Finite Group Databasebeta

← #28507366

Subgroups — #28507366

Order 15104.

492 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 99 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#21429292C211yesmaximal, characteristic
2C4×#14273642C211yesmaximal, characteristic
3#21422220C212yesmaximal
4#21422220C212yesmaximal
5C59×#1799C211yesmaximal, characteristic
6C2×#14273719C411yescharacteristic, complemented
7C2×#14273642C2212yes—
8#16837325—21no—
9#16837325—21no—
10#16837305C2211yescharacteristic
11#16837305C2211yescharacteristic
12#16837275—21no—
13#16837275—21no—
14#16837271C411yescharacteristic, complemented
15#16837267C411yescharacteristic, complemented
16C4×C4⋊Dic59C411yescharacteristic, complemented
17C4×C22⋊Dic59—21no—
18#16837169—21no—
19C2×C4×D4×C59C2211yescharacteristic
20C2×#99×C59C2212yes—
21C59×#380—21no—
22C59×#342—21no—
23#14273719—41no—
24#14273719—41no—
25C2×C4⋊Dic59—21no—
26C2×C4⋊Dic59C2×C411yescharacteristic
27C2×C4⋊Dic59C2×C411yescharacteristic
28C2×Dic59⋊C4—21no—
29C2×Dic59⋊C4—21no—
30C2×Dic59⋊C4—21no—
31C2×Dic59⋊C4—21no—
32C2×C22⋊Dic59—22no—
33C2×C22⋊Dic59—21no—
34C2×C22⋊Dic59—21no—
35C2×C22⋊Dic59—21no—
36C2×C22⋊Dic59—21no—
37C2×C4×Dic59—41no—
38C2×C4×Dic59C2×C411yescharacteristic
39C2×C4×Dic59C2×C411yescharacteristic
40#14273642—22no—
41#14273642Q814yes—
42C59⋊#95—21no—
43C59⋊#95—21no—
44C59⋊#95—21no—
45C59⋊#95—41no—
46C22×D4×C59C2311yescharacteristic
47C23×C4×C59D411yescharacteristic
48C23×C4×C59D411yescharacteristic
49C4×D4×C59—41no—
50C4×D4×C59—41no—
51C2×C4⋊C4×C59—41no—
52C2×C4⋊C4×C59C2×C411yescharacteristic
53C2×C22⋊C4×C59—22no—
54C2×C22⋊C4×C59D411yescharacteristic
55C2×C22⋊C4×C59D411yescharacteristic
56C2×C42×C59C2×C411yescharacteristic
57#99×C59—24no—
58#95×C59—41no—
59C24×C59C2×D411yescharacteristic
60C24×C59C2×D411yescharacteristic
61C2×D4×C59—41no—
62C2×D4×C59—41no—
63C2×D4×C59C2×Q812yes—
64C2×D4×C59#4112yes—
65C22×C4×C59—21no—
66C22×C4×C59—21no—
67C22×C4×C59—21no—
68C22×C4×C59—21no—
69C22×C4×C59—42no—
70C22×C4×C59—81no—
71C22×C4×C59—81no—
72C22×C4×C59C22⋊C411yescharacteristic
73C22×C4×C59C22⋊C411yescharacteristic
74C22×C4×C59C22⋊C411yescharacteristic
75C22×C4×C59C22⋊C411yescharacteristic
76C22×C4×C59C22×C411yescharacteristic
77C22×Dic59—21no—
78C22×Dic59—21no—
79C22×Dic59—21no—
80C22×Dic59—21no—
81C22×Dic59—42no—
82C4⋊C4×C59—41no—
83C4⋊C4×C59—82no—
84C22⋊C4×C59—41no—
85C22⋊C4×C59—41no—
86C22⋊C4×C59—42no—
87C22⋊C4×C59—42no—
88C42×C59—21no—
89C42×C59—21no—
90C22⋊Dic59—24no—
91C22⋊Dic59—42no—
92C22⋊Dic59—42no—
93C22⋊Dic59—42no—
94C22⋊Dic59—81no—
95C22⋊Dic59—81no—
96C4⋊Dic59—42no—
97C4⋊Dic59—41no—
98C4⋊Dic59—41no—
99Dic59⋊C4—42no—
100Dic59⋊C4—41no—
101Dic59⋊C4—41no—
102Dic59⋊C4—81no—
103Dic59⋊C4—81no—
104C4×Dic59—41no—
105C4×Dic59—41no—
106C4×Dic59—82no—
107C23×C59—21no—
108C23×C59—21no—
109C23×C59—21no—
110C23×C59—21no—
111C23×C59—41no—
112C23×C59—41no—
113C23×C59—41no—
114C23×C59—41no—
115C23×C59C2×C22⋊C411yescharacteristic
116C23×C59C22⋊Q812yes—
117C23×C59#12411yescharacteristic
118C23×C59C4⋊Q811yescharacteristic
119D4×C59—42no—
120D4×C59—42no—
121C2×C4×C59—21no—
122C2×C4×C59—21no—
123C2×C4×C59—21no—
124C2×C4×C59—21no—
125C2×C4×C59—21no—
126C2×C4×C59—41no—
127C2×C4×C59—42no—
128C2×C4×C59—42no—
129C2×C4×C59—41no—
130C2×C4×C59—41no—
131C2×C4×C59—41no—
132C2×C4×C59—81no—
133C2×C4×C59—81no—
134C2×C4×C59—82no—
135C2×C4×C59—82no—
136C2×C4×C59—82no—
137C2×C4×C59#9911yescharacteristic
138C2×C4×C59#9911yescharacteristic
139C2×C4×C59C4×Q812yes—
140C2×Dic59—22no—
141C2×Dic59—41no—
142C2×Dic59—41no—
143C2×Dic59—41no—
144C2×Dic59—41no—
145C2×Dic59—41no—
146C2×Dic59—44no—
147C2×Dic59—41no—
148C2×Dic59—41no—
149C2×Dic59—41no—
150C2×Dic59—41no—
151C2×Dic59—41no—
152C2×Dic59—41no—
153C2×Dic59—82no—
154C2×Dic59—82no—
155#13300—591nomaximal, Sylow
156C22×C59—21no—
157C22×C59—21no—
158C22×C59—22no—
159C22×C59—21no—
160C22×C59—21no—
161C22×C59—42no—
162C22×C59—41no—
163C22×C59—41no—
164C22×C59—41no—
165C22×C59—41no—
166C22×C59—41no—
167C22×C59—41no—
168C22×C59—81no—
169C22×C59—81no—
170C22×C59#39111yescharacteristic
171C22×C59C2×#9911yescharacteristic
172C22×C59#41011yescharacteristic
173C4×C59—21no—
174C4×C59—21no—
175C4×C59—22no—
176C4×C59—41no—
177C4×C59—41no—
178C4×C59—41no—
179C4×C59—162no—
180Dic59—42no—
181Dic59—81no—
182Dic59—84no—
183Dic59—81no—
184Dic59—81no—
185#2566—591no—
186#1799D5911yescharacteristic
187C4×#99—591no—
188#1393—592no—
189#1393—592no—
190C2×C59—21no—
191C2×C59—21no—
192C2×C59—42no—
193C2×C59—41no—
194C2×C59—41no—
195C2×C59#199611yescharacteristic
196C2×C59#212611yescharacteristic
197C2×C59#212711yescharacteristic
198C2×#125—591no—
199C2×C4×D4C2×D5911yescharacteristic
200C2×#99—592no—
201C2×#99C2×D5912yes—
202#396—1181no—
203#396—1181no—
204#390—591no—
205#390—591no—
206#385—1181no—
207#385—1181no—
208#384—591no—
209#382—591no—
210#380—21no—
211C4×C4⋊C4—591no—
212C4×C22⋊C4—1181no—
213#342—21no—
214#342—1181no—
215C59#1330011yescharacteristic, complemented, Sylow
216C22×D4C22×D5911yescharacteristic
217C23×C4D23611yescharacteristic
218C23×C4C22⋊D5911yescharacteristic
219#125—2361no—
220#125—2361no—
221C4×D4—41no—
222C4×D4—41no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—591no—
225C2×C4⋊C4—591no—
226C2×C4⋊C4—1181no—
227C2×C4⋊C4—1181no—
228C2×C4⋊C4—1181no—
229C2×C4⋊C4—1181no—
230C2×C4⋊C4—1181no—
231C2×C4⋊C4C4×D5911yescharacteristic
232C2×C22⋊C4—22no—
233C2×C22⋊C4—1182no—
234C2×C22⋊C4—1181no—
235C2×C22⋊C4—1181no—
236C2×C22⋊C4—1181no—
237C2×C22⋊C4—1181no—
238C2×C22⋊C4D23611yescharacteristic
239C2×C22⋊C4C22⋊D5911yescharacteristic
240C2×C42—591no—
241C2×C42—591no—
242C2×C42—2361no—
243C2×C42C4×D5911yescharacteristic
244#99—24no—
245#99—594no—
246#99—1182no—
247#95—41no—
248#95—1181no—
249#95—1181no—
250#95—1181no—
251#95—2361no—
252C24C2×D23611yescharacteristic
253C24C2×C22⋊D5911yescharacteristic
254C2×D4—41no—
255C2×D4—41no—
256C2×D4#1174568712yes—
257C2×D4Q8×D5912yes—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—42no—
263C22×C4—81no—
264C22×C4—81no—
265C22×C4—1181no—
266C22×C4—1181no—
267C22×C4—1181no—
268C22×C4—1181no—
269C22×C4—2362no—
270C22×C4(C2×D59)⋊C411yescharacteristic
271C22×C4(C2×D59)⋊C411yescharacteristic
272C22×C4(C2×D59)⋊C411yescharacteristic
273C22×C4(C2×D59)⋊C411yescharacteristic
274C22×C4C2×C4×D5911yescharacteristic
275C4⋊C4—41no—
276C4⋊C4—82no—
277C4⋊C4—4721no—
278C4⋊C4—4721no—
279C4⋊C4—2361no—
280C4⋊C4—2361no—
281C4⋊C4—2362no—
282C4⋊C4—2362no—
283C4⋊C4—2361no—
284C4⋊C4—2361no—
285C22⋊C4—41no—
286C22⋊C4—41no—
287C22⋊C4—42no—
288C22⋊C4—42no—
289C22⋊C4—1184no—
290C22⋊C4—4721no—
291C22⋊C4—4721no—
292C22⋊C4—2362no—
293C22⋊C4—2362no—
294C22⋊C4—2362no—
295C42—21no—
296C42—21no—
297C42—4722no—
298C42—2361no—
299C42—2361no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23C2×(C2×D59)⋊C411yescharacteristic
309C23#1427370712yes—
310C23#1427371411yescharacteristic
311C23#1427372511yescharacteristic
312D4—42no—
313D4—42no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—41no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4—82no—
330C2×C4—1182no—
331C2×C4—4722no—
332C2×C4—4722no—
333C2×C4—2364no—
334C2×C4—2361no—
335C2×C4—2361no—
336C2×C4—2361no—
337C2×C4—2361no—
338C2×C4—2361no—
339C2×C4—2361no—
340C2×C4—2361no—
341C2×C4—2361no—
342C2×C4—2361no—
343C2×C4—2361no—
344C2×C4—2361no—
345C2×C4#1427364111yescharacteristic
346C2×C4#1427364111yescharacteristic
347C2×C4#1427369212yes—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—22no—
352C22—21no—
353C22—42no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—81no—
361C22—81no—
362C22#1683730711yescharacteristic
363C22C2×#1427364111yescharacteristic
364C22#1683737511yescharacteristic
365C4—21no—
366C4—21no—
367C4—22no—
368C4—41no—
369C4—41no—
370C4—41no—
371C4—162no—
372C4—4721no—
373C4—4724no—
374C4—4721no—
375C4—4721no—
376C4—2362no—
377C2—21no—
378C2—21no—
379C2—41no—
380C2—42no—
381C2—41no—
382C2#2142508011yescharacteristic
383C2#2142575911yescharacteristic
384C2#2142576311yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC22×C4×C59C22×C4
FrattiniC22×D4C22×D59
FittingC59×#1799C2
socleC22×C59C2×#99
prime radical, cosocleC22×D4×C59C23
hypercenter#1799D59
hypocenterC59#13300
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1799

Complements

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

NH ≅ G/Ntype
6. C2×#14273719C4semidirect
14. #16837271C4semidirect
15. #16837267C4semidirect
16. C4×C4⋊Dic59C4semidirect
215. C59#13300semidirect

Maximal subgroups

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

subgroupindex
#214292922
C4×#142736422
#214222202
C59×#17992
#1330059