Finite Group Databasebeta

← #28499148

Subgroups — #28499148

Order 15104.

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

normal subgroups only

This group has 500 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#21429662C211yesmaximal, characteristic
2#21422430C211yesmaximal, characteristic, complemented
3#21422119C212yesmaximal, complemented
4#21422092C212yesmaximal
5C59×#1740C211yesmaximal, characteristic, complemented
6C2×C4×D236C411yescharacteristic, complemented
7#16837459—21no—
8#16837341C411yescharacteristic, complemented
9#16837338C2211yescharacteristic
10#16837319C411yescharacteristic, complemented
11#16837311C2211yescharacteristic
12#16837293—21no—
13#16837281—21no—
14#16837278—21no—
15#16837273—21no—
16#16837270C411yescharacteristic, complemented
17#16837128—21no—
18#16837126C2212yes—
19C2×#117×C59—21no—
20C2×#117×C59C2211yescharacteristic
21C59×#342—21no—
22C59×#342C2212yes—
23C22×D236C2×C411yescharacteristic
24C22×C22⋊D59—21no—
25C22×C4×D59—21no—
26C4×D236—41no—
27C4×D236—41no—
28C2×C4⋊Dic59C2×C411yescharacteristic
29C2×Dic59⋊C4—21no—
30C2×C22⋊Dic59—21no—
31C2×(C2×D59)⋊C4—21no—
32C2×(C2×D59)⋊C4—21no—
33C2×(C2×D59)⋊C4—21no—
34C2×(C2×D59)⋊C4—21no—
35C4⋊(C59⋊C8)—41no—
36C22⋊(C59⋊C8)—22no—
37#14273638—41no—
38C59⋊#95—21no—
39C59⋊#95—21no—
40C59⋊#95—21no—
41C59⋊#95—21no—
42C59⋊#95C2×C411yescharacteristic
43C59⋊#95C2×C411yescharacteristic
44C23×C4×C59D411yescharacteristic
45#117×C59—24no—
46#117×C59—41no—
47#117×C59—41no—
48#117×C59—42no—
49C2×C4⋊C4×C59—41no—
50C2×C4⋊C4×C59C2×C411yescharacteristic
51C2×C4⋊C4×C59D411yescharacteristic
52C2×C22⋊C4×C59—22no—
53C2×C22⋊C4×C59D411yescharacteristic
54C2×C22⋊C4×C59C2311yescharacteristic
55C2×C42×C59—41no—
56C2×C42×C59C2×C411yescharacteristic
57C2×C42×C59D411yescharacteristic
58C24×C59C2×D411yescharacteristic
59C23×D59—21no—
60C22×C4×C59—21no—
61C22×C4×C59—21no—
62C22×C4×C59—21no—
63C22×C4×C59—21no—
64C22×C4×C59—22no—
65C22×C4×C59—42no—
66C22×C4×C59C22⋊C411yescharacteristic
67C22×C4×C59C22⋊C411yescharacteristic
68C22×C4×C59C22⋊C411yescharacteristic
69C22×C4×C59C22⋊C411yescharacteristic
70C22×C4×C59C22×C411yescharacteristic
71C22×C4×C59C2×D411yescharacteristic
72C2×C22⋊D59—42no—
73C2×C22⋊D59—41no—
74C2×C22⋊D59—41no—
75C2×C22⋊D59—81no—
76C22×Dic59—21no—
77C22×Dic59—21no—
78C22×Dic59—21no—
79C2×D236—41no—
80C2×D236—42no—
81C2×C4×D59—42no—
82C2×C4×D59—81no—
83C2×C4×D59—81no—
84C4⋊C4×C59—41no—
85C4⋊C4×C59—41no—
86C4⋊C4×C59—44no—
87C22⋊C4×C59—24no—
88C22⋊C4×C59—41no—
89C22⋊C4×C59—41no—
90C22⋊C4×C59—42no—
91C42×C59—21no—
92C42×C59—21no—
93C42×C59—21no—
94C42×C59—21no—
95C42×C59—44no—
96C22⋊Dic59—81no—
97(C2×D59)⋊C4—42no—
98(C2×D59)⋊C4—41no—
99(C2×D59)⋊C4—41no—
100(C2×D59)⋊C4—42no—
101(C2×D59)⋊C4—81no—
102C4⋊Dic59—41no—
103Dic59⋊C4—81no—
104C2×C59⋊C8—42no—
105C23×C59—21no—
106C23×C59—21no—
107C23×C59—41no—
108C23×C59—41no—
109C23×C59C2×C22⋊C411yescharacteristic
110C23×C59AGL(1,ℤ/8)11yescharacteristic
111C23×C59#13711yescharacteristic
112C22×D59—22no—
113C22×D59—41no—
114C22×D59—41no—
115C22×D59—41no—
116C22×D59—41no—
117C22×D59—41no—
118C22×D59—41no—
119C2×C4×C59—22no—
120C2×C4×C59—22no—
121C2×C4×C59—22no—
122C2×C4×C59—22no—
123C2×C4×C59—21no—
124C2×C4×C59—21no—
125C2×C4×C59—21no—
126C2×C4×C59—21no—
127C2×C4×C59—21no—
128C2×C4×C59—21no—
129C2×C4×C59—21no—
130C2×C4×C59—21no—
131C2×C4×C59—41no—
132C2×C4×C59—41no—
133C2×C4×C59—41no—
134C2×C4×C59—41no—
135C2×C4×C59—44no—
136C2×C4×C59—82no—
137C2×C4×C59—82no—
138C2×C4×C59#9911yescharacteristic
139C2×C4×C59#9911yescharacteristic
140C2×C4×C59#10412yes—
141C22⋊D59—82no—
142C22⋊D59—82no—
143C2×Dic59—41no—
144C2×Dic59—41no—
145C2×Dic59—41no—
146C2×Dic59—41no—
147C2×Dic59—41no—
148C2×Dic59—41no—
149C2×Dic59—41no—
150C2×Dic59—41no—
151C2×Dic59—41no—
152D236—42no—
153D236—42no—
154C4×D59—84no—
155C59⋊C8—82no—
156#11754—591nomaximal, Sylow
157C22×C59—21no—
158C22×C59—21no—
159C22×C59—21no—
160C22×C59—21no—
161C22×C59—41no—
162C22×C59—41no—
163C22×C59—41no—
164C22×C59—41no—
165C22×C59—81no—
166C22×C59C2×#9911yescharacteristic
167C22×C59#41711yescharacteristic
168C22×C59C2×#10411yescharacteristic
169C2×D59—42no—
170C2×D59—42no—
171C2×D59—42no—
172C2×D59—81no—
173C2×D59—81no—
174C2×D59—81no—
175C2×D59—81no—
176C4×C59—21no—
177C4×C59—21no—
178C4×C59—22no—
179C4×C59—41no—
180C4×C59—41no—
181C4×C59—41no—
182C4×C59—41no—
183C4×C59—84no—
184Dic59—81no—
185Dic59—81no—
186Dic59—81no—
187#2598—591no—
188#1740D5911yescharacteristic, complemented
189#1464—591no—
190#1352—592no—
191#1343—592no—
192C2×C59—21no—
193C2×C59—21no—
194C2×C59—41no—
195C2×C59—41no—
196C2×C59#150411yescharacteristic
197C2×C59#212011yescharacteristic
198C2×C59#212811yescharacteristic
199D59—82no—
200C2×C4×D4—591no—
201C2×#117—21no—
202C2×#117C2×D5911yescharacteristic
203#432—1181no—
204#400—591no—
205#399—591no—
206#394—591no—
207#392—591no—
208#388—1181no—
209#386—1181no—
210#386—1181no—
211#385—1181no—
212#383—591no—
213#342—21no—
214#342C2×D5912yes—
215#324—1181no—
216#323—592no—
217C59#1175411yescharacteristic, complemented, Sylow
218C22×D4—591no—
219C22×D4—1181no—
220C23×C4—1181no—
221C23×C4C22⋊D5911yescharacteristic
222C4×D4—2361no—
223C4×D4—2361no—
224#117—24no—
225#117—41no—
226#117—41no—
227#117—42no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4—591no—
230C2×C4⋊C4—1181no—
231C2×C4⋊C4C4×D5911yescharacteristic
232C2×C4⋊C4D23611yescharacteristic
233C2×C22⋊C4—22no—
234C2×C22⋊C4—1181no—
235C2×C22⋊C4—1181no—
236C2×C22⋊C4—1181no—
237C2×C22⋊C4—1181no—
238C2×C22⋊C4—1181no—
239C2×C22⋊C4C22⋊D5911yescharacteristic
240C2×C22⋊C4C22×D5911yescharacteristic
241C2×C42—41no—
242C2×C42C4×D5911yescharacteristic
243C2×C42D23611yescharacteristic
244C4⋊C8—2361no—
245C22⋊C8—1182no—
246#97—2361no—
247#95—591no—
248#95—591no—
249#95—1181no—
250#95—1181no—
251#95—1181no—
252#95—1181no—
253C24—1181no—
254C24C2×C22⋊D5911yescharacteristic
255C2×D4—4721no—
256C2×D4—2361no—
257C2×D4—2362no—
258C2×D4—2362no—
259C2×D4—2361no—
260C2×D4—2361no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—42no—
267C22×C4—1181no—
268C22×C4—1181no—
269C22×C4—1181no—
270C22×C4—4721no—
271C22×C4—4721no—
272C22×C4—2362no—
273C22×C4(C2×D59)⋊C411yescharacteristic
274C22×C4(C2×D59)⋊C411yescharacteristic
275C22×C4(C2×D59)⋊C411yescharacteristic
276C22×C4(C2×D59)⋊C411yescharacteristic
277C22×C4C2×C4×D5911yescharacteristic
278C22×C4C2×D23611yescharacteristic
279C2×C8—2362no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—44no—
283C4⋊C4—4721no—
284C4⋊C4—2361no—
285C22⋊C4—24no—
286C22⋊C4—41no—
287C22⋊C4—41no—
288C22⋊C4—42no—
289C22⋊C4—4721no—
290C22⋊C4—4721no—
291C22⋊C4—2362no—
292C22⋊C4—2361no—
293C22⋊C4—2361no—
294C22⋊C4—2362no—
295C42—21no—
296C42—21no—
297C42—21no—
298C42—21no—
299C42—44no—
300C23—21no—
301C23—21no—
302C23—41no—
303C23—41no—
304C23—1182no—
305C23—2361no—
306C23—2361no—
307C23—2361no—
308C23—2361no—
309C23—2361no—
310C23—2361no—
311C23C2×(C2×D59)⋊C411yescharacteristic
312C23#1427375311yescharacteristic
313C23#1427375811yescharacteristic
314D4—4722no—
315D4—4722no—
316D4—2362no—
317D4—2362no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—44no—
335C2×C4—82no—
336C2×C4—82no—
337C2×C4—4724no—
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—2361no—
347C2×C4#1427364111yescharacteristic
348C2×C4#1427364111yescharacteristic
349C2×C4#1427365512yes—
350C8—4722no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—81no—
360C22—4721no—
361C22—4721no—
362C22—4721no—
363C22—4721no—
364C22—2362no—
365C22—2362no—
366C22—2362no—
367C22C2×#1427364111yescharacteristic
368C22#1683740111yescharacteristic
369C22C2×#1427365511yescharacteristic
370C4—21no—
371C4—21no—
372C4—22no—
373C4—41no—
374C4—41no—
375C4—41no—
376C4—41no—
377C4—84no—
378C4—4721no—
379C4—4721no—
380C4—4721no—
381C2—21no—
382C2—21no—
383C2—41no—
384C2—41no—
385C2—4722no—
386C2#2142259811yescharacteristic
387C2#2142573511yescharacteristic
388C2#2142576711yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC22×C4×C59C22×C4
FrattiniC2×C22⋊C4C22×D59
FittingC59×#1740C2
socleC22×C59C2×#99
prime radical#16837311C22
cosocleC2×C22⋊C4×C59C23
hypercenter#1740D59
hypocenterC59#11754
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1740

Complements

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

NH ≅ G/Ntype
2. #21422430C2semidirect
3. #21422119C2semidirect
5. C59×#1740C2semidirect
6. C2×C4×D236C4semidirect
8. #16837341C4semidirect
10. #16837319C4semidirect
16. #16837270C4semidirect
188. #1740D59semidirect
217. C59#11754semidirect

Maximal subgroups

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

subgroupindex
#214296622
#214224302
#214221192
#214220922
C59×#17402
#1175459