Finite Group Databasebeta

← #28499166

Subgroups — #28499166

Order 15104.

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

normal subgroups only

This group has 700 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#21429027C211yesmaximal, characteristic
2C2×#16837169C211yesmaximal, characteristic, complemented
3#21422121C212yesmaximal, complemented
4#21422094C212yesmaximal, complemented
5C59×#1462C211yesmaximal, characteristic, complemented
6C2×#14273710—21no—
7C2×#14273701C411yescharacteristic, complemented
8C2×#14273696—21no—
9C22×C22⋊Dic59—21no—
10#16837338C411yescharacteristic, complemented
11#16837338C2211yescharacteristic
12#16837311C411yescharacteristic, complemented
13#16837311C2211yescharacteristic
14#16837302—21no—
15#16837302—21no—
16C4×C22⋊Dic59C411yescharacteristic, complemented
17#16837169—21no—
18#16837169C2212yes—
19C22×C22⋊C4×C59C2211yescharacteristic
20C59×#407—21no—
21C59×#323—21no—
22C59×#323C2212yes—
23C22×D236C2×C411yescharacteristic, complemented
24C22×C22⋊D59—21no—
25C22×C22⋊D59—21no—
26C22×C22⋊D59—21no—
27C23×Dic59—41no—
28#14273710—44no—
29#14273701—41no—
30#14273701—41no—
31#14273696—44no—
32C2×C4⋊Dic59C2×C411yescharacteristic, complemented
33C2×Dic59⋊C4—21no—
34C2×Dic59⋊C4—21no—
35C2×C22⋊Dic59—21no—
36C2×C22⋊Dic59—24no—
37C2×C22⋊Dic59—22no—
38C2×C22⋊Dic59—21no—
39C2×C22⋊Dic59—21no—
40C2×C22⋊Dic59—41no—
41C2×C22⋊Dic59—42no—
42C2×(C2×D59)⋊C4—21no—
43C2×(C2×D59)⋊C4—21no—
44C2×(C2×D59)⋊C4—21no—
45C2×C4×Dic59—21no—
46C59⋊#95C2×C411yescharacteristic, complemented
47C59⋊#95C2×C411yescharacteristic, complemented
48C25×C59D411yescharacteristic
49C23×C4×C59C2×C411yescharacteristic
50C23×C4×C59D411yescharacteristic
51C2×M16×C59—41no—
52C2×C22⋊C4×C59—41no—
53C2×C22⋊C4×C59—41no—
54C2×C22⋊C4×C59C2×C411yescharacteristic
55C2×C22⋊C4×C59D411yescharacteristic
56C2×C22⋊C4×C59D411yescharacteristic
57C2×C22⋊C4×C59C2311yescharacteristic
58C22⋊C8×C59—22no—
59C22⋊C8×C59—41no—
60C24×C59—21no—
61C24×C59—21no—
62C24×C59—22no—
63C24×C59—21no—
64C24×C59—21no—
65C24×C59—42no—
66C24×C59—81no—
67C24×C59C22⋊C411yescharacteristic
68C24×C59C22⋊C411yescharacteristic
69C24×C59C2×D411yescharacteristic
70C23×D59—21no—
71C22×C4×C59—21no—
72C22×C4×C59—21no—
73C22×C4×C59—21no—
74C22×C4×C59—21no—
75C22×C4×C59—42no—
76C22×C4×C59—81no—
77C22×C4×C59C22⋊C411yescharacteristic
78C22×C4×C59C22⋊C411yescharacteristic
79C22×C4×C59C22×C411yescharacteristic
80C22×C4×C59C2×D411yescharacteristic
81C2×C22⋊D59—42no—
82C2×C22⋊D59—42no—
83C2×C22⋊D59—42no—
84C2×C22⋊D59—42no—
85C2×C22⋊D59—41no—
86C2×C22⋊D59—41no—
87C2×C22⋊D59—42no—
88C2×C22⋊D59—41no—
89C2×C22⋊D59—41no—
90C2×C22⋊D59—81no—
91C2×C22⋊D59—81no—
92C22×Dic59—21no—
93C22×Dic59—21no—
94C22×Dic59—21no—
95C22×Dic59—44no—
96C22×Dic59—42no—
97C22×Dic59—82no—
98C22×Dic59—82no—
99C2×D236—41no—
100C2×D236—81no—
101M16×C59—82no—
102C2×C8×C59—42no—
103C22⋊C4×C59—41no—
104C22⋊C4×C59—41no—
105C22⋊C4×C59—41no—
106C22⋊C4×C59—41no—
107C22⋊Dic59—24no—
108C22⋊Dic59—24no—
109C22⋊Dic59—42no—
110C22⋊Dic59—42no—
111C22⋊Dic59—44no—
112C22⋊Dic59—42no—
113(C2×D59)⋊C4—42no—
114(C2×D59)⋊C4—42no—
115(C2×D59)⋊C4—42no—
116C4⋊Dic59—41no—
117Dic59⋊C4—42no—
118Dic59⋊C4—42no—
119C4×Dic59—81no—
120C23×C59—22no—
121C23×C59—22no—
122C23×C59—22no—
123C23×C59—22no—
124C23×C59—22no—
125C23×C59—21no—
126C23×C59—21no—
127C23×C59—21no—
128C23×C59—21no—
129C23×C59—21no—
130C23×C59—21no—
131C23×C59—41no—
132C23×C59—42no—
133C23×C59—41no—
134C23×C59—41no—
135C23×C59—41no—
136C23×C59—41no—
137C23×C59—42no—
138C23×C59—81no—
139C23×C59—81no—
140C23×C59—84no—
141C23×C59—82no—
142C23×C59—82no—
143C23×C59#9911yescharacteristic
144C23×C59#9911yescharacteristic
145C23×C59#10412yes—
146C23×C59C2×C22⋊C411yescharacteristic
147C23×C59AGL(1,ℤ/8)11yescharacteristic
148C23×C59#13711yescharacteristic
149C22×D59—41no—
150C22×D59—41no—
151C22×D59—41no—
152C22×D59—41no—
153C22×D59—41no—
154C22×D59—41no—
155C22×D59—41no—
156C2×C4×C59—21no—
157C2×C4×C59—21no—
158C2×C4×C59—21no—
159C2×C4×C59—21no—
160C2×C4×C59—42no—
161C2×C4×C59—41no—
162C2×C4×C59—41no—
163C2×C4×C59—41no—
164C2×C4×C59—41no—
165C2×C4×C59—81no—
166C2×C4×C59—81no—
167C2×C4×C59—81no—
168C22⋊D59—84no—
169C22⋊D59—82no—
170C22⋊D59—82no—
171C22⋊D59—82no—
172C22⋊D59—82no—
173C2×Dic59—22no—
174C2×Dic59—22no—
175C2×Dic59—41no—
176C2×Dic59—41no—
177C2×Dic59—41no—
178C2×Dic59—41no—
179C2×Dic59—41no—
180C2×Dic59—44no—
181C2×Dic59—41no—
182C2×Dic59—41no—
183C2×Dic59—88no—
184C2×Dic59—82no—
185C2×Dic59—82no—
186D236—82no—
187C8×C59—82no—
188#11757—591nomaximal, Sylow
189C22×C59—21no—
190C22×C59—21no—
191C22×C59—21no—
192C22×C59—21no—
193C22×C59—21no—
194C22×C59—21no—
195C22×C59—21no—
196C22×C59—21no—
197C22×C59—22no—
198C22×C59—21no—
199C22×C59—42no—
200C22×C59—42no—
201C22×C59—42no—
202C22×C59—41no—
203C22×C59—41no—
204C22×C59—44no—
205C22×C59—44no—
206C22×C59—41no—
207C22×C59—41no—
208C22×C59—41no—
209C22×C59—41no—
210C22×C59—81no—
211C22×C59—81no—
212C22×C59—82no—
213C22×C59—82no—
214C22×C59#35312yes—
215C22×C59C2×#9911yescharacteristic
216C22×C59#41711yescharacteristic
217C22×C59C2×#10411yescharacteristic
218C2×D59—81no—
219C2×D59—81no—
220C2×D59—81no—
221C2×D59—81no—
222C2×D59—81no—
223C2×D59—81no—
224C2×D59—81no—
225C4×C59—41no—
226C4×C59—41no—
227C4×C59—41no—
228C4×C59—41no—
229Dic59—42no—
230Dic59—42no—
231Dic59—84no—
232Dic59—81no—
233#2539—591no—
234C2×#342—591no—
235#1462D5911yescharacteristic, complemented
236#1352—592no—
237#1343—592no—
238C2×C59—21no—
239C2×C59—21no—
240C2×C59—22no—
241C2×C59—41no—
242C2×C59—41no—
243C2×C59—42no—
244C2×C59—41no—
245C2×C59#150411yescharacteristic
246C2×C59#212111yescharacteristic
247C2×C59C2×#35311yescharacteristic
248D59—161no—
249C2×#123—1181no—
250C2×#121—591no—
251C2×#120—1181no—
252C22×C22⋊C4—1181no—
253C22×C22⋊C4C2×D5911yescharacteristic
254#407—21no—
255#399—591no—
256#399—591no—
257#392—591no—
258#392—591no—
259#390—1181no—
260#390—1181no—
261C4×C22⋊C4—591no—
262#342—592no—
263#342—1181no—
264#323—21no—
265#323C2×D5912yes—
266C59#1175711yescharacteristic, complemented, Sylow
267C25D23611yescharacteristic
268C22×D4—591no—
269C22×D4—1181no—
270C22×D4—1181no—
271C22×D4—1181no—
272C23×C4—2361no—
273C23×C4C4×D5911yescharacteristic
274C23×C4C22⋊D5911yescharacteristic
275C2×M16—41no—
276#123—2364no—
277#121—2361no—
278#121—2361no—
279#120—2364no—
280C2×C4⋊C4—591no—
281C2×C4⋊C4—1181no—
282C2×C4⋊C4—1181no—
283C2×C22⋊C4—41no—
284C2×C22⋊C4—41no—
285C2×C22⋊C4—1181no—
286C2×C22⋊C4—1184no—
287C2×C22⋊C4—1182no—
288C2×C22⋊C4—1181no—
289C2×C22⋊C4—1181no—
290C2×C22⋊C4—1181no—
291C2×C22⋊C4—1181no—
292C2×C22⋊C4—1181no—
293C2×C22⋊C4—2361no—
294C2×C22⋊C4—2362no—
295C2×C22⋊C4C4×D5911yescharacteristic
296C2×C22⋊C4D23611yescharacteristic
297C2×C22⋊C4C22⋊D5911yescharacteristic
298C2×C22⋊C4C22×D5911yescharacteristic
299C2×C42—1181no—
300C22⋊C8—22no—
301C22⋊C8—41no—
302#95—591no—
303#95—591no—
304C24—21no—
305C24—21no—
306C24—22no—
307C24—21no—
308C24—21no—
309C24—42no—
310C24—81no—
311C24—1181no—
312C24(C2×D59)⋊C411yescharacteristic
313C24(C2×D59)⋊C411yescharacteristic
314C24C2×D23611yescharacteristic
315C2×D4—4721no—
316C2×D4—4721no—
317C2×D4—4721no—
318C2×D4—2361no—
319C2×D4—2362no—
320C2×D4—2362no—
321C2×D4—2362no—
322C2×D4—2362no—
323C2×D4—2361no—
324C2×D4—2361no—
325C2×D4—2362no—
326C2×D4—2361no—
327C2×D4—2361no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—42no—
333C22×C4—81no—
334C22×C4—1181no—
335C22×C4—1181no—
336C22×C4—1181no—
337C22×C4—4722no—
338C22×C4—4722no—
339C22×C4—2364no—
340C22×C4—2362no—
341C22×C4(C2×D59)⋊C411yescharacteristic
342C22×C4(C2×D59)⋊C411yescharacteristic
343C22×C4C2×C4×D5911yescharacteristic
344C22×C4C2×C22⋊D5911yescharacteristic
345M16—82no—
346C2×C8—42no—
347C4⋊C4—2361no—
348C4⋊C4—2362no—
349C4⋊C4—2362no—
350C22⋊C4—41no—
351C22⋊C4—41no—
352C22⋊C4—41no—
353C22⋊C4—41no—
354C22⋊C4—1184no—
355C22⋊C4—1184no—
356C22⋊C4—2362no—
357C22⋊C4—2362no—
358C22⋊C4—2362no—
359C22⋊C4—2362no—
360C22⋊C4—2362no—
361C22⋊C4—2364no—
362C22⋊C4—2362no—
363C42—4721no—
364C23—22no—
365C23—22no—
366C23—22no—
367C23—22no—
368C23—21no—
369C23—21no—
370C23—22no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—41no—
376C23—42no—
377C23—41no—
378C23—41no—
379C23—41no—
380C23—41no—
381C23—42no—
382C23—81no—
383C23—81no—
384C23—84no—
385C23—82no—
386C23—82no—
387C23—2361no—
388C23—2361no—
389C23—2361no—
390C23—2361no—
391C23—2361no—
392C23—2361no—
393C23—2361no—
394C23#1427364111yescharacteristic
395C23#1427364111yescharacteristic
396C23#1427365312yes—
397C23C2×(C2×D59)⋊C411yescharacteristic
398C23#1427375511yescharacteristic
399C23#1427376011yescharacteristic
400D4—4722no—
401D4—4722no—
402D4—4722no—
403D4—4724no—
404D4—4722no—
405D4—4722no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—42no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—1182no—
419C2×C4—1182no—
420C2×C4—4728no—
421C2×C4—4722no—
422C2×C4—4722no—
423C2×C4—2364no—
424C2×C4—2361no—
425C2×C4—2361no—
426C2×C4—2361no—
427C2×C4—2361no—
428C2×C4—2361no—
429C2×C4—2361no—
430C2×C4—2361no—
431C8—82no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—22no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—42no—
443C22—42no—
444C22—42no—
445C22—41no—
446C22—41no—
447C22—44no—
448C22—44no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—81no—
454C22—81no—
455C22—82no—
456C22—82no—
457C22—4721no—
458C22—4721no—
459C22—4721no—
460C22—4721no—
461C22—4721no—
462C22—4721no—
463C22—4721no—
464C22#1683719412yes—
465C22C2×#1427364111yescharacteristic
466C22#1683740411yescharacteristic
467C22C2×#1427365311yescharacteristic
468C4—41no—
469C4—41no—
470C4—41no—
471C4—41no—
472C4—4724no—
473C4—4721no—
474C4—2362no—
475C4—2362no—
476C2—21no—
477C2—21no—
478C2—22no—
479C2—41no—
480C2—41no—
481C2—42no—
482C2—41no—
483C2—9441no—
484C2#2142260111yescharacteristic
485C2#2142574311yescharacteristic
486C2C2×#1683719411yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC22×C4×C59C22×C4
FrattiniC2×C22⋊C4C22×D59
FittingC59×#1462C2
socleC22×C59C2×#99
prime radical#21429027C2
cosocleC2×C22⋊C4×C59C23
hypercenter#1462D59
hypocenterC59#11757
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ #1462

Complements

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

NH ≅ G/Ntype
2. C2×#16837169C2semidirect
3. #21422121C2semidirect
4. #21422094C2semidirect
5. C59×#1462C2semidirect
7. C2×#14273701C4semidirect
10. #16837338C4semidirect
12. #16837311C4semidirect
16. C4×C22⋊Dic59C4semidirect
23. C22×D236C2×C4semidirect
32. C2×C4⋊Dic59C2×C4semidirect
46. C59⋊#95C2×C4semidirect
47. C59⋊#95C2×C4semidirect
235. #1462D59semidirect
266. C59#11757semidirect

Maximal subgroups

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

subgroupindex
#214290272
C2×#168371692
#214221212
#214220942
C59×#14622
#1175759