Finite Group Databasebeta

← #28483648

Subgroups — #28483648

Order 15104.

502 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 502 conjugacy classes of subgroups — too many to draw. Only its 99 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#21434541C211yesmaximal, characteristic
2#21422758C211yesmaximal, characteristic, complemented
3#21422564C212yesmaximal, complemented
4#21421994C212yesmaximal, complemented
5C59×#2980C211yesmaximal, characteristic, complemented
6C2×#14273747—22no—
7#16837952C2211yescharacteristic
8#16837797C412yes—
9#16837794—21no—
10#16837792C412yes—
11#16837789—21no—
12#16837455C2212yes—
13C2×#14273638C2211yescharacteristic
14#16837151—21no—
15#16837146—21no—
16#16837129—24no—
17C59×#532C2211yescharacteristic
18C59×#435—24no—
19C59×#432C2212yes—
20C59×#408—22no—
21C2×#11745685—21no—
22#14273747—44no—
23#14273747—44no—
24C2×#11745672—22no—
25C2×#11745670—22no—
26C2×#11745671—22no—
27C2×#11745669—22no—
28C22×C59⋊C8—21no—
29#14273727C2×C411yescharacteristic
30#14273724C2×C411yescharacteristic
31#14273715—21no—
32#14273701—22no—
33C4×D236—22no—
34C4⋊(C59⋊C8)—22no—
35C22⋊(C59⋊C8)—22no—
36#14273638—21no—
37#14273638C2×C412yes—
38C4×C59⋊C8—22no—
39C2×#41×C59D412yes—
40C2×M16×C59—22no—
41C22×C8×C59—22no—
42C4⋊Q8×C59D411yescharacteristic
43#127×C59D411yescharacteristic
44#124×C59C2×C412yes—
45#121×C59—22no—
46C4×D4×C59—22no—
47C2×C42×C59C2311yescharacteristic
48C4⋊C8×C59D414yes—
49C22⋊C8×C59—24no—
50C22⋊C8×C59—22no—
51#97×C59—22no—
52#41×C59—42no—
53#41×C59—42no—
54C2×Q8×C59C22⋊C412yes—
55C2×D4×C59—22no—
56C2×D4×C59—22no—
57C2×D4×C59C22⋊C412yes—
58C22×C4×C59—22no—
59C22×C4×C59C2×D411yescharacteristic
60C22×C4×C59#4112yes—
61C2×C22⋊D59—41no—
62#11745685—42no—
63#11745685—42no—
64C2×D236—21no—
65C2×D236—22no—
66C2×C4×D59—41no—
67C2×Dic118—21no—
68M16×C59—82no—
69C2×C8×C59—24no—
70C2×C8×C59—82no—
71C4⋊C4×C59—22no—
72C22⋊C4×C59—24no—
73C42×C59C22×C411yescharacteristic
74C42×C59C2×D411yescharacteristic
75C42×C59C2×D412yes—
76#11745672—44no—
77#11745671—44no—
78#11745670—44no—
79#11745669—44no—
80(C2×D59)⋊C4—42no—
81C4⋊Dic59—22no—
82C2×C59⋊C8—24no—
83C2×C59⋊C8—22no—
84C2×C59⋊C8—42no—
85C23×C59—22no—
86C23×C59#12311yescharacteristic
87C22×D59—41no—
88Q8×C59—42no—
89D4×C59—42no—
90D4×C59—82no—
91D4×C59—82no—
92C2×C4×C59—22no—
93C2×C4×C59—22no—
94C2×C4×C59—42no—
95C2×C4×C59—42no—
96C2×C4×C59—42no—
97C2×C4×C59C2×C22⋊C411yescharacteristic
98C2×C4×C59C4×D412yes—
99C2×C4×C59#12011yescharacteristic
100C2×C4×C59#12112yes—
101C22⋊D59—81no—
102C22⋊D59—81no—
103C2×Dic59—41no—
104D236—42no—
105D236—44no—
106C4×D59—82no—
107Dic118—42no—
108C8×C59—44no—
109C59⋊C8—24no—
110C59⋊C8—82no—
111#9007—591nomaximal, Sylow
112C22×C59—22no—
113C22×C59—21no—
114C22×C59—44no—
115C22×C59—42no—
116C22×C59#38611yescharacteristic
117C2×D59—82no—
118C2×D59—81no—
119C4×C59—22no—
120C4×C59—22no—
121C4×C59—21no—
122C4×C59—42no—
123C4×C59#45312yes—
124Dic59—81no—
125#3150—591no—
126#2980D5911yescharacteristic, complemented
127#1543—591no—
128#1496—592no—
129#1302—592no—
130C2×C59—41no—
131C2×C59—82no—
132C2×C59#188112yes—
133C2×C59#188711yescharacteristic
134D59—161no—
135C2×#135—1182no—
136#532—591no—
137#532C2×D5911yescharacteristic
138#497—592no—
139#497—1181no—
140#496—592no—
141#496—1181no—
142#435—24no—
143#432C2×D5912yes—
144#431—592no—
145#408—22no—
146C2×#97—591no—
147#333—1181no—
148#331—1181no—
149D4⋊C8—1184no—
150C59#900711yescharacteristic, complemented, Sylow
151C2×#41—1181no—
152C2×#41C22⋊D5912yes—
153#135—2364no—
154#135—2364no—
155C2×Q16—1182no—
156C2×SD16—1182no—
157C2×SD16—1182no—
158C2×D8—1182no—
159C2×M16—22no—
160C22×C8—22no—
161C22×C8—1181no—
162C4⋊Q8—591no—
163C4⋊Q8D23611yescharacteristic
164#127—591no—
165#127D23611yescharacteristic
166#124—1181no—
167#124C4×D5912yes—
168#121—22no—
169#121—1182no—
170C4×D4—22no—
171C4×D4—1182no—
172C2×C42C22×D5911yescharacteristic
173C4⋊C8—1182no—
174C4⋊C8C22⋊D5914yes—
175C22⋊C8—24no—
176C22⋊C8—22no—
177C22⋊C8—1182no—
178#97—22no—
179#97—592no—
180#97—1181no—
181C4×C8—1182no—
182#41—42no—
183#41—42no—
184#41—2362no—
185#41—2362no—
186C2×Q8—1181no—
187C2×Q8(C2×D59)⋊C412yes—
188C2×D4—22no—
189C2×D4—22no—
190C2×D4—1181no—
191C2×D4—1182no—
192C2×D4—2361no—
193C2×D4(C2×D59)⋊C412yes—
194C22×C4—22no—
195C22×C4—2361no—
196C22×C4#1174568512yes—
197C22×C4C2×C22⋊D5911yescharacteristic
198Q16—2364no—
199SD16—2364no—
200SD16—2364no—
201D8—2364no—
202M16—82no—
203C2×C8—24no—
204C2×C8—82no—
205C2×C8—1184no—
206C2×C8—1182no—
207C2×C8—2362no—
208C4⋊C4—22no—
209C4⋊C4—1182no—
210C22⋊C4—24no—
211C22⋊C4—2362no—
212C42C2×C4×D5911yescharacteristic
213C42C2×D23611yescharacteristic
214C42C2×C22⋊D5912yes—
215C23—22no—
216C23—2361no—
217C23#1427371011yescharacteristic
218Q8—42no—
219Q8—2362no—
220D4—42no—
221D4—82no—
222D4—82no—
223D4—4721no—
224D4—4721no—
225D4—2362no—
226D4—2364no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—4722no—
233C2×C4—2361no—
234C2×C4C2×(C2×D59)⋊C411yescharacteristic
235C2×C4C4×C22⋊D5912yes—
236C2×C4#1427369611yescharacteristic
237C2×C4#1427370112yes—
238C8—44no—
239C8—1184no—
240C8—4722no—
241C22—22no—
242C22—21no—
243C22—44no—
244C22—42no—
245C22—4722no—
246C22—4721no—
247C22#1683728111yescharacteristic
248C4—22no—
249C4—22no—
250C4—21no—
251C4—42no—
252C4—4721no—
253C4#1683757012yes—
254C2—41no—
255C2—82no—
256C2—9441no—
257C2#2142445812yes—
258C2#2142449411yescharacteristic
subgroupG/H
centerC2×C4#14273696
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#2980C2
socleC22×C59#386
prime radical#21434541C2
cosocleC2×C42×C59C23
hypercenter#2980D59
hypocenterC59#9007
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2980

Complements

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

NH ≅ G/Ntype
2. #21422758C2semidirect
3. #21422564C2semidirect
4. #21421994C2semidirect
5. C59×#2980C2semidirect
126. #2980D59semidirect
150. C59#9007semidirect

Maximal subgroups

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

subgroupindex
#214345412
#214227582
#214225642
#214219942
C59×#29802
#900759