Finite Group Databasebeta

← #28484734

Subgroups — #28484734

Order 15104.

560 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 113 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#21429911C211yesmaximal, characteristic
2#21426713C211yesmaximal, characteristic
3#21422668C211yesmaximal, characteristic
4C2×#16837127C211yesmaximal, characteristic
5#21421964C211yesmaximal, characteristic
6#21421963C211yesmaximal, characteristic
7C59×#1520C211yesmaximal, characteristic
8C2×#14273683—21no—
9C22×Dic59⋊C4C2211yescharacteristic
10C2×C22⋊(C59⋊C8)C2211yescharacteristic
11#16837340C412yescomplemented
12#16837335—22no—
13#16837321C412yescomplemented
14#16837299—22no—
15#16837297—22no—
16#16837274—21no—
17#16837263—21no—
18#16837262C2211yescharacteristic
19#16837262C2211yescharacteristic
20#16837260—21no—
21#16837139—22no—
22#16837127—22no—
23C2×C22⋊Q8×C59C2211yescharacteristic
24C2×C22⋊C8×C59C2211yescharacteristic
25C59×#380C2211yescharacteristic
26C59×#328—22no—
27C23×Dic59D411yescharacteristic
28C23×Dic59D411yescharacteristic
29C22×C59⋊C8—21no—
30#14273683—42no—
31#14273683—42no—
32C2×C4⋊Dic59—21no—
33C2×C4⋊Dic59—21no—
34C2×Dic59⋊C4—21no—
35C2×Dic59⋊C4—21no—
36C2×Dic59⋊C4—21no—
37C2×Dic59⋊C4—21no—
38C2×Dic59⋊C4—22no—
39C2×Dic59⋊C4—22no—
40C2×Dic59⋊C4—22no—
41C2×Dic59⋊C4C2×C412yes—
42C2×Dic59⋊C4D412yes—
43C2×C22⋊Dic59—21no—
44C2×C22⋊Dic59—21no—
45C2×C4×Dic59—22no—
46C22⋊(C59⋊C8)—21no—
47C22⋊(C59⋊C8)—21no—
48C59⋊#95—22no—
49C59⋊#95—22no—
50C59⋊#95—22no—
51C59⋊#95—22no—
52C59⋊#95C2×C412yes—
53C59⋊#95D412yes—
54C22×Q8×C59C2×C411yescharacteristic
55C23×C4×C59C2311yescharacteristic
56C22×C8×C59—21no—
57C22⋊Q8×C59—21no—
58C22⋊Q8×C59—21no—
59C22⋊Q8×C59—21no—
60C22⋊Q8×C59—21no—
61C2×C4⋊C4×C59—21no—
62C2×C4⋊C4×C59—21no—
63C2×C4⋊C4×C59C2×C411yescharacteristic
64C2×C22⋊C4×C59—21no—
65C2×C22⋊C4×C59—21no—
66C22⋊C8×C59—21no—
67C22⋊C8×C59—21no—
68#95×C59D412yes—
69C24×C59C2×D411yescharacteristic
70C2×Q8×C59—21no—
71C2×Q8×C59—21no—
72C2×Q8×C59—81no—
73C22×C4×C59—21no—
74C22×C4×C59—21no—
75C22×C4×C59—22no—
76C22×C4×C59—81no—
77C22×C4×C59C22×C411yescharacteristic
78C22×C4×C59C2×D411yescharacteristic
79C22×C4×C59C2×D411yescharacteristic
80C22×C4×C59C2×D411yescharacteristic
81C22×C4×C59#4111yescharacteristic
82C22×C4×C59#4111yescharacteristic
83C22×Dic59—21no—
84C22×Dic59—21no—
85C22×Dic59—21no—
86C22×Dic59—21no—
87C22×Dic59—21no—
88C22×Dic59—21no—
89C22×Dic59—22no—
90C22×Dic59—22no—
91C22×Dic59—81no—
92C22×Dic59C22⋊C412yes—
93C22×Dic59C22⋊C412yes—
94C22×Dic59SD1612yes—
95C22×Dic59Q1612yes—
96C2×C8×C59—41no—
97C2×C8×C59—81no—
98C4⋊C4×C59—21no—
99C4⋊C4×C59—21no—
100C4⋊C4×C59—41no—
101C4⋊C4×C59—41no—
102C4⋊C4×C59—81no—
103C22⋊C4×C59—41no—
104C22⋊C4×C59—41no—
105C22⋊C4×C59—41no—
106C22⋊C4×C59—41no—
107C22⋊Dic59—41no—
108C22⋊Dic59—41no—
109C22⋊Dic59—41no—
110C22⋊Dic59—41no—
111C4⋊Dic59—81no—
112C4⋊Dic59—81no—
113Dic59⋊C4—22no—
114Dic59⋊C4—22no—
115Dic59⋊C4—22no—
116Dic59⋊C4—22no—
117Dic59⋊C4—44no—
118Dic59⋊C4—82no—
119Dic59⋊C4—82no—
120C4×Dic59—44no—
121C2×C59⋊C8—41no—
122C2×C59⋊C8—81no—
123C23×C59—41no—
124C23×C59—41no—
125C23×C59C2×C22⋊C411yescharacteristic
126C23×C59#12011yescharacteristic
127C23×C59#12311yescharacteristic
128C23×C59C2×SD1611yescharacteristic
129C23×C59C2×Q1611yescharacteristic
130C23×C59AGL(1,ℤ/8)11yescharacteristic
131C23×C59#13711yescharacteristic
132Q8×C59—81no—
133Q8×C59—81no—
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—42no—
141C2×C4×C59—42no—
142C2×C4×C59—42no—
143C2×C4×C59—42no—
144C2×C4×C59—41no—
145C2×C4×C59—41no—
146C2×C4×C59—42no—
147C2×C4×C59C4×D411yescharacteristic
148C2×C4×C59C4×D411yescharacteristic
149C2×C4×C59#12111yescharacteristic
150C2×C4×C59#12111yescharacteristic
151C2×Dic59—22no—
152C2×Dic59—22no—
153C2×Dic59—22no—
154C2×Dic59—22no—
155C2×Dic59—22no—
156C2×Dic59—22no—
157C2×Dic59—22no—
158C2×Dic59—42no—
159C2×Dic59—42no—
160C2×Dic59—42no—
161C2×Dic59—42no—
162C2×Dic59—42no—
163C2×Dic59—42no—
164C2×Dic59—42no—
165C2×Dic59—42no—
166C2×Dic59—81no—
167C2×Dic59#10314yes—
168C8×C59—81no—
169C59⋊C8—81no—
170#9200—591nomaximal, Sylow
171C22×C59—21no—
172C22×C59—21no—
173C22×C59—21no—
174C22×C59—21no—
175C22×C59—21no—
176C22×C59—21no—
177C22×C59—41no—
178C22×C59—41no—
179C22×C59—41no—
180C22×C59—41no—
181C22×C59#38611yescharacteristic
182C22×C59C2×#10311yescharacteristic
183C22×C59#41711yescharacteristic
184C22×C59#45011yescharacteristic
185C22×C59#45111yescharacteristic
186C22×C59#48311yescharacteristic
187C22×C59#48411yescharacteristic
188C4×C59—21no—
189C4×C59—21no—
190C4×C59—21no—
191C4×C59—21no—
192C4×C59—82no—
193C4×C59—81no—
194C4×C59—81no—
195Dic59—24no—
196Dic59—42no—
197Dic59—82no—
198Dic59—82no—
199#2627—591no—
200#2316—591no—
201#1520—591no—
202#1520D5911yescharacteristic
203C2×#324—591no—
204#1289—591no—
205#1289—591no—
206C2×C59—21no—
207C2×C59—21no—
208C2×C59—21no—
209C2×C59—21no—
210C2×C59—21no—
211C2×C59—21no—
212C2×C59#187211yescharacteristic
213C2×C59#191211yescharacteristic
214C2×C59#191311yescharacteristic
215C2×C22⋊Q8C2×D5911yescharacteristic
216C2×#117—1181no—
217C22×C4⋊C4—591no—
218C2×C22⋊C8—591no—
219C2×C22⋊C8C2×D5911yescharacteristic
220#400—592no—
221#398—1182no—
222#395—592no—
223#389—1182no—
224#389—1182no—
225#385—1181no—
226#381—1181no—
227#380—591no—
228#380—591no—
229#380—1181no—
230#380C2×D5911yescharacteristic
231#328—22no—
232#328—1182no—
233#324—1182no—
234C59#920011yescharacteristic, complemented, Sylow
235C22×Q8C4×D5911yescharacteristic
236C23×C4—591no—
237C23×C4—591no—
238C23×C4C22×D5911yescharacteristic
239C22×C8—21no—
240C22×C8—1181no—
241C22⋊Q8—21no—
242C22⋊Q8—21no—
243C22⋊Q8—21no—
244C22⋊Q8—21no—
245#117—2362no—
246#117—2362no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—592no—
250C2×C4⋊C4—592no—
251C2×C4⋊C4—1181no—
252C2×C4⋊C4—1181no—
253C2×C4⋊C4—1181no—
254C2×C4⋊C4—1181no—
255C2×C4⋊C4—1182no—
256C2×C4⋊C4—1182no—
257C2×C4⋊C4—1181no—
258C2×C4⋊C4—1181no—
259C2×C4⋊C4—1182no—
260C2×C4⋊C4C4×D5911yescharacteristic
261C2×C22⋊C4—21no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—1181no—
264C2×C22⋊C4—1181no—
265C2×C42—1182no—
266C22⋊C8—21no—
267C22⋊C8—21no—
268C22⋊C8—1181no—
269C22⋊C8—1181no—
270#95—592no—
271#95—592no—
272#95—1182no—
273#95—1182no—
274#95—1182no—
275#95—1182no—
276#95C22⋊D5912yes—
277C24D4×D5911yescharacteristic
278C2×Q8—21no—
279C2×Q8—21no—
280C2×Q8—81no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—81no—
285C22×C4—592no—
286C22×C4—592no—
287C22×C4—592no—
288C22×C4—592no—
289C22×C4—1181no—
290C22×C4—1181no—
291C22×C4—1181no—
292C22×C4—1181no—
293C22×C4—1181no—
294C22×C4—1181no—
295C22×C4—1182no—
296C22×C4—1182no—
297C22×C4—4721no—
298C22×C4C2×C4×D5911yescharacteristic
299C22×C4#1174568511yescharacteristic
300C22×C4D4×D5911yescharacteristic
301C22×C4D4×D5911yescharacteristic
302C22×C4#1174568711yescharacteristic
303C22×C4C2×C22⋊D5911yescharacteristic
304C2×C8—41no—
305C2×C8—81no—
306C2×C8—4721no—
307C2×C8—2361no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—81no—
313C4⋊C4—1182no—
314C4⋊C4—1182no—
315C4⋊C4—1182no—
316C4⋊C4—1182no—
317C4⋊C4—4722no—
318C4⋊C4—4722no—
319C4⋊C4—4721no—
320C4⋊C4—4721no—
321C4⋊C4—2364no—
322C22⋊C4—41no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C22⋊C4—41no—
326C22⋊C4—2361no—
327C22⋊C4—2361no—
328C22⋊C4—2361no—
329C22⋊C4—2361no—
330C42—2364no—
331C23—41no—
332C23—41no—
333C23C22⋊C4×D5911yescharacteristic
334C23#1427369511yescharacteristic
335C23#1427370911yescharacteristic
336C23SD16×D5911yescharacteristic
337C23Q16×D5911yescharacteristic
338C23#1427375411yescharacteristic
339C23#1427375911yescharacteristic
340Q8—81no—
341Q8—81no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—42no—
355C2×C4—594no—
356C2×C4—1182no—
357C2×C4—1182no—
358C2×C4—1182no—
359C2×C4—1182no—
360C2×C4—1182no—
361C2×C4—1182no—
362C2×C4—1182no—
363C2×C4—4721no—
364C2×C4—2362no—
365C2×C4—2362no—
366C2×C4—2362no—
367C2×C4—2362no—
368C2×C4—2362no—
369C2×C4—2362no—
370C2×C4—2362no—
371C2×C4—2362no—
372C2×C4(C22⋊D59)⋊C411yescharacteristic
373C2×C4C4×C22⋊D5911yescharacteristic
374C2×C4#1427369811yescharacteristic
375C2×C4#1427370011yescharacteristic
376C8—81no—
377C8—4721no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22#1683727811yescharacteristic
389C22#103×D5911yescharacteristic
390C22#1683740211yescharacteristic
391C22#1683755111yescharacteristic
392C22#1683755611yescharacteristic
393C22#1683773111yescharacteristic
394C22#1683773611yescharacteristic
395C4—21no—
396C4—21no—
397C4—21no—
398C4—21no—
399C4—82no—
400C4—81no—
401C4—81no—
402C4—1184no—
403C4—4722no—
404C4—4722no—
405C4—2362no—
406C2—21no—
407C2—21no—
408C2—21no—
409C2—21no—
410C2—21no—
411C2—21no—
412C2#2142440011yescharacteristic
413C2#2142463011yescharacteristic
414C2#2142463511yescharacteristic
subgroupG/H
centerC22#16837278
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×#1520C2
socleC22×C59#386
prime radicalC24×C59C2×D4
cosocleC23×C4×C59C23
hypercenter#1520D59
hypocenterC59#9200
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1520

Complements

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

NH ≅ G/Ntype
11. #16837340C4semidirect
13. #16837321C4semidirect
234. C59#9200semidirect

Maximal subgroups

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

subgroupindex
#214299112
#214267132
#214226682
C2×#168371272
#214219642
#214219632
C59×#15202
#920059