Finite Group Databasebeta

← #28477601

Subgroups — #28477601

Order 15104.

460 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 87 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#21425429C211yesmaximal, characteristic, complemented
2#21425342C211yesmaximal, characteristic, complemented
3#21425232C211yesmaximal, characteristic
4#21421919C211yesmaximal, characteristic, complemented
5#21421919C211yesmaximal, characteristic, complemented
6#21421917C211yesmaximal, characteristic
7C59×#2724C211yesmaximal, characteristic
8C2×#14273727C2211yescharacteristic
9C2×#14273658—21no—
10C2×#14273650—21no—
11C2×#14273650—21no—
12C2×#14273648—21no—
13#16837164—21no—
14#16837164—21no—
15#16837157C2211yescharacteristic
16#16837157C2211yescharacteristic
17#16837157C2211yescharacteristic
18C2×#123×C59—21no—
19C59×#398—21no—
20C59×#397—21no—
21C59×#397—21no—
22C59×#390C2211yescharacteristic
23C59×#389C2211yescharacteristic
24C59×#389C2211yescharacteristic
25C59×#385—21no—
26C59×#385—21no—
27C22×Dic118—21no—
28C22×C59⋊C8—21no—
29C22×C59⋊C8—21no—
30C22×C59⋊C8—21no—
31#14273727—42no—
32C2×C4⋊Dic59—21no—
33C2×C4⋊Dic59—21no—
34#14273658—42no—
35#14273650—42no—
36#14273650—42no—
37#14273648—42no—
38C22×D4×C59D411yescharacteristic
39C23×C4×C59—21no—
40#123×C59—42no—
41#123×C59—42no—
42C2×C4⋊C4×C59—21no—
43C2×C4⋊C4×C59—21no—
44C2×C4⋊C4×C59—21no—
45C2×C4⋊C4×C59D411yescharacteristic
46C2×C4⋊C4×C59D411yescharacteristic
47C2×C4⋊C4×C59D411yescharacteristic
48C2×C4⋊C4×C59D411yescharacteristic
49C2×C4⋊C4×C59D411yescharacteristic
50C2×C22⋊C4×C59—21no—
51C2×C22⋊C4×C59—21no—
52C2×C22⋊C4×C59—21no—
53C2×C22⋊C4×C59—21no—
54C2×C22⋊C4×C59—21no—
55C2×C42×C59C2311yescharacteristic
56#95×C59—21no—
57#95×C59—21no—
58#95×C59—21no—
59#95×C59—21no—
60#95×C59—21no—
61C24×C59—21no—
62C2×D4×C59—21no—
63C2×D4×C59—41no—
64C2×D4×C59—41no—
65C2×D4×C59SD1612yes—
66C22×C4×C59—21no—
67C22×C4×C59—21no—
68C22×C4×C59—21no—
69C22×C4×C59—21no—
70C22×C4×C59—21no—
71C22×C4×C59—42no—
72C22×C4×C59—81no—
73C22×C4×C59—81no—
74C22×C4×C59C2×D411yescharacteristic
75C22×C4×C59C2×D411yescharacteristic
76C22×C4×C59C2×D411yescharacteristic
77C22×Dic59—41no—
78C2×Dic118—22no—
79C2×Dic118—41no—
80C2×Dic118—41no—
81C2×Dic118—41no—
82C2×Dic118—41no—
83C2×Dic118—41no—
84C4⋊C4×C59—21no—
85C4⋊C4×C59—21no—
86C4⋊C4×C59—21no—
87C4⋊C4×C59—21no—
88C4⋊C4×C59—41no—
89C4⋊C4×C59—41no—
90C4⋊C4×C59—41no—
91C4⋊C4×C59—41no—
92C4⋊C4×C59—41no—
93C4⋊C4×C59—41no—
94C4⋊C4×C59—41no—
95C4⋊C4×C59—81no—
96C22⋊C4×C59—42no—
97C22⋊C4×C59—42no—
98C22⋊C4×C59—42no—
99C22⋊C4×C59—41no—
100C22⋊C4×C59—41no—
101C22⋊C4×C59—41no—
102C22⋊C4×C59—41no—
103C42×C59—41no—
104C4⋊Dic59—22no—
105C4⋊Dic59—22no—
106C4⋊Dic59—41no—
107C4⋊Dic59—41no—
108C2×C59⋊C8—22no—
109C2×C59⋊C8—41no—
110C2×C59⋊C8—41no—
111C2×C59⋊C8—41no—
112C2×C59⋊C8—41no—
113C2×C59⋊C8—41no—
114C2×C59⋊C8—41no—
115C2×C59⋊C8—41no—
116C2×C59⋊C8—41no—
117C23×C59—22no—
118C23×C59—41no—
119C23×C59—41no—
120C23×C59—41no—
121C23×C59—41no—
122C23×C59—41no—
123C23×C59—41no—
124C23×C59#12011yescharacteristic
125D4×C59—42no—
126D4×C59—81no—
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—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—84no—
149C2×C4×C59C2×SD1611yescharacteristic
150C2×C4×C59#13511yescharacteristic
151C2×C4×C59#13511yescharacteristic
152C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
153C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
154C2×C4×C59#13711yescharacteristic
155C2×Dic59—42no—
156C2×Dic59—81no—
157C2×Dic59—81no—
158Dic118—42no—
159Dic118—42no—
160Dic118—81no—
161Dic118—81no—
162C59⋊C8—42no—
163C59⋊C8—81no—
164C59⋊C8—81no—
165#7770—591nomaximal, Sylow
166C22×C59—42no—
167C22×C59—42no—
168C22×C59—42no—
169C22×C59—81no—
170C22×C59—81no—
171C22×C59—81no—
172C22×C59—81no—
173C22×C59#44911yescharacteristic
174C22×C59#45011yescharacteristic
175C22×C59#45211yescharacteristic
176C22×C59#45411yescharacteristic
177C22×C59#45511yescharacteristic
178C22×C59#45611yescharacteristic
179C22×C59B(4,𝔽2)11yescharacteristic
180C4×C59—21no—
181C4×C59—21no—
182C4×C59—21no—
183C4×C59—21no—
184C4×C59—21no—
185C4×C59—21no—
186C4×C59—81no—
187C4×C59—81no—
188C4×C59—81no—
189C4×C59—81no—
190C4×C59—81no—
191Dic59—82no—
192#2724D5911yescharacteristic
193#2058—591no—
194#2043—591no—
195#2023—591no—
196#1271—591no—
197#1271—591no—
198#1270—591no—
199C2×C59—82no—
200C2×C59#159711yescharacteristic
201C2×C59#160111yescharacteristic
202C2×C59#160311yescharacteristic
203C2×C59#163511yescharacteristic
204C2×C59#164011yescharacteristic
205C2×C59#165411yescharacteristic
206C2×C59#166411yescharacteristic
207C2×C4⋊Q8—591no—
208C2×#123—21no—
209C2×#106—1181no—
210C2×#103—1181no—
211C2×#103—1181no—
212C2×#102—1181no—
213#398—21no—
214#397—21no—
215#397—21no—
216#390C2×D5911yescharacteristic
217#389C2×D5911yescharacteristic
218#389C2×D5911yescharacteristic
219#385—21no—
220#385—21no—
221#340—1181no—
222#340—1181no—
223#336—591no—
224#336—591no—
225#336—591no—
226C59#777011yescharacteristic, complemented, Sylow
227C22×Q8—1181no—
228C22×D4C22⋊D5911yescharacteristic
229C23×C4—21no—
230C22×C8—1181no—
231C22×C8—1181no—
232C22×C8—1181no—
233C4⋊Q8—2362no—
234#123—42no—
235#123—42no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—1181no—
240C2×C4⋊C4—1181no—
241C2×C4⋊C4C22⋊D5911yescharacteristic
242C2×C4⋊C4C22⋊D5911yescharacteristic
243C2×C4⋊C4C22⋊D5911yescharacteristic
244C2×C4⋊C4C22⋊D5911yescharacteristic
245C2×C4⋊C4C22⋊D5911yescharacteristic
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C42C22×D5911yescharacteristic
252#106—2362no—
253#103—2362no—
254#103—2362no—
255#102—2362no—
256#95—21no—
257#95—21no—
258#95—21no—
259#95—21no—
260#95—21no—
261C24—21no—
262C2×Q8—1182no—
263C2×Q8—2361no—
264C2×Q8—2361no—
265C2×Q8—2361no—
266C2×Q8—2361no—
267C2×Q8—2361no—
268C2×D4—21no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4#1174567012yes—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—42no—
278C22×C4—81no—
279C22×C4—81no—
280C22×C4—2361no—
281C22×C4C2×C22⋊D5911yescharacteristic
282C22×C4C2×C22⋊D5911yescharacteristic
283C22×C4C2×C22⋊D5911yescharacteristic
284C2×C8—1182no—
285C2×C8—2361no—
286C2×C8—2361no—
287C2×C8—2361no—
288C2×C8—2361no—
289C2×C8—2361no—
290C2×C8—2361no—
291C2×C8—2361no—
292C2×C8—2361no—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—81no—
305C4⋊C4—1182no—
306C4⋊C4—1182no—
307C4⋊C4—2361no—
308C4⋊C4—2361no—
309C22⋊C4—42no—
310C22⋊C4—42no—
311C22⋊C4—42no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C42—41no—
317C23—22no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23—41no—
324C23#1427369611yescharacteristic
325Q8—4721no—
326Q8—4721no—
327Q8—2362no—
328Q8—2362no—
329D4—42no—
330D4—81no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
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—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—84no—
353C2×C4—4721no—
354C2×C4—4721no—
355C2×C4—2362no—
356C2×C4C2×#1174567011yescharacteristic
357C2×C4#1427374711yescharacteristic
358C2×C4#1427374711yescharacteristic
359C2×C4#1427375211yescharacteristic
360C2×C4#1427375311yescharacteristic
361C2×C4#1427375711yescharacteristic
362C8—4721no—
363C8—4721no—
364C8—2362no—
365C22—42no—
366C22—42no—
367C22—42no—
368C22—81no—
369C22—81no—
370C22—81no—
371C22—81no—
372C22#1683754711yescharacteristic
373C22#1683755311yescharacteristic
374C22#1683756411yescharacteristic
375C22#1683757511yescharacteristic
376C22#1683758011yescharacteristic
377C22#1683758511yescharacteristic
378C22#1683758911yescharacteristic
379C4—21no—
380C4—21no—
381C4—21no—
382C4—21no—
383C4—21no—
384C4—21no—
385C4—81no—
386C4—81no—
387C4—81no—
388C4—81no—
389C4—81no—
390C4—4722no—
391C2—82no—
392C2#2142302511yescharacteristic
393C2#2142304811yescharacteristic
394C2#2142305911yescharacteristic
395C2#2142324711yescharacteristic
396C2#2142328211yescharacteristic
397C2#2142338011yescharacteristic
398C2#2142343011yescharacteristic
subgroupG/H
centerC23#14273696
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
FittingC59×#2724C2
socleC23×C59#120
prime radicalC22×D4×C59D4
hypercenter#2724D59
hypocenterC59#7770
LCSG ▷ C2×C42×C59 ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2724

Complements

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

NH ≅ G/Ntype
1. #21425429C2semidirect
2. #21425342C2semidirect
4. #21421919C2semidirect
5. #21421919C2semidirect
226. C59#7770semidirect

Maximal subgroups

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

subgroupindex
#214254292
#214253422
#214252322
#214219192
#214219172
C59×#27242
#777059