Finite Group Databasebeta

← #28487818

Subgroups — #28487818

Order 15104.

742 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
1#21432397C211yesmaximal, characteristic
2C2×#16837140C212yesmaximal, complemented
3C2×#16837137C212yesmaximal
4C4×C59⋊#95C211yesmaximal, characteristic, complemented
5C59×#1797C211yesmaximal, characteristic, complemented
6C2×#14273715C411yescharacteristic, complemented
7C2×#14273706C2211yescharacteristic
8C2×#14273701C2211yescharacteristic
9C22×C4×Dic59—21no—
10#16837308C411yescharacteristic, complemented
11#16837304C411yescharacteristic, complemented
12#16837293—21no—
13#16837293—21no—
14#16837281—21no—
15#16837281—21no—
16#16837268C411yescharacteristic, complemented
17C2×C59⋊#95—21no—
18C2×C59⋊#95C2212yes—
19#16837140—24no—
20#16837137—24no—
21C22×C42×C59C2211yescharacteristic
22C2×C4⋊C8×C59—21no—
23C2×C22⋊C8×C59C2212yes—
24C59×#336—21no—
25C22×Dic118C2×C411yescharacteristic
26C22×D236C2×C411yescharacteristic
27C22×C22⋊D59—21no—
28C23×Dic59—22no—
29#14273715—42no—
30#14273706—22no—
31#14273706—22no—
32#14273701—22no—
33#14273701—22no—
34C2×C4⋊Dic59C2×C411yescharacteristic
35C2×C4⋊Dic59C2×C411yescharacteristic
36C2×Dic59⋊C4—21no—
37C2×C22⋊Dic59—21no—
38C2×(C2×D59)⋊C4—21no—
39C2×(C2×D59)⋊C4—21no—
40C2×(C2×D59)⋊C4—21no—
41C2×(C2×D59)⋊C4—21no—
42C2×(C2×D59)⋊C4—21no—
43C2×C4×Dic59—24no—
44C2×C4×Dic59—42no—
45C2×C4×Dic59—42no—
46C59⋊#95—24no—
47C59⋊#95—21no—
48C59⋊#95—21no—
49C59⋊#95—21no—
50C59⋊#95—21no—
51C59⋊#95—22no—
52C59⋊#95D414yes—
53C23×C4×C59D411yescharacteristic
54C23×C4×C59D411yescharacteristic
55C23×C4×C59C2311yescharacteristic
56C22×C8×C59—22no—
57C2×C42×C59—42no—
58C2×C42×C59C2×C411yescharacteristic
59C2×C42×C59C2×C411yescharacteristic
60C2×C42×C59D411yescharacteristic
61C2×C42×C59D411yescharacteristic
62C4⋊C8×C59—42no—
63C22⋊C8×C59—22no—
64C22⋊C8×C59—22no—
65C24×C59C2×D411yescharacteristic
66C23×D59—21no—
67C22×C4×C59—21no—
68C22×C4×C59—21no—
69C22×C4×C59—22no—
70C22×C4×C59—22no—
71C22×C4×C59—22no—
72C22×C4×C59—22no—
73C22×C4×C59—41no—
74C22×C4×C59—41no—
75C22×C4×C59—41no—
76C22×C4×C59C22⋊C411yescharacteristic
77C22×C4×C59C22⋊C411yescharacteristic
78C22×C4×C59C22⋊C411yescharacteristic
79C22×C4×C59C22⋊C411yescharacteristic
80C22×C4×C59C22×C411yescharacteristic
81C22×C4×C59C2×D411yescharacteristic
82C22×C4×C59C2×D412yes—
83C22×C4×C59#4112yes—
84C2×C22⋊D59—42no—
85C2×C22⋊D59—42no—
86C2×C22⋊D59—41no—
87C2×C22⋊D59—41no—
88C22×Dic59—24no—
89C22×Dic59—24no—
90C22×Dic59—24no—
91C22×Dic59—21no—
92C22×Dic59—21no—
93C22×Dic59—21no—
94C22×Dic59—42no—
95C22×Dic59—42no—
96C22×Dic59—42no—
97C22×Dic59—42no—
98C2×D236—22no—
99C2×D236—41no—
100C2×D236—41no—
101C2×Dic118—22no—
102C2×Dic118—41no—
103C2×Dic118—41no—
104C2×C8×C59—42no—
105C2×C8×C59—42no—
106C2×C8×C59—42no—
107C42×C59—22no—
108C42×C59—22no—
109C42×C59—41no—
110C42×C59—41no—
111C22⋊Dic59—42no—
112(C2×D59)⋊C4—42no—
113(C2×D59)⋊C4—42no—
114(C2×D59)⋊C4—42no—
115(C2×D59)⋊C4—42no—
116(C2×D59)⋊C4—41no—
117(C2×D59)⋊C4—41no—
118C4⋊Dic59—22no—
119C4⋊Dic59—22no—
120Dic59⋊C4—41no—
121Dic59⋊C4—41no—
122C4×Dic59—48no—
123C23×C59—21no—
124C23×C59—21no—
125C23×C59—21no—
126C23×C59—21no—
127C23×C59C2×C22⋊C411yescharacteristic
128C23×C59#12112yes—
129C23×C59#13512yes—
130C23×C59AGL(1,ℤ/8)11yescharacteristic
131C23×C59#13711yescharacteristic
132C22×D59—41no—
133C22×D59—41no—
134C22×D59—41no—
135C22×D59—41no—
136C22×D59—41no—
137C22×D59—41no—
138C22×D59—41no—
139C2×C4×C59—24no—
140C2×C4×C59—24no—
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—21no—
154C2×C4×C59—42no—
155C2×C4×C59—42no—
156C2×C4×C59—44no—
157C2×C4×C59#9912yes—
158C2×C4×C59#10412yes—
159C2×C4×C59#10412yes—
160C2×C4×C59C4×D412yes—
161C2×C4×C59#12411yescharacteristic
162C2×C4×C59#12711yescharacteristic
163C22⋊D59—82no—
164C22⋊D59—82no—
165C2×Dic59—28no—
166C2×Dic59—44no—
167C2×Dic59—44no—
168C2×Dic59—44no—
169C2×Dic59—44no—
170C2×Dic59—44no—
171C2×Dic59—44no—
172C2×Dic59—41no—
173C2×Dic59—41no—
174C2×Dic59—41no—
175C2×Dic59—41no—
176C2×Dic59—41no—
177C2×Dic59—41no—
178C2×Dic59—41no—
179C2×Dic59—41no—
180C2×Dic59—41no—
181D236—82no—
182Dic118—82no—
183C8×C59—82no—
184#9713—591nomaximal, Sylow
185C22×C59—22no—
186C22×C59—22no—
187C22×C59—22no—
188C22×C59—22no—
189C22×C59—21no—
190C22×C59—21no—
191C22×C59—21no—
192C22×C59—21no—
193C22×C59#39011yescharacteristic
194C22×C59C2×#9911yescharacteristic
195C22×C59#41011yescharacteristic
196C22×C59#41611yescharacteristic
197C22×C59#41711yescharacteristic
198C22×C59C2×#10411yescharacteristic
199C22×C59C2×#10411yescharacteristic
200C22×C59#46312yes—
201C22×C59#46412yes—
202C2×D59—81no—
203C2×D59—81no—
204C2×D59—81no—
205C2×D59—81no—
206C2×D59—81no—
207C2×D59—81no—
208C2×D59—81no—
209C4×C59—21no—
210C4×C59—22no—
211C4×C59—21no—
212C4×C59—22no—
213C4×C59—22no—
214C4×C59—22no—
215C4×C59—41no—
216Dic59—48no—
217Dic59—81no—
218Dic59—81no—
219Dic59—81no—
220#2850—591no—
221#1797D5911yescharacteristic, complemented
222C2×#328—592no—
223C2×#327—592no—
224C4×#95—591no—
225C2×C59—22no—
226C2×C59—22no—
227C2×C59#150011yescharacteristic
228C2×C59#150111yescharacteristic
229C2×C59#150411yescharacteristic
230C2×C59#150911yescharacteristic
231C2×C59#196211yescharacteristic
232C2×C59#196411yescharacteristic
233C2×C59#198311yescharacteristic
234D59—161no—
235C2×#124—591no—
236C2×C22⋊Q8—591no—
237C2×#121—591no—
238C22×C42—1181no—
239C22×C42C2×D5911yescharacteristic
240C2×C4⋊C8—21no—
241C2×C22⋊C8C2×D5912yes—
242#391—591no—
243#390—591no—
244#388—1181no—
245#388—1181no—
246#386—1181no—
247#386—1181no—
248#382—591no—
249C2×#95—592no—
250C2×#95—1181no—
251#336—21no—
252#328—1184no—
253#327—1184no—
254C59#971311yescharacteristic, complemented, Sylow
255C22×Q8—591no—
256C22×D4—591no—
257C22×D4—1181no—
258C23×C4—1182no—
259C23×C4D23611yescharacteristic
260C23×C4D23611yescharacteristic
261C23×C4C22×D5911yescharacteristic
262C22×C8—22no—
263#124—2362no—
264C22⋊Q8—1182no—
265C22⋊Q8—1182no—
266#121—1182no—
267#121—1182no—
268C2×C4⋊C4—591no—
269C2×C4⋊C4—591no—
270C2×C4⋊C4—1181no—
271C2×C22⋊C4—1181no—
272C2×C22⋊C4—1181no—
273C2×C22⋊C4—1181no—
274C2×C22⋊C4—1181no—
275C2×C22⋊C4—1181no—
276C2×C22⋊C4—1181no—
277C2×C42—42no—
278C2×C42—1184no—
279C2×C42—2362no—
280C2×C42—2362no—
281C2×C42C4×D5911yescharacteristic
282C2×C42C4×D5911yescharacteristic
283C2×C42C22⋊D5911yescharacteristic
284C2×C42C22⋊D5911yescharacteristic
285C4⋊C8—42no—
286C22⋊C8—22no—
287C22⋊C8—22no—
288#95—594no—
289#95—1184no—
290#95—1181no—
291#95—1181no—
292#95—1181no—
293#95—1181no—
294#95—1182no—
295C24—1181no—
296C24C2×D23611yescharacteristic
297C2×Q8—1182no—
298C2×Q8—2361no—
299C2×Q8—2361no—
300C2×D4—1182no—
301C2×D4—2361no—
302C2×D4—2361no—
303C2×D4—2362no—
304C2×D4—2362no—
305C2×D4—2361no—
306C2×D4—2361no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—1184no—
317C22×C4—1184no—
318C22×C4—1184no—
319C22×C4—1181no—
320C22×C4—1181no—
321C22×C4—1181no—
322C22×C4—2362no—
323C22×C4—2362no—
324C22×C4—2362no—
325C22×C4—2362no—
326C22×C4(C2×D59)⋊C411yescharacteristic
327C22×C4(C2×D59)⋊C411yescharacteristic
328C22×C4(C2×D59)⋊C411yescharacteristic
329C22×C4(C2×D59)⋊C411yescharacteristic
330C22×C4C2×C4×D5911yescharacteristic
331C22×C4D4×D5912yes—
332C22×C4#1174568912yes—
333C22×C4C2×C22⋊D5911yescharacteristic
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—42no—
337C4⋊C4—1182no—
338C4⋊C4—1182no—
339C4⋊C4—2361no—
340C4⋊C4—2361no—
341C22⋊C4—2362no—
342C22⋊C4—2362no—
343C22⋊C4—2362no—
344C22⋊C4—2362no—
345C22⋊C4—2362no—
346C22⋊C4—2361no—
347C22⋊C4—2361no—
348C42—22no—
349C42—22no—
350C42—41no—
351C42—41no—
352C42—2368no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—2361no—
358C23—2361no—
359C23—2361no—
360C23—2361no—
361C23—2361no—
362C23—2361no—
363C23—2361no—
364C23C2×(C2×D59)⋊C411yescharacteristic
365C23#1427370212yes—
366C23#1427374912yes—
367C23#1427375511yescharacteristic
368C23#1427376011yescharacteristic
369Q8—4722no—
370D4—4722no—
371D4—4722no—
372D4—4722no—
373C2×C4—24no—
374C2×C4—24no—
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—21no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—44no—
391C2×C4—1188no—
392C2×C4—2361no—
393C2×C4—2361no—
394C2×C4—2361no—
395C2×C4—2361no—
396C2×C4—2361no—
397C2×C4—2361no—
398C2×C4—2361no—
399C2×C4—2361no—
400C2×C4—2361no—
401C2×C4—2364no—
402C2×C4—2364no—
403C2×C4—2364no—
404C2×C4—2364no—
405C2×C4—2364no—
406C2×C4—2364no—
407C2×C4#1427364112yes—
408C2×C4#1427365312yes—
409C2×C4#1427365312yes—
410C2×C4#1427368912yes—
411C2×C4#1427371611yescharacteristic
412C2×C4#1427372311yescharacteristic
413C8—82no—
414C22—21no—
415C22—21no—
416C22—22no—
417C22—21no—
418C22—21no—
419C22—22no—
420C22—22no—
421C22—22no—
422C22—4721no—
423C22—4721no—
424C22—4721no—
425C22—4721no—
426C22—4721no—
427C22—4721no—
428C22—4721no—
429C22#1683730311yescharacteristic
430C22C2×#1427364111yescharacteristic
431C22#1683737511yescharacteristic
432C22#1683739911yescharacteristic
433C22#1683740411yescharacteristic
434C22C2×#1427365311yescharacteristic
435C22C2×#1427365311yescharacteristic
436C22#1683761812yes—
437C22#1683762512yes—
438C4—22no—
439C4—22no—
440C4—21no—
441C4—22no—
442C4—21no—
443C4—22no—
444C4—41no—
445C4—4721no—
446C4—4721no—
447C4—4721no—
448C4—2368no—
449C2—22no—
450C2—22no—
451C2—9441no—
452C2#2142258311yescharacteristic
453C2#2142258811yescharacteristic
454C2#2142260111yescharacteristic
455C2#2142262211yescharacteristic
456C2#2142492311yescharacteristic
457C2#2142493111yescharacteristic
458C2#2142502611yescharacteristic
subgroupG/H
centerC23C2×(C2×D59)⋊C4
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×#1797C2
socleC23×C59C2×C22⋊C4
prime radicalC2×#14273701C22
cosocleC23×C4×C59C23
hypercenter#1797D59
hypocenterC59#9713
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1797

Complements

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

NH ≅ G/Ntype
2. C2×#16837140C2semidirect
4. C4×C59⋊#95C2semidirect
5. C59×#1797C2semidirect
6. C2×#14273715C4semidirect
10. #16837308C4semidirect
11. #16837304C4semidirect
16. #16837268C4semidirect
221. #1797D59semidirect
254. C59#9713semidirect

Maximal subgroups

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

subgroupindex
#214323972
C2×#168371402
C2×#168371372
C4×C59⋊#952
C59×#17972
#971359