Finite Group Databasebeta

← #28491907

Subgroups — #28491907

Order 15104.

438 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 75 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#21430293C211yesmaximal, characteristic, complemented
2#21422625C211yesmaximal, characteristic
3#21422612C211yesmaximal, characteristic
4#21422002C211yesmaximal, characteristic, complemented
5#21421982C211yesmaximal, characteristic, complemented
6#21421949C211yesmaximal, characteristic, complemented
7C59×#1964C211yesmaximal, characteristic
8#16837366C2211yescharacteristic
9#16837366C2211yescharacteristic
10#16837338—21no—
11#16837335—21no—
12#16837335—21no—
13#16837325—21no—
14#16837299—21no—
15#16837285—21no—
16#16837265C2211yescharacteristic
17C4×C22⋊Dic59C2211yescharacteristic
18#16837136—21no—
19#16837136—21no—
20#16837136—21no—
21#16837136—21no—
22C2×#121×C59C2211yescharacteristic
23C2×#99×C59—21no—
24C2×#99×C59—21no—
25C59×#380C2211yescharacteristic
26C4×C22⋊C4×C59C2211yescharacteristic
27C2×#11745664—21no—
28C2×#11745664—21no—
29C2×C4⋊Dic59—21no—
30C2×C4⋊Dic59—21no—
31C2×Dic59⋊C4—21no—
32C2×Dic59⋊C4—21no—
33C2×Dic59⋊C4—21no—
34C2×Dic59⋊C4—21no—
35C2×C22⋊Dic59—21no—
36C2×C22⋊Dic59—21no—
37C2×C4×Dic59—21no—
38C22⋊(C59⋊C8)—21no—
39C22⋊(C59⋊C8)—21no—
40C22⋊(C59⋊C8)—21no—
41C22⋊(C59⋊C8)—21no—
42C59⋊#95—21no—
43C59⋊#95—21no—
44C59⋊#95—21no—
45C59⋊#95D411yescharacteristic
46C59⋊#95D411yescharacteristic
47C59⋊#95D411yescharacteristic
48C59⋊#95D411yescharacteristic
49C22×D4×C59—21no—
50C22×D4×C59Q811yescharacteristic
51C23×C4×C59C2311yescharacteristic
52#121×C59—21no—
53#121×C59—21no—
54#121×C59—21no—
55#121×C59—21no—
56C2×C4⋊C4×C59—21no—
57C2×C4⋊C4×C59Q811yescharacteristic
58C2×C22⋊C4×C59—21no—
59C2×C22⋊C4×C59—21no—
60C2×C22⋊C4×C59D411yescharacteristic
61C2×C22⋊C4×C59D411yescharacteristic
62C2×C42×C59—21no—
63#99×C59—42no—
64#99×C59—42no—
65#95×C59—21no—
66#95×C59—21no—
67C24×C59—21no—
68C24×C59C2×D411yescharacteristic
69C2×D4×C59—21no—
70C2×D4×C59—21no—
71C2×D4×C59—41no—
72C2×D4×C59—41no—
73C2×D4×C59—41no—
74C2×D4×C59—41no—
75C2×D4×C59—41no—
76C2×D4×C59—41no—
77C2×D4×C59—81no—
78C22×C4×C59—21no—
79C22×C4×C59—21no—
80C22×C4×C59—21no—
81C22×C4×C59—41no—
82C22×C4×C59—81no—
83C22×C4×C59C2×D411yescharacteristic
84C22×C4×C59C2×D411yescharacteristic
85C22×C4×C59C2×Q811yescharacteristic
86C22×C4×C59#4111yescharacteristic
87C22×C4×C59#4111yescharacteristic
88C22×C4×C59#4111yescharacteristic
89C22×Dic59—21no—
90C22×Dic59—21no—
91C22×Dic59—21no—
92C22×Dic59—21no—
93C4⋊C4×C59—21no—
94C4⋊C4×C59—21no—
95C4⋊C4×C59—81no—
96C22⋊C4×C59—22no—
97C22⋊C4×C59—41no—
98C22⋊C4×C59—41no—
99C22⋊C4×C59—41no—
100C22⋊C4×C59—41no—
101C22⋊C4×C59—41no—
102C42×C59—81no—
103C22⋊Dic59—42no—
104C22⋊Dic59—81no—
105C4⋊Dic59—42no—
106C4⋊Dic59—81no—
107Dic59⋊C4—42no—
108Dic59⋊C4—81no—
109Dic59⋊C4—81no—
110Dic59⋊C4—81no—
111C4×Dic59—81no—
112#11745664—81no—
113#11745664—81no—
114C2×C59⋊C8—41no—
115C2×C59⋊C8—41no—
116C23×C59—21no—
117C23×C59—21no—
118C23×C59—41no—
119C23×C59—41no—
120C23×C59—41no—
121C23×C59—41no—
122C23×C59—41no—
123C23×C59—81no—
124C23×C59—81no—
125C23×C59#12011yescharacteristic
126C23×C59C22⋊Q811yescharacteristic
127C23×C59#12411yescharacteristic
128D4×C59—81no—
129D4×C59—81no—
130D4×C59—81no—
131D4×C59—81no—
132D4×C59—81no—
133D4×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—41no—
141C2×C4×C59—41no—
142C2×C4×C59—41no—
143C2×C4×C59—41no—
144C2×C4×C59—42no—
145C2×C4×C59—41no—
146C2×C4×C59—41no—
147C2×C4×C59—41no—
148C2×C4×C59—41no—
149C2×C4×C59—41no—
150C2×C4×C59—81no—
151C2×C4×C59—81no—
152C2×C4×C59C22⋊Q811yescharacteristic
153C2×C4×C59C22⋊Q811yescharacteristic
154C2×C4×C59#12411yescharacteristic
155C2×C4×C59#12411yescharacteristic
156C2×Dic59—22no—
157C2×Dic59—41no—
158C2×Dic59—41no—
159C2×Dic59—41no—
160C2×Dic59—41no—
161C2×Dic59—41no—
162C2×Dic59—41no—
163C2×Dic59—41no—
164C2×Dic59—41no—
165C2×Dic59—41no—
166C2×Dic59—41no—
167C2×Dic59—41no—
168C59⋊C8—81no—
169C59⋊C8—81no—
170#10384—591nomaximal, Sylow
171C22×C59—21no—
172C22×C59—21no—
173C22×C59—21no—
174C22×C59—21no—
175C22×C59—21no—
176C22×C59—21no—
177C22×C59—21no—
178C22×C59—21no—
179C22×C59—81no—
180C22×C59—81no—
181C22×C59—81no—
182C22×C59—81no—
183C22×C59—81no—
184C22×C59—161no—
185C22×C59—161no—
186C22×C59#39311yescharacteristic
187C22×C59#45311yescharacteristic
188C22×C59#45411yescharacteristic
189C4×C59—21no—
190C4×C59—21no—
191C4×C59—21no—
192C4×C59—21no—
193C4×C59—81no—
194C4×C59—81no—
195C4×C59—82no—
196C4×C59—81no—
197Dic59—42no—
198Dic59—81no—
199Dic59—81no—
200Dic59—81no—
201#2667—591no—
202#1964D5911yescharacteristic
203#1510—591no—
204#1507—591no—
205#1305—591no—
206#1297—591no—
207#1284—591no—
208C2×C59—21no—
209C2×C59—21no—
210C2×C59—41no—
211C2×C59—41no—
212C2×C59—161no—
213C2×C59#202711yescharacteristic
214C2×C59#203111yescharacteristic
215C2×C59#203211yescharacteristic
216C2×#121C2×D5911yescharacteristic
217C2×#99—21no—
218C2×#99—21no—
219#407—591no—
220#407—591no—
221#399—1181no—
222#398—1181no—
223#398—1181no—
224#396—1181no—
225#389—1181no—
226#387—1181no—
227#381—591no—
228#380C2×D5911yescharacteristic
229C4×C22⋊C4—591no—
230C4×C22⋊C4C2×D5911yescharacteristic
231#327—1181no—
232#327—1181no—
233#327—1181no—
234#327—1181no—
235C59#1038411yescharacteristic, complemented, Sylow
236C22×D4—21no—
237C22×D4Dic11811yescharacteristic
238C23×C4C22×D5911yescharacteristic
239C2×M16—1181no—
240C2×M16—1181no—
241#121—21no—
242#121—21no—
243#121—21no—
244#121—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—1181no—
247C2×C4⋊C4—1181no—
248C2×C4⋊C4—1181no—
249C2×C4⋊C4—1181no—
250C2×C4⋊C4—1181no—
251C2×C4⋊C4—1181no—
252C2×C4⋊C4Dic11811yescharacteristic
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—1181no—
256C2×C22⋊C4—1181no—
257C2×C22⋊C4C22⋊D5911yescharacteristic
258C2×C22⋊C4C22⋊D5911yescharacteristic
259C2×C42—21no—
260C2×C42—1181no—
261#99—42no—
262#99—42no—
263C22⋊C8—1181no—
264C22⋊C8—1181no—
265C22⋊C8—1181no—
266C22⋊C8—1181no—
267#95—21no—
268#95—21no—
269#95—591no—
270#95—591no—
271#95—591no—
272#95—591no—
273#95—1181no—
274#95—1181no—
275#95—1181no—
276C24—21no—
277C24C2×C22⋊D5911yescharacteristic
278C2×D4—21no—
279C2×D4—21no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—41no—
285C2×D4—41no—
286C2×D4—81no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—41no—
291C22×C4—81no—
292C22×C4—1181no—
293C22×C4—1181no—
294C22×C4—1181no—
295C22×C4—1181no—
296C22×C4C2×Dic11811yescharacteristic
297C22×C4#1174568511yescharacteristic
298C22×C4D4×D5911yescharacteristic
299C22×C4D4×D5911yescharacteristic
300C22×C4#1174568711yescharacteristic
301C22×C4#1174568711yescharacteristic
302M16—4721no—
303M16—4721no—
304C2×C8—2361no—
305C2×C8—2361no—
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4—81no—
309C4⋊C4—4721no—
310C4⋊C4—4721no—
311C4⋊C4—4721no—
312C4⋊C4—4721no—
313C4⋊C4—2362no—
314C4⋊C4—2362no—
315C22⋊C4—22no—
316C22⋊C4—41no—
317C22⋊C4—41no—
318C22⋊C4—41no—
319C22⋊C4—41no—
320C22⋊C4—41no—
321C22⋊C4—4721no—
322C22⋊C4—2362no—
323C42—81no—
324C42—4721no—
325C23—21no—
326C23—21no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—81no—
333C23—81no—
334C23#1427369511yescharacteristic
335C23#1427370611yescharacteristic
336C23#1427371411yescharacteristic
337D4—81no—
338D4—81no—
339D4—81no—
340D4—81no—
341D4—81no—
342D4—81no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—1182no—
362C2×C4—2361no—
363C2×C4—2361no—
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#1427370411yescharacteristic
374C2×C4#1427370411yescharacteristic
375C2×C4#1427371311yescharacteristic
376C2×C4#1427371311yescharacteristic
377C8—4721no—
378C8—4721no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—81no—
388C22—81no—
389C22—81no—
390C22—81no—
391C22—81no—
392C22—161no—
393C22—161no—
394C22#1683731411yescharacteristic
395C22#1683756911yescharacteristic
396C22#1683757411yescharacteristic
397C4—21no—
398C4—21no—
399C4—21no—
400C4—21no—
401C4—81no—
402C4—81no—
403C4—82no—
404C4—81no—
405C4—4721no—
406C4—4721no—
407C4—4721no—
408C4—2362no—
409C2—21no—
410C2—21no—
411C2—41no—
412C2—41no—
413C2—161no—
414C2#2142525211yescharacteristic
415C2#2142527011yescharacteristic
416C2#2142527311yescharacteristic
subgroupG/H
centerC22#16837314
commutator, cosocleC23×C4×C59C23
FrattiniC23×C4C22×D59
FittingC59×#1964C2
socleC22×C59#393
prime radicalC2×#121×C59C22
hypercenter#1964D59
hypocenterC59#10384
LCSG ▷ C23×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1964

Complements

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

NH ≅ G/Ntype
1. #21430293C2semidirect
4. #21422002C2semidirect
5. #21421982C2semidirect
6. #21421949C2semidirect
235. C59#10384semidirect

Maximal subgroups

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

subgroupindex
#214302932
#214226252
#214226122
#214220022
#214219822
#214219492
C59×#19642
#1038459