Finite Group Databasebeta

← #28501464

Subgroups — #28501464

Order 15104.

460 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 63 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#21425257C211yesmaximal, characteristic, complemented
2#21423035C212yesmaximal, complemented
3#21422099C212yesmaximal, complemented
4#21422030C211yesmaximal, characteristic, complemented
5C59×#2733C211yesmaximal, characteristic
6#16838033C2211yescharacteristic
7#16837553—22no—
8#16837411—21no—
9#16837171C2211yescharacteristic
10#16837162—21no—
11#16837138—22no—
12#16837128C2212yes—
13C2×#124×C59—21no—
14C59×#394C2212yes—
15C59×#392—22no—
16C59×#391C2211yescharacteristic
17C59×#390—21no—
18C59×#388—22no—
19C2×#11745670—42no—
20C2×#11745664—21no—
21#14273726—41no—
22#14273718—41no—
23#14273706—22no—
24#14273686—21no—
25#14273651—41no—
26#14273651—41no—
27#14273648—42no—
28C22⋊(C59⋊C8)—22no—
29C22×Q8×C59D411yescharacteristic
30C22×D4×C59—21no—
31C22×D4×C59—21no—
32C22×D4×C59D412yes—
33#124×C59—42no—
34#124×C59—42no—
35C2×C4⋊C4×C59—21no—
36C2×C4⋊C4×C59C2311yescharacteristic
37C2×C22⋊C4×C59—22no—
38C2×C22⋊C4×C59—22no—
39C2×C22⋊C4×C59—22no—
40C2×C22⋊C4×C59—22no—
41C2×C42×C59—21no—
42C2×C42×C59D411yescharacteristic
43#95×C59—22no—
44#95×C59D412yes—
45C24×C59—22no—
46C2×Q8×C59—21no—
47C2×Q8×C59—41no—
48C2×Q8×C59—41no—
49C2×Q8×C59SD1612yescomplemented
50C2×D4×C59—42no—
51C2×D4×C59—42no—
52C2×D4×C59—42no—
53C2×D4×C59—82no—
54C2×D4×C59—81no—
55C2×D4×C59—81no—
56C2×D4×C59—81no—
57C2×D4×C59—81no—
58C22×C4×C59—21no—
59C22×C4×C59—21no—
60C22×C4×C59—22no—
61C22×C4×C59C2×D411yescharacteristic
62C22×C4×C59C2×D412yes—
63C2×Dic118—22no—
64C4⋊C4×C59—41no—
65C4⋊C4×C59—81no—
66C22⋊C4×C59—44no—
67C22⋊C4×C59—44no—
68C22⋊C4×C59—42no—
69C22⋊C4×C59—42no—
70C22⋊C4×C59—82no—
71C42×C59—42no—
72C42×C59—41no—
73C22⋊Dic59—41no—
74#11745670—84no—
75C4⋊Dic59—22no—
76C4⋊Dic59—21no—
77C4⋊Dic59—21no—
78Dic59⋊C4—42no—
79C4×Dic59—41no—
80#11745664—41no—
81#11745664—41no—
82C2×C59⋊C8—42no—
83C2×C59⋊C8—41no—
84C23×C59—42no—
85C23×C59—42no—
86C23×C59—42no—
87C23×C59—42no—
88C23×C59—42no—
89C23×C59—42no—
90C23×C59—42no—
91C23×C59#12011yescharacteristic
92Q8×C59—42no—
93Q8×C59—81no—
94D4×C59—44no—
95D4×C59—82no—
96D4×C59—84no—
97D4×C59—84no—
98C2×C4×C59—21no—
99C2×C4×C59—21no—
100C2×C4×C59—22no—
101C2×C4×C59—22no—
102C2×C4×C59—22no—
103C2×C4×C59—41no—
104C2×C4×C59—41no—
105C2×C4×C59—41no—
106C2×C4×C59—41no—
107C2×C4×C59—42no—
108C2×C4×C59—42no—
109C2×C4×C59—42no—
110C2×C4×C59—42no—
111C2×C4×C59C2×SD1611yescharacteristic
112C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
113C2×Dic59—41no—
114C2×Dic59—41no—
115Dic118—82no—
116C59⋊C8—81no—
117C59⋊C8—82no—
118#12212—591nomaximal, Sylow
119C22×C59—21no—
120C22×C59—21no—
121C22×C59—82no—
122C22×C59—82no—
123C22×C59—82no—
124C22×C59—82no—
125C22×C59—82no—
126C22×C59—82no—
127C22×C59—82no—
128C22×C59#45011yescharacteristic
129C22×C59#45511yescharacteristic
130C22×C59B(4,𝔽2)11yescharacteristic
131C4×C59—21no—
132C4×C59—21no—
133C4×C59—42no—
134C4×C59—42no—
135C4×C59—42no—
136C4×C59—82no—
137Dic59—81no—
138Dic59—81no—
139#2733D5911yescharacteristic
140#2028—591no—
141#1599—592no—
142#1345—592no—
143#1319—591no—
144C2×C59—21no—
145C2×C59—21no—
146C2×C59—162no—
147C2×C59#160111yescharacteristic
148C2×C59#219911yescharacteristic
149C2×C59#220411yescharacteristic
150#544—591no—
151C2×#124—21no—
152#450—1182no—
153#419—1181no—
154#394C2×D5912yes—
155#392—22no—
156#391C2×D5911yescharacteristic
157#390—21no—
158#388—22no—
159#343—591no—
160#339—1181no—
161#328—1182no—
162#324—592no—
163C59#1221211yescharacteristic, complemented, Sylow
164C22×Q8C22⋊D5911yescharacteristic
165C22×D4—21no—
166C22×D4—21no—
167C22×D4C22⋊D5912yes—
168C2×SD16—2362no—
169C2×M16—1181no—
170C4⋊Q8—2361no—
171#125—2361no—
172#124—42no—
173#124—42no—
174C22⋊Q8—1182no—
175#117—1181no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4C22×D5911yescharacteristic
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C42—21no—
183C2×C42C22⋊D5911yescharacteristic
184#103—2361no—
185#103—2361no—
186#102—2362no—
187C22⋊C8—1182no—
188#95—22no—
189#95C22⋊D5912yes—
190C24—22no—
191C2×Q8—21no—
192C2×Q8—41no—
193C2×Q8—41no—
194C2×Q8—1182no—
195C2×Q8#1174567012yescomplemented
196C2×D4—42no—
197C2×D4—42no—
198C2×D4—42no—
199C2×D4—82no—
200C2×D4—81no—
201C2×D4—81no—
202C2×D4—81no—
203C2×D4—81no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—22no—
207C22×C4C2×C22⋊D5911yescharacteristic
208C22×C4C2×C22⋊D5912yes—
209SD16—4724no—
210M16—2361no—
211M16—2361no—
212C2×C8—2362no—
213C2×C8—2361no—
214C4⋊C4—41no—
215C4⋊C4—81no—
216C4⋊C4—1182no—
217C4⋊C4—1181no—
218C4⋊C4—1181no—
219C4⋊C4—2362no—
220C22⋊C4—44no—
221C22⋊C4—44no—
222C22⋊C4—42no—
223C22⋊C4—42no—
224C22⋊C4—82no—
225C22⋊C4—2361no—
226C42—41no—
227C42—42no—
228C42—2361no—
229C23—42no—
230C23—42no—
231C23—42no—
232C23—42no—
233C23—42no—
234C23—42no—
235C23—42no—
236C23#1427369611yescharacteristic
237Q8—42no—
238Q8—81no—
239Q8—4722no—
240D4—44no—
241D4—82no—
242D4—84no—
243D4—84no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—2361no—
258C2×C4—2361no—
259C2×C4C2×#1174567011yescharacteristic
260C2×C4#1427375311yescharacteristic
261C8—4721no—
262C8—4722no—
263C22—21no—
264C22—21no—
265C22—82no—
266C22—82no—
267C22—82no—
268C22—82no—
269C22—82no—
270C22—82no—
271C22—82no—
272C22#1683755311yescharacteristic
273C22#1683758011yescharacteristic
274C22#1683758911yescharacteristic
275C4—21no—
276C4—21no—
277C4—42no—
278C4—42no—
279C4—42no—
280C4—82no—
281C4—4721no—
282C4—4721no—
283C2—21no—
284C2—21no—
285C2—162no—
286C2#2142304811yescharacteristic
287C2#2142609711yescharacteristic
288C2#2142612211yescharacteristic
subgroupG/H
centerC22#16837589
commutator, cosocleC2×C4⋊C4×C59C23
FrattiniC2×C4⋊C4C22×D59
Fitting, prime radicalC59×#2733C2
socleC22×C59B(4,𝔽2)
hypercenter#2733D59
hypocenterC59#12212
LCSG ▷ C2×C4⋊C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2733

Complements

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

NH ≅ G/Ntype
1. #21425257C2semidirect
2. #21423035C2semidirect
3. #21422099C2semidirect
4. #21422030C2semidirect
49. C2×Q8×C59SD16semidirect
163. C59#12212semidirect
195. C2×Q8#11745670semidirect

Maximal subgroups

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

subgroupindex
#214252572
#214230352
#214220992
#214220302
C59×#27332
#1221259