Finite Group Databasebeta

← #28486543

Subgroups — #28486543

Order 15104.

542 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 131 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#21429001C211yesmaximal, characteristic
2#21424092C211yesmaximal, characteristic, complemented
3#21422522C212yesmaximal
4#21421955C212yesmaximal, complemented
5C59×#2982C211yesmaximal, characteristic, complemented
6C2×#14273703—21no—
7C2×C4×D236C2211yescharacteristic
8#16837426C2212yes—
9C2×#14273646—21no—
10#16837332C412yes—
11#16837319C412yes—
12#16837293—22no—
13#16837268C2211yescharacteristic
14#16837259—21no—
15#16837171—21no—
16#16837131—24no—
17C2×C4×Q8×C59C2211yescharacteristic
18C8×Q8×C59—24no—
19C59×#431—22no—
20C59×#431C2212yes—
21C22×D236C2×C411yescharacteristic
22C22×C4×D59—21no—
23C2×#11745664—22no—
24#14273703—42no—
25#14273703—42no—
26C4×D236—22no—
27C4×D236—22no—
28C2×C4⋊Dic59—21no—
29C2×C4⋊Dic59C2×C411yescharacteristic
30C2×Dic59⋊C4—22no—
31C2×(C2×D59)⋊C4—22no—
32C2×(C2×D59)⋊C4—22no—
33C2×(C2×D59)⋊C4—21no—
34C4⋊(C59⋊C8)—22no—
35C4⋊(C59⋊C8)—22no—
36#14273646—42no—
37C59⋊#95—22no—
38C59⋊#95—21no—
39C59⋊#95C2×C412yes—
40C22×Q8×C59D411yescharacteristic
41C4×Q8×C59—22no—
42C4×Q8×C59—22no—
43C2×C4⋊C4×C59C2×C412yes—
44C2×C4⋊C4×C59D411yescharacteristic
45C2×C42×C59D412yes—
46C2×C42×C59C2311yescharacteristic
47C4⋊C8×C59—24no—
48C4⋊C8×C59—22no—
49C22⋊C8×C59—22no—
50C4×C8×C59—24no—
51C4×C8×C59D414yes—
52C23×D59—21no—
53C2×Q8×C59—22no—
54C2×Q8×C59—22no—
55C2×Q8×C59D812yes—
56C2×Q8×C59SD1612yes—
57C22×C4×C59C22⋊C412yes—
58C22×C4×C59C22⋊C412yes—
59C22×C4×C59C22×C411yescharacteristic
60C22×C4×C59C2×D411yescharacteristic
61C22×C4×C59C2×D411yescharacteristic
62C22×Dic59—22no—
63C22×Dic59—21no—
64C2×D236—22no—
65C2×D236—42no—
66C2×C4×D59—42no—
67C2×C4×D59—41no—
68C2×C4×D59—41no—
69C2×C4×D59—42no—
70C2×C8×C59—24no—
71C4⋊C4×C59—24no—
72C4⋊C4×C59—22no—
73C42×C59—22no—
74C42×C59—22no—
75C42×C59C2×D412yes—
76C42×C59#4112yes—
77(C2×D59)⋊C4—44no—
78(C2×D59)⋊C4—44no—
79(C2×D59)⋊C4—41no—
80(C2×D59)⋊C4—41no—
81C4⋊Dic59—22no—
82C4⋊Dic59—41no—
83C4⋊Dic59—41no—
84Dic59⋊C4—44no—
85#11745664—42no—
86#11745664—42no—
87C2×C59⋊C8—42no—
88C23×C59C2×C22⋊C411yescharacteristic
89C22×D59—22no—
90C22×D59—41no—
91C22×D59—41no—
92C22×D59—41no—
93C22×D59—41no—
94C22×D59—41no—
95C22×D59—41no—
96Q8×C59—24no—
97Q8×C59—24no—
98C2×C4×C59—22no—
99C2×C4×C59—22no—
100C2×C4×C59—22no—
101C2×C4×C59—41no—
102C2×C4×C59—41no—
103C2×C4×C59—42no—
104C2×C4×C59#10214yes—
105C2×C4×C59C4×D412yes—
106C2×C4×C59#12011yescharacteristic
107C2×C4×C59#12112yes—
108C2×C4×C59#12311yescharacteristic
109C2×C4×C59C2×D811yescharacteristic
110C2×C4×C59C2×SD1611yescharacteristic
111C2×C4×C59#13512yes—
112C2×Dic59—42no—
113C2×Dic59—42no—
114C2×Dic59—41no—
115C2×Dic59—41no—
116C2×Dic59—42no—
117C2×Dic59—41no—
118D236—24no—
119D236—42no—
120C4×D59—82no—
121C4×D59—82no—
122C8×C59—44no—
123C59⋊C8—82no—
124#9486—591nomaximal, Sylow
125C22×C59—21no—
126C22×C59—21no—
127C22×C59#38611yescharacteristic
128C22×C59C2×#10211yescharacteristic
129C22×C59#41611yescharacteristic
130C2×D59—42no—
131C2×D59—42no—
132C2×D59—42no—
133C2×D59—81no—
134C2×D59—81no—
135C2×D59—81no—
136C2×D59—81no—
137C4×C59—22no—
138C4×C59—24no—
139C4×C59—21no—
140C4×C59—21no—
141C4×C59—42no—
142C4×C59#46512yes—
143C4×C59#46612yes—
144Dic59—82no—
145Dic59—81no—
146#2982D5911yescharacteristic, complemented
147#2536—591no—
148#1801—591no—
149#1486—592no—
150#1286—592no—
151C2×C59—21no—
152C2×C59—21no—
153C2×C59#189111yescharacteristic
154C2×C59#189311yescharacteristic
155C2×C59#191211yescharacteristic
156D59—82no—
157C2×C22⋊Q8—1181no—
158C2×C4×Q8C2×D5911yescharacteristic
159C2×C4×D4—591no—
160C8×Q8—24no—
161#431—22no—
162#431C2×D5912yes—
163#423—592no—
164C2×#101—1181no—
165#398—592no—
166#394—592no—
167#388—1182no—
168#382—591no—
169#380—1181no—
170#343—1181no—
171D4⋊C8—1184no—
172C59#948611yescharacteristic, complemented, Sylow
173C22×Q8D23611yescharacteristic
174C22×D4—591no—
175C23×C4—1181no—
176C2×M16—1182no—
177C22⋊Q8—2362no—
178C22⋊Q8—2362no—
179C4×Q8—22no—
180C4×Q8—22no—
181C4×D4—1182no—
182C4×D4—1182no—
183C2×C4⋊C4—591no—
184C2×C4⋊C4—1182no—
185C2×C4⋊C4—1181no—
186C2×C4⋊C4C4×D5912yes—
187C2×C4⋊C4D23611yescharacteristic
188C2×C22⋊C4—1182no—
189C2×C22⋊C4—1182no—
190C2×C22⋊C4—1181no—
191C2×C42C22⋊D5912yes—
192C2×C42C22×D5911yescharacteristic
193C4⋊C8—24no—
194C4⋊C8—22no—
195C4⋊C8—1182no—
196C4⋊C8—1182no—
197#101—2362no—
198C22⋊C8—22no—
199C4×C8—24no—
200C4×C8C22⋊D5914yes—
201#95—592no—
202#95—1182no—
203#95—1181no—
204C24—1181no—
205C2×Q8—22no—
206C2×Q8—22no—
207C2×Q8#1174566012yes—
208C2×Q8D47212yes—
209C2×D4—1182no—
210C2×D4—2362no—
211C22×C4—1182no—
212C22×C4—1181no—
213C22×C4—2362no—
214C22×C4—2361no—
215C22×C4—2361no—
216C22×C4—2362no—
217C22×C4(C2×D59)⋊C412yes—
218C22×C4(C2×D59)⋊C412yes—
219C22×C4C2×C4×D5911yescharacteristic
220C22×C4C2×D23611yescharacteristic
221C22×C4C2×C22⋊D5911yescharacteristic
222M16—2362no—
223M16—2362no—
224C2×C8—24no—
225C2×C8—2362no—
226C4⋊C4—24no—
227C4⋊C4—22no—
228C4⋊C4—1182no—
229C4⋊C4—2364no—
230C4⋊C4—2361no—
231C4⋊C4—2361no—
232C22⋊C4—2364no—
233C22⋊C4—2364no—
234C22⋊C4—2361no—
235C22⋊C4—2361no—
236C42—22no—
237C42—22no—
238C42#1174568512yes—
239C42C2×C22⋊D5912yes—
240C23—1182no—
241C23—2361no—
242C23—2361no—
243C23—2361no—
244C23—2361no—
245C23—2361no—
246C23—2361no—
247C23C2×(C2×D59)⋊C411yescharacteristic
248Q8—24no—
249Q8—24no—
250D4—1184no—
251D4—2362no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—4722no—
259C2×C4—4722no—
260C2×C4—2362no—
261C2×C4—2362no—
262C2×C4—2362no—
263C2×C4—2361no—
264C2×C4—2361no—
265C2×C4—2361no—
266C2×C4#1427364914yes—
267C2×C4C4×C22⋊D5912yes—
268C2×C4#1427369611yescharacteristic
269C2×C4#1427370112yes—
270C2×C4#1427371011yescharacteristic
271C2×C4C2×D47211yescharacteristic
272C2×C4C2×#1174566011yescharacteristic
273C2×C4#1427374912yes—
274C8—44no—
275C8—4722no—
276C22—21no—
277C22—21no—
278C22—4721no—
279C22—4721no—
280C22—4721no—
281C22—4721no—
282C22—2362no—
283C22—2362no—
284C22—2362no—
285C22#1683728111yescharacteristic
286C22C2×#1427364911yescharacteristic
287C22#1683739911yescharacteristic
288C4—22no—
289C4—24no—
290C4—21no—
291C4—21no—
292C4—42no—
293C4—4722no—
294C4—4721no—
295C4#1683763112yes—
296C4#1683763812yes—
297C2—21no—
298C2—21no—
299C2—4722no—
300C2#2142451411yescharacteristic
301C2#2142452411yescharacteristic
302C2#2142463311yescharacteristic
subgroupG/H
centerC22#16837281
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#2982C2
socleC22×C59#386
prime radicalC22×D236C2×C4
cosocleC2×C42×C59C23
hypercenter#2982D59
hypocenterC59#9486
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2982

Complements

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

NH ≅ G/Ntype
2. #21424092C2semidirect
4. #21421955C2semidirect
5. C59×#2982C2semidirect
146. #2982D59semidirect
172. C59#9486semidirect

Maximal subgroups

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

subgroupindex
#214290012
#214240922
#214225222
#214219552
C59×#29822
#948659