Finite Group Databasebeta

← #28499555

Subgroups — #28499555

Order 15104.

506 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 506 conjugacy classes of subgroups — too many to draw. Only its 83 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#21428370C211yesmaximal, characteristic
2C2×#16837126C211yesmaximal, characteristic
3#21422119C212yesmaximal, complemented
4#21422095C212yesmaximal, complemented
5C59×#1819C211yesmaximal, characteristic
6C2×#14273686C411yescharacteristic, complemented
7C2×C22⋊(C59⋊C8)—21no—
8#16837338C411yescharacteristic, complemented
9#16837338C2211yescharacteristic
10#16837327C411yescharacteristic, complemented
11#16837327C2211yescharacteristic
12#16837290—21no—
13#16837290—21no—
14#16837275—21no—
15#16837265C411yescharacteristic, complemented
16#16837258—21no—
17#16837126—21no—
18#16837126C2212yes—
19C22×C22⋊C4×C59C2211yescharacteristic
20C59×#379—21no—
21C59×#342—21no—
22C59×#342C2212yes—
23C22×C59⋊C8—41no—
24#14273686—41no—
25#14273686—41no—
26C2×C4⋊Dic59C2×C411yescharacteristic
27C2×C4⋊Dic59C2×C411yescharacteristic
28C2×Dic59⋊C4—21no—
29C2×Dic59⋊C4—21no—
30C2×C22⋊Dic59—21no—
31C2×C22⋊Dic59—21no—
32C2×C22⋊Dic59—21no—
33C2×C22⋊Dic59—21no—
34C2×C22⋊Dic59—21no—
35C2×C22⋊Dic59—21no—
36C2×C4×Dic59—21no—
37C22⋊(C59⋊C8)—22no—
38C22⋊(C59⋊C8)—41no—
39C59⋊#95—21no—
40C59⋊#95—21no—
41C59⋊#95—21no—
42C59⋊#95C2×C411yescharacteristic
43C59⋊#95C2×C411yescharacteristic
44C25×C59D411yescharacteristic
45C23×C4×C59C2×C411yescharacteristic
46C23×C4×C59D411yescharacteristic
47C2×C22⋊C4×C59—22no—
48C2×C22⋊C4×C59—41no—
49C2×C22⋊C4×C59—41no—
50C2×C22⋊C4×C59—41no—
51C2×C22⋊C4×C59—41no—
52C2×C22⋊C4×C59C2×C411yescharacteristic
53C2×C22⋊C4×C59D411yescharacteristic
54C2×C22⋊C4×C59D411yescharacteristic
55C2×C22⋊C4×C59C2311yescharacteristic
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×C59C2×D411yescharacteristic
65C22×C4×C59—21no—
66C22×C4×C59—21no—
67C22×C4×C59—21no—
68C22×C4×C59—21no—
69C22×C4×C59—42no—
70C22×C4×C59—42no—
71C22×C4×C59—81no—
72C22×C4×C59C22⋊C411yescharacteristic
73C22×C4×C59C22⋊C411yescharacteristic
74C22×C4×C59C22×C411yescharacteristic
75C22×C4×C59C2×D411yescharacteristic
76C22×Dic59—21no—
77C22×Dic59—21no—
78C22×Dic59—21no—
79C22×Dic59—21no—
80C22⋊C4×C59—41no—
81C22⋊C4×C59—41no—
82C22⋊C4×C59—41no—
83C22⋊C4×C59—41no—
84C22⋊C4×C59—42no—
85C22⋊C4×C59—42no—
86C22⋊C4×C59—82no—
87C22⋊C4×C59—82no—
88C22⋊Dic59—42no—
89C22⋊Dic59—42no—
90C22⋊Dic59—41no—
91C22⋊Dic59—41no—
92C22⋊Dic59—42no—
93C22⋊Dic59—81no—
94C22⋊Dic59—81no—
95C4⋊Dic59—41no—
96C4⋊Dic59—41no—
97Dic59⋊C4—81no—
98Dic59⋊C4—81no—
99C4×Dic59—42no—
100C2×C59⋊C8—42no—
101C2×C59⋊C8—82no—
102C23×C59—22no—
103C23×C59—21no—
104C23×C59—21no—
105C23×C59—21no—
106C23×C59—21no—
107C23×C59—21no—
108C23×C59—21no—
109C23×C59—21no—
110C23×C59—41no—
111C23×C59—42no—
112C23×C59—42no—
113C23×C59—41no—
114C23×C59—41no—
115C23×C59—41no—
116C23×C59—41no—
117C23×C59—42no—
118C23×C59—42no—
119C23×C59—81no—
120C23×C59—81no—
121C23×C59—84no—
122C23×C59—84no—
123C23×C59#9911yescharacteristic
124C23×C59#9911yescharacteristic
125C23×C59C2×C22⋊C411yescharacteristic
126C23×C59#13512yes—
127C2×C4×C59—21no—
128C2×C4×C59—21no—
129C2×C4×C59—21no—
130C2×C4×C59—21no—
131C2×C4×C59—42no—
132C2×C4×C59—42no—
133C2×C4×C59—42no—
134C2×C4×C59—41no—
135C2×C4×C59—41no—
136C2×C4×C59—41no—
137C2×C4×C59—41no—
138C2×C4×C59—81no—
139C2×C4×C59—82no—
140C2×C4×C59—82no—
141C2×C4×C59—82no—
142C2×Dic59—41no—
143C2×Dic59—41no—
144C2×Dic59—41no—
145C2×Dic59—41no—
146C2×Dic59—41no—
147C2×Dic59—41no—
148C2×Dic59—41no—
149C2×Dic59—41no—
150C2×Dic59—41no—
151C2×Dic59—41no—
152C2×Dic59—41no—
153C2×Dic59—41no—
154C59⋊C8—82no—
155#11841—591nomaximal, Sylow
156C22×C59—21no—
157C22×C59—21no—
158C22×C59—21no—
159C22×C59—21no—
160C22×C59—21no—
161C22×C59—21no—
162C22×C59—22no—
163C22×C59—21no—
164C22×C59—42no—
165C22×C59—42no—
166C22×C59—42no—
167C22×C59—42no—
168C22×C59—41no—
169C22×C59—41no—
170C22×C59—41no—
171C22×C59—41no—
172C22×C59—41no—
173C22×C59—81no—
174C22×C59—84no—
175C22×C59—81no—
176C22×C59—84no—
177C22×C59#35212yes—
178C22×C59C2×#9911yescharacteristic
179C22×C59#41611yescharacteristic
180C22×C59#42111yescharacteristic
181C4×C59—41no—
182C4×C59—41no—
183C4×C59—41no—
184C4×C59—41no—
185C4×C59—162no—
186Dic59—81no—
187Dic59—81no—
188Dic59—81no—
189Dic59—81no—
190#2482—591no—
191#1819D5911yescharacteristic
192C2×#323—591no—
193#1352—592no—
194#1344—592no—
195C2×C59—21no—
196C2×C59—21no—
197C2×C59—22no—
198C2×C59—42no—
199C2×C59—41no—
200C2×C59—81no—
201C2×C59#151611yescharacteristic
202C2×C59C2×#35211yescharacteristic
203C2×C59#212711yescharacteristic
204C2×#117—591no—
205C22×C22⋊C4C2×D5911yescharacteristic
206C2×C22⋊C8—1181no—
207#399—591no—
208#399—591no—
209#397—591no—
210#397—591no—
211#388—1181no—
212#388—1181no—
213#385—1181no—
214#381—591no—
215#379—21no—
216#379—1181no—
217#342—21no—
218#342C2×D5912yes—
219#323—592no—
220#323—1181no—
221C59#1184111yescharacteristic, complemented, Sylow
222C25C22⋊D5911yescharacteristic
223C23×C4C4×D5911yescharacteristic
224C23×C4D23611yescharacteristic
225C22×C8—2361no—
226#117—2361no—
227#117—2361no—
228C2×C4⋊C4—591no—
229C2×C4⋊C4—591no—
230C2×C4⋊C4—1181no—
231C2×C4⋊C4—1181no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—41no—
234C2×C22⋊C4—41no—
235C2×C22⋊C4—41no—
236C2×C22⋊C4—41no—
237C2×C22⋊C4—1181no—
238C2×C22⋊C4—1181no—
239C2×C22⋊C4—1181no—
240C2×C22⋊C4—1181no—
241C2×C22⋊C4—1181no—
242C2×C22⋊C4—1181no—
243C2×C22⋊C4C4×D5911yescharacteristic
244C2×C22⋊C4D23611yescharacteristic
245C2×C22⋊C4C22⋊D5911yescharacteristic
246C2×C22⋊C4C22×D5911yescharacteristic
247C2×C42—1181no—
248C22⋊C8—1182no—
249C22⋊C8—2361no—
250#95—591no—
251#95—591no—
252#95—1181no—
253#95—1181no—
254#95—1181no—
255C24—21no—
256C24—21no—
257C24—22no—
258C24—41no—
259C24—42no—
260C24—81no—
261C24(C2×D59)⋊C411yescharacteristic
262C24(C2×D59)⋊C411yescharacteristic
263C24C2×C22⋊D5911yescharacteristic
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—81no—
271C22×C4—1181no—
272C22×C4—1181no—
273C22×C4—1181no—
274C22×C4—1181no—
275C22×C4(C2×D59)⋊C411yescharacteristic
276C22×C4(C2×D59)⋊C411yescharacteristic
277C22×C4C2×C4×D5911yescharacteristic
278C22×C4C2×D23611yescharacteristic
279C2×C8—4722no—
280C2×C8—2362no—
281C4⋊C4—4721no—
282C4⋊C4—4721no—
283C4⋊C4—2361no—
284C4⋊C4—2361no—
285C22⋊C4—41no—
286C22⋊C4—41no—
287C22⋊C4—41no—
288C22⋊C4—41no—
289C22⋊C4—42no—
290C22⋊C4—42no—
291C22⋊C4—82no—
292C22⋊C4—82no—
293C22⋊C4—4721no—
294C22⋊C4—4721no—
295C22⋊C4—2362no—
296C22⋊C4—2362no—
297C22⋊C4—2361no—
298C22⋊C4—2361no—
299C22⋊C4—2362no—
300C42—2362no—
301C23—21no—
302C23—21no—
303C23—22no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—41no—
310C23—42no—
311C23—42no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—42no—
317C23—42no—
318C23—81no—
319C23—81no—
320C23—84no—
321C23—84no—
322C23#1427364111yescharacteristic
323C23#1427364111yescharacteristic
324C23C2×(C2×D59)⋊C411yescharacteristic
325C23#1427374712yes—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—81no—
338C2×C4—82no—
339C2×C4—82no—
340C2×C4—82no—
341C2×C4—2361no—
342C2×C4—2361no—
343C2×C4—2361no—
344C2×C4—2361no—
345C2×C4—2361no—
346C2×C4—2361no—
347C2×C4—2361no—
348C2×C4—2361no—
349C2×C4—2361no—
350C2×C4—2361no—
351C2×C4—2361no—
352C2×C4—2361no—
353C8—4722no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—22no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—42no—
363C22—42no—
364C22—42no—
365C22—42no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—81no—
372C22—84no—
373C22—81no—
374C22—84no—
375C22#1683719112yes—
376C22C2×#1427364111yescharacteristic
377C22#1683739611yescharacteristic
378C22#1683742011yescharacteristic
379C4—41no—
380C4—41no—
381C4—41no—
382C4—41no—
383C4—162no—
384C4—4721no—
385C4—4721no—
386C4—4721no—
387C4—4721no—
388C2—21no—
389C2—21no—
390C2—22no—
391C2—42no—
392C2—41no—
393C2—81no—
394C2#2142265011yescharacteristic
395C2C2×#1683719111yescharacteristic
396C2#2142576311yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC22×C4×C59C22×C4
FrattiniC2×C22⋊C4C22×D59
FittingC59×#1819C2
socleC22×C59C2×#99
prime radicalC25×C59D4
cosocleC2×C22⋊C4×C59C23
hypercenter#1819D59
hypocenterC59#11841
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ #1819

Complements

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

NH ≅ G/Ntype
3. #21422119C2semidirect
4. #21422095C2semidirect
6. C2×#14273686C4semidirect
8. #16837338C4semidirect
10. #16837327C4semidirect
15. #16837265C4semidirect
221. C59#11841semidirect

Maximal subgroups

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

subgroupindex
#214283702
C2×#168371262
#214221192
#214220952
C59×#18192
#1184159