Finite Group Databasebeta

← #28501599

Subgroups — #28501599

Order 15104.

448 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 59 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#21425441C211yesmaximal, characteristic, complemented
2#21423069C211yesmaximal, characteristic, complemented
3#21423035C211yesmaximal, characteristic
4#21422126C211yesmaximal, characteristic, complemented
5#21422099C211yesmaximal, characteristic
6#21422042C211yesmaximal, characteristic, complemented
7C59×#2680C211yesmaximal, characteristic
8#16838033C2211yescharacteristic
9#16837558—21no—
10#16837553—21no—
11#16837442—21no—
12#16837171C2211yescharacteristic
13#16837162—21no—
14#16837138—21no—
15#16837138—21no—
16#16837128C2211yescharacteristic
17#16837128C2211yescharacteristic
18C2×#123×C59—21no—
19C2×C22⋊Q8×C59—21no—
20C59×#398—21no—
21C59×#397—21no—
22C59×#395C2211yescharacteristic
23C59×#394C2211yescharacteristic
24C59×#388—21no—
25C59×#385—21no—
26C59×#384C2211yescharacteristic
27C2×#11745672—41no—
28C2×#11745670—41no—
29C2×#11745664—21no—
30#14273726—41no—
31#14273718—41no—
32#14273706—21no—
33#14273706—21no—
34#14273686—21no—
35#14273660—41no—
36#14273658—41no—
37#14273651—41no—
38#14273648—41no—
39C22⋊(C59⋊C8)—21no—
40C22⋊(C59⋊C8)—21no—
41C22×Q8×C59D411yescharacteristic
42C22×D4×C59D411yescharacteristic
43C23×C4×C59—21no—
44#123×C59—44no—
45C22⋊Q8×C59—44no—
46C2×C4⋊C4×C59—21no—
47C2×C4⋊C4×C59—21no—
48C2×C4⋊C4×C59—21no—
49C2×C4⋊C4×C59—21no—
50C2×C4⋊C4×C59—21no—
51C2×C4⋊C4×C59D411yescharacteristic
52C2×C4⋊C4×C59C2311yescharacteristic
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×C59D411yescharacteristic
59#95×C59—21no—
60#95×C59—21no—
61#95×C59—21no—
62#95×C59D411yescharacteristic
63#95×C59D411yescharacteristic
64C24×C59—21no—
65C2×Q8×C59—42no—
66C2×Q8×C59—41no—
67C2×D4×C59—41no—
68C2×D4×C59—42no—
69C22×C4×C59—21no—
70C22×C4×C59—21no—
71C22×C4×C59—21no—
72C22×C4×C59—21no—
73C22×C4×C59—21no—
74C22×C4×C59—42no—
75C22×C4×C59—42no—
76C22×C4×C59—81no—
77C22×C4×C59C2×D411yescharacteristic
78C22×C4×C59C2×D411yescharacteristic
79C22×C4×C59C2×D411yescharacteristic
80C2×Dic118—21no—
81C2×Dic118—21no—
82C4⋊C4×C59—41no—
83C4⋊C4×C59—42no—
84C4⋊C4×C59—42no—
85C4⋊C4×C59—42no—
86C4⋊C4×C59—42no—
87C4⋊C4×C59—42no—
88C4⋊C4×C59—41no—
89C22⋊C4×C59—24no—
90C22⋊C4×C59—42no—
91C22⋊C4×C59—42no—
92C22⋊C4×C59—42no—
93C22⋊C4×C59—41no—
94C22⋊C4×C59—41no—
95C42×C59—21no—
96C42×C59—21no—
97C22⋊Dic59—41no—
98#11745672—82no—
99#11745670—82no—
100C4⋊Dic59—21no—
101C4⋊Dic59—21no—
102C4⋊Dic59—21no—
103C4⋊Dic59—21no—
104Dic59⋊C4—41no—
105Dic59⋊C4—41no—
106C4×Dic59—41no—
107#11745664—22no—
108#11745664—41no—
109C2×C59⋊C8—41no—
110C2×C59⋊C8—41no—
111C2×C59⋊C8—41no—
112C23×C59—22no—
113C23×C59—41no—
114C23×C59—41no—
115C23×C59—41no—
116C23×C59—41no—
117C23×C59—41no—
118C23×C59—41no—
119C23×C59#12011yescharacteristic
120Q8×C59—42no—
121Q8×C59—81no—
122D4×C59—42no—
123D4×C59—81no—
124C2×C4×C59—21no—
125C2×C4×C59—21no—
126C2×C4×C59—22no—
127C2×C4×C59—22no—
128C2×C4×C59—21no—
129C2×C4×C59—21no—
130C2×C4×C59—41no—
131C2×C4×C59—41no—
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—84no—
146C2×C4×C59#13511yescharacteristic
147C2×C4×C59#13711yescharacteristic
148C2×Dic59—41no—
149C2×Dic59—41no—
150Dic118—81no—
151Dic118—81no—
152C59⋊C8—42no—
153C59⋊C8—81no—
154C59⋊C8—81no—
155#12234—591nomaximal, Sylow
156C22×C59—21no—
157C22×C59—21no—
158C22×C59—42no—
159C22×C59—42no—
160C22×C59—42no—
161C22×C59—81no—
162C22×C59—81no—
163C22×C59—81no—
164C22×C59—81no—
165C22×C59#45211yescharacteristic
166C22×C59#45611yescharacteristic
167C22×C59B(4,𝔽2)11yescharacteristic
168C4×C59—21no—
169C4×C59—21no—
170C4×C59—42no—
171C4×C59—41no—
172C4×C59—41no—
173C4×C59—42no—
174C4×C59—81no—
175C4×C59—81no—
176C4×C59—81no—
177Dic59—81no—
178Dic59—81no—
179#2680D5911yescharacteristic
180#2060—591no—
181#1605—591no—
182#1599—591no—
183#1354—591no—
184#1345—591no—
185#1325—591no—
186C2×C59—21no—
187C2×C59—21no—
188C2×C59—82no—
189C2×C59#160311yescharacteristic
190C2×C59#219611yescharacteristic
191C2×C59#220311yescharacteristic
192#544—591no—
193C2×#123—21no—
194C2×C22⋊Q8—21no—
195#451—1181no—
196#450—1181no—
197#428—1181no—
198#398—21no—
199#397—21no—
200#395C2×D5911yescharacteristic
201#394C2×D5911yescharacteristic
202#388—21no—
203#385—21no—
204#384C2×D5911yescharacteristic
205#343—591no—
206#339—1181no—
207#328—1181no—
208#328—1181no—
209#324—591no—
210#324—591no—
211C59#1223411yescharacteristic, complemented, Sylow
212C22×Q8C22⋊D5911yescharacteristic
213C22×D4C22⋊D5911yescharacteristic
214C23×C4—21no—
215C2×Q16—2361no—
216C2×SD16—2361no—
217C2×M16—1181no—
218C4⋊Q8—2361no—
219#125—2361no—
220#123—44no—
221C22⋊Q8—44no—
222C22⋊Q8—1181no—
223C22⋊Q8—1181no—
224#117—1181no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4C22⋊D5911yescharacteristic
231C2×C4⋊C4C22×D5911yescharacteristic
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C42C22⋊D5911yescharacteristic
238#107—2361no—
239#106—2361no—
240#103—2361no—
241#102—2361no—
242C22⋊C8—1181no—
243C22⋊C8—1181no—
244#95—21no—
245#95—21no—
246#95—21no—
247#95C22⋊D5911yescharacteristic
248#95C22⋊D5911yescharacteristic
249C24—21no—
250C2×Q8—42no—
251C2×Q8—41no—
252C2×Q8—1181no—
253C2×Q8—1181no—
254C2×D4—41no—
255C2×D4—42no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—81no—
264C22×C4C2×C22⋊D5911yescharacteristic
265C22×C4C2×C22⋊D5911yescharacteristic
266C22×C4C2×C22⋊D5911yescharacteristic
267Q16—4722no—
268SD16—4722no—
269M16—1182no—
270M16—2361no—
271C2×C8—2361no—
272C2×C8—2361no—
273C2×C8—2361no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—1181no—
282C4⋊C4—1181no—
283C4⋊C4—1181no—
284C4⋊C4—1181no—
285C4⋊C4—2361no—
286C4⋊C4—2361no—
287C22⋊C4—24no—
288C22⋊C4—42no—
289C22⋊C4—42no—
290C22⋊C4—42no—
291C22⋊C4—41no—
292C22⋊C4—41no—
293C22⋊C4—2361no—
294C42—21no—
295C42—21no—
296C42—2361no—
297C23—22no—
298C23—41no—
299C23—41no—
300C23—41no—
301C23—41no—
302C23—41no—
303C23—41no—
304C23#1427369611yescharacteristic
305Q8—42no—
306Q8—81no—
307Q8—4721no—
308Q8—4721no—
309D4—42no—
310D4—81no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—84no—
333C2×C4—2361no—
334C2×C4—2361no—
335C2×C4#1427374711yescharacteristic
336C2×C4#1427375711yescharacteristic
337C8—4721no—
338C8—4721no—
339C8—2362no—
340C22—21no—
341C22—21no—
342C22—42no—
343C22—42no—
344C22—42no—
345C22—81no—
346C22—81no—
347C22—81no—
348C22—81no—
349C22#1683756411yescharacteristic
350C22#1683758511yescharacteristic
351C22#1683758911yescharacteristic
352C4—21no—
353C4—21no—
354C4—42no—
355C4—41no—
356C4—41no—
357C4—42no—
358C4—81no—
359C4—81no—
360C4—81no—
361C4—4721no—
362C4—4721no—
363C2—21no—
364C2—21no—
365C2—82no—
366C2#2142305911yescharacteristic
367C2#2142608011yescharacteristic
368C2#2142611711yescharacteristic
subgroupG/H
centerC22#16837589
commutator, cosocleC2×C4⋊C4×C59C23
FrattiniC2×C4⋊C4C22×D59
FittingC59×#2680C2
socleC22×C59B(4,𝔽2)
prime radicalC22×D4×C59D4
hypercenter#2680D59
hypocenterC59#12234
LCSG ▷ C2×C4⋊C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2680

Complements

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

NH ≅ G/Ntype
1. #21425441C2semidirect
2. #21423069C2semidirect
4. #21422126C2semidirect
6. #21422042C2semidirect
211. C59#12234semidirect

Maximal subgroups

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

subgroupindex
#214254412
#214230692
#214230352
#214221262
#214220992
#214220422
C59×#26802
#1223459