Finite Group Databasebeta

← #28498235

Subgroups — #28498235

Order 15104.

492 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 91 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#21422423C211yesmaximal, characteristic
2C2×#16837126C211yesmaximal, characteristic
3#21422044C212yesmaximal, complemented
4#21422026C212yesmaximal, complemented
5C59×#2483C211yesmaximal, characteristic
6#16837366—21no—
7C2×C22⋊(C59⋊C8)—21no—
8#16837126—21no—
9#16837126—21no—
10#16837126C2212yes—
11#16837126C2212yes—
12C22×C22⋊C4×C59C411yescharacteristic
13C22×C22⋊C4×C59C2211yescharacteristic
14C59×#397—21no—
15C59×#397—21no—
16C59×#385—21no—
17C59×#385C411yescharacteristic
18C59×#385C2211yescharacteristic
19C59×#381—21no—
20C59×#381C411yescharacteristic
21C59×#381C2211yescharacteristic
22C59×#379C411yescharacteristic
23C2×#11745664—41no—
24C22×C59⋊C8—41no—
25C22⋊(C59⋊C8)—22no—
26C22⋊(C59⋊C8)—22no—
27C22⋊(C59⋊C8)—41no—
28C22⋊(C59⋊C8)—41no—
29C25×C59C2×C411yescharacteristic
30C23×C4×C59C2×C411yescharacteristic
31C23×C4×C59C2×C411yescharacteristic
32C23×C4×C59D411yescharacteristic
33C23×C4×C59D411yescharacteristic
34C2×C4⋊C4×C59—21no—
35C2×C4⋊C4×C59—21no—
36C2×C22⋊C4×C59—21no—
37C2×C22⋊C4×C59—21no—
38C2×C22⋊C4×C59—21no—
39C2×C22⋊C4×C59—21no—
40C2×C22⋊C4×C59—41no—
41C2×C22⋊C4×C59—41no—
42C2×C22⋊C4×C59—41no—
43C2×C22⋊C4×C59—41no—
44C2×C22⋊C4×C59C2×C411yescharacteristic
45C2×C22⋊C4×C59C2×C411yescharacteristic
46C2×C22⋊C4×C59C2×C411yescharacteristic
47C2×C22⋊C4×C59D411yescharacteristic
48C2×C22⋊C4×C59D411yescharacteristic
49C2×C22⋊C4×C59C2311yescharacteristic
50#95×C59—21no—
51#95×C59—21no—
52#95×C59—21no—
53#95×C59—21no—
54#95×C59—21no—
55#95×C59—21no—
56C24×C59—21no—
57C24×C59—21no—
58C24×C59—22no—
59C24×C59—41no—
60C24×C59—42no—
61C24×C59—81no—
62C24×C59C22⋊C411yescharacteristic
63C24×C59C22⋊C411yescharacteristic
64C24×C59C22×C411yescharacteristic
65C22×C4×C59—21no—
66C22×C4×C59—21no—
67C22×C4×C59—21no—
68C22×C4×C59—21no—
69C22×C4×C59—21no—
70C22×C4×C59—21no—
71C22×C4×C59—21no—
72C22×C4×C59—21no—
73C22×C4×C59—22no—
74C22×C4×C59—42no—
75C22×C4×C59—41no—
76C22×C4×C59—41no—
77C22×C4×C59—41no—
78C22×C4×C59—81no—
79C22×C4×C59—81no—
80C22×C4×C59C22⋊C411yescharacteristic
81C22×C4×C59C22⋊C411yescharacteristic
82C22×C4×C59C2×D411yescharacteristic
83C22×C4×C59C2×D411yescharacteristic
84C4⋊C4×C59—81no—
85C4⋊C4×C59—81no—
86C22⋊C4×C59—22no—
87C22⋊C4×C59—41no—
88C22⋊C4×C59—41no—
89C22⋊C4×C59—41no—
90C22⋊C4×C59—41no—
91C22⋊C4×C59—42no—
92C22⋊C4×C59—42no—
93C22⋊C4×C59—41no—
94C22⋊C4×C59—41no—
95C22⋊C4×C59—41no—
96C22⋊C4×C59—81no—
97#11745664—82no—
98C2×C59⋊C8—42no—
99C2×C59⋊C8—42no—
100C2×C59⋊C8—82no—
101C23×C59—22no—
102C23×C59—21no—
103C23×C59—21no—
104C23×C59—21no—
105C23×C59—21no—
106C23×C59—21no—
107C23×C59—21no—
108C23×C59—21no—
109C23×C59—41no—
110C23×C59—42no—
111C23×C59—42no—
112C23×C59—41no—
113C23×C59—41no—
114C23×C59—42no—
115C23×C59—41no—
116C23×C59—41no—
117C23×C59—42no—
118C23×C59—81no—
119C23×C59—81no—
120C23×C59—84no—
121C23×C59—84no—
122C23×C59#9911yescharacteristic
123C23×C59#9911yescharacteristic
124C23×C59C2×C22⋊C411yescharacteristic
125C23×C59#13112yes—
126C2×C4×C59—21no—
127C2×C4×C59—21no—
128C2×C4×C59—21no—
129C2×C4×C59—21no—
130C2×C4×C59—42no—
131C2×C4×C59—42no—
132C2×C4×C59—41no—
133C2×C4×C59—41no—
134C2×C4×C59—41no—
135C2×C4×C59—41no—
136C2×C4×C59—41no—
137C2×C4×C59—41no—
138C2×C4×C59—41no—
139C2×C4×C59—41no—
140C2×C4×C59—41no—
141C2×C4×C59—41no—
142C2×C4×C59—41no—
143C2×C4×C59—41no—
144C2×C4×C59—41no—
145C2×C4×C59—41no—
146C2×C4×C59—41no—
147C2×C4×C59—41no—
148C2×C4×C59—81no—
149C2×C4×C59—81no—
150C2×C4×C59—82no—
151C2×C4×C59—82no—
152C59⋊C8—82no—
153C59⋊C8—82no—
154#11547—591nomaximal, Sylow
155C22×C59—21no—
156C22×C59—21no—
157C22×C59—21no—
158C22×C59—21no—
159C22×C59—21no—
160C22×C59—21no—
161C22×C59—22no—
162C22×C59—21no—
163C22×C59—42no—
164C22×C59—42no—
165C22×C59—42no—
166C22×C59—42no—
167C22×C59—41no—
168C22×C59—41no—
169C22×C59—41no—
170C22×C59—41no—
171C22×C59—41no—
172C22×C59—84no—
173C22×C59—81no—
174C22×C59—81no—
175C22×C59—84no—
176C22×C59#35212yes—
177C22×C59#40811yescharacteristic
178C22×C59C2×#9911yescharacteristic
179C22×C59#41311yescharacteristic
180C4×C59—41no—
181C4×C59—41no—
182C4×C59—41no—
183C4×C59—41no—
184C4×C59—81no—
185C4×C59—81no—
186C4×C59—81no—
187C4×C59—81no—
188#2483D5911yescharacteristic
189#1462—591no—
190C2×#323—591no—
191#1326—592no—
192#1317—592no—
193C2×C59—21no—
194C2×C59—21no—
195C2×C59—22no—
196C2×C59—41no—
197C2×C59—42no—
198C2×C59—81no—
199C2×C59#146811yescharacteristic
200C2×C59C2×#35211yescharacteristic
201C2×C59#212111yescharacteristic
202C22×C22⋊C4Dic5911yescharacteristic
203C22×C22⋊C4C2×D5911yescharacteristic
204#407—1181no—
205C2×C22⋊C8—1181no—
206#397—21no—
207#397—21no—
208#385—21no—
209#385Dic5911yescharacteristic
210#385C2×D5911yescharacteristic
211#381—21no—
212#381Dic5911yescharacteristic
213#381C2×D5911yescharacteristic
214#379Dic5911yescharacteristic
215#323—592no—
216#323—592no—
217#323—1181no—
218#323—1181no—
219C59#1154711yescharacteristic, complemented, Sylow
220C25C2×Dic5911yescharacteristic
221C23×C4C2×Dic5911yescharacteristic
222C23×C4C2×Dic5911yescharacteristic
223C23×C4C22⋊D5911yescharacteristic
224C23×C4C22⋊D5911yescharacteristic
225C2×M16—2361no—
226C22×C8—2361no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—41no—
234C2×C22⋊C4—41no—
235C2×C22⋊C4—41no—
236C2×C22⋊C4—41no—
237C2×C22⋊C4C2×Dic5911yescharacteristic
238C2×C22⋊C4C2×Dic5911yescharacteristic
239C2×C22⋊C4C2×Dic5911yescharacteristic
240C2×C22⋊C4C22⋊D5911yescharacteristic
241C2×C22⋊C4C22⋊D5911yescharacteristic
242C2×C22⋊C4C22×D5911yescharacteristic
243C22⋊C8—1182no—
244C22⋊C8—1182no—
245C22⋊C8—2361no—
246C22⋊C8—2361no—
247#95—21no—
248#95—21no—
249#95—21no—
250#95—21no—
251#95—21no—
252#95—21no—
253C24—21no—
254C24—21no—
255C24—22no—
256C24—41no—
257C24—42no—
258C24—81no—
259C24C22⋊Dic5911yescharacteristic
260C24C22⋊Dic5911yescharacteristic
261C24C22×Dic5911yescharacteristic
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—22no—
271C22×C4—42no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—81no—
276C22×C4—81no—
277C22×C4C22⋊Dic5911yescharacteristic
278C22×C4C22⋊Dic5911yescharacteristic
279C22×C4C2×C22⋊D5911yescharacteristic
280C22×C4C2×C22⋊D5911yescharacteristic
281M16—4722no—
282C2×C8—4722no—
283C2×C8—2362no—
284C2×C8—2362no—
285C4⋊C4—81no—
286C4⋊C4—81no—
287C22⋊C4—22no—
288C22⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—41no—
291C22⋊C4—41no—
292C22⋊C4—42no—
293C22⋊C4—42no—
294C22⋊C4—41no—
295C22⋊C4—41no—
296C22⋊C4—41no—
297C22⋊C4—81no—
298C23—21no—
299C23—21no—
300C23—22no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—41no—
307C23—42no—
308C23—42no—
309C23—41no—
310C23—41no—
311C23—42no—
312C23—41no—
313C23—41no—
314C23—42no—
315C23—81no—
316C23—81no—
317C23—84no—
318C23—84no—
319C23#1427364211yescharacteristic
320C23#1427364211yescharacteristic
321C23C2×C22⋊Dic5911yescharacteristic
322C23#1427373412yes—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—82no—
348C2×C4—82no—
349C8—4722no—
350C8—4722no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—22no—
358C22—21no—
359C22—42no—
360C22—42no—
361C22—42no—
362C22—42no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—84no—
369C22—81no—
370C22—81no—
371C22—84no—
372C22#1683719212yes—
373C22#1683736911yescharacteristic
374C22C2×#1427364211yescharacteristic
375C22#1683738611yescharacteristic
376C4—41no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—81no—
381C4—81no—
382C4—81no—
383C4—81no—
384C2—21no—
385C2—21no—
386C2—22no—
387C2—41no—
388C2—42no—
389C2—81no—
390C2#2142244411yescharacteristic
391C2C2×#1683719211yescharacteristic
392C2#2142574211yescharacteristic
subgroupG/H
centerC22C2×#14273642
commutatorC24×C59C22×C4
FrattiniC2×C22⋊C4C22×D59
FittingC59×#2483C2
socleC22×C59C2×#99
prime radicalC25×C59C2×C4
cosocleC2×C22⋊C4×C59C23
hypercenter#2483D59
hypocenterC59#11547
LCSG ▷ C24×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ #2483

Complements

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

NH ≅ G/Ntype
3. #21422044C2semidirect
4. #21422026C2semidirect
219. C59#11547semidirect

Maximal subgroups

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

subgroupindex
#214224232
C2×#168371262
#214220442
#214220262
C59×#24832
#1154759