Finite Group Databasebeta

← #28484475

Subgroups — #28484475

Order 15104.

648 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 135 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
1C2×#16837270C211yesmaximal, characteristic, complemented
2#21424164C211yesmaximal, characteristic, complemented
3C2×#16837136C212yesmaximal
4#21421974C212yesmaximal, complemented
5C59×#2854C211yesmaximal, characteristic
6C2×C22⋊(C59⋊C8)C2212yes—
7#16837270—24no—
8C2×C59⋊#95—21no—
9C2×C59⋊#95—21no—
10C2×C59⋊#95C2212yes—
11#16837157—22no—
12#16837136—24no—
13C2×#123×C59—21no—
14C2×#121×C59C2211yescharacteristic
15C22×C42×C59C2211yescharacteristic
16C59×#390C412yescomplemented
17C59×#389C412yescomplemented
18C59×#388—22no—
19C59×#386—21no—
20C59×#380C2211yescharacteristic
21C23×Dic59—22no—
22C22×C59⋊C8—22no—
23C22⋊(C59⋊C8)—22no—
24C22⋊(C59⋊C8)—22no—
25C59⋊#95—24no—
26C59⋊#95—24no—
27C59⋊#95—22no—
28C59⋊#95D414yes—
29C22×D4×C59—21no—
30C22×D4×C59C2×C411yescharacteristic
31C23×C4×C59D411yescharacteristic
32C23×C4×C59D411yescharacteristic
33C23×C4×C59C2311yescharacteristic
34#123×C59—44no—
35#121×C59—22no—
36#121×C59—22no—
37C2×C4⋊C4×C59—22no—
38C2×C4⋊C4×C59C2×C411yescharacteristic
39C2×C4⋊C4×C59C2×C412yes—
40C2×C22⋊C4×C59—21no—
41C2×C22⋊C4×C59—22no—
42C2×C22⋊C4×C59—22no—
43C2×C22⋊C4×C59—21no—
44C2×C42×C59—22no—
45C2×C42×C59—22no—
46C2×C42×C59C2×C412yes—
47C2×C42×C59D412yes—
48#95×C59—22no—
49#95×C59—21no—
50C24×C59—21no—
51C24×C59C2×D411yescharacteristic
52C2×D4×C59—22no—
53C2×D4×C59—42no—
54C2×D4×C59—41no—
55C2×D4×C59—41no—
56C2×D4×C59—42no—
57C2×D4×C59—42no—
58C22×C4×C59—21no—
59C22×C4×C59—21no—
60C22×C4×C59—21no—
61C22×C4×C59—21no—
62C22×C4×C59—22no—
63C22×C4×C59—22no—
64C22×C4×C59—22no—
65C22×C4×C59—21no—
66C22×C4×C59—42no—
67C22×C4×C59—42no—
68C22×C4×C59C22⋊C412yes—
69C22×C4×C59C22⋊C412yes—
70C22×C4×C59C22×C411yescharacteristic
71C22×C4×C59C2×D411yescharacteristic
72C22×C4×C59C2×D412yes—
73C22×C4×C59#4112yes—
74C22×Dic59—24no—
75C22×Dic59—44no—
76C22×Dic59—44no—
77C22×Dic59—42no—
78C22×Dic59—42no—
79C4⋊C4×C59—22no—
80C4⋊C4×C59—44no—
81C4⋊C4×C59—42no—
82C22⋊C4×C59—44no—
83C22⋊C4×C59—44no—
84C22⋊C4×C59—42no—
85C22⋊C4×C59—42no—
86C42×C59—24no—
87C42×C59—24no—
88C2×C59⋊C8—44no—
89C2×C59⋊C8—42no—
90C23×C59—22no—
91C23×C59—22no—
92C23×C59—41no—
93C23×C59—42no—
94C23×C59—41no—
95C23×C59—41no—
96C23×C59—42no—
97C23×C59C2×C22⋊C411yescharacteristic
98C23×C59#12011yescharacteristic
99C23×C59#12311yescharacteristic
100C23×C59#13512yes—
101C23×C59AGL(1,ℤ/8)12yes—
102D4×C59—82no—
103D4×C59—82no—
104D4×C59—82no—
105C2×C4×C59—22no—
106C2×C4×C59—22no—
107C2×C4×C59—22no—
108C2×C4×C59—22no—
109C2×C4×C59—22no—
110C2×C4×C59—22no—
111C2×C4×C59—22no—
112C2×C4×C59—22no—
113C2×C4×C59—42no—
114C2×C4×C59—42no—
115C2×C4×C59—42no—
116C2×C4×C59—42no—
117C2×C4×C59—42no—
118C2×C4×C59—42no—
119C2×C4×C59—41no—
120C2×C4×C59—42no—
121C2×C4×C59—41no—
122C2×C4×C59—42no—
123C2×C4×C59—41no—
124C2×C4×C59—42no—
125C2×C4×C59C4×D412yes—
126C2×C4×C59#12112yes—
127C2×Dic59—44no—
128C2×Dic59—44no—
129C2×Dic59—44no—
130C2×Dic59—84no—
131C2×Dic59—84no—
132C59⋊C8—82no—
133#9157—591nomaximal, Sylow
134C22×C59—21no—
135C22×C59—21no—
136C22×C59—21no—
137C22×C59—21no—
138C22×C59—24no—
139C22×C59—24no—
140C22×C59—81no—
141C22×C59—81no—
142C22×C59—81no—
143C22×C59—82no—
144C22×C59—82no—
145C22×C59#38611yescharacteristic
146C22×C59#41012yes—
147C22×C59#41611yescharacteristic
148C22×C59#41811yescharacteristic
149C22×C59#42112yes—
150C22×C59#44912yes—
151C22×C59#48212yes—
152C4×C59—22no—
153C4×C59—22no—
154C4×C59—42no—
155C4×C59—42no—
156C4×C59—81no—
157C4×C59—82no—
158Dic59—84no—
159#2854D5911yescharacteristic
160C2×#383—591no—
161#1817—591no—
162C2×#327—592no—
163#1294—592no—
164C2×C59—22no—
165C2×C59—22no—
166C2×C59—161no—
167C2×C59#151412yes—
168C2×C59#152112yes—
169C2×C59#187511yescharacteristic
170C2×C59#189711yescharacteristic
171C2×C59#189911yescharacteristic
172C2×#123—21no—
173C2×#121C2×D5911yescharacteristic
174C22×C42C2×D5911yescharacteristic
175C2×C22⋊C8—592no—
176#390Dic5912yescomplemented
177#389Dic5912yescomplemented
178#388—22no—
179#386—21no—
180#383—1184no—
181#380C2×D5911yescharacteristic
182C2×#95—592no—
183C2×#95—1181no—
184C2×#95—1181no—
185#336—1182no—
186#327—1184no—
187C59#915711yescharacteristic, complemented, Sylow
188C22×D4—21no—
189C22×D4C2×Dic5911yescharacteristic
190C23×C4—1182no—
191C23×C4C22⋊D5911yescharacteristic
192C23×C4C22⋊D5911yescharacteristic
193C23×C4C22×D5911yescharacteristic
194C22×C8—1182no—
195#123—44no—
196#121—22no—
197#121—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4C2×Dic5911yescharacteristic
200C2×C4⋊C4C2×Dic5912yes—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—21no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42C2×Dic5912yes—
208C2×C42C22⋊D5912yes—
209C22⋊C8—1182no—
210C22⋊C8—1182no—
211#95—22no—
212#95—21no—
213#95—594no—
214#95—1184no—
215#95—1184no—
216#95—1182no—
217C24—21no—
218C24C2×C22⋊D5911yescharacteristic
219C2×D4—22no—
220C2×D4—42no—
221C2×D4—41no—
222C2×D4—41no—
223C2×D4—42no—
224C2×D4—42no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—21no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—1184no—
236C22×C4—2364no—
237C22×C4—2364no—
238C22×C4—2362no—
239C22×C4—2362no—
240C22×C4C22⋊Dic5912yes—
241C22×C4C22⋊Dic5912yes—
242C22×C4D4×D5912yes—
243C22×C4#1174568712yes—
244C22×C4C22×Dic5911yescharacteristic
245C22×C4C2×C22⋊D5911yescharacteristic
246C2×C8—2364no—
247C2×C8—2362no—
248C4⋊C4—22no—
249C4⋊C4—42no—
250C4⋊C4—44no—
251C22⋊C4—44no—
252C22⋊C4—44no—
253C22⋊C4—42no—
254C22⋊C4—42no—
255C42—24no—
256C42—24no—
257C23—22no—
258C23—22no—
259C23—42no—
260C23—41no—
261C23—41no—
262C23—42no—
263C23—41no—
264C23C2×C22⋊Dic5911yescharacteristic
265C23#1427369511yescharacteristic
266C23#1427370811yescharacteristic
267C23#1427374712yes—
268C23#1427375312yes—
269D4—82no—
270D4—82no—
271D4—82no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—41no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—42no—
291C2×C4—41no—
292C2×C4—4724no—
293C2×C4—4724no—
294C2×C4—2364no—
295C2×C4—2364no—
296C2×C4—2364no—
297C2×C4D4×Dic5912yes—
298C2×C4#1427369912yes—
299C8—4722no—
300C22—24no—
301C22—24no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—81no—
307C22—82no—
308C22—82no—
309C22—81no—
310C22—81no—
311C22#1683728211yescharacteristic
312C22#1683737612yes—
313C22#1683739811yescharacteristic
314C22#1683740711yescharacteristic
315C22#1683741912yes—
316C22#1683754512yes—
317C22#1683772312yes—
318C4—22no—
319C4—22no—
320C4—42no—
321C4—42no—
322C4—81no—
323C4—82no—
324C4—4724no—
325C2—22no—
326C2—22no—
327C2—161no—
328C2#2142264312yes—
329C2#2142267312yes—
330C2#2142441911yescharacteristic
331C2#2142454511yescharacteristic
332C2#2142455511yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic59
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×#2854C2
socleC23×C59C2×C22⋊C4
prime radicalC2×#121×C59C22
cosocleC23×C4×C59C23
hypercenter#2854D59
hypocenterC59#9157
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2854

Complements

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

NH ≅ G/Ntype
1. C2×#16837270C2semidirect
2. #21424164C2semidirect
4. #21421974C2semidirect
16. C59×#390C4semidirect
17. C59×#389C4semidirect
176. #390Dic59semidirect
177. #389Dic59semidirect
187. C59#9157semidirect

Maximal subgroups

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

subgroupindex
C2×#168372702
#214241642
C2×#168371362
#214219742
C59×#28542
#915759