Finite Group Databasebeta

← #28562816

Subgroups — #28562816

Order 15104.

1058 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 1058 conjugacy classes of subgroups — too many to draw. Only its 183 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#21434917C212yesmaximal, complemented
2#21429254C211yesmaximal, characteristic
3#21423087C214yesmaximal, complemented
4#21423053C214yesmaximal, complemented
5#21422616C212yesmaximal, complemented
6#21422423C211yesmaximal, characteristic
7C2×C59×#543C211yesmaximal, characteristic
8C2×#14273706C2211yescharacteristic
9#16837732—22no—
10#16837730C2214yescomplemented
11#16837658—24no—
12#16837558—22no—
13#16837557C2214yescomplemented
14#16837366C2212yes—
15#16837327C2212yes—
16#16837314C2212yes—
17#16837296—22no—
18#16837265—21no—
19#16837258—21no—
20C4×C22⋊Dic59C2211yescharacteristic
21#16837139C2218yescomplemented
22#16837126C2214yes—
23C59×#543—24no—
24C59×#543—22no—
25C59×#543C2214yes—
26C2×C22⋊Q8×C59—22no—
27C2×C22⋊Q8×C59—21no—
28C2×C22⋊Q8×C59—22no—
29C2×C22⋊Q8×C59C2212yes—
30C22×C22⋊C4×C59—21no—
31C22×C22⋊C4×C59C2211yescharacteristic
32C22×Dic118C2311yescharacteristic
33C2×#11745672—24no—
34C2×#11745664—22no—
35#14273706—22no—
36#14273706—22no—
37C2×C4⋊Dic59C2311yescharacteristic
38C2×Dic59⋊C4—21no—
39C2×C22⋊Dic59—21no—
40C2×C22⋊Dic59—21no—
41C2×C22⋊Dic59—22no—
42C2×C22⋊Dic59—22no—
43C2×C4×Dic59—21no—
44#14273658—24no—
45#14273651—24no—
46#14273650—24no—
47C22⋊(C59⋊C8)—22no—
48C22⋊(C59⋊C8)C2314yes—
49C59⋊#95—22no—
50C59⋊#95—22no—
51C59⋊#95C2312yes—
52C25×C59D411yescharacteristic
53C22×Q8×C59—21no—
54C22×Q8×C59D412yes—
55C23×C4×C59—21no—
56C23×C4×C59—21no—
57C23×C4×C59D411yescharacteristic
58C23×C4×C59C2311yescharacteristic
59C22⋊Q8×C59—28no—
60C22⋊Q8×C59—28no—
61C22⋊Q8×C59—24no—
62C22⋊Q8×C59—28no—
63C22⋊Q8×C59—24no—
64C22⋊Q8×C59—22no—
65C22⋊Q8×C59—22no—
66C22⋊Q8×C59—28no—
67C22⋊Q8×C59D414yes—
68C22⋊Q8×C59C2314yes—
69C2×C4⋊C4×C59—21no—
70C2×C4⋊C4×C59—22no—
71C2×C4⋊C4×C59—22no—
72C2×C4⋊C4×C59D412yes—
73C2×C22⋊C4×C59—22no—
74C2×C22⋊C4×C59—24no—
75C2×C22⋊C4×C59—22no—
76C2×C22⋊C4×C59—24no—
77C2×C22⋊C4×C59—22no—
78C2×C22⋊C4×C59—22no—
79C2×C22⋊C4×C59D412yes—
80C2×C22⋊C4×C59C2312yes—
81C24×C59—21no—
82C24×C59—21no—
83C24×C59—24no—
84C24×C59—42no—
85C24×C59—81no—
86C24×C59C2×D411yescharacteristic
87C24×C59C2×D412yes—
88C2×Q8×C59—24no—
89C2×Q8×C59—22no—
90C2×Q8×C59—82no—
91C2×Q8×C59—82no—
92C2×Q8×C59C2×D414yes—
93C22×C4×C59—22no—
94C22×C4×C59—21no—
95C22×C4×C59—22no—
96C22×C4×C59—24no—
97C22×C4×C59—24no—
98C22×C4×C59—21no—
99C22×C4×C59—42no—
100C22×C4×C59—81no—
101C22×C4×C59—82no—
102C22×C4×C59—82no—
103C22×C4×C59C2×D412yes—
104C22×C4×C59C2×D411yescharacteristic
105C22×C4×C59C2×D412yes—
106C22×C4×C59C2×D412yes—
107C22×C4×C59C2411yescharacteristic
108C22×Dic59—22no—
109C22×Dic59—21no—
110C22×Dic59—21no—
111C2×Dic118—22no—
112C2×Dic118—81no—
113C4⋊C4×C59—24no—
114C4⋊C4×C59—28no—
115C4⋊C4×C59—24no—
116C4⋊C4×C59—22no—
117C4⋊C4×C59—24no—
118C4⋊C4×C59C2×D414yes—
119C22⋊C4×C59—28no—
120C22⋊C4×C59—24no—
121C22⋊C4×C59—24no—
122C22⋊C4×C59—24no—
123C22⋊C4×C59—24no—
124C22⋊Dic59—42no—
125C22⋊Dic59—41no—
126C22⋊Dic59—41no—
127C22⋊Dic59—44no—
128C22⋊Dic59—82no—
129#11745672—84no—
130C4⋊Dic59—22no—
131Dic59⋊C4—42no—
132C4×Dic59—81no—
133#11745664—82no—
134C2×C59⋊C8—24no—
135C23×C59—24no—
136C23×C59—24no—
137C23×C59—22no—
138C23×C59—22no—
139C23×C59—22no—
140C23×C59—22no—
141C23×C59—41no—
142C23×C59—42no—
143C23×C59—42no—
144C23×C59—42no—
145C23×C59—42no—
146C23×C59—81no—
147C23×C59—81no—
148C23×C59—88no—
149C23×C59#12011yescharacteristic
150C23×C59#12011yescharacteristic
151C23×C59#13714yes—
152C23×C59C22×D411yescharacteristic
153Q8×C59—44no—
154Q8×C59—82no—
155Q8×C59—84no—
156C2×C4×C59—24no—
157C2×C4×C59—22no—
158C2×C4×C59—24no—
159C2×C4×C59—42no—
160C2×C4×C59—41no—
161C2×C4×C59—41no—
162C2×C4×C59—42no—
163C2×C4×C59—44no—
164C2×C4×C59—42no—
165C2×C4×C59—44no—
166C2×C4×C59—42no—
167C2×C4×C59—81no—
168C2×C4×C59—84no—
169C2×C4×C59—84no—
170C2×C4×C59#12012yes—
171C2×C4×C59C22×D412yes—
172C2×Dic59—42no—
173C2×Dic59—42no—
174C2×Dic59—42no—
175C2×Dic59—41no—
176C2×Dic59—41no—
177C2×Dic59—42no—
178C2×Dic59—42no—
179Dic118—44no—
180C59⋊C8—44no—
181#18319—591nomaximal, Sylow
182C22×C59—21no—
183C22×C59—21no—
184C22×C59—21no—
185C22×C59—21no—
186C22×C59—21no—
187C22×C59—24no—
188C22×C59—21no—
189C22×C59—21no—
190C22×C59—44no—
191C22×C59—44no—
192C22×C59—48no—
193C22×C59—42no—
194C22×C59—42no—
195C22×C59—82no—
196C22×C59—84no—
197C22×C59#45512yes—
198C22×C59C2×#12011yescharacteristic
199C22×C59C2×#13712yes—
200C4×C59—22no—
201C4×C59—22no—
202C4×C59—42no—
203C4×C59—44no—
204C4×C59—44no—
205Dic59—82no—
206Dic59—81no—
207Dic59—81no—
208C2×#543D5911yescharacteristic
209#3195—592no—
210#2561—591no—
211#1608—594no—
212#1602—594no—
213#1508—592no—
214#1462—591no—
215C2×C59—22no—
216C2×C59—22no—
217C2×C59—41no—
218C2×C59—44no—
219C2×C59#300411yescharacteristic
220C2×C59C2×#45511yescharacteristic
221C2×C59#302611yescharacteristic
222#543—24no—
223#543—22no—
224#543C2×D5914yes—
225C2×C22⋊Q8—22no—
226C2×C22⋊Q8—21no—
227C2×C22⋊Q8—22no—
228C2×C22⋊Q8—591no—
229C2×C22⋊Q8C2×D5912yes—
230C22×C22⋊C4—21no—
231C22×C22⋊C4C2×D5911yescharacteristic
232#483—594no—
233#483—1182no—
234#470—1184no—
235#451—594no—
236#451—1182no—
237#407—592no—
238#397—592no—
239#393—592no—
240#388—1182no—
241#381—1181no—
242#379—1181no—
243C4×C22⋊C4—591no—
244#328—598no—
245#323—594no—
246C59#1831911yescharacteristic, complemented, Sylow
247C25C22⋊D5911yescharacteristic
248C22×Q8—21no—
249C22×Q8—591no—
250C22×Q8C22⋊D5912yes—
251C23×C4—21no—
252C23×C4—21no—
253C23×C4C22⋊D5911yescharacteristic
254C23×C4C22×D5911yescharacteristic
255C2×Q16—1184no—
256C2×M16—1182no—
257C22⋊Q8—28no—
258C22⋊Q8—28no—
259C22⋊Q8—24no—
260C22⋊Q8—28no—
261C22⋊Q8—24no—
262C22⋊Q8—22no—
263C22⋊Q8—22no—
264C22⋊Q8—28no—
265C22⋊Q8—1182no—
266C22⋊Q8—1182no—
267C22⋊Q8C22⋊D5914yes—
268C22⋊Q8C22×D5914yes—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—591no—
273C2×C4⋊C4—1181no—
274C2×C4⋊C4C22⋊D5912yes—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—1182no—
282C2×C22⋊C4—1181no—
283C2×C22⋊C4—1181no—
284C2×C22⋊C4—1182no—
285C2×C22⋊C4C22⋊D5912yes—
286C2×C22⋊C4C22×D5912yes—
287C2×C42—1181no—
288#106—1184no—
289#103—1184no—
290#103—1184no—
291C22⋊C8—594no—
292C22⋊C8—1182no—
293#95—592no—
294#95—1182no—
295#95—1182no—
296C24—21no—
297C24—21no—
298C24—24no—
299C24—42no—
300C24—81no—
301C24C2×C22⋊D5911yescharacteristic
302C24C2×C22⋊D5912yes—
303C2×Q8—24no—
304C2×Q8—22no—
305C2×Q8—82no—
306C2×Q8—82no—
307C2×Q8—1182no—
308C2×Q8—4721no—
309C2×Q8C2×C22⋊D5914yes—
310C22×C4—22no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—21no—
316C22×C4—42no—
317C22×C4—81no—
318C22×C4—82no—
319C22×C4—82no—
320C22×C4—1182no—
321C22×C4—1181no—
322C22×C4—1181no—
323C22×C4C2×C22⋊D5912yes—
324C22×C4C2×C22⋊D5911yescharacteristic
325C22×C4C2×C22⋊D5912yes—
326C22×C4C2×C22⋊D5912yes—
327C22×C4C23×D5911yescharacteristic
328Q16—4724no—
329M16—4722no—
330C2×C8—1184no—
331C4⋊C4—24no—
332C4⋊C4—28no—
333C4⋊C4—24no—
334C4⋊C4—22no—
335C4⋊C4—24no—
336C4⋊C4—1182no—
337C4⋊C4—2362no—
338C4⋊C4C2×C22⋊D5914yes—
339C22⋊C4—28no—
340C22⋊C4—24no—
341C22⋊C4—24no—
342C22⋊C4—24no—
343C22⋊C4—24no—
344C22⋊C4—4722no—
345C22⋊C4—2362no—
346C22⋊C4—2361no—
347C22⋊C4—2361no—
348C22⋊C4—2364no—
349C42—4721no—
350C23—24no—
351C23—24no—
352C23—22no—
353C23—22no—
354C23—22no—
355C23—22no—
356C23—41no—
357C23—42no—
358C23—42no—
359C23—42no—
360C23—42no—
361C23—81no—
362C23—81no—
363C23—88no—
364C23#1427369611yescharacteristic
365C23#1427369611yescharacteristic
366C23#1427375814yes—
367C23C22×C22⋊D5911yescharacteristic
368Q8—44no—
369Q8—82no—
370Q8—84no—
371Q8—2364no—
372C2×C4—24no—
373C2×C4—22no—
374C2×C4—24no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—41no—
379C2×C4—44no—
380C2×C4—42no—
381C2×C4—44no—
382C2×C4—42no—
383C2×C4—81no—
384C2×C4—84no—
385C2×C4—84no—
386C2×C4—2362no—
387C2×C4—2362no—
388C2×C4—2362no—
389C2×C4—2361no—
390C2×C4—2361no—
391C2×C4—2362no—
392C2×C4—2362no—
393C2×C4#1427369612yes—
394C2×C4C22×C22⋊D5912yes—
395C8—2364no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—24no—
404C22—44no—
405C22—44no—
406C22—48no—
407C22—42no—
408C22—42no—
409C22—82no—
410C22—84no—
411C22#1683758012yes—
412C22C2×#1427369611yescharacteristic
413C22C2×#1427375812yes—
414C4—22no—
415C4—22no—
416C4—42no—
417C4—44no—
418C4—44no—
419C4—4722no—
420C4—4721no—
421C4—4721no—
422C2—22no—
423C2—22no—
424C2—41no—
425C2—44no—
426C2#2143333011yescharacteristic
427C2C2×#1683758011yescharacteristic
428C2#2143349311yescharacteristic
subgroupG/H
centerC22C2×#14273696
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC2×C59×#543C2
socleC22×C59C2×#120
prime radicalC25×C59D4
hypercenterC2×#543D59
hypocenterC59#18319
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#543

Complements

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

NH ≅ G/Ntype
1. #21434917C2semidirect
3. #21423087C2semidirect
4. #21423053C2semidirect
5. #21422616C2semidirect
10. #16837730C22semidirect
13. #16837557C22semidirect
21. #16837139C22semidirect
246. C59#18319semidirect

Maximal subgroups

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

subgroupindex
#214349172
#214292542
#214230872
#214230532
#214226162
#214224232
C2×C59×#5432
#1831959