Finite Group Databasebeta

← #28499033

Subgroups — #28499033

Order 15104.

486 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 91 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
1C4×#14273642C211yesmaximal, characteristic, complemented
2#21422417C211yesmaximal, characteristic, complemented
3#21422120C212yesmaximal, complemented
4#21422093C212yesmaximal, complemented
5C59×#2635C211yesmaximal, characteristic
6#16837455—21no—
7C2×#14273642—21no—
8C4×C22⋊Dic59—21no—
9#16837169C2212yes—
10#16837126—21no—
11#16837126C2212yes—
12C2×#121×C59C411yescharacteristic, complemented
13C2×C4×D4×C59C2211yescharacteristic
14C59×#400C411yescharacteristic, complemented
15C59×#399C2211yescharacteristic
16C59×#394—21no—
17C59×#394C411yescharacteristic, complemented
18C59×#392—21no—
19C59×#392C2211yescharacteristic
20C59×#388—21no—
21C59×#386—21no—
22C59×#381C411yescharacteristic, complemented
23C2×C22⋊Dic59—22no—
24C2×C4×Dic59—41no—
25C4⋊(C59⋊C8)—41no—
26#14273642—24no—
27C22⋊(C59⋊C8)—22no—
28C4×C59⋊C8—41no—
29C59⋊#95—41no—
30C22×D4×C59—21no—
31C22×D4×C59—21no—
32C22×D4×C59C2×C411yescharacteristic
33C22×D4×C59D411yescharacteristic
34C23×C4×C59C2×C411yescharacteristic
35C23×C4×C59D411yescharacteristic
36#121×C59—41no—
37#121×C59—41no—
38C4×D4×C59—41no—
39C4×D4×C59—41no—
40C2×C4⋊C4×C59—21no—
41C2×C4⋊C4×C59C2×C411yescharacteristic
42C2×C4⋊C4×C59C2×C411yescharacteristic
43C2×C22⋊C4×C59—21no—
44C2×C22⋊C4×C59—21no—
45C2×C22⋊C4×C59—21no—
46C2×C22⋊C4×C59—21no—
47C2×C22⋊C4×C59—21no—
48C2×C22⋊C4×C59—21no—
49C2×C22⋊C4×C59D411yescharacteristic
50C2×C22⋊C4×C59C2311yescharacteristic
51C2×C42×C59D411yescharacteristic
52#95×C59—21no—
53#95×C59—21no—
54#95×C59—21no—
55#95×C59C2×C411yescharacteristic
56#95×C59C2×C411yescharacteristic
57C24×C59—21no—
58C24×C59C22⋊C411yescharacteristic
59C24×C59C2×D411yescharacteristic
60C2×D4×C59—22no—
61C2×D4×C59—22no—
62C2×D4×C59—41no—
63C2×D4×C59—41no—
64C2×D4×C59—42no—
65C2×D4×C59—42no—
66C2×D4×C59—81no—
67C2×D4×C59—81no—
68C2×D4×C59—81no—
69C2×D4×C59—81no—
70C2×D4×C59—81no—
71C22×C4×C59—21no—
72C22×C4×C59—21no—
73C22×C4×C59—21no—
74C22×C4×C59—21no—
75C22×C4×C59—21no—
76C22×C4×C59—21no—
77C22×C4×C59—21no—
78C22×C4×C59—81no—
79C22×C4×C59—81no—
80C22×C4×C59C22⋊C411yescharacteristic
81C22×C4×C59C22⋊C411yescharacteristic
82C22×C4×C59C22⋊C411yescharacteristic
83C22×C4×C59C22×C411yescharacteristic
84C22×C4×C59C2×D411yescharacteristic
85C22×Dic59—42no—
86C4⋊C4×C59—41no—
87C4⋊C4×C59—41no—
88C4⋊C4×C59—81no—
89C22⋊C4×C59—41no—
90C22⋊C4×C59—41no—
91C22⋊C4×C59—41no—
92C22⋊C4×C59—41no—
93C22⋊C4×C59—42no—
94C22⋊C4×C59—81no—
95C22⋊C4×C59—81no—
96C22⋊C4×C59—81no—
97C22⋊C4×C59—81no—
98C42×C59—21no—
99C42×C59—21no—
100C22⋊Dic59—24no—
101C22⋊Dic59—42no—
102C4×Dic59—82no—
103C2×C59⋊C8—42no—
104C23×C59—21no—
105C23×C59—21no—
106C23×C59—21no—
107C23×C59—21no—
108C23×C59—21no—
109C23×C59—41no—
110C23×C59—41no—
111C23×C59—41no—
112C23×C59—41no—
113C23×C59—41no—
114C23×C59—41no—
115C23×C59—41no—
116C23×C59—41no—
117C23×C59—41no—
118C23×C59—41no—
119C23×C59—41no—
120C23×C59C2×C22⋊C411yescharacteristic
121C23×C59#13512yes—
122D4×C59—44no—
123D4×C59—82no—
124D4×C59—84no—
125D4×C59—84no—
126C2×C4×C59—21no—
127C2×C4×C59—21no—
128C2×C4×C59—21no—
129C2×C4×C59—21no—
130C2×C4×C59—21no—
131C2×C4×C59—41no—
132C2×C4×C59—42no—
133C2×C4×C59—41no—
134C2×C4×C59—41no—
135C2×C4×C59—41no—
136C2×C4×C59—41no—
137C2×C4×C59—41no—
138C2×C4×C59—41no—
139C2×C4×C59—41no—
140C2×C4×C59—41no—
141C2×C4×C59—41no—
142C2×C4×C59—41no—
143C2×C4×C59—41no—
144C2×C4×C59—41no—
145C2×C4×C59—81no—
146C2×C4×C59—81no—
147C2×C4×C59—81no—
148C2×C4×C59#9911yescharacteristic
149C2×C4×C59#9911yescharacteristic
150C2×C4×C59#10412yes—
151C2×Dic59—44no—
152C2×Dic59—82no—
153C2×Dic59—82no—
154C59⋊C8—82no—
155#11729—591nomaximal, Sylow
156C22×C59—21no—
157C22×C59—21no—
158C22×C59—22no—
159C22×C59—42no—
160C22×C59—41no—
161C22×C59—41no—
162C22×C59—41no—
163C22×C59—41no—
164C22×C59—41no—
165C22×C59—81no—
166C22×C59—81no—
167C22×C59—81no—
168C22×C59—81no—
169C22×C59—81no—
170C22×C59—81no—
171C22×C59—81no—
172C22×C59—81no—
173C22×C59—81no—
174C22×C59—81no—
175C22×C59C2×#9911yescharacteristic
176C22×C59#41611yescharacteristic
177C22×C59C2×#10411yescharacteristic
178C4×C59—21no—
179C4×C59—21no—
180C4×C59—22no—
181C4×C59—41no—
182C4×C59—41no—
183C4×C59—41no—
184C4×C59—81no—
185C4×C59—81no—
186C4×C59—81no—
187Dic59—84no—
188#2635D5911yescharacteristic
189C4×#99—591no—
190#1460—591no—
191#1352—592no—
192#1343—592no—
193C2×C59—21no—
194C2×C59—21no—
195C2×C59—42no—
196C2×C59—81no—
197C2×C59—161no—
198C2×C59#150111yescharacteristic
199C2×C59#212111yescharacteristic
200C2×C59#212811yescharacteristic
201C2×#121Dic5911yescharacteristic, complemented
202C2×C4×D4C2×D5911yescharacteristic
203#431—1181no—
204C2×#99—1181no—
205#400Dic5911yescharacteristic, complemented
206#399C2×D5911yescharacteristic
207#394—21no—
208#394Dic5911yescharacteristic, complemented
209#392—21no—
210#392C2×D5911yescharacteristic
211#388—21no—
212#386—21no—
213#381Dic5911yescharacteristic, complemented
214C4×C22⋊C4—1181no—
215#342—592no—
216#323—592no—
217#323—1181no—
218C59#1172911yescharacteristic, complemented, Sylow
219C22×D4—21no—
220C22×D4—21no—
221C22×D4C2×Dic5911yescharacteristic
222C22×D4C22⋊D5911yescharacteristic
223C23×C4C2×Dic5911yescharacteristic
224C23×C4C22⋊D5911yescharacteristic
225#121—41no—
226#121—41no—
227C4×D4—41no—
228C4×D4—41no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4C2×Dic5911yescharacteristic
231C2×C4⋊C4C2×Dic5911yescharacteristic
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—1182no—
239C2×C22⋊C4C22⋊D5911yescharacteristic
240C2×C22⋊C4C22×D5911yescharacteristic
241C2×C42—2361no—
242C2×C42C22⋊D5911yescharacteristic
243C4⋊C8—2361no—
244#99—1184no—
245C22⋊C8—1182no—
246C4×C8—2361no—
247#95—21no—
248#95—21no—
249#95—21no—
250#95—2361no—
251#95C2×Dic5911yescharacteristic
252#95C2×Dic5911yescharacteristic
253C24—21no—
254C24C22⋊Dic5911yescharacteristic
255C24C2×C22⋊D5911yescharacteristic
256C2×D4—22no—
257C2×D4—22no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—42no—
261C2×D4—42no—
262C2×D4—81no—
263C2×D4—81no—
264C2×D4—81no—
265C2×D4—81no—
266C2×D4—81no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—81no—
275C22×C4—81no—
276C22×C4—2362no—
277C22×C4C22⋊Dic5911yescharacteristic
278C22×C4C22⋊Dic5911yescharacteristic
279C22×C4C22⋊Dic5911yescharacteristic
280C22×C4C22×Dic5911yescharacteristic
281C22×C4C2×C22⋊D5911yescharacteristic
282C2×C8—2362no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—81no—
286C22⋊C4—41no—
287C22⋊C4—41no—
288C22⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—42no—
291C22⋊C4—81no—
292C22⋊C4—81no—
293C22⋊C4—81no—
294C22⋊C4—81no—
295C22⋊C4—1184no—
296C22⋊C4—2362no—
297C42—21no—
298C42—21no—
299C42—4722no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—41no—
309C23—41no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23C2×C22⋊Dic5911yescharacteristic
317C23#1427374712yes—
318D4—44no—
319D4—82no—
320D4—84no—
321D4—84no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—41no—
328C2×C4—42no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—4722no—
345C2×C4—4722no—
346C2×C4—2364no—
347C2×C4#1427364211yescharacteristic
348C2×C4#1427364211yescharacteristic
349C2×C4#1427365412yes—
350C8—4722no—
351C22—21no—
352C22—21no—
353C22—22no—
354C22—42no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—81no—
361C22—81no—
362C22—81no—
363C22—81no—
364C22—81no—
365C22—81no—
366C22—81no—
367C22—81no—
368C22—81no—
369C22—81no—
370C22C2×#1427364211yescharacteristic
371C22#1683739811yescharacteristic
372C22C2×#1427365411yescharacteristic
373C4—21no—
374C4—21no—
375C4—22no—
376C4—41no—
377C4—41no—
378C4—41no—
379C4—81no—
380C4—81no—
381C4—81no—
382C4—4724no—
383C2—21no—
384C2—21no—
385C2—42no—
386C2—81no—
387C2—161no—
388C2#2142258711yescharacteristic
389C2#2142574211yescharacteristic
390C2#2142576811yescharacteristic
subgroupG/H
centerC22C2×#14273642
commutatorC22×C4×C59C22×C4
FrattiniC2×C22⋊C4C22×D59
Fitting, prime radicalC59×#2635C2
socleC22×C59C2×#99
cosocleC2×C22⋊C4×C59C23
hypercenter#2635D59
hypocenterC59#11729
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2635

Complements

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

NH ≅ G/Ntype
1. C4×#14273642C2semidirect
2. #21422417C2semidirect
3. #21422120C2semidirect
4. #21422093C2semidirect
12. C2×#121×C59C4semidirect
14. C59×#400C4semidirect
17. C59×#394C4semidirect
22. C59×#381C4semidirect
201. C2×#121Dic59semidirect
205. #400Dic59semidirect
208. #394Dic59semidirect
213. #381Dic59semidirect
218. C59#11729semidirect

Maximal subgroups

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

subgroupindex
C4×#142736422
#214224172
#214221202
#214220932
C59×#26352
#1172959