Finite Group Databasebeta

← #28481676

Subgroups — #28481676

Order 15104.

456 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21424134C211yesmaximal, characteristic, complemented
2#21424121C211yesmaximal, characteristic, complemented
3#21422639C212yesmaximal
4#21422001C212yesmaximal, complemented
5C59×#2929C211yesmaximal, characteristic
6C2×#14273663—21no—
7C2×#14273659—21no—
8#16837366C2212yes—
9#16837262C2212yes—
10#16837167—21no—
11#16837165—21no—
12#16837139—22no—
13#16837136—22no—
14C2×#131×C59C411yescharacteristic, complemented
15C23×C8×C59C2211yescharacteristic
16C59×#525C2211yescharacteristic
17C59×#435—21no—
18C59×#435—21no—
19C8×D4×C59—21no—
20C8×D4×C59—21no—
21C59×#424C411yescharacteristic, complemented
22C59×#408C411yescharacteristic, complemented
23C59×#407C2211yescharacteristic
24C59×#405C411yescharacteristic, complemented
25C2×#11745664—22no—
26C2×C4⋊Dic59—22no—
27C2×C22⋊Dic59—22no—
28#14273663—42no—
29#14273659—42no—
30C22⋊(C59⋊C8)—22no—
31C22⋊(C59⋊C8)—22no—
32C59⋊#95—22no—
33C2×#41×C59C2×C411yescharacteristic
34C23×C4×C59C2311yescharacteristic
35#131×C59—42no—
36C2×M16×C59—21no—
37C2×M16×C59C2×C411yescharacteristic
38C2×M16×C59C2×C411yescharacteristic
39C22×C8×C59—21no—
40C22×C8×C59—42no—
41C22×C8×C59C2×C411yescharacteristic
42C22×C8×C59C2×C411yescharacteristic
43C22×C8×C59D411yescharacteristic
44C22×C8×C59D411yescharacteristic
45C22×C8×C59D411yescharacteristic
46C22×C8×C59D411yescharacteristic
47#123×C59—21no—
48C22⋊Q8×C59D411yescharacteristic
49C22⋊Q8×C59Q811yescharacteristic
50#121×C59D411yescharacteristic
51#121×C59Q811yescharacteristic
52#120×C59—21no—
53C4×D4×C59—21no—
54C4×D4×C59—21no—
55#117×C59C2×C411yescharacteristic
56C4⋊C8×C59—21no—
57C4⋊C8×C59—21no—
58C22⋊C8×C59—21no—
59C22⋊C8×C59—21no—
60C22⋊C8×C59—21no—
61C22⋊C8×C59—21no—
62#97×C59—21no—
63C4×C8×C59—21no—
64C24×C59#4111yescharacteristic
65#41×C59—41no—
66#41×C59—41no—
67C2×Q8×C59C4⋊C411yescharacteristic
68C2×D4×C59—21no—
69C2×D4×C59—21no—
70C2×D4×C59—21no—
71C2×D4×C59C4⋊C411yescharacteristic
72C22×C4×C59—21no—
73C22×C4×C59—41no—
74C22×C4×C59—41no—
75C22×C4×C59C22×C411yescharacteristic
76C22×C4×C59C2×D411yescharacteristic
77C22×C4×C59C2×D411yescharacteristic
78C22×C4×C59C2×D411yescharacteristic
79C22×C4×C59C2×Q811yescharacteristic
80C22×C4×C59#4111yescharacteristic
81C22×Dic59—42no—
82M16×C59—42no—
83M16×C59—41no—
84M16×C59—41no—
85C2×C8×C59—22no—
86C2×C8×C59—22no—
87C2×C8×C59—22no—
88C2×C8×C59—21no—
89C2×C8×C59—21no—
90C2×C8×C59—41no—
91C2×C8×C59—41no—
92C2×C8×C59—41no—
93C2×C8×C59—42no—
94C2×C8×C59C22⋊C411yescharacteristic
95C2×C8×C59C22⋊C411yescharacteristic
96C2×C8×C59C22⋊C411yescharacteristic
97C2×C8×C59C22⋊C411yescharacteristic
98C4⋊C4×C59—21no—
99C4⋊C4×C59—21no—
100C4⋊C4×C59C4⋊C411yescharacteristic
101C4⋊C4×C59C4⋊C411yescharacteristic
102C22⋊C4×C59—21no—
103C22⋊C4×C59—21no—
104C22⋊C4×C59—21no—
105C22⋊C4×C59—21no—
106C22⋊C4×C59—21no—
107C42×C59—21no—
108C22⋊Dic59—82no—
109C4⋊Dic59—42no—
110C4⋊Dic59—42no—
111#11745664—42no—
112#11745664—42no—
113C2×C59⋊C8—42no—
114C23×C59—22no—
115C23×C59—21no—
116C23×C59—41no—
117C23×C59—41no—
118C23×C59#11711yescharacteristic
119C23×C59#12111yescharacteristic
120C23×C59C22⋊Q811yescharacteristic
121Q8×C59—41no—
122D4×C59—41no—
123D4×C59—42no—
124D4×C59—42no—
125D4×C59—42no—
126C2×C4×C59—21no—
127C2×C4×C59—21no—
128C2×C4×C59—21no—
129C2×C4×C59—21no—
130C2×C4×C59—21no—
131C2×C4×C59—21no—
132C2×C4×C59—21no—
133C2×C4×C59—21no—
134C2×C4×C59—21no—
135C2×C4×C59—41no—
136C2×C4×C59—41no—
137C2×C4×C59—41no—
138C2×C4×C59—41no—
139C2×C4×C59—41no—
140C2×C4×C59C2×C22⋊C411yescharacteristic
141C2×C4×C59C2×C4⋊C411yescharacteristic
142C2×C4×C59#12111yescharacteristic
143C2×C4×C59C22⋊Q811yescharacteristic
144C2×Dic59—82no—
145C2×Dic59—82no—
146C2×Dic59—82no—
147C8×C59—21no—
148C8×C59—21no—
149C8×C59—21no—
150C8×C59—41no—
151C8×C59—41no—
152C8×C59#9912yes—
153C59⋊C8—82no—
154#8629—591nomaximal, Sylow
155C22×C59—21no—
156C22×C59—21no—
157C22×C59—21no—
158C22×C59—21no—
159C22×C59—21no—
160C22×C59—21no—
161C22×C59—21no—
162C22×C59—21no—
163C22×C59—44no—
164C22×C59—41no—
165C22×C59—41no—
166C22×C59—41no—
167C22×C59#38011yescharacteristic
168C22×C59#47112yes—
169C4×C59—21no—
170C4×C59—21no—
171C4×C59—21no—
172C4×C59—41no—
173C4×C59—41no—
174C4×C59—41no—
175C4×C59C2×#9911yescharacteristic
176C4×C59#41011yescharacteristic
177Dic59—162no—
178#2929D5911yescharacteristic
179#1810—591no—
180#1807—591no—
181#1513—592no—
182#1305—592no—
183C2×C59—22no—
184C2×C59—41no—
185C2×C59—41no—
186C2×C59—81no—
187C2×C59#179911yescharacteristic
188C2×C59#181411yescharacteristic
189C2×C59#181511yescharacteristic
190C2×#131Dic5911yescharacteristic, complemented
191C23×C8C2×D5911yescharacteristic
192#525C2×D5911yescharacteristic
193#435—21no—
194#435—21no—
195C8×D4—21no—
196C8×D4—21no—
197C2×#108—1181no—
198C2×#106—1181no—
199#424Dic5911yescharacteristic, complemented
200#408Dic5911yescharacteristic, complemented
201#407—592no—
202#407C2×D5911yescharacteristic
203#405Dic5911yescharacteristic, complemented
204#380—592no—
205#341—1181no—
206#340—1181no—
207#328—1182no—
208#327—1182no—
209C59#862911yescharacteristic, complemented, Sylow
210C2×#41C2×Dic5911yescharacteristic
211C23×C4C22×D5911yescharacteristic
212#131—42no—
213C2×M16—21no—
214C2×M16—1182no—
215C2×M16C2×Dic5911yescharacteristic
216C2×M16C2×Dic5911yescharacteristic
217C22×C8—21no—
218C22×C8—42no—
219C22×C8C2×Dic5911yescharacteristic
220C22×C8C2×Dic5911yescharacteristic
221C22×C8C22⋊D5911yescharacteristic
222C22×C8C22⋊D5911yescharacteristic
223C22×C8C22⋊D5911yescharacteristic
224C22×C8C22⋊D5911yescharacteristic
225#123—21no—
226C22⋊Q8Dic11811yescharacteristic
227C22⋊Q8D23611yescharacteristic
228#121Dic11811yescharacteristic
229#121D23611yescharacteristic
230#120—21no—
231C4×D4—21no—
232C4×D4—21no—
233#117C2×Dic5911yescharacteristic
234C2×C4⋊C4—1182no—
235C2×C22⋊C4—1182no—
236#108—2362no—
237#106—2362no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C22⋊C8—21no—
241C22⋊C8—21no—
242C22⋊C8—21no—
243C22⋊C8—21no—
244C22⋊C8—1182no—
245C22⋊C8—1182no—
246#97—21no—
247C4×C8—21no—
248#95—1182no—
249C24#1174568511yescharacteristic
250#41—41no—
251#41—41no—
252C2×Q8C4⋊Dic5911yescharacteristic
253C2×D4—21no—
254C2×D4—21no—
255C2×D4—21no—
256C2×D4C4⋊Dic5911yescharacteristic
257C22×C4—21no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—2362no—
261C22×C4C2×Dic11811yescharacteristic
262C22×C4C2×D23611yescharacteristic
263C22×C4#1174568511yescharacteristic
264C22×C4C22×Dic5911yescharacteristic
265C22×C4C2×C22⋊D5911yescharacteristic
266C22×C4C2×C22⋊D5911yescharacteristic
267M16—42no—
268M16—41no—
269M16—41no—
270M16—2362no—
271M16—2362no—
272C2×C8—22no—
273C2×C8—22no—
274C2×C8—22no—
275C2×C8—21no—
276C2×C8—21no—
277C2×C8—41no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8—42no—
281C2×C8—2362no—
282C2×C8C22⋊Dic5911yescharacteristic
283C2×C8C22⋊Dic5911yescharacteristic
284C2×C8C22⋊Dic5911yescharacteristic
285C2×C8C22⋊Dic5911yescharacteristic
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—2362no—
289C4⋊C4—2362no—
290C4⋊C4C4⋊Dic5911yescharacteristic
291C4⋊C4C4⋊Dic5911yescharacteristic
292C22⋊C4—21no—
293C22⋊C4—21no—
294C22⋊C4—21no—
295C22⋊C4—21no—
296C22⋊C4—21no—
297C22⋊C4—4722no—
298C42—21no—
299C23—22no—
300C23—21no—
301C23—41no—
302C23—41no—
303C23#1427368611yescharacteristic
304C23#1427370111yescharacteristic
305C23#1427370611yescharacteristic
306Q8—41no—
307D4—41no—
308D4—42no—
309D4—42no—
310D4—42no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—4722no—
326C2×C4—4722no—
327C2×C4—4722no—
328C2×C4C2×C22⋊Dic5911yescharacteristic
329C2×C4C2×C4⋊Dic5911yescharacteristic
330C2×C4#1427370111yescharacteristic
331C2×C4#1427370611yescharacteristic
332C8—21no—
333C8—21no—
334C8—21no—
335C8—41no—
336C8—41no—
337C8—4722no—
338C8#1427364212yes—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—44no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22#1683726211yescharacteristic
352C22#1683766512yes—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—9442no—
360C4C2×#1427364211yescharacteristic
361C4#1683737611yescharacteristic
362C2—22no—
363C2—41no—
364C2—41no—
365C2—81no—
366C2#2142408411yescharacteristic
367C2#2142415011yescharacteristic
368C2#2142415511yescharacteristic
subgroupG/H
centerC22#16837262
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×#2929C2
socleC22×C59#380
prime radicalC59×#525C22
cosocleC23×C4×C59C23
hypercenter#2929D59
hypocenterC59#8629
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2929

Complements

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

NH ≅ G/Ntype
1. #21424134C2semidirect
2. #21424121C2semidirect
4. #21422001C2semidirect
14. C2×#131×C59C4semidirect
21. C59×#424C4semidirect
22. C59×#408C4semidirect
24. C59×#405C4semidirect
190. C2×#131Dic59semidirect
199. #424Dic59semidirect
200. #408Dic59semidirect
203. #405Dic59semidirect
209. C59#8629semidirect

Maximal subgroups

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

subgroupindex
#214241342
#214241212
#214226392
#214220012
C59×#29292
#862959