Finite Group Databasebeta

← #28499389

Subgroups — #28499389

Order 15104.

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

normal subgroups only

This group has 488 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#21424179C211yesmaximal, characteristic, complemented
2#21422430C211yesmaximal, characteristic, complemented
3#21422099C214yesmaximal, complemented
4C59×#2619C211yesmaximal, characteristic
5#16837459—21no—
6#16837366—21no—
7#16837171—21no—
8#16837171C2212yes—
9#16837128C2212yes—
10#16837126—21no—
11C2×#127×C59C411yescharacteristic
12C2×#121×C59—21no—
13C2×#117×C59C2211yescharacteristic
14C59×#394—21no—
15C59×#394C2212yes—
16C59×#392—21no—
17C59×#392C412yes—
18C59×#390—21no—
19C59×#382C411yescharacteristic
20C2×#11745664—22no—
21C4⋊(C59⋊C8)—41no—
22C22⋊(C59⋊C8)—22no—
23C22⋊(C59⋊C8)—41no—
24C22⋊(C59⋊C8)—41no—
25#14273638—41no—
26C22×D4×C59—22no—
27C22×D4×C59—21no—
28C22×D4×C59—21no—
29C22×D4×C59C2×C412yes—
30C23×C4×C59D411yescharacteristic
31#127×C59—41no—
32#127×C59—41no—
33#121×C59—44no—
34#117×C59—41no—
35#117×C59—41no—
36C2×C4⋊C4×C59—21no—
37C2×C4⋊C4×C59D411yescharacteristic
38C2×C4⋊C4×C59C2311yescharacteristic
39C2×C22⋊C4×C59—22no—
40C2×C22⋊C4×C59—22no—
41C2×C22⋊C4×C59—22no—
42C2×C22⋊C4×C59C2×C411yescharacteristic
43C2×C22⋊C4×C59D411yescharacteristic
44C2×C42×C59C2×C411yescharacteristic
45C2×C42×C59D411yescharacteristic
46#95×C59—21no—
47#95×C59C2×C412yes—
48C24×C59—22no—
49C24×C59C22⋊C411yescharacteristic
50C2×D4×C59—42no—
51C2×D4×C59—42no—
52C2×D4×C59—42no—
53C2×D4×C59—42no—
54C2×D4×C59—44no—
55C2×D4×C59—41no—
56C2×D4×C59—41no—
57C2×D4×C59—82no—
58C2×D4×C59—82no—
59C2×D4×C59—82no—
60C2×D4×C59—82no—
61C22×C4×C59—21no—
62C22×C4×C59—21no—
63C22×C4×C59—22no—
64C22×C4×C59—22no—
65C22×C4×C59—41no—
66C22×C4×C59C22⋊C411yescharacteristic
67C22×C4×C59C22⋊C411yescharacteristic
68C22×C4×C59C22⋊C411yescharacteristic
69C22×C4×C59C22×C411yescharacteristic
70C22×C4×C59C2×D411yescharacteristic
71C22×C4×C59C2×D411yescharacteristic
72C4⋊C4×C59—41no—
73C4⋊C4×C59—41no—
74C4⋊C4×C59—42no—
75C22⋊C4×C59—41no—
76C22⋊C4×C59—41no—
77C22⋊C4×C59—44no—
78C22⋊C4×C59—82no—
79C22⋊C4×C59—82no—
80C42×C59—21no—
81C42×C59—21no—
82C42×C59—21no—
83C42×C59—21no—
84#11745664—42no—
85#11745664—42no—
86C2×C59⋊C8—42no—
87C2×C59⋊C8—42no—
88C23×C59—21no—
89C23×C59—21no—
90C23×C59—21no—
91C23×C59—41no—
92C23×C59—41no—
93C23×C59—42no—
94C23×C59—42no—
95C23×C59—42no—
96C23×C59—42no—
97C23×C59—42no—
98C23×C59—42no—
99C23×C59—42no—
100C23×C59C2×C22⋊C411yescharacteristic
101D4×C59—82no—
102D4×C59—82no—
103D4×C59—84no—
104D4×C59—84no—
105D4×C59—88no—
106C2×C4×C59—21no—
107C2×C4×C59—21no—
108C2×C4×C59—21no—
109C2×C4×C59—21no—
110C2×C4×C59—21no—
111C2×C4×C59—21no—
112C2×C4×C59—21no—
113C2×C4×C59—21no—
114C2×C4×C59—42no—
115C2×C4×C59—42no—
116C2×C4×C59—41no—
117C2×C4×C59—41no—
118C2×C4×C59—41no—
119C2×C4×C59—41no—
120C2×C4×C59—42no—
121C2×C4×C59—42no—
122C2×C4×C59—42no—
123C2×C4×C59#9911yescharacteristic
124C2×C4×C59#9911yescharacteristic
125C2×C4×C59AGL(1,ℤ/8)12yes—
126C59⋊C8—82no—
127C59⋊C8—82no—
128#11806—591nomaximal, Sylow
129C22×C59—21no—
130C22×C59—21no—
131C22×C59—42no—
132C22×C59—41no—
133C22×C59—41no—
134C22×C59—41no—
135C22×C59—81no—
136C22×C59—82no—
137C22×C59—82no—
138C22×C59—82no—
139C22×C59—82no—
140C22×C59—82no—
141C22×C59—82no—
142C22×C59—82no—
143C22×C59C2×#9911yescharacteristic
144C22×C59#41811yescharacteristic
145C22×C59#42111yescharacteristic
146C4×C59—22no—
147C4×C59—21no—
148C4×C59—21no—
149C4×C59—41no—
150C4×C59—41no—
151C4×C59—41no—
152C4×C59—41no—
153C4×C59—82no—
154#2619D5911yescharacteristic
155#1822—591no—
156#1464—591no—
157#1345—594no—
158C2×C59—21no—
159C2×C59—21no—
160C2×C59—81no—
161C2×C59—162no—
162C2×C59#151011yescharacteristic
163C2×C59#212211yescharacteristic
164C2×C59#213411yescharacteristic
165C2×#127Dic5911yescharacteristic
166C2×#121—21no—
167C2×#117C2×D5911yescharacteristic
168#432—1181no—
169#407—1181no—
170#394—21no—
171#394C2×D5912yes—
172#392—21no—
173#392Dic5912yes—
174#390—21no—
175#382Dic5911yescharacteristic
176#343—592no—
177#343—1181no—
178#324—592no—
179#323—1181no—
180C59#1180611yescharacteristic, complemented, Sylow
181C22×D4—22no—
182C22×D4—21no—
183C22×D4—21no—
184C22×D4C2×Dic5912yes—
185C23×C4C22⋊D5911yescharacteristic
186C2×M16—1182no—
187#127—41no—
188#127—41no—
189#121—44no—
190#117—41no—
191#117—41no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4C22⋊D5911yescharacteristic
194C2×C4⋊C4C22×D5911yescharacteristic
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4C2×Dic5911yescharacteristic
199C2×C22⋊C4C22⋊D5911yescharacteristic
200C2×C42C2×Dic5911yescharacteristic
201C2×C42C22⋊D5911yescharacteristic
202C4⋊C8—2361no—
203C22⋊C8—1182no—
204C22⋊C8—2361no—
205C22⋊C8—2361no—
206#97—2361no—
207#95—21no—
208#95C2×Dic5912yes—
209C24—22no—
210C24C22⋊Dic5911yescharacteristic
211C2×D4—42no—
212C2×D4—42no—
213C2×D4—42no—
214C2×D4—42no—
215C2×D4—44no—
216C2×D4—41no—
217C2×D4—41no—
218C2×D4—82no—
219C2×D4—82no—
220C2×D4—82no—
221C2×D4—82no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—41no—
227C22×C4C22⋊Dic5911yescharacteristic
228C22×C4C22⋊Dic5911yescharacteristic
229C22×C4C22⋊Dic5911yescharacteristic
230C22×C4C22×Dic5911yescharacteristic
231C22×C4C2×C22⋊D5911yescharacteristic
232C22×C4C2×C22⋊D5911yescharacteristic
233M16—2362no—
234M16—2362no—
235C2×C8—2362no—
236C2×C8—2362no—
237C4⋊C4—41no—
238C4⋊C4—41no—
239C4⋊C4—42no—
240C22⋊C4—41no—
241C22⋊C4—41no—
242C22⋊C4—44no—
243C22⋊C4—82no—
244C22⋊C4—82no—
245C42—21no—
246C42—21no—
247C42—21no—
248C42—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—41no—
253C23—41no—
254C23—42no—
255C23—42no—
256C23—42no—
257C23—42no—
258C23—42no—
259C23—42no—
260C23—42no—
261C23C2×C22⋊Dic5911yescharacteristic
262D4—82no—
263D4—82no—
264D4—84no—
265D4—84no—
266D4—88no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4#1427364211yescharacteristic
285C2×C4#1427364211yescharacteristic
286C2×C4#1427375312yes—
287C8—4722no—
288C8—4722no—
289C22—21no—
290C22—21no—
291C22—42no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22—81no—
296C22—82no—
297C22—82no—
298C22—82no—
299C22—82no—
300C22—82no—
301C22—82no—
302C22—82no—
303C22C2×#1427364211yescharacteristic
304C22#1683740711yescharacteristic
305C22#1683741911yescharacteristic
306C4—22no—
307C4—21no—
308C4—21no—
309C4—41no—
310C4—41no—
311C4—41no—
312C4—41no—
313C4—82no—
314C2—21no—
315C2—21no—
316C2—81no—
317C2—162no—
318C2#2142262511yescharacteristic
319C2#2142574611yescharacteristic
320C2#2142578811yescharacteristic
subgroupG/H
centerC22C2×#14273642
commutatorC22×C4×C59C22×C4
FrattiniC2×C4⋊C4C22×D59
Fitting, prime radicalC59×#2619C2
socleC22×C59C2×#99
cosocleC2×C4⋊C4×C59C23
hypercenter#2619D59
hypocenterC59#11806
LCSG ▷ C22×C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2619

Complements

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

NH ≅ G/Ntype
1. #21424179C2semidirect
2. #21422430C2semidirect
3. #21422099C2semidirect
180. C59#11806semidirect

Maximal subgroups

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

subgroupindex
#214241792
#214224302
#214220992
C59×#26192
#1180659