Finite Group Databasebeta

← #28479586

Subgroups — #28479586

Order 15104.

804 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#14273701C211yesmaximal, characteristic
2C2×#16837137C212yesmaximal
3C4×C59⋊#95C211yesmaximal, characteristic, complemented
4#21421976C212yesmaximal, complemented
5C59×#1749C211yesmaximal, characteristic, complemented
6C2×#14273701C2211yescharacteristic
7C2×C4×D236C411yescharacteristic, complemented
8C2×C4×C22⋊D59—21no—
9C22×C4×Dic59—21no—
10#16837304C411yescharacteristic, complemented
11#16837300C411yescharacteristic, complemented
12#16837293—21no—
13#16837281—21no—
14#16837262C2211yescharacteristic
15C4×C4⋊Dic59C411yescharacteristic, complemented
16C4×(C2×D59)⋊C4—21no—
17C2×C59⋊#95—21no—
18C2×C59⋊#95C2212yes—
19#16837137C418yescomplemented
20C22×C42×C59C2211yescharacteristic
21C2×C22⋊C8×C59C2212yes—
22C2×#97×C59—21no—
23C59×#336—21no—
24C22×D236C2×C411yescharacteristic
25C22×C22⋊D59—21no—
26C23×Dic59—22no—
27C22×C4×D59—21no—
28#14273701—22no—
29#14273701C2×C414yes—
30C4×D236—42no—
31C4×C22⋊D59—44no—
32C2×C4⋊Dic59C2×C411yescharacteristic
33C2×C4⋊Dic59C2×C411yescharacteristic
34C2×C4⋊Dic59C2×C411yescharacteristic
35C2×Dic59⋊C4—21no—
36C2×C22⋊Dic59—21no—
37C2×(C2×D59)⋊C4—21no—
38C2×(C2×D59)⋊C4—21no—
39C2×(C2×D59)⋊C4—21no—
40C2×(C2×D59)⋊C4—21no—
41C2×C4×Dic59—24no—
42C2×C4×Dic59—21no—
43C2×C4×Dic59—42no—
44C2×C4×Dic59—42no—
45C59⋊#95—24no—
46C59⋊#95—21no—
47C59⋊#95—21no—
48C59⋊#95—21no—
49C59⋊#95—22no—
50C59⋊#95C2×C414yescomplemented
51C23×C4×C59D411yescharacteristic
52C23×C4×C59D411yescharacteristic
53C23×C4×C59C2311yescharacteristic
54C22×C8×C59—22no—
55C2×C42×C59—42no—
56C2×C42×C59C2×C411yescharacteristic
57C2×C42×C59C2×C411yescharacteristic
58C2×C42×C59D411yescharacteristic
59C2×C42×C59D411yescharacteristic
60C22⋊C8×C59—22no—
61C22⋊C8×C59C2×C414yescomplemented
62#97×C59—42no—
63C24×C59C2×D411yescharacteristic
64C23×D59—21no—
65C22×C4×C59—21no—
66C22×C4×C59—21no—
67C22×C4×C59—22no—
68C22×C4×C59—22no—
69C22×C4×C59—22no—
70C22×C4×C59—22no—
71C22×C4×C59—41no—
72C22×C4×C59—41no—
73C22×C4×C59—41no—
74C22×C4×C59C22⋊C411yescharacteristic
75C22×C4×C59C22⋊C411yescharacteristic
76C22×C4×C59C22⋊C411yescharacteristic
77C22×C4×C59C22⋊C411yescharacteristic
78C22×C4×C59C22×C411yescharacteristic
79C22×C4×C59C22×C412yes—
80C22×C4×C59C2×D411yescharacteristic
81C22×C4×C59#4112yes—
82C2×C22⋊D59—24no—
83C2×C22⋊D59—42no—
84C2×C22⋊D59—41no—
85C2×C22⋊D59—41no—
86C22×Dic59—24no—
87C22×Dic59—24no—
88C22×Dic59—21no—
89C22×Dic59—24no—
90C22×Dic59—21no—
91C22×Dic59—21no—
92C22×Dic59—42no—
93C22×Dic59—42no—
94C22×Dic59—42no—
95C22×Dic59—42no—
96C2×D236—21no—
97C2×D236—41no—
98C2×D236—41no—
99C2×D236C4212yescomplemented
100C2×C4×D59—42no—
101C2×C4×D59—42no—
102C2×C4×D59—42no—
103C2×C8×C59—24no—
104C2×C8×C59—42no—
105C2×C8×C59—42no—
106C42×C59—22no—
107C42×C59—22no—
108C42×C59—41no—
109C42×C59—41no—
110C22⋊Dic59—42no—
111(C2×D59)⋊C4—24no—
112(C2×D59)⋊C4—42no—
113(C2×D59)⋊C4—42no—
114(C2×D59)⋊C4—42no—
115C4⋊Dic59—21no—
116C4⋊Dic59—41no—
117C4⋊Dic59—41no—
118C4⋊Dic59C4212yescomplemented
119Dic59⋊C4—42no—
120C4×Dic59—48no—
121C4×Dic59—42no—
122C23×C59—21no—
123C23×C59—21no—
124C23×C59—21no—
125C23×C59—21no—
126C23×C59C2×C22⋊C411yescharacteristic
127C23×C59C4×D412yes—
128C23×C59AGL(1,ℤ/8)11yescharacteristic
129C23×C59AGL(1,ℤ/8)12yes—
130C23×C59#13711yescharacteristic
131C22×D59—22no—
132C22×D59—41no—
133C22×D59—41no—
134C22×D59—41no—
135C22×D59—41no—
136C22×D59—41no—
137C22×D59—41no—
138C2×C4×C59—24no—
139C2×C4×C59—24no—
140C2×C4×C59—21no—
141C2×C4×C59—21no—
142C2×C4×C59—21no—
143C2×C4×C59—21no—
144C2×C4×C59—21no—
145C2×C4×C59—21no—
146C2×C4×C59—21no—
147C2×C4×C59—21no—
148C2×C4×C59—21no—
149C2×C4×C59—21no—
150C2×C4×C59—21no—
151C2×C4×C59—21no—
152C2×C4×C59—21no—
153C2×C4×C59—42no—
154C2×C4×C59—42no—
155C2×C4×C59—44no—
156C2×C4×C59#9912yes—
157C2×C4×C59#10412yes—
158C2×C4×C59#10412yes—
159C2×C4×C59C2×C4211yescharacteristic
160C2×C4×C59#11711yescharacteristic
161C2×C4×C59C4×D412yes—
162C22⋊D59—44no—
163C22⋊D59—44no—
164C2×Dic59—28no—
165C2×Dic59—22no—
166C2×Dic59—44no—
167C2×Dic59—44no—
168C2×Dic59—44no—
169C2×Dic59—44no—
170C2×Dic59—44no—
171C2×Dic59—41no—
172C2×Dic59—41no—
173C2×Dic59—44no—
174C2×Dic59—41no—
175C2×Dic59—41no—
176C2×Dic59—41no—
177C2×Dic59—41no—
178C2×Dic59—41no—
179C2×Dic59—41no—
180D236—42no—
181D236—42no—
182C4×D59—84no—
183C8×C59—44no—
184#8218—591nomaximal, Sylow
185C22×C59—22no—
186C22×C59—22no—
187C22×C59—21no—
188C22×C59—21no—
189C22×C59—22no—
190C22×C59—22no—
191C22×C59—21no—
192C22×C59—21no—
193C22×C59C4×C22⋊C411yescharacteristic
194C22×C59C2×#9911yescharacteristic
195C22×C59#41011yescharacteristic
196C22×C59#41711yescharacteristic
197C22×C59#41811yescharacteristic
198C22×C59C2×#10411yescharacteristic
199C22×C59C2×#10411yescharacteristic
200C22×C59#44012yes—
201C22×C59#44212yes—
202C2×D59—42no—
203C2×D59—42no—
204C2×D59—42no—
205C2×D59—81no—
206C2×D59—81no—
207C2×D59—81no—
208C2×D59—81no—
209C4×C59—22no—
210C4×C59—21no—
211C4×C59—21no—
212C4×C59—22no—
213C4×C59—22no—
214C4×C59—22no—
215C4×C59—41no—
216Dic59—48no—
217Dic59—42no—
218Dic59—81no—
219Dic59—81no—
220C4×#121—591no—
221#1749D5911yescharacteristic, complemented
222C2×#327—592no—
223C4×#95—591no—
224#1294—592no—
225C2×C59—22no—
226C2×C59—22no—
227C2×C59#150411yescharacteristic
228C2×C59#150511yescharacteristic
229C2×C59#150711yescharacteristic
230C2×C59#150911yescharacteristic
231C2×C59C4×#9911yescharacteristic
232C2×C59C4×#10411yescharacteristic
233C2×C59#176211yescharacteristic
234D59—82no—
235C2×#121—591no—
236C2×C4×D4—591no—
237C2×C4×D4—1181no—
238C22×C42—1181no—
239C22×C42C2×D5911yescharacteristic
240C2×C22⋊C8C2×D5912yes—
241C2×#97—21no—
242#390—591no—
243#389—591no—
244#388—1181no—
245#386—1181no—
246#380—591no—
247C4×C4⋊C4—591no—
248C4×C22⋊C4—1181no—
249C2×#95—592no—
250C2×#95—1181no—
251#336—21no—
252#327—598no—
253C59#821811yescharacteristic, complemented, Sylow
254C22×D4—591no—
255C22×D4—1181no—
256C23×C4—1182no—
257C23×C4—1181no—
258C23×C4D23611yescharacteristic
259C23×C4D23611yescharacteristic
260C23×C4C22×D5911yescharacteristic
261C22×C8—22no—
262#121—594no—
263#121—1182no—
264C4×D4—2362no—
265C4×D4—2364no—
266C2×C4⋊C4—591no—
267C2×C4⋊C4—591no—
268C2×C4⋊C4—591no—
269C2×C4⋊C4—1181no—
270C2×C22⋊C4—1181no—
271C2×C22⋊C4—1181no—
272C2×C22⋊C4—1181no—
273C2×C22⋊C4—1181no—
274C2×C22⋊C4—1181no—
275C2×C42—42no—
276C2×C42—1184no—
277C2×C42—1181no—
278C2×C42—2362no—
279C2×C42—2362no—
280C2×C42C4×D5911yescharacteristic
281C2×C42C4×D5911yescharacteristic
282C2×C42C22⋊D5911yescharacteristic
283C2×C42C22⋊D5911yescharacteristic
284C22⋊C8—22no—
285C22⋊C8C4×D5914yescomplemented
286#97—42no—
287#95—594no—
288#95—1184no—
289#95—1181no—
290#95—1181no—
291#95—1182no—
292#95—1181no—
293C24—1181no—
294C24C2×D23611yescharacteristic
295C2×D4—592no—
296C2×D4—1184no—
297C2×D4—1181no—
298C2×D4—2361no—
299C2×D4—2361no—
300C2×D4—2362no—
301C2×D4—2361no—
302C2×D4—2361no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—1184no—
313C22×C4—1184no—
314C22×C4—1181no—
315C22×C4—1184no—
316C22×C4—1181no—
317C22×C4—1181no—
318C22×C4—2362no—
319C22×C4—2362no—
320C22×C4—2362no—
321C22×C4—2362no—
322C22×C4—2362no—
323C22×C4—2362no—
324C22×C4—2362no—
325C22×C4(C2×D59)⋊C411yescharacteristic
326C22×C4(C2×D59)⋊C411yescharacteristic
327C22×C4(C2×D59)⋊C411yescharacteristic
328C22×C4(C2×D59)⋊C411yescharacteristic
329C22×C4C2×C4×D5911yescharacteristic
330C22×C4C2×C4×D5912yes—
331C22×C4#1174568512yes—
332C22×C4C2×C22⋊D5911yescharacteristic
333C2×C8—24no—
334C2×C8—42no—
335C2×C8—42no—
336C4⋊C4—592no—
337C4⋊C4—1181no—
338C4⋊C4—2361no—
339C4⋊C4—2361no—
340C4⋊C4—2362no—
341C22⋊C4—1184no—
342C22⋊C4—2362no—
343C22⋊C4—2362no—
344C22⋊C4—2362no—
345C22⋊C4—2362no—
346C42—22no—
347C42—22no—
348C42—41no—
349C42—41no—
350C42—2368no—
351C42—2362no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—1182no—
357C23—2361no—
358C23—2361no—
359C23—2361no—
360C23—2361no—
361C23—2361no—
362C23—2361no—
363C23C2×(C2×D59)⋊C411yescharacteristic
364C23C4×D23612yes—
365C23#1427375511yescharacteristic
366C23#1427375512yes—
367C23#1427376011yescharacteristic
368D4—2362no—
369D4—2362no—
370D4—2364no—
371D4—2364no—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—44no—
390C2×C4—1188no—
391C2×C4—1182no—
392C2×C4—4724no—
393C2×C4—2361no—
394C2×C4—2361no—
395C2×C4—2361no—
396C2×C4—2361no—
397C2×C4—2361no—
398C2×C4—2361no—
399C2×C4—2364no—
400C2×C4—2364no—
401C2×C4—2364no—
402C2×C4—2364no—
403C2×C4—2364no—
404C2×C4—2361no—
405C2×C4—2361no—
406C2×C4—2364no—
407C2×C4#1427364112yes—
408C2×C4#1427365312yes—
409C2×C4#1427365312yes—
410C2×C4C42×D5911yescharacteristic
411C2×C4#1427368411yescharacteristic
412C2×C4C4×C22⋊D5912yes—
413C8—44no—
414C22—21no—
415C22—21no—
416C22—22no—
417C22—22no—
418C22—22no—
419C22—21no—
420C22—21no—
421C22—22no—
422C22—4721no—
423C22—4721no—
424C22—4721no—
425C22—4721no—
426C22—2362no—
427C22—2362no—
428C22—2362no—
429C22C4×(C2×D59)⋊C411yescharacteristic
430C22C2×#1427364111yescharacteristic
431C22#1683737511yescharacteristic
432C22#1683740411yescharacteristic
433C22#1683740811yescharacteristic
434C22C2×#1427365311yescharacteristic
435C22C2×#1427365311yescharacteristic
436C22#1683750412yes—
437C22#1683751512yes—
438C4—22no—
439C4—22no—
440C4—22no—
441C4—21no—
442C4—21no—
443C4—22no—
444C4—41no—
445C4—4721no—
446C4—4721no—
447C4—2368no—
448C4—2362no—
449C2—22no—
450C2—22no—
451C2—4722no—
452C2#2142260111yescharacteristic
453C2#2142260511yescharacteristic
454C2#2142261311yescharacteristic
455C2#2142262211yescharacteristic
456C2C4×#1427364111yescharacteristic
457C2C4×#1427365311yescharacteristic
458C2#2142392211yescharacteristic
subgroupG/H
centerC23C2×(C2×D59)⋊C4
commutatorC2×C4×C59C2×C42
FrattiniC23×C4C22×D59
FittingC59×#1749C2
socleC23×C59C2×C22⋊C4
prime radicalC2×#14273701C22
cosocleC23×C4×C59C23
hypercenter#1749D59
hypocenterC59#8218
LCSG ▷ C2×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1749

Complements

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

NH ≅ G/Ntype
3. C4×C59⋊#95C2semidirect
4. #21421976C2semidirect
5. C59×#1749C2semidirect
7. C2×C4×D236C4semidirect
10. #16837304C4semidirect
11. #16837300C4semidirect
15. C4×C4⋊Dic59C4semidirect
19. #16837137C4semidirect
50. C59⋊#95C2×C4semidirect
61. C22⋊C8×C59C2×C4semidirect
99. C2×D236C42semidirect
118. C4⋊Dic59C42semidirect
221. #1749D59semidirect
253. C59#8218semidirect
285. C22⋊C8C4×D59semidirect

Maximal subgroups

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

subgroupindex
C4×#142737012
C2×#168371372
C4×C59⋊#952
#214219762
C59×#17492
#821859