Finite Group Databasebeta

← #28507361

Subgroups — #28507361

Order 15104.

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

normal subgroups only

This group has 500 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#21429267C211yesmaximal, characteristic
2#21424082C211yesmaximal, characteristic
3#99×Dic59C211yesmaximal, characteristic
4#21422221C211yesmaximal, characteristic
5#21422220C211yesmaximal, characteristic
6#21422220C211yesmaximal, characteristic
7C59×#1393C211yesmaximal, characteristic
8C2×#14273726C411yescharacteristic, complemented
9C2×D4×Dic59C2211yescharacteristic
10C2×#14273642C2211yescharacteristic
11C2×#14273642C2211yescharacteristic
12#16837314—21no—
13#16837314—21no—
14#16837305C2211yescharacteristic
15#16837268C411yescharacteristic, complemented
16#16837267C411yescharacteristic, complemented
17#16837260—21no—
18#16837260—21no—
19#16837260—21no—
20C4×Dic59⋊C4C411yescharacteristic, complemented
21C22⋊C4×Dic59—21no—
22#16837168—21no—
23#16837168—21no—
24C2×#99×C59C2211yescharacteristic
25C2×#99×C59C2211yescharacteristic
26C59×#390C2211yescharacteristic
27C22×Dic118—21no—
28C23×Dic59D411yescharacteristic
29C23×Dic59D411yescharacteristic
30#14273726—41no—
31#14273726—41no—
32D4×Dic59—41no—
33D4×Dic59—41no—
34C2×C4⋊Dic59C2×C411yescharacteristic
35C2×C4⋊Dic59C2×C411yescharacteristic
36C2×Dic59⋊C4—21no—
37C2×Dic59⋊C4—21no—
38C2×Dic59⋊C4—41no—
39C2×C22⋊Dic59—21no—
40C2×C22⋊Dic59—21no—
41C2×C22⋊Dic59—21no—
42C2×C22⋊Dic59—21no—
43C2×C22⋊Dic59D411yescharacteristic
44C2×C22⋊Dic59D411yescharacteristic
45C2×C4×Dic59—41no—
46C2×C4×Dic59C2×C411yescharacteristic
47C2×C4×Dic59C2×C411yescharacteristic
48#14273642—21no—
49#14273642—21no—
50#14273642—21no—
51#14273642Q812yescomplemented
52C59⋊#95—21no—
53C59⋊#95—21no—
54C59⋊#95—21no—
55C59⋊#95—21no—
56C59⋊#95—21no—
57C59⋊#95—41no—
58C59⋊#95—41no—
59C22×D4×C59C2311yescharacteristic
60C2×C4⋊C4×C59C2×C411yescharacteristic
61C2×C22⋊C4×C59—21no—
62C2×C22⋊C4×C59—21no—
63C2×C22⋊C4×C59—21no—
64C2×C22⋊C4×C59—21no—
65C2×C42×C59C2×C411yescharacteristic
66#99×C59—21no—
67#99×C59—21no—
68#99×C59—21no—
69#99×C59Q812yescomplemented
70C24×C59C2×D411yescharacteristic
71C24×C59C2×D411yescharacteristic
72C2×D4×C59—41no—
73C2×D4×C59—41no—
74C2×D4×C59C2×Q811yescharacteristic
75C2×D4×C59C2×Q811yescharacteristic
76C2×D4×C59#4111yescharacteristic
77C2×D4×C59#4111yescharacteristic
78C22×C4×C59—21no—
79C22×C4×C59—21no—
80C22×C4×C59—41no—
81C22×C4×C59—41no—
82C22×C4×C59C22×C411yescharacteristic
83C22×Dic59—21no—
84C22×Dic59—21no—
85C22×Dic59—21no—
86C22×Dic59—21no—
87C22×Dic59—21no—
88C22×Dic59—21no—
89C22×Dic59—42no—
90C22×Dic59—42no—
91C22×Dic59—41no—
92C22×Dic59—41no—
93C22×Dic59C22⋊C411yescharacteristic
94C22×Dic59C22⋊C411yescharacteristic
95C22×Dic59C22⋊C411yescharacteristic
96C22×Dic59C22⋊C411yescharacteristic
97C2×Dic118—42no—
98C2×Dic118—81no—
99C2×Dic118—81no—
100C4⋊C4×C59—41no—
101C22⋊C4×C59—22no—
102C22⋊C4×C59—41no—
103C22⋊C4×C59—41no—
104C22⋊C4×C59—41no—
105C22⋊C4×C59—81no—
106C22⋊C4×C59—81no—
107C42×C59—41no—
108C22⋊Dic59—22no—
109C22⋊Dic59—41no—
110C22⋊Dic59—41no—
111C22⋊Dic59—41no—
112C22⋊Dic59—41no—
113C22⋊Dic59—41no—
114C22⋊Dic59—81no—
115C22⋊Dic59—81no—
116C4⋊Dic59—41no—
117C4⋊Dic59—41no—
118Dic59⋊C4—42no—
119Dic59⋊C4—41no—
120Dic59⋊C4—41no—
121Dic59⋊C4—82no—
122C4×Dic59—21no—
123C4×Dic59—21no—
124C4×Dic59—41no—
125C4×Dic59—82no—
126C23×C59—21no—
127C23×C59—21no—
128C23×C59—21no—
129C23×C59—21no—
130C23×C59—41no—
131C23×C59—41no—
132C23×C59—41no—
133C23×C59—41no—
134C23×C59C2×C22⋊C411yescharacteristic
135C23×C59C22⋊Q811yescharacteristic
136C23×C59C22⋊Q811yescharacteristic
137C23×C59#12411yescharacteristic
138C23×C59C4⋊Q811yescharacteristic
139D4×C59—41no—
140D4×C59—41no—
141D4×C59—41no—
142D4×C59—41no—
143C2×C4×C59—21no—
144C2×C4×C59—21no—
145C2×C4×C59—41no—
146C2×C4×C59—42no—
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×C59—81no—
156C2×C4×C59—81no—
157C2×C4×C59C4×Q811yescharacteristic
158C2×C4×C59C4×Q811yescharacteristic
159C2×Dic59—22no—
160C2×Dic59—21no—
161C2×Dic59—21no—
162C2×Dic59—21no—
163C2×Dic59—42no—
164C2×Dic59—42no—
165C2×Dic59—41no—
166C2×Dic59—41no—
167C2×Dic59—41no—
168C2×Dic59—41no—
169C2×Dic59—42no—
170C2×Dic59—41no—
171C2×Dic59—41no—
172C2×Dic59—41no—
173C2×Dic59—41no—
174C2×Dic59—41no—
175C2×Dic59—81no—
176C2×Dic59—81no—
177C2×Dic59—81no—
178C2×Dic59—81no—
179C2×Dic59—81no—
180C2×Dic59—81no—
181C2×Dic59—81no—
182C2×Dic59#9911yescharacteristic
183C2×Dic59#9911yescharacteristic
184Dic118—84no—
185#13299—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—41no—
197C22×C59—41no—
198C22×C59—41no—
199C22×C59—41no—
200C22×C59—81no—
201C22×C59—81no—
202C22×C59#39111yescharacteristic
203C22×C59C2×#9911yescharacteristic
204C22×C59#41011yescharacteristic
205C4×C59—21no—
206C4×C59—21no—
207C4×C59—81no—
208C4×C59—82no—
209C4×C59—81no—
210C4×C59—161no—
211Dic59—21no—
212Dic59—41no—
213Dic59—41no—
214Dic59—41no—
215Dic59—81no—
216Dic59—82no—
217Dic59—81no—
218Dic59—161no—
219Dic59#35412yes—
220#2563—591no—
221#1799—591no—
222C4×#99—591no—
223#1393—591no—
224#1393—591no—
225#1393—591no—
226#1393D5911yescharacteristic
227C2×C59—21no—
228C2×C59—21no—
229C2×C59—41no—
230C2×C59—41no—
231C2×C59—41no—
232C2×C59—41no—
233C2×C59#199611yescharacteristic
234C2×C59C2×#35411yescharacteristic
235C2×C59#212811yescharacteristic
236C2×C4⋊Q8—591no—
237C2×C4×D4—591no—
238C2×#99—591no—
239C2×#99—591no—
240C2×#99C2×D5911yescharacteristic
241C2×#99C2×D5911yescharacteristic
242#393—1181no—
243#393—1181no—
244#390—591no—
245#390C2×D5911yescharacteristic
246#382—591no—
247#382—591no—
248#380—1181no—
249#380—1181no—
250#380—1181no—
251C4×C4⋊C4—591no—
252C4×C22⋊C4—1181no—
253#342—1181no—
254#342—1181no—
255C59#1329911yescharacteristic, complemented, Sylow
256C22×Q8—1181no—
257C22×D4C22×D5911yescharacteristic
258C23×C4—591no—
259C23×C4—591no—
260C4⋊Q8—2361no—
261C4⋊Q8—2361no—
262C4×D4—2361no—
263C4×D4—2361no—
264C2×C4⋊C4—591no—
265C2×C4⋊C4—591no—
266C2×C4⋊C4—1181no—
267C2×C4⋊C4—1181no—
268C2×C4⋊C4—2361no—
269C2×C4⋊C4C4×D5911yescharacteristic
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—591no—
275C2×C22⋊C4—591no—
276C2×C22⋊C4—1181no—
277C2×C22⋊C4—1181no—
278C2×C22⋊C4—1181no—
279C2×C22⋊C4—1181no—
280C2×C42—591no—
281C2×C42—591no—
282C2×C42—2361no—
283C2×C42C4×D5911yescharacteristic
284#99—21no—
285#99—21no—
286#99—21no—
287#99—592no—
288#99—1181no—
289#99—1181no—
290#99—1181no—
291#99Dic11812yescomplemented
292#95—1181no—
293#95—1181no—
294#95—1181no—
295#95—1181no—
296#95—1181no—
297#95—2361no—
298#95—2361no—
299C24D4×D5911yescharacteristic
300C24D4×D5911yescharacteristic
301C2×Q8—4721no—
302C2×Q8—4721no—
303C2×Q8—2362no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4C2×Dic11811yescharacteristic
307C2×D4#1174568511yescharacteristic
308C2×D4#1174568711yescharacteristic
309C2×D4Q8×D5911yescharacteristic
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—591no—
315C22×C4—591no—
316C22×C4—591no—
317C22×C4—591no—
318C22×C4—1181no—
319C22×C4—1181no—
320C22×C4—1181no—
321C22×C4—1181no—
322C22×C4—1181no—
323C22×C4—1181no—
324C22×C4—2362no—
325C22×C4—2362no—
326C22×C4—2361no—
327C22×C4—2361no—
328C22×C4C2×C4×D5911yescharacteristic
329C4⋊C4—41no—
330C4⋊C4—4722no—
331C4⋊C4—2361no—
332C4⋊C4—2361no—
333C4⋊C4—2361no—
334C4⋊C4—2361no—
335C4⋊C4—2362no—
336C22⋊C4—22no—
337C22⋊C4—41no—
338C22⋊C4—41no—
339C22⋊C4—41no—
340C22⋊C4—81no—
341C22⋊C4—81no—
342C22⋊C4—1182no—
343C22⋊C4—4721no—
344C22⋊C4—4721no—
345C22⋊C4—2361no—
346C22⋊C4—2361no—
347C22⋊C4—2361no—
348C22⋊C4—2361no—
349C22⋊C4—2361no—
350C42—41no—
351C42—1181no—
352C42—1181no—
353C42—4722no—
354C42—2361no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23C22⋊C4×D5911yescharacteristic
364C23#1427370411yescharacteristic
365C23#1427370511yescharacteristic
366C23#1427371311yescharacteristic
367C23#1427372611yescharacteristic
368Q8—4724no—
369D4—41no—
370D4—41no—
371D4—41no—
372D4—41no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—42no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—591no—
388C2×C4—591no—
389C2×C4—1182no—
390C2×C4—1181no—
391C2×C4—1181no—
392C2×C4—1181no—
393C2×C4—4721no—
394C2×C4—4721no—
395C2×C4—4721no—
396C2×C4—4721no—
397C2×C4—4721no—
398C2×C4—4721no—
399C2×C4—4721no—
400C2×C4—2362no—
401C2×C4—2362no—
402C2×C4—2362no—
403C2×C4—2361no—
404C2×C4—2361no—
405C2×C4—2361no—
406C2×C4—2361no—
407C2×C4—2361no—
408C2×C4—2361no—
409C2×C4—2361no—
410C2×C4—2361no—
411C2×C4—2361no—
412C2×C4#1427369211yescharacteristic
413C2×C4C4×Dic11811yescharacteristic
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—41no—
421C22—41no—
422C22—41no—
423C22—41no—
424C22—41no—
425C22—41no—
426C22—41no—
427C22—41no—
428C22—81no—
429C22—81no—
430C22#1683730611yescharacteristic
431C22#99×D5911yescharacteristic
432C22#1683737411yescharacteristic
433C4—21no—
434C4—21no—
435C4—81no—
436C4—82no—
437C4—81no—
438C4—161no—
439C4—592no—
440C4—1181no—
441C4—9441no—
442C4—4721no—
443C4—4722no—
444C4—4721no—
445C4—2361no—
446C4—2361no—
447C4—2361no—
448C2—21no—
449C2—21no—
450C2—41no—
451C2—41no—
452C2—41no—
453C2—41no—
454C2#2142507911yescharacteristic
455C2#354×D5911yescharacteristic
456C2#2142576611yescharacteristic
subgroupG/H
centerC22#99×D59
commutatorC22×C4×C59C22×C4
FrattiniC22×D4C22×D59
FittingC59×#1393C2
socleC22×C59C2×#99
prime radical, cosocleC22×D4×C59C23
hypercenter#1393D59
hypocenterC59#13299
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ #1393

Complements

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

NH ≅ G/Ntype
8. C2×#14273726C4semidirect
15. #16837268C4semidirect
16. #16837267C4semidirect
20. C4×Dic59⋊C4C4semidirect
51. #14273642Q8semidirect
69. #99×C59Q8semidirect
255. C59#13299semidirect
291. #99Dic118semidirect

Maximal subgroups

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

subgroupindex
#214292672
#214240822
#99×Dic592
#214222212
#214222202
C59×#13932
#1329959