Finite Group Databasebeta

← #28477494

Subgroups — #28477494

Order 15104.

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

normal subgroups only

This group has 504 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#21425429C211yesmaximal, characteristic, complemented
2#21425237C211yesmaximal, characteristic, complemented
3#21425178C211yesmaximal, characteristic
4#21421922C211yesmaximal, characteristic, complemented
5#21421919C211yesmaximal, characteristic, complemented
6#21421917C211yesmaximal, characteristic
7C59×#2732C211yesmaximal, characteristic
8C22×#11745670—21no—
9C2×#14273727C2211yescharacteristic
10C2×#14273658—21no—
11C2×#14273651—21no—
12C2×#14273650—21no—
13#16837164—21no—
14#16837164—21no—
15#16837157C2211yescharacteristic
16#16837157C2211yescharacteristic
17#16837157C2211yescharacteristic
18C2×C22⋊Q8×C59—21no—
19C59×#399—21no—
20C59×#398—21no—
21C59×#396—21no—
22C59×#391C2211yescharacteristic
23C59×#390C2211yescharacteristic
24C59×#389C2211yescharacteristic
25C59×#386—21no—
26C59×#385—21no—
27C22×Dic118—21no—
28C2×#11745670—42no—
29C2×#11745670—42no—
30C2×#11745670—42no—
31C22×C59⋊C8—21no—
32C22×C59⋊C8—21no—
33C22×C59⋊C8—21no—
34#14273727—42no—
35C2×C4⋊Dic59—21no—
36C2×C4⋊Dic59—21no—
37#14273658—42no—
38#14273651—42no—
39#14273650—42no—
40C22×Q8×C59D411yescharacteristic
41C22×D4×C59D411yescharacteristic
42C23×C4×C59—21no—
43C22⋊Q8×C59—42no—
44C22⋊Q8×C59—42no—
45C2×C4⋊C4×C59—21no—
46C2×C4⋊C4×C59—21no—
47C2×C4⋊C4×C59—21no—
48C2×C4⋊C4×C59—21no—
49C2×C4⋊C4×C59D411yescharacteristic
50C2×C4⋊C4×C59D411yescharacteristic
51C2×C4⋊C4×C59D411yescharacteristic
52C2×C4⋊C4×C59D411yescharacteristic
53C2×C22⋊C4×C59—21no—
54C2×C22⋊C4×C59—21no—
55C2×C22⋊C4×C59—21no—
56C2×C22⋊C4×C59—21no—
57C2×C22⋊C4×C59—21no—
58C2×C42×C59C2311yescharacteristic
59#95×C59—21no—
60#95×C59—21no—
61#95×C59—21no—
62#95×C59—21no—
63C24×C59—21no—
64C2×Q8×C59—21no—
65C2×Q8×C59—41no—
66C2×Q8×C59—41no—
67C2×Q8×C59SD1612yescomplemented
68C2×D4×C59—21no—
69C2×D4×C59—41no—
70C2×D4×C59—41no—
71C2×D4×C59SD1612yes—
72C22×C4×C59—21no—
73C22×C4×C59—21no—
74C22×C4×C59—21no—
75C22×C4×C59—21no—
76C22×C4×C59—21no—
77C22×C4×C59—42no—
78C22×C4×C59—81no—
79C22×C4×C59—81no—
80C22×C4×C59C2×D411yescharacteristic
81C22×C4×C59C2×D411yescharacteristic
82C22×C4×C59C2×D411yescharacteristic
83C22×Dic59—41no—
84C2×Dic118—22no—
85C2×Dic118—41no—
86C2×Dic118—41no—
87C2×Dic118—41no—
88C2×Dic118—41no—
89C2×Dic118—41no—
90C4⋊C4×C59—21no—
91C4⋊C4×C59—21no—
92C4⋊C4×C59—42no—
93C4⋊C4×C59—42no—
94C4⋊C4×C59—41no—
95C4⋊C4×C59—41no—
96C4⋊C4×C59—41no—
97C4⋊C4×C59—41no—
98C4⋊C4×C59—41no—
99C4⋊C4×C59—81no—
100C22⋊C4×C59—42no—
101C22⋊C4×C59—41no—
102C22⋊C4×C59—41no—
103C22⋊C4×C59—41no—
104C22⋊C4×C59—41no—
105C22⋊C4×C59—81no—
106C22⋊C4×C59—81no—
107C42×C59—41no—
108#11745670—48no—
109C4⋊Dic59—22no—
110C4⋊Dic59—22no—
111C4⋊Dic59—41no—
112C4⋊Dic59—41no—
113C2×C59⋊C8—22no—
114C2×C59⋊C8—22no—
115C2×C59⋊C8—41no—
116C2×C59⋊C8—41no—
117C2×C59⋊C8—41no—
118C2×C59⋊C8—41no—
119C2×C59⋊C8—41no—
120C2×C59⋊C8—41no—
121C2×C59⋊C8—41no—
122C23×C59—22no—
123C23×C59—41no—
124C23×C59—41no—
125C23×C59—41no—
126C23×C59—41no—
127C23×C59—41no—
128C23×C59—41no—
129C23×C59#12011yescharacteristic
130Q8×C59—42no—
131Q8×C59—81no—
132D4×C59—42no—
133D4×C59—81no—
134C2×C4×C59—21no—
135C2×C4×C59—21no—
136C2×C4×C59—21no—
137C2×C4×C59—21no—
138C2×C4×C59—21no—
139C2×C4×C59—21no—
140C2×C4×C59—22no—
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—41no—
149C2×C4×C59—41no—
150C2×C4×C59—41no—
151C2×C4×C59—41no—
152C2×C4×C59—41no—
153C2×C4×C59—41no—
154C2×C4×C59—41no—
155C2×C4×C59—84no—
156C2×C4×C59C2×SD1611yescharacteristic
157C2×C4×C59C2×SD1611yescharacteristic
158C2×C4×C59#13511yescharacteristic
159C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
160C2×C4×C59#13711yescharacteristic
161C2×C4×C59#13711yescharacteristic
162C2×Dic59—42no—
163C2×Dic59—81no—
164C2×Dic59—81no—
165Dic118—42no—
166Dic118—42no—
167Dic118—81no—
168Dic118—81no—
169C59⋊C8—42no—
170C59⋊C8—42no—
171C59⋊C8—81no—
172#7753—591nomaximal, Sylow
173C22×C59—42no—
174C22×C59—42no—
175C22×C59—42no—
176C22×C59—81no—
177C22×C59—81no—
178C22×C59—81no—
179C22×C59—81no—
180C22×C59#44811yescharacteristic
181C22×C59#45011yescharacteristic
182C22×C59#45211yescharacteristic
183C22×C59#45311yescharacteristic
184C22×C59#45511yescharacteristic
185C22×C59#45611yescharacteristic
186C22×C59B(4,𝔽2)11yescharacteristic
187C4×C59—21no—
188C4×C59—21no—
189C4×C59—21no—
190C4×C59—21no—
191C4×C59—21no—
192C4×C59—21no—
193C4×C59—42no—
194C4×C59—81no—
195C4×C59—81no—
196C4×C59—81no—
197C4×C59—81no—
198Dic59—82no—
199#2732D5911yescharacteristic
200#2058—591no—
201#2024—591no—
202#2013—591no—
203#1272—591no—
204#1271—591no—
205#1270—591no—
206C2×C59—82no—
207C2×C59#159611yescharacteristic
208C2×C59#160111yescharacteristic
209C2×C59#160311yescharacteristic
210C2×C59#163111yescharacteristic
211C2×C59#164311yescharacteristic
212C2×C59#165111yescharacteristic
213C2×C59#166111yescharacteristic
214C22×SD16—1181no—
215C2×C4⋊Q8—591no—
216C2×C22⋊Q8—21no—
217C2×#106—1181no—
218C2×#103—1181no—
219C2×#103—1181no—
220#399—21no—
221#398—21no—
222#396—21no—
223#391C2×D5911yescharacteristic
224#390C2×D5911yescharacteristic
225#389C2×D5911yescharacteristic
226#386—21no—
227#385—21no—
228#340—1181no—
229#340—1181no—
230#336—591no—
231#336—591no—
232#336—591no—
233C59#775311yescharacteristic, complemented, Sylow
234C22×Q8—1181no—
235C22×Q8C22⋊D5911yescharacteristic
236C22×D4C22⋊D5911yescharacteristic
237C23×C4—21no—
238C2×SD16—2362no—
239C2×SD16—2362no—
240C2×SD16—2362no—
241C22×C8—1181no—
242C22×C8—1181no—
243C22×C8—1181no—
244C4⋊Q8—2362no—
245C22⋊Q8—42no—
246C22⋊Q8—42no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—1181no—
252C2×C4⋊C4—1181no—
253C2×C4⋊C4C22⋊D5911yescharacteristic
254C2×C4⋊C4C22⋊D5911yescharacteristic
255C2×C4⋊C4C22⋊D5911yescharacteristic
256C2×C4⋊C4C22⋊D5911yescharacteristic
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—21no—
262C2×C42C22×D5911yescharacteristic
263#106—2362no—
264#103—2362no—
265#103—2362no—
266#95—21no—
267#95—21no—
268#95—21no—
269#95—21no—
270C24—21no—
271C2×Q8—21no—
272C2×Q8—41no—
273C2×Q8—41no—
274C2×Q8—1182no—
275C2×Q8—2361no—
276C2×Q8—2361no—
277C2×Q8—2361no—
278C2×Q8—2361no—
279C2×Q8—2361no—
280C2×Q8#1174567012yescomplemented
281C2×D4—21no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4#1174567112yes—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—42no—
291C22×C4—81no—
292C22×C4—81no—
293C22×C4—2361no—
294C22×C4C2×C22⋊D5911yescharacteristic
295C22×C4C2×C22⋊D5911yescharacteristic
296C22×C4C2×C22⋊D5911yescharacteristic
297SD16—2368no—
298C2×C8—1182no—
299C2×C8—1182no—
300C2×C8—2361no—
301C2×C8—2361no—
302C2×C8—2361no—
303C2×C8—2361no—
304C2×C8—2361no—
305C2×C8—2361no—
306C2×C8—2361no—
307C4⋊C4—21no—
308C4⋊C4—21no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—81no—
317C4⋊C4—1182no—
318C4⋊C4—1182no—
319C4⋊C4—2361no—
320C4⋊C4—2361no—
321C22⋊C4—42no—
322C22⋊C4—41no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C22⋊C4—41no—
326C22⋊C4—81no—
327C22⋊C4—81no—
328C42—41no—
329C23—22no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23#1427369611yescharacteristic
337Q8—42no—
338Q8—81no—
339Q8—4721no—
340Q8—4721no—
341Q8—2362no—
342Q8—2362no—
343D4—42no—
344D4—81no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—22no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—84no—
367C2×C4—4721no—
368C2×C4—4721no—
369C2×C4—2362no—
370C2×C4C2×#1174567111yescharacteristic
371C2×C4C2×#1174567011yescharacteristic
372C2×C4#1427374711yescharacteristic
373C2×C4#1427375311yescharacteristic
374C2×C4#1427375711yescharacteristic
375C2×C4#1427375811yescharacteristic
376C8—4721no—
377C8—2362no—
378C8—2362no—
379C22—42no—
380C22—42no—
381C22—42no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22—81no—
386C22#1683754111yescharacteristic
387C22#1683755311yescharacteristic
388C22#1683756411yescharacteristic
389C22#1683757011yescharacteristic
390C22#1683758011yescharacteristic
391C22#1683758511yescharacteristic
392C22#1683758911yescharacteristic
393C4—21no—
394C4—21no—
395C4—21no—
396C4—21no—
397C4—21no—
398C4—21no—
399C4—42no—
400C4—81no—
401C4—81no—
402C4—81no—
403C4—81no—
404C4—4722no—
405C2—82no—
406C2#2142301911yescharacteristic
407C2#2142304811yescharacteristic
408C2#2142305911yescharacteristic
409C2#2142322311yescharacteristic
410C2#2142330311yescharacteristic
411C2#2142335911yescharacteristic
412C2#2142341311yescharacteristic
subgroupG/H
centerC23#14273696
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
FittingC59×#2732C2
socleC23×C59#120
prime radicalC22×D4×C59D4
hypercenter#2732D59
hypocenterC59#7753
LCSG ▷ C2×C42×C59 ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2732

Complements

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

NH ≅ G/Ntype
1. #21425429C2semidirect
2. #21425237C2semidirect
4. #21421922C2semidirect
5. #21421919C2semidirect
67. C2×Q8×C59SD16semidirect
233. C59#7753semidirect
280. C2×Q8#11745670semidirect

Maximal subgroups

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

subgroupindex
#214254292
#214252372
#214251782
#214219222
#214219192
#214219172
C59×#27322
#775359