Finite Group Databasebeta

← #28501957

Subgroups — #28501957

Order 15104.

310 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 71 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#21426291C212yesmaximal
2#21425584C211yesmaximal, characteristic
3#21422108C212yesmaximal
4#21422036C211yesmaximal, characteristic
5C59×#3249C211yesmaximal, characteristic
6C2×#14273661C2211yescharacteristic
7#16837300—21no—
8#16837229—22no—
9#16837206—22no—
10C22⋊(C59⋊C16)C2212yes—
11#16837165—21no—
12#16837165C2211yescharacteristic
13C59×#533—21no—
14C59×#477—22no—
15C59×#467—21no—
16C59×#467C2212yes—
17C59×#462—22no—
18C2×C4⋊C8×C59C2211yescharacteristic
19C2×C4⋊Dic59—21no—
20C2×C4⋊Dic59—21no—
21C2×Dic59⋊C4—41no—
22C2×C59⋊C16—22no—
23#14273661—22no—
24C59⋊#95—41no—
25C2×Q16×C59—21no—
26C2×Q16×C59D412yes—
27C22×C8×C59D411yescharacteristic
28C22×C8×C59C2311yescharacteristic
29C4⋊Q8×C59—21no—
30C4⋊Q8×C59—21no—
31#125×C59—41no—
32C22⋊Q8×C59—22no—
33C22⋊Q8×C59—41no—
34C4×Q8×C59—22no—
35C4×Q8×C59—41no—
36#117×C59—41no—
37C2×C42×C59D411yescharacteristic
38#107×C59—21no—
39#107×C59D412yes—
40C4⋊C8×C59—22no—
41#103×C59—22no—
42#103×C59—22no—
43C2×Q8×C59—22no—
44C2×Q8×C59—41no—
45C22×C4×C59D811yescharacteristic
46C22×C4×C59D811yescharacteristic
47C22×C4×C59C2×D411yescharacteristic
48C22×Dic59—41no—
49C22×Dic59—41no—
50Q16×C59—42no—
51Q16×C59—81no—
52C2×C8×C59—21no—
53C2×C8×C59—41no—
54C2×C8×C59—41no—
55C2×C8×C59C2×D412yes—
56C4⋊C4×C59—22no—
57C4⋊C4×C59—41no—
58C4⋊C4×C59—41no—
59C4⋊C4×C59—42no—
60C4⋊C4×C59—41no—
61C4⋊C4×C59—42no—
62C22⋊C4×C59—42no—
63C42×C59—21no—
64C42×C59—21no—
65C42×C59—42no—
66C4⋊Dic59—22no—
67C4⋊Dic59—41no—
68C4⋊Dic59—41no—
69C4⋊Dic59—41no—
70Dic59⋊C4—82no—
71C59⋊C16—42no—
72C23×C59C2×D811yescharacteristic
73Q8×C59—42no—
74Q8×C59—42no—
75Q8×C59—82no—
76C2×C4×C59—21no—
77C2×C4×C59—21no—
78C2×C4×C59—21no—
79C2×C4×C59—21no—
80C2×C4×C59—21no—
81C2×C4×C59—41no—
82C2×C4×C59—42no—
83C2×C4×C59—42no—
84C2×C4×C59SD3212yes—
85C2×C4×C59#12011yescharacteristic
86C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
87C2×Dic59—42no—
88C2×Dic59—81no—
89C2×Dic59—81no—
90C2×Dic59—81no—
91C2×Dic59—81no—
92C2×Dic59—81no—
93C8×C59—22no—
94C8×C59—41no—
95#12289—591nomaximal, Sylow
96C22×C59—21no—
97C22×C59—21no—
98C22×C59#44711yescharacteristic
99C22×C59C2×SD3211yescharacteristic
100C22×C59#51011yescharacteristic
101C4×C59—21no—
102C4×C59—22no—
103C4×C59—41no—
104C4×C59—82no—
105C4×C59—82no—
106C4×C59#45611yescharacteristic
107C4×C59#45811yescharacteristic
108Dic59—82no—
109Dic59—161no—
110#3249D5911yescharacteristic
111#2235—592no—
112#2086—591no—
113#1348—592no—
114#1322—591no—
115C2×C59—21no—
116C2×C59—21no—
117C2×C59#161011yescharacteristic
118C2×C59#218511yescharacteristic
119C2×C59#219111yescharacteristic
120#533—21no—
121#477—22no—
122#467—21no—
123#467C2×D5912yes—
124#462—22no—
125C2×#107—591no—
126C2×C4⋊C8C2×D5911yescharacteristic
127#389—1181no—
128#367—1182no—
129#358—1182no—
130C22⋊C16—592no—
131#340—591no—
132#340—1181no—
133C59#1228911yescharacteristic, complemented, Sylow
134C2×Q16—21no—
135C2×Q16C22⋊D5912yes—
136C22×C8C22⋊D5911yescharacteristic
137C22×C8C22×D5911yescharacteristic
138C4⋊Q8—21no—
139C4⋊Q8—21no—
140#125—41no—
141C22⋊Q8—22no—
142C22⋊Q8—41no—
143C4×Q8—22no—
144C4×Q8—41no—
145#117—41no—
146C2×C4⋊C4—1181no—
147C2×C4⋊C4—1181no—
148C2×C4⋊C4—2361no—
149C2×C42C22⋊D5911yescharacteristic
150C2×C16—1182no—
151#107—21no—
152#107—1182no—
153#107C22⋊D5912yes—
154C4⋊C8—22no—
155#103—22no—
156#103—22no—
157#95—2361no—
158C2×Q8—22no—
159C2×Q8—41no—
160C22×C4—2361no—
161C22×C4—2361no—
162C22×C4#1174566911yescharacteristic
163C22×C4#1174566911yescharacteristic
164C22×C4C2×C22⋊D5911yescharacteristic
165Q16—42no—
166Q16—81no—
167C2×C8—21no—
168C2×C8—41no—
169C2×C8—41no—
170C2×C8C2×C22⋊D5912yes—
171C4⋊C4—22no—
172C4⋊C4—41no—
173C4⋊C4—41no—
174C4⋊C4—42no—
175C4⋊C4—41no—
176C4⋊C4—42no—
177C4⋊C4—1182no—
178C4⋊C4—4722no—
179C4⋊C4—2361no—
180C4⋊C4—2361no—
181C4⋊C4—2361no—
182C22⋊C4—42no—
183C42—21no—
184C42—21no—
185C42—42no—
186C16—2362no—
187C23C2×#1174566911yescharacteristic
188Q8—42no—
189Q8—42no—
190Q8—82no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—41no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—4721no—
200C2×C4—4721no—
201C2×C4—4721no—
202C2×C4—4721no—
203C2×C4—4721no—
204C2×C4—2362no—
205C2×C4#1427367012yes—
206C2×C4#1427369611yescharacteristic
207C2×C4#1427375311yescharacteristic
208C8—22no—
209C8—41no—
210C22—21no—
211C22—21no—
212C22#1683753611yescharacteristic
213C22C2×#1427367011yescharacteristic
214C22#1683784311yescharacteristic
215C4—22no—
216C4—21no—
217C4—41no—
218C4—82no—
219C4—82no—
220C4—9441no—
221C4—4722no—
222C4#1683758511yescharacteristic
223C4#1683759211yescharacteristic
224C2—21no—
225C2—21no—
226C2#2142309911yescharacteristic
227C2#2142601311yescharacteristic
228C2#2142604711yescharacteristic
subgroupG/H
centerC22#16837536
commutator, cosocleC22×C8×C59C23
FrattiniC22×C8C22×D59
FittingC59×#3249C2
socleC22×C59#447
prime radicalC23×C59C2×D8
hypercenter#3249D59
hypocenterC59#12289
LCSG ▷ C22×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C8 ◁ #3249

Complements

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

NH ≅ G/Ntype
133. C59#12289semidirect

Maximal subgroups

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

subgroupindex
#214262912
#214255842
#214221082
#214220362
C59×#32492
#1228959