Finite Group Databasebeta

← #28489542

Subgroups — #28489542

Order 15104.

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

normal subgroups only

This group has 566 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#21434127C212yesmaximal
2#21424928C211yesmaximal, characteristic
3#21422525C212yesmaximal, complemented
4#21421925C211yesmaximal, characteristic, complemented
5C59×#2399C211yesmaximal, characteristic, complemented
6#16837952C2211yescharacteristic
7#16837720—22no—
8#16837687—22no—
9#16837602—22no—
10#16837541—22no—
11C4×#11745671—24no—
12#16837455C2212yes—
13#16837426C2211yescharacteristic
14C2×#14273655—22no—
15Q8⋊(C59⋊C8)—24no—
16C2×C4×Q8×C59C2212yes—
17C59×#398—22no—
18C59×#395—22no—
19C4×C4⋊C4×C59—22no—
20C59×#376C2211yescharacteristic
21C2×#11745685—21no—
22C2×#11745671—24no—
23C2×#11745664—21no—
24#14273727D411yescharacteristic
25#14273724D411yescharacteristic
26#14273715—21no—
27#14273701—22no—
28C4×D236—22no—
29#14273658—24no—
30C4⋊(C59⋊C8)—22no—
31C4⋊(C59⋊C8)—22no—
32#14273655—44no—
33#14273651—24no—
34#14273647—24no—
35C22⋊(C59⋊C8)—22no—
36C4×C59⋊C8D414yes—
37C22×Q8×C59D412yes—
38C4×Q8×C59—22no—
39C4×Q8×C59—22no—
40C4×Q8×C59—22no—
41C4×Q8×C59—22no—
42C2×C4⋊C4×C59—24no—
43C2×C4⋊C4×C59—22no—
44C2×C4⋊C4×C59D412yes—
45C2×C42×C59—22no—
46C2×C42×C59D412yes—
47C2×C42×C59C2311yescharacteristic
48#95×C59—22no—
49#95×C59—22no—
50#95×C59—22no—
51C2×Q8×C59—22no—
52C2×Q8×C59—22no—
53C2×Q8×C59—44no—
54C22×C4×C59—22no—
55C22×C4×C59—22no—
56C22×C4×C59—22no—
57C22×C4×C59C2×D412yes—
58C22×C4×C59C2×D411yescharacteristic
59C2×C22⋊D59—41no—
60#11745685—42no—
61#11745685—42no—
62C2×D236—22no—
63C2×D236—21no—
64C2×C4×D59—41no—
65C2×Dic118—21no—
66C4⋊C4×C59—22no—
67C4⋊C4×C59—22no—
68C4⋊C4×C59—48no—
69C4⋊C4×C59—42no—
70C4⋊C4×C59—42no—
71C42×C59—22no—
72C42×C59—42no—
73C42×C59—42no—
74C42×C59SD1614yescomplemented
75C42×C59C2×D411yescharacteristic
76C42×C59C2×D412yes—
77C42×C59#4111yescharacteristic
78#11745671—48no—
79(C2×D59)⋊C4—42no—
80C4⋊Dic59—22no—
81#11745664—42no—
82C2×C59⋊C8—24no—
83C2×C59⋊C8—41no—
84C23×C59#12011yescharacteristic
85C22×D59—41no—
86Q8×C59—24no—
87Q8×C59—24no—
88Q8×C59—42no—
89Q8×C59—42no—
90C2×C4×C59—24no—
91C2×C4×C59—24no—
92C2×C4×C59—42no—
93C2×C4×C59—44no—
94C2×C4×C59—42no—
95C2×C4×C59—42no—
96C2×C4×C59—44no—
97C2×C4×C59—42no—
98C2×C4×C59#12012yes—
99C2×C4×C59#12111yescharacteristic
100C2×C4×C59#12112yes—
101C2×C4×C59#12711yescharacteristic
102C2×C4×C59C2×SD1612yes—
103C2×C4×C59#13712yes—
104C22⋊D59—82no—
105C2×Dic59—41no—
106D236—44no—
107D236—42no—
108C4×D59—81no—
109C4×D59—81no—
110Dic118—42no—
111C59⋊C8—44no—
112C59⋊C8—81no—
113#10014—591nomaximal, Sylow
114C22×C59—22no—
115C22×C59#39211yescharacteristic
116C22×C59#44812yes—
117C2×D59—82no—
118C2×D59—81no—
119C4×C59—22no—
120C4×C59—22no—
121C4×C59—44no—
122C4×C59—44no—
123C4×C59—82no—
124C4×C59#46112yes—
125C4×C59#49211yescharacteristic
126C4×C59#49711yescharacteristic
127Dic59—81no—
128#3103—592no—
129#2399D5911yescharacteristic, complemented
130#1963—591no—
131#1487—592no—
132#1274—591no—
133C2×C59—22no—
134C2×C59#201012yes—
135C2×C59#201311yescharacteristic
136D59—161no—
137#532—591no—
138C2×C4×Q8C2×D5912yes—
139#481—1182no—
140#475—1182no—
141#460—1182no—
142#448—1182no—
143C4×SD16—1184no—
144#431—592no—
145#423—591no—
146C2×#104—1182no—
147#398—22no—
148#395—22no—
149C4×C4⋊C4—22no—
150#376C2×D5911yescharacteristic
151Q8⋊C8—1184no—
152C59#1001411yescharacteristic, complemented, Sylow
153C2×#41—1181no—
154C22×Q8C22⋊D5912yes—
155C2×SD16—1184no—
156C2×M16—1181no—
157C4⋊Q8—591no—
158#127—591no—
159#124—1181no—
160#121—1182no—
161C4×Q8—22no—
162C4×Q8—22no—
163C4×Q8—22no—
164C4×Q8—22no—
165C4×D4—1182no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4C22⋊D5912yes—
169C2×C42—22no—
170C2×C42C22⋊D5912yes—
171C2×C42C22×D5911yescharacteristic
172#106—1184no—
173C4⋊C8—1182no—
174C4⋊C8—1182no—
175#104—2364no—
176#103—1184no—
177#102—1184no—
178C22⋊C8—1182no—
179C4×C8—594no—
180#95—22no—
181#95—22no—
182#95—22no—
183#41—2362no—
184#41—2362no—
185C2×Q8—22no—
186C2×Q8—22no—
187C2×Q8—44no—
188C2×Q8—1181no—
189C2×D4—1182no—
190C2×D4—1181no—
191C2×D4—2361no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—2361no—
196C22×C4C2×C22⋊D5912yes—
197C22×C4C2×C22⋊D5911yescharacteristic
198SD16—2368no—
199M16—2362no—
200C2×C8—1184no—
201C2×C8—2361no—
202C4⋊C4—22no—
203C4⋊C4—22no—
204C4⋊C4—48no—
205C4⋊C4—42no—
206C4⋊C4—42no—
207C4⋊C4—1182no—
208C22⋊C4—2362no—
209C42—22no—
210C42—42no—
211C42—42no—
212C42#1174567114yescomplemented
213C42D4×D5911yescharacteristic
214C42D4×D5912yes—
215C42#1174568711yescharacteristic
216C23—2361no—
217C23#1427369611yescharacteristic
218Q8—24no—
219Q8—24no—
220Q8—42no—
221Q8—42no—
222Q8—2362no—
223D4—4722no—
224D4—2364no—
225D4—2362no—
226C2×C4—24no—
227C2×C4—24no—
228C2×C4—42no—
229C2×C4—44no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—44no—
233C2×C4—42no—
234C2×C4—4721no—
235C2×C4—4721no—
236C2×C4—2361no—
237C2×C4#1427369512yes—
238C2×C4#1427369811yescharacteristic
239C2×C4#1427369912yes—
240C2×C4#1427372311yescharacteristic
241C2×C4C2×#1174567112yes—
242C2×C4#1427375812yes—
243C8—4721no—
244C8—2364no—
245C22—22no—
246C22—4722no—
247C22—4721no—
248C22#1683731011yescharacteristic
249C22#1683754112yes—
250C4—22no—
251C4—22no—
252C4—44no—
253C4—44no—
254C4—82no—
255C4—4721no—
256C4#1683760412yes—
257C4#1683777611yescharacteristic
258C4#1683779411yescharacteristic
259C2—22no—
260C2—9441no—
261C2#2142516112yes—
262C2#2142517911yescharacteristic
subgroupG/H
centerC2×C4#14273723
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
FittingC59×#2399C2
socleC22×C59#392
prime radical#16837952C22
hypercenter#2399D59
hypocenterC59#10014
LCSG ▷ C2×C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2399

Complements

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

NH ≅ G/Ntype
3. #21422525C2semidirect
4. #21421925C2semidirect
5. C59×#2399C2semidirect
74. C42×C59SD16semidirect
129. #2399D59semidirect
152. C59#10014semidirect
212. C42#11745671semidirect

Maximal subgroups

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

subgroupindex
#214341272
#214249282
#214225252
#214219252
C59×#23992
#1001459