Finite Group Databasebeta

← #28477481

Subgroups — #28477481

Order 15104.

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

normal subgroups only

This group has 492 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#21425424C212yesmaximal, complemented
2#21425178C211yesmaximal, characteristic
3#21421919C212yesmaximal, complemented
4#21421917C211yesmaximal, characteristic
5C59×#2724C211yesmaximal, characteristic
6C22×#11745670—21no—
7C2×#14273727C2211yescharacteristic
8C2×#14273660—22no—
9C2×#14273650—21no—
10#16837164—22no—
11#16837157C2212yes—
12#16837157C2211yescharacteristic
13C2×#123×C59—21no—
14C59×#398—21no—
15C59×#397—22no—
16C59×#390C2211yescharacteristic
17C59×#389C2212yes—
18C59×#385—22no—
19C22×Dic118—21no—
20C2×#11745670—42no—
21C2×#11745670—42no—
22C2×#11745670—42no—
23C22×C59⋊C8—21no—
24C22×C59⋊C8—22no—
25#14273727—42no—
26C2×C4⋊Dic59—22no—
27#14273660—44no—
28#14273650—42no—
29C22×D4×C59D411yescharacteristic
30C23×C4×C59—21no—
31#123×C59—42no—
32#123×C59—42no—
33C2×C4⋊C4×C59—22no—
34C2×C4⋊C4×C59—21no—
35C2×C4⋊C4×C59D412yes—
36C2×C4⋊C4×C59D411yescharacteristic
37C2×C4⋊C4×C59D412yes—
38C2×C22⋊C4×C59—22no—
39C2×C22⋊C4×C59—21no—
40C2×C22⋊C4×C59—22no—
41C2×C42×C59C2311yescharacteristic
42#95×C59—22no—
43#95×C59—21no—
44#95×C59—22no—
45C24×C59—21no—
46C2×D4×C59—21no—
47C2×D4×C59—41no—
48C2×D4×C59—41no—
49C2×D4×C59D812yes—
50C22×C4×C59—22no—
51C22×C4×C59—21no—
52C22×C4×C59—22no—
53C22×C4×C59—42no—
54C22×C4×C59—81no—
55C22×C4×C59—81no—
56C22×C4×C59C2×D411yescharacteristic
57C22×C4×C59C2×D412yes—
58C22×Dic59—41no—
59C2×Dic118—22no—
60C2×Dic118—41no—
61C2×Dic118—41no—
62C2×Dic118—41no—
63C2×Dic118—41no—
64C2×Dic118—41no—
65C4⋊C4×C59—22no—
66C4⋊C4×C59—22no—
67C4⋊C4×C59—41no—
68C4⋊C4×C59—42no—
69C4⋊C4×C59—42no—
70C4⋊C4×C59—42no—
71C4⋊C4×C59—81no—
72C22⋊C4×C59—42no—
73C22⋊C4×C59—44no—
74C22⋊C4×C59—42no—
75C22⋊C4×C59—42no—
76C42×C59—41no—
77#11745670—48no—
78C4⋊Dic59—24no—
79C4⋊Dic59—42no—
80C2×C59⋊C8—22no—
81C2×C59⋊C8—24no—
82C2×C59⋊C8—42no—
83C2×C59⋊C8—41no—
84C2×C59⋊C8—41no—
85C2×C59⋊C8—42no—
86C23×C59—22no—
87C23×C59—42no—
88C23×C59—41no—
89C23×C59—42no—
90C23×C59—41no—
91C23×C59#12011yescharacteristic
92D4×C59—42no—
93D4×C59—81no—
94C2×C4×C59—21no—
95C2×C4×C59—22no—
96C2×C4×C59—22no—
97C2×C4×C59—21no—
98C2×C4×C59—42no—
99C2×C4×C59—42no—
100C2×C4×C59—42no—
101C2×C4×C59—42no—
102C2×C4×C59—41no—
103C2×C4×C59—42no—
104C2×C4×C59—42no—
105C2×C4×C59—42no—
106C2×C4×C59—84no—
107C2×C4×C59C2×D811yescharacteristic
108C2×C4×C59#13512yes—
109C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
110C2×C4×C59#13712yes—
111C2×Dic59—42no—
112C2×Dic59—81no—
113C2×Dic59—81no—
114Dic118—42no—
115Dic118—42no—
116Dic118—81no—
117Dic118—81no—
118C59⋊C8—44no—
119C59⋊C8—42no—
120#7751—591nomaximal, Sylow
121C22×C59—42no—
122C22×C59—44no—
123C22×C59—82no—
124C22×C59—81no—
125C22×C59—81no—
126C22×C59#44711yescharacteristic
127C22×C59#45212yes—
128C22×C59#45311yescharacteristic
129C22×C59#45612yes—
130C22×C59B(4,𝔽2)11yescharacteristic
131C4×C59—21no—
132C4×C59—21no—
133C4×C59—22no—
134C4×C59—22no—
135C4×C59—82no—
136C4×C59—81no—
137C4×C59—82no—
138Dic59—82no—
139#2724D5911yescharacteristic
140#2057—592no—
141#2013—591no—
142#1271—592no—
143#1270—591no—
144C2×C59—82no—
145C2×C59#159511yescharacteristic
146C2×C59#160312yes—
147C2×C59#163011yescharacteristic
148C2×C59#164712yes—
149C2×C59#166011yescharacteristic
150C22×SD16—1181no—
151C2×C4⋊Q8—591no—
152C2×#123—21no—
153C2×#107—1182no—
154C2×#103—1181no—
155#398—21no—
156#397—22no—
157#390C2×D5911yescharacteristic
158#389C2×D5912yes—
159#385—22no—
160#340—1182no—
161#336—592no—
162#336—591no—
163C59#775111yescharacteristic, complemented, Sylow
164C22×Q8—1181no—
165C22×D4C22⋊D5911yescharacteristic
166C23×C4—21no—
167C2×SD16—2362no—
168C2×SD16—2362no—
169C2×SD16—2362no—
170C22×C8—1181no—
171C22×C8—1182no—
172C4⋊Q8—2362no—
173#123—42no—
174#123—42no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—1182no—
178C2×C4⋊C4C22⋊D5912yes—
179C2×C4⋊C4C22⋊D5912yes—
180C2×C4⋊C4C22⋊D5911yescharacteristic
181C2×C22⋊C4—22no—
182C2×C22⋊C4—21no—
183C2×C22⋊C4—22no—
184C2×C42C22×D5911yescharacteristic
185#107—2364no—
186#103—2362no—
187#95—22no—
188#95—21no—
189#95—22no—
190C24—21no—
191C2×Q8—1182no—
192C2×Q8—2361no—
193C2×Q8—2361no—
194C2×Q8—2361no—
195C2×Q8—2361no—
196C2×Q8—2361no—
197C2×D4—21no—
198C2×D4—41no—
199C2×D4—41no—
200C2×D4#1174566912yes—
201C22×C4—22no—
202C22×C4—21no—
203C22×C4—22no—
204C22×C4—42no—
205C22×C4—81no—
206C22×C4—81no—
207C22×C4—2361no—
208C22×C4C2×C22⋊D5911yescharacteristic
209C22×C4C2×C22⋊D5912yes—
210SD16—2368no—
211C2×C8—1182no—
212C2×C8—1184no—
213C2×C8—2362no—
214C2×C8—2361no—
215C2×C8—2361no—
216C2×C8—2362no—
217C4⋊C4—22no—
218C4⋊C4—22no—
219C4⋊C4—41no—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—81no—
224C4⋊C4—1184no—
225C4⋊C4—2362no—
226C22⋊C4—42no—
227C22⋊C4—44no—
228C22⋊C4—42no—
229C22⋊C4—42no—
230C42—41no—
231C23—22no—
232C23—42no—
233C23—41no—
234C23—42no—
235C23—41no—
236C23#1427369611yescharacteristic
237Q8—4721no—
238Q8—4721no—
239Q8—2362no—
240Q8—2362no—
241D4—42no—
242D4—81no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—21no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—41no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—84no—
256C2×C4—4721no—
257C2×C4—4721no—
258C2×C4—2362no—
259C2×C4C2×#1174566911yescharacteristic
260C2×C4#1427374712yes—
261C2×C4#1427375311yescharacteristic
262C2×C4#1427375712yes—
263C8—2364no—
264C8—2362no—
265C22—42no—
266C22—44no—
267C22—81no—
268C22—81no—
269C22—82no—
270C22#1683753611yescharacteristic
271C22#1683756412yes—
272C22#1683757011yescharacteristic
273C22#1683758512yes—
274C22#1683758911yescharacteristic
275C4—21no—
276C4—21no—
277C4—22no—
278C4—22no—
279C4—81no—
280C4—82no—
281C4—82no—
282C4—4722no—
283C2—82no—
284C2#2142301411yescharacteristic
285C2#2142305912yes—
286C2#2142321711yescharacteristic
287C2#2142333112yes—
288C2#2142340711yescharacteristic
subgroupG/H
centerC23#14273696
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
FittingC59×#2724C2
socleC23×C59#120
prime radicalC22×D4×C59D4
hypercenter#2724D59
hypocenterC59#7751
LCSG ▷ C2×C42×C59 ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2724

Complements

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

NH ≅ G/Ntype
1. #21425424C2semidirect
3. #21421919C2semidirect
163. C59#7751semidirect

Maximal subgroups

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

subgroupindex
#214254242
#214251782
#214219192
#214219172
C59×#27242
#775159