Finite Group Databasebeta

← #28486544

Subgroups — #28486544

Order 15104.

542 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 135 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#21433177C211yesmaximal, characteristic, complemented
2#21424092C211yesmaximal, characteristic, complemented
3#21422521C212yesmaximal
4#21421954C212yesmaximal, complemented
5C59×#2536C211yesmaximal, characteristic
6Q8×C59⋊C8—24no—
7#16837455—22no—
8#16837455C2212yes—
9#16837426C2212yes—
10C2×#14273646—21no—
11#16837171—21no—
12D4⋊(C59⋊C8)—24no—
13C2×C22⋊Q8×C59—21no—
14C2×C4×Q8×C59C2211yescharacteristic
15C2×C4×D4×C59C2211yescharacteristic
16C59×#398C412yes—
17C59×#394C412yes—
18C59×#388—22no—
19C59×#382C2211yescharacteristic
20C59×#380—21no—
21C2×#11745664—22no—
22C4⋊(C59⋊C8)—24no—
23C4⋊(C59⋊C8)—22no—
24C4⋊(C59⋊C8)—22no—
25C4⋊(C59⋊C8)—22no—
26#14273646—42no—
27C22⋊(C59⋊C8)—22no—
28C4×C59⋊C8—24no—
29C4×C59⋊C8D414yes—
30C22×Q8×C59D411yescharacteristic
31C22×D4×C59C2×C411yescharacteristic
32C23×C4×C59—21no—
33C22⋊Q8×C59—42no—
34C22⋊Q8×C59—42no—
35C4×Q8×C59—22no—
36C4×Q8×C59—22no—
37C4×D4×C59—22no—
38C4×D4×C59—22no—
39C2×C4⋊C4×C59—22no—
40C2×C4⋊C4×C59—21no—
41C2×C4⋊C4×C59C2×C411yescharacteristic
42C2×C4⋊C4×C59C2×C412yes—
43C2×C4⋊C4×C59D411yescharacteristic
44C2×C22⋊C4×C59—22no—
45C2×C22⋊C4×C59—22no—
46C2×C22⋊C4×C59—21no—
47C2×C42×C59D412yes—
48C2×C42×C59C2311yescharacteristic
49#95×C59—22no—
50#95×C59—21no—
51#95×C59C2×C412yes—
52C24×C59—21no—
53C2×Q8×C59—22no—
54C2×Q8×C59—22no—
55C2×Q8×C59D812yes—
56C2×Q8×C59SD1612yes—
57C2×D4×C59—22no—
58C2×D4×C59—42no—
59C22×C4×C59—22no—
60C22×C4×C59—21no—
61C22×C4×C59—42no—
62C22×C4×C59—41no—
63C22×C4×C59—41no—
64C22×C4×C59—42no—
65C22×C4×C59C22⋊C412yes—
66C22×C4×C59C22⋊C412yes—
67C22×C4×C59C22×C411yescharacteristic
68C22×C4×C59C2×D411yescharacteristic
69C22×C4×C59C2×D411yescharacteristic
70C4⋊C4×C59—24no—
71C4⋊C4×C59—22no—
72C4⋊C4×C59—22no—
73C4⋊C4×C59—44no—
74C4⋊C4×C59—41no—
75C4⋊C4×C59—41no—
76C22⋊C4×C59—44no—
77C22⋊C4×C59—44no—
78C22⋊C4×C59—41no—
79C22⋊C4×C59—41no—
80C42×C59—22no—
81C42×C59—22no—
82C42×C59C2×D412yes—
83C42×C59#4112yes—
84#11745664—42no—
85#11745664—42no—
86C2×C59⋊C8—24no—
87C2×C59⋊C8—42no—
88C23×C59—22no—
89C23×C59—41no—
90C23×C59—41no—
91C23×C59—41no—
92C23×C59—41no—
93C23×C59—41no—
94C23×C59—41no—
95C23×C59C2×C22⋊C411yescharacteristic
96Q8×C59—24no—
97Q8×C59—24no—
98D4×C59—24no—
99D4×C59—42no—
100C2×C4×C59—22no—
101C2×C4×C59—22no—
102C2×C4×C59—22no—
103C2×C4×C59—41no—
104C2×C4×C59—41no—
105C2×C4×C59—42no—
106C2×C4×C59—42no—
107C2×C4×C59—42no—
108C2×C4×C59—41no—
109C2×C4×C59—42no—
110C2×C4×C59—41no—
111C2×C4×C59—41no—
112C2×C4×C59—82no—
113C2×C4×C59—82no—
114C2×C4×C59#10214yes—
115C2×C4×C59C4×D412yes—
116C2×C4×C59#12011yescharacteristic
117C2×C4×C59#12112yes—
118C2×C4×C59#12311yescharacteristic
119C2×C4×C59C2×D811yescharacteristic
120C2×C4×C59C2×SD1611yescharacteristic
121C2×C4×C59#13512yes—
122C59⋊C8—44no—
123C59⋊C8—82no—
124#9486—591nomaximal, Sylow
125C22×C59—21no—
126C22×C59—21no—
127C22×C59—42no—
128C22×C59—42no—
129C22×C59—42no—
130C22×C59—81no—
131C22×C59—81no—
132C22×C59—81no—
133C22×C59—81no—
134C22×C59#38611yescharacteristic
135C22×C59C2×#10211yescharacteristic
136C22×C59#41611yescharacteristic
137C4×C59—22no—
138C4×C59—24no—
139C4×C59—21no—
140C4×C59—21no—
141C4×C59—42no—
142C4×C59—82no—
143C4×C59—81no—
144C4×C59#46512yes—
145C4×C59#46612yes—
146#2982—591no—
147#2536D5911yescharacteristic
148#1801—591no—
149#1486—592no—
150#1286—592no—
151C2×C59—21no—
152C2×C59—21no—
153C2×C59—82no—
154C2×C59#189111yescharacteristic
155C2×C59#189311yescharacteristic
156C2×C59#191211yescharacteristic
157C2×C22⋊Q8—21no—
158C2×C4×Q8C2×D5911yescharacteristic
159C2×C4×D4C2×D5911yescharacteristic
160C8×Q8—1184no—
161#431—592no—
162#431—1182no—
163#423—592no—
164C2×#101—1181no—
165#398Dic5912yes—
166#394Dic5912yes—
167#388—22no—
168#382C2×D5911yescharacteristic
169#380—21no—
170#343—1181no—
171D4⋊C8—1184no—
172C59#948611yescharacteristic, complemented, Sylow
173C22×Q8C22⋊D5911yescharacteristic
174C22×D4C2×Dic5911yescharacteristic
175C23×C4—21no—
176C2×M16—1182no—
177C22⋊Q8—42no—
178C22⋊Q8—42no—
179C4×Q8—22no—
180C4×Q8—22no—
181C4×D4—22no—
182C4×D4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4C2×Dic5911yescharacteristic
186C2×C4⋊C4C2×Dic5912yes—
187C2×C4⋊C4C22⋊D5911yescharacteristic
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—21no—
191C2×C42C22⋊D5912yes—
192C2×C42C22×D5911yescharacteristic
193C4⋊C8—1184no—
194C4⋊C8—1182no—
195C4⋊C8—1182no—
196C4⋊C8—1182no—
197#101—2362no—
198C22⋊C8—1182no—
199C4×C8—594no—
200C4×C8—1184no—
201#95—22no—
202#95—21no—
203#95C2×Dic5912yes—
204C24—21no—
205C2×Q8—22no—
206C2×Q8—22no—
207C2×Q8#1174566912yes—
208C2×Q8#1174567012yes—
209C2×D4—22no—
210C2×D4—42no—
211C22×C4—22no—
212C22×C4—21no—
213C22×C4—42no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—42no—
217C22×C4C22⋊Dic5912yes—
218C22×C4C22⋊Dic5912yes—
219C22×C4C22×Dic5911yescharacteristic
220C22×C4C2×C22⋊D5911yescharacteristic
221C22×C4C2×C22⋊D5911yescharacteristic
222M16—2362no—
223M16—2362no—
224C2×C8—1184no—
225C2×C8—2362no—
226C4⋊C4—24no—
227C4⋊C4—22no—
228C4⋊C4—22no—
229C4⋊C4—44no—
230C4⋊C4—41no—
231C4⋊C4—41no—
232C22⋊C4—44no—
233C22⋊C4—44no—
234C22⋊C4—41no—
235C22⋊C4—41no—
236C42—22no—
237C42—22no—
238C42D4×D5912yes—
239C42#1174568712yes—
240C23—22no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23C2×C22⋊Dic5911yescharacteristic
248Q8—24no—
249Q8—24no—
250D4—24no—
251D4—42no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—82no—
265C2×C4—82no—
266C2×C4#1427364814yes—
267C2×C4D4×Dic5912yes—
268C2×C4#1427369511yescharacteristic
269C2×C4#1427369912yes—
270C2×C4#1427370811yescharacteristic
271C2×C4C2×#1174566911yescharacteristic
272C2×C4C2×#1174567011yescharacteristic
273C2×C4#1427374712yes—
274C8—4722no—
275C8—2364no—
276C22—21no—
277C22—21no—
278C22—42no—
279C22—42no—
280C22—42no—
281C22—81no—
282C22—81no—
283C22—81no—
284C22—81no—
285C22#1683728211yescharacteristic
286C22C2×#1427364811yescharacteristic
287C22#1683739811yescharacteristic
288C4—22no—
289C4—24no—
290C4—21no—
291C4—21no—
292C4—42no—
293C4—82no—
294C4—81no—
295C4#1683762912yes—
296C4#1683763612yes—
297C2—21no—
298C2—21no—
299C2—82no—
300C2#2142451511yescharacteristic
301C2#2142452511yescharacteristic
302C2#2142463411yescharacteristic
subgroupG/H
centerC22#16837282
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#2536C2
socleC22×C59#386
prime radicalC22×D4×C59C2×C4
cosocleC2×C42×C59C23
hypercenter#2536D59
hypocenterC59#9486
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2536

Complements

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

NH ≅ G/Ntype
1. #21433177C2semidirect
2. #21424092C2semidirect
4. #21421954C2semidirect
172. C59#9486semidirect

Maximal subgroups

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

subgroupindex
#214331772
#214240922
#214225212
#214219542
C59×#25362
#948659