Finite Group Databasebeta

← #28492329

Subgroups — #28492329

Order 15104.

376 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 101 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#21424092C211yesmaximal, characteristic
2#21422870C211yesmaximal, characteristic
3#21422781C211yesmaximal, characteristic
4#21422777C211yesmaximal, characteristic
5#21422491C211yesmaximal, characteristic
6#21421993C211yesmaximal, characteristic
7C2×C59×#475C211yesmaximal, characteristic
8#16837455C2211yescharacteristic
9#16837455C2211yescharacteristic
10#16837426C2211yescharacteristic
11#16837426C2211yescharacteristic
12C2×#14273646—21no—
13#16837171—21no—
14#16837154—22no—
15#16837151—22no—
16#16837148—22no—
17Q8⋊(C59⋊C8)—22no—
18C2×C4⋊Q8×C59C2211yescharacteristic
19C2×C4×Q8×C59C2211yescharacteristic
20C59×#475—22no—
21C59×#475—22no—
22C2×#106×C59—21no—
23C2×C4⋊C8×C59C2211yescharacteristic
24C2×#103×C59—21no—
25C2×#11745664—21no—
26C2×#11745664—21no—
27C4⋊(C59⋊C8)—21no—
28C4⋊(C59⋊C8)—21no—
29C4⋊(C59⋊C8)—21no—
30C4⋊(C59⋊C8)—21no—
31C4⋊(C59⋊C8)—21no—
32C4⋊(C59⋊C8)D412yes—
33#14273646—42no—
34C22⋊(C59⋊C8)—21no—
35C22⋊(C59⋊C8)—21no—
36C4×C59⋊C8—21no—
37C4×C59⋊C8—21no—
38C22×Q8×C59—21no—
39C22×Q8×C59D411yescharacteristic
40C22×C8×C59—21no—
41C4⋊Q8×C59—21no—
42C4⋊Q8×C59—21no—
43C4⋊Q8×C59—21no—
44C4⋊Q8×C59D412yes—
45C4×Q8×C59—21no—
46C4×Q8×C59—21no—
47C4×Q8×C59—21no—
48C4×Q8×C59—21no—
49C2×C4⋊C4×C59—21no—
50C2×C4⋊C4×C59—21no—
51C2×C4⋊C4×C59—21no—
52C2×C4⋊C4×C59D411yescharacteristic
53C2×C42×C59—21no—
54C2×C42×C59C2311yescharacteristic
55#106×C59—24no—
56C4⋊C8×C59—21no—
57C4⋊C8×C59D412yes—
58#103×C59—24no—
59C2×Q8×C59—21no—
60C2×Q8×C59—21no—
61C2×Q8×C59—21no—
62C2×Q8×C59—22no—
63C2×Q8×C59—21no—
64C2×Q8×C59—21no—
65C2×Q8×C59—41no—
66C2×Q8×C59—82no—
67C2×Q8×C59Q1612yes—
68C22×C4×C59—21no—
69C22×C4×C59—21no—
70C22×C4×C59—41no—
71C22×C4×C59C2×D411yescharacteristic
72C22×C4×C59#4111yescharacteristic
73C22×C4×C59#4111yescharacteristic
74C2×C8×C59—22no—
75C2×C8×C59—41no—
76C2×C8×C59—41no—
77C4⋊C4×C59—22no—
78C4⋊C4×C59—21no—
79C4⋊C4×C59—22no—
80C4⋊C4×C59—21no—
81C4⋊C4×C59—41no—
82C4⋊C4×C59—41no—
83C4⋊C4×C59—41no—
84C4⋊C4×C59—41no—
85C42×C59—41no—
86C42×C59—41no—
87C42×C59C2×D411yescharacteristic
88C42×C59C2×D411yescharacteristic
89C42×C59C2×D411yescharacteristic
90C42×C59#4111yescharacteristic
91#11745664—41no—
92#11745664—41no—
93#11745664—41no—
94#11745664—41no—
95C2×C59⋊C8—22no—
96C2×C59⋊C8—41no—
97C2×C59⋊C8—41no—
98C2×C59⋊C8—41no—
99C23×C59#12311yescharacteristic
100Q8×C59—22no—
101Q8×C59—22no—
102Q8×C59—41no—
103Q8×C59—41no—
104Q8×C59—82no—
105Q8×C59—82no—
106C2×C4×C59—22no—
107C2×C4×C59—22no—
108C2×C4×C59—41no—
109C2×C4×C59—41no—
110C2×C4×C59—41no—
111C2×C4×C59—41no—
112C2×C4×C59—42no—
113C2×C4×C59—82no—
114C2×C4×C59—81no—
115C2×C4×C59#12011yescharacteristic
116C2×C4×C59#12111yescharacteristic
117C2×C4×C59#12111yescharacteristic
118C2×C4×C59#12111yescharacteristic
119C2×C4×C59#12311yescharacteristic
120C2×C4×C59#12411yescharacteristic
121C2×C4×C59C2×Q1611yescharacteristic
122C2×C4×C59#13511yescharacteristic
123C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
124C2×C4×C59#13711yescharacteristic
125C8×C59—42no—
126C59⋊C8—42no—
127C59⋊C8—81no—
128C59⋊C8—81no—
129C59⋊C8—81no—
130#10451—591nomaximal, Sylow
131C22×C59—21no—
132C22×C59—21no—
133C22×C59#39411yescharacteristic
134C22×C59#48411yescharacteristic
135C22×C59#48511yescharacteristic
136C4×C59—21no—
137C4×C59—21no—
138C4×C59—21no—
139C4×C59—42no—
140C4×C59—81no—
141C4×C59—82no—
142C4×C59#44911yescharacteristic
143C4×C59#45111yescharacteristic
144C4×C59#46711yescharacteristic
145C4×C59#46811yescharacteristic
146C4×C59#47112yes—
147C2×#475D5911yescharacteristic
148#1801—591no—
149#1571—591no—
150#1549—591no—
151#1548—591no—
152#1479—591no—
153#1301—591no—
154C2×C59—21no—
155C2×C59—21no—
156C2×C59#203611yescharacteristic
157C2×C59#204611yescharacteristic
158C2×C59#204911yescharacteristic
159C2×C4⋊Q8C2×D5911yescharacteristic
160C2×C4×Q8C2×D5911yescharacteristic
161#475—22no—
162#475—22no—
163#431—591no—
164#431—591no—
165C2×#106—21no—
166#423—591no—
167#423—591no—
168C2×C4⋊C8C2×D5911yescharacteristic
169C2×#103—21no—
170C2×#101—1181no—
171#343—1181no—
172#335—1182no—
173#333—1182no—
174#332—1182no—
175Q8⋊C8—1182no—
176C59#1045111yescharacteristic, complemented, Sylow
177C22×Q8—21no—
178C22×Q8C22⋊D5911yescharacteristic
179C2×M16—1181no—
180C2×M16—1181no—
181C22×C8—21no—
182C4⋊Q8—21no—
183C4⋊Q8—21no—
184C4⋊Q8—21no—
185C4⋊Q8C22⋊D5912yes—
186C4×Q8—21no—
187C4×Q8—21no—
188C4×Q8—21no—
189C4×Q8—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4C22⋊D5911yescharacteristic
194C2×C42—21no—
195C2×C42C22×D5911yescharacteristic
196#106—24no—
197C4⋊C8—21no—
198C4⋊C8—592no—
199C4⋊C8—1181no—
200C4⋊C8—1181no—
201C4⋊C8—1181no—
202C4⋊C8—1181no—
203C4⋊C8—1181no—
204C4⋊C8C22⋊D5912yes—
205#103—24no—
206#101—2362no—
207C22⋊C8—1181no—
208C22⋊C8—1181no—
209C4×C8—1181no—
210C4×C8—1181no—
211C2×Q8—21no—
212C2×Q8—21no—
213C2×Q8—21no—
214C2×Q8—22no—
215C2×Q8—21no—
216C2×Q8—21no—
217C2×Q8—41no—
218C2×Q8—82no—
219C2×Q8#1174567212yes—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—41no—
223C22×C4#1174568711yescharacteristic
224C22×C4#1174568711yescharacteristic
225C22×C4C2×C22⋊D5911yescharacteristic
226M16—2361no—
227M16—2361no—
228M16—2361no—
229M16—2361no—
230C2×C8—22no—
231C2×C8—41no—
232C2×C8—41no—
233C2×C8—1182no—
234C2×C8—2361no—
235C2×C8—2361no—
236C2×C8—2361no—
237C4⋊C4—22no—
238C4⋊C4—21no—
239C4⋊C4—22no—
240C4⋊C4—21no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C42—41no—
246C42—41no—
247C42D4×D5911yescharacteristic
248C42#1174568711yescharacteristic
249C42C2×C22⋊D5911yescharacteristic
250C42C2×C22⋊D5911yescharacteristic
251C23#1427370811yescharacteristic
252Q8—22no—
253Q8—22no—
254Q8—41no—
255Q8—41no—
256Q8—82no—
257Q8—82no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—42no—
265C2×C4—82no—
266C2×C4—81no—
267C2×C4#1427369611yescharacteristic
268C2×C4#1427369811yescharacteristic
269C2×C4#1427369911yescharacteristic
270C2×C4#1427369911yescharacteristic
271C2×C4#1427370811yescharacteristic
272C2×C4#1427371411yescharacteristic
273C2×C4C2×#1174567211yescharacteristic
274C2×C4#1427374711yescharacteristic
275C2×C4#1427375211yescharacteristic
276C2×C4#1427375811yescharacteristic
277C8—42no—
278C8—4721no—
279C8—4721no—
280C8—4721no—
281C8—2362no—
282C22—21no—
283C22—21no—
284C22#1683731611yescharacteristic
285C22#1683773511yescharacteristic
286C22#1683774011yescharacteristic
287C4—21no—
288C4—21no—
289C4—21no—
290C4—42no—
291C4—82no—
292C4—81no—
293C4#1683754711yescharacteristic
294C4#1683755811yescharacteristic
295C4#1683764111yescharacteristic
296C4#1683765011yescharacteristic
297C4#1683766212yes—
298C2—21no—
299C2—21no—
300C2#2142529611yescharacteristic
301C2#2142535411yescharacteristic
302C2#2142537311yescharacteristic
subgroupG/H
centerC22#16837316
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
FittingC2×C59×#475C2
socleC22×C59#394
prime radicalC23×C59#123
hypercenterC2×#475D59
hypocenterC59#10451
LCSG ▷ C2×C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#475

Complements

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

NH ≅ G/Ntype
176. C59#10451semidirect

Maximal subgroups

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

subgroupindex
#214240922
#214228702
#214227812
#214227772
#214224912
#214219932
C2×C59×#4752
#1045159