Finite Group Databasebeta

← #28486102

Subgroups — #28486102

Order 15104.

570 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 570 conjugacy classes of subgroups — too many to draw. Only its 115 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#21433137C211yesmaximal, characteristic
2#21429049C211yesmaximal, characteristic
3#21422782C212yesmaximal, complemented
4#21421997C212yesmaximal, complemented
5C59×#1800C211yesmaximal, characteristic
6C2×#14273727C2211yescharacteristic
7C2×#14273706—22no—
8#16837480—24no—
9#16837455C2212yes—
10#16837366—22no—
11#16837338C414yes—
12#16837305—21no—
13#16837271—21no—
14C4×C4⋊Dic59C2211yescharacteristic
15#16837147—24no—
16C2×C4×D4×C59C2211yescharacteristic
17C59×#423C2212yes—
18C2×#100×C59—21no—
19C59×#343—21no—
20C22×Dic118—21no—
21C2×#11745664—24no—
22#14273727—22no—
23#14273727—22no—
24#14273706—48no—
25C2×C4⋊Dic59—21no—
26C2×C4⋊Dic59—21no—
27C2×C4⋊Dic59C2×C412yes—
28C2×Dic59⋊C4—24no—
29C2×C22⋊Dic59—24no—
30C2×C4×Dic59—21no—
31C4⋊(C59⋊C8)D414yes—
32C22⋊(C59⋊C8)—24no—
33C22⋊(C59⋊C8)—22no—
34C4×C59⋊C8—22no—
35C59⋊#95C2×C412yes—
36C22×D4×C59D411yescharacteristic
37C23×C4×C59D412yes—
38C2×M16×C59—22no—
39C4×D4×C59—22no—
40C4×D4×C59—22no—
41C2×C4⋊C4×C59D411yescharacteristic
42C2×C22⋊C4×C59C2×C412yes—
43C2×C42×C59C2311yescharacteristic
44C4⋊C8×C59—22no—
45C4⋊C8×C59—22no—
46#100×C59—42no—
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—44no—
62C2×Dic118—41no—
63C2×Dic118—41no—
64M16×C59—42no—
65M16×C59—42no—
66C2×C8×C59—42no—
67C4⋊C4×C59—22no—
68C22⋊C4×C59—24no—
69C42×C59C2×D412yes—
70C42×C59#4112yes—
71C22⋊Dic59—48no—
72C4⋊Dic59—24no—
73C4⋊Dic59—24no—
74C4⋊Dic59—42no—
75Dic59⋊C4—48no—
76C4×Dic59—41no—
77C4×Dic59—41no—
78#11745664—84no—
79C2×C59⋊C8—24no—
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)11yescharacteristic
99C2×C4×C59#13712yes—
100C2×C4×C59#13711yescharacteristic
101C2×Dic59—24no—
102C2×Dic59—42no—
103C2×Dic59—42no—
104C2×Dic59—44no—
105C2×Dic59—42no—
106Dic118—82no—
107Dic118—82no—
108C8×C59—82no—
109C59⋊C8—44no—
110#9403—591nomaximal, Sylow
111C22×C59—21no—
112C22×C59—21no—
113C22×C59—44no—
114C22×C59—42no—
115C22×C59—42no—
116C22×C59—42no—
117C22×C59—82no—
118C22×C59#38611yescharacteristic
119C22×C59#41711yescharacteristic
120C22×C59#41911yescharacteristic
121C4×C59—21no—
122C4×C59—22no—
123C4×C59—42no—
124C4×C59#45312yes—
125C4×C59#46812yes—
126C4×C59#47012yes—
127Dic59—44no—
128Dic59—82no—
129#2975—591no—
130#2542—591no—
131#1800D5911yescharacteristic
132#1549—592no—
133#1303—592no—
134C2×C59—21no—
135C2×C59—21no—
136C2×C59—82no—
137C2×C59#188711yescharacteristic
138C2×C59#189411yescharacteristic
139C2×C59#191511yescharacteristic
140C2×C4⋊Q8—591no—
141C2×C22⋊Q8—1182no—
142C2×C4×D4C2×D5911yescharacteristic
143#436—1184no—
144#431—592no—
145#423C2×D5912yes—
146C2×#100—21no—
147#407—1182no—
148#399—594no—
149#390—1181no—
150#384—1181no—
151C4×C4⋊C4—591no—
152#343—21no—
153#332—1184no—
154C59#940311yescharacteristic, complemented, Sylow
155C22×Q8—1181no—
156C22×D4C22⋊D5911yescharacteristic
157C23×C4D23612yes—
158C2×M16—22no—
159C2×M16—1184no—
160C4⋊Q8—1182no—
161C4⋊Q8—1182no—
162C22⋊Q8—2368no—
163C4×D4—22no—
164C4×D4—22no—
165C2×C4⋊C4—592no—
166C2×C4⋊C4—1181no—
167C2×C4⋊C4—1184no—
168C2×C4⋊C4—1181no—
169C2×C4⋊C4C22⋊D5911yescharacteristic
170C2×C22⋊C4—1184no—
171C2×C22⋊C4C4×D5912yes—
172C2×C42—1181no—
173C2×C42C22×D5911yescharacteristic
174C4⋊C8—22no—
175C4⋊C8—22no—
176C4⋊C8—594no—
177#100—42no—
178C22⋊C8—1184no—
179C22⋊C8—1182no—
180C4×C8—1182no—
181#95—592no—
182C24(C2×D59)⋊C412yes—
183C2×Q8—2364no—
184C2×Q8—2361no—
185C2×Q8—2361no—
186C2×D4—22no—
187C2×D4—21no—
188C2×D4—22no—
189C2×D4—21no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—44no—
193C22×C4—1182no—
194C22×C4—1182no—
195C22×C4(C2×D59)⋊C412yes—
196C22×C4C2×C4×D5911yescharacteristic
197C22×C4C2×D23611yescharacteristic
198C22×C4C2×C22⋊D5911yescharacteristic
199M16—42no—
200M16—42no—
201M16—4724no—
202C2×C8—42no—
203C2×C8—1184no—
204C4⋊C4—22no—
205C4⋊C4—1184no—
206C4⋊C4—1184no—
207C4⋊C4—2368no—
208C4⋊C4—2362no—
209C22⋊C4—24no—
210C22⋊C4—2368no—
211C42—2361no—
212C42—2361no—
213C42D4×D5912yes—
214C42#1174568712yes—
215C23—22no—
216C23—22no—
217C23—22no—
218C23—42no—
219C23—42no—
220C23C2×(C2×D59)⋊C411yescharacteristic
221Q8—4722no—
222Q8—4722no—
223D4—42no—
224D4—42no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—41no—
228C2×C4—44no—
229C2×C4—42no—
230C2×C4—82no—
231C2×C4—1184no—
232C2×C4—2364no—
233C2×C4—2362no—
234C2×C4—2362no—
235C2×C4—2362no—
236C2×C4(C22⋊D59)⋊C412yes—
237C2×C4#1427369711yescharacteristic
238C2×C4#1427369912yes—
239C2×C4#1427371111yescharacteristic
240C2×C4#1427375311yescharacteristic
241C2×C4#1427375712yes—
242C2×C4#1427375811yescharacteristic
243C8—82no—
244C8—2364no—
245C22—21no—
246C22—21no—
247C22—44no—
248C22—42no—
249C22—42no—
250C22—42no—
251C22—82no—
252C22#1683727911yescharacteristic
253C22#1683740111yescharacteristic
254C22#1683741011yescharacteristic
255C4—22no—
256C4—21no—
257C4—42no—
258C4—4722no—
259C4—2364no—
260C4#1683757212yes—
261C4#1683764612yes—
262C4#1683765812yes—
263C2—21no—
264C2—21no—
265C2—82no—
266C2#2142449211yescharacteristic
267C2#2142452711yescharacteristic
268C2#2142464611yescharacteristic
subgroupG/H
centerC22#16837279
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1800C2
socleC22×C59#386
prime radicalC22×D4×C59D4
cosocleC2×C42×C59C23
hypercenter#1800D59
hypocenterC59#9403
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1800

Complements

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

NH ≅ G/Ntype
3. #21422782C2semidirect
4. #21421997C2semidirect
154. C59#9403semidirect

Maximal subgroups

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

subgroupindex
#214331372
#214290492
#214227822
#214219972
C59×#18002
#940359