Finite Group Databasebeta

← #28499019

Subgroups — #28499019

Order 15104.

606 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 606 conjugacy classes of subgroups — too many to draw. Only its 87 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#21427375C211yesmaximal, characteristic
2C4×#14273642C211yesmaximal, characteristic, complemented
3#21422121C212yesmaximal, complemented
4#21422094C212yesmaximal, complemented
5C59×#1459C211yesmaximal, characteristic, complemented
6C2×C4×D236C411yescharacteristic, complemented
7C2×C4×C22⋊D59—21no—
8C2×C4×C22⋊D59—21no—
9C2×#14273642—21no—
10#16837338C411yescharacteristic, complemented
11#16837338C2211yescharacteristic
12#16837337—21no—
13#16837311C411yescharacteristic, complemented
14#16837311C2211yescharacteristic
15#16837310—21no—
16C4×C22⋊Dic59—21no—
17#16837248C411yescharacteristic, complemented
18#16837169C2212yes—
19C2×C4×D4×C59C2211yescharacteristic
20C2×C22⋊C8×C59—21no—
21C59×#324—21no—
22C59×#323C2212yes—
23C22×D236C2×C411yescharacteristic
24C22×C22⋊D59—21no—
25C22×C22⋊D59—21no—
26C22×C4×D59—21no—
27C4×D236—41no—
28C4×D236—41no—
29C4×C22⋊D59—44no—
30C4×C22⋊D59—44no—
31C2×C4⋊Dic59C2×C411yescharacteristic
32C2×Dic59⋊C4—21no—
33C2×Dic59⋊C4—21no—
34C2×C22⋊Dic59—22no—
35C2×C22⋊Dic59—21no—
36C2×C22⋊Dic59—21no—
37C2×(C2×D59)⋊C4—21no—
38C2×(C2×D59)⋊C4—21no—
39C2×(C2×D59)⋊C4—21no—
40C2×C4×Dic59—21no—
41C2×C4×Dic59—41no—
42#14273642—24no—
43C59⋊#95—21no—
44C59⋊#95—41no—
45C59⋊#95C2×C411yescharacteristic
46C59⋊#95C2×C411yescharacteristic
47C22×D4×C59D411yescharacteristic
48C23×C4×C59D411yescharacteristic
49C23×C4×C59D411yescharacteristic
50C22×C8×C59—41no—
51C4×D4×C59—41no—
52C4×D4×C59—41no—
53C2×C4⋊C4×C59D411yescharacteristic
54C2×C22⋊C4×C59C2×C411yescharacteristic
55C2×C22⋊C4×C59C2311yescharacteristic
56C2×C42×C59C2×C411yescharacteristic
57C22⋊C8×C59—22no—
58C22⋊C8×C59—41no—
59C24×C59C22⋊C411yescharacteristic
60C24×C59C2×D411yescharacteristic
61C23×D59—21no—
62C2×D4×C59—22no—
63C2×D4×C59—22no—
64C2×D4×C59—41no—
65C22×C4×C59—21no—
66C22×C4×C59—21no—
67C22×C4×C59—21no—
68C22×C4×C59—21no—
69C22×C4×C59—42no—
70C22×C4×C59—42no—
71C22×C4×C59C22⋊C411yescharacteristic
72C22×C4×C59C22⋊C411yescharacteristic
73C22×C4×C59C22⋊C411yescharacteristic
74C22×C4×C59C22×C411yescharacteristic
75C22×C4×C59C2×D411yescharacteristic
76C2×C22⋊D59—42no—
77C2×C22⋊D59—42no—
78C2×C22⋊D59—42no—
79C2×C22⋊D59—42no—
80C2×C22⋊D59—42no—
81C2×C22⋊D59—42no—
82C22×Dic59—21no—
83C22×Dic59—21no—
84C22×Dic59—21no—
85C22×Dic59—42no—
86C2×D236—41no—
87C2×D236—42no—
88C2×C4×D59—42no—
89C2×C4×D59—42no—
90C2×C4×D59—41no—
91C2×C4×D59—41no—
92C2×C8×C59—42no—
93C2×C8×C59—82no—
94C4⋊C4×C59—41no—
95C22⋊C4×C59—41no—
96C22⋊C4×C59—41no—
97C42×C59—21no—
98C42×C59—21no—
99C22⋊Dic59—24no—
100C22⋊Dic59—42no—
101C22⋊Dic59—42no—
102C22⋊Dic59—42no—
103(C2×D59)⋊C4—42no—
104(C2×D59)⋊C4—42no—
105(C2×D59)⋊C4—42no—
106C4⋊Dic59—41no—
107Dic59⋊C4—42no—
108Dic59⋊C4—42no—
109C4×Dic59—24no—
110C4×Dic59—82no—
111C23×C59—21no—
112C23×C59—21no—
113C23×C59—21no—
114C23×C59—21no—
115C23×C59—41no—
116C23×C59—41no—
117C23×C59—41no—
118C23×C59—41no—
119C23×C59#10412yes—
120C23×C59C2×C22⋊C411yescharacteristic
121C23×C59#13512yes—
122C22×D59—22no—
123C22×D59—41no—
124C22×D59—41no—
125C22×D59—41no—
126C22×D59—41no—
127C22×D59—41no—
128C22×D59—41no—
129D4×C59—44no—
130C2×C4×C59—22no—
131C2×C4×C59—21no—
132C2×C4×C59—21no—
133C2×C4×C59—21no—
134C2×C4×C59—21no—
135C2×C4×C59—21no—
136C2×C4×C59—21no—
137C2×C4×C59—42no—
138C2×C4×C59—42no—
139C2×C4×C59—41no—
140C2×C4×C59—41no—
141C2×C4×C59—41no—
142C2×C4×C59—81no—
143C2×C4×C59—81no—
144C2×C4×C59#9911yescharacteristic
145C2×C4×C59#9911yescharacteristic
146C22⋊D59—48no—
147C22⋊D59—48no—
148C2×Dic59—22no—
149C2×Dic59—22no—
150C2×Dic59—41no—
151C2×Dic59—41no—
152C2×Dic59—41no—
153C2×Dic59—41no—
154C2×Dic59—41no—
155C2×Dic59—41no—
156C2×Dic59—44no—
157C2×Dic59—41no—
158C2×Dic59—82no—
159C2×Dic59—82no—
160D236—42no—
161D236—42no—
162C4×D59—82no—
163C4×D59—82no—
164C8×C59—82no—
165#11726—591nomaximal, Sylow
166C22×C59—21no—
167C22×C59—21no—
168C22×C59—22no—
169C22×C59—22no—
170C22×C59—42no—
171C22×C59—42no—
172C22×C59—41no—
173C22×C59—41no—
174C22×C59—41no—
175C22×C59—41no—
176C22×C59—81no—
177C22×C59—81no—
178C22×C59C2×#9911yescharacteristic
179C22×C59#41611yescharacteristic
180C22×C59C2×#10411yescharacteristic
181C2×D59—42no—
182C2×D59—42no—
183C2×D59—42no—
184C2×D59—81no—
185C2×D59—81no—
186C2×D59—81no—
187C2×D59—81no—
188C4×C59—21no—
189C4×C59—41no—
190C4×C59—41no—
191C4×C59—41no—
192C4×C59—41no—
193C4×C59#35312yes—
194Dic59—42no—
195Dic59—42no—
196Dic59—84no—
197Dic59—81no—
198#2397—591no—
199C4×#99—591no—
200#1459D5911yescharacteristic, complemented
201#1352—592no—
202#1343—592no—
203C2×C59—21no—
204C2×C59—21no—
205C2×C59—42no—
206C2×C59—42no—
207C2×C59#150111yescharacteristic
208C2×C59#212011yescharacteristic
209C2×C59C2×#35311yescharacteristic
210D59—82no—
211C2×C4×D4—591no—
212C2×C4×D4—1181no—
213C2×C4×D4—1181no—
214C2×C4×D4C2×D5911yescharacteristic
215C2×#99—1181no—
216C2×C22⋊C8—21no—
217#399—591no—
218#399—591no—
219#399—1181no—
220#392—591no—
221#392—591no—
222#392—1181no—
223C4×C22⋊C4—1181no—
224#376—591no—
225#342—592no—
226#324—21no—
227#323C2×D5912yes—
228C59#1172611yescharacteristic, complemented, Sylow
229C22×D4—591no—
230C22×D4—1181no—
231C22×D4—1181no—
232C22×D4D23611yescharacteristic
233C23×C4—1181no—
234C23×C4D23611yescharacteristic
235C23×C4C22⋊D5911yescharacteristic
236C22×C8—41no—
237C4×D4—41no—
238C4×D4—41no—
239C4×D4—2361no—
240C4×D4—2361no—
241C4×D4—2364no—
242C4×D4—2364no—
243C2×C4⋊C4—591no—
244C2×C4⋊C4—1181no—
245C2×C4⋊C4—1181no—
246C2×C4⋊C4C22⋊D5911yescharacteristic
247C2×C22⋊C4—1182no—
248C2×C22⋊C4—1181no—
249C2×C22⋊C4—1181no—
250C2×C22⋊C4—1181no—
251C2×C22⋊C4—1181no—
252C2×C22⋊C4—1181no—
253C2×C22⋊C4C4×D5911yescharacteristic
254C2×C22⋊C4C22×D5911yescharacteristic
255C2×C42—1181no—
256C2×C42—2361no—
257C2×C42C4×D5911yescharacteristic
258#99—1184no—
259C22⋊C8—22no—
260C22⋊C8—41no—
261#95—591no—
262#95—591no—
263#95—1181no—
264#95—2361no—
265C24—1181no—
266C24(C2×D59)⋊C411yescharacteristic
267C24C2×D23611yescharacteristic
268C2×D4—22no—
269C2×D4—22no—
270C2×D4—41no—
271C2×D4—2361no—
272C2×D4—2362no—
273C2×D4—2362no—
274C2×D4—2362no—
275C2×D4—2362no—
276C2×D4—2362no—
277C2×D4—2362no—
278C2×D4—2362no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—1181no—
286C22×C4—1181no—
287C22×C4—1181no—
288C22×C4—2362no—
289C22×C4—2362no—
290C22×C4—2362no—
291C22×C4—2361no—
292C22×C4—2361no—
293C22×C4(C2×D59)⋊C411yescharacteristic
294C22×C4(C2×D59)⋊C411yescharacteristic
295C22×C4(C2×D59)⋊C411yescharacteristic
296C22×C4C2×C4×D5911yescharacteristic
297C22×C4C2×C22⋊D5911yescharacteristic
298C2×C8—42no—
299C2×C8—82no—
300C4⋊C4—41no—
301C4⋊C4—2361no—
302C4⋊C4—2362no—
303C4⋊C4—2362no—
304C22⋊C4—41no—
305C22⋊C4—41no—
306C22⋊C4—1184no—
307C22⋊C4—2362no—
308C22⋊C4—2362no—
309C22⋊C4—2362no—
310C22⋊C4—2362no—
311C22⋊C4—2362no—
312C22⋊C4—2362no—
313C42—21no—
314C42—21no—
315C42—1184no—
316C42—4722no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—41no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—1182no—
326C23—2361no—
327C23—2361no—
328C23—2361no—
329C23—2361no—
330C23—2361no—
331C23—2361no—
332C23#1427365312yes—
333C23C2×(C2×D59)⋊C411yescharacteristic
334C23#1427374912yes—
335D4—44no—
336D4—2362no—
337D4—2362no—
338D4—2368no—
339D4—2368no—
340C2×C4—22no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—1182no—
355C2×C4—1182no—
356C2×C4—4722no—
357C2×C4—4722no—
358C2×C4—4722no—
359C2×C4—4722no—
360C2×C4—2364no—
361C2×C4—2361no—
362C2×C4—2361no—
363C2×C4—2361no—
364C2×C4—2361no—
365C2×C4—2361no—
366C2×C4—2361no—
367C2×C4—2361no—
368C2×C4#1427364111yescharacteristic
369C2×C4#1427364111yescharacteristic
370C8—82no—
371C22—21no—
372C22—21no—
373C22—22no—
374C22—22no—
375C22—42no—
376C22—42no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—41no—
381C22—81no—
382C22—81no—
383C22—4721no—
384C22—4721no—
385C22—4721no—
386C22—4721no—
387C22—2362no—
388C22—2362no—
389C22—2362no—
390C22C2×#1427364111yescharacteristic
391C22#1683739911yescharacteristic
392C22C2×#1427365311yescharacteristic
393C4—21no—
394C4—41no—
395C4—41no—
396C4—41no—
397C4—41no—
398C4—4724no—
399C4—4721no—
400C4—2362no—
401C4—2362no—
402C4#1683719412yes—
403C2—21no—
404C2—21no—
405C2—42no—
406C2—42no—
407C2—4722no—
408C2#2142258811yescharacteristic
409C2#2142573811yescharacteristic
410C2C2×#1683719411yescharacteristic
subgroupG/H
centerC2×C4#14273641
commutatorC22×C4×C59C22×C4
FrattiniC2×C22⋊C4C22×D59
FittingC59×#1459C2
socleC22×C59C2×#99
prime radical#21427375C2
cosocleC2×C22⋊C4×C59C23
hypercenter#1459D59
hypocenterC59#11726
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1459

Complements

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

NH ≅ G/Ntype
2. C4×#14273642C2semidirect
3. #21422121C2semidirect
4. #21422094C2semidirect
5. C59×#1459C2semidirect
6. C2×C4×D236C4semidirect
10. #16837338C4semidirect
13. #16837311C4semidirect
17. #16837248C4semidirect
200. #1459D59semidirect
228. C59#11726semidirect

Maximal subgroups

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

subgroupindex
#214273752
C4×#142736422
#214221212
#214220942
C59×#14592
#1172659