Finite Group Databasebeta

← #28484439

Subgroups — #28484439

Order 15104.

1088 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 1088 conjugacy classes of subgroups — too many to draw. Only its 147 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#21432378C211yesmaximal, characteristic
2#21424006C211yesmaximal, characteristic, complemented
3C2×#16837135C212yesmaximal, complemented
4#21421975C212yesmaximal, complemented
5C2×C4×C22⋊C4×C59C211yesmaximal, characteristic, complemented
6C2×#14273701C412yescomplemented
7C2×#14273701C2211yescharacteristic
8C2×#14273696—21no—
9C2×C22⋊(C59⋊C8)—22no—
10C2×C22⋊(C59⋊C8)C2212yes—
11#16837281—22no—
12#16837281—21no—
13#16837262C412yescomplemented
14#16837262C2211yescharacteristic
15#16837135—24no—
16C24×C4×C59C2211yescharacteristic
17C22×C22⋊C4×C59—21no—
18C22×C42×C59—21no—
19C4×C22⋊C4×C59—24no—
20C2×#95×C59C2212yes—
21C22×D236C2×C411yescharacteristic
22C22×C22⋊D59—22no—
23C22×C22⋊D59—21no—
24C22×C59⋊C8—22no—
25#14273701—22no—
26#14273701—22no—
27#14273701—44no—
28#14273696—44no—
29C2×C4⋊Dic59C2×C411yescharacteristic
30C2×C4⋊Dic59C2×C412yes—
31C2×C22⋊Dic59—22no—
32C2×C22⋊Dic59—21no—
33C2×(C2×D59)⋊C4—22no—
34C2×(C2×D59)⋊C4—21no—
35C22⋊(C59⋊C8)—22no—
36C22⋊(C59⋊C8)—22no—
37C22⋊(C59⋊C8)—44no—
38C59⋊#95—22no—
39C59⋊#95—21no—
40C25×C59D411yescharacteristic
41C23×C4×C59—22no—
42C23×C4×C59—22no—
43C23×C4×C59—22no—
44C23×C4×C59—44no—
45C23×C4×C59C2×C412yes—
46C23×C4×C59D412yes—
47C23×C4×C59D411yescharacteristic
48C23×C4×C59C2311yescharacteristic
49C2×C22⋊C4×C59—24no—
50C2×C22⋊C4×C59—28no—
51C2×C42×C59—24no—
52C2×C42×C59—44no—
53#95×C59—22no—
54#95×C59D414yescomplemented
55C24×C59—24no—
56C24×C59—21no—
57C24×C59—21no—
58C24×C59—22no—
59C24×C59—22no—
60C24×C59—42no—
61C24×C59C22⋊C412yes—
62C24×C59C2×D411yescharacteristic
63C23×D59—21no—
64C22×C4×C59—21no—
65C22×C4×C59—21no—
66C22×C4×C59—24no—
67C22×C4×C59—28no—
68C22×C4×C59—24no—
69C22×C4×C59—24no—
70C22×C4×C59—24no—
71C22×C4×C59—24no—
72C22×C4×C59—24no—
73C22×C4×C59—42no—
74C22×C4×C59—42no—
75C22×C4×C59—42no—
76C22×C4×C59—48no—
77C22×C4×C59—48no—
78C22×C4×C59—44no—
79C22×C4×C59—42no—
80C22×C4×C59—42no—
81C22×C4×C59C22⋊C412yes—
82C22×C4×C59C22×C411yescharacteristic
83C22×C4×C59C2×D411yescharacteristic
84C22×C4×C59C2×D412yes—
85C22×C4×C59#4112yes—
86C2×C22⋊D59—44no—
87C2×C22⋊D59—44no—
88C2×C22⋊D59—44no—
89C2×C22⋊D59—41no—
90C2×C22⋊D59—41no—
91C2×C22⋊D59—42no—
92C2×C22⋊D59—42no—
93C22×Dic59—21no—
94C22×Dic59—22no—
95C2×D236—22no—
96C2×D236—42no—
97C22⋊C4×C59—216no—
98C42×C59—48no—
99C22⋊Dic59—44no—
100C22⋊Dic59—42no—
101(C2×D59)⋊C4—44no—
102(C2×D59)⋊C4—42no—
103C4⋊Dic59—22no—
104C4⋊Dic59—42no—
105C2×C59⋊C8—44no—
106C2×C59⋊C8—42no—
107C23×C59—22no—
108C23×C59—22no—
109C23×C59—24no—
110C23×C59—24no—
111C23×C59—22no—
112C23×C59—22no—
113C23×C59—22no—
114C23×C59—22no—
115C23×C59—22no—
116C23×C59—22no—
117C23×C59—48no—
118C23×C59—48no—
119C23×C59—42no—
120C23×C59—42no—
121C23×C59—42no—
122C23×C59—42no—
123C23×C59#10414yes—
124C23×C59C2×C22⋊C411yescharacteristic
125C23×C59#12011yescharacteristic
126C23×C59#12311yescharacteristic
127C23×C59#13512yes—
128C23×C59AGL(1,ℤ/8)12yes—
129C22×D59—41no—
130C22×D59—42no—
131C22×D59—41no—
132C22×D59—41no—
133C22×D59—42no—
134C2×C4×C59—24no—
135C2×C4×C59—24no—
136C2×C4×C59—22no—
137C2×C4×C59—22no—
138C2×C4×C59—22no—
139C2×C4×C59—22no—
140C2×C4×C59—28no—
141C2×C4×C59—42no—
142C2×C4×C59—42no—
143C2×C4×C59—48no—
144C2×C4×C59—48no—
145C2×C4×C59—44no—
146C2×C4×C59—48no—
147C2×C4×C59—48no—
148C2×C4×C59—44no—
149C2×C4×C59#9914yes—
150C2×C4×C59C4×D412yes—
151C2×C4×C59#12112yes—
152C22⋊D59—88no—
153C22⋊D59—82no—
154C22⋊D59—82no—
155C2×Dic59—41no—
156C2×Dic59—42no—
157C2×Dic59—41no—
158C2×Dic59—41no—
159C2×Dic59—42no—
160C2×Dic59—42no—
161D236—82no—
162C59⋊C8—82no—
163#9150—591nomaximal, Sylow
164C22×C59—21no—
165C22×C59—21no—
166C22×C59—21no—
167C22×C59—21no—
168C22×C59—24no—
169C22×C59—24no—
170C22×C59—24no—
171C22×C59—24no—
172C22×C59—24no—
173C22×C59—48no—
174C22×C59—48no—
175C22×C59—42no—
176C22×C59—42no—
177C22×C59—42no—
178C22×C59—42no—
179C22×C59#38611yescharacteristic
180C22×C59C2×#9912yes—
181C22×C59#41611yescharacteristic
182C22×C59#41811yescharacteristic
183C22×C59C2×#10412yes—
184C22×C59#44912yes—
185C22×C59#48212yes—
186C2×D59—81no—
187C2×D59—81no—
188C2×D59—81no—
189C2×D59—82no—
190C2×D59—82no—
191C4×C59—22no—
192C4×C59—22no—
193C4×C59—24no—
194C4×C59—48no—
195Dic59—81no—
196Dic59—82no—
197#2847—591no—
198C2×C4×C22⋊C4D5911yescharacteristic, complemented
199#1779—591no—
200C2×#327—592no—
201#1294—592no—
202C2×C59—22no—
203C2×C59—22no—
204C2×C59—44no—
205C2×C59#150112yes—
206C2×C59#150712yes—
207C2×C59#187511yescharacteristic
208C2×C59#189611yescharacteristic
209C2×C59#189811yescharacteristic
210D59—161no—
211C24×C4C2×D5911yescharacteristic
212C2×#121—591no—
213C2×#121—592no—
214C2×#120—1181no—
215C22×C22⋊C4—21no—
216C22×C42—21no—
217C2×C22⋊C8—592no—
218C2×C22⋊C8—1182no—
219#386—1182no—
220#386—1181no—
221#380—592no—
222#380—591no—
223C4×C22⋊C4—24no—
224C2×#95C2×D5912yes—
225#327—1184no—
226C59#915011yescharacteristic, complemented, Sylow
227C25C22⋊D5911yescharacteristic
228C22×D4—591no—
229C22×D4—1182no—
230C22×D4—1181no—
231C23×C4—22no—
232C23×C4—22no—
233C23×C4—22no—
234C23×C4—44no—
235C23×C4C4×D5912yes—
236C23×C4D23612yes—
237C23×C4C22⋊D5911yescharacteristic
238C23×C4C22×D5911yescharacteristic
239C22×C8—1182no—
240#121—1182no—
241#121—1182no—
242#121—2364no—
243#120—2364no—
244C2×C4⋊C4—591no—
245C2×C4⋊C4—592no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—1182no—
249C2×C22⋊C4—1181no—
250C2×C22⋊C4—1182no—
251C2×C22⋊C4—1181no—
252C2×C42—24no—
253C2×C42—44no—
254C22⋊C8—1182no—
255C22⋊C8—1182no—
256C22⋊C8—2364no—
257#95—22no—
258#95—1182no—
259#95—1181no—
260#95C22⋊D5914yescomplemented
261C24—24no—
262C24—21no—
263C24—21no—
264C24—22no—
265C24—22no—
266C24—42no—
267C24—1181no—
268C24(C2×D59)⋊C412yes—
269C24C2×C22⋊D5911yescharacteristic
270C2×D4—1182no—
271C2×D4—2362no—
272C2×D4—2364no—
273C2×D4—2364no—
274C2×D4—2364no—
275C2×D4—2361no—
276C2×D4—2361no—
277C2×D4—2362no—
278C2×D4—2362no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—24no—
282C22×C4—28no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—48no—
292C22×C4—48no—
293C22×C4—44no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—1181no—
297C22×C4—1182no—
298C22×C4(C2×D59)⋊C412yes—
299C22×C4C2×C4×D5911yescharacteristic
300C22×C4C2×D23611yescharacteristic
301C22×C4#1174568512yes—
302C22×C4C2×C22⋊D5912yes—
303C2×C8—2364no—
304C2×C8—2362no—
305C4⋊C4—1182no—
306C4⋊C4—2362no—
307C22⋊C4—216no—
308C22⋊C4—2364no—
309C22⋊C4—2362no—
310C22⋊C4—2364no—
311C22⋊C4—2362no—
312C42—48no—
313C23—22no—
314C23—22no—
315C23—24no—
316C23—24no—
317C23—22no—
318C23—22no—
319C23—22no—
320C23—22no—
321C23—22no—
322C23—22no—
323C23—48no—
324C23—48no—
325C23—42no—
326C23—42no—
327C23—42no—
328C23—42no—
329C23—2361no—
330C23—2361no—
331C23—2362no—
332C23—2361no—
333C23—2362no—
334C23#1427365514yes—
335C23C2×(C2×D59)⋊C411yescharacteristic
336C23#1427369611yescharacteristic
337C23#1427371011yescharacteristic
338C23#1427374712yes—
339C23#1427375212yes—
340D4—4722no—
341D4—4728no—
342D4—4722no—
343D4—4722no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—28no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—48no—
354C2×C4—48no—
355C2×C4—44no—
356C2×C4—48no—
357C2×C4—48no—
358C2×C4—44no—
359C2×C4—2361no—
360C2×C4—2362no—
361C2×C4—2361no—
362C2×C4—2362no—
363C2×C4—2361no—
364C2×C4—2362no—
365C2×C4#1427364114yes—
366C2×C4C4×C22⋊D5912yes—
367C2×C4#1427370112yes—
368C8—4722no—
369C22—24no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—24no—
375C22—24no—
376C22—24no—
377C22—24no—
378C22—48no—
379C22—48no—
380C22—42no—
381C22—42no—
382C22—42no—
383C22—42no—
384C22—4722no—
385C22—4721no—
386C22—4721no—
387C22—4721no—
388C22—4722no—
389C22#1683728111yescharacteristic
390C22C2×#1427364112yes—
391C22#1683739611yescharacteristic
392C22#1683740611yescharacteristic
393C22C2×#1427365512yes—
394C22#1683754712yes—
395C22#1683772612yes—
396C4—22no—
397C4—24no—
398C4—22no—
399C4—48no—
400C4—4722no—
401C4—4721no—
402C2—22no—
403C2—22no—
404C2—44no—
405C2—9441no—
406C2#2142258512yes—
407C2#2142261112yes—
408C2#2142441811yescharacteristic
409C2#2142453911yescharacteristic
410C2#2142454911yescharacteristic
subgroupG/H
centerC23C2×(C2×D59)⋊C4
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC2×C4×C22⋊C4×C59C2
socleC23×C59C2×C22⋊C4
prime radical#21432378C2
cosocleC23×C4×C59C23
hypercenterC2×C4×C22⋊C4D59
hypocenterC59#9150
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C24×C4 ◁ C2×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. #21424006C2semidirect
3. C2×#16837135C2semidirect
4. #21421975C2semidirect
5. C2×C4×C22⋊C4×C59C2semidirect
6. C2×#14273701C4semidirect
13. #16837262C4semidirect
54. #95×C59D4semidirect
198. C2×C4×C22⋊C4D59semidirect
226. C59#9150semidirect
260. #95C22⋊D59semidirect

Maximal subgroups

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

subgroupindex
#214323782
#214240062
C2×#168371352
#214219752
C2×C4×C22⋊C4×C592
#915059