Finite Group Databasebeta

← #28482829

Subgroups — #28482829

Order 15104.

364 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 107 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#21433033C211yesmaximal, characteristic
2#21424820C211yesmaximal, characteristic
3#21422872C211yesmaximal, characteristic
4#21422594C211yesmaximal, characteristic
5#21422500C211yesmaximal, characteristic
6#21421940C211yesmaximal, characteristic
7C59×#1303C211yesmaximal, characteristic
8#16837959C2211yescharacteristic
9#16837531—22no—
10#16837531—22no—
11#16837455C2211yescharacteristic
12#16837430—21no—
13C2×C4⋊(C59⋊C8)—21no—
14C2×C4⋊(C59⋊C8)C2211yescharacteristic
15#16837360—21no—
16C2×#14273638C2211yescharacteristic
17#16837157—21no—
18#16837154—22no—
19#16837138C412yes—
20#16837138C412yes—
21#16837134—22no—
22C59×#431C2211yescharacteristic
23C59×#423C2211yescharacteristic
24C4×C4⋊C4×C59C2211yescharacteristic
25C22×C59⋊C8—21no—
26C22×C59⋊C8D411yescharacteristic
27C22×C59⋊C8D411yescharacteristic
28#14273727Q811yescharacteristic
29#14273727Q811yescharacteristic
30#14273719—21no—
31#14273706—21no—
32#14273706C2×C412yes—
33C4×Dic118—21no—
34C4×Dic118D412yes—
35#14273686—21no—
36C4⋊(C59⋊C8)—21no—
37C4⋊(C59⋊C8)—21no—
38C4⋊(C59⋊C8)—21no—
39C4⋊(C59⋊C8)—42no—
40Dic59⋊C8—24no—
41C22⋊(C59⋊C8)C2×C412yes—
42#14273638—21no—
43#14273638—21no—
44#14273637—22no—
45C4×C59⋊C8—21no—
46C8×Dic59—22no—
47C2×M16×C59—21no—
48C2×C4⋊C4×C59—21no—
49C2×C42×C59—21no—
50C2×C42×C59C2311yescharacteristic
51C4⋊C8×C59—21no—
52C4⋊C8×C59—21no—
53C4⋊C8×C59D412yes—
54C22⋊C8×C59—21no—
55C4×C8×C59—21no—
56#95×C59C2×C411yescharacteristic
57#95×C59C2×C411yescharacteristic
58C22×C4×C59—21no—
59C22×C4×C59—21no—
60C22×C4×C59C22×C411yescharacteristic
61C22×C4×C59C2×D411yescharacteristic
62C22×C4×C59#4111yescharacteristic
63C2×Dic118—21no—
64C2×Dic118C4⋊C412yes—
65M16×C59—41no—
66M16×C59—41no—
67C2×C8×C59—22no—
68C2×C8×C59—41no—
69C4⋊C4×C59—42no—
70C42×C59—41no—
71C42×C59—41no—
72C42×C59C2×D411yescharacteristic
73C42×C59C2×D411yescharacteristic
74C42×C59C2×Q811yescharacteristic
75C42×C59#4111yescharacteristic
76C22⋊Dic59—22no—
77C4⋊Dic59—21no—
78C4⋊Dic59—21no—
79C4⋊Dic59—21no—
80C4⋊Dic59C4⋊C412yes—
81Dic59⋊C4—22no—
82Dic59⋊C4—22no—
83C4×Dic59—22no—
84C2×C59⋊C8—21no—
85C2×C59⋊C8—21no—
86C2×C59⋊C8—22no—
87C2×C59⋊C8—21no—
88C2×C59⋊C8—21no—
89C2×C59⋊C8—21no—
90C2×C59⋊C8—41no—
91C2×C59⋊C8—41no—
92C2×C59⋊C8—41no—
93C2×C59⋊C8Q1612yes—
94C23×C59C4×D411yescharacteristic
95C2×C4×C59—22no—
96C2×C4×C59—41no—
97C2×C4×C59—41no—
98C2×C4×C59—41no—
99C2×C4×C59—41no—
100C2×C4×C59—41no—
101C2×C4×C59—41no—
102C2×C4×C59—41no—
103C2×C4×C59C2×C4⋊C411yescharacteristic
104C2×C4×C59C4×D411yescharacteristic
105C2×C4×C59#12011yescharacteristic
106C2×C4×C59C22⋊Q811yescharacteristic
107C2×C4×C59C22⋊Q811yescharacteristic
108C2×C4×C59#12311yescharacteristic
109C2×C4×C59C2×Q1611yescharacteristic
110C2×C4×C59#13511yescharacteristic
111C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
112C2×C4×C59#13711yescharacteristic
113C2×Dic59—22no—
114C2×Dic59—22no—
115Dic118—42no—
116Dic118—81no—
117C8×C59—42no—
118C8×C59—81no—
119C59⋊C8—22no—
120C59⋊C8—42no—
121C59⋊C8—41no—
122C59⋊C8—81no—
123#8879—591nomaximal, Sylow
124C22×C59—21no—
125C22×C59—21no—
126C22×C59#38511yescharacteristic
127C22×C59C4×Q1611yescharacteristic
128C22×C59#44011yescharacteristic
129C4×C59—21no—
130C4×C59—21no—
131C4×C59—21no—
132C4×C59—21no—
133C4×C59—42no—
134C4×C59—81no—
135C4×C59#44911yescharacteristic
136C4×C59#45111yescharacteristic
137C4×C59#47711yescharacteristic
138C4×C59#47911yescharacteristic
139Dic59—42no—
140Dic59—42no—
141#2961—591no—
142#1940—591no—
143#1571—591no—
144#1503—591no—
145#1481—591no—
146#1303D5911yescharacteristic
147#1281—591no—
148C2×C59—21no—
149C2×C59—21no—
150C2×C59#186111yescharacteristic
151C2×C59#186311yescharacteristic
152C2×C59#186811yescharacteristic
153#533—591no—
154#446—1182no—
155#446—1182no—
156#431—591no—
157#431C2×D5911yescharacteristic
158#424—1181no—
159#423C2×D5911yescharacteristic
160C2×C4⋊C8—591no—
161C2×C4⋊C8—1181no—
162#405—1181no—
163C2×#97—591no—
164C4×C4⋊C4C2×D5911yescharacteristic
165#336—1181no—
166#335—1182no—
167#328—592no—
168#328—592no—
169Q8⋊C8—1182no—
170C59#887911yescharacteristic, complemented, Sylow
171C2×M16—21no—
172C22×C8—591no—
173C22×C8—591no—
174C22×C8—1181no—
175C4⋊Q8—591no—
176C4⋊Q8—591no—
177#125—1181no—
178C22⋊Q8—592no—
179C22⋊Q8—1181no—
180C4×Q8—592no—
181C4×Q8—1181no—
182#117—1181no—
183C2×C4⋊C4—21no—
184C2×C42—21no—
185C2×C42C22×D5911yescharacteristic
186C4⋊C8—21no—
187C4⋊C8—21no—
188C4⋊C8—1184no—
189C4⋊C8—1181no—
190C4⋊C8—1181no—
191C4⋊C8—1181no—
192C4⋊C8—2362no—
193C4⋊C8D23612yes—
194C22⋊C8—21no—
195C22⋊C8—592no—
196#97—1182no—
197#97—1181no—
198#97—1181no—
199C4×C8—21no—
200C4×C8—1182no—
201C4×C8—1181no—
202#95C4×D5911yescharacteristic
203#95C4×D5911yescharacteristic
204C2×Q8—592no—
205C2×Q8—1181no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4C2×C4×D5911yescharacteristic
209C22×C4D4×D5911yescharacteristic
210C22×C4#1174568711yescharacteristic
211M16—41no—
212M16—41no—
213C2×C8—22no—
214C2×C8—41no—
215C2×C8—592no—
216C2×C8—1181no—
217C2×C8—1181no—
218C2×C8—1182no—
219C2×C8—1181no—
220C2×C8—1181no—
221C2×C8—1181no—
222C2×C8—2361no—
223C2×C8—2361no—
224C2×C8—2361no—
225C4⋊C4—42no—
226C4⋊C4—592no—
227C4⋊C4—1182no—
228C4⋊C4—1181no—
229C4⋊C4—1181no—
230C4⋊C4—1181no—
231C4⋊C4—1182no—
232C22⋊C4—1182no—
233C42—41no—
234C42—41no—
235C42—1182no—
236C42C2×D23611yescharacteristic
237C42#1174568511yescharacteristic
238C42D4×D5911yescharacteristic
239C42Q8×D5911yescharacteristic
240C23(C22⋊D59)⋊C411yescharacteristic
241Q8—4721no—
242Q8—2362no—
243C2×C4—22no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—1182no—
252C2×C4—1182no—
253C2×C4C4⋊C4×D5911yescharacteristic
254C2×C4C4×D23611yescharacteristic
255C2×C4#1427369711yescharacteristic
256C2×C4#1427370511yescharacteristic
257C2×C4#1427370711yescharacteristic
258C2×C4#1427370911yescharacteristic
259C2×C4Q16×D5911yescharacteristic
260C2×C4#1427374811yescharacteristic
261C2×C4#1427375111yescharacteristic
262C2×C4#1427375611yescharacteristic
263C8—42no—
264C8—81no—
265C8—1182no—
266C8—4721no—
267C8—2362no—
268C8—2361no—
269C22—21no—
270C22—21no—
271C22#1683727611yescharacteristic
272C22#1683749411yescharacteristic
273C22#1683749911yescharacteristic
274C4—21no—
275C4—21no—
276C4—21no—
277C4—21no—
278C4—42no—
279C4—81no—
280C4—2362no—
281C4—2362no—
282C4#1683754811yescharacteristic
283C4#1683755911yescharacteristic
284C4#1683769611yescharacteristic
285C4#1683771211yescharacteristic
286C2—21no—
287C2—21no—
288C2#2142433711yescharacteristic
289C2#2142435111yescharacteristic
290C2#2142438311yescharacteristic
subgroupG/H
centerC2×C4#14273697
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1303C2
socleC22×C59#385
prime radicalC23×C59C4×D4
cosocleC2×C42×C59C23
hypercenter#1303D59
hypocenterC59#8879
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1303

Complements

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

NH ≅ G/Ntype
170. C59#8879semidirect

Maximal subgroups

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

subgroupindex
#214330332
#214248202
#214228722
#214225942
#214225002
#214219402
C59×#13032
#887959