Finite Group Databasebeta

← #28486009

Subgroups — #28486009

Order 15104.

640 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 640 conjugacy classes of subgroups — too many to draw. Only its 131 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#21429052C211yesmaximal, characteristic
2#21424084C211yesmaximal, characteristic, complemented
3#21422665C212yesmaximal, complemented
4#21421976C212yesmaximal, complemented
5C59×#2972C211yesmaximal, characteristic, complemented
6C2×#14273701C2211yescharacteristic
7C2×#14273699—21no—
8C2×#14273642—21no—
9#16837332C411yescharacteristic, complemented
10#16837327C411yescharacteristic, complemented
11#16837319C411yescharacteristic, complemented
12#16837311C411yescharacteristic, complemented
13#16837293—21no—
14#16837281—21no—
15#16837279—21no—
16#16837262C2212yes—
17C2×C59⋊#95C2211yescharacteristic
18#16837169—21no—
19#16837137—24no—
20C2×C4×D4×C59C2211yescharacteristic
21C8×D4×C59—24no—
22C59×#431—21no—
23C2×C22⋊C8×C59—21no—
24C2×C22⋊C8×C59C2212yes—
25C22×D236C2×C411yescharacteristic
26C22×C22⋊D59—21no—
27C22×C22⋊D59—21no—
28C23×Dic59—21no—
29#14273701—22no—
30#14273701—22no—
31#14273699—42no—
32#14273699—42no—
33C2×C4⋊Dic59—22no—
34C2×C4⋊Dic59C2×C411yescharacteristic
35C2×Dic59⋊C4—21no—
36C2×C22⋊Dic59—21no—
37C2×C22⋊Dic59—22no—
38C2×(C2×D59)⋊C4—21no—
39C2×(C2×D59)⋊C4—21no—
40C2×(C2×D59)⋊C4—21no—
41C2×(C2×D59)⋊C4—21no—
42#14273642—42no—
43C59⋊#95—21no—
44C59⋊#95—21no—
45C59⋊#95—21no—
46C59⋊#95—22no—
47C59⋊#95C2×C411yescharacteristic
48C59⋊#95C2×C411yescharacteristic
49C22×D4×C59D411yescharacteristic
50C23×C4×C59D411yescharacteristic
51C23×C4×C59C2311yescharacteristic
52C22×C8×C59—22no—
53C22×C8×C59—22no—
54C4×D4×C59—22no—
55C4×D4×C59—22no—
56C2×C4⋊C4×C59C2×C411yescharacteristic
57C2×C22⋊C4×C59C2×C411yescharacteristic
58C2×C22⋊C4×C59D411yescharacteristic
59C2×C42×C59D411yescharacteristic
60C4⋊C8×C59—22no—
61C22⋊C8×C59—22no—
62C22⋊C8×C59—22no—
63C22⋊C8×C59D414yescomplemented
64C4×C8×C59—22no—
65C24×C59C22⋊C411yescharacteristic
66C24×C59C2×D411yescharacteristic
67C23×D59—21no—
68C2×D4×C59—22no—
69C2×D4×C59—22no—
70C2×D4×C59D812yes—
71C2×D4×C59SD1612yes—
72C22×C4×C59—21no—
73C22×C4×C59—21no—
74C22×C4×C59—81no—
75C22×C4×C59—81no—
76C22×C4×C59C22⋊C411yescharacteristic
77C22×C4×C59C22⋊C411yescharacteristic
78C22×C4×C59C22⋊C411yescharacteristic
79C22×C4×C59C22×C411yescharacteristic
80C22×C4×C59C2×D412yes—
81C22×C4×C59C2×D411yescharacteristic
82C22×C4×C59#4112yes—
83C2×C22⋊D59—42no—
84C2×C22⋊D59—42no—
85C2×C22⋊D59—42no—
86C2×C22⋊D59—41no—
87C2×C22⋊D59—41no—
88C2×C22⋊D59—42no—
89C2×C22⋊D59—81no—
90C22×Dic59—21no—
91C22×Dic59—21no—
92C22×Dic59—21no—
93C22×Dic59—42no—
94C22×Dic59—41no—
95C22×Dic59—41no—
96C22×Dic59—42no—
97C22×Dic59—81no—
98C2×D236—22no—
99C2×D236—81no—
100C2×C8×C59—24no—
101C2×C8×C59—82no—
102C2×C8×C59—82no—
103C4⋊C4×C59—22no—
104C22⋊C4×C59—22no—
105C22⋊C4×C59—22no—
106C42×C59—21no—
107C42×C59—21no—
108C22⋊Dic59—42no—
109C22⋊Dic59—42no—
110C22⋊Dic59—42no—
111(C2×D59)⋊C4—42no—
112(C2×D59)⋊C4—42no—
113(C2×D59)⋊C4—41no—
114(C2×D59)⋊C4—41no—
115(C2×D59)⋊C4—81no—
116C4⋊Dic59—22no—
117C4⋊Dic59—82no—
118Dic59⋊C4—42no—
119C23×C59—21no—
120C23×C59—21no—
121C23×C59—41no—
122C23×C59—41no—
123C23×C59—41no—
124C23×C59—41no—
125C23×C59#10212yes—
126C23×C59C2×C22⋊C411yescharacteristic
127C23×C59#12112yes—
128C23×C59C2×D811yescharacteristic
129C23×C59C2×SD1611yescharacteristic
130C23×C59#13512yes—
131C22×D59—41no—
132C22×D59—41no—
133C22×D59—41no—
134C22×D59—41no—
135C22×D59—41no—
136C22×D59—41no—
137C22×D59—41no—
138D4×C59—24no—
139D4×C59—24no—
140C2×C4×C59—21no—
141C2×C4×C59—21no—
142C2×C4×C59—21no—
143C2×C4×C59—21no—
144C2×C4×C59—42no—
145C2×C4×C59—42no—
146C2×C4×C59—41no—
147C2×C4×C59—41no—
148C2×C4×C59—42no—
149C2×C4×C59—41no—
150C2×C4×C59—41no—
151C2×C4×C59—81no—
152C2×C4×C59#10212yes—
153C2×C4×C59C4×D412yes—
154C2×C4×C59#12011yescharacteristic
155C2×C4×C59#12311yescharacteristic
156C22⋊D59—84no—
157C22⋊D59—82no—
158C22⋊D59—82no—
159C2×Dic59—22no—
160C2×Dic59—41no—
161C2×Dic59—41no—
162C2×Dic59—41no—
163C2×Dic59—41no—
164C2×Dic59—41no—
165C2×Dic59—41no—
166C2×Dic59—41no—
167C2×Dic59—41no—
168C2×Dic59—82no—
169C2×Dic59—82no—
170C2×Dic59—82no—
171C2×Dic59—82no—
172C2×Dic59—82no—
173D236—82no—
174C8×C59—44no—
175#9384—591nomaximal, Sylow
176C22×C59—21no—
177C22×C59—22no—
178C22×C59—21no—
179C22×C59—22no—
180C22×C59—22no—
181C22×C59—41no—
182C22×C59—41no—
183C22×C59—42no—
184C22×C59—42no—
185C22×C59—41no—
186C22×C59—41no—
187C22×C59—81no—
188C22×C59#38611yescharacteristic
189C22×C59C2×#10211yescharacteristic
190C22×C59#41611yescharacteristic
191C22×C59#46512yes—
192C22×C59#46612yes—
193C2×D59—81no—
194C2×D59—81no—
195C2×D59—81no—
196C2×D59—81no—
197C2×D59—81no—
198C2×D59—81no—
199C2×D59—81no—
200C4×C59—22no—
201C4×C59—21no—
202C4×C59—22no—
203C4×C59—21no—
204C4×C59—41no—
205C4×C59—41no—
206Dic59—42no—
207Dic59—81no—
208Dic59—81no—
209Dic59—162no—
210#2972D5911yescharacteristic, complemented
211#2543—591no—
212#1799—591no—
213#1519—592no—
214#1294—592no—
215C2×C59—21no—
216C2×C59—22no—
217C2×C59—21no—
218C2×C59—22no—
219C2×C59—42no—
220C2×C59#188411yescharacteristic
221C2×C59#189311yescharacteristic
222C2×C59#191311yescharacteristic
223D59—161no—
224C2×#121—591no—
225C2×#121—1181no—
226C2×C4×D4C2×D5911yescharacteristic
227C8×D4—24no—
228#431—21no—
229C2×#99—1181no—
230C2×C22⋊C8—21no—
231C2×C22⋊C8C2×D5912yes—
232#398—591no—
233#397—591no—
234#394—591no—
235#392—591no—
236#388—1181no—
237#386—1181no—
238#386—1181no—
239#380—592no—
240C2×#95—591no—
241#342—1181no—
242#327—1184no—
243C59#938411yescharacteristic, complemented, Sylow
244C22×D4—591no—
245C22×D4—1181no—
246C22×D4—1181no—
247C22×D4D23611yescharacteristic
248C23×C4—1181no—
249C23×C4C22⋊D5911yescharacteristic
250C23×C4C22×D5911yescharacteristic
251C22×C8—22no—
252C22×C8—22no—
253#121—1182no—
254#121—1182no—
255#121—2362no—
256#121—2362no—
257C4×D4—22no—
258C4×D4—22no—
259C2×C4⋊C4—591no—
260C2×C4⋊C4—1182no—
261C2×C4⋊C4—1181no—
262C2×C4⋊C4C4×D5911yescharacteristic
263C2×C22⋊C4—1181no—
264C2×C22⋊C4—1181no—
265C2×C22⋊C4—1181no—
266C2×C22⋊C4—1181no—
267C2×C22⋊C4—1181no—
268C2×C22⋊C4—1182no—
269C2×C22⋊C4C4×D5911yescharacteristic
270C2×C22⋊C4D23611yescharacteristic
271C2×C42C22⋊D5911yescharacteristic
272C4⋊C8—22no—
273#99—2362no—
274C22⋊C8—22no—
275C22⋊C8—22no—
276C22⋊C8C22⋊D5914yescomplemented
277C4×C8—22no—
278#95—591no—
279#95—591no—
280#95—1181no—
281#95—1181no—
282#95—1181no—
283#95—1182no—
284C24—1181no—
285C24(C2×D59)⋊C411yescharacteristic
286C24C2×D23611yescharacteristic
287C2×D4—22no—
288C2×D4—22no—
289C2×D4—1182no—
290C2×D4—4721no—
291C2×D4—4721no—
292C2×D4—2362no—
293C2×D4—2362no—
294C2×D4—2362no—
295C2×D4—2361no—
296C2×D4—2361no—
297C2×D4—2362no—
298C2×D4#1174566012yes—
299C2×D4D47212yes—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—81no—
303C22×C4—81no—
304C22×C4—1181no—
305C22×C4—1181no—
306C22×C4—1181no—
307C22×C4—4721no—
308C22×C4—2362no—
309C22×C4—2362no—
310C22×C4—2361no—
311C22×C4—2361no—
312C22×C4(C2×D59)⋊C411yescharacteristic
313C22×C4(C2×D59)⋊C411yescharacteristic
314C22×C4(C2×D59)⋊C411yescharacteristic
315C22×C4C2×C4×D5911yescharacteristic
316C22×C4#1174568512yes—
317C22×C4C2×C22⋊D5912yes—
318C22×C4C2×C22⋊D5911yescharacteristic
319C2×C8—24no—
320C2×C8—82no—
321C2×C8—82no—
322C4⋊C4—22no—
323C4⋊C4—1182no—
324C4⋊C4—4722no—
325C4⋊C4—2362no—
326C22⋊C4—22no—
327C22⋊C4—22no—
328C22⋊C4—4721no—
329C22⋊C4—2362no—
330C22⋊C4—2362no—
331C22⋊C4—2362no—
332C22⋊C4—2362no—
333C22⋊C4—2362no—
334C22⋊C4—2361no—
335C22⋊C4—2361no—
336C42—21no—
337C42—21no—
338C23—21no—
339C23—21no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23—2361no—
345C23—2361no—
346C23—2361no—
347C23—2361no—
348C23—2361no—
349C23—2361no—
350C23—2361no—
351C23#1427364912yes—
352C23C2×(C2×D59)⋊C411yescharacteristic
353C23#1427370112yes—
354C23C2×D47211yescharacteristic
355C23C2×#1174566011yescharacteristic
356C23#1427374912yes—
357D4—24no—
358D4—24no—
359D4—4722no—
360D4—4722no—
361D4—4722no—
362D4—4724no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—81no—
375C2×C4—1182no—
376C2×C4—4722no—
377C2×C4—4722no—
378C2×C4—4722no—
379C2×C4—4722no—
380C2×C4—4722no—
381C2×C4—2361no—
382C2×C4—2361no—
383C2×C4—2361no—
384C2×C4—2361no—
385C2×C4—2361no—
386C2×C4—2361no—
387C2×C4—2361no—
388C2×C4—2361no—
389C2×C4#1427364912yes—
390C2×C4C4×C22⋊D5912yes—
391C2×C4#1427369611yescharacteristic
392C2×C4#1427371011yescharacteristic
393C8—44no—
394C22—21no—
395C22—22no—
396C22—21no—
397C22—22no—
398C22—22no—
399C22—41no—
400C22—41no—
401C22—42no—
402C22—42no—
403C22—41no—
404C22—41no—
405C22—81no—
406C22—4721no—
407C22—4721no—
408C22—4721no—
409C22—4721no—
410C22—4721no—
411C22—4721no—
412C22—4721no—
413C22#1683728111yescharacteristic
414C22C2×#1427364911yescharacteristic
415C22#1683739911yescharacteristic
416C22#1683763112yes—
417C22#1683763812yes—
418C4—22no—
419C4—21no—
420C4—21no—
421C4—22no—
422C4—41no—
423C4—41no—
424C4—9442no—
425C4—4721no—
426C4—4721no—
427C4—2362no—
428C2—22no—
429C2—21no—
430C2—21no—
431C2—22no—
432C2—42no—
433C2—9441no—
434C2#2142447911yescharacteristic
435C2#2142452411yescharacteristic
436C2#2142463811yescharacteristic
subgroupG/H
centerC22#16837281
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×#2972C2
socleC22×C59#386
prime radical#21429052C2
cosocleC23×C4×C59C23
hypercenter#2972D59
hypocenterC59#9384
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2972

Complements

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

NH ≅ G/Ntype
2. #21424084C2semidirect
3. #21422665C2semidirect
4. #21421976C2semidirect
5. C59×#2972C2semidirect
9. #16837332C4semidirect
10. #16837327C4semidirect
11. #16837319C4semidirect
12. #16837311C4semidirect
63. C22⋊C8×C59D4semidirect
210. #2972D59semidirect
243. C59#9384semidirect
276. C22⋊C8C22⋊D59semidirect

Maximal subgroups

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

subgroupindex
#214290522
#214240842
#214226652
#214219762
C59×#29722
#938459