Finite Group Databasebeta

← #28491778

Subgroups — #28491778

Order 15104.

454 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 97 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#21432474C211yesmaximal, characteristic, complemented
2#21424924C211yesmaximal, characteristic
3#21422657C212yesmaximal
4#21421955C212yesmaximal, complemented
5C59×#1276C211yesmaximal, characteristic, complemented
6#16837952C2211yescharacteristic
7C2×#14273727—21no—
8#16837426C2211yescharacteristic
9C2×#14273655—21no—
10C2×#14273655—21no—
11#16837308—24no—
12#16837268—21no—
13#16837268C2212yes—
14#16837137—24no—
15C59×#431C2212yes—
16C43×C59C2211yescharacteristic
17C2×#11745685—21no—
18C22×Dic118—22no—
19C2×#11745664—21no—
20#14273727—42no—
21#14273727—42no—
22#14273727Q811yescharacteristic, complemented
23#14273724Q811yescharacteristic, complemented
24#14273715—21no—
25#14273701—22no—
26C4×D236—22no—
27C2×C4⋊Dic59—22no—
28C2×C4⋊Dic59—22no—
29C4⋊(C59⋊C8)—22no—
30#14273655—42no—
31#14273655—42no—
32C59⋊#95—24no—
33C59⋊#95—22no—
34C59⋊#95D414yes—
35C2×C42×C59—22no—
36C2×C42×C59D411yescharacteristic
37C2×C42×C59D411yescharacteristic
38C2×C42×C59C2311yescharacteristic
39C4⋊C8×C59—22no—
40C22⋊C8×C59—22no—
41C4×C8×C59—22no—
42C22×C4×C59—21no—
43C22×C4×C59—21no—
44C22×C4×C59C2×D412yes—
45C22×C4×C59C2×D411yescharacteristic
46C2×C22⋊D59—41no—
47C22×Dic59—24no—
48#11745685—41no—
49#11745685—41no—
50#11745685—41no—
51#11745685—41no—
52C2×D236—21no—
53C2×D236—22no—
54C2×C4×D59—41no—
55C2×Dic118—21no—
56C2×Dic118—44no—
57C2×Dic118—48no—
58C2×C8×C59—42no—
59C42×C59—21no—
60C42×C59—21no—
61C42×C59—44no—
62C42×C59D812yes—
63C42×C59SD1612yes—
64C42×C59C2×Q811yescharacteristic
65C42×C59#4112yes—
66C42×C59#4111yescharacteristic
67(C2×D59)⋊C4—42no—
68C4⋊Dic59—22no—
69C4⋊Dic59—44no—
70C4⋊Dic59—42no—
71C4⋊Dic59—42no—
72#11745664—41no—
73#11745664—41no—
74C2×C59⋊C8—41no—
75C23×C59#12011yescharacteristic
76C22×D59—41no—
77C2×C4×C59—24no—
78C2×C4×C59—22no—
79C2×C4×C59—22no—
80C2×C4×C59—42no—
81C2×C4×C59—42no—
82C2×C4×C59C22⋊Q812yes—
83C2×C4×C59C22⋊Q811yescharacteristic
84C2×C4×C59#12412yes—
85C2×C4×C59#12411yescharacteristic
86C2×C4×C59C2×D811yescharacteristic
87C2×C4×C59C2×SD1611yescharacteristic
88C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
89C2×C4×C59#13711yescharacteristic
90C22⋊D59—81no—
91C22⋊D59—81no—
92C2×Dic59—48no—
93C2×Dic59—41no—
94C2×Dic59—44no—
95D236—41no—
96D236—41no—
97D236—82no—
98C4×D59—81no—
99C4×D59—81no—
100Dic118—41no—
101Dic118—41no—
102Dic118—88no—
103C8×C59—82no—
104C59⋊C8—81no—
105#10362—591nomaximal, Sylow
106C22×C59—22no—
107C22×C59#39311yescharacteristic
108C22×C59#44711yescharacteristic
109C22×C59#44811yescharacteristic
110C2×D59—81no—
111C2×D59—82no—
112C4×C59—24no—
113C4×C59—22no—
114C4×C59—22no—
115C4×C59—42no—
116C4×C59#47411yescharacteristic
117C4×C59#47611yescharacteristic
118C4×C59#48611yescharacteristic
119C4×C59#48911yescharacteristic
120Dic59—84no—
121Dic59—81no—
122#2862—591no—
123#1962—591no—
124#1517—592no—
125#1286—592no—
126#1276D5911yescharacteristic, complemented
127C2×C59—21no—
128C2×C59—21no—
129C2×C59#202311yescharacteristic
130C2×C59#203112yes—
131D59—161no—
132#532—591no—
133C2×C4⋊Q8—1181no—
134#431C2×D5912yes—
135#423—591no—
136C2×#104—1181no—
137C2×#104—1181no—
138#391—1184no—
139#382—592no—
140#382—1181no—
141C43C2×D5911yescharacteristic
142#327—1184no—
143C59#1036211yescharacteristic, complemented, Sylow
144C2×#41—1181no—
145C22×Q8—1182no—
146C2×M16—1181no—
147C4⋊Q8—591no—
148C4⋊Q8—2362no—
149C4⋊Q8—2362no—
150#127—591no—
151#124—1181no—
152#121—1182no—
153C4×D4—1182no—
154C2×C4⋊C4—1182no—
155C2×C4⋊C4—1182no—
156C2×C42—22no—
157C2×C42D23611yescharacteristic
158C2×C42D23611yescharacteristic
159C2×C42C22×D5911yescharacteristic
160C4⋊C8—22no—
161C4⋊C8—1182no—
162#104—2362no—
163#104—2362no—
164C22⋊C8—22no—
165C4×C8—22no—
166#95—594no—
167#95—1184no—
168#95—1182no—
169#41—2361no—
170#41—2361no—
171#41—2361no—
172#41—2361no—
173C2×Q8—1181no—
174C2×Q8—2364no—
175C2×Q8—2368no—
176C2×D4—1182no—
177C2×D4—1181no—
178C2×D4—2361no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—1184no—
182C22×C4—2361no—
183C22×C4C2×D23611yescharacteristic
184C22×C4D4×D5912yes—
185M16—2361no—
186M16—2361no—
187C2×C8—42no—
188C2×C8—2361no—
189C4⋊C4—1182no—
190C4⋊C4—2364no—
191C4⋊C4—2362no—
192C4⋊C4—2362no—
193C22⋊C4—2362no—
194C42—21no—
195C42—21no—
196C42—44no—
197C42#1174566012yes—
198C42D47212yes—
199C42#1174568512yes—
200C42#1174568711yescharacteristic
201C42Q8×D5911yescharacteristic
202C23—2361no—
203C23#1427369711yescharacteristic
204Q8—4728no—
205Q8—2361no—
206Q8—2361no—
207D4—4722no—
208D4—4721no—
209D4—4721no—
210D4—2361no—
211D4—2361no—
212C2×C4—24no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—4721no—
218C2×C4—4721no—
219C2×C4—2368no—
220C2×C4—2364no—
221C2×C4—2361no—
222C2×C4#1427370512yes—
223C2×C4#1427370711yescharacteristic
224C2×C4#1427371312yes—
225C2×C4#1427371511yescharacteristic
226C2×C4C2×D47211yescharacteristic
227C2×C4C2×#1174566011yescharacteristic
228C2×C4#1427375511yescharacteristic
229C2×C4#1427376011yescharacteristic
230C8—82no—
231C8—4721no—
232C22—22no—
233C22—4721no—
234C22—4722no—
235C22#1683731511yescharacteristic
236C22#1683753811yescharacteristic
237C22#1683754311yescharacteristic
238C4—24no—
239C4—22no—
240C4—22no—
241C4—42no—
242C4—4724no—
243C4—4721no—
244C4#1683768311yescharacteristic
245C4#1683769311yescharacteristic
246C4#1683774911yescharacteristic
247C4#1683776211yescharacteristic
248C2—21no—
249C2—21no—
250C2—9441no—
251C2#2142523311yescharacteristic
252C2#2142526912yes—
subgroupG/H
centerC22#16837315
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
FittingC59×#1276C2
socleC22×C59#393
prime radical#16837952C22
hypercenter#1276D59
hypocenterC59#10362
LCSG ▷ C2×C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1276

Complements

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

NH ≅ G/Ntype
1. #21432474C2semidirect
4. #21421955C2semidirect
5. C59×#1276C2semidirect
22. #14273727Q8semidirect
23. #14273724Q8semidirect
126. #1276D59semidirect
143. C59#10362semidirect

Maximal subgroups

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

subgroupindex
#214324742
#214249242
#214226572
#214219552
C59×#12762
#1036259