Finite Group Databasebeta

← #28501606

Subgroups — #28501606

Order 15104.

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

normal subgroups only

This group has 382 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#21425446C211yesmaximal, characteristic, complemented
2#21423137C211yesmaximal, characteristic, complemented
3#21423035C211yesmaximal, characteristic
4#21422123C211yesmaximal, characteristic, complemented
5#21422099C211yesmaximal, characteristic
6#21422040C211yesmaximal, characteristic, complemented
7C59×#2782C211yesmaximal, characteristic
8#16838033C2211yescharacteristic
9#16837737—21no—
10#16837553—21no—
11#16837442—21no—
12#16837171C2211yescharacteristic
13#16837162—21no—
14#16837138—21no—
15#16837138—21no—
16#16837128C2211yescharacteristic
17#16837128C2211yescharacteristic
18C59×#400—21no—
19C59×#400C2211yescharacteristic
20C59×#397—21no—
21C59×#394C2211yescharacteristic
22C59×#388—21no—
23C59×#386—21no—
24C59×#385—21no—
25C59×#385—21no—
26C59×#382C2211yescharacteristic
27C2×#11745670—41no—
28C2×#11745664—21no—
29#14273726—41no—
30#14273718—41no—
31#14273706—21no—
32#14273706—21no—
33#14273686—21no—
34#14273660—41no—
35#14273660—41no—
36#14273658—41no—
37#14273650—41no—
38#14273648—41no—
39C22⋊(C59⋊C8)—21no—
40C22⋊(C59⋊C8)—21no—
41C22×D4×C59D411yescharacteristic
42C23×C4×C59—21no—
43C2×C4⋊C4×C59—21no—
44C2×C4⋊C4×C59—21no—
45C2×C4⋊C4×C59D411yescharacteristic
46C2×C4⋊C4×C59D411yescharacteristic
47C2×C4⋊C4×C59C2311yescharacteristic
48C2×C22⋊C4×C59—21no—
49C2×C22⋊C4×C59—21no—
50C2×C22⋊C4×C59—21no—
51C2×C22⋊C4×C59—21no—
52C2×C22⋊C4×C59—21no—
53C2×C42×C59D411yescharacteristic
54#95×C59—21no—
55#95×C59—21no—
56#95×C59—21no—
57#95×C59—21no—
58#95×C59—21no—
59#95×C59—21no—
60#95×C59D411yescharacteristic
61#95×C59D411yescharacteristic
62C24×C59—21no—
63C2×D4×C59—41no—
64C2×D4×C59—42no—
65C22×C4×C59—21no—
66C22×C4×C59—21no—
67C22×C4×C59—21no—
68C22×C4×C59—21no—
69C22×C4×C59—21no—
70C22×C4×C59—81no—
71C22×C4×C59—81no—
72C22×C4×C59—81no—
73C22×C4×C59C2×D411yescharacteristic
74C22×C4×C59C2×D411yescharacteristic
75C22×C4×C59C2×D411yescharacteristic
76C2×Dic118—21no—
77C2×Dic118—21no—
78C4⋊C4×C59—41no—
79C4⋊C4×C59—41no—
80C4⋊C4×C59—41no—
81C4⋊C4×C59—81no—
82C4⋊C4×C59—81no—
83C22⋊C4×C59—42no—
84C22⋊C4×C59—42no—
85C22⋊C4×C59—41no—
86C22⋊C4×C59—41no—
87C22⋊C4×C59—81no—
88C22⋊C4×C59—81no—
89C42×C59—21no—
90C42×C59—21no—
91C22⋊Dic59—41no—
92#11745670—82no—
93C4⋊Dic59—21no—
94C4⋊Dic59—21no—
95C4⋊Dic59—21no—
96C4⋊Dic59—21no—
97Dic59⋊C4—41no—
98Dic59⋊C4—41no—
99C4×Dic59—41no—
100#11745664—22no—
101#11745664—41no—
102C2×C59⋊C8—41no—
103C2×C59⋊C8—41no—
104C2×C59⋊C8—41no—
105C23×C59—22no—
106C23×C59—41no—
107C23×C59—41no—
108C23×C59—41no—
109C23×C59—41no—
110C23×C59—41no—
111C23×C59—41no—
112C23×C59#12011yescharacteristic
113D4×C59—42no—
114D4×C59—81no—
115C2×C4×C59—21no—
116C2×C4×C59—21no—
117C2×C4×C59—21no—
118C2×C4×C59—21no—
119C2×C4×C59—41no—
120C2×C4×C59—41no—
121C2×C4×C59—41no—
122C2×C4×C59—41no—
123C2×C4×C59—41no—
124C2×C4×C59—41no—
125C2×C4×C59—41no—
126C2×C4×C59—41no—
127C2×C4×C59—41no—
128C2×C4×C59—41no—
129C2×C4×C59—41no—
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—84no—
137C2×C4×C59#13511yescharacteristic
138C2×C4×C59#13711yescharacteristic
139C2×Dic59—41no—
140C2×Dic59—41no—
141Dic118—81no—
142Dic118—81no—
143C59⋊C8—42no—
144C59⋊C8—81no—
145C59⋊C8—81no—
146#12235—591nomaximal, Sylow
147C22×C59—21no—
148C22×C59—21no—
149C22×C59—42no—
150C22×C59—42no—
151C22×C59—42no—
152C22×C59—81no—
153C22×C59—81no—
154C22×C59—81no—
155C22×C59—81no—
156C22×C59#45211yescharacteristic
157C22×C59#45611yescharacteristic
158C22×C59B(4,𝔽2)11yescharacteristic
159C4×C59—21no—
160C4×C59—21no—
161C4×C59—41no—
162C4×C59—41no—
163C4×C59—81no—
164C4×C59—81no—
165C4×C59—81no—
166C4×C59—81no—
167C4×C59—81no—
168Dic59—81no—
169Dic59—81no—
170#2782D5911yescharacteristic
171#2061—591no—
172#1617—591no—
173#1599—591no—
174#1353—591no—
175#1345—591no—
176#1324—591no—
177C2×C59—21no—
178C2×C59—21no—
179C2×C59—82no—
180C2×C59#160311yescharacteristic
181C2×C59#219711yescharacteristic
182C2×C59#220211yescharacteristic
183#544—591no—
184#484—1181no—
185#450—1181no—
186#428—1181no—
187#400—21no—
188#400C2×D5911yescharacteristic
189#397—21no—
190#394C2×D5911yescharacteristic
191#388—21no—
192#386—21no—
193#385—21no—
194#385—21no—
195#382C2×D5911yescharacteristic
196#343—591no—
197#339—1181no—
198#328—1181no—
199#328—1181no—
200#324—591no—
201#324—591no—
202C59#1223511yescharacteristic, complemented, Sylow
203C22×D4C22⋊D5911yescharacteristic
204C23×C4—21no—
205C2×SD16—2361no—
206C2×M16—1181no—
207C4⋊Q8—2361no—
208#125—2361no—
209C22⋊Q8—1181no—
210C22⋊Q8—1181no—
211#117—1181no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4C22⋊D5911yescharacteristic
215C2×C4⋊C4C22⋊D5911yescharacteristic
216C2×C4⋊C4C22×D5911yescharacteristic
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C42C22⋊D5911yescharacteristic
223#107—2361no—
224#107—2361no—
225#106—2361no—
226#103—2361no—
227#102—2361no—
228C22⋊C8—1181no—
229C22⋊C8—1181no—
230#95—21no—
231#95—21no—
232#95—21no—
233#95—21no—
234#95—21no—
235#95—21no—
236#95C22⋊D5911yescharacteristic
237#95C22⋊D5911yescharacteristic
238C24—21no—
239C2×Q8—1181no—
240C2×Q8—1181no—
241C2×D4—41no—
242C2×D4—42no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—81no—
249C22×C4—81no—
250C22×C4—81no—
251C22×C4C2×C22⋊D5911yescharacteristic
252C22×C4C2×C22⋊D5911yescharacteristic
253C22×C4C2×C22⋊D5911yescharacteristic
254SD16—4722no—
255M16—1182no—
256M16—2361no—
257C2×C8—2361no—
258C2×C8—2361no—
259C2×C8—2361no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—81no—
264C4⋊C4—81no—
265C4⋊C4—1181no—
266C4⋊C4—1181no—
267C4⋊C4—1181no—
268C4⋊C4—1181no—
269C4⋊C4—2361no—
270C4⋊C4—2361no—
271C22⋊C4—42no—
272C22⋊C4—42no—
273C22⋊C4—41no—
274C22⋊C4—41no—
275C22⋊C4—81no—
276C22⋊C4—81no—
277C22⋊C4—2361no—
278C42—21no—
279C42—21no—
280C42—2361no—
281C23—22no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23#1427369611yescharacteristic
289Q8—4721no—
290Q8—4721no—
291D4—42no—
292D4—81no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—84no—
315C2×C4—2361no—
316C2×C4—2361no—
317C2×C4#1427374711yescharacteristic
318C2×C4#1427375711yescharacteristic
319C8—4721no—
320C8—4721no—
321C8—2362no—
322C22—21no—
323C22—21no—
324C22—42no—
325C22—42no—
326C22—42no—
327C22—81no—
328C22—81no—
329C22—81no—
330C22—81no—
331C22#1683756411yescharacteristic
332C22#1683758511yescharacteristic
333C22#1683758911yescharacteristic
334C4—21no—
335C4—21no—
336C4—41no—
337C4—41no—
338C4—81no—
339C4—81no—
340C4—81no—
341C4—81no—
342C4—81no—
343C4—4721no—
344C4—4721no—
345C2—21no—
346C2—21no—
347C2—82no—
348C2#2142305911yescharacteristic
349C2#2142608611yescharacteristic
350C2#2142611311yescharacteristic
subgroupG/H
centerC22#16837589
commutator, cosocleC2×C4⋊C4×C59C23
FrattiniC2×C4⋊C4C22×D59
FittingC59×#2782C2
socleC22×C59B(4,𝔽2)
prime radicalC22×D4×C59D4
hypercenter#2782D59
hypocenterC59#12235
LCSG ▷ C2×C4⋊C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2782

Complements

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

NH ≅ G/Ntype
1. #21425446C2semidirect
2. #21423137C2semidirect
4. #21422123C2semidirect
6. #21422040C2semidirect
202. C59#12235semidirect

Maximal subgroups

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

subgroupindex
#214254462
#214231372
#214230352
#214221232
#214220992
#214220402
C59×#27822
#1223559