Finite Group Databasebeta

← #28491774

Subgroups — #28491774

Order 15104.

454 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 101 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#21424924C211yesmaximal, characteristic
2#21422655C212yesmaximal
3#21421954C212yesmaximal, complemented
4#21421929C211yesmaximal, characteristic, complemented
5C59×#2862C211yesmaximal, characteristic, complemented
6#16837952C2211yescharacteristic
7#16837455C2212yes—
8#16837426C2211yescharacteristic
9C2×#14273655—21no—
10C2×#14273655—21no—
11#16837135—24no—
12C2×C4⋊Q8×C59—21no—
13C59×#391—24no—
14C59×#382—21no—
15C59×#382C2212yes—
16C43×C59C2211yescharacteristic
17C2×#11745685—21no—
18C2×#11745664—21no—
19#14273727Q811yescharacteristic, complemented
20#14273724Q811yescharacteristic, complemented
21#14273715—21no—
22#14273701—22no—
23C4×D236—22no—
24C4⋊(C59⋊C8)—22no—
25C4⋊(C59⋊C8)—22no—
26#14273655—42no—
27#14273655—42no—
28C22⋊(C59⋊C8)—22no—
29C4×C59⋊C8—22no—
30C22×Q8×C59—22no—
31C4⋊Q8×C59—42no—
32C4⋊Q8×C59—42no—
33C2×C4⋊C4×C59—22no—
34C2×C4⋊C4×C59—22no—
35C2×C42×C59—22no—
36C2×C42×C59D411yescharacteristic
37C2×C42×C59D411yescharacteristic
38C2×C42×C59C2311yescharacteristic
39#95×C59—24no—
40#95×C59—22no—
41#95×C59D414yes—
42C2×Q8×C59—44no—
43C2×Q8×C59—48no—
44C22×C4×C59—21no—
45C22×C4×C59—21no—
46C22×C4×C59—24no—
47C22×C4×C59C2×D412yes—
48C22×C4×C59C2×D411yescharacteristic
49C2×C22⋊D59—41no—
50#11745685—41no—
51#11745685—41no—
52#11745685—41no—
53#11745685—41no—
54C2×D236—21no—
55C2×D236—22no—
56C2×C4×D59—41no—
57C2×Dic118—21no—
58C4⋊C4×C59—44no—
59C4⋊C4×C59—42no—
60C4⋊C4×C59—42no—
61C42×C59—21no—
62C42×C59—21no—
63C42×C59—44no—
64C42×C59D812yes—
65C42×C59SD1612yes—
66C42×C59C2×Q811yescharacteristic
67C42×C59#4111yescharacteristic
68C42×C59#4112yes—
69(C2×D59)⋊C4—42no—
70C4⋊Dic59—22no—
71#11745664—41no—
72#11745664—41no—
73C2×C59⋊C8—42no—
74C2×C59⋊C8—41no—
75C23×C59#12011yescharacteristic
76C22×D59—41no—
77Q8×C59—88no—
78C2×C4×C59—24no—
79C2×C4×C59—22no—
80C2×C4×C59—22no—
81C2×C4×C59—42no—
82C2×C4×C59—42no—
83C2×C4×C59—48no—
84C2×C4×C59—44no—
85C2×C4×C59C22⋊Q812yes—
86C2×C4×C59C22⋊Q811yescharacteristic
87C2×C4×C59#12412yes—
88C2×C4×C59#12411yescharacteristic
89C2×C4×C59C2×D811yescharacteristic
90C2×C4×C59C2×SD1611yescharacteristic
91C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
92C2×C4×C59#13711yescharacteristic
93C22⋊D59—81no—
94C22⋊D59—81no—
95C2×Dic59—41no—
96D236—41no—
97D236—41no—
98D236—82no—
99C4×D59—81no—
100C4×D59—81no—
101Dic118—41no—
102Dic118—41no—
103C59⋊C8—82no—
104C59⋊C8—81no—
105#10362—591nomaximal, Sylow
106C22×C59—22no—
107C22×C59#39311yescharacteristic
108C22×C59#44711yescharacteristic
109C22×C59#44811yescharacteristic
110C2×D59—81no—
111C2×D59—82no—
112C4×C59—24no—
113C4×C59—22no—
114C4×C59—22no—
115C4×C59—42no—
116C4×C59—84no—
117C4×C59#47411yescharacteristic
118C4×C59#47611yescharacteristic
119C4×C59#48611yescharacteristic
120C4×C59#48911yescharacteristic
121Dic59—81no—
122#2862D5911yescharacteristic, complemented
123#1962—591no—
124#1517—592no—
125#1286—592no—
126#1276—591no—
127C2×C59—21no—
128C2×C59—21no—
129C2×C59#202311yescharacteristic
130C2×C59#203112yes—
131D59—161no—
132#532—591no—
133C2×C4⋊Q8—21no—
134#431—592no—
135#423—591no—
136C2×#104—1181no—
137C2×#104—1181no—
138#391—24no—
139#382—21no—
140#382C2×D5912yes—
141C43C2×D5911yescharacteristic
142#327—1184no—
143C59#1036211yescharacteristic, complemented, Sylow
144C2×#41—1181no—
145C22×Q8—22no—
146C2×M16—1181no—
147C4⋊Q8—42no—
148C4⋊Q8—42no—
149C4⋊Q8—591no—
150#127—591no—
151#124—1181no—
152#121—1182no—
153C4×D4—1182no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C42—22no—
157C2×C42C22⋊D5911yescharacteristic
158C2×C42C22⋊D5911yescharacteristic
159C2×C42C22×D5911yescharacteristic
160C4⋊C8—1182no—
161C4⋊C8—1182no—
162#104—2362no—
163#104—2362no—
164C22⋊C8—1182no—
165C4×C8—1182no—
166#95—24no—
167#95—22no—
168#95C22⋊D5914yes—
169#41—2361no—
170#41—2361no—
171#41—2361no—
172#41—2361no—
173C2×Q8—44no—
174C2×Q8—48no—
175C2×Q8—1181no—
176C2×D4—1182no—
177C2×D4—1181no—
178C2×D4—2361no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—24no—
182C22×C4—2361no—
183C22×C4C2×C22⋊D5912yes—
184C22×C4C2×C22⋊D5911yescharacteristic
185M16—2361no—
186M16—2361no—
187C2×C8—2362no—
188C2×C8—2361no—
189C4⋊C4—44no—
190C4⋊C4—42no—
191C4⋊C4—42no—
192C4⋊C4—1182no—
193C22⋊C4—2362no—
194C42—21no—
195C42—21no—
196C42—44no—
197C42#1174566912yes—
198C42#1174567112yes—
199C42Q8×D5911yescharacteristic
200C42#1174568911yescharacteristic
201C42#1174568912yes—
202C23—2361no—
203C23#1427369611yescharacteristic
204Q8—88no—
205Q8—2361no—
206Q8—2361no—
207D4—4722no—
208D4—4721no—
209D4—4721no—
210D4—2361no—
211D4—2361no—
212C2×C4—24no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—48no—
218C2×C4—44no—
219C2×C4—4721no—
220C2×C4—4721no—
221C2×C4—2361no—
222C2×C4#1427370312yes—
223C2×C4#1427370311yescharacteristic
224C2×C4#1427371612yes—
225C2×C4#1427371611yescharacteristic
226C2×C4C2×#1174566911yescharacteristic
227C2×C4C2×#1174567111yescharacteristic
228C2×C4#1427375311yescharacteristic
229C2×C4#1427375811yescharacteristic
230C8—4722no—
231C8—4721no—
232C22—22no—
233C22—4721no—
234C22—4722no—
235C22#1683731311yescharacteristic
236C22#1683753611yescharacteristic
237C22#1683754111yescharacteristic
238C4—24no—
239C4—22no—
240C4—22no—
241C4—42no—
242C4—84no—
243C4—4721no—
244C4#1683767911yescharacteristic
245C4#1683768911yescharacteristic
246C4#1683775011yescharacteristic
247C4#1683776411yescharacteristic
248C2—21no—
249C2—21no—
250C2—9441no—
251C2#2142522911yescharacteristic
252C2#2142526512yes—
subgroupG/H
centerC22#16837313
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
FittingC59×#2862C2
socleC22×C59#393
prime radical#16837952C22
hypercenter#2862D59
hypocenterC59#10362
LCSG ▷ C2×C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2862

Complements

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

NH ≅ G/Ntype
3. #21421954C2semidirect
4. #21421929C2semidirect
5. C59×#2862C2semidirect
19. #14273727Q8semidirect
20. #14273724Q8semidirect
122. #2862D59semidirect
143. C59#10362semidirect

Maximal subgroups

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

subgroupindex
#214249242
#214226552
#214219542
#214219292
C59×#28622
#1036259