Finite Group Databasebeta

← #28490715

Subgroups — #28490715

Order 15104.

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

normal subgroups only

This group has 528 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#21435240C211yesmaximal, characteristic, complemented
2#21433893C211yesmaximal, characteristic
3#21422768C211yesmaximal, characteristic, complemented
4#21422498C211yesmaximal, characteristic, complemented
5#21422417C211yesmaximal, characteristic
6#21421993C211yesmaximal, characteristic, complemented
7C2×C59×#461C211yesmaximal, characteristic
8#16837959C2211yescharacteristic
9#16837798—21no—
10#16837754—21no—
11#16837732—22no—
12#16837702—22no—
13#16837692—21no—
14#16837684—22no—
15#16837682—21no—
16#16837455C2211yescharacteristic
17#16837455C2211yescharacteristic
18#16837442—21no—
19#16837426C2211yescharacteristic
20#16837407—21no—
21#16837148—22no—
22#16837132—22no—
23#16837126—22no—
24C22×SD16×C59—21no—
25C2×C4⋊Q8×C59C2211yescharacteristic
26C2×C4×D4×C59C2211yescharacteristic
27C59×#461—22no—
28C59×#461—22no—
29C2×C4⋊C8×C59C2211yescharacteristic
30C2×#103×C59—21no—
31C2×#11745664—21no—
32#14273727D411yescharacteristic
33#14273727D411yescharacteristic
34#14273719—21no—
35#14273706—21no—
36#14273706—21no—
37C4×Dic118—21no—
38C4×Dic118—21no—
39#14273686—21no—
40#14273660—22no—
41#14273658—24no—
42#14273658—22no—
43C4⋊(C59⋊C8)—21no—
44C4⋊(C59⋊C8)—21no—
45C4⋊(C59⋊C8)—21no—
46C4⋊(C59⋊C8)D412yes—
47#14273650—24no—
48#14273648—22no—
49#14273648—22no—
50C22⋊(C59⋊C8)—21no—
51C22⋊(C59⋊C8)D412yes—
52C4×C59⋊C8—21no—
53C4×C59⋊C8—21no—
54C22×Q8×C59—21no—
55C22×D4×C59D411yescharacteristic
56C23×C4×C59—21no—
57C2×SD16×C59—24no—
58C2×SD16×C59—42no—
59C2×SD16×C59—42no—
60C22×C8×C59—21no—
61C4⋊Q8×C59—21no—
62C4⋊Q8×C59—21no—
63C4⋊Q8×C59—21no—
64C4⋊Q8×C59D412yes—
65C4×D4×C59—21no—
66C4×D4×C59—21no—
67C4×D4×C59—21no—
68C4×D4×C59—21no—
69C2×C4⋊C4×C59—21no—
70C2×C4⋊C4×C59—21no—
71C2×C4⋊C4×C59D411yescharacteristic
72C2×C22⋊C4×C59—21no—
73C2×C42×C59C2311yescharacteristic
74C4⋊C8×C59—21no—
75C4⋊C8×C59D412yes—
76#103×C59—24no—
77C24×C59—21no—
78C2×Q8×C59—22no—
79C2×Q8×C59—41no—
80C2×Q8×C59—41no—
81C2×Q8×C59—41no—
82C2×Q8×C59—41no—
83C2×Q8×C59—41no—
84C2×D4×C59—21no—
85C2×D4×C59—21no—
86C2×D4×C59—21no—
87C2×D4×C59—21no—
88C2×D4×C59—21no—
89C2×D4×C59SD1612yes—
90C22×C4×C59—21no—
91C22×C4×C59—41no—
92C22×C4×C59—41no—
93C22×C4×C59—41no—
94C22×C4×C59—82no—
95C22×C4×C59C2×D411yescharacteristic
96C22×C4×C59C2×D411yescharacteristic
97C22×C4×C59#4111yescharacteristic
98C2×Dic118—21no—
99C2×Dic118—21no—
100SD16×C59—48no—
101C2×C8×C59—22no—
102C2×C8×C59—41no—
103C2×C8×C59—41no—
104C4⋊C4×C59—21no—
105C4⋊C4×C59—22no—
106C4⋊C4×C59—21no—
107C4⋊C4×C59—22no—
108C4⋊C4×C59—41no—
109C4⋊C4×C59—41no—
110C22⋊C4×C59—41no—
111C22⋊C4×C59—41no—
112C42×C59C2×D411yescharacteristic
113C42×C59C2×D411yescharacteristic
114C42×C59C2×D411yescharacteristic
115C42×C59C2×D411yescharacteristic
116C22⋊Dic59—41no—
117C4⋊Dic59—21no—
118C4⋊Dic59—21no—
119C4⋊Dic59—21no—
120C4⋊Dic59—21no—
121Dic59⋊C4—41no—
122Dic59⋊C4—41no—
123C4×Dic59—41no—
124#11745664—41no—
125#11745664—41no—
126C2×C59⋊C8—22no—
127C2×C59⋊C8—22no—
128C2×C59⋊C8—41no—
129C23×C59—22no—
130C23×C59—41no—
131C23×C59—41no—
132C23×C59—81no—
133C23×C59—81no—
134C23×C59#12111yescharacteristic
135Q8×C59—42no—
136Q8×C59—42no—
137Q8×C59—81no—
138Q8×C59—81no—
139D4×C59—22no—
140D4×C59—22no—
141D4×C59—41no—
142D4×C59—41no—
143C2×C4×C59—22no—
144C2×C4×C59—41no—
145C2×C4×C59—41no—
146C2×C4×C59—42no—
147C2×C4×C59—82no—
148C2×C4×C59—81no—
149C2×C4×C59—81no—
150C2×C4×C59—82no—
151C2×C4×C59—81no—
152C2×C4×C59#12011yescharacteristic
153C2×C4×C59#12011yescharacteristic
154C2×C4×C59#12011yescharacteristic
155C2×C4×C59#12111yescharacteristic
156C2×C4×C59#12111yescharacteristic
157C2×C4×C59#12711yescharacteristic
158C2×C4×C59C2×SD1611yescharacteristic
159C2×C4×C59#13511yescharacteristic
160C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
161C2×C4×C59#13711yescharacteristic
162C2×Dic59—41no—
163C2×Dic59—41no—
164Dic118—42no—
165Dic118—42no—
166C8×C59—42no—
167C59⋊C8—42no—
168C59⋊C8—42no—
169C59⋊C8—81no—
170#10186—591nomaximal, Sylow
171C22×C59—21no—
172C22×C59—21no—
173C22×C59—42no—
174C22×C59—82no—
175C22×C59—82no—
176C22×C59—81no—
177C22×C59—81no—
178C22×C59#39211yescharacteristic
179C22×C59#46511yescharacteristic
180C22×C59#46811yescharacteristic
181C4×C59—21no—
182C4×C59—21no—
183C4×C59—21no—
184C4×C59—82no—
185C4×C59—81no—
186C4×C59#44811yescharacteristic
187C4×C59#44911yescharacteristic
188C4×C59#45011yescharacteristic
189C4×C59#45211yescharacteristic
190C4×C59#47112yes—
191Dic59—81no—
192Dic59—81no—
193#3229—591no—
194#3077—591no—
195C2×#461D5911yescharacteristic
196#1546—591no—
197#1481—591no—
198#1460—591no—
199#1301—591no—
200C2×C59—21no—
201C2×C59—21no—
202C2×C59—82no—
203C2×C59#200011yescharacteristic
204C2×C59#200911yescharacteristic
205C2×C59#201911yescharacteristic
206C22×SD16—21no—
207#533—591no—
208C2×C4⋊Q8C2×D5911yescharacteristic
209C2×C4×D4C2×D5911yescharacteristic
210#498—1181no—
211#487—1181no—
212#483—1182no—
213#478—1182no—
214#476—1181no—
215#475—1182no—
216#474—1181no—
217#461—22no—
218#461—22no—
219#431—591no—
220#431—591no—
221#428—1181no—
222#423—591no—
223C2×C4⋊C8C2×D5911yescharacteristic
224#418—1181no—
225C2×#103—21no—
226#332—1182no—
227Q8⋊C8—1182no—
228#323—1182no—
229C59#1018611yescharacteristic, complemented, Sylow
230C22×Q8—21no—
231C22×D4C22⋊D5911yescharacteristic
232C23×C4—21no—
233C2×SD16—24no—
234C2×SD16—42no—
235C2×SD16—42no—
236C2×M16—1181no—
237C22×C8—21no—
238C4⋊Q8—21no—
239C4⋊Q8—21no—
240C4⋊Q8—21no—
241C4⋊Q8—591no—
242C4⋊Q8—591no—
243C4⋊Q8C22⋊D5912yes—
244#125—1181no—
245C22⋊Q8—1181no—
246C22⋊Q8—1181no—
247C4×Q8—1181no—
248C4×Q8—1181no—
249C4×D4—21no—
250C4×D4—21no—
251C4×D4—21no—
252C4×D4—21no—
253#117—1181no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4C22⋊D5911yescharacteristic
257C2×C22⋊C4—21no—
258C2×C42C22×D5911yescharacteristic
259#107—1182no—
260#106—1182no—
261#106—1184no—
262C4⋊C8—21no—
263C4⋊C8—592no—
264C4⋊C8—1181no—
265C4⋊C8—1181no—
266C4⋊C8—1181no—
267C4⋊C8C22⋊D5912yes—
268#103—24no—
269#103—1184no—
270#102—1182no—
271#102—1182no—
272C22⋊C8—592no—
273C22⋊C8—1181no—
274C4×C8—1181no—
275C4×C8—1181no—
276C24—21no—
277C2×Q8—22no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×Q8—41no—
283C2×Q8—1181no—
284C2×Q8—1181no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—21no—
290C2×D4#1174567012yes—
291C22×C4—21no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—82no—
296C22×C4D4×D5911yescharacteristic
297C22×C4#1174568711yescharacteristic
298C22×C4C2×C22⋊D5911yescharacteristic
299SD16—48no—
300M16—2361no—
301M16—2361no—
302C2×C8—22no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—1182no—
306C2×C8—1182no—
307C2×C8—2361no—
308C4⋊C4—22no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—22no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—1181no—
315C4⋊C4—1181no—
316C4⋊C4—1181no—
317C4⋊C4—1181no—
318C4⋊C4—2361no—
319C4⋊C4—2361no—
320C22⋊C4—41no—
321C22⋊C4—41no—
322C22⋊C4—2361no—
323C42—2361no—
324C42D4×D5911yescharacteristic
325C42D4×D5911yescharacteristic
326C42C2×C22⋊D5911yescharacteristic
327C42C2×C22⋊D5911yescharacteristic
328C23—22no—
329C23—41no—
330C23—41no—
331C23—81no—
332C23—81no—
333C23#1427369911yescharacteristic
334Q8—42no—
335Q8—42no—
336Q8—81no—
337Q8—81no—
338Q8—2362no—
339Q8—2362no—
340D4—22no—
341D4—22no—
342D4—41no—
343D4—41no—
344C2×C4—22no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—82no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—82no—
352C2×C4—81no—
353C2×C4—2361no—
354C2×C4—2361no—
355C2×C4#1427369511yescharacteristic
356C2×C4#1427369511yescharacteristic
357C2×C4#1427369611yescharacteristic
358C2×C4#1427369811yescharacteristic
359C2×C4#1427369911yescharacteristic
360C2×C4#1427372311yescharacteristic
361C2×C4C2×#1174567011yescharacteristic
362C2×C4#1427374711yescharacteristic
363C2×C4#1427375311yescharacteristic
364C2×C4#1427375711yescharacteristic
365C8—42no—
366C8—4721no—
367C8—2362no—
368C8—2362no—
369C22—21no—
370C22—21no—
371C22—42no—
372C22—82no—
373C22—82no—
374C22—81no—
375C22—81no—
376C22#1683731011yescharacteristic
377C22#1683762911yescharacteristic
378C22#1683764611yescharacteristic
379C4—21no—
380C4—21no—
381C4—21no—
382C4—81no—
383C4—82no—
384C4—4721no—
385C4—4721no—
386C4#1683754011yescharacteristic
387C4#1683754511yescharacteristic
388C4#1683755311yescharacteristic
389C4#1683756411yescharacteristic
390C4#1683766212yes—
391C2—21no—
392C2—21no—
393C2—82no—
394C2#2142509911yescharacteristic
395C2#2142515411yescharacteristic
396C2#2142520811yescharacteristic
subgroupG/H
centerC22#16837310
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
FittingC2×C59×#461C2
socleC22×C59#392
prime radicalC22×D4×C59D4
hypercenterC2×#461D59
hypocenterC59#10186
LCSG ▷ C2×C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#461

Complements

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

NH ≅ G/Ntype
1. #21435240C2semidirect
3. #21422768C2semidirect
4. #21422498C2semidirect
6. #21421993C2semidirect
229. C59#10186semidirect

Maximal subgroups

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

subgroupindex
#214352402
#214338932
#214227682
#214224982
#214224172
#214219932
C2×C59×#4612
#1018659