Finite Group Databasebeta

← #28506973

Subgroups — #28506973

Order 15104.

536 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 111 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×#16837192C214yesmaximal
2#21424084C211yesmaximal, characteristic
3C4×#14273644C211yesmaximal, characteristic
4C59×#2486C211yesmaximal, characteristic
5C2×#14273644C2212yes—
6C2×#14273642C2212yes—
7C4×#11745664—21no—
8#16837262—21no—
9#16837192—28no—
10#16837171—21no—
11#16837169—21no—
12C2×#123×C59C2212yes—
13C2×C4×D4×C59C2211yescharacteristic
14C22×C4⋊C4×C59C411yescharacteristic, complemented
15C59×#398—21no—
16C59×#397—21no—
17C59×#389—21no—
18C59×#381C412yescomplemented
19C59×#380—21no—
20C59×#380C411yescharacteristic, complemented
21C2×#11745664—22no—
22C2×C4⋊Dic59—41no—
23C2×C22⋊Dic59—22no—
24#14273644—22no—
25#14273644D414yes—
26#14273642—24no—
27#14273638—41no—
28C4×C59⋊C8—41no—
29C59⋊#95—41no—
30C22×D4×C59C2311yescharacteristic
31C23×C4×C59C2×C412yes—
32C23×C4×C59C2×C411yescharacteristic
33C23×C4×C59D411yescharacteristic
34#123×C59—24no—
35#123×C59—24no—
36C4×D4×C59—41no—
37C4×D4×C59—41no—
38C2×C4⋊C4×C59—21no—
39C2×C4⋊C4×C59—21no—
40C2×C4⋊C4×C59—21no—
41C2×C4⋊C4×C59—22no—
42C2×C4⋊C4×C59—41no—
43C2×C4⋊C4×C59—41no—
44C2×C4⋊C4×C59D411yescharacteristic
45C2×C22⋊C4×C59—22no—
46C2×C22⋊C4×C59C2×C412yes—
47C2×C22⋊C4×C59C2×C411yescharacteristic
48C2×C22⋊C4×C59D411yescharacteristic
49C2×C42×C59D411yescharacteristic
50#95×C59—22no—
51#95×C59—21no—
52#95×C59—22no—
53C24×C59C22×C411yescharacteristic
54C24×C59C2×D411yescharacteristic
55C2×D4×C59—41no—
56C2×D4×C59—41no—
57C2×D4×C59C2×D412yes—
58C2×D4×C59#4112yes—
59C22×C4×C59—21no—
60C22×C4×C59—21no—
61C22×C4×C59—21no—
62C22×C4×C59—21no—
63C22×C4×C59—22no—
64C22×C4×C59—22no—
65C22×C4×C59—24no—
66C22×C4×C59—42no—
67C22×C4×C59—42no—
68C22×C4×C59—42no—
69C22×C4×C59—42no—
70C22×C4×C59C22⋊C411yescharacteristic
71C22×C4×C59C22⋊C411yescharacteristic
72C22×C4×C59C22⋊C411yescharacteristic
73C22×C4×C59C22⋊C411yescharacteristic
74C22×C4×C59C2×D411yescharacteristic
75C22×Dic59—42no—
76C4⋊C4×C59—41no—
77C4⋊C4×C59—42no—
78C4⋊C4×C59—42no—
79C4⋊C4×C59—41no—
80C4⋊C4×C59—41no—
81C4⋊C4×C59—44no—
82C22⋊C4×C59—24no—
83C22⋊C4×C59—41no—
84C22⋊C4×C59—41no—
85C22⋊C4×C59—42no—
86C22⋊C4×C59—42no—
87C42×C59—21no—
88C42×C59—21no—
89C22⋊Dic59—42no—
90C22⋊Dic59—42no—
91C4⋊Dic59—82no—
92#11745664—24no—
93#11745664—42no—
94C2×C59⋊C8—42no—
95C23×C59—21no—
96C23×C59—21no—
97C23×C59—21no—
98C23×C59—21no—
99C23×C59—41no—
100C23×C59—41no—
101C23×C59—41no—
102C23×C59—41no—
103C23×C59C2×C22⋊C411yescharacteristic
104C23×C59C4×D412yes—
105C23×C59#12411yescharacteristic
106C23×C59#12711yescharacteristic
107D4×C59—42no—
108D4×C59—42no—
109C2×C4×C59—21no—
110C2×C4×C59—21no—
111C2×C4×C59—22no—
112C2×C4×C59—21no—
113C2×C4×C59—21no—
114C2×C4×C59—21no—
115C2×C4×C59—42no—
116C2×C4×C59—42no—
117C2×C4×C59—42no—
118C2×C4×C59—41no—
119C2×C4×C59—41no—
120C2×C4×C59—41no—
121C2×C4×C59—42no—
122C2×C4×C59—42no—
123C2×C4×C59—42no—
124C2×C4×C59—42no—
125C2×C4×C59—42no—
126C2×C4×C59—42no—
127C2×C4×C59—81no—
128C2×C4×C59—84no—
129C2×C4×C59#9911yescharacteristic
130C2×C4×C59#9911yescharacteristic
131C2×C4×C59#12112yes—
132C2×Dic59—82no—
133C2×Dic59—82no—
134C2×Dic59—82no—
135C59⋊C8—44no—
136#13217—591nomaximal, Sylow
137C22×C59—21no—
138C22×C59—21no—
139C22×C59—21no—
140C22×C59—22no—
141C22×C59—21no—
142C22×C59—42no—
143C22×C59—41no—
144C22×C59—41no—
145C22×C59—42no—
146C22×C59—41no—
147C22×C59—41no—
148C22×C59—41no—
149C22×C59—41no—
150C22×C59—81no—
151C22×C59#39011yescharacteristic
152C22×C59C2×#9911yescharacteristic
153C22×C59#41011yescharacteristic
154C4×C59—21no—
155C4×C59—22no—
156C4×C59—41no—
157C4×C59—41no—
158C4×C59—41no—
159C4×C59—82no—
160C4×C59—82no—
161C4×C59#35212yescomplemented
162Dic59—162no—
163#2486D5911yescharacteristic
164C2×#352—594no—
165#1799—591no—
166C4×#100—591no—
167C2×C59—21no—
168C2×C59—21no—
169C2×C59—41no—
170C2×C59—42no—
171C2×C59—41no—
172C2×C59#196411yescharacteristic
173C2×C59C2×#35211yescharacteristic
174C2×C59#212311yescharacteristic
175C2×#123C2×D5912yes—
176C2×C4×D4C2×D5911yescharacteristic
177C22×C4⋊C4Dic5911yescharacteristic, complemented
178C2×#100—592no—
179C2×#99—592no—
180C4×M16—1181no—
181#398—21no—
182#397—21no—
183#389—21no—
184#381Dic5912yescomplemented
185#380—21no—
186#380—1181no—
187#380Dic5911yescharacteristic, complemented
188#352—1188no—
189#343—1181no—
190#342—1181no—
191C59#1321711yescharacteristic, complemented, Sylow
192C22×D4C22×D5911yescharacteristic
193C23×C4C2×Dic5912yes—
194C23×C4C2×Dic5911yescharacteristic
195C23×C4C22⋊D5911yescharacteristic
196C2×M16—1182no—
197#123—24no—
198#123—24no—
199C4×D4—41no—
200C4×D4—41no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—2361no—
208C2×C4⋊C4C22⋊D5911yescharacteristic
209C2×C22⋊C4—22no—
210C2×C22⋊C4—1182no—
211C2×C22⋊C4C2×Dic5912yes—
212C2×C22⋊C4C2×Dic5911yescharacteristic
213C2×C22⋊C4C22⋊D5911yescharacteristic
214C2×C42C22⋊D5911yescharacteristic
215#100—594no—
216#100—1182no—
217#99—1184no—
218#97—2361no—
219C4×C8—2361no—
220#95—22no—
221#95—21no—
222#95—22no—
223#95—2361no—
224C24C22×Dic5911yescharacteristic
225C24C2×C22⋊D5911yescharacteristic
226C2×D4—41no—
227C2×D4—41no—
228C2×D4D4×D5912yes—
229C2×D4#1174568712yes—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—24no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—2362no—
242C22×C4C22⋊Dic5911yescharacteristic
243C22×C4C22⋊Dic5911yescharacteristic
244C22×C4C22⋊Dic5911yescharacteristic
245C22×C4C22⋊Dic5911yescharacteristic
246C22×C4C2×C22⋊D5911yescharacteristic
247M16—1184no—
248M16—2362no—
249C2×C8—2362no—
250C4⋊C4—41no—
251C4⋊C4—42no—
252C4⋊C4—42no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—44no—
256C4⋊C4—4722no—
257C22⋊C4—24no—
258C22⋊C4—41no—
259C22⋊C4—41no—
260C22⋊C4—42no—
261C22⋊C4—42no—
262C22⋊C4—2362no—
263C22⋊C4—2362no—
264C42—21no—
265C42—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—41no—
271C23—41no—
272C23—41no—
273C23—41no—
274C23C2×C22⋊Dic5911yescharacteristic
275C23D4×Dic5912yes—
276C23#1427371411yescharacteristic
277C23#1427372311yescharacteristic
278D4—42no—
279D4—42no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—81no—
299C2×C4—84no—
300C2×C4—4722no—
301C2×C4—4722no—
302C2×C4—4722no—
303C2×C4#1427364211yescharacteristic
304C2×C4#1427364211yescharacteristic
305C2×C4#1427369812yes—
306C8—2364no—
307C22—21no—
308C22—21no—
309C22—22no—
310C22—21no—
311C22—21no—
312C22—42no—
313C22—41no—
314C22—41no—
315C22—42no—
316C22—41no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22—81no—
321C22#1683730511yescharacteristic
322C22C2×#1427364211yescharacteristic
323C22#1683737611yescharacteristic
324C4—22no—
325C4—21no—
326C4—41no—
327C4—41no—
328C4—41no—
329C4—82no—
330C4—82no—
331C4—9442no—
332C4#1683719212yescomplemented
333C2—21no—
334C2—21no—
335C2—41no—
336C2—42no—
337C2—41no—
338C2#2142493311yescharacteristic
339C2C2×#1683719211yescharacteristic
340C2#2142575011yescharacteristic
subgroupG/H
centerC22C2×#14273642
commutatorC24×C59C22×C4
FrattiniC22×D4C22×D59
FittingC59×#2486C2
socleC22×C59C2×#99
prime radical, cosocleC22×D4×C59C23
hypercenter#2486D59
hypocenterC59#13217
LCSG ▷ C24×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2486

Complements

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

NH ≅ G/Ntype
14. C22×C4⋊C4×C59C4semidirect
18. C59×#381C4semidirect
20. C59×#380C4semidirect
161. C4×C59#352semidirect
177. C22×C4⋊C4Dic59semidirect
184. #381Dic59semidirect
187. #380Dic59semidirect
191. C59#13217semidirect
332. C4#16837192semidirect

Maximal subgroups

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

subgroupindex
C2×#168371922
#214240842
C4×#142736442
C59×#24862
#1321759