Finite Group Databasebeta

← #28486082

Subgroups — #28486082

Order 15104.

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

normal subgroups only

This group has 514 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#21429269C211yesmaximal, characteristic
3#21422778C212yesmaximal, complemented
4#21421997C212yesmaximal, complemented
5C59×#1800C211yesmaximal, characteristic
6C2×#14273727C2211yescharacteristic
7#16837480—24no—
8#16837455C2212yes—
9#16837366—22no—
10#16837314C414yes—
11#16837305—21no—
12#16837267—21no—
13#16837262—22no—
14C4×C4⋊Dic59C2211yescharacteristic
15#16837150—24no—
16C2×C4×D4×C59C2211yescharacteristic
17C59×#423C2212yes—
18C2×#100×C59—21no—
19C59×#343—21no—
20C22×Dic118C2×C412yes—
21C2×#11745664—24no—
22#14273727—22no—
23#14273727—22no—
24C2×C4⋊Dic59—21no—
25C2×C4⋊Dic59—21no—
26C2×C4⋊Dic59—21no—
27C2×C22⋊Dic59—24no—
28C2×C4×Dic59—21no—
29C4⋊(C59⋊C8)D414yes—
30C22⋊(C59⋊C8)—24no—
31C22⋊(C59⋊C8)—22no—
32C4×C59⋊C8—22no—
33C59⋊#95—24no—
34C59⋊#95C2×C412yes—
35C22×D4×C59D411yescharacteristic
36C23×C4×C59D412yes—
37C2×M16×C59—22no—
38C4×D4×C59—22no—
39C4×D4×C59—22no—
40C2×C4⋊C4×C59D411yescharacteristic
41C2×C22⋊C4×C59C2×C412yes—
42C2×C42×C59C2311yescharacteristic
43C4⋊C8×C59—22no—
44C4⋊C8×C59—22no—
45#100×C59—42no—
46C24×C59C22⋊C412yes—
47C2×D4×C59—22no—
48C2×D4×C59—21no—
49C2×D4×C59—22no—
50C2×D4×C59—21no—
51C22×C4×C59—22no—
52C22×C4×C59—22no—
53C22×C4×C59—44no—
54C22×C4×C59C22⋊C412yes—
55C22×C4×C59C22×C411yescharacteristic
56C22×C4×C59C2×D411yescharacteristic
57C22×C4×C59C2×D411yescharacteristic
58C22×Dic59—22no—
59C22×Dic59—22no—
60C2×Dic118—24no—
61C2×Dic118—82no—
62M16×C59—42no—
63M16×C59—42no—
64C2×C8×C59—42no—
65C4⋊C4×C59—22no—
66C22⋊C4×C59—24no—
67C42×C59C2×D412yes—
68C42×C59#4112yes—
69C22⋊Dic59—84no—
70C4⋊Dic59—42no—
71C4⋊Dic59—41no—
72C4⋊Dic59—42no—
73C4⋊Dic59—41no—
74C4×Dic59—41no—
75C4×Dic59—41no—
76#11745664—84no—
77C2×C59⋊C8—24no—
78C23×C59—22no—
79C23×C59—22no—
80C23×C59—22no—
81C23×C59—42no—
82C23×C59—42no—
83C23×C59C2×C22⋊C411yescharacteristic
84D4×C59—42no—
85D4×C59—42no—
86C2×C4×C59—22no—
87C2×C4×C59—22no—
88C2×C4×C59—41no—
89C2×C4×C59—44no—
90C2×C4×C59—42no—
91C2×C4×C59—82no—
92C2×C4×C59C4×D412yes—
93C2×C4×C59#12011yescharacteristic
94C2×C4×C59#12112yes—
95C2×C4×C59#12311yescharacteristic
96C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
97C2×C4×C59#13712yes—
98C2×C4×C59#13711yescharacteristic
99C2×Dic59—42no—
100C2×Dic59—42no—
101C2×Dic59—44no—
102C2×Dic59—42no—
103C2×Dic59—42no—
104Dic118—84no—
105C8×C59—82no—
106C59⋊C8—44no—
107#9399—591nomaximal, Sylow
108C22×C59—21no—
109C22×C59—21no—
110C22×C59—44no—
111C22×C59—42no—
112C22×C59—42no—
113C22×C59—42no—
114C22×C59—82no—
115C22×C59#38611yescharacteristic
116C22×C59#41711yescharacteristic
117C22×C59#41911yescharacteristic
118C4×C59—21no—
119C4×C59—22no—
120C4×C59—42no—
121C4×C59#45512yes—
122C4×C59#46812yes—
123C4×C59#47012yes—
124Dic59—82no—
125Dic59—82no—
126#2975—591no—
127#2563—591no—
128#1800D5911yescharacteristic
129#1548—592no—
130#1303—592no—
131C2×C59—21no—
132C2×C59—21no—
133C2×C59—82no—
134C2×C59#188511yescharacteristic
135C2×C59#189411yescharacteristic
136C2×C59#191411yescharacteristic
137C2×C4⋊Q8—591no—
138C2×C4×D4C2×D5911yescharacteristic
139#436—1184no—
140#431—592no—
141#423C2×D5912yes—
142C2×#100—21no—
143#407—1182no—
144#393—594no—
145#390—1181no—
146#382—1181no—
147#380—1182no—
148C4×C4⋊C4—591no—
149#343—21no—
150#333—1184no—
151C59#939911yescharacteristic, complemented, Sylow
152C22×Q8—592no—
153C22×D4C22⋊D5911yescharacteristic
154C23×C4D23612yes—
155C2×M16—22no—
156C2×M16—1184no—
157C4⋊Q8—1182no—
158C4⋊Q8—1182no—
159C4×D4—22no—
160C4×D4—22no—
161C2×C4⋊C4—1181no—
162C2×C4⋊C4—1181no—
163C2×C4⋊C4—1181no—
164C2×C4⋊C4C22⋊D5911yescharacteristic
165C2×C22⋊C4—1184no—
166C2×C22⋊C4C4×D5912yes—
167C2×C42—1181no—
168C2×C42C22×D5911yescharacteristic
169C4⋊C8—22no—
170C4⋊C8—22no—
171C4⋊C8—594no—
172#100—42no—
173C22⋊C8—1184no—
174C22⋊C8—1182no—
175C4×C8—1182no—
176#95—592no—
177#95—1184no—
178C24(C2×D59)⋊C412yes—
179C2×Q8—1184no—
180C2×Q8—4722no—
181C2×D4—22no—
182C2×D4—21no—
183C2×D4—22no—
184C2×D4—21no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—44no—
188C22×C4—1182no—
189C22×C4—1182no—
190C22×C4(C2×D59)⋊C412yes—
191C22×C4C2×C4×D5911yescharacteristic
192C22×C4C2×D23611yescharacteristic
193C22×C4C2×C22⋊D5911yescharacteristic
194M16—42no—
195M16—42no—
196M16—4724no—
197C2×C8—42no—
198C2×C8—1184no—
199C4⋊C4—22no—
200C4⋊C4—2362no—
201C4⋊C4—2361no—
202C4⋊C4—2362no—
203C4⋊C4—2361no—
204C22⋊C4—24no—
205C22⋊C4—4724no—
206C42—2361no—
207C42—2361no—
208C42D4×D5912yes—
209C42#1174568712yes—
210C23—22no—
211C23—22no—
212C23—22no—
213C23—42no—
214C23—42no—
215C23C2×(C2×D59)⋊C411yescharacteristic
216Q8—4724no—
217D4—42no—
218D4—42no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—41no—
222C2×C4—44no—
223C2×C4—42no—
224C2×C4—82no—
225C2×C4—2362no—
226C2×C4—2364no—
227C2×C4—2362no—
228C2×C4—2362no—
229C2×C4—2362no—
230C2×C4(C22⋊D59)⋊C412yes—
231C2×C4#1427369711yescharacteristic
232C2×C4#1427369912yes—
233C2×C4#1427371111yescharacteristic
234C2×C4#1427375311yescharacteristic
235C2×C4#1427375712yes—
236C2×C4#1427375811yescharacteristic
237C8—82no—
238C8—2364no—
239C22—21no—
240C22—21no—
241C22—44no—
242C22—42no—
243C22—42no—
244C22—42no—
245C22—82no—
246C22#1683727911yescharacteristic
247C22#1683740111yescharacteristic
248C22#1683741011yescharacteristic
249C4—22no—
250C4—21no—
251C4—42no—
252C4—4722no—
253C4—4722no—
254C4#1683758212yes—
255C4#1683764612yes—
256C4#1683765812yes—
257C2—21no—
258C2—21no—
259C2—82no—
260C2#2142448211yescharacteristic
261C2#2142452711yescharacteristic
262C2#2142464111yescharacteristic
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#9399
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. #21422778C2semidirect
4. #21421997C2semidirect
151. C59#9399semidirect

Maximal subgroups

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

subgroupindex
#214331372
#214292692
#214227782
#214219972
C59×#18002
#939959