Finite Group Databasebeta

← #28489547

Subgroups — #28489547

Order 15104.

696 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 696 conjugacy classes of subgroups — too many to draw. Only its 105 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#21434104C212yesmaximal, complemented
2#21425077C211yesmaximal, characteristic, complemented
3#21422521C212yesmaximal, complemented
4#21421925C211yesmaximal, characteristic, complemented
5C59×#2397C211yesmaximal, characteristic
6#16837959C2211yescharacteristic
7#16837732—22no—
8#16837692—22no—
9#16837607—22no—
10#16837553—22no—
11C4×#11745670—24no—
12#16837455C2212yes—
13#16837426C2211yescharacteristic
14#16837162—22no—
15D4⋊(C59⋊C8)—24no—
16C2×C4×D4×C59—22no—
17C2×C4×D4×C59C2212yes—
18C59×#399—22no—
19C59×#392—22no—
20C59×#376C2211yescharacteristic
21C2×#11745670—24no—
22C2×#11745664—21no—
23#14273727D412yescomplemented
24#14273719—21no—
25#14273706—22no—
26C4×Dic118—22no—
27#14273686—21no—
28#14273658—24no—
29C4⋊(C59⋊C8)—22no—
30C4⋊(C59⋊C8)—22no—
31#14273650—24no—
32#14273648—24no—
33C22⋊(C59⋊C8)—22no—
34C4×C59⋊C8D414yescomplemented
35C22×D4×C59—22no—
36C22×D4×C59D412yes—
37C23×C4×C59—22no—
38C4×D4×C59—22no—
39C4×D4×C59—22no—
40C4×D4×C59—22no—
41C4×D4×C59—22no—
42C4×D4×C59—44no—
43C4×D4×C59—44no—
44C2×C4⋊C4×C59—22no—
45C2×C4⋊C4×C59D412yes—
46C2×C22⋊C4×C59—24no—
47C2×C22⋊C4×C59—22no—
48C2×C42×C59D412yes—
49C2×C42×C59C2311yescharacteristic
50#95×C59—22no—
51C24×C59—22no—
52C2×D4×C59—22no—
53C2×D4×C59—22no—
54C2×D4×C59—44no—
55C2×D4×C59—44no—
56C2×D4×C59—48no—
57C22×C4×C59—22no—
58C22×C4×C59—22no—
59C22×C4×C59—42no—
60C22×C4×C59—42no—
61C22×C4×C59—44no—
62C22×C4×C59—44no—
63C22×C4×C59C2×D412yes—
64C22×C4×C59C2×D411yescharacteristic
65C2×Dic118—22no—
66C4⋊C4×C59—22no—
67C4⋊C4×C59—22no—
68C4⋊C4×C59—44no—
69C22⋊C4×C59—48no—
70C22⋊C4×C59—42no—
71C22⋊C4×C59—42no—
72C42×C59—22no—
73C42×C59SD1614yescomplemented
74C42×C59C2×D411yescharacteristic
75C42×C59C2×D412yes—
76C42×C59#4111yescharacteristic
77C22⋊Dic59—41no—
78#11745670—48no—
79C4⋊Dic59—22no—
80C4⋊Dic59—22no—
81Dic59⋊C4—42no—
82C4×Dic59—41no—
83#11745664—42no—
84C2×C59⋊C8—24no—
85C2×C59⋊C8—41no—
86C23×C59—24no—
87C23×C59—44no—
88C23×C59—42no—
89C23×C59—42no—
90C23×C59—44no—
91C23×C59#12011yescharacteristic
92D4×C59—24no—
93D4×C59—24no—
94D4×C59—42no—
95D4×C59—42no—
96D4×C59—416no—
97C2×C4×C59—24no—
98C2×C4×C59—42no—
99C2×C4×C59—44no—
100C2×C4×C59—42no—
101C2×C4×C59—44no—
102C2×C4×C59—84no—
103C2×C4×C59—82no—
104C2×C4×C59—82no—
105C2×C4×C59#12012yes—
106C2×C4×C59#12111yescharacteristic
107C2×C4×C59#12112yes—
108C2×C4×C59#12711yescharacteristic
109C2×C4×C59C2×SD1612yes—
110C2×C4×C59AGL(1,ℤ/8)12yes—
111C2×Dic59—42no—
112Dic118—44no—
113C59⋊C8—44no—
114C59⋊C8—81no—
115#10015—591nomaximal, Sylow
116C22×C59—22no—
117C22×C59—48no—
118C22×C59—44no—
119C22×C59—84no—
120C22×C59—82no—
121C22×C59—82no—
122C22×C59#39211yescharacteristic
123C22×C59#45012yes—
124C4×C59—22no—
125C4×C59—22no—
126C4×C59—44no—
127C4×C59—82no—
128C4×C59#46012yes—
129C4×C59#49211yescharacteristic
130C4×C59#49611yescharacteristic
131Dic59—82no—
132#3101—592no—
133#2397D5911yescharacteristic
134#1995—591no—
135#1486—592no—
136#1274—591no—
137C2×C59—22no—
138C2×C59—84no—
139C2×C59#201012yes—
140C2×C59#201411yescharacteristic
141#533—591no—
142C2×C4×D4—22no—
143C2×C4×D4C2×D5912yes—
144#483—1182no—
145#476—1182no—
146#461—1182no—
147#450—1182no—
148C4×SD16—1184no—
149#431—592no—
150#423—591no—
151#399—22no—
152#392—22no—
153#376C2×D5911yescharacteristic
154#339—1182no—
155D4⋊C8—1184no—
156C59#1001511yescharacteristic, complemented, Sylow
157C22×D4—22no—
158C22×D4C22⋊D5912yes—
159C23×C4—22no—
160C2×SD16—1184no—
161C2×M16—1181no—
162C4⋊Q8—592no—
163#125—1181no—
164C22⋊Q8—1182no—
165C4×Q8—1182no—
166C4×D4—22no—
167C4×D4—22no—
168C4×D4—22no—
169C4×D4—22no—
170C4×D4—44no—
171C4×D4—44no—
172#117—1181no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4C22⋊D5912yes—
175C2×C22⋊C4—24no—
176C2×C22⋊C4—22no—
177C2×C42C22⋊D5912yes—
178C2×C42C22×D5911yescharacteristic
179#106—1184no—
180C4⋊C8—1182no—
181C4⋊C8—1182no—
182#103—1184no—
183#102—1184no—
184C22⋊C8—1182no—
185C4×C8—594no—
186#95—22no—
187C24—22no—
188C2×Q8—1182no—
189C2×D4—22no—
190C2×D4—22no—
191C2×D4—44no—
192C2×D4—44no—
193C2×D4—48no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—44no—
199C22×C4—44no—
200C22×C4C2×C22⋊D5912yes—
201C22×C4C2×C22⋊D5911yescharacteristic
202SD16—2368no—
203M16—2362no—
204C2×C8—1184no—
205C2×C8—2361no—
206C4⋊C4—22no—
207C4⋊C4—22no—
208C4⋊C4—44no—
209C4⋊C4—1182no—
210C4⋊C4—1182no—
211C4⋊C4—2362no—
212C22⋊C4—48no—
213C22⋊C4—42no—
214C22⋊C4—42no—
215C22⋊C4—2361no—
216C42—22no—
217C42—2361no—
218C42#1174567014yescomplemented
219C42D4×D5911yescharacteristic
220C42D4×D5912yes—
221C42#1174568711yescharacteristic
222C23—24no—
223C23—44no—
224C23—42no—
225C23—44no—
226C23—42no—
227C23#1427369611yescharacteristic
228Q8—2364no—
229D4—24no—
230D4—24no—
231D4—42no—
232D4—42no—
233D4—416no—
234C2×C4—24no—
235C2×C4—42no—
236C2×C4—44no—
237C2×C4—44no—
238C2×C4—42no—
239C2×C4—82no—
240C2×C4—82no—
241C2×C4—84no—
242C2×C4—2362no—
243C2×C4#1427369512yes—
244C2×C4#1427369811yescharacteristic
245C2×C4#1427369912yes—
246C2×C4#1427372311yescharacteristic
247C2×C4C2×#1174567012yes—
248C2×C4#1427375312yes—
249C8—4721no—
250C8—2364no—
251C22—22no—
252C22—48no—
253C22—44no—
254C22—84no—
255C22—82no—
256C22—82no—
257C22#1683731011yescharacteristic
258C22#1683755312yes—
259C4—22no—
260C4—22no—
261C4—44no—
262C4—82no—
263C4—4722no—
264C4#1683759912yes—
265C4#1683777411yescharacteristic
266C4#1683778911yescharacteristic
267C2—22no—
268C2—84no—
269C2#2142516112yes—
270C2#2142518411yescharacteristic
subgroupG/H
centerC2×C4#14273723
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
Fitting, prime radicalC59×#2397C2
socleC22×C59#392
hypercenter#2397D59
hypocenterC59#10015
LCSG ▷ C2×C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2397

Complements

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

NH ≅ G/Ntype
1. #21434104C2semidirect
2. #21425077C2semidirect
3. #21422521C2semidirect
4. #21421925C2semidirect
23. #14273727D4semidirect
34. C4×C59⋊C8D4semidirect
73. C42×C59SD16semidirect
156. C59#10015semidirect
218. C42#11745670semidirect

Maximal subgroups

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

subgroupindex
#214341042
#214250772
#214225212
#214219252
C59×#23972
#1001559