Finite Group Databasebeta

← #28486094

Subgroups — #28486094

Order 15104.

534 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 119 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#21429088C211yesmaximal, characteristic
2#21424088C211yesmaximal, characteristic
3#21422767C212yesmaximal, complemented
4#21421998C212yesmaximal, complemented
5C59×#2975C211yesmaximal, characteristic
6C2×#14273727C2211yescharacteristic
7C2×#14273714—21no—
8#16837426C2212yes—
9C2×#14273644—21no—
10#16837327C414yes—
11#16837307—21no—
12#16837262—22no—
13C4×C4⋊Dic59C2211yescharacteristic
14#16837171—21no—
15#16837147—24no—
16C2×C4×D4×C59C2211yescharacteristic
17C59×#436—24no—
18C59×#431C2212yes—
19C59×#407—22no—
20C22×Dic118—21no—
21C2×#11745664—22no—
22#14273727—22no—
23#14273727—22no—
24#14273714—44no—
25C2×C4⋊Dic59—21no—
26C2×C4⋊Dic59—21no—
27C2×C4⋊Dic59C2×C412yes—
28C2×C22⋊Dic59—24no—
29C2×C4×Dic59—21no—
30C4⋊(C59⋊C8)—22no—
31C4⋊(C59⋊C8)—22no—
32#14273644—42no—
33C59⋊#95—24no—
34C59⋊#95C2×C412yes—
35C22×D4×C59D411yescharacteristic
36C23×C4×C59D412yes—
37C2×M16×C59—24no—
38C4×D4×C59—22no—
39C4×D4×C59—22no—
40C2×C4⋊C4×C59D411yescharacteristic
41C2×C22⋊C4×C59C2×C412yes—
42C2×C42×C59C2311yescharacteristic
43C4⋊C8×C59D414yes—
44C22⋊C8×C59—24no—
45C22⋊C8×C59—22no—
46C4×C8×C59—22no—
47C24×C59C22⋊C412yes—
48C2×D4×C59—22no—
49C2×D4×C59—21no—
50C2×D4×C59—22no—
51C2×D4×C59—21no—
52C22×C4×C59—22no—
53C22×C4×C59—22no—
54C22×C4×C59—44no—
55C22×C4×C59C22⋊C412yes—
56C22×C4×C59C22×C411yescharacteristic
57C22×C4×C59C2×D411yescharacteristic
58C22×C4×C59C2×D411yescharacteristic
59C22×Dic59—22no—
60C22×Dic59—22no—
61C2×Dic118—42no—
62C2×Dic118—42no—
63C2×Dic118—41no—
64C2×Dic118—41no—
65M16×C59—84no—
66C2×C8×C59—24no—
67C4⋊C4×C59—22no—
68C22⋊C4×C59—24no—
69C42×C59C2×D412yes—
70C42×C59#4112yes—
71C22⋊Dic59—48no—
72C4⋊Dic59—24no—
73C4⋊Dic59—42no—
74C4⋊Dic59—41no—
75C4⋊Dic59—41no—
76C4×Dic59—42no—
77#11745664—42no—
78#11745664—42no—
79C2×C59⋊C8—42no—
80C23×C59—22no—
81C23×C59—22no—
82C23×C59—22no—
83C23×C59—42no—
84C23×C59—42no—
85C23×C59C2×C22⋊C411yescharacteristic
86D4×C59—42no—
87D4×C59—42no—
88C2×C4×C59—22no—
89C2×C4×C59—22no—
90C2×C4×C59—41no—
91C2×C4×C59—44no—
92C2×C4×C59—42no—
93C2×C4×C59—82no—
94C2×C4×C59C4×D412yes—
95C2×C4×C59#12011yescharacteristic
96C2×C4×C59#12112yes—
97C2×C4×C59#12311yescharacteristic
98C2×C4×C59AGL(1,ℤ/8)12yes—
99C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
100C2×C4×C59#13711yescharacteristic
101C2×Dic59—42no—
102C2×Dic59—42no—
103C2×Dic59—42no—
104C2×Dic59—42no—
105C2×Dic59—42no—
106C2×Dic59—42no—
107Dic118—82no—
108Dic118—82no—
109C8×C59—44no—
110C59⋊C8—82no—
111#9401—591nomaximal, Sylow
112C22×C59—21no—
113C22×C59—21no—
114C22×C59—44no—
115C22×C59—42no—
116C22×C59—42no—
117C22×C59—42no—
118C22×C59—82no—
119C22×C59#38611yescharacteristic
120C22×C59#41711yescharacteristic
121C22×C59#41811yescharacteristic
122C4×C59—21no—
123C4×C59—22no—
124C4×C59—42no—
125C4×C59#45512yes—
126C4×C59#46812yes—
127C4×C59#46912yes—
128Dic59—82no—
129Dic59—82no—
130#2975D5911yescharacteristic
131#2547—591no—
132#1800—591no—
133#1546—592no—
134#1303—592no—
135C2×C59—21no—
136C2×C59—21no—
137C2×C59—82no—
138C2×C59#188511yescharacteristic
139C2×C59#189511yescharacteristic
140C2×C59#191511yescharacteristic
141C2×C4⋊Q8—591no—
142C2×#124—1181no—
143C2×C4×D4C2×D5911yescharacteristic
144#436—24no—
145#431C2×D5912yes—
146#423—592no—
147C2×#100—1181no—
148#407—22no—
149#397—594no—
150#391—1181no—
151#380—1182no—
152C4×C4⋊C4—591no—
153#343—1181no—
154#332—1184no—
155C59#940111yescharacteristic, complemented, Sylow
156C22×Q8—1181no—
157C22×D4D23611yescharacteristic
158C23×C4C22⋊D5912yes—
159C2×M16—24no—
160C2×M16—1182no—
161C4⋊Q8—1182no—
162C4⋊Q8—1182no—
163#124—2364no—
164C4×D4—22no—
165C4×D4—22no—
166C2×C4⋊C4—592no—
167C2×C4⋊C4—1181no—
168C2×C4⋊C4—1181no—
169C2×C4⋊C4D23611yescharacteristic
170C2×C22⋊C4—1184no—
171C2×C22⋊C4C4×D5912yes—
172C2×C42—1181no—
173C2×C42C22×D5911yescharacteristic
174C4⋊C8—1182no—
175C4⋊C8—1182no—
176C4⋊C8C22⋊D5914yes—
177#100—2362no—
178C22⋊C8—24no—
179C22⋊C8—22no—
180C4×C8—22no—
181#95—592no—
182#95—1184no—
183C24(C2×D59)⋊C412yes—
184C2×Q8—2362no—
185C2×Q8—2362no—
186C2×Q8—2361no—
187C2×Q8—2361no—
188C2×D4—22no—
189C2×D4—21no—
190C2×D4—22no—
191C2×D4—21no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—44no—
195C22×C4—1182no—
196C22×C4—1182no—
197C22×C4(C2×D59)⋊C412yes—
198C22×C4C2×C4×D5911yescharacteristic
199C22×C4C2×D23611yescharacteristic
200C22×C4C2×C22⋊D5911yescharacteristic
201M16—84no—
202M16—2362no—
203M16—2362no—
204C2×C8—24no—
205C2×C8—2362no—
206C4⋊C4—22no—
207C4⋊C4—1184no—
208C4⋊C4—2362no—
209C4⋊C4—2361no—
210C4⋊C4—2361no—
211C22⋊C4—24no—
212C22⋊C4—2368no—
213C42—2362no—
214C42#1174568512yes—
215C42C2×C22⋊D5912yes—
216C23—22no—
217C23—22no—
218C23—22no—
219C23—42no—
220C23—42no—
221C23C2×(C2×D59)⋊C411yescharacteristic
222Q8—4722no—
223Q8—4722no—
224D4—42no—
225D4—42no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—41no—
229C2×C4—44no—
230C2×C4—42no—
231C2×C4—82no—
232C2×C4—2362no—
233C2×C4—2362no—
234C2×C4—2362no—
235C2×C4—2362no—
236C2×C4—2362no—
237C2×C4—2362no—
238C2×C4C4×C22⋊D5912yes—
239C2×C4#1427369611yescharacteristic
240C2×C4#1427370112yes—
241C2×C4#1427371011yescharacteristic
242C2×C4#1427375512yes—
243C2×C4#1427375511yescharacteristic
244C2×C4#1427376011yescharacteristic
245C8—44no—
246C8—4722no—
247C22—21no—
248C22—21no—
249C22—44no—
250C22—42no—
251C22—42no—
252C22—42no—
253C22—82no—
254C22#1683728111yescharacteristic
255C22#1683740411yescharacteristic
256C22#1683740811yescharacteristic
257C4—22no—
258C4—21no—
259C4—42no—
260C4—4722no—
261C4—4722no—
262C4#1683758012yes—
263C4#1683764812yes—
264C4#1683765512yes—
265C2—21no—
266C2—21no—
267C2—82no—
268C2#2142448411yescharacteristic
269C2#2142453411yescharacteristic
270C2#2142464811yescharacteristic
subgroupG/H
centerC22#16837281
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#2975C2
socleC22×C59#386
prime radicalC22×D4×C59D4
cosocleC2×C42×C59C23
hypercenter#2975D59
hypocenterC59#9401
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2975

Complements

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

NH ≅ G/Ntype
3. #21422767C2semidirect
4. #21421998C2semidirect
155. C59#9401semidirect

Maximal subgroups

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

subgroupindex
#214290882
#214240882
#214227672
#214219982
C59×#29752
#940159