Finite Group Databasebeta

← #28504943

Subgroups — #28504943

Order 15104.

914 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 914 conjugacy classes of subgroups — too many to draw. Only its 115 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#21424011C212yesmaximal
2#21422221C214yesmaximal, complemented
3C2×C59×#390C211yesmaximal, characteristic
4C2×#14273642—22no—
5C2×#14273642C2214yes—
6#16837258—22no—
7C23×D4×C59C2211yescharacteristic
8C22×C4⋊C4×C59C411yescharacteristic, complemented
9C22×C22⋊C4×C59—22no—
10C22×C42×C59C411yescharacteristic, complemented
11C59×#390—22no—
12C59×#390C412yescomplemented
13C59×#390C2212yes—
14C2×C22⋊Dic59—24no—
15C2×C22⋊Dic59—42no—
16C2×C22⋊Dic59—42no—
17#14273642—28no—
18#14273642—44no—
19C25×C59D412yes—
20C22×D4×C59—24no—
21C22×D4×C59—42no—
22C22×D4×C59—41no—
23C22×D4×C59—41no—
24C22×D4×C59C2×C411yescharacteristic
25C22×D4×C59D412yes—
26C22×D4×C59C2311yescharacteristic
27C23×C4×C59—21no—
28C23×C4×C59—21no—
29C23×C4×C59C2×C411yescharacteristic
30C2×C4⋊C4×C59—21no—
31C2×C4⋊C4×C59—41no—
32C2×C4⋊C4×C59—41no—
33C2×C4⋊C4×C59C2×C412yes—
34C2×C22⋊C4×C59—24no—
35C2×C22⋊C4×C59—24no—
36C2×C22⋊C4×C59—48no—
37C2×C42×C59—21no—
38C2×C42×C59—42no—
39C2×C42×C59C2×C412yes—
40C24×C59—24no—
41C24×C59—22no—
42C24×C59—22no—
43C24×C59—42no—
44C24×C59—44no—
45C24×C59—44no—
46C24×C59C22⋊C411yescharacteristic
47C24×C59C22⋊C412yes—
48C24×C59C2×D412yes—
49C2×D4×C59—24no—
50C2×D4×C59—24no—
51C2×D4×C59—22no—
52C2×D4×C59—41no—
53C2×D4×C59—41no—
54C2×D4×C59—42no—
55C2×D4×C59—41no—
56C2×D4×C59—41no—
57C2×D4×C59—48no—
58C2×D4×C59—48no—
59C2×D4×C59#4114yes—
60C22×C4×C59—21no—
61C22×C4×C59—21no—
62C22×C4×C59—22no—
63C22×C4×C59—22no—
64C22×C4×C59—24no—
65C22×C4×C59—22no—
66C22×C4×C59—22no—
67C22×C4×C59—42no—
68C22×C4×C59—42no—
69C22×C4×C59—44no—
70C22×C4×C59—41no—
71C22×C4×C59—41no—
72C22×C4×C59C22⋊C411yescharacteristic
73C22×C4×C59C22×C411yescharacteristic
74C22×Dic59—44no—
75C4⋊C4×C59—42no—
76C4⋊C4×C59—42no—
77C22⋊C4×C59—48no—
78C22⋊C4×C59—48no—
79C42×C59—42no—
80C42×C59—42no—
81C22⋊Dic59—44no—
82C22⋊Dic59—44no—
83C22⋊Dic59—84no—
84C22⋊Dic59—84no—
85C23×C59—22no—
86C23×C59—22no—
87C23×C59—21no—
88C23×C59—21no—
89C23×C59—22no—
90C23×C59—22no—
91C23×C59—22no—
92C23×C59—22no—
93C23×C59—22no—
94C23×C59—44no—
95C23×C59—44no—
96C23×C59—42no—
97C23×C59—42no—
98C23×C59—42no—
99C23×C59—42no—
100C23×C59—416no—
101C23×C59—44no—
102C23×C59—44no—
103C23×C59—82no—
104C23×C59—82no—
105C23×C59—88no—
106C23×C59#9911yescharacteristic
107C23×C59#9911yescharacteristic
108C23×C59C2×C22⋊C411yescharacteristic
109C23×C59#12314yes—
110D4×C59—44no—
111D4×C59—48no—
112D4×C59—44no—
113C2×C4×C59—24no—
114C2×C4×C59—24no—
115C2×C4×C59—21no—
116C2×C4×C59—21no—
117C2×C4×C59—21no—
118C2×C4×C59—21no—
119C2×C4×C59—24no—
120C2×C4×C59—48no—
121C2×C4×C59—42no—
122C2×C4×C59—42no—
123C2×C4×C59—44no—
124C2×C4×C59—42no—
125C2×C4×C59—82no—
126C2×C4×C59—82no—
127C2×C4×C59#9912yes—
128C2×C4×C59#11712yes—
129C2×Dic59—84no—
130C2×Dic59—84no—
131C2×Dic59—84no—
132#12856—591nomaximal, Sylow
133C22×C59—24no—
134C22×C59—24no—
135C22×C59—21no—
136C22×C59—21no—
137C22×C59—21no—
138C22×C59—21no—
139C22×C59—21no—
140C22×C59—21no—
141C22×C59—24no—
142C22×C59—44no—
143C22×C59—44no—
144C22×C59—42no—
145C22×C59—42no—
146C22×C59—42no—
147C22×C59—42no—
148C22×C59—42no—
149C22×C59—82no—
150C22×C59—88no—
151C22×C59—88no—
152C22×C59—82no—
153C22×C59#35312yes—
154C22×C59#35412yes—
155C22×C59#38111yescharacteristic
156C22×C59C2×#9911yescharacteristic
157C22×C59C2×#9911yescharacteristic
158C4×C59—22no—
159C4×C59—22no—
160C4×C59—44no—
161C4×C59—82no—
162Dic59—164no—
163C2×#390D5911yescharacteristic
164#1781—592no—
165#1393—594no—
166C2×C59—21no—
167C2×C59—21no—
168C2×C59—22no—
169C2×C59—22no—
170C2×C59—44no—
171C2×C59—82no—
172C2×C59#181911yescharacteristic
173C2×C59C2×#35311yescharacteristic
174C2×C59C2×#35411yescharacteristic
175C23×D4C2×D5911yescharacteristic
176C22×C4⋊C4Dic5911yescharacteristic, complemented
177C22×C22⋊C4—22no—
178C22×C42Dic5911yescharacteristic, complemented
179C2×#99—594no—
180C2×#99—1182no—
181#390—22no—
182#390Dic5912yescomplemented
183#390C2×D5912yes—
184#379—1182no—
185C59#1285611yescharacteristic, complemented, Sylow
186C25C22⋊D5912yes—
187C22×D4—24no—
188C22×D4—42no—
189C22×D4—41no—
190C22×D4—41no—
191C22×D4C2×Dic5911yescharacteristic
192C22×D4C22⋊D5912yes—
193C22×D4C22×D5911yescharacteristic
194C23×C4—21no—
195C23×C4—21no—
196C23×C4C2×Dic5911yescharacteristic
197C2×C4⋊C4—21no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4C2×Dic5912yes—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—48no—
204C2×C22⋊C4—1184no—
205C2×C22⋊C4—2362no—
206C2×C22⋊C4—2362no—
207C2×C42—21no—
208C2×C42—42no—
209C2×C42C2×Dic5912yes—
210#99—1188no—
211#99—2364no—
212C24—24no—
213C24—22no—
214C24—22no—
215C24—42no—
216C24—44no—
217C24—44no—
218C24C22⋊Dic5912yes—
219C24C22⋊Dic5911yescharacteristic
220C24C2×C22⋊D5912yes—
221C2×D4—24no—
222C2×D4—24no—
223C2×D4—22no—
224C2×D4—41no—
225C2×D4—41no—
226C2×D4—42no—
227C2×D4—41no—
228C2×D4—41no—
229C2×D4—48no—
230C2×D4—48no—
231C2×D4#1174568514yes—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—24no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—44no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—2364no—
245C22×C4C22⋊Dic5911yescharacteristic
246C22×C4C22×Dic5911yescharacteristic
247C4⋊C4—42no—
248C4⋊C4—42no—
249C22⋊C4—48no—
250C22⋊C4—48no—
251C22⋊C4—4724no—
252C22⋊C4—4724no—
253C22⋊C4—2364no—
254C22⋊C4—2364no—
255C42—42no—
256C42—42no—
257C23—22no—
258C23—22no—
259C23—21no—
260C23—21no—
261C23—22no—
262C23—22no—
263C23—22no—
264C23—22no—
265C23—22no—
266C23—44no—
267C23—44no—
268C23—42no—
269C23—42no—
270C23—42no—
271C23—42no—
272C23—416no—
273C23—44no—
274C23—44no—
275C23—82no—
276C23—82no—
277C23—88no—
278C23#1427364211yescharacteristic
279C23#1427364211yescharacteristic
280C23C2×C22⋊Dic5911yescharacteristic
281C23#1427371014yes—
282D4—44no—
283D4—48no—
284D4—44no—
285C2×C4—24no—
286C2×C4—24no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—24no—
292C2×C4—48no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—44no—
296C2×C4—42no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4—4724no—
300C2×C4—4724no—
301C2×C4—4724no—
302C2×C4#1427364212yes—
303C2×C4#1427368612yes—
304C22—24no—
305C22—24no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—24no—
313C22—44no—
314C22—44no—
315C22—42no—
316C22—42no—
317C22—42no—
318C22—42no—
319C22—42no—
320C22—82no—
321C22—88no—
322C22—88no—
323C22—82no—
324C22#1683719512yes—
325C22#1683719712yes—
326C22#1683726511yescharacteristic
327C22C2×#1427364211yescharacteristic
328C22C2×#1427364211yescharacteristic
329C4—22no—
330C4—22no—
331C4—44no—
332C4—82no—
333C4—9444no—
334C2—21no—
335C2—21no—
336C2—22no—
337C2—22no—
338C2—44no—
339C2—82no—
340C2#2142417011yescharacteristic
341C2C2×#1683719511yescharacteristic
342C2C2×#1683719711yescharacteristic
subgroupG/H
centerC22C2×#14273642
commutatorC22×C4×C59C22×C4
FrattiniC22×D4C22×D59
FittingC2×C59×#390C2
socleC22×C59C2×#99
prime radicalC23×D4×C59C22
cosocleC22×D4×C59C23
hypercenterC2×#390D59
hypocenterC59#12856
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ C2×#390

Complements

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

NH ≅ G/Ntype
2. #21422221C2semidirect
8. C22×C4⋊C4×C59C4semidirect
10. C22×C42×C59C4semidirect
12. C59×#390C4semidirect
176. C22×C4⋊C4Dic59semidirect
178. C22×C42Dic59semidirect
182. #390Dic59semidirect
185. C59#12856semidirect

Maximal subgroups

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

subgroupindex
#214240112
#214222212
C2×C59×#3902
#1285659