Finite Group Databasebeta

← #28499568

Subgroups — #28499568

Order 15104.

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

normal subgroups only

This group has 620 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#21429027C211yesmaximal, characteristic
2#21424170C211yesmaximal, characteristic, complemented
3#21422121C212yesmaximal, complemented
4#21422094C212yesmaximal, complemented
5C2×C59×#323C211yesmaximal, characteristic, complemented
6C2×#14273710—21no—
7C2×#14273701C411yescharacteristic, complemented
8C2×#14273696—21no—
9#16837338C411yescharacteristic, complemented
10#16837338C2211yescharacteristic
11#16837311C411yescharacteristic, complemented
12#16837311C2211yescharacteristic
13#16837302—21no—
14#16837302—21no—
15#16837258—21no—
16C4×C22⋊Dic59C411yescharacteristic, complemented
17#16837169—21no—
18#16837169C2212yes—
19C22×C22⋊C4×C59C2211yescharacteristic
20C2×C22⋊C8×C59—21no—
21C59×#323—21no—
22C59×#323C2212yes—
23C22×D236C2×C411yescharacteristic
24C22×C22⋊D59—21no—
25C22×C22⋊D59—21no—
26C22×C22⋊D59—21no—
27#14273710—44no—
28#14273701—41no—
29#14273701—41no—
30#14273696—44no—
31C2×C4⋊Dic59C2×C411yescharacteristic
32C2×Dic59⋊C4—21no—
33C2×Dic59⋊C4—21no—
34C2×C22⋊Dic59—21no—
35C2×C22⋊Dic59—21no—
36C2×C22⋊Dic59—21no—
37C2×C22⋊Dic59—22no—
38C2×C22⋊Dic59—41no—
39C2×C22⋊Dic59—41no—
40C2×(C2×D59)⋊C4—21no—
41C2×(C2×D59)⋊C4—21no—
42C2×(C2×D59)⋊C4—21no—
43C2×C4×Dic59—21no—
44C59⋊#95C2×C411yescharacteristic
45C59⋊#95C2×C411yescharacteristic
46C25×C59D411yescharacteristic
47C23×C4×C59C2×C411yescharacteristic
48C23×C4×C59D411yescharacteristic
49C22×C8×C59—41no—
50C2×C22⋊C4×C59—41no—
51C2×C22⋊C4×C59—41no—
52C2×C22⋊C4×C59C2×C411yescharacteristic
53C2×C22⋊C4×C59D411yescharacteristic
54C2×C22⋊C4×C59D411yescharacteristic
55C2×C22⋊C4×C59C2311yescharacteristic
56C22⋊C8×C59—22no—
57C22⋊C8×C59—41no—
58C24×C59—21no—
59C24×C59—21no—
60C24×C59—22no—
61C24×C59—41no—
62C24×C59—42no—
63C24×C59—81no—
64C24×C59C22⋊C411yescharacteristic
65C24×C59C22⋊C411yescharacteristic
66C24×C59C2×D411yescharacteristic
67C23×D59—21no—
68C22×C4×C59—21no—
69C22×C4×C59—21no—
70C22×C4×C59—21no—
71C22×C4×C59—21no—
72C22×C4×C59—42no—
73C22×C4×C59—81no—
74C22×C4×C59C22⋊C411yescharacteristic
75C22×C4×C59C22⋊C411yescharacteristic
76C22×C4×C59C22×C411yescharacteristic
77C22×C4×C59C2×D411yescharacteristic
78C2×C22⋊D59—42no—
79C2×C22⋊D59—42no—
80C2×C22⋊D59—42no—
81C2×C22⋊D59—42no—
82C2×C22⋊D59—42no—
83C2×C22⋊D59—41no—
84C2×C22⋊D59—41no—
85C2×C22⋊D59—42no—
86C2×C22⋊D59—81no—
87C2×C22⋊D59—81no—
88C22×Dic59—21no—
89C22×Dic59—21no—
90C22×Dic59—21no—
91C22×Dic59—42no—
92C2×D236—41no—
93C2×D236—81no—
94C2×C8×C59—42no—
95C2×C8×C59—82no—
96C22⋊C4×C59—41no—
97C22⋊C4×C59—41no—
98C22⋊C4×C59—41no—
99C22⋊C4×C59—41no—
100C22⋊Dic59—24no—
101C22⋊Dic59—42no—
102C22⋊Dic59—42no—
103C22⋊Dic59—42no—
104C22⋊Dic59—42no—
105C22⋊Dic59—82no—
106C22⋊Dic59—82no—
107(C2×D59)⋊C4—42no—
108(C2×D59)⋊C4—42no—
109(C2×D59)⋊C4—42no—
110C4⋊Dic59—41no—
111Dic59⋊C4—42no—
112Dic59⋊C4—42no—
113C4×Dic59—81no—
114C23×C59—22no—
115C23×C59—21no—
116C23×C59—21no—
117C23×C59—21no—
118C23×C59—21no—
119C23×C59—21no—
120C23×C59—21no—
121C23×C59—21no—
122C23×C59—41no—
123C23×C59—42no—
124C23×C59—42no—
125C23×C59—41no—
126C23×C59—41no—
127C23×C59—41no—
128C23×C59—41no—
129C23×C59—42no—
130C23×C59—42no—
131C23×C59—81no—
132C23×C59—81no—
133C23×C59—84no—
134C23×C59—84no—
135C23×C59#9911yescharacteristic
136C23×C59#9911yescharacteristic
137C23×C59C2×C22⋊C411yescharacteristic
138C23×C59#13512yes—
139C22×D59—41no—
140C22×D59—41no—
141C22×D59—41no—
142C22×D59—41no—
143C22×D59—41no—
144C22×D59—41no—
145C22×D59—41no—
146C2×C4×C59—21no—
147C2×C4×C59—21no—
148C2×C4×C59—21no—
149C2×C4×C59—21no—
150C2×C4×C59—42no—
151C2×C4×C59—41no—
152C2×C4×C59—41no—
153C2×C4×C59—41no—
154C2×C4×C59—41no—
155C2×C4×C59—81no—
156C2×C4×C59—81no—
157C2×C4×C59—81no—
158C22⋊D59—84no—
159C22⋊D59—84no—
160C22⋊D59—82no—
161C22⋊D59—82no—
162C2×Dic59—22no—
163C2×Dic59—22no—
164C2×Dic59—41no—
165C2×Dic59—41no—
166C2×Dic59—41no—
167C2×Dic59—41no—
168C2×Dic59—41no—
169C2×Dic59—41no—
170C2×Dic59—41no—
171C2×Dic59—82no—
172C2×Dic59—82no—
173C2×Dic59—82no—
174D236—82no—
175C8×C59—82no—
176#11843—591nomaximal, Sylow
177C22×C59—21no—
178C22×C59—21no—
179C22×C59—21no—
180C22×C59—21no—
181C22×C59—21no—
182C22×C59—21no—
183C22×C59—22no—
184C22×C59—21no—
185C22×C59—42no—
186C22×C59—42no—
187C22×C59—42no—
188C22×C59—42no—
189C22×C59—41no—
190C22×C59—41no—
191C22×C59—41no—
192C22×C59—41no—
193C22×C59—41no—
194C22×C59—81no—
195C22×C59—84no—
196C22×C59—81no—
197C22×C59—84no—
198C22×C59#35312yes—
199C22×C59C2×#9911yescharacteristic
200C22×C59#41611yescharacteristic
201C22×C59#42111yescharacteristic
202C2×D59—81no—
203C2×D59—81no—
204C2×D59—81no—
205C2×D59—81no—
206C2×D59—81no—
207C2×D59—81no—
208C2×D59—81no—
209C4×C59—41no—
210C4×C59—41no—
211C4×C59—41no—
212C4×C59—41no—
213Dic59—42no—
214Dic59—42no—
215Dic59—81no—
216Dic59—162no—
217#2539—591no—
218#1819—591no—
219C2×#323D5911yescharacteristic, complemented
220#1352—592no—
221#1343—592no—
222C2×C59—21no—
223C2×C59—21no—
224C2×C59—22no—
225C2×C59—42no—
226C2×C59—41no—
227C2×C59—81no—
228C2×C59#151611yescharacteristic
229C2×C59#212111yescharacteristic
230C2×C59C2×#35311yescharacteristic
231D59—161no—
232C2×#123—1181no—
233C2×#121—591no—
234C2×#120—1181no—
235C22×C22⋊C4C2×D5911yescharacteristic
236C2×C22⋊C8—21no—
237#399—591no—
238#399—591no—
239#392—591no—
240#392—591no—
241#390—1181no—
242#390—1181no—
243#379—1181no—
244C4×C22⋊C4—591no—
245#342—592no—
246#342—1181no—
247#323—21no—
248#323C2×D5912yes—
249C59#1184311yescharacteristic, complemented, Sylow
250C25D23611yescharacteristic
251C22×D4—591no—
252C22×D4—1181no—
253C22×D4—1181no—
254C22×D4—1181no—
255C23×C4C4×D5911yescharacteristic
256C23×C4C22⋊D5911yescharacteristic
257C22×C8—41no—
258#123—2364no—
259#121—2361no—
260#121—2361no—
261#120—2364no—
262C2×C4⋊C4—591no—
263C2×C4⋊C4—1181no—
264C2×C4⋊C4—1181no—
265C2×C22⋊C4—41no—
266C2×C22⋊C4—41no—
267C2×C22⋊C4—1181no—
268C2×C22⋊C4—1181no—
269C2×C22⋊C4—1181no—
270C2×C22⋊C4—1181no—
271C2×C22⋊C4—1181no—
272C2×C22⋊C4—1181no—
273C2×C22⋊C4—1182no—
274C2×C22⋊C4—2361no—
275C2×C22⋊C4—2361no—
276C2×C22⋊C4C4×D5911yescharacteristic
277C2×C22⋊C4D23611yescharacteristic
278C2×C22⋊C4C22⋊D5911yescharacteristic
279C2×C22⋊C4C22×D5911yescharacteristic
280C2×C42—1181no—
281C22⋊C8—22no—
282C22⋊C8—41no—
283#95—591no—
284#95—591no—
285C24—21no—
286C24—21no—
287C24—22no—
288C24—41no—
289C24—42no—
290C24—81no—
291C24—1181no—
292C24(C2×D59)⋊C411yescharacteristic
293C24(C2×D59)⋊C411yescharacteristic
294C24C2×D23611yescharacteristic
295C2×D4—4721no—
296C2×D4—4721no—
297C2×D4—4721no—
298C2×D4—2361no—
299C2×D4—2362no—
300C2×D4—2362no—
301C2×D4—2362no—
302C2×D4—2362no—
303C2×D4—2362no—
304C2×D4—2361no—
305C2×D4—2361no—
306C2×D4—2362no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—42no—
312C22×C4—81no—
313C22×C4—1181no—
314C22×C4—1181no—
315C22×C4—1181no—
316C22×C4—2362no—
317C22×C4(C2×D59)⋊C411yescharacteristic
318C22×C4(C2×D59)⋊C411yescharacteristic
319C22×C4C2×C4×D5911yescharacteristic
320C22×C4C2×C22⋊D5911yescharacteristic
321C2×C8—42no—
322C2×C8—82no—
323C4⋊C4—2361no—
324C4⋊C4—2362no—
325C4⋊C4—2362no—
326C22⋊C4—41no—
327C22⋊C4—41no—
328C22⋊C4—41no—
329C22⋊C4—41no—
330C22⋊C4—1184no—
331C22⋊C4—4722no—
332C22⋊C4—4722no—
333C22⋊C4—2362no—
334C22⋊C4—2362no—
335C22⋊C4—2362no—
336C22⋊C4—2362no—
337C22⋊C4—2362no—
338C22⋊C4—2362no—
339C22⋊C4—2362no—
340C42—4721no—
341C23—21no—
342C23—21no—
343C23—22no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—41no—
350C23—42no—
351C23—42no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—42no—
357C23—42no—
358C23—81no—
359C23—81no—
360C23—84no—
361C23—84no—
362C23—2361no—
363C23—2361no—
364C23—2361no—
365C23—2361no—
366C23—2361no—
367C23—2361no—
368C23—2361no—
369C23#1427364111yescharacteristic
370C23#1427364111yescharacteristic
371C23C2×(C2×D59)⋊C411yescharacteristic
372C23#1427374912yes—
373D4—4722no—
374D4—4722no—
375D4—4722no—
376D4—4724no—
377D4—4724no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—42no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—1182no—
391C2×C4—1182no—
392C2×C4—4722no—
393C2×C4—4722no—
394C2×C4—4722no—
395C2×C4—2361no—
396C2×C4—2361no—
397C2×C4—2361no—
398C2×C4—2361no—
399C2×C4—2361no—
400C2×C4—2361no—
401C2×C4—2361no—
402C8—82no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—22no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—42no—
412C22—42no—
413C22—42no—
414C22—42no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22—41no—
420C22—81no—
421C22—84no—
422C22—81no—
423C22—84no—
424C22—4721no—
425C22—4721no—
426C22—4721no—
427C22—4721no—
428C22—4721no—
429C22—4721no—
430C22—4721no—
431C22#1683719412yes—
432C22C2×#1427364111yescharacteristic
433C22#1683739911yescharacteristic
434C22#1683741811yescharacteristic
435C4—41no—
436C4—41no—
437C4—41no—
438C4—41no—
439C4—9442no—
440C4—4721no—
441C4—2362no—
442C4—2362no—
443C2—21no—
444C2—21no—
445C2—22no—
446C2—42no—
447C2—41no—
448C2—81no—
449C2—9441no—
450C2#2142265311yescharacteristic
451C2#2142574311yescharacteristic
452C2C2×#1683719411yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC22×C4×C59C22×C4
FrattiniC2×C22⋊C4C22×D59
FittingC2×C59×#323C2
socleC22×C59C2×#99
prime radical#21429027C2
cosocleC2×C22⋊C4×C59C23
hypercenterC2×#323D59
hypocenterC59#11843
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ C2×#323

Complements

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

NH ≅ G/Ntype
2. #21424170C2semidirect
3. #21422121C2semidirect
4. #21422094C2semidirect
5. C2×C59×#323C2semidirect
7. C2×#14273701C4semidirect
9. #16837338C4semidirect
11. #16837311C4semidirect
16. C4×C22⋊Dic59C4semidirect
219. C2×#323D59semidirect
249. C59#11843semidirect

Maximal subgroups

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

subgroupindex
#214290272
#214241702
#214221212
#214220942
C2×C59×#3232
#1184359