Finite Group Databasebeta

← #28477643

Subgroups — #28477643

Order 15104.

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

normal subgroups only

This group has 446 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×#2727C211yesmaximal, 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×#126×C59—21no—
19C59×#399—21no—
20C59×#396—21no—
21C59×#394—21no—
22C59×#390C2211yescharacteristic
23C59×#389—21no—
24C59×#389C2211yescharacteristic
25C59×#389C2211yescharacteristic
26C59×#388—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
39#126×C59—42no—
40#126×C59—42no—
41C2×C4⋊C4×C59—21no—
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×C22⋊C4×C59—21no—
56C2×C42×C59—21no—
57C2×C42×C59C2311yescharacteristic
58#95×C59—21no—
59#95×C59—21no—
60#95×C59—21no—
61C24×C59—21no—
62C2×D4×C59—21no—
63C2×D4×C59—21no—
64C2×D4×C59—41no—
65C2×D4×C59—41no—
66C22×C4×C59—21no—
67C22×C4×C59—21no—
68C22×C4×C59—21no—
69C22×C4×C59—21no—
70C22×C4×C59—21no—
71C22×C4×C59C2×D411yescharacteristic
72C22×C4×C59C2×D411yescharacteristic
73C22×C4×C59C2×D411yescharacteristic
74C22×Dic59—41no—
75C2×Dic118—22no—
76C2×Dic118—41no—
77C2×Dic118—41no—
78C2×Dic118—41no—
79C2×Dic118—41no—
80C2×Dic118—41no—
81C4⋊C4×C59—21no—
82C4⋊C4×C59—21no—
83C4⋊C4×C59—21no—
84C4⋊C4×C59—42no—
85C4⋊C4×C59—41no—
86C4⋊C4×C59—41no—
87C4⋊C4×C59—41no—
88C4⋊C4×C59—41no—
89C4⋊C4×C59—41no—
90C4⋊C4×C59—41no—
91C4⋊C4×C59—41no—
92C4⋊C4×C59—81no—
93C4⋊C4×C59SD1612yes—
94C22⋊C4×C59—42no—
95C22⋊C4×C59—42no—
96C22⋊C4×C59—41no—
97C22⋊C4×C59—41no—
98C22⋊C4×C59—41no—
99C22⋊C4×C59—41no—
100C22⋊C4×C59—81no—
101C22⋊C4×C59—81no—
102C42×C59—41no—
103C42×C59—42no—
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—41no—
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—22no—
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×C59C2×SD1611yescharacteristic
149C2×C4×C59#13511yescharacteristic
150C2×C4×C59#13511yescharacteristic
151C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
152C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
153C2×C4×C59#13711yescharacteristic
154C2×Dic59—42no—
155C2×Dic59—81no—
156C2×Dic59—81no—
157Dic118—42no—
158Dic118—42no—
159Dic118—81no—
160Dic118—81no—
161C59⋊C8—42no—
162C59⋊C8—81no—
163C59⋊C8—81no—
164#7776—591nomaximal, Sylow
165C22×C59—81no—
166C22×C59—81no—
167C22×C59—81no—
168C22×C59—81no—
169C22×C59—81no—
170C22×C59—81no—
171C22×C59—81no—
172C22×C59#44911yescharacteristic
173C22×C59#45011yescharacteristic
174C22×C59#45211yescharacteristic
175C22×C59#45411yescharacteristic
176C22×C59#45511yescharacteristic
177C22×C59#45611yescharacteristic
178C22×C59B(4,𝔽2)11yescharacteristic
179C4×C59—21no—
180C4×C59—21no—
181C4×C59—21no—
182C4×C59—21no—
183C4×C59—21no—
184C4×C59—21no—
185C4×C59—42no—
186C4×C59—81no—
187C4×C59—81no—
188C4×C59—81no—
189C4×C59—81no—
190Dic59—82no—
191#2727D5911yescharacteristic
192#2058—591no—
193#2043—591no—
194#2023—591no—
195#1271—591no—
196#1271—591no—
197#1270—591no—
198C2×C59—161no—
199C2×C59#159711yescharacteristic
200C2×C59#160411yescharacteristic
201C2×C59#160611yescharacteristic
202C2×C59#163511yescharacteristic
203C2×C59#164211yescharacteristic
204C2×C59#165211yescharacteristic
205C2×C59#166411yescharacteristic
206C2×C4⋊Q8—591no—
207C2×#126—21no—
208C2×#106—1181no—
209C2×#103—1181no—
210C2×#103—1181no—
211C2×#102—1181no—
212#399—21no—
213#396—21no—
214#394—21no—
215#390C2×D5911yescharacteristic
216#389—21no—
217#389C2×D5911yescharacteristic
218#389C2×D5911yescharacteristic
219#388—21no—
220#340—1181no—
221#340—1181no—
222#336—591no—
223#336—591no—
224#336—591no—
225C59#777611yescharacteristic, complemented, Sylow
226C22×Q8—1181no—
227C22×D4C22⋊D5911yescharacteristic
228C22×C8—1181no—
229C22×C8—1181no—
230C22×C8—1181no—
231C4⋊Q8—2362no—
232#126—42no—
233#126—42no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—1181no—
239C2×C4⋊C4—1181no—
240C2×C4⋊C4C22⋊D5911yescharacteristic
241C2×C4⋊C4C22⋊D5911yescharacteristic
242C2×C4⋊C4C22⋊D5911yescharacteristic
243C2×C4⋊C4C22⋊D5911yescharacteristic
244C2×C4⋊C4C22⋊D5911yescharacteristic
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C42—21no—
252C2×C42C22×D5911yescharacteristic
253#106—2362no—
254#103—2362no—
255#103—2362no—
256#102—2362no—
257#95—21no—
258#95—21no—
259#95—21no—
260C24—21no—
261C2×Q8—1182no—
262C2×Q8—2361no—
263C2×Q8—2361no—
264C2×Q8—2361no—
265C2×Q8—2361no—
266C2×Q8—2361no—
267C2×D4—21no—
268C2×D4—21no—
269C2×D4—41no—
270C2×D4—41no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—2361no—
277C22×C4C2×C22⋊D5911yescharacteristic
278C22×C4C2×C22⋊D5911yescharacteristic
279C22×C4C2×C22⋊D5911yescharacteristic
280C2×C8—1182no—
281C2×C8—2361no—
282C2×C8—2361no—
283C2×C8—2361no—
284C2×C8—2361no—
285C2×C8—2361no—
286C2×C8—2361no—
287C2×C8—2361no—
288C2×C8—2361no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—42no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—81no—
301C4⋊C4—1182no—
302C4⋊C4—1182no—
303C4⋊C4—2361no—
304C4⋊C4—2361no—
305C4⋊C4#1174567012yes—
306C22⋊C4—42no—
307C22⋊C4—42no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—81no—
313C22⋊C4—81no—
314C42—41no—
315C42—42no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23#1427369611yescharacteristic
324Q8—4721no—
325Q8—4721no—
326Q8—2362no—
327Q8—2362no—
328D4—42no—
329D4—81no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—22no—
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—4721no—
352C2×C4—4721no—
353C2×C4—2362no—
354C2×C4C2×#1174567011yescharacteristic
355C2×C4#1427374711yescharacteristic
356C2×C4#1427374711yescharacteristic
357C2×C4#1427375211yescharacteristic
358C2×C4#1427375311yescharacteristic
359C2×C4#1427375711yescharacteristic
360C8—4721no—
361C8—4721no—
362C8—2362no—
363C22—81no—
364C22—81no—
365C22—81no—
366C22—81no—
367C22—81no—
368C22—81no—
369C22—81no—
370C22#1683754711yescharacteristic
371C22#1683755311yescharacteristic
372C22#1683756411yescharacteristic
373C22#1683757511yescharacteristic
374C22#1683758011yescharacteristic
375C22#1683758511yescharacteristic
376C22#1683758911yescharacteristic
377C4—21no—
378C4—21no—
379C4—21no—
380C4—21no—
381C4—21no—
382C4—21no—
383C4—42no—
384C4—81no—
385C4—81no—
386C4—81no—
387C4—81no—
388C4—4722no—
389C2—161no—
390C2#2142302511yescharacteristic
391C2#2142306511yescharacteristic
392C2#2142307611yescharacteristic
393C2#2142324711yescharacteristic
394C2#2142329611yescharacteristic
395C2#2142336611yescharacteristic
396C2#2142343011yescharacteristic
subgroupG/H
centerC23#14273696
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
FittingC59×#2727C2
socleC23×C59#120
prime radicalC22×D4×C59D4
hypercenter#2727D59
hypocenterC59#7776
LCSG ▷ C2×C42×C59 ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2727

Complements

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

NH ≅ G/Ntype
1. #21425429C2semidirect
2. #21425342C2semidirect
4. #21421919C2semidirect
5. #21421919C2semidirect
225. C59#7776semidirect

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×#27272
#777659