Finite Group Databasebeta

← #28499736

Subgroups — #28499736

Order 15104.

764 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 764 conjugacy classes of subgroups — too many to draw. Only its 99 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#21427452C211yesmaximal, characteristic
2#21424088C211yesmaximal, characteristic, complemented
3#21422100C214yesmaximal, complemented
4C59×#1459C211yesmaximal, characteristic, complemented
5C23×D236C411yescharacteristic
6C2×#14273701—21no—
7C2×#14273695—21no—
8#16837426—21no—
9C2×#14273644—21no—
10#16837319C2212yes—
11#16837311C412yes—
12#16837304—21no—
13#16837262C411yescharacteristic
14#16837257—21no—
15#16837171C2212yescomplemented
16C2×C4×D4×C59C2211yescharacteristic
17C2×C22⋊C8×C59—21no—
18C59×#324C2212yescomplemented
19C59×#323—21no—
20C24×D59—21no—
21C22×D236—21no—
22C22×D236—44no—
23C22×D236—41no—
24C22×D236—41no—
25C22×D236C2×C412yes—
26C22×C22⋊D59—22no—
27C2×#11745664—22no—
28#14273701—44no—
29#14273695—42no—
30#14273695—42no—
31C2×C4⋊Dic59—21no—
32C2×C22⋊Dic59—21no—
33C2×(C2×D59)⋊C4—22no—
34C2×(C2×D59)⋊C4—22no—
35C2×(C2×D59)⋊C4—22no—
36C4⋊(C59⋊C8)—41no—
37C4⋊(C59⋊C8)—41no—
38#14273644—42no—
39C59⋊#95C2×C412yes—
40C22×D4×C59D411yescharacteristic
41C23×C4×C59C2×C411yescharacteristic
42C23×C4×C59D411yescharacteristic
43C22×C8×C59—41no—
44C4×D4×C59—41no—
45C4×D4×C59—41no—
46C2×C4⋊C4×C59C2311yescharacteristic
47C2×C22⋊C4×C59C2×C411yescharacteristic
48C2×C22⋊C4×C59D411yescharacteristic
49C2×C42×C59D411yescharacteristic
50C22⋊C8×C59—22no—
51C22⋊C8×C59—41no—
52C24×C59C22⋊C411yescharacteristic
53C24×C59C22⋊C411yescharacteristic
54C23×D59—22no—
55C23×D59—44no—
56C23×D59—42no—
57C23×D59—42no—
58C23×D59—42no—
59C23×D59—82no—
60C2×D4×C59—41no—
61C2×D4×C59—41no—
62C2×D4×C59D812yes—
63C2×D4×C59SD1612yes—
64C22×C4×C59—21no—
65C22×C4×C59—21no—
66C22×C4×C59—21no—
67C22×C4×C59—21no—
68C22×C4×C59—42no—
69C22×C4×C59—42no—
70C22×C4×C59C22⋊C411yescharacteristic
71C22×C4×C59C22⋊C411yescharacteristic
72C22×C4×C59C22×C411yescharacteristic
73C22×C4×C59C2×D411yescharacteristic
74C22×C4×C59C2×D411yescharacteristic
75C2×C22⋊D59—44no—
76C2×C22⋊D59—44no—
77C2×C22⋊D59—82no—
78C22×Dic59—22no—
79C2×D236—24no—
80C2×D236—42no—
81C2×D236—42no—
82C2×D236—48no—
83C2×D236—44no—
84C2×D236—41no—
85C2×D236—41no—
86C2×D236—44no—
87C2×D236—44no—
88C2×C8×C59—42no—
89C2×C8×C59—82no—
90C4⋊C4×C59—41no—
91C22⋊C4×C59—41no—
92C22⋊C4×C59—41no—
93C42×C59—21no—
94C42×C59—21no—
95C22⋊Dic59—42no—
96(C2×D59)⋊C4—44no—
97(C2×D59)⋊C4—42no—
98(C2×D59)⋊C4—42no—
99(C2×D59)⋊C4—82no—
100C4⋊Dic59—42no—
101#11745664—42no—
102#11745664—42no—
103C2×C59⋊C8—42no—
104C23×C59—21no—
105C23×C59—21no—
106C23×C59—21no—
107C23×C59—21no—
108C23×C59—41no—
109C23×C59—41no—
110C23×C59—41no—
111C23×C59—41no—
112C23×C59#10212yes—
113C23×C59#10212yes—
114C23×C59C2×C22⋊C411yescharacteristic
115C22×D59—24no—
116C22×D59—42no—
117C22×D59—42no—
118C22×D59—42no—
119C22×D59—48no—
120C22×D59—42no—
121C22×D59—42no—
122C22×D59—42no—
123C22×D59—88no—
124C22×D59—88no—
125C22×D59—84no—
126C22×D59—84no—
127D4×C59—42no—
128D4×C59—42no—
129C2×C4×C59—21no—
130C2×C4×C59—21no—
131C2×C4×C59—22no—
132C2×C4×C59—21no—
133C2×C4×C59—21no—
134C2×C4×C59—21no—
135C2×C4×C59—42no—
136C2×C4×C59—42no—
137C2×C4×C59—42no—
138C2×C4×C59—41no—
139C2×C4×C59—41no—
140C2×C4×C59—41no—
141C2×C4×C59—81no—
142C2×C4×C59#9911yescharacteristic
143C2×C4×C59#9911yescharacteristic
144C2×C4×C59C2×D811yescharacteristic
145C2×C4×C59C2×SD1611yescharacteristic
146C22⋊D59—88no—
147C2×Dic59—42no—
148C2×Dic59—42no—
149C2×Dic59—42no—
150D236—44no—
151D236—44no—
152D236—44no—
153D236—44no—
154C8×C59—82no—
155C59⋊C8—82no—
156#11880—591nomaximal, Sylow
157C22×C59—21no—
158C22×C59—21no—
159C22×C59—22no—
160C22×C59—22no—
161C22×C59—42no—
162C22×C59—42no—
163C22×C59—42no—
164C22×C59—41no—
165C22×C59—41no—
166C22×C59—41no—
167C22×C59—41no—
168C22×C59—81no—
169C22×C59C2×#9911yescharacteristic
170C22×C59C2×#10211yescharacteristic
171C22×C59#42111yescharacteristic
172C2×D59—44no—
173C2×D59—44no—
174C2×D59—44no—
175C2×D59—82no—
176C2×D59—82no—
177C2×D59—816no—
178C2×D59—82no—
179C2×D59—82no—
180C4×C59—21no—
181C4×C59—21no—
182C4×C59—21no—
183C4×C59—41no—
184C4×C59—41no—
185C4×C59—41no—
186C4×C59AΣL(1,𝔽16)12yes—
187Dic59—82no—
188#2408—591no—
189#1800—591no—
190#1459D5911yescharacteristic, complemented
191#1345—594no—
192C2×C59—21no—
193C2×C59—21no—
194C2×C59—42no—
195C2×C59—42no—
196C2×C59#151911yescharacteristic
197C2×C59C2×AΣL(1,𝔽16)11yescharacteristic
198C2×C59#213411yescharacteristic
199D59—84no—
200C23×D4—591no—
201C2×#121—1181no—
202C2×#120—1181no—
203C2×C4×D4C2×D5911yescharacteristic
204#423—1181no—
205C2×#100—1181no—
206C2×C22⋊C8—21no—
207#394—592no—
208#392—592no—
209#390—1181no—
210#380—591no—
211#379—1181no—
212#343—592no—
213#324C2×D5912yescomplemented
214#323—21no—
215C59#1188011yescharacteristic, complemented, Sylow
216C25—1181no—
217C22×D4—592no—
218C22×D4—1182no—
219C22×D4—1181no—
220C22×D4—2361no—
221C22×D4—2361no—
222C22×D4—2364no—
223C22×D4D23611yescharacteristic
224C23×C4C4×D5911yescharacteristic
225C23×C4C22⋊D5911yescharacteristic
226C2×M16—1182no—
227C22×C8—41no—
228#121—2364no—
229#120—2362no—
230#120—2362no—
231C4×D4—41no—
232C4×D4—41no—
233C2×C4⋊C4—1181no—
234C2×C4⋊C4C22×D5911yescharacteristic
235C2×C22⋊C4—1182no—
236C2×C22⋊C4—1181no—
237C2×C22⋊C4—1182no—
238C2×C22⋊C4—1182no—
239C2×C22⋊C4C4×D5911yescharacteristic
240C2×C22⋊C4C22⋊D5911yescharacteristic
241C2×C42D23611yescharacteristic
242C4⋊C8—2361no—
243C4⋊C8—2361no—
244#100—2362no—
245C22⋊C8—22no—
246C22⋊C8—41no—
247#95—592no—
248C24—1182no—
249C24—4722no—
250C24—2362no—
251C24—2362no—
252C24—2364no—
253C24—2362no—
254C24(C2×D59)⋊C411yescharacteristic
255C24(C2×D59)⋊C411yescharacteristic
256C2×D4—41no—
257C2×D4—41no—
258C2×D4—1184no—
259C2×D4—4722no—
260C2×D4—2362no—
261C2×D4—2362no—
262C2×D4—2368no—
263C2×D4—2364no—
264C2×D4—2364no—
265C2×D4—2364no—
266C2×D4—2364no—
267C2×D4—2364no—
268C2×D4—2361no—
269C2×D4—2361no—
270C2×D4#1174566012yes—
271C2×D4D47212yes—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—1182no—
279C22×C4(C2×D59)⋊C411yescharacteristic
280C22×C4(C2×D59)⋊C411yescharacteristic
281C22×C4C2×C4×D5911yescharacteristic
282C22×C4C2×D23611yescharacteristic
283C22×C4C2×C22⋊D5911yescharacteristic
284M16—2362no—
285M16—2362no—
286C2×C8—42no—
287C2×C8—82no—
288C2×C8—2362no—
289C4⋊C4—41no—
290C4⋊C4—2362no—
291C22⋊C4—41no—
292C22⋊C4—41no—
293C22⋊C4—4722no—
294C22⋊C4—2364no—
295C22⋊C4—2362no—
296C22⋊C4—2362no—
297C22⋊C4—2362no—
298C42—21no—
299C42—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—1184no—
309C23—4724no—
310C23—4724no—
311C23—4728no—
312C23—4728no—
313C23—2362no—
314C23—2362no—
315C23—2362no—
316C23—2368no—
317C23—2362no—
318C23—2362no—
319C23—2362no—
320C23#1427364912yes—
321C23#1427364912yes—
322C23C2×(C2×D59)⋊C411yescharacteristic
323D4—42no—
324D4—42no—
325D4—4728no—
326D4—2364no—
327D4—2364no—
328D4—2364no—
329D4—2364no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—22no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—81no—
343C2×C4—2362no—
344C2×C4—2362no—
345C2×C4—2362no—
346C2×C4#1427364111yescharacteristic
347C2×C4#1427364111yescharacteristic
348C2×C4C2×D47211yescharacteristic
349C2×C4C2×#1174566011yescharacteristic
350C8—82no—
351C8—4722no—
352C22—21no—
353C22—21no—
354C22—22no—
355C22—22no—
356C22—42no—
357C22—42no—
358C22—42no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—81no—
364C22—4722no—
365C22—4722no—
366C22—47216no—
367C22—4722no—
368C22—4722no—
369C22—2364no—
370C22—2364no—
371C22—2364no—
372C22C2×#1427364111yescharacteristic
373C22C2×#1427364911yescharacteristic
374C22#1683741811yescharacteristic
375C4—21no—
376C4—21no—
377C4—21no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4—4722no—
382C4#1683718812yes—
383C2—21no—
384C2—21no—
385C2—42no—
386C2—42no—
387C2—4724no—
388C2#2142266511yescharacteristic
389C2C2×#1683718811yescharacteristic
390C2#2142578711yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC22×C4×C59C22×C4
FrattiniC2×C4⋊C4C22×D59
FittingC59×#1459C2
socleC22×C59C2×#99
prime radical#21427452C2
cosocleC2×C4⋊C4×C59C23
hypercenter#1459D59
hypocenterC59#11880
LCSG ▷ C22×C4×C59 ▷ C23×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. #21424088C2semidirect
3. #21422100C2semidirect
4. C59×#1459C2semidirect
15. #16837171C22semidirect
18. C59×#324C22semidirect
190. #1459D59semidirect
213. #324C2×D59semidirect
215. C59#11880semidirect

Maximal subgroups

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

subgroupindex
#214274522
#214240882
#214221002
C59×#14592
#1188059