Finite Group Databasebeta

← #28499719

Subgroups — #28499719

Order 15104.

522 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 522 conjugacy classes of subgroups — too many to draw. Only its 95 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#21429052C211yesmaximal, characteristic
2#21424084C211yesmaximal, characteristic, complemented
3#21422097C212yesmaximal, complemented
4#21422094C212yesmaximal, complemented
5C59×#1460C211yesmaximal, characteristic, complemented
6C2×#14273701C411yescharacteristic, complemented
7C2×#14273699—21no—
8C2×#14273642—21no—
9#16837332C411yescharacteristic, complemented
10#16837327C2211yescharacteristic
11#16837319C411yescharacteristic, complemented
12#16837311C2211yescharacteristic
13#16837293—21no—
14#16837281—21no—
15#16837279—21no—
16#16837262—21no—
17C2×C59⋊#95C411yescharacteristic, complemented
18#16837169C2212yes—
19C2×C4×D4×C59C2211yescharacteristic
20C59×#431—21no—
21C59×#323—21no—
22C59×#323C2212yes—
23C22×D236C2×C411yescharacteristic
24C22×C22⋊D59—21no—
25C22×C22⋊D59—21no—
26C23×Dic59—21no—
27#14273701—41no—
28#14273701—41no—
29#14273699—42no—
30#14273699—42no—
31C2×C4⋊Dic59—41no—
32C2×C4⋊Dic59C2×C411yescharacteristic
33C2×Dic59⋊C4—21no—
34C2×C22⋊Dic59—21no—
35C2×C22⋊Dic59—22no—
36C2×(C2×D59)⋊C4—21no—
37C2×(C2×D59)⋊C4—21no—
38C2×(C2×D59)⋊C4—21no—
39C2×(C2×D59)⋊C4—21no—
40#14273642—42no—
41C59⋊#95—21no—
42C59⋊#95—21no—
43C59⋊#95—21no—
44C59⋊#95—41no—
45C59⋊#95C2×C411yescharacteristic
46C59⋊#95C2×C411yescharacteristic
47C22×D4×C59D411yescharacteristic
48C23×C4×C59C2×C411yescharacteristic
49C23×C4×C59D411yescharacteristic
50C4×D4×C59—41no—
51C4×D4×C59—41no—
52C2×C4⋊C4×C59C2×C411yescharacteristic
53C2×C22⋊C4×C59D411yescharacteristic
54C2×C22⋊C4×C59C2311yescharacteristic
55C2×C42×C59D411yescharacteristic
56C4⋊C8×C59—41no—
57C22⋊C8×C59—22no—
58C4×C8×C59—41no—
59C24×C59C22⋊C411yescharacteristic
60C24×C59C2×D411yescharacteristic
61C23×D59—21no—
62C2×D4×C59—41no—
63C2×D4×C59—41no—
64C2×D4×C59D812yes—
65C2×D4×C59SD1612yes—
66C22×C4×C59—21no—
67C22×C4×C59—21no—
68C22×C4×C59—21no—
69C22×C4×C59—21no—
70C22×C4×C59—81no—
71C22×C4×C59—81no—
72C22×C4×C59C22⋊C411yescharacteristic
73C22×C4×C59C22⋊C411yescharacteristic
74C22×C4×C59C22⋊C411yescharacteristic
75C22×C4×C59C22×C411yescharacteristic
76C22×C4×C59C2×D411yescharacteristic
77C2×C22⋊D59—42no—
78C2×C22⋊D59—42no—
79C2×C22⋊D59—42no—
80C2×C22⋊D59—42no—
81C2×C22⋊D59—41no—
82C2×C22⋊D59—41no—
83C2×C22⋊D59—81no—
84C22×Dic59—21no—
85C22×Dic59—21no—
86C22×Dic59—21no—
87C22×Dic59—42no—
88C22×Dic59—42no—
89C22×Dic59—42no—
90C22×Dic59—81no—
91C2×D236—41no—
92C2×D236—81no—
93C2×C8×C59—42no—
94C4⋊C4×C59—41no—
95C22⋊C4×C59—41no—
96C22⋊C4×C59—41no—
97C42×C59—21no—
98C42×C59—21no—
99C22⋊Dic59—41no—
100C22⋊Dic59—41no—
101C22⋊Dic59—42no—
102C22⋊Dic59—42no—
103(C2×D59)⋊C4—42no—
104(C2×D59)⋊C4—41no—
105(C2×D59)⋊C4—41no—
106(C2×D59)⋊C4—42no—
107(C2×D59)⋊C4—81no—
108C4⋊Dic59—41no—
109C4⋊Dic59—82no—
110Dic59⋊C4—42no—
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#10212yes—
120C23×C59C2×C22⋊C411yescharacteristic
121C23×C59C2×D811yescharacteristic
122C23×C59C2×SD1611yescharacteristic
123C22×D59—41no—
124C22×D59—41no—
125C22×D59—41no—
126C22×D59—41no—
127C22×D59—41no—
128C22×D59—41no—
129C22×D59—41no—
130D4×C59—42no—
131D4×C59—42no—
132C2×C4×C59—21no—
133C2×C4×C59—21no—
134C2×C4×C59—21no—
135C2×C4×C59—21no—
136C2×C4×C59—21no—
137C2×C4×C59—41no—
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—81no—
145C2×C4×C59#9911yescharacteristic
146C2×C4×C59#9911yescharacteristic
147C2×C4×C59#10212yes—
148C22⋊D59—82no—
149C22⋊D59—82no—
150C22⋊D59—84no—
151C2×Dic59—22no—
152C2×Dic59—41no—
153C2×Dic59—41no—
154C2×Dic59—41no—
155C2×Dic59—41no—
156C2×Dic59—41no—
157C2×Dic59—41no—
158C2×Dic59—41no—
159C2×Dic59—41no—
160C2×Dic59—84no—
161C2×Dic59—82no—
162C2×Dic59—82no—
163C2×Dic59—82no—
164D236—82no—
165C8×C59—82no—
166#11876—591nomaximal, Sylow
167C22×C59—21no—
168C22×C59—21no—
169C22×C59—21no—
170C22×C59—22no—
171C22×C59—21no—
172C22×C59—42no—
173C22×C59—41no—
174C22×C59—41no—
175C22×C59—42no—
176C22×C59—41no—
177C22×C59—41no—
178C22×C59—41no—
179C22×C59—41no—
180C22×C59—81no—
181C22×C59C2×#9911yescharacteristic
182C22×C59C2×#10211yescharacteristic
183C22×C59#42111yescharacteristic
184C2×D59—81no—
185C2×D59—81no—
186C2×D59—81no—
187C2×D59—81no—
188C2×D59—81no—
189C2×D59—81no—
190C2×D59—81no—
191C4×C59—21no—
192C4×C59—21no—
193C4×C59—22no—
194C4×C59—41no—
195C4×C59—41no—
196C4×C59—41no—
197Dic59—42no—
198Dic59—81no—
199Dic59—81no—
200Dic59—162no—
201#2543—591no—
202#1799—591no—
203#1460D5911yescharacteristic, complemented
204#1344—592no—
205#1343—592no—
206C2×C59—21no—
207C2×C59—21no—
208C2×C59—41no—
209C2×C59—42no—
210C2×C59—41no—
211C2×C59#151911yescharacteristic
212C2×C59#212111yescharacteristic
213C2×C59#212611yescharacteristic
214D59—161no—
215C2×#121—591no—
216C2×#121—1181no—
217C2×C4×D4C2×D5911yescharacteristic
218#431—21no—
219C2×#99—1181no—
220#398—591no—
221#397—591no—
222#394—591no—
223#392—591no—
224#388—1181no—
225#386—1181no—
226#386—1181no—
227#380—1181no—
228C2×#95—591no—
229#342—592no—
230#323—21no—
231#323C2×D5912yes—
232C59#1187611yescharacteristic, complemented, Sylow
233C22×D4—591no—
234C22×D4—1181no—
235C22×D4—1181no—
236C22×D4D23611yescharacteristic
237C23×C4—1181no—
238C23×C4C4×D5911yescharacteristic
239C23×C4D23611yescharacteristic
240#121—2362no—
241#121—2362no—
242#121—2361no—
243#121—2361no—
244C4×D4—41no—
245C4×D4—41no—
246C2×C4⋊C4—591no—
247C2×C4⋊C4—1181no—
248C2×C4⋊C4—2361no—
249C2×C4⋊C4C4×D5911yescharacteristic
250C2×C22⋊C4—1181no—
251C2×C22⋊C4—1181no—
252C2×C22⋊C4—1181no—
253C2×C22⋊C4—1181no—
254C2×C22⋊C4—1181no—
255C2×C22⋊C4—1182no—
256C2×C22⋊C4C22⋊D5911yescharacteristic
257C2×C22⋊C4C22×D5911yescharacteristic
258C2×C42C22⋊D5911yescharacteristic
259C4⋊C8—41no—
260#99—2362no—
261C22⋊C8—22no—
262C4×C8—41no—
263#95—591no—
264#95—591no—
265#95—1181no—
266#95—1181no—
267#95—1181no—
268#95—2361no—
269C24—1181no—
270C24(C2×D59)⋊C411yescharacteristic
271C24C2×D23611yescharacteristic
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—4721no—
275C2×D4—4721no—
276C2×D4—2361no—
277C2×D4—2362no—
278C2×D4—2362no—
279C2×D4—2362no—
280C2×D4—2362no—
281C2×D4—2361no—
282C2×D4—2361no—
283C2×D4#1174566012yes—
284C2×D4D47212yes—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—81no—
290C22×C4—81no—
291C22×C4—1181no—
292C22×C4—1181no—
293C22×C4—1181no—
294C22×C4—4721no—
295C22×C4—2362no—
296C22×C4—2362no—
297C22×C4—2362no—
298C22×C4(C2×D59)⋊C411yescharacteristic
299C22×C4(C2×D59)⋊C411yescharacteristic
300C22×C4(C2×D59)⋊C411yescharacteristic
301C22×C4C2×C4×D5911yescharacteristic
302C22×C4C2×C22⋊D5911yescharacteristic
303C2×C8—42no—
304C4⋊C4—41no—
305C4⋊C4—4722no—
306C4⋊C4—2361no—
307C4⋊C4—2362no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—4721no—
311C22⋊C4—2362no—
312C22⋊C4—2362no—
313C22⋊C4—2362no—
314C22⋊C4—2361no—
315C22⋊C4—2361no—
316C22⋊C4—2361no—
317C22⋊C4—2361no—
318C22⋊C4—2362no—
319C42—21no—
320C42—21no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—2361no—
330C23—2361no—
331C23—2361no—
332C23—2361no—
333C23—2361no—
334C23—2361no—
335C23—2361no—
336C23#1427364912yes—
337C23C2×(C2×D59)⋊C411yescharacteristic
338C23C2×D47211yescharacteristic
339C23C2×#1174566011yescharacteristic
340D4—42no—
341D4—42no—
342D4—4722no—
343D4—4724no—
344D4—4722no—
345D4—4722no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—1182no—
360C2×C4—4722no—
361C2×C4—4722no—
362C2×C4—4722no—
363C2×C4—4724no—
364C2×C4—2361no—
365C2×C4—2361no—
366C2×C4—2361no—
367C2×C4—2361no—
368C2×C4—2361no—
369C2×C4—2361no—
370C2×C4—2361no—
371C2×C4—2361no—
372C2×C4#1427364111yescharacteristic
373C2×C4#1427364111yescharacteristic
374C2×C4#1427364912yes—
375C8—82no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—22no—
380C22—21no—
381C22—42no—
382C22—41no—
383C22—41no—
384C22—42no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—81no—
390C22—4721no—
391C22—4721no—
392C22—4721no—
393C22—4721no—
394C22—4721no—
395C22—4721no—
396C22—4721no—
397C22C2×#1427364111yescharacteristic
398C22C2×#1427364911yescharacteristic
399C22#1683741811yescharacteristic
400C4—22no—
401C4—21no—
402C4—21no—
403C4—41no—
404C4—41no—
405C4—41no—
406C4—9442no—
407C4—4721no—
408C4—4721no—
409C4—2362no—
410C2—21no—
411C2—21no—
412C2—41no—
413C2—42no—
414C2—41no—
415C2—9441no—
416C2#2142266511yescharacteristic
417C2#2142574311yescharacteristic
418C2#2142575911yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC22×C4×C59C22×C4
FrattiniC2×C22⋊C4C22×D59
FittingC59×#1460C2
socleC22×C59C2×#99
prime radical#21429052C2
cosocleC2×C22⋊C4×C59C23
hypercenter#1460D59
hypocenterC59#11876
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1460

Complements

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

NH ≅ G/Ntype
2. #21424084C2semidirect
3. #21422097C2semidirect
4. #21422094C2semidirect
5. C59×#1460C2semidirect
6. C2×#14273701C4semidirect
9. #16837332C4semidirect
11. #16837319C4semidirect
17. C2×C59⋊#95C4semidirect
203. #1460D59semidirect
232. C59#11876semidirect

Maximal subgroups

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

subgroupindex
#214290522
#214240842
#214220972
#214220942
C59×#14602
#1187659