Finite Group Databasebeta

← #28484237

Subgroups — #28484237

Order 15104.

560 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 115 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#21433125C211yesmaximal, characteristic
2#21426761C211yesmaximal, characteristic
3C2×#16837127C212yesmaximal, complemented
4#21421943C212yesmaximal, complemented
5C59×#1459C211yesmaximal, characteristic
6C2×#14273708C411yescharacteristic
7C22×Dic59⋊C4C2211yescharacteristic
8#16837474—24no—
9#16837459—21no—
10#16837366—21no—
11C2×C22⋊(C59⋊C8)C2212yes—
12#16837297—21no—
13#16837296—21no—
14#16837286—21no—
15#16837282C411yescharacteristic
16#16837275—21no—
17#16837274C411yescharacteristic
18#16837263C411yescharacteristic
19C4×C22⋊Dic59C2211yescharacteristic
20#16837127—24no—
21C2×C4×D4×C59C2211yescharacteristic
22C2×C22⋊C8×C59C2212yes—
23C59×#324—21no—
24C59×#323—21no—
25C23×Dic59C2×C411yescharacteristic
26C23×Dic59C2×C411yescharacteristic
27C2×#11745664—22no—
28C22×C59⋊C8—22no—
29#14273708—22no—
30#14273708—22no—
31C2×Dic59⋊C4—21no—
32C2×Dic59⋊C4—22no—
33C2×Dic59⋊C4—21no—
34C2×Dic59⋊C4—21no—
35C2×Dic59⋊C4—22no—
36C2×Dic59⋊C4—21no—
37C2×C22⋊Dic59—21no—
38C2×C22⋊Dic59—21no—
39C2×C22⋊Dic59C2×C411yescharacteristic
40C2×C22⋊Dic59C2×C411yescharacteristic
41C2×C4×Dic59—21no—
42C4⋊(C59⋊C8)—22no—
43C22⋊(C59⋊C8)—22no—
44C22⋊(C59⋊C8)—22no—
45C22⋊(C59⋊C8)D414yes—
46#14273638—22no—
47C59⋊#95—21no—
48C59⋊#95—21no—
49C59⋊#95—21no—
50C59⋊#95—21no—
51C59⋊#95—21no—
52C22×D4×C59C2×C411yescharacteristic
53C23×C4×C59D411yescharacteristic
54C23×C4×C59C2311yescharacteristic
55C22×C8×C59—22no—
56C4×D4×C59—22no—
57C4×D4×C59—22no—
58C2×C4⋊C4×C59D411yescharacteristic
59C2×C22⋊C4×C59C2×C411yescharacteristic
60C2×C22⋊C4×C59D411yescharacteristic
61C2×C42×C59D411yescharacteristic
62C22⋊C8×C59—22no—
63C22⋊C8×C59—22no—
64C24×C59C22⋊C411yescharacteristic
65C24×C59C22×C411yescharacteristic
66C2×D4×C59—22no—
67C2×D4×C59—21no—
68C2×D4×C59—21no—
69C2×D4×C59—22no—
70C22×C4×C59—21no—
71C22×C4×C59—21no—
72C22×C4×C59—81no—
73C22×C4×C59—81no—
74C22×C4×C59C22⋊C411yescharacteristic
75C22×C4×C59C22⋊C411yescharacteristic
76C22×C4×C59C22⋊C411yescharacteristic
77C22×C4×C59C2×D412yes—
78C22×C4×C59C2×D411yescharacteristic
79C22×C4×C59C2×D411yescharacteristic
80C22×C4×C59#4112yes—
81C22×Dic59—21no—
82C22×Dic59—21no—
83C22×Dic59—22no—
84C22×Dic59—21no—
85C22×Dic59—22no—
86C22×Dic59—21no—
87C22×Dic59—21no—
88C22×Dic59—22no—
89C22×Dic59—81no—
90C22×Dic59M1612yescomplemented
91C2×C8×C59—42no—
92C2×C8×C59—82no—
93C4⋊C4×C59—22no—
94C22⋊C4×C59—22no—
95C22⋊C4×C59—22no—
96C42×C59—21no—
97C42×C59—21no—
98C22⋊Dic59—24no—
99C22⋊Dic59—21no—
100C22⋊Dic59—22no—
101C22⋊Dic59—81no—
102C22⋊Dic59M1612yescomplemented
103Dic59⋊C4—24no—
104Dic59⋊C4—41no—
105Dic59⋊C4—42no—
106Dic59⋊C4—41no—
107Dic59⋊C4—81no—
108C4×Dic59—81no—
109#11745664—82no—
110C2×C59⋊C8—24no—
111C2×C59⋊C8—82no—
112C23×C59—21no—
113C23×C59—21no—
114C23×C59—21no—
115C23×C59—41no—
116C23×C59—41no—
117C23×C59—41no—
118C23×C59—41no—
119C23×C59C2×C22⋊C411yescharacteristic
120C23×C59C4×D412yes—
121C23×C59C2×M1611yescharacteristic
122C23×C59C2×M1611yescharacteristic
123C23×C59#13112yes—
124D4×C59—42no—
125D4×C59—42no—
126C2×C4×C59—21no—
127C2×C4×C59—21no—
128C2×C4×C59—21no—
129C2×C4×C59—21no—
130C2×C4×C59—21no—
131C2×C4×C59—42no—
132C2×C4×C59—41no—
133C2×C4×C59—42no—
134C2×C4×C59—41no—
135C2×C4×C59—42no—
136C2×C4×C59—41no—
137C2×C4×C59—41no—
138C2×C4×C59—81no—
139C2×C4×C59#12011yescharacteristic
140C2×C4×C59#12112yes—
141C2×C4×C59#12311yescharacteristic
142C2×Dic59—22no—
143C2×Dic59—22no—
144C2×Dic59—24no—
145C2×Dic59—41no—
146C2×Dic59—41no—
147C2×Dic59—41no—
148C2×Dic59—41no—
149C2×Dic59—42no—
150C2×Dic59—41no—
151C2×Dic59—41no—
152C2×Dic59—41no—
153C2×Dic59—41no—
154C2×Dic59—41no—
155C2×Dic59—41no—
156C2×Dic59—82no—
157C8×C59—82no—
158C59⋊C8—44no—
159#9110—591nomaximal, Sylow
160C22×C59—21no—
161C22×C59—22no—
162C22×C59—21no—
163C22×C59—22no—
164C22×C59—41no—
165C22×C59—41no—
166C22×C59—42no—
167C22×C59—42no—
168C22×C59—41no—
169C22×C59—41no—
170C22×C59—41no—
171C22×C59—81no—
172C22×C59#38611yescharacteristic
173C22×C59#40711yescharacteristic
174C22×C59#40811yescharacteristic
175C22×C59#43512yes—
176C22×C59#43612yes—
177C4×C59—21no—
178C4×C59—22no—
179C4×C59—21no—
180C4×C59—41no—
181C4×C59—41no—
182C4×C59—41no—
183Dic59—42no—
184Dic59—42no—
185Dic59—81no—
186Dic59—81no—
187#2973—591no—
188#2322—591no—
189#1459D5911yescharacteristic
190C2×#324—592no—
191#1282—592no—
192C2×C59—21no—
193C2×C59—21no—
194C2×C59—22no—
195C2×C59—22no—
196C2×C59—81no—
197C2×C59#187011yescharacteristic
198C2×C59#191311yescharacteristic
199C2×C59#191511yescharacteristic
200C2×#123—591no—
201C2×C4×D4C2×D5911yescharacteristic
202C22×C4⋊C4—591no—
203#435—1184no—
204#432—1181no—
205#407—1181no—
206C2×C22⋊C8—592no—
207C2×C22⋊C8C2×D5912yes—
208#389—1181no—
209#388—1181no—
210#387—1181no—
211#386—591no—
212#385—591no—
213#385—1181no—
214#381—591no—
215C4×C22⋊C4—591no—
216#324—21no—
217#324—1184no—
218#323—21no—
219C59#911011yescharacteristic, complemented, Sylow
220C22×D4C4×D5911yescharacteristic
221C23×C4—591no—
222C23×C4—591no—
223C23×C4D23611yescharacteristic
224C23×C4C22×D5911yescharacteristic
225C2×M16—1182no—
226C22×C8—22no—
227C22×C8—1182no—
228#123—1182no—
229#123—1182no—
230C4×D4—22no—
231C4×D4—22no—
232C2×C4⋊C4—1181no—
233C2×C4⋊C4—1181no—
234C2×C4⋊C4—1181no—
235C2×C4⋊C4—1182no—
236C2×C4⋊C4—1182no—
237C2×C4⋊C4—1181no—
238C2×C4⋊C4C22⋊D5911yescharacteristic
239C2×C22⋊C4—591no—
240C2×C22⋊C4—591no—
241C2×C22⋊C4—1181no—
242C2×C22⋊C4—1181no—
243C2×C22⋊C4C4×D5911yescharacteristic
244C2×C22⋊C4C22⋊D5911yescharacteristic
245C2×C42—1181no—
246C2×C42D23611yescharacteristic
247C4⋊C8—1182no—
248C22⋊C8—22no—
249C22⋊C8—22no—
250C22⋊C8—594no—
251C22⋊C8—1182no—
252C22⋊C8—1182no—
253#97—1182no—
254#95—1181no—
255#95—1181no—
256#95—1181no—
257#95—1181no—
258#95—1181no—
259C24(C2×D59)⋊C411yescharacteristic
260C24C2×C4×D5911yescharacteristic
261C2×D4—22no—
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—22no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—81no—
268C22×C4—81no—
269C22×C4—592no—
270C22×C4—1181no—
271C22×C4—1181no—
272C22×C4—1182no—
273C22×C4—1181no—
274C22×C4—1182no—
275C22×C4—1182no—
276C22×C4—1181no—
277C22×C4—1181no—
278C22×C4—4721no—
279C22×C4(C2×D59)⋊C411yescharacteristic
280C22×C4(C2×D59)⋊C411yescharacteristic
281C22×C4(C2×D59)⋊C411yescharacteristic
282C22×C4C2×D23611yescharacteristic
283C22×C4D4×D5912yes—
284C22×C4#1174568712yes—
285C22×C4C2×C22⋊D5911yescharacteristic
286M16—4722no—
287C2×C8—42no—
288C2×C8—82no—
289C2×C8—1184no—
290C2×C8—4722no—
291C4⋊C4—22no—
292C4⋊C4—1184no—
293C4⋊C4—4721no—
294C4⋊C4—2361no—
295C4⋊C4—2362no—
296C4⋊C4—2361no—
297C22⋊C4—22no—
298C22⋊C4—22no—
299C22⋊C4—592no—
300C22⋊C4—1184no—
301C22⋊C4—1181no—
302C22⋊C4—1182no—
303C22⋊C4—4721no—
304C42—21no—
305C42—21no—
306C42—4721no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23C2×(C2×D59)⋊C411yescharacteristic
315C23(C22⋊D59)⋊C412yes—
316C23M16×D5911yescharacteristic
317C23M16×D5911yescharacteristic
318C23#1427373312yes—
319D4—42no—
320D4—42no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—41no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—41no—
330C2×C4—42no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—81no—
334C2×C4—1184no—
335C2×C4—1182no—
336C2×C4—1182no—
337C2×C4—4722no—
338C2×C4—2361no—
339C2×C4—2361no—
340C2×C4—2361no—
341C2×C4—2361no—
342C2×C4—2361no—
343C2×C4—2361no—
344C2×C4—2361no—
345C2×C4—2361no—
346C2×C4—2362no—
347C2×C4—2361no—
348C2×C4—2361no—
349C2×C4#1427369711yescharacteristic
350C2×C4#1427369912yes—
351C2×C4#1427371111yescharacteristic
352C8—82no—
353C8—2364no—
354C22—21no—
355C22—22no—
356C22—21no—
357C22—22no—
358C22—41no—
359C22—41no—
360C22—42no—
361C22—42no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—81no—
366C22#1683727911yescharacteristic
367C22#1683736511yescharacteristic
368C22#1683736811yescharacteristic
369C22#1683747612yes—
370C22#1683747712yes—
371C4—22no—
372C4—21no—
373C4—21no—
374C4—41no—
375C4—41no—
376C4—41no—
377C4—4721no—
378C4—4721no—
379C4—2362no—
380C4—2362no—
381C2—21no—
382C2—21no—
383C2—22no—
384C2—22no—
385C2—81no—
386C2#2142439111yescharacteristic
387C2#2142463611yescharacteristic
388C2#2142464611yescharacteristic
subgroupG/H
centerC22#16837279
commutatorC24×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×#1459C2
socleC22×C59#386
prime radicalC22×D4×C59C2×C4
cosocleC23×C4×C59C23
hypercenter#1459D59
hypocenterC59#9110
LCSG ▷ C24×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1459

Complements

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

NH ≅ G/Ntype
3. C2×#16837127C2semidirect
4. #21421943C2semidirect
90. C22×Dic59M16semidirect
102. C22⋊Dic59M16semidirect
219. C59#9110semidirect

Maximal subgroups

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

subgroupindex
#214331252
#214267612
C2×#168371272
#214219432
C59×#14592
#911059