteristic
2#21424132C211yesmaximal, characteristic, complemented
3#21424119C211yesmaximal, characteristic, complemented
4#21422640C211yesmaximal, characteristic
5#21422637C211yesmaximal, characteristic
6#21422001C211yesmaximal, characteristic, complemented
7C59×#1305C211yesmaximal, characteristic, complemented
8C2×#14273735C411yescharacteristic, complemented
9C23×C59⋊C8C2211yescharacteristic
10#16837913C2211yescharacteristic
11#16837470—21no—
12#16837470—21no—
13(C22⋊D59)⋊C8—21no—
14(C22⋊D59)⋊C8—21no—
15C2×#14273662—21no—
16C2×#14273658—21no—
17#16837430C411yescharacteristic, complemented
18#16837370C411yescharacteristic, complemented
19#16837366C2211yescharacteristic
20#16837360C411yescharacteristic, complemented
21#16837262C2211yescharacteristic
22#16837166—21no—
23#16837164—21no—
24#16837140—21no—
25#16837138—21no—
26#16837137—21no—
27#16837135—21no—
28C59×#407C2211yescharacteristic
29C59×#407C2211yescharacteristic
30C59×#380C2211yescharacteristic
31C2×#11745685C2×C411yescharacteristic
32#14273735—42no—
33C2×#11745664—21no—
34C2×#11745659—21no—
35C22×C59⋊C8—42no—
36C22×C59⋊C8C2×C411yescharacteristic
37C22×C59⋊C8C2×C411yescharacteristic
38C22×C59⋊C8D411yescharacteristic
39C22×C59⋊C8D411yescharacteristic
40C22×C59⋊C8D411yescharacteristic
41C22×C59⋊C8D411yescharacteristic
42C2×C8×D59—21no—
43#14273710—21no—
44#14273706D411yescharacteristic
45#14273706Q811yescharacteristic
46#14273701D411yescharacteristic
47#14273701Q811yescharacteristic
48#14273696—21no—
49C4×C22⋊D59—21no—
50C4×C22⋊D59—21no—
51#14273686C2×C411yescharacteristic
52C2×C4⋊Dic59—21no—
53C2×C22⋊Dic59—21no—
54#14273662—42no—
55#14273658—42no—
56Dic59⋊C8—21no—
57Dic59⋊C8—21no—
58C22⋊(C59⋊C8)—21no—
59C22⋊(C59⋊C8)—21no—
60(C2×D59)⋊C8—21no—
61(C2×D59)⋊C8—21no—
62#14273637—21no—
63C8×Dic59—21no—
64C59⋊#95—21no—
65C23×C4×C59C2311yescharacteristic
66C2×M16×C59—21no—
67C2×M16×C59C2×C411yescharacteristic
68C2×M16×C59C2×C411yescharacteristic
69C2×C4⋊C4×C59—21no—
70C2×C22⋊C4×C59—21no—
71C22⋊C8×C59—21no—
72C22⋊C8×C59—21no—
73C22⋊C8×C59—21no—
74C22⋊C8×C59—21no—
75#95×C59—21no—
76C24×C59#4111yescharacteristic
77C22×C4×C59—41no—
78C22×C4×C59—41no—
79C22×C4×C59—41no—
80C22×C4×C59C22×C411yescharacteristic
81C22×C4×C59C2×D411yescharacteristic
82C22×C4×C59C2×D411yescharacteristic
83C22×C4×C59C2×D411yescharacteristic
84C22×C4×C59C2×Q811yescharacteristic
85C22×C4×C59#4111yescharacteristic
86C2×C22⋊D59—21no—
87C2×C22⋊D59—21no—
88C2×C22⋊D59—21no—
89C22×Dic59—41no—
90#11745685—41no—
91#11745685—41no—
92C2×D236C4⋊C411yescharacteristic
93C2×C4×D59—21no—
94C2×Dic118C4⋊C411yescharacteristic
95M16×C59—41no—
96M16×C59—41no—
97M16×C59—41no—
98M16×C59—41no—
99C2×C8×C59—21no—
100C2×C8×C59—21no—
101C2×C8×C59—41no—
102C4⋊C4×C59—41no—
103C4⋊C4×C59—41no—
104C22⋊C4×C59—81no—
105C22⋊Dic59—21no—
106C22⋊Dic59—21no—
107C22⋊Dic59—21no—
108C22⋊Dic59—81no—
109(C2×D59)⋊C4—21no—
110(C2×D59)⋊C4—21no—
111C4⋊Dic59—41no—
112C4⋊Dic59—41no—
113C4⋊Dic59C4⋊C411yescharacteristic
114C4⋊Dic59C4⋊C411yescharacteristic
115Dic59⋊C4—21no—
116Dic59⋊C4—21no—
117C4×Dic59—21no—
118#11745664—41no—
119#11745664—41no—
120C2×C59⋊C8—22no—
121C2×C59⋊C8—22no—
122C2×C59⋊C8—22no—
123C2×C59⋊C8—41no—
124C2×C59⋊C8—41no—
125C2×C59⋊C8—41no—
126C2×C59⋊C8—41no—
127C2×C59⋊C8C22⋊C411yescharacteristic
128C2×C59⋊C8C22⋊C411yescharacteristic
129C2×C59⋊C8C22⋊C411yescharacteristic
130C2×C59⋊C8C22⋊C411yescharacteristic
131#11745659—42no—
132C8×D59—42no—
133C23×C59—21no—
134C23×C59—21no—
135C23×C59—41no—
136C23×C59—41no—
137C23×C59#11711yescharacteristic
138C23×C59#12111yescharacteristic
139C23×C59C22⋊Q811yescharacteristic
140C22×D59—21no—
141C2×C4×C59—21no—
142C2×C4×C59—21no—
143C2×C4×C59—21no—
144C2×C4×C59—21no—
145C2×C4×C59—21no—
146C2×C4×C59—21no—
147C2×C4×C59—41no—
148C2×C4×C59—41no—
149C2×C4×C59—41no—
150C2×C4×C59—81no—
151C2×C4×C59—81no—
152C2×C4×C59—81no—
153C2×C4×C59C2×C22⋊C411yescharacteristic
154C2×C4×C59C2×C4⋊C411yescharacteristic
155C2×C4×C59#12111yescharacteristic
156C2×C4×C59C22⋊Q811yescharacteristic
157C22⋊D59—42no—
158C22⋊D59—42no—
159C22⋊D59—41no—
160C22⋊D59—41no—
161C2×Dic59—21no—
162C2×Dic59—21no—
163C2×Dic59—21no—
164C2×Dic59—81no—
165C2×Dic59—81no—
166C2×Dic59—81no—
167D236—41no—
168C4×D59—41no—
169C4×D59—41no—
170Dic118—41no—
171C8×C59—41no—
172C8×C59—41no—
173C8×C59—81no—
174C59⋊C8—21no—
175C59⋊C8—21no—
176C59⋊C8—21no—
177C59⋊C8—81no—
178C59⋊C8#9912yes—
179#8629—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—42no—
189C22×C59—42no—
190C22×C59#38011yescharacteristic
191C22×C59#47111yescharacteristic
192C22×C59#47111yescharacteristic
193C2×D59—41no—
194C2×D59—41no—
195C2×D59—41no—
196C4×C59—21no—
197C4×C59—21no—
198C4×C59—21no—
199C4×C59—161no—
200C4×C59C2×#9911yescharacteristic
201C4×C59#41011yescharacteristic
202Dic59—41no—
203Dic59—41no—
204Dic59—41no—
205Dic59—161no—
206#2929—591no—
207#1810—591no—
208#1807—591no—
209#1513—591no—
210#1513—591no—
211#1305—591no—
212#1305D5911yescharacteristic, complemented
213C2×C59—21no—
214C2×C59—21no—
215C2×C59—41no—
216C2×C59—41no—
217C2×C59#179911yescharacteristic
218C2×C59#181411yescharacteristic
219C2×C59#181511yescharacteristic
220D59—81no—
221C2×#131—591no—
222C23×C8—591no—
223#525—591no—
224#435—1181no—
225#435—1181no—
226C8×D4—1181no—
227C8×D4—1181no—
228C2×#108—1181no—
229C2×#106—1181no—
230#424—591no—
231#408—591no—
232#407—591no—
233#407C2×D5911yescharacteristic
234#407C2×D5911yescharacteristic
235#405—591no—
236#380—591no—
237#380C2×D5911yescharacteristic
238#341—1181no—
239#340—1181no—
240#328—1181no—
241#328—1181no—
242#327—1181no—
243#327—1181no—
244C59#862911yescharacteristic, complemented, Sylow
245C2×#41—591no—
246C23×C4C22×D5911yescharacteristic
247#131—2362no—
248C2×M16—21no—
249C2×M16—1181no—
250C2×M16—1181no—
251C2×M16C4×D5911yescharacteristic
252C2×M16C4×D5911yescharacteristic
253C22×C8—591no—
254C22×C8—591no—
255C22×C8—591no—
256C22×C8—591no—
257C22×C8—591no—
258C22×C8—591no—
259C22×C8—1181no—
260C22×C8—2362no—
261#123—1181no—
262C22⋊Q8—591no—
263C22⋊Q8—591no—
264#121—591no—
265#121—591no—
266#120—1181no—
267C4×D4—1181no—
268C4×D4—1181no—
269#117—591no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—1181no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—1181no—
274#108—2362no—
275#106—2362no—
276C4⋊C8—1181no—
277C4⋊C8—1181no—
278C22⋊C8—21no—
279C22⋊C8—21no—
280C22⋊C8—21no—
281C22⋊C8—21no—
282C22⋊C8—1181no—
283C22⋊C8—1181no—
284C22⋊C8—1181no—
285C22⋊C8—1181no—
286#97—1181no—
287C4×C8—1181no—
288#95—21no—
289#95—1181no—
290C24#1174568511yescharacteristic
291#41—2361no—
292#41—2361no—
293C2×Q8—591no—
294C2×D4—591no—
295C2×D4—1181no—
296C2×D4—1181no—
297C2×D4—1181no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—1181no—
302C22×C4—2361no—
303C22×C4C2×C4×D5911yescharacteristic
304C22×C4#1174568511yescharacteristic
305C22×C4D4×D5911yescharacteristic
306C22×C4D4×D5911yescharacteristic
307C22×C4D4×D5911yescharacteristic
308C22×C4Q8×D5911yescharacteristic
309M16—41no—
310M16—41no—
311M16—41no—
312M16—41no—
313M16—2362no—
314M16—2361no—
315M16—2361no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—41no—
319C2×C8—591no—
320C2×C8—591no—
321C2×C8—591no—
322C2×C8—591no—
323C2×C8—1182no—
324C2×C8—1182no—
325C2×C8—1182no—
326C2×C8—2362no—
327C2×C8—2361no—
328C2×C8—2361no—
329C2×C8—2361no—
330C2×C8—2361no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—591no—
334C4⋊C4—591no—
335C4⋊C4—1181no—
336C4⋊C4—1181no—
337C4⋊C4—2361no—
338C4⋊C4—2361no—
339C22⋊C4—81no—
340C22⋊C4—1181no—
341C22⋊C4—1181no—
342C22⋊C4—1181no—
343C22⋊C4—1181no—
344C22⋊C4—1181no—
345C22⋊C4—4721no—
346C42—1181no—
347C23—21no—
348C23—21no—
349C23—41no—
350C23—41no—
351C23—1181no—
352C23#1427368411yescharacteristic
353C23#1427370011yescharacteristic
354C23#1427370511yescharacteristic
355Q8—2361no—
356D4—2361no—
357D4—2362no—
358D4—2362no—
359D4—2361no—
360D4—2361no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—1181no—
374C2×C4—1181no—
375C2×C4—1181no—
376C2×C4—4721no—
377C2×C4—4721no—
378C2×C4—4721no—
379C2×C4—2361no—
380C2×C4—2361no—
381C2×C4C22⋊C4×D5911yescharacteristic
382C2×C4C4⋊C4×D5911yescharacteristic
383C2×C4#1427370011yescharacteristic
384C2×C4#1427370511yescharacteristic
385C8—41no—
386C8—41no—
387C8—81no—
388C8—592no—
389C8—1181no—
390C8—1181no—
391C8—1181no—
392C8—4721no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—42no—
402C22—42no—
403C22—2361no—
404C22—2361no—
405C22—2361no—
406C22#1683726111yescharacteristic
407C22#1683766411yescharacteristic
408C22#1683766811yescharacteristic
409C4—21no—
410C4—21no—
411C4—21no—
412C4—161no—
413C4—9441no—
414C4—2361no—
415C4—2361no—
416C4—2361no—
417C4#99×D5911yescharacteristic
418C4#1683737411yescharacteristic
419C2—21no—
420C2—21no—
421C2—41no—
422C2—41no—
423C2—4721no—
424C2#2142408311yescharacteristic
425C2#2142414911yescharacteristic
426C2#2142415411yescharacteristic
subgroupG/H
centerC22#16837261
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×#1305C2
socleC22×C59#380
prime radical#16837913C22
cosocleC23×C4×C59C23
hypercenter#1305D59
hypocenterC59#8629
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1305

Complements

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

NH ≅ G/Ntype
2. #21424132C2semidirect
3. #21424119C2semidirect
6. #21422001C2semidirect
7. C59×#1305C2semidirect
8. C2×#14273735C4semidirect
17. #16837430C4semidirect
18. #16837370C4semidirect
20. #16837360C4semidirect
212. #1305D59semidirect
244. C59#8629semidirect

Maximal subgroups

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

subgroupindex
#214327862
#214241322
#214241192
#214226402
#214226372
#214220012
C59×#13052
#862959