Finite Group Databasebeta

← #28500746

Subgroups — #28500746

Order 15104.

458 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 59 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#21432133C211yesmaximal, characteristic, complemented
2#21423082C212yesmaximal
3#21422095C212yesmaximal, complemented
4#21422026C211yesmaximal, characteristic, complemented
5C59×#2022C211yesmaximal, characteristic
6C2×#14273706—21no—
7#16837713—22no—
8#16837342—22no—
9#16837327—21no—
10#16837327C2212yes—
11#16837323—22no—
12#16837136—22no—
13#16837126C2212yes—
14C59×#534C2211yescharacteristic
15C2×#123×C59—21no—
16C2×#99×C59—21no—
17C59×#381C2211yescharacteristic
18C59×#342C2211yescharacteristic
19C22×Dic118—21no—
20#14273706—42no—
21#14273706—42no—
22C2×C4⋊Dic59—21no—
23C2×C4⋊Dic59—21no—
24C2×C4⋊Dic59D412yes—
25C2×Dic59⋊C4—22no—
26C2×C22⋊Dic59—22no—
27#14273660—42no—
28#14273648—42no—
29C22⋊(C59⋊C8)—22no—
30C59⋊#95—22no—
31C59⋊#95—21no—
32C59⋊#95—22no—
33C59⋊#95D412yes—
34C22×D4×C59—21no—
35C23×C4×C59D411yescharacteristic
36#123×C59—42no—
37#123×C59—41no—
38#123×C59—42no—
39#121×C59—22no—
40#120×C59—41no—
41C2×C4⋊C4×C59—41no—
42C2×C22⋊C4×C59—21no—
43C2×C22⋊C4×C59—41no—
44C2×C22⋊C4×C59D411yescharacteristic
45C2×C22⋊C4×C59C2311yescharacteristic
46#99×C59—42no—
47#95×C59—22no—
48C24×C59—21no—
49C24×C59C2×D411yescharacteristic
50C2×D4×C59—22no—
51C2×D4×C59—41no—
52C2×D4×C59—41no—
53C2×D4×C59—41no—
54C2×D4×C59—41no—
55C2×D4×C59—41no—
56C2×D4×C59—41no—
57C22×C4×C59—21no—
58C22×C4×C59—21no—
59C22×C4×C59—21no—
60C22×C4×C59—21no—
61C22×C4×C59—41no—
62C22×C4×C59—41no—
63C22×C4×C59—41no—
64C22×C4×C59D812yes—
65C22×C4×C59C2×D412yes—
66C22×Dic59—22no—
67C22×Dic59—22no—
68C2×Dic118—42no—
69C2×Dic118—81no—
70C2×Dic118—81no—
71C4⋊C4×C59—22no—
72C4⋊C4×C59—44no—
73C22⋊C4×C59—22no—
74C22⋊C4×C59—41no—
75C22⋊C4×C59—42no—
76C22⋊C4×C59—44no—
77C22⋊C4×C59—41no—
78C22⋊C4×C59—41no—
79C22⋊Dic59—42no—
80C22⋊Dic59—42no—
81C4⋊Dic59—42no—
82C4⋊Dic59—42no—
83C4⋊Dic59—81no—
84Dic59⋊C4—44no—
85C2×C59⋊C8—42no—
86C23×C59—21no—
87C23×C59—21no—
88C23×C59—42no—
89C23×C59—41no—
90C23×C59—41no—
91C23×C59—41no—
92C23×C59—82no—
93C23×C59#12011yescharacteristic
94C23×C59C2×D811yescharacteristic
95C23×C59AGL(1,ℤ/8)11yescharacteristic
96D4×C59—82no—
97D4×C59—81no—
98D4×C59—81no—
99D4×C59—81no—
100D4×C59—81no—
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—41no—
113C2×C4×C59—81no—
114C2×C4×C59—82no—
115C2×Dic59—44no—
116C2×Dic59—42no—
117C2×Dic59—42no—
118C2×Dic59—42no—
119C2×Dic59—42no—
120Dic118—84no—
121C59⋊C8—82no—
122#12095—591nomaximal, Sylow
123C22×C59—22no—
124C22×C59—21no—
125C22×C59—21no—
126C22×C59—41no—
127C22×C59—41no—
128C22×C59—42no—
129C22×C59—81no—
130C22×C59—82no—
131C22×C59—81no—
132C22×C59—82no—
133C22×C59—161no—
134C22×C59#44711yescharacteristic
135C22×C59#45311yescharacteristic
136C22×C59B(4,𝔽2)11yescharacteristic
137C4×C59—42no—
138C4×C59—42no—
139C4×C59—81no—
140C4×C59—82no—
141C4×C59—81no—
142Dic59—82no—
143Dic59—82no—
144#2817—591no—
145#2022D5911yescharacteristic
146#1607—592no—
147#1344—592no—
148#1317—591no—
149C2×C59—21no—
150C2×C59—21no—
151C2×C59—41no—
152C2×C59—41no—
153C2×C59—161no—
154C2×C59#159511yescharacteristic
155C2×C59#220112yes—
156#534C2×D5911yescharacteristic
157C2×#123—21no—
158C2×C22⋊Q8—1181no—
159#480—1182no—
160C2×#99—21no—
161#400—1182no—
162#397—592no—
163#397—1181no—
164#395—1182no—
165#381C2×D5911yescharacteristic
166#342C2×D5911yescharacteristic
167#327—1182no—
168#323—592no—
169C59#1209511yescharacteristic, complemented, Sylow
170C22×Q8—1181no—
171C22×D4—21no—
172C23×C4C22⋊D5911yescharacteristic
173#123—42no—
174#123—41no—
175#123—42no—
176C22⋊Q8—2362no—
177C22⋊Q8—2362no—
178#121—22no—
179#120—41no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4—592no—
182C2×C4⋊C4—1181no—
183C2×C4⋊C4—1182no—
184C2×C4⋊C4—1181no—
185C2×C22⋊C4—21no—
186C2×C22⋊C4—41no—
187C2×C22⋊C4—1182no—
188C2×C22⋊C4C22⋊D5911yescharacteristic
189C2×C22⋊C4C22×D5911yescharacteristic
190#107—2362no—
191#102—2362no—
192#99—42no—
193C22⋊C8—1182no—
194#95—22no—
195#95—592no—
196#95—1182no—
197#95—1181no—
198#95—1182no—
199C24—21no—
200C24C2×C22⋊D5911yescharacteristic
201C2×Q8—4721no—
202C2×Q8—4721no—
203C2×Q8—2362no—
204C2×D4—22no—
205C2×D4—41no—
206C2×D4—41no—
207C2×D4—41no—
208C2×D4—41no—
209C2×D4—41no—
210C2×D4—41no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—1182no—
219C22×C4—1182no—
220C22×C4#1174566912yes—
221C22×C4D4×D5912yes—
222C2×C8—2362no—
223C4⋊C4—22no—
224C4⋊C4—44no—
225C4⋊C4—4721no—
226C4⋊C4—2362no—
227C4⋊C4—2364no—
228C4⋊C4—2362no—
229C22⋊C4—22no—
230C22⋊C4—41no—
231C22⋊C4—42no—
232C22⋊C4—44no—
233C22⋊C4—41no—
234C22⋊C4—41no—
235C22⋊C4—2362no—
236C22⋊C4—2362no—
237C23—21no—
238C23—21no—
239C23—42no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23—82no—
244C23#1427369511yescharacteristic
245C23C2×#1174566911yescharacteristic
246C23#1427375211yescharacteristic
247Q8—4724no—
248D4—82no—
249D4—81no—
250D4—81no—
251D4—81no—
252D4—81no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—41no—
261C2×C4—42no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—81no—
266C2×C4—82no—
267C2×C4—2364no—
268C2×C4—2362no—
269C2×C4—2362no—
270C2×C4—2362no—
271C2×C4—2362no—
272C8—4722no—
273C22—22no—
274C22—21no—
275C22—21no—
276C22—41no—
277C22—41no—
278C22—42no—
279C22—81no—
280C22—82no—
281C22—81no—
282C22—82no—
283C22—161no—
284C22#1683753511yescharacteristic
285C22#1683756911yescharacteristic
286C22#1683758811yescharacteristic
287C4—42no—
288C4—42no—
289C4—81no—
290C4—82no—
291C4—81no—
292C4—4722no—
293C4—4722no—
294C2—21no—
295C2—21no—
296C2—41no—
297C2—41no—
298C2—161no—
299C2#2142301311yescharacteristic
300C2#2142610712yes—
subgroupG/H
centerC22#16837588
commutator, cosocleC2×C22⋊C4×C59C23
FrattiniC2×C22⋊C4C22×D59
FittingC59×#2022C2
socleC22×C59B(4,𝔽2)
prime radicalC59×#534C22
hypercenter#2022D59
hypocenterC59#12095
LCSG ▷ C2×C22⋊C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #2022

Complements

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

NH ≅ G/Ntype
1. #21432133C2semidirect
3. #21422095C2semidirect
4. #21422026C2semidirect
169. C59#12095semidirect

Maximal subgroups

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

subgroupindex
#214321332
#214230822
#214220952
#214220262
C59×#20222
#1209559