Finite Group Databasebeta

← #28491836

Subgroups — #28491836

Order 15104.

508 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 93 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#21429108C211yesmaximal, characteristic, complemented
2#21424931C211yesmaximal, characteristic
3#21422663C211yesmaximal, characteristic
4C2×#16837137C211yesmaximal, characteristic
5#21421974C211yesmaximal, characteristic, complemented
6#21421942C211yesmaximal, characteristic, complemented
7C59×#1289C211yesmaximal, characteristic, complemented
8C2×#14273704—21no—
9C2×#14273701C2211yescharacteristic
10C2×#14273641—21no—
11C2×#14273641—21no—
12C2×C22⋊(C59⋊C8)C2211yescharacteristic
13#16837335—21no—
14#16837322—21no—
15#16837306—21no—
16#16837299—21no—
17#16837274—21no—
18C4×C22⋊Dic59C2211yescharacteristic
19C2×C59⋊#95C2211yescharacteristic
20#16837137—21no—
21#16837137—21no—
22#16837135—21no—
23#16837135—21no—
24C2×C22⋊C8×C59C2211yescharacteristic
25C59×#380C2211yescharacteristic
26C59×#380C2211yescharacteristic
27C22×Dic118—21no—
28C22×D236Q811yescharacteristic, complemented
29C22×C22⋊D59—21no—
30C23×Dic59—21no—
31C22×C59⋊C8—21no—
32#14273704—42no—
33#14273704—42no—
34#14273701—21no—
35#14273701—21no—
36#14273701—21no—
37#14273701—21no—
38C2×C4⋊Dic59—21no—
39C2×C4⋊Dic59Q811yescharacteristic, complemented
40C2×Dic59⋊C4—21no—
41C2×Dic59⋊C4—21no—
42C2×Dic59⋊C4—21no—
43C2×Dic59⋊C4—21no—
44C2×C22⋊Dic59D411yescharacteristic
45C2×C22⋊Dic59D411yescharacteristic
46C2×(C2×D59)⋊C4—21no—
47C2×C4×Dic59—21no—
48#14273641—42no—
49#14273641—42no—
50C22⋊(C59⋊C8)—21no—
51C22⋊(C59⋊C8)—21no—
52C59⋊#95—21no—
53C59⋊#95—21no—
54C59⋊#95—21no—
55C59⋊#95—21no—
56C59⋊#95D411yescharacteristic
57C59⋊#95D411yescharacteristic
58C23×C4×C59C2311yescharacteristic
59C22×C8×C59—21no—
60C2×C4⋊C4×C59—21no—
61C2×C4⋊C4×C59—21no—
62C2×C22⋊C4×C59—21no—
63C2×C22⋊C4×C59—21no—
64C22⋊C8×C59—21no—
65C22⋊C8×C59—21no—
66#95×C59—21no—
67#95×C59D411yescharacteristic
68#95×C59D411yescharacteristic
69C24×C59C2×D411yescharacteristic
70C23×D59—21no—
71C22×C4×C59—21no—
72C22×C4×C59—21no—
73C22×C4×C59—41no—
74C22×C4×C59—81no—
75C22×C4×C59C2×D411yescharacteristic
76C22×C4×C59C2×D411yescharacteristic
77C22×C4×C59C2×Q811yescharacteristic
78C22×C4×C59#4111yescharacteristic
79C22×C4×C59#4111yescharacteristic
80C22×C4×C59#4111yescharacteristic
81C2×C22⋊D59—41no—
82C2×C22⋊D59—41no—
83C2×C22⋊D59—41no—
84C2×C22⋊D59—41no—
85C2×C22⋊D59—41no—
86C2×C22⋊D59—41no—
87C22×Dic59—21no—
88C22×Dic59—21no—
89C22×Dic59—21no—
90C22×Dic59—21no—
91C22×Dic59—21no—
92C22×Dic59—42no—
93C22×Dic59—41no—
94C22×Dic59—41no—
95C22×Dic59—81no—
96C2×D236—21no—
97C2×D236—21no—
98C2×D236—81no—
99C2×Dic118—42no—
100C2×Dic118—42no—
101C2×Dic118—81no—
102C2×C8×C59—41no—
103C2×C8×C59—81no—
104C4⋊C4×C59—81no—
105C4⋊C4×C59—81no—
106C22⋊C4×C59—41no—
107C22⋊C4×C59—41no—
108C22⋊C4×C59—41no—
109C22⋊C4×C59—41no—
110C22⋊Dic59—21no—
111C22⋊Dic59—21no—
112C22⋊Dic59D812yes—
113C22⋊Dic59SD1612yes—
114(C2×D59)⋊C4—41no—
115(C2×D59)⋊C4—41no—
116C4⋊Dic59—21no—
117C4⋊Dic59—21no—
118C4⋊Dic59—42no—
119Dic59⋊C4—42no—
120Dic59⋊C4—42no—
121Dic59⋊C4—42no—
122Dic59⋊C4—81no—
123C4×Dic59—81no—
124C2×C59⋊C8—41no—
125C2×C59⋊C8—81no—
126C23×C59—41no—
127C23×C59—41no—
128C23×C59#12011yescharacteristic
129C23×C59C22⋊Q811yescharacteristic
130C23×C59#12411yescharacteristic
131C23×C59C2×D811yescharacteristic
132C23×C59C2×SD1611yescharacteristic
133C23×C59AGL(1,ℤ/8)11yescharacteristic
134C23×C59#13711yescharacteristic
135C22×D59—41no—
136C22×D59—41no—
137C22×D59—41no—
138C22×D59—81no—
139C22×D59—81no—
140C2×C4×C59—41no—
141C2×C4×C59—41no—
142C2×C4×C59—41no—
143C2×C4×C59—41no—
144C2×C4×C59—41no—
145C2×C4×C59—41no—
146C2×C4×C59—41no—
147C2×C4×C59—41no—
148C2×C4×C59—41no—
149C2×C4×C59—41no—
150C2×C4×C59—41no—
151C2×C4×C59—41no—
152C2×C4×C59—81no—
153C2×C4×C59—81no—
154C2×C4×C59—81no—
155C2×C4×C59C22⋊Q811yescharacteristic
156C2×C4×C59C22⋊Q811yescharacteristic
157C2×C4×C59#12411yescharacteristic
158C2×C4×C59#12411yescharacteristic
159C22⋊D59—81no—
160C22⋊D59—81no—
161C22⋊D59—81no—
162C22⋊D59—81no—
163C2×Dic59—22no—
164C2×Dic59—22no—
165C2×Dic59—22no—
166C2×Dic59—41no—
167C2×Dic59—41no—
168C2×Dic59—41no—
169C2×Dic59—41no—
170C2×Dic59—41no—
171C2×Dic59—41no—
172C2×Dic59—41no—
173C2×Dic59—41no—
174C2×Dic59—41no—
175C2×Dic59—41no—
176C2×Dic59—41no—
177C2×Dic59—41no—
178C2×Dic59—82no—
179C2×Dic59—82no—
180D236—81no—
181D236—81no—
182Dic118—84no—
183C8×C59—81no—
184C59⋊C8—81no—
185#10374—591nomaximal, Sylow
186C22×C59—21no—
187C22×C59—21no—
188C22×C59—21no—
189C22×C59—21no—
190C22×C59—21no—
191C22×C59—21no—
192C22×C59—41no—
193C22×C59—41no—
194C22×C59—41no—
195C22×C59—41no—
196C22×C59#39311yescharacteristic
197C22×C59#44711yescharacteristic
198C22×C59#44811yescharacteristic
199C22×C59#47411yescharacteristic
200C22×C59#47611yescharacteristic
201C22×C59#48611yescharacteristic
202C22×C59#48911yescharacteristic
203C2×D59—81no—
204C2×D59—81no—
205C2×D59—81no—
206C2×D59—161no—
207C2×D59—161no—
208C4×C59—21no—
209C4×C59—21no—
210C4×C59—21no—
211C4×C59—21no—
212C4×C59—81no—
213C4×C59—81no—
214C4×C59—161no—
215Dic59—42no—
216Dic59—42no—
217Dic59—42no—
218Dic59—81no—
219Dic59—81no—
220#2549—591no—
221#1964—591no—
222#1519—591no—
223C2×#327—591no—
224#1294—591no—
225#1289D5911yescharacteristic, complemented
226#1282—591no—
227C2×C59—21no—
228C2×C59—21no—
229C2×C59—21no—
230C2×C59—21no—
231C2×C59—21no—
232C2×C59—21no—
233C2×C59#202311yescharacteristic
234C2×C59#203111yescharacteristic
235C2×C59#203211yescharacteristic
236D59—161no—
237C2×C22⋊Q8—1181no—
238C2×#121—591no—
239C2×#99—1181no—
240C2×#99—1181no—
241C2×C22⋊C8—591no—
242C2×C22⋊C8C2×D5911yescharacteristic
243#398—1181no—
244#395—1181no—
245#391—1181no—
246#389—1181no—
247#385—1181no—
248#380C2×D5911yescharacteristic
249#380C2×D5911yescharacteristic
250C4×C22⋊C4—591no—
251C2×#95—591no—
252#327—1181no—
253#327—1181no—
254#327—1181no—
255#327—1181no—
256C59#1037411yescharacteristic, complemented, Sylow
257C22×Q8—1181no—
258C22×D4—591no—
259C22×D4—1181no—
260C23×C4—1181no—
261C23×C4C22×D5911yescharacteristic
262C22×C8—21no—
263C22×C8—1181no—
264C22⋊Q8—2362no—
265C22⋊Q8—2362no—
266#121—1181no—
267#121—1181no—
268#121—1181no—
269#121—1181no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—591no—
273C2×C4⋊C4—1181no—
274C2×C4⋊C4—1181no—
275C2×C4⋊C4—1181no—
276C2×C4⋊C4—1181no—
277C2×C4⋊C4—1181no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—591no—
281C2×C22⋊C4—591no—
282C2×C22⋊C4—1181no—
283C2×C42—1181no—
284#99—2362no—
285#99—2362no—
286C22⋊C8—21no—
287C22⋊C8—21no—
288C22⋊C8—1181no—
289C22⋊C8—1181no—
290#95—21no—
291#95—591no—
292#95—591no—
293#95—1181no—
294#95—1181no—
295#95—1181no—
296#95—1181no—
297#95D23611yescharacteristic
298#95D23611yescharacteristic
299C24—1181no—
300C24D4×D5911yescharacteristic
301C2×Q8—4721no—
302C2×Q8—2362no—
303C2×Q8—2362no—
304C2×D4—1181no—
305C2×D4—1181no—
306C2×D4—4721no—
307C2×D4—2361no—
308C2×D4—2361no—
309C2×D4—2361no—
310C2×D4—2361no—
311C2×D4—2361no—
312C2×D4—2361no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—41no—
316C22×C4—81no—
317C22×C4—1181no—
318C22×C4—1181no—
319C22×C4—1181no—
320C22×C4—1181no—
321C22×C4—1181no—
322C22×C4—4721no—
323C22×C4—2362no—
324C22×C4—2361no—
325C22×C4—2361no—
326C22×C4C2×D23611yescharacteristic
327C22×C4#1174568511yescharacteristic
328C22×C4#1174568511yescharacteristic
329C22×C4D4×D5911yescharacteristic
330C22×C4#1174568711yescharacteristic
331C22×C4Q8×D5911yescharacteristic
332C2×C8—41no—
333C2×C8—81no—
334C2×C8—4721no—
335C2×C8—2361no—
336C4⋊C4—81no—
337C4⋊C4—81no—
338C4⋊C4—1181no—
339C4⋊C4—1181no—
340C4⋊C4—4721no—
341C4⋊C4—2362no—
342C4⋊C4—2362no—
343C4⋊C4—2362no—
344C4⋊C4—2362no—
345C22⋊C4—41no—
346C22⋊C4—41no—
347C22⋊C4—41no—
348C22⋊C4—41no—
349C22⋊C4—592no—
350C22⋊C4—592no—
351C22⋊C4—1181no—
352C22⋊C4—1181no—
353C22⋊C4—2361no—
354C22⋊C4—2361no—
355C42—4721no—
356C23—41no—
357C23—41no—
358C23—4721no—
359C23—4721no—
360C23—2361no—
361C23—2361no—
362C23—2361no—
363C23#1427369711yescharacteristic
364C23#1427370511yescharacteristic
365C23#1427371311yescharacteristic
366C23D8×D5911yescharacteristic
367C23SD16×D5911yescharacteristic
368C23#1427375411yescharacteristic
369C23#1427375911yescharacteristic
370Q8—4724no—
371D4—4721no—
372D4—4721no—
373D4—4721no—
374D4—4721no—
375D4—4721no—
376D4—4721no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—1182no—
393C2×C4—1182no—
394C2×C4—1182no—
395C2×C4—4722no—
396C2×C4—4722no—
397C2×C4—2361no—
398C2×C4—2361no—
399C2×C4—2361no—
400C2×C4—2361no—
401C2×C4—2361no—
402C2×C4—2361no—
403C2×C4—2361no—
404C2×C4—2361no—
405C2×C4—2361no—
406C2×C4—2361no—
407C2×C4—2361no—
408C2×C4—2361no—
409C2×C4#1427370511yescharacteristic
410C2×C4#1427370711yescharacteristic
411C2×C4#1427371311yescharacteristic
412C2×C4#1427371511yescharacteristic
413C8—81no—
414C8—4721no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—41no—
422C22—41no—
423C22—41no—
424C22—41no—
425C22—9441no—
426C22—9441no—
427C22—4721no—
428C22—4721no—
429C22—4721no—
430C22#1683731511yescharacteristic
431C22#1683753711yescharacteristic
432C22#1683754211yescharacteristic
433C22#1683768111yescharacteristic
434C22#1683769111yescharacteristic
435C22#1683774711yescharacteristic
436C22#1683776311yescharacteristic
437C4—21no—
438C4—21no—
439C4—21no—
440C4—21no—
441C4—81no—
442C4—81no—
443C4—161no—
444C4—4721no—
445C4—4721no—
446C4—2362no—
447C4—2362no—
448C4—2362no—
449C2—21no—
450C2—21no—
451C2—21no—
452C2—21no—
453C2—21no—
454C2—21no—
455C2—9441no—
456C2#2142523111yescharacteristic
457C2#2142526711yescharacteristic
458C2#2142527411yescharacteristic
subgroupG/H
centerC22#16837315
commutator, cosocleC23×C4×C59C23
FrattiniC23×C4C22×D59
FittingC59×#1289C2
socleC22×C59#393
prime radicalC2×#14273701C22
hypercenter#1289D59
hypocenterC59#10374
LCSG ▷ C23×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1289

Complements

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

NH ≅ G/Ntype
1. #21429108C2semidirect
5. #21421974C2semidirect
6. #21421942C2semidirect
7. C59×#1289C2semidirect
28. C22×D236Q8semidirect
39. C2×C4⋊Dic59Q8semidirect
225. #1289D59semidirect
256. C59#10374semidirect

Maximal subgroups

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

subgroupindex
#214291082
#214249312
#214226632
C2×#168371372
#214219742
#214219422
C59×#12892
#1037459