Finite Group Databasebeta

← #28482062

Subgroups — #28482062

Order 15104.

940 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21424006C211yesmaximal, characteristic
2#21422319C211yesmaximal, characteristic
3#21421974C214yesmaximal, complemented
4C2×C59×#380C211yesmaximal, characteristic
5C22×C22⋊Dic59—21no—
6C2×C22⋊(C59⋊C8)—21no—
7C2×C22⋊(C59⋊C8)—21no—
8C2×C22⋊(C59⋊C8)C2212yes—
9C2×C59⋊#95—21no—
10C2×C59⋊#95C2212yes—
11C24×C4×C59C2211yescharacteristic
12C22×C4⋊C4×C59C411yescharacteristic, complemented
13C22×C4⋊C4×C59C411yescharacteristic, complemented
14C22×C22⋊C4×C59—21no—
15C59×#380—21no—
16C59×#380—21no—
17C59×#380C412yescomplemented
18C59×#380C2212yes—
19C2×#95×C59—21no—
20C23×Dic59—22no—
21C22×C59⋊C8—22no—
22C2×C22⋊Dic59—24no—
23C2×C22⋊Dic59—42no—
24C2×C22⋊Dic59—42no—
25C22⋊(C59⋊C8)—24no—
26C22⋊(C59⋊C8)—42no—
27C22⋊(C59⋊C8)—42no—
28C59⋊#95—24no—
29C59⋊#95—42no—
30C25×C59D411yescharacteristic
31C23×C4×C59—24no—
32C23×C4×C59—21no—
33C23×C4×C59—21no—
34C23×C4×C59—42no—
35C23×C4×C59—42no—
36C23×C4×C59C2×C411yescharacteristic
37C23×C4×C59C2×C411yescharacteristic
38C23×C4×C59D411yescharacteristic
39C23×C4×C59D411yescharacteristic
40C23×C4×C59D411yescharacteristic
41C23×C4×C59C2311yescharacteristic
42C2×C4⋊C4×C59—21no—
43C2×C4⋊C4×C59—21no—
44C2×C4⋊C4×C59—42no—
45C2×C4⋊C4×C59—42no—
46C2×C4⋊C4×C59C2×C412yes—
47C2×C4⋊C4×C59C2×C412yes—
48C2×C22⋊C4×C59—22no—
49C2×C22⋊C4×C59—22no—
50C2×C22⋊C4×C59—42no—
51C2×C22⋊C4×C59—42no—
52#95×C59—22no—
53#95×C59—22no—
54C24×C59—22no—
55C24×C59—22no—
56C24×C59—22no—
57C24×C59—22no—
58C24×C59—41no—
59C24×C59—41no—
60C24×C59C22⋊C411yescharacteristic
61C24×C59C22⋊C411yescharacteristic
62C24×C59D812yes—
63C24×C59SD1612yes—
64C24×C59C2×D411yescharacteristic
65C22×C4×C59—22no—
66C22×C4×C59—22no—
67C22×C4×C59—22no—
68C22×C4×C59—22no—
69C22×C4×C59—22no—
70C22×C4×C59—22no—
71C22×C4×C59—22no—
72C22×C4×C59—22no—
73C22×C4×C59—22no—
74C22×C4×C59—22no—
75C22×C4×C59—22no—
76C22×C4×C59—22no—
77C22×C4×C59—41no—
78C22×C4×C59—41no—
79C22×C4×C59—41no—
80C22×C4×C59—41no—
81C22×C4×C59—41no—
82C22×C4×C59—41no—
83C22×C4×C59—41no—
84C22×C4×C59—416no—
85C22×C4×C59—42no—
86C22×C4×C59—42no—
87C22×C4×C59—42no—
88C22×C4×C59—42no—
89C22×C4×C59—42no—
90C22×C4×C59—42no—
91C22×C4×C59C22⋊C411yescharacteristic
92C22×C4×C59C22⋊C411yescharacteristic
93C22×C4×C59C22×C411yescharacteristic
94C22×C4×C59C2×D411yescharacteristic
95C22×C4×C59#4114yes—
96C22×Dic59—24no—
97C22×Dic59—42no—
98C22×Dic59—42no—
99C22×Dic59—42no—
100C22×Dic59—42no—
101C22×Dic59—42no—
102C22×Dic59—42no—
103C4⋊C4×C59—42no—
104C4⋊C4×C59—42no—
105C4⋊C4×C59—42no—
106C4⋊C4×C59—42no—
107C22⋊C4×C59—44no—
108C22⋊C4×C59—42no—
109C22⋊C4×C59—42no—
110C22⋊Dic59—48no—
111C2×C59⋊C8—42no—
112C2×C59⋊C8—42no—
113C2×C59⋊C8—42no—
114C23×C59—21no—
115C23×C59—21no—
116C23×C59—21no—
117C23×C59—21no—
118C23×C59—21no—
119C23×C59—21no—
120C23×C59—21no—
121C23×C59—21no—
122C23×C59—24no—
123C23×C59—24no—
124C23×C59—21no—
125C23×C59—21no—
126C23×C59—21no—
127C23×C59—21no—
128C23×C59—21no—
129C23×C59—21no—
130C23×C59—24no—
131C23×C59—24no—
132C23×C59—42no—
133C23×C59—42no—
134C23×C59—44no—
135C23×C59—44no—
136C23×C59—44no—
137C23×C59—44no—
138C23×C59#9912yes—
139C23×C59#10212yes—
140C23×C59#10212yes—
141C23×C59#10412yes—
142C23×C59C2×C22⋊C411yescharacteristic
143C23×C59#12312yes—
144C23×C59C2×D811yescharacteristic
145C23×C59C2×SD1611yescharacteristic
146C23×C59#13712yes—
147C2×C4×C59—28no—
148C2×C4×C59—21no—
149C2×C4×C59—21no—
150C2×C4×C59—21no—
151C2×C4×C59—21no—
152C2×C4×C59—21no—
153C2×C4×C59—21no—
154C2×C4×C59—21no—
155C2×C4×C59—21no—
156C2×C4×C59—21no—
157C2×C4×C59—44no—
158C2×C4×C59—44no—
159C2×C4×C59—44no—
160C2×C4×C59—44no—
161C2×C4×C59—44no—
162C2×C4×C59—42no—
163C2×C4×C59—42no—
164C2×C4×C59—42no—
165C2×C4×C59—42no—
166C2×C4×C59—42no—
167C2×C4×C59—42no—
168C2×C4×C59—84no—
169C2×C4×C59—84no—
170C2×C4×C59#9912yes—
171C2×C4×C59#11712yes—
172C2×C4×C59#12312yes—
173C2×Dic59—44no—
174C2×Dic59—44no—
175C2×Dic59—44no—
176C2×Dic59—82no—
177C2×Dic59—82no—
178C2×Dic59—82no—
179C2×Dic59—82no—
180C59⋊C8—82no—
181#8712—591nomaximal, Sylow
182C22×C59—22no—
183C22×C59—22no—
184C22×C59—22no—
185C22×C59—22no—
186C22×C59—22no—
187C22×C59—21no—
188C22×C59—21no—
189C22×C59—22no—
190C22×C59—22no—
191C22×C59—22no—
192C22×C59—22no—
193C22×C59—21no—
194C22×C59—22no—
195C22×C59—21no—
196C22×C59—22no—
197C22×C59—22no—
198C22×C59—22no—
199C22×C59—48no—
200C22×C59—48no—
201C22×C59—41no—
202C22×C59—41no—
203C22×C59—41no—
204C22×C59—41no—
205C22×C59#32714yes—
206C22×C59#38111yescharacteristic
207C22×C59C2×#9911yescharacteristic
208C22×C59C2×#9911yescharacteristic
209C22×C59C2×#10211yescharacteristic
210C22×C59#41911yescharacteristic
211C22×C59C2×#10411yescharacteristic
212C22×C59#42111yescharacteristic
213C22×C59#48012yes—
214C22×C59#48312yes—
215C4×C59—22no—
216C4×C59—22no—
217C4×C59—22no—
218C4×C59—41no—
219C4×C59—82no—
220C4×C59—82no—
221Dic59—84no—
222C2×#380D5911yescharacteristic
223#1779—591no—
224C22⋊#95—591no—
225#1294—594no—
226C2×C59—22no—
227C2×C59—24no—
228C2×C59—22no—
229C2×C59—42no—
230C2×C59C2×#32711yescharacteristic
231C2×C59#150811yescharacteristic
232C2×C59#151111yescharacteristic
233C2×C59#151911yescharacteristic
234C2×C59#181911yescharacteristic
235C2×C59#182311yescharacteristic
236C2×C59#182611yescharacteristic
237C24×C4C2×D5911yescharacteristic
238C22×C4⋊C4Dic5911yescharacteristic, complemented
239C22×C4⋊C4Dic5911yescharacteristic, complemented
240C22×C22⋊C4—21no—
241C22×C22⋊C4—1181no—
242C2×C22⋊C8—592no—
243C2×C22⋊C8—1181no—
244C2×C22⋊C8—1181no—
245#380—21no—
246#380—21no—
247#380Dic5912yescomplemented
248#380C2×D5912yes—
249C2×#95—21no—
250C2×#95—592no—
251C2×#95—1181no—
252C59#871211yescharacteristic, complemented, Sylow
253C25C22⋊D5911yescharacteristic
254C23×C4—24no—
255C23×C4—21no—
256C23×C4—21no—
257C23×C4—42no—
258C23×C4—42no—
259C23×C4—1182no—
260C23×C4C2×Dic5911yescharacteristic
261C23×C4C2×Dic5911yescharacteristic
262C23×C4C22⋊D5911yescharacteristic
263C23×C4C22⋊D5911yescharacteristic
264C23×C4C22⋊D5911yescharacteristic
265C23×C4C22×D5911yescharacteristic
266C22×C8—1182no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—42no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4C2×Dic5912yes—
272C2×C4⋊C4C2×Dic5912yes—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—42no—
276C2×C22⋊C4—42no—
277C2×C22⋊C4—1184no—
278C2×C22⋊C4—2362no—
279C2×C22⋊C4—2362no—
280C22⋊C8—1184no—
281C22⋊C8—2362no—
282C22⋊C8—2362no—
283#95—22no—
284#95—22no—
285#95—1184no—
286#95—2362no—
287C24—22no—
288C24—22no—
289C24—22no—
290C24—22no—
291C24—41no—
292C24—41no—
293C24#1174566912yes—
294C24#1174567012yes—
295C24C22⋊Dic5911yescharacteristic
296C24C22⋊Dic5911yescharacteristic
297C24C2×C22⋊D5911yescharacteristic
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—416no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—1184no—
325C22×C4—2362no—
326C22×C4—2362no—
327C22×C4—2362no—
328C22×C4—2362no—
329C22×C4—2362no—
330C22×C4—2362no—
331C22×C4C22⋊Dic5911yescharacteristic
332C22×C4C22⋊Dic5911yescharacteristic
333C22×C4#1174568514yes—
334C22×C4C22×Dic5911yescharacteristic
335C22×C4C2×C22⋊D5911yescharacteristic
336C2×C8—2362no—
337C2×C8—2362no—
338C2×C8—2362no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C22⋊C4—44no—
344C22⋊C4—42no—
345C22⋊C4—42no—
346C22⋊C4—2368no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—24no—
356C23—24no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—24no—
361C23—24no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23—42no—
366C23—42no—
367C23—44no—
368C23—44no—
369C23—44no—
370C23—44no—
371C23#1427364212yes—
372C23#1427364812yes—
373C23#1427364812yes—
374C23#1427365412yes—
375C23C2×C22⋊Dic5911yescharacteristic
376C23#1427371012yes—
377C23C2×#1174566911yescharacteristic
378C23C2×#1174567011yescharacteristic
379C23#1427375812yes—
380C2×C4—28no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—84no—
402C2×C4—84no—
403C2×C4—4722no—
404C2×C4—4722no—
405C2×C4—4722no—
406C2×C4—4722no—
407C2×C4—2364no—
408C2×C4—2364no—
409C2×C4—2364no—
410C2×C4#1427364212yes—
411C2×C4#1427368612yes—
412C2×C4#1427371012yes—
413C8—4722no—
414C22—22no—
415C22—22no—
416C22—22no—
417C22—22no—
418C22—22no—
419C22—22no—
420C22—22no—
421C22—22no—
422C22—22no—
423C22—22no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—22no—
429C22—22no—
430C22—22no—
431C22—48no—
432C22—48no—
433C22—41no—
434C22—41no—
435C22—41no—
436C22—41no—
437C22#1683713614yes—
438C22#1683726511yescharacteristic
439C22C2×#1427364211yescharacteristic
440C22C2×#1427364211yescharacteristic
441C22C2×#1427364811yescharacteristic
442C22#1683741111yescharacteristic
443C22C2×#1427365411yescharacteristic
444C22#1683741911yescharacteristic
445C22#1683771512yes—
446C22#1683773212yes—
447C4—22no—
448C4—22no—
449C4—22no—
450C4—41no—
451C4—82no—
452C4—82no—
453C4—4724no—
454C2—22no—
455C2—24no—
456C2—22no—
457C2—42no—
458C2C2×#1683713611yescharacteristic
459C2#2142261611yescharacteristic
460C2#2142262911yescharacteristic
461C2#2142266411yescharacteristic
462C2#2142417011yescharacteristic
463C2#2142418211yescharacteristic
464C2#2142419511yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic59
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC2×C59×#380C2
socleC23×C59C2×C22⋊C4
prime radicalC25×C59D4
cosocleC23×C4×C59C23
hypercenterC2×#380D59
hypocenterC59#8712
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C24×C4 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
3. #21421974C2semidirect
12. C22×C4⋊C4×C59C4semidirect
13. C22×C4⋊C4×C59C4semidirect
17. C59×#380C4semidirect
238. C22×C4⋊C4Dic59semidirect
239. C22×C4⋊C4Dic59semidirect
247. #380Dic59semidirect
252. C59#8712semidirect

Maximal subgroups

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

subgroupindex
#214240062
#214223192
#214219742
C2×C59×#3802
#871259