Finite Group Databasebeta

← #28499281

Subgroups — #28499281

Order 15104.

482 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 87 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#21429095C211yesmaximal, characteristic
2C4×#14273646C211yesmaximal, characteristic, complemented
3#21422100C214yesmaximal, complemented
4C59×#1467C211yesmaximal, characteristic, complemented
5C2×#14273724C411yescharacteristic
6C2×#14273716—21no—
7C2×#14273646—21no—
8C4×#11745664—21no—
9#16837319C412yes—
10#16837319C2212yes—
11#16837304—21no—
12#16837304—21no—
13#16837303—21no—
14C4×C4⋊Dic59C411yescharacteristic
15#16837171C2212yes—
16C2×C4×Q8×C59C2211yescharacteristic
17C59×#432—21no—
18C59×#324—21no—
19C59×#324C2212yes—
20C22×D236—21no—
21C22×D236—21no—
22C22×D236C2×C412yes—
23C2×#11745664—22no—
24#14273724—41no—
25#14273724—41no—
26#14273716—44no—
27C2×C4⋊Dic59—21no—
28C2×(C2×D59)⋊C4—22no—
29C2×(C2×D59)⋊C4—22no—
30C2×(C2×D59)⋊C4—22no—
31C2×(C2×D59)⋊C4—22no—
32C2×C4×Dic59—21no—
33#14273646—24no—
34#14273638—41no—
35C4×C59⋊C8—41no—
36C59⋊#95C2×C412yes—
37C22×Q8×C59D411yescharacteristic
38C4×Q8×C59—41no—
39C4×Q8×C59—41no—
40C2×C4⋊C4×C59C2×C411yescharacteristic
41C2×C4⋊C4×C59D411yescharacteristic
42C2×C4⋊C4×C59C2311yescharacteristic
43C2×C42×C59C2×C411yescharacteristic
44C2×C42×C59D411yescharacteristic
45C2×C42×C59D411yescharacteristic
46C4⋊C8×C59—41no—
47C22⋊C8×C59—22no—
48#97×C59—41no—
49C23×D59—22no—
50C2×Q8×C59—22no—
51C2×Q8×C59—22no—
52C2×Q8×C59—41no—
53C22×C4×C59C22⋊C411yescharacteristic
54C22×C4×C59C22⋊C411yescharacteristic
55C22×C4×C59C22⋊C411yescharacteristic
56C22×C4×C59C22⋊C411yescharacteristic
57C22×C4×C59C22×C411yescharacteristic
58C22×C4×C59C2×D411yescharacteristic
59C22×C4×C59C2×D411yescharacteristic
60C22×Dic59—22no—
61C2×D236—42no—
62C2×D236—42no—
63C2×D236—42no—
64C2×D236—42no—
65C2×D236—42no—
66C2×D236—41no—
67C2×D236—41no—
68C2×D236—42no—
69C2×D236—82no—
70C2×C8×C59—42no—
71C4⋊C4×C59—41no—
72C4⋊C4×C59—41no—
73C4⋊C4×C59—41no—
74C42×C59—21no—
75C42×C59—21no—
76C42×C59—21no—
77C42×C59—21no—
78C42×C59—21no—
79C42×C59—21no—
80(C2×D59)⋊C4—44no—
81(C2×D59)⋊C4—44no—
82(C2×D59)⋊C4—82no—
83(C2×D59)⋊C4—82no—
84C4⋊Dic59—42no—
85C4×Dic59—42no—
86#11745664—24no—
87#11745664—42no—
88C2×C59⋊C8—42no—
89C23×C59C2×C22⋊C411yescharacteristic
90C22×D59—42no—
91C22×D59—42no—
92C22×D59—42no—
93C22×D59—42no—
94C22×D59—42no—
95C22×D59—42no—
96C22×D59—42no—
97Q8×C59—44no—
98C2×C4×C59—21no—
99C2×C4×C59—21no—
100C2×C4×C59—21no—
101C2×C4×C59—21no—
102C2×C4×C59—22no—
103C2×C4×C59—21no—
104C2×C4×C59—21no—
105C2×C4×C59—21no—
106C2×C4×C59—21no—
107C2×C4×C59—21no—
108C2×C4×C59—21no—
109C2×C4×C59—41no—
110C2×C4×C59—41no—
111C2×C4×C59—41no—
112C2×C4×C59#9911yescharacteristic
113C2×C4×C59#9911yescharacteristic
114C2×C4×C59#10412yes—
115C2×C4×C59AGL(1,ℤ/8)12yes—
116C2×Dic59—42no—
117C2×Dic59—42no—
118C2×Dic59—42no—
119D236—82no—
120D236—84no—
121D236—84no—
122D236—82no—
123C8×C59—82no—
124C59⋊C8—44no—
125#11781—591nomaximal, Sylow
126C22×C59—21no—
127C22×C59—21no—
128C22×C59C2×#9911yescharacteristic
129C22×C59#41811yescharacteristic
130C22×C59C2×#10411yescharacteristic
131C2×D59—82no—
132C2×D59—82no—
133C2×D59—82no—
134C2×D59—82no—
135C2×D59—82no—
136C2×D59—82no—
137C2×D59—82no—
138C4×C59—21no—
139C4×C59—22no—
140C4×C59—22no—
141C4×C59—41no—
142C4×C59—41no—
143C4×C59—41no—
144C4×C59—41no—
145C4×C59#35512yes—
146Dic59—82no—
147#2548—591no—
148C4×#101—591no—
149#1467D5911yescharacteristic, complemented
150#1345—594no—
151C2×C59—21no—
152C2×C59—21no—
153C2×C59#150711yescharacteristic
154C2×C59#212211yescharacteristic
155C2×C59C2×#35511yescharacteristic
156D59—162no—
157C2×#127—591no—
158C2×#124—1181no—
159C2×C4×Q8C2×D5911yescharacteristic
160#432—21no—
161C2×#101—1181no—
162C4×M16—1181no—
163#394—592no—
164#394—592no—
165#390—1181no—
166#390—1181no—
167#390—1181no—
168C4×C4⋊C4—591no—
169#343—592no—
170#324—21no—
171#324C2×D5912yes—
172C59#1178111yescharacteristic, complemented, Sylow
173C22×Q8D23611yescharacteristic
174C22×D4—592no—
175C22×D4—1181no—
176C22×D4—1181no—
177C2×M16—1182no—
178#127—2361no—
179#127—2361no—
180#124—2364no—
181C4×Q8—41no—
182C4×Q8—41no—
183C2×C4⋊C4—1181no—
184C2×C4⋊C4C4×D5911yescharacteristic
185C2×C4⋊C4C22⋊D5911yescharacteristic
186C2×C4⋊C4C22×D5911yescharacteristic
187C2×C22⋊C4—1182no—
188C2×C22⋊C4—1182no—
189C2×C22⋊C4—1182no—
190C2×C22⋊C4—1182no—
191C2×C42—1181no—
192C2×C42C4×D5911yescharacteristic
193C2×C42D23611yescharacteristic
194C2×C42C22⋊D5911yescharacteristic
195C4⋊C8—41no—
196#101—1184no—
197C22⋊C8—22no—
198#97—41no—
199#97—2361no—
200C4×C8—2361no—
201#95—592no—
202C24—1182no—
203C2×Q8—22no—
204C2×Q8—22no—
205C2×Q8—41no—
206C2×D4—4722no—
207C2×D4—2362no—
208C2×D4—2362no—
209C2×D4—2362no—
210C2×D4—2362no—
211C2×D4—2362no—
212C2×D4—2361no—
213C2×D4—2361no—
214C2×D4—2362no—
215C22×C4—1182no—
216C22×C4(C2×D59)⋊C411yescharacteristic
217C22×C4(C2×D59)⋊C411yescharacteristic
218C22×C4(C2×D59)⋊C411yescharacteristic
219C22×C4(C2×D59)⋊C411yescharacteristic
220C22×C4C2×C4×D5911yescharacteristic
221C22×C4C2×D23611yescharacteristic
222C22×C4C2×C22⋊D5911yescharacteristic
223M16—1184no—
224M16—2362no—
225C2×C8—42no—
226C2×C8—2362no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—41no—
230C4⋊C4—2362no—
231C22⋊C4—4722no—
232C22⋊C4—4722no—
233C22⋊C4—2364no—
234C22⋊C4—2364no—
235C42—21no—
236C42—21no—
237C42—21no—
238C42—21no—
239C42—21no—
240C42—21no—
241C42—2362no—
242C23—2362no—
243C23—2362no—
244C23—2362no—
245C23—2362no—
246C23—2362no—
247C23—2362no—
248C23—2362no—
249C23C2×(C2×D59)⋊C411yescharacteristic
250Q8—44no—
251D4—4724no—
252D4—4722no—
253D4—4724no—
254D4—4722no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—2362no—
270C2×C4—2362no—
271C2×C4—2362no—
272C2×C4#1427364111yescharacteristic
273C2×C4#1427364111yescharacteristic
274C2×C4#1427365312yes—
275C2×C4#1427375512yes—
276C8—82no—
277C8—2364no—
278C22—21no—
279C22—21no—
280C22—4722no—
281C22—4722no—
282C22—4722no—
283C22—4722no—
284C22—4722no—
285C22—4722no—
286C22—4722no—
287C22C2×#1427364111yescharacteristic
288C22#1683740811yescharacteristic
289C22C2×#1427365311yescharacteristic
290C4—22no—
291C4—22no—
292C4—21no—
293C4—41no—
294C4—41no—
295C4—41no—
296C4—41no—
297C4—4722no—
298C4#1683719812yes—
299C2—21no—
300C2—21no—
301C2—9442no—
302C2#2142261311yescharacteristic
303C2#2142574511yescharacteristic
304C2C2×#1683719811yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC22×C4×C59C22×C4
FrattiniC2×C4⋊C4C22×D59
FittingC59×#1467C2
socleC22×C59C2×#99
prime radicalC2×#14273724C4
cosocleC2×C4⋊C4×C59C23
hypercenter#1467D59
hypocenterC59#11781
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1467

Complements

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

NH ≅ G/Ntype
2. C4×#14273646C2semidirect
3. #21422100C2semidirect
4. C59×#1467C2semidirect
149. #1467D59semidirect
172. C59#11781semidirect

Maximal subgroups

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

subgroupindex
#214290952
C4×#142736462
#214221002
C59×#14672
#1178159