Finite Group Databasebeta

← #28484738

Subgroups — #28484738

Order 15104.

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

normal subgroups only

This group has 560 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
1#21426714C211yesmaximal, characteristic
2#21422668C212yesmaximal
3C2×#16837128C211yesmaximal, characteristic
4#21421963C212yesmaximal
5C59×#2627C211yesmaximal, characteristic
6C2×#14273686—21no—
7C2×C22⋊(C59⋊C8)C2212yes—
8#16837301—24no—
9#16837262—21no—
10#16837262C2212yes—
11#16837139—24no—
12#16837128—22no—
13C2×C22⋊Q8×C59C2211yescharacteristic
14C22×C4⋊C4×C59C2211yescharacteristic
15C59×#400C412yescomplemented
16C59×#398—22no—
17C59×#395C412yescomplemented
18C59×#385—21no—
19C59×#381—21no—
20C59×#380C2211yescharacteristic
21C22×C59⋊C8—22no—
22#14273686—42no—
23#14273686—42no—
24C2×C4⋊Dic59—22no—
25C2×C4⋊Dic59—22no—
26C2×C4⋊Dic59—22no—
27C2×C22⋊Dic59—22no—
28C2×C4×Dic59—22no—
29C22⋊(C59⋊C8)—22no—
30C22⋊(C59⋊C8)—22no—
31C59⋊#95—22no—
32C59⋊#95D414yes—
33C22×Q8×C59C2×C411yescharacteristic
34C23×C4×C59D411yescharacteristic
35C23×C4×C59D411yescharacteristic
36C23×C4×C59C2311yescharacteristic
37C22⋊Q8×C59—22no—
38C22⋊Q8×C59—22no—
39C2×C4⋊C4×C59—22no—
40C2×C4⋊C4×C59—22no—
41C2×C4⋊C4×C59—21no—
42C2×C4⋊C4×C59—22no—
43C2×C4⋊C4×C59—21no—
44C2×C4⋊C4×C59C2×C411yescharacteristic
45C2×C4⋊C4×C59C2×C412yes—
46C2×C4⋊C4×C59D412yes—
47C2×C22⋊C4×C59—21no—
48C2×C22⋊C4×C59—21no—
49#95×C59—22no—
50#95×C59—22no—
51#95×C59—22no—
52#95×C59C2×C412yes—
53C24×C59C2×D411yescharacteristic
54C2×Q8×C59—22no—
55C2×Q8×C59—81no—
56C22×C4×C59—21no—
57C22×C4×C59—21no—
58C22×C4×C59—21no—
59C22×C4×C59—21no—
60C22×C4×C59—21no—
61C22×C4×C59—21no—
62C22×C4×C59—21no—
63C22×C4×C59—22no—
64C22×C4×C59—21no—
65C22×C4×C59—81no—
66C22×C4×C59—81no—
67C22×C4×C59C22⋊C412yes—
68C22×C4×C59C22⋊C412yes—
69C22×C4×C59SD1612yes—
70C22×C4×C59Q1612yes—
71C22×C4×C59C22×C411yescharacteristic
72C22×C4×C59C2×D412yes—
73C22×C4×C59C2×D411yescharacteristic
74C22×C4×C59#4112yes—
75C22×Dic59—24no—
76C4⋊C4×C59—24no—
77C4⋊C4×C59—24no—
78C4⋊C4×C59—22no—
79C4⋊C4×C59—42no—
80C4⋊C4×C59—81no—
81C4⋊C4×C59—82no—
82C22⋊C4×C59—41no—
83C22⋊C4×C59—41no—
84C22⋊C4×C59—41no—
85C22⋊C4×C59—41no—
86C22⋊Dic59—42no—
87C22⋊Dic59—42no—
88C4⋊Dic59—44no—
89C4⋊Dic59—82no—
90C4⋊Dic59—82no—
91C4×Dic59—44no—
92C2×C59⋊C8—42no—
93C2×C59⋊C8—82no—
94C23×C59—41no—
95C23×C59—41no—
96C23×C59C2×C22⋊C411yescharacteristic
97C23×C59#12011yescharacteristic
98C23×C59#12311yescharacteristic
99C23×C59C2×SD1611yescharacteristic
100C23×C59C2×Q1611yescharacteristic
101C23×C59AGL(1,ℤ/8)11yescharacteristic
102C23×C59#13711yescharacteristic
103Q8×C59—82no—
104C2×C4×C59—22no—
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—42no—
112C2×C4×C59—42no—
113C2×C4×C59—41no—
114C2×C4×C59—41no—
115C2×C4×C59—42no—
116C2×C4×C59—42no—
117C2×C4×C59—42no—
118C2×C4×C59—42no—
119C2×C4×C59—42no—
120C2×C4×C59—42no—
121C2×C4×C59—41no—
122C2×C4×C59—41no—
123C2×C4×C59—42no—
124C2×C4×C59—81no—
125C2×C4×C59#10314yes—
126C2×C4×C59C4×D412yes—
127C2×C4×C59#12112yes—
128C2×Dic59—44no—
129C2×Dic59—44no—
130C2×Dic59—44no—
131C59⋊C8—82no—
132#9200—591nomaximal, Sylow
133C22×C59—21no—
134C22×C59—21no—
135C22×C59—21no—
136C22×C59—21no—
137C22×C59—21no—
138C22×C59—21no—
139C22×C59—41no—
140C22×C59—41no—
141C22×C59—41no—
142C22×C59—41no—
143C22×C59#38611yescharacteristic
144C22×C59C2×#10311yescharacteristic
145C22×C59#41711yescharacteristic
146C22×C59#45011yescharacteristic
147C22×C59#45111yescharacteristic
148C22×C59#48311yescharacteristic
149C22×C59#48411yescharacteristic
150C4×C59—24no—
151C4×C59—22no—
152C4×C59—22no—
153C4×C59—42no—
154C4×C59—82no—
155C4×C59—81no—
156C4×C59—81no—
157Dic59—84no—
158#2627D5911yescharacteristic
159#2316—591no—
160#1520—592no—
161C2×#324—591no—
162#1289—592no—
163C2×C59—21no—
164C2×C59—21no—
165C2×C59—21no—
166C2×C59—21no—
167C2×C59—21no—
168C2×C59—21no—
169C2×C59#187211yescharacteristic
170C2×C59#191211yescharacteristic
171C2×C59#191311yescharacteristic
172C2×C22⋊Q8C2×D5911yescharacteristic
173C2×#117—1181no—
174C22×C4⋊C4C2×D5911yescharacteristic
175C2×C22⋊C8—592no—
176#400Dic5912yescomplemented
177#398—22no—
178#395Dic5912yescomplemented
179#389—1184no—
180#385—21no—
181#381—21no—
182#380—592no—
183#380—1181no—
184#380C2×D5911yescharacteristic
185#328—1184no—
186#324—1182no—
187C59#920011yescharacteristic, complemented, Sylow
188C22×Q8C2×Dic5911yescharacteristic
189C23×C4C22⋊D5911yescharacteristic
190C23×C4C22⋊D5911yescharacteristic
191C23×C4C22×D5911yescharacteristic
192C22×C8—1182no—
193C22⋊Q8—22no—
194C22⋊Q8—22no—
195#117—2362no—
196#117—2362no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—1182no—
203C2×C4⋊C4—1182no—
204C2×C4⋊C4—1182no—
205C2×C4⋊C4C2×Dic5911yescharacteristic
206C2×C4⋊C4C2×Dic5912yes—
207C2×C4⋊C4C22⋊D5912yes—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—1182no—
211C2×C42—1182no—
212C22⋊C8—1182no—
213C22⋊C8—1182no—
214#95—22no—
215#95—22no—
216#95—22no—
217#95—594no—
218#95—1182no—
219#95C2×Dic5912yes—
220C24C2×C22⋊D5911yescharacteristic
221C2×Q8—22no—
222C2×Q8—81no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—22no—
231C22×C4—21no—
232C22×C4—81no—
233C22×C4—81no—
234C22×C4—1184no—
235C22×C4#1174567112yes—
236C22×C4#1174567212yes—
237C22×C4C22⋊Dic5912yes—
238C22×C4C22⋊Dic5912yes—
239C22×C4D4×D5912yes—
240C22×C4#1174568712yes—
241C22×C4C22×Dic5911yescharacteristic
242C22×C4C2×C22⋊D5911yescharacteristic
243C2×C8—4722no—
244C2×C8—2362no—
245C4⋊C4—24no—
246C4⋊C4—24no—
247C4⋊C4—22no—
248C4⋊C4—42no—
249C4⋊C4—81no—
250C4⋊C4—82no—
251C4⋊C4—4722no—
252C4⋊C4—4722no—
253C4⋊C4—2364no—
254C22⋊C4—41no—
255C22⋊C4—41no—
256C22⋊C4—41no—
257C22⋊C4—41no—
258C22⋊C4—2362no—
259C22⋊C4—2362no—
260C42—2364no—
261C23—41no—
262C23—41no—
263C23C2×C22⋊Dic5911yescharacteristic
264C23#1427369511yescharacteristic
265C23#1427370811yescharacteristic
266C23C2×#1174567111yescharacteristic
267C23C2×#1174567211yescharacteristic
268C23#1427375211yescharacteristic
269C23#1427375711yescharacteristic
270Q8—82no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—81no—
292C2×C4—2364no—
293C2×C4—2364no—
294C2×C4—2364no—
295C2×C4#1427365114yes—
296C2×C4D4×Dic5912yes—
297C2×C4#1427369912yes—
298C8—4722no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—41no—
306C22—41no—
307C22—41no—
308C22—41no—
309C22#1683728211yescharacteristic
310C22C2×#1427365111yescharacteristic
311C22#1683740311yescharacteristic
312C22#1683755211yescharacteristic
313C22#1683755711yescharacteristic
314C22#1683773011yescharacteristic
315C22#1683773511yescharacteristic
316C4—24no—
317C4—22no—
318C4—22no—
319C4—42no—
320C4—82no—
321C4—81no—
322C4—81no—
323C4—4724no—
324C2—21no—
325C2—21no—
326C2—21no—
327C2—21no—
328C2—21no—
329C2—21no—
330C2#2142440411yescharacteristic
331C2#2142463411yescharacteristic
332C2#2142463911yescharacteristic
subgroupG/H
centerC22#16837282
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×#2627C2
socleC22×C59#386
prime radicalC24×C59C2×D4
cosocleC23×C4×C59C23
hypercenter#2627D59
hypocenterC59#9200
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2627

Complements

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

NH ≅ G/Ntype
15. C59×#400C4semidirect
17. C59×#395C4semidirect
176. #400Dic59semidirect
178. #395Dic59semidirect
187. C59#9200semidirect

Maximal subgroups

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

subgroupindex
#214267142
#214226682
C2×#168371282
#214219632
C59×#26272
#920059