Finite Group Databasebeta

← #28498296

Subgroups — #28498296

Order 15104.

534 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 83 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#21428982C211yesmaximal, characteristic
2#21422437C211yesmaximal, characteristic, complemented
3#21422025C212yesmaximal, complemented
4#21422023C212yesmaximal, complemented
5C59×#1459C211yesmaximal, characteristic, complemented
6C2×#14273708C411yescharacteristic
7C2×#14273695C411yescharacteristic
8C22×(C2×D59)⋊C4C411yescharacteristic
9#16837459—21no—
10#16837324—21no—
11#16837318—21no—
12#16837293—21no—
13#16837281—21no—
14#16837263C2211yescharacteristic
15#16837262C411yescharacteristic
16#16837257C2211yescharacteristic
17#16837126—21no—
18#16837126C2212yes—
19C2×C4×D4×C59C2211yescharacteristic
20C2×C22⋊C8×C59—21no—
21C59×#324—21no—
22C59×#323C2212yes—
23C24×D59C2×C411yescharacteristic
24C22×C22⋊D59—21no—
25C23×Dic59C2×C411yescharacteristic
26#14273708—42no—
27#14273695—42no—
28C2×C4⋊Dic59—21no—
29C2×Dic59⋊C4—21no—
30C2×C22⋊Dic59—21no—
31C2×C22⋊Dic59C2×C411yescharacteristic
32C2×C22⋊Dic59C2×C411yescharacteristic
33C2×(C2×D59)⋊C4—21no—
34C2×(C2×D59)⋊C4—21no—
35C2×(C2×D59)⋊C4—21no—
36C2×(C2×D59)⋊C4—21no—
37C2×(C2×D59)⋊C4—21no—
38C2×(C2×D59)⋊C4—41no—
39C2×(C2×D59)⋊C4—41no—
40C4⋊(C59⋊C8)—41no—
41C22⋊(C59⋊C8)—22no—
42#14273638—41no—
43C59⋊#95—21no—
44C59⋊#95—21no—
45C59⋊#95—21no—
46C22×D4×C59C2×C411yescharacteristic
47C23×C4×C59C2×C411yescharacteristic
48C23×C4×C59D411yescharacteristic
49C22×C8×C59—41no—
50C4×D4×C59—41no—
51C4×D4×C59—41no—
52C2×C4⋊C4×C59D411yescharacteristic
53C2×C22⋊C4×C59D411yescharacteristic
54C2×C22⋊C4×C59C2311yescharacteristic
55C2×C42×C59D411yescharacteristic
56C22⋊C8×C59—22no—
57C22⋊C8×C59—41no—
58C24×C59C22⋊C411yescharacteristic
59C24×C59C22×C411yescharacteristic
60C23×D59—21no—
61C23×D59—22no—
62C23×D59—42no—
63C23×D59—41no—
64C23×D59—41no—
65C23×D59—81no—
66C2×D4×C59—22no—
67C2×D4×C59—22no—
68C2×D4×C59—41no—
69C22×C4×C59—21no—
70C22×C4×C59—21no—
71C22×C4×C59—21no—
72C22×C4×C59—21no—
73C22×C4×C59—81no—
74C22×C4×C59—81no—
75C22×C4×C59C22⋊C411yescharacteristic
76C22×C4×C59C22⋊C411yescharacteristic
77C22×C4×C59C22⋊C411yescharacteristic
78C22×C4×C59C2×D411yescharacteristic
79C22×C4×C59C2×D411yescharacteristic
80C2×C22⋊D59—42no—
81C2×C22⋊D59—42no—
82C2×C22⋊D59—81no—
83C22×Dic59—21no—
84C22×Dic59—21no—
85C22×Dic59—21no—
86C22×Dic59—22no—
87C22×Dic59—41no—
88C22×Dic59—41no—
89C2×C8×C59—42no—
90C2×C8×C59—82no—
91C4⋊C4×C59—41no—
92C22⋊C4×C59—41no—
93C22⋊C4×C59—41no—
94C42×C59—21no—
95C42×C59—21no—
96C22⋊Dic59—42no—
97C22⋊Dic59—41no—
98C22⋊Dic59—41no—
99(C2×D59)⋊C4—42no—
100(C2×D59)⋊C4—42no—
101(C2×D59)⋊C4—42no—
102(C2×D59)⋊C4—41no—
103(C2×D59)⋊C4—41no—
104(C2×D59)⋊C4—81no—
105C4⋊Dic59—81no—
106Dic59⋊C4—42no—
107C2×C59⋊C8—42no—
108C23×C59—21no—
109C23×C59—21no—
110C23×C59—21no—
111C23×C59—21no—
112C23×C59—41no—
113C23×C59—41no—
114C23×C59—41no—
115C23×C59—41no—
116C23×C59#10012yes—
117C23×C59C2×C22⋊C411yescharacteristic
118C23×C59#13112yes—
119C22×D59—22no—
120C22×D59—41no—
121C22×D59—41no—
122C22×D59—41no—
123C22×D59—42no—
124C22×D59—41no—
125C22×D59—41no—
126C22×D59—41no—
127C22×D59—42no—
128C22×D59—84no—
129C22×D59—84no—
130C22×D59—82no—
131C22×D59—82no—
132D4×C59—44no—
133C2×C4×C59—21no—
134C2×C4×C59—21no—
135C2×C4×C59—21no—
136C2×C4×C59—21no—
137C2×C4×C59—21no—
138C2×C4×C59—21no—
139C2×C4×C59—41no—
140C2×C4×C59—42no—
141C2×C4×C59—41no—
142C2×C4×C59—41no—
143C2×C4×C59—41no—
144C2×C4×C59—81no—
145C2×C4×C59—81no—
146C2×C4×C59—81no—
147C2×C4×C59#9911yescharacteristic
148C2×C4×C59#9911yescharacteristic
149C22⋊D59—84no—
150C2×Dic59—41no—
151C2×Dic59—41no—
152C2×Dic59—41no—
153C2×Dic59—41no—
154C2×Dic59—41no—
155C2×Dic59—41no—
156C2×Dic59—41no—
157C2×Dic59—41no—
158C2×Dic59—41no—
159C2×Dic59—82no—
160C8×C59—82no—
161C59⋊C8—82no—
162#11560—591nomaximal, Sylow
163C22×C59—21no—
164C22×C59—21no—
165C22×C59—22no—
166C22×C59—22no—
167C22×C59—42no—
168C22×C59—42no—
169C22×C59—42no—
170C22×C59—41no—
171C22×C59—41no—
172C22×C59—41no—
173C22×C59—41no—
174C22×C59—81no—
175C22×C59#40811yescharacteristic
176C22×C59C2×#9911yescharacteristic
177C22×C59C2×#10011yescharacteristic
178C2×D59—42no—
179C2×D59—42no—
180C2×D59—42no—
181C2×D59—81no—
182C2×D59—81no—
183C2×D59—84no—
184C2×D59—81no—
185C2×D59—81no—
186C2×D59—84no—
187C4×C59—21no—
188C4×C59—21no—
189C4×C59—41no—
190C4×C59—41no—
191C4×C59—41no—
192C4×C59—41no—
193Dic59—81no—
194Dic59—81no—
195Dic59—81no—
196#2534—591no—
197#1466—591no—
198#1459D5911yescharacteristic, complemented
199#1317—592no—
200#1316—592no—
201C2×C59—21no—
202C2×C59—21no—
203C2×C59—42no—
204C2×C59—42no—
205C2×C59#147111yescharacteristic
206C2×C59#212011yescharacteristic
207C2×C59#212111yescharacteristic
208D59—82no—
209C2×#123—591no—
210C2×#120—591no—
211C2×C4×D4C2×D5911yescharacteristic
212C22×C22⋊C4—591no—
213#432—1181no—
214C2×C22⋊C8—21no—
215#396—1181no—
216#394—1181no—
217#388—1181no—
218#386—1181no—
219#381—591no—
220#380—591no—
221#379—591no—
222#324—21no—
223#323—592no—
224#323—1181no—
225#323C2×D5912yes—
226C59#1156011yescharacteristic, complemented, Sylow
227C25—591no—
228C22×D4—1181no—
229C22×D4C4×D5911yescharacteristic
230C23×C4—591no—
231C23×C4C4×D5911yescharacteristic
232C23×C4C22⋊D5911yescharacteristic
233C22×C8—41no—
234#123—2362no—
235#120—2362no—
236C4×D4—41no—
237C4×D4—41no—
238C2×C4⋊C4—1181no—
239C2×C4⋊C4—1181no—
240C2×C4⋊C4C22⋊D5911yescharacteristic
241C2×C22⋊C4—591no—
242C2×C22⋊C4—591no—
243C2×C22⋊C4—1181no—
244C2×C22⋊C4—1181no—
245C2×C22⋊C4—1181no—
246C2×C22⋊C4—1181no—
247C2×C22⋊C4—1181no—
248C2×C22⋊C4—1181no—
249C2×C22⋊C4—2361no—
250C2×C22⋊C4—2361no—
251C2×C22⋊C4D23611yescharacteristic
252C2×C22⋊C4C22×D5911yescharacteristic
253C2×C42D23611yescharacteristic
254C4⋊C8—2361no—
255C22⋊C8—22no—
256C22⋊C8—41no—
257C22⋊C8—1182no—
258#97—2361no—
259#95—1181no—
260#95—1181no—
261#95—1181no—
262C24—1181no—
263C24—1182no—
264C24—4721no—
265C24—2361no—
266C24—2361no—
267C24—2362no—
268C24(C2×D59)⋊C411yescharacteristic
269C24C2×C4×D5911yescharacteristic
270C2×D4—22no—
271C2×D4—22no—
272C2×D4—41no—
273C2×D4—4721no—
274C2×D4—2362no—
275C2×D4—2362no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—81no—
281C22×C4—81no—
282C22×C4—1182no—
283C22×C4—1181no—
284C22×C4—1181no—
285C22×C4—1181no—
286C22×C4—2361no—
287C22×C4—2361no—
288C22×C4(C2×D59)⋊C411yescharacteristic
289C22×C4(C2×D59)⋊C411yescharacteristic
290C22×C4(C2×D59)⋊C411yescharacteristic
291C22×C4C2×D23611yescharacteristic
292C22×C4C2×C22⋊D5911yescharacteristic
293C2×C8—42no—
294C2×C8—82no—
295C2×C8—2362no—
296C4⋊C4—41no—
297C4⋊C4—4721no—
298C4⋊C4—2362no—
299C22⋊C4—41no—
300C22⋊C4—41no—
301C22⋊C4—4721no—
302C22⋊C4—2362no—
303C22⋊C4—2362no—
304C22⋊C4—2362no—
305C22⋊C4—2362no—
306C22⋊C4—2361no—
307C22⋊C4—2361no—
308C22⋊C4—2361no—
309C22⋊C4—2361no—
310C42—21no—
311C42—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—1182no—
321C23—4722no—
322C23—4722no—
323C23—4724no—
324C23—4724no—
325C23—2361no—
326C23—2361no—
327C23—2361no—
328C23—2362no—
329C23—2362no—
330C23—2361no—
331C23—2361no—
332C23—2361no—
333C23#1427364312yes—
334C23C2×(C2×D59)⋊C411yescharacteristic
335C23#1427373312yes—
336D4—44no—
337D4—4724no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—4722no—
353C2×C4—2361no—
354C2×C4—2361no—
355C2×C4—2361no—
356C2×C4—2361no—
357C2×C4—2361no—
358C2×C4—2361no—
359C2×C4—2361no—
360C2×C4—2361no—
361C2×C4—2361no—
362C2×C4#1427364111yescharacteristic
363C2×C4#1427364111yescharacteristic
364C8—82no—
365C8—4722no—
366C22—21no—
367C22—21no—
368C22—22no—
369C22—22no—
370C22—42no—
371C22—42no—
372C22—42no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—81no—
378C22—4721no—
379C22—4721no—
380C22—4724no—
381C22—4724no—
382C22—4721no—
383C22—4721no—
384C22—2362no—
385C22—2362no—
386C22—2362no—
387C22#1683736811yescharacteristic
388C22C2×#1427364111yescharacteristic
389C22C2×#1427364311yescharacteristic
390C4—21no—
391C4—21no—
392C4—41no—
393C4—41no—
394C4—41no—
395C4—41no—
396C4—4721no—
397C4—4721no—
398C4—4721no—
399C2—21no—
400C2—21no—
401C2—42no—
402C2—42no—
403C2—4722no—
404C2#2142245511yescharacteristic
405C2#2142573811yescharacteristic
406C2#2142574011yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC24×C59C22×C4
FrattiniC2×C22⋊C4C22×D59
FittingC59×#1459C2
socleC22×C59C2×#99
prime radicalC2×#14273695C4
cosocleC2×C22⋊C4×C59C23
hypercenter#1459D59
hypocenterC59#11560
LCSG ▷ C24×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1459

Complements

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

NH ≅ G/Ntype
2. #21422437C2semidirect
3. #21422025C2semidirect
4. #21422023C2semidirect
5. C59×#1459C2semidirect
198. #1459D59semidirect
226. C59#11560semidirect

Maximal subgroups

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

subgroupindex
#214289822
#214224372
#214220252
#214220232
C59×#14592
#1156059