Finite Group Databasebeta

← #28491788

Subgroups — #28491788

Order 15104.

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

normal subgroups only

This group has 438 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#21432459C211yesmaximal, characteristic, complemented
2#21424924C211yesmaximal, characteristic
3#21422657C212yesmaximal
4#21421955C212yesmaximal, complemented
5C59×#1276C211yesmaximal, characteristic, complemented
6#16837952C2211yescharacteristic
7C2×#14273719—21no—
8#16837426C2211yescharacteristic
9C2×#14273655—21no—
10C2×#14273655—21no—
11#16837300—24no—
12#16837272—21no—
13#16837268C2212yes—
14#16837137—24no—
15C59×#431C2212yes—
16C43×C59C2211yescharacteristic
17C2×#11745685—21no—
18C2×#11745664—21no—
19#14273727Q811yescharacteristic
20#14273724Q811yescharacteristic
21#14273719—42no—
22#14273719—42no—
23#14273715—21no—
24#14273701—22no—
25C4×D236—22no—
26C2×C4⋊Dic59—24no—
27C2×C4⋊Dic59—22no—
28C2×Dic59⋊C4—24no—
29C4⋊(C59⋊C8)—22no—
30#14273655—42no—
31#14273655—42no—
32C59⋊#95—22no—
33C59⋊#95D414yes—
34C2×C42×C59—22no—
35C2×C42×C59D411yescharacteristic
36C2×C42×C59D411yescharacteristic
37C2×C42×C59C2311yescharacteristic
38C4⋊C8×C59—22no—
39C22⋊C8×C59—22no—
40C4×C8×C59—22no—
41C22×C4×C59—21no—
42C22×C4×C59—21no—
43C22×C4×C59C2×D412yes—
44C22×C4×C59C2×D411yescharacteristic
45C2×C22⋊D59—41no—
46C22×Dic59—24no—
47#11745685—41no—
48#11745685—41no—
49#11745685—41no—
50#11745685—41no—
51C2×D236—21no—
52C2×D236—22no—
53C2×C4×D59—41no—
54C2×Dic118—21no—
55C2×C8×C59—42no—
56C42×C59—21no—
57C42×C59—21no—
58C42×C59—44no—
59C42×C59D812yes—
60C42×C59SD1612yes—
61C42×C59C2×Q811yescharacteristic
62C42×C59#4112yes—
63C42×C59#4111yescharacteristic
64(C2×D59)⋊C4—42no—
65C4⋊Dic59—22no—
66C4⋊Dic59—48no—
67C4⋊Dic59—42no—
68C4⋊Dic59—42no—
69Dic59⋊C4—48no—
70#11745664—41no—
71#11745664—41no—
72C2×C59⋊C8—41no—
73C23×C59#12011yescharacteristic
74C22×D59—41no—
75C2×C4×C59—24no—
76C2×C4×C59—22no—
77C2×C4×C59—22no—
78C2×C4×C59—42no—
79C2×C4×C59—42no—
80C2×C4×C59C22⋊Q812yes—
81C2×C4×C59C22⋊Q811yescharacteristic
82C2×C4×C59#12412yes—
83C2×C4×C59#12411yescharacteristic
84C2×C4×C59C2×D811yescharacteristic
85C2×C4×C59C2×SD1611yescharacteristic
86C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
87C2×C4×C59#13711yescharacteristic
88C22⋊D59—81no—
89C22⋊D59—81no—
90C2×Dic59—48no—
91C2×Dic59—41no—
92C2×Dic59—44no—
93D236—41no—
94D236—41no—
95D236—82no—
96C4×D59—81no—
97C4×D59—81no—
98Dic118—41no—
99Dic118—41no—
100C8×C59—82no—
101C59⋊C8—81no—
102#10364—591nomaximal, Sylow
103C22×C59—22no—
104C22×C59#39311yescharacteristic
105C22×C59#44711yescharacteristic
106C22×C59#44811yescharacteristic
107C2×D59—81no—
108C2×D59—82no—
109C4×C59—24no—
110C4×C59—22no—
111C4×C59—22no—
112C4×C59—42no—
113C4×C59#47411yescharacteristic
114C4×C59#47611yescharacteristic
115C4×C59#48611yescharacteristic
116C4×C59#48911yescharacteristic
117Dic59—84no—
118Dic59—81no—
119#2859—591no—
120#1962—591no—
121#1517—592no—
122#1286—592no—
123#1276D5911yescharacteristic, complemented
124C2×C59—21no—
125C2×C59—21no—
126C2×C59#202311yescharacteristic
127C2×C59#203112yes—
128D59—161no—
129#532—591no—
130C2×#125—1181no—
131#431C2×D5912yes—
132#423—591no—
133C2×#104—1181no—
134C2×#104—1181no—
135#389—1184no—
136#384—1181no—
137#382—592no—
138C43C2×D5911yescharacteristic
139#327—1184no—
140C59#1036411yescharacteristic, complemented, Sylow
141C2×#41—1181no—
142C2×M16—1181no—
143C4⋊Q8—591no—
144#127—591no—
145#125—2362no—
146#125—2362no—
147#124—1181no—
148#121—1182no—
149C4×D4—1182no—
150C2×C4⋊C4—1184no—
151C2×C4⋊C4—1184no—
152C2×C4⋊C4—1182no—
153C2×C42—22no—
154C2×C42D23611yescharacteristic
155C2×C42D23611yescharacteristic
156C2×C42C22×D5911yescharacteristic
157C4⋊C8—22no—
158C4⋊C8—1182no—
159#104—2362no—
160#104—2362no—
161C22⋊C8—22no—
162C4×C8—22no—
163#95—594no—
164#95—1182no—
165#41—2361no—
166#41—2361no—
167#41—2361no—
168#41—2361no—
169C2×Q8—1181no—
170C2×D4—1182no—
171C2×D4—1181no—
172C2×D4—2361no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—1184no—
176C22×C4—2361no—
177C22×C4C2×D23611yescharacteristic
178C22×C4D4×D5912yes—
179M16—2361no—
180M16—2361no—
181C2×C8—42no—
182C2×C8—2361no—
183C4⋊C4—1182no—
184C4⋊C4—2368no—
185C4⋊C4—2368no—
186C4⋊C4—2362no—
187C4⋊C4—2362no—
188C22⋊C4—2362no—
189C42—21no—
190C42—21no—
191C42—44no—
192C42#1174566012yes—
193C42D47212yes—
194C42#1174568512yes—
195C42#1174568711yescharacteristic
196C42Q8×D5911yescharacteristic
197C23—2361no—
198C23#1427369711yescharacteristic
199Q8—2361no—
200Q8—2361no—
201D4—4722no—
202D4—4721no—
203D4—4721no—
204D4—2361no—
205D4—2361no—
206C2×C4—24no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—4721no—
212C2×C4—4721no—
213C2×C4—2368no—
214C2×C4—2364no—
215C2×C4—2361no—
216C2×C4#1427370512yes—
217C2×C4#1427370711yescharacteristic
218C2×C4#1427371312yes—
219C2×C4#1427371511yescharacteristic
220C2×C4C2×D47211yescharacteristic
221C2×C4C2×#1174566011yescharacteristic
222C2×C4#1427375511yescharacteristic
223C2×C4#1427376011yescharacteristic
224C8—82no—
225C8—4721no—
226C22—22no—
227C22—4721no—
228C22—4722no—
229C22#1683731511yescharacteristic
230C22#1683753811yescharacteristic
231C22#1683754311yescharacteristic
232C4—24no—
233C4—22no—
234C4—22no—
235C4—42no—
236C4—4724no—
237C4—4721no—
238C4#1683768311yescharacteristic
239C4#1683769311yescharacteristic
240C4#1683774911yescharacteristic
241C4#1683776211yescharacteristic
242C2—21no—
243C2—21no—
244C2—9441no—
245C2#2142523311yescharacteristic
246C2#2142526912yes—
subgroupG/H
centerC22#16837315
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
FittingC59×#1276C2
socleC22×C59#393
prime radical#16837952C22
hypercenter#1276D59
hypocenterC59#10364
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. #21432459C2semidirect
4. #21421955C2semidirect
5. C59×#1276C2semidirect
123. #1276D59semidirect
140. C59#10364semidirect

Maximal subgroups

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

subgroupindex
#214324592
#214249242
#214226572
#214219552
C59×#12762
#1036459