Finite Group Databasebeta

← #28501895

Subgroups — #28501895

Order 15104.

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

normal subgroups only

This group has 344 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, complemented
2#21425519C211yesmaximal, characteristic
3#21422108C212yesmaximal, complemented
4#21422020C211yesmaximal, characteristic
5C59×#3189C211yesmaximal, characteristic
6C2×#14273661C2211yescharacteristic
7#16837262—21no—
8#16837229—22no—
9#16837206—22no—
10C22⋊(C59⋊C16)C2212yes—
11#16837165—21no—
12#16837165C2211yescharacteristic
13C2×C22⋊Q8×C59—21no—
14C59×#484—22no—
15C59×#467—21no—
16C59×#467C2212yes—
17C59×#451—22no—
18C2×C22⋊C8×C59C2211yescharacteristic
19C2×C4⋊Dic59—21no—
20C2×C4⋊Dic59—21no—
21C2×C22⋊Dic59—41no—
22C2×C59⋊C16—22no—
23#14273661—22no—
24C59⋊#95—41no—
25C22×Q8×C59—21no—
26C23×C4×C59D411yescharacteristic
27C2×Q16×C59—21no—
28C2×Q16×C59D412yes—
29C22×C8×C59D411yescharacteristic
30C22×C8×C59C2311yescharacteristic
31C22⋊Q8×C59—22no—
32C22⋊Q8×C59—22no—
33C22⋊Q8×C59—41no—
34C22⋊Q8×C59—41no—
35C2×C4⋊C4×C59—21no—
36C2×C4⋊C4×C59—41no—
37C2×C22⋊C4×C59—41no—
38#107×C59—21no—
39#107×C59D412yes—
40#103×C59—22no—
41#103×C59—22no—
42C22⋊C8×C59—22no—
43C24×C59D811yescharacteristic
44C2×Q8×C59—22no—
45C2×Q8×C59—41no—
46C2×Q8×C59—81no—
47C2×Q8×C59—81no—
48C22×C4×C59—21no—
49C22×C4×C59—21no—
50C22×C4×C59—41no—
51C22×C4×C59—41no—
52C22×C4×C59—41no—
53C22×C4×C59—41no—
54C22×C4×C59D811yescharacteristic
55C22×C4×C59C2×D411yescharacteristic
56C22×Dic59—41no—
57C22×Dic59—41no—
58Q16×C59—42no—
59Q16×C59—81no—
60C2×C8×C59—21no—
61C2×C8×C59—41no—
62C2×C8×C59—41no—
63C2×C8×C59C2×D412yes—
64C4⋊C4×C59—22no—
65C4⋊C4×C59—41no—
66C4⋊C4×C59—42no—
67C4⋊C4×C59—42no—
68C22⋊C4×C59—42no—
69C22⋊C4×C59—42no—
70C22⋊Dic59—82no—
71C4⋊Dic59—41no—
72C4⋊Dic59—41no—
73C4⋊Dic59—41no—
74C4⋊Dic59—41no—
75C59⋊C16—42no—
76C23×C59—21no—
77C23×C59—21no—
78C23×C59—41no—
79C23×C59—41no—
80C23×C59SD3212yes—
81C23×C59C2×D811yescharacteristic
82Q8×C59—42no—
83Q8×C59—42no—
84Q8×C59—82no—
85C2×C4×C59—21no—
86C2×C4×C59—21no—
87C2×C4×C59—21no—
88C2×C4×C59—41no—
89C2×C4×C59—41no—
90C2×C4×C59—41no—
91C2×C4×C59—42no—
92C2×C4×C59—42no—
93C2×C4×C59—81no—
94C2×C4×C59—82no—
95C2×C4×C59—82no—
96C2×C4×C59#12011yescharacteristic
97C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
98C2×Dic59—81no—
99C2×Dic59—81no—
100C2×Dic59—81no—
101C2×Dic59—81no—
102C2×Dic59—81no—
103C2×Dic59—81no—
104C8×C59—22no—
105C8×C59—41no—
106#12279—591nomaximal, Sylow
107C22×C59—21no—
108C22×C59—21no—
109C22×C59—22no—
110C22×C59—42no—
111C22×C59—41no—
112C22×C59—41no—
113C22×C59—81no—
114C22×C59#44711yescharacteristic
115C22×C59C2×SD3211yescharacteristic
116C22×C59#51011yescharacteristic
117C4×C59—21no—
118C4×C59—41no—
119C4×C59—82no—
120C4×C59—82no—
121C4×C59#45511yescharacteristic
122C4×C59#45811yescharacteristic
123Dic59—161no—
124Dic59—161no—
125#3189D5911yescharacteristic
126#2235—592no—
127#2075—591no—
128#1348—592no—
129#1314—591no—
130C2×C59—21no—
131C2×C59—21no—
132C2×C59—42no—
133C2×C59#160711yescharacteristic
134C2×C59#218511yescharacteristic
135C2×C59#219111yescharacteristic
136C2×C22⋊Q8—21no—
137#484—22no—
138#467—21no—
139#467C2×D5912yes—
140#451—22no—
141C2×#107—591no—
142C2×C22⋊C8C2×D5911yescharacteristic
143#380—1181no—
144#367—1182no—
145#358—1182no—
146C22⋊C16—592no—
147#340—591no—
148#340—1181no—
149C59#1227911yescharacteristic, complemented, Sylow
150C22×Q8—21no—
151C23×C4C22⋊D5911yescharacteristic
152C2×Q16—21no—
153C2×Q16C22⋊D5912yes—
154C22×C8C22⋊D5911yescharacteristic
155C22×C8C22×D5911yescharacteristic
156C22⋊Q8—22no—
157C22⋊Q8—22no—
158C22⋊Q8—41no—
159C22⋊Q8—41no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—1181no—
163C2×C4⋊C4—1181no—
164C2×C22⋊C4—41no—
165C2×C22⋊C4—2361no—
166C2×C16—1182no—
167#107—21no—
168#107—1182no—
169#107C22⋊D5912yes—
170#103—22no—
171#103—22no—
172C22⋊C8—22no—
173#95—2361no—
174C24#1174566911yescharacteristic
175C2×Q8—22no—
176C2×Q8—41no—
177C2×Q8—81no—
178C2×Q8—81no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—2361no—
186C22×C4—2361no—
187C22×C4#1174566911yescharacteristic
188C22×C4C2×C22⋊D5911yescharacteristic
189Q16—42no—
190Q16—81no—
191C2×C8—21no—
192C2×C8—41no—
193C2×C8—41no—
194C2×C8C2×C22⋊D5912yes—
195C4⋊C4—22no—
196C4⋊C4—41no—
197C4⋊C4—42no—
198C4⋊C4—42no—
199C4⋊C4—2361no—
200C4⋊C4—2361no—
201C4⋊C4—2361no—
202C4⋊C4—2361no—
203C22⋊C4—42no—
204C22⋊C4—42no—
205C22⋊C4—4722no—
206C16—2362no—
207C23—21no—
208C23—21no—
209C23—41no—
210C23—41no—
211C23#1427367012yes—
212C23C2×#1174566911yescharacteristic
213Q8—42no—
214Q8—42no—
215Q8—82no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—81no—
225C2×C4—82no—
226C2×C4—82no—
227C2×C4—4721no—
228C2×C4—4721no—
229C2×C4—4721no—
230C2×C4—4721no—
231C2×C4—4721no—
232C2×C4—4721no—
233C2×C4#1427369611yescharacteristic
234C2×C4#1427375311yescharacteristic
235C8—22no—
236C8—41no—
237C22—21no—
238C22—21no—
239C22—22no—
240C22—42no—
241C22—41no—
242C22—41no—
243C22—81no—
244C22#1683753611yescharacteristic
245C22C2×#1427367011yescharacteristic
246C22#1683784311yescharacteristic
247C4—21no—
248C4—41no—
249C4—82no—
250C4—82no—
251C4—9441no—
252C4—9441no—
253C4#1683758011yescharacteristic
254C4#1683759211yescharacteristic
255C2—21no—
256C2—21no—
257C2—42no—
258C2#2142308211yescharacteristic
259C2#2142601311yescharacteristic
260C2#2142604711yescharacteristic
subgroupG/H
centerC22#16837536
commutator, cosocleC22×C8×C59C23
FrattiniC22×C8C22×D59
FittingC59×#3189C2
socleC22×C59#447
prime radicalC24×C59D8
hypercenter#3189D59
hypocenterC59#12279
LCSG ▷ C22×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C8 ◁ #3189

Complements

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

NH ≅ G/Ntype
1. #21426291C2semidirect
3. #21422108C2semidirect
149. C59#12279semidirect

Maximal subgroups

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

subgroupindex
#214262912
#214255192
#214221082
#214220202
C59×#31892
#1227959