Finite Group Databasebeta

← #28506173

Subgroups — #28506173

Order 15104.

722 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 722 conjugacy classes of subgroups — too many to draw. Only its 107 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#21427427C211yesmaximal, characteristic
2C2×#16837193C212yesmaximal, complemented
3#21424014C211yesmaximal, characteristic
4#21422215C212yesmaximal, complemented
5C2×C59×#413C211yesmaximal, characteristic
6C2×#14273708C411yescharacteristic, complemented
7C2×#14273708C2211yescharacteristic
8C22×Dic59⋊C4C411yescharacteristic, complemented
9C2×#14273642—21no—
10C2×#14273642C2212yes—
11#16837321—21no—
12#16837316—21no—
13#16837282C411yescharacteristic, complemented
14#16837282C2211yescharacteristic
15#16837275—21no—
16#16837275—21no—
17#16837265—21no—
18#16837265C411yescharacteristic, complemented
19#16837193—24no—
20C22×#41×C59C2211yescharacteristic
21C22×M16×C59—21no—
22C59×#413—24no—
23C2×#101×C59—21no—
24C2×#100×C59C2212yes—
25C23×Dic59C2×C411yescharacteristic
26C23×Dic59C2×C411yescharacteristic
27#14273708—22no—
28#14273708—22no—
29#14273708—42no—
30C2×Dic59⋊C4—21no—
31C2×Dic59⋊C4—21no—
32C2×Dic59⋊C4—21no—
33C2×Dic59⋊C4—21no—
34C2×Dic59⋊C4—41no—
35C2×Dic59⋊C4—41no—
36C2×C22⋊Dic59—21no—
37C2×C22⋊Dic59—21no—
38C2×C22⋊Dic59—21no—
39C2×C22⋊Dic59—21no—
40C2×C22⋊Dic59—22no—
41C2×C22⋊Dic59C2×C411yescharacteristic
42C2×C22⋊Dic59C2×C411yescharacteristic
43#14273642—22no—
44#14273642—22no—
45#14273642—42no—
46C59⋊#95—21no—
47C59⋊#95—21no—
48C59⋊#95—21no—
49C59⋊#95—21no—
50C59⋊#95—41no—
51C59⋊#95—41no—
52C2×#41×C59—22no—
53C2×#41×C59—22no—
54C2×#41×C59—41no—
55C2×#41×C59—41no—
56C2×#41×C59—42no—
57C22×Q8×C59D411yescharacteristic
58C22×D4×C59C2×C411yescharacteristic
59C22×D4×C59D411yescharacteristic
60C22×D4×C59C2311yescharacteristic
61C23×C4×C59C2×C411yescharacteristic
62C23×C4×C59D411yescharacteristic
63C23×C4×C59D411yescharacteristic
64C2×M16×C59—24no—
65C2×M16×C59—22no—
66C2×M16×C59—42no—
67C2×M16×C59—41no—
68C22×C8×C59—41no—
69#101×C59—24no—
70#100×C59—22no—
71#100×C59D414yes—
72C24×C59C22⋊C411yescharacteristic
73C24×C59C22×C411yescharacteristic
74C24×C59C2×D411yescharacteristic
75#41×C59—44no—
76#41×C59—44no—
77#41×C59—44no—
78#41×C59—44no—
79C2×Q8×C59—21no—
80C2×Q8×C59—21no—
81C2×Q8×C59—41no—
82C2×Q8×C59—41no—
83C2×D4×C59—22no—
84C2×D4×C59—22no—
85C2×D4×C59—22no—
86C2×D4×C59—22no—
87C2×D4×C59—41no—
88C2×D4×C59—41no—
89C2×D4×C59—41no—
90C2×D4×C59—41no—
91C2×D4×C59C2×D412yes—
92C2×D4×C59#4112yes—
93C22×C4×C59—22no—
94C22×C4×C59—21no—
95C22×C4×C59—21no—
96C22×C4×C59—21no—
97C22×C4×C59—21no—
98C22×C4×C59—22no—
99C22×C4×C59—22no—
100C22×C4×C59—21no—
101C22×C4×C59—21no—
102C22×C4×C59—21no—
103C22×C4×C59—21no—
104C22×C4×C59—41no—
105C22×C4×C59—41no—
106C22×C4×C59C22⋊C411yescharacteristic
107C22×C4×C59C22⋊C411yescharacteristic
108C22×C4×C59C22⋊C411yescharacteristic
109C22×C4×C59C2×D411yescharacteristic
110C22×Dic59—21no—
111C22×Dic59—21no—
112C22×Dic59—21no—
113C22×Dic59—21no—
114C22×Dic59—22no—
115C22×Dic59—42no—
116C22×Dic59—41no—
117C22×Dic59—41no—
118C22×Dic59—41no—
119C22×Dic59—42no—
120M16×C59—24no—
121M16×C59—44no—
122M16×C59—42no—
123C2×C8×C59—44no—
124C2×C8×C59—42no—
125C22⋊Dic59—22no—
126C22⋊Dic59—42no—
127C22⋊Dic59—42no—
128C22⋊Dic59—42no—
129C22⋊Dic59—42no—
130C22⋊Dic59—41no—
131C22⋊Dic59—42no—
132C22⋊Dic59—42no—
133Dic59⋊C4—42no—
134Dic59⋊C4—42no—
135Dic59⋊C4—41no—
136Dic59⋊C4—41no—
137Dic59⋊C4—41no—
138Dic59⋊C4—41no—
139C23×C59—21no—
140C23×C59—21no—
141C23×C59—21no—
142C23×C59—21no—
143C23×C59—21no—
144C23×C59—21no—
145C23×C59—41no—
146C23×C59—41no—
147C23×C59—41no—
148C23×C59—41no—
149C23×C59—41no—
150C23×C59—41no—
151C23×C59#9912yes—
152C23×C59C2×C22⋊C411yescharacteristic
153C23×C59C4×D412yes—
154C23×C59#12112yes—
155Q8×C59—42no—
156Q8×C59—42no—
157D4×C59—42no—
158D4×C59—42no—
159D4×C59—44no—
160D4×C59—44no—
161C2×C4×C59—22no—
162C2×C4×C59—22no—
163C2×C4×C59—22no—
164C2×C4×C59—22no—
165C2×C4×C59—21no—
166C2×C4×C59—21no—
167C2×C4×C59—21no—
168C2×C4×C59—21no—
169C2×C4×C59—21no—
170C2×C4×C59—21no—
171C2×C4×C59—21no—
172C2×C4×C59—21no—
173C2×C4×C59—21no—
174C2×C4×C59—21no—
175C2×C4×C59—42no—
176C2×C4×C59—42no—
177C2×C4×C59—42no—
178C2×C4×C59—42no—
179C2×C4×C59#9911yescharacteristic
180C2×C4×C59#9911yescharacteristic
181C2×C4×C59#12011yescharacteristic
182C2×C4×C59#12311yescharacteristic
183C2×Dic59—22no—
184C2×Dic59—41no—
185C2×Dic59—42no—
186C2×Dic59—41no—
187C2×Dic59—42no—
188C2×Dic59—41no—
189C2×Dic59—41no—
190C2×Dic59—41no—
191C2×Dic59—41no—
192C2×Dic59—41no—
193C2×Dic59—41no—
194C2×Dic59—41no—
195C2×Dic59—41no—
196C2×Dic59—41no—
197C2×Dic59—82no—
198C2×Dic59—82no—
199C2×Dic59—82no—
200C2×Dic59—82no—
201C8×C59—44no—
202#13081—591nomaximal, Sylow
203C22×C59—21no—
204C22×C59—21no—
205C22×C59—22no—
206C22×C59—22no—
207C22×C59—22no—
208C22×C59—42no—
209C22×C59—42no—
210C22×C59—42no—
211C22×C59—42no—
212C22×C59—41no—
213C22×C59—41no—
214C22×C59—41no—
215C22×C59—41no—
216C22×C59—41no—
217C22×C59—41no—
218C22×C59—81no—
219C22×C59—81no—
220C22×C59#38611yescharacteristic
221C22×C59C2×#9911yescharacteristic
222C22×C59C2×#9911yescharacteristic
223C4×C59—21no—
224C4×C59—21no—
225C4×C59—21no—
226C4×C59—21no—
227C4×C59—41no—
228C4×C59—41no—
229Dic59—42no—
230Dic59—81no—
231Dic59—81no—
232Dic59—81no—
233Dic59—162no—
234C2×#413D5911yescharacteristic
235#2406—591no—
236C2×#352—592no—
237#1782—591no—
238#1390—592no—
239C2×C59—21no—
240C2×C59—21no—
241C2×C59—42no—
242C2×C59—42no—
243C2×C59—42no—
244C2×C59#189811yescharacteristic
245C2×C59#212011yescharacteristic
246C2×C59#212311yescharacteristic
247C22×#41C2×D5911yescharacteristic
248C22×M16—21no—
249C2×#123—591no—
250C2×#123—591no—
251C22×C4⋊C4—591no—
252#413—24no—
253C2×#101—21no—
254C2×#100C2×D5912yes—
255C2×#99—592no—
256C2×#99—1181no—
257#395—1181no—
258#394—1181no—
259#386—591no—
260#386—591no—
261#385—1181no—
262#385—1181no—
263#381—591no—
264#381—1181no—
265#352—1184no—
266C59#1308111yescharacteristic, complemented, Sylow
267C2×#41—22no—
268C2×#41—22no—
269C2×#41—41no—
270C2×#41—41no—
271C2×#41—42no—
272C22×Q8C22⋊D5911yescharacteristic
273C22×D4C4×D5911yescharacteristic
274C22×D4D23611yescharacteristic
275C22×D4C22×D5911yescharacteristic
276C23×C4—591no—
277C23×C4—591no—
278C23×C4C4×D5911yescharacteristic
279C23×C4D23611yescharacteristic
280C23×C4C22⋊D5911yescharacteristic
281C2×M16—24no—
282C2×M16—22no—
283C2×M16—42no—
284C2×M16—41no—
285C22×C8—41no—
286#123—1182no—
287#123—1182no—
288#123—2362no—
289C2×C4⋊C4—1181no—
290C2×C4⋊C4—1181no—
291C2×C4⋊C4—1181no—
292C2×C4⋊C4—1181no—
293C2×C4⋊C4—2361no—
294C2×C4⋊C4—2361no—
295C2×C22⋊C4—591no—
296C2×C22⋊C4—591no—
297C2×C22⋊C4—1181no—
298C2×C22⋊C4—1181no—
299C2×C22⋊C4—1181no—
300C2×C22⋊C4—1181no—
301C2×C22⋊C4—1182no—
302#101—24no—
303#100—22no—
304#100C22⋊D5914yes—
305#99—1182no—
306#99—1182no—
307#99—2362no—
308#95—1181no—
309#95—1181no—
310#95—1181no—
311#95—1181no—
312#95—2361no—
313#95—2361no—
314C24(C2×D59)⋊C411yescharacteristic
315C24C2×C4×D5911yescharacteristic
316C24C2×D23611yescharacteristic
317#41—44no—
318#41—44no—
319#41—44no—
320#41—44no—
321C2×Q8—21no—
322C2×Q8—21no—
323C2×Q8—41no—
324C2×Q8—41no—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4—41no—
330C2×D4—41no—
331C2×D4—41no—
332C2×D4—41no—
333C2×D4#1174568512yes—
334C2×D4C2×C22⋊D5912yes—
335C22×C4—22no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—1181no—
349C22×C4—1182no—
350C22×C4—1181no—
351C22×C4—1181no—
352C22×C4—1181no—
353C22×C4—2362no—
354C22×C4—2362no—
355C22×C4—2361no—
356C22×C4—2361no—
357C22×C4—2361no—
358C22×C4(C2×D59)⋊C411yescharacteristic
359C22×C4(C2×D59)⋊C411yescharacteristic
360C22×C4(C2×D59)⋊C411yescharacteristic
361C22×C4C2×C22⋊D5911yescharacteristic
362M16—24no—
363M16—44no—
364M16—42no—
365C2×C8—44no—
366C2×C8—42no—
367C4⋊C4—2362no—
368C4⋊C4—2362no—
369C4⋊C4—2361no—
370C4⋊C4—2361no—
371C4⋊C4—2361no—
372C4⋊C4—2361no—
373C22⋊C4—1182no—
374C22⋊C4—2362no—
375C22⋊C4—2362no—
376C22⋊C4—2362no—
377C22⋊C4—2362no—
378C22⋊C4—2362no—
379C22⋊C4—2362no—
380C22⋊C4—2361no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—41no—
388C23—41no—
389C23—41no—
390C23—41no—
391C23—41no—
392C23—41no—
393C23#1427364112yes—
394C23C2×(C2×D59)⋊C411yescharacteristic
395C23C4×C22⋊D5912yes—
396C23#1427370112yes—
397Q8—42no—
398Q8—42no—
399D4—42no—
400D4—42no—
401D4—44no—
402D4—44no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—1182no—
422C2×C4—4722no—
423C2×C4—4722no—
424C2×C4—4722no—
425C2×C4—4722no—
426C2×C4—2361no—
427C2×C4—2361no—
428C2×C4—2361no—
429C2×C4—2361no—
430C2×C4—2361no—
431C2×C4—2361no—
432C2×C4—2361no—
433C2×C4—2361no—
434C2×C4—2361no—
435C2×C4—2361no—
436C2×C4—2362no—
437C2×C4—2361no—
438C2×C4—2362no—
439C2×C4#1427364111yescharacteristic
440C2×C4#1427364111yescharacteristic
441C2×C4#1427369611yescharacteristic
442C2×C4#1427371011yescharacteristic
443C8—44no—
444C22—21no—
445C22—21no—
446C22—22no—
447C22—22no—
448C22—22no—
449C22—42no—
450C22—42no—
451C22—42no—
452C22—42no—
453C22—41no—
454C22—41no—
455C22—41no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—81no—
460C22—81no—
461C22#1683728111yescharacteristic
462C22C2×#1427364111yescharacteristic
463C22C2×#1427364111yescharacteristic
464C4—21no—
465C4—21no—
466C4—21no—
467C4—21no—
468C4—41no—
469C4—41no—
470C4—9442no—
471C4—4721no—
472C4—4721no—
473C4—4721no—
474C4—2362no—
475C2—21no—
476C2—21no—
477C2—42no—
478C2—42no—
479C2—42no—
480C2#2142454911yescharacteristic
481C2#2142573811yescharacteristic
482C2#2142575111yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC24×C59C22×C4
FrattiniC22×D4C22×D59
FittingC2×C59×#413C2
socleC22×C59C2×#99
prime radicalC22×#41×C59C22
cosocleC22×D4×C59C23
hypercenterC2×#413D59
hypocenterC59#13081
LCSG ▷ C24×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ C2×#413

Complements

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

NH ≅ G/Ntype
2. C2×#16837193C2semidirect
4. #21422215C2semidirect
6. C2×#14273708C4semidirect
8. C22×Dic59⋊C4C4semidirect
13. #16837282C4semidirect
18. #16837265C4semidirect
266. C59#13081semidirect

Maximal subgroups

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

subgroupindex
#214274272
C2×#168371932
#214240142
#214222152
C2×C59×#4132
#1308159