Finite Group Databasebeta

← #28504946

Subgroups — #28504946

Order 15104.

670 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 670 conjugacy classes of subgroups — too many to draw. Only its 107 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#21424014C212yesmaximal
2#21422221C214yesmaximal, complemented
3C59×#2297C211yesmaximal, characteristic
4C2×#14273642—22no—
5C2×#14273642C2214yes—
6#16837265—22no—
7C22×#41×C59C2211yescharacteristic
8C2×#117×C59C411yescharacteristic, complemented
9C22×C42×C59C411yescharacteristic, complemented
10C59×#391C412yescomplemented
11C59×#390C2212yes—
12C59×#386—22no—
13C4×C22⋊C4×C59—22no—
14C2×C22⋊Dic59—24no—
15#14273642—28no—
16#14273642—44no—
17C59⋊#95—44no—
18C2×#41×C59—24no—
19C2×#41×C59—42no—
20C2×#41×C59—42no—
21C22×Q8×C59C2×C411yescharacteristic
22C22×D4×C59D412yes—
23C22×D4×C59C2311yescharacteristic
24C23×C4×C59—21no—
25C23×C4×C59C2×C411yescharacteristic
26C23×C4×C59D412yes—
27#117×C59—42no—
28C2×C4⋊C4×C59C2×C412yes—
29C2×C22⋊C4×C59—21no—
30C2×C22⋊C4×C59—24no—
31C2×C42×C59—21no—
32C2×C42×C59—21no—
33C2×C42×C59—42no—
34C2×C42×C59C2×C412yes—
35#95×C59—24no—
36C24×C59C22⋊C411yescharacteristic
37C24×C59C2×D412yes—
38#41×C59—416no—
39C2×Q8×C59—22no—
40C2×Q8×C59—41no—
41C2×Q8×C59—41no—
42C2×D4×C59—24no—
43C2×D4×C59—24no—
44C2×D4×C59—42no—
45C2×D4×C59—41no—
46C2×D4×C59—41no—
47C2×D4×C59#4114yes—
48C22×C4×C59—22no—
49C22×C4×C59—22no—
50C22×C4×C59—22no—
51C22×C4×C59—22no—
52C22×C4×C59—24no—
53C22×C4×C59—24no—
54C22×C4×C59—22no—
55C22×C4×C59—42no—
56C22×C4×C59—44no—
57C22×C4×C59—42no—
58C22×C4×C59C22⋊C412yes—
59C22×C4×C59C22⋊C411yescharacteristic
60C22×C4×C59C22×C411yescharacteristic
61C22×Dic59—44no—
62C4⋊C4×C59—42no—
63C22⋊C4×C59—48no—
64C22⋊C4×C59—42no—
65C42×C59—42no—
66C42×C59—42no—
67C42×C59—42no—
68C22⋊Dic59—44no—
69C22⋊Dic59—44no—
70C23×C59—21no—
71C23×C59—21no—
72C23×C59—22no—
73C23×C59—22no—
74C23×C59—42no—
75C23×C59—44no—
76C23×C59#9912yes—
77C23×C59C2×C22⋊C411yescharacteristic
78C23×C59#12314yes—
79Q8×C59—44no—
80D4×C59—44no—
81D4×C59—48no—
82C2×C4×C59—28no—
83C2×C4×C59—21no—
84C2×C4×C59—21no—
85C2×C4×C59—21no—
86C2×C4×C59—21no—
87C2×C4×C59—24no—
88C2×C4×C59—22no—
89C2×C4×C59—22no—
90C2×C4×C59—22no—
91C2×C4×C59—48no—
92C2×C4×C59—42no—
93C2×C4×C59—48no—
94C2×C4×C59—42no—
95C2×C4×C59—44no—
96C2×C4×C59—42no—
97C2×C4×C59#9912yes—
98C2×C4×C59#11712yes—
99C2×Dic59—84no—
100C2×Dic59—88no—
101#12857—591nomaximal, Sylow
102C22×C59—22no—
103C22×C59—24no—
104C22×C59—22no—
105C22×C59—42no—
106C22×C59—44no—
107C22×C59—42no—
108C22×C59—44no—
109C22×C59—42no—
110C22×C59—82no—
111C22×C59#38111yescharacteristic
112C22×C59C2×#9911yescharacteristic
113C22×C59C2×#9911yescharacteristic
114C4×C59—22no—
115C4×C59—22no—
116C4×C59—44no—
117C4×C59—42no—
118C4×C59—82no—
119Dic59—164no—
120#2297D5911yescharacteristic
121#1782—592no—
122#1393—594no—
123C2×C59—21no—
124C2×C59—21no—
125C2×C59—44no—
126C2×C59—42no—
127C2×C59#181911yescharacteristic
128C2×C59#212612yes—
129C22×#41C2×D5911yescharacteristic
130C2×#117Dic5911yescharacteristic, complemented
131C22×C42Dic5911yescharacteristic, complemented
132C2×#99—594no—
133C2×#99—1182no—
134#391Dic5912yescomplemented
135#390C2×D5912yes—
136#386—22no—
137#381—1182no—
138C4×C22⋊C4—22no—
139C59#1285711yescharacteristic, complemented, Sylow
140C2×#41—24no—
141C2×#41—42no—
142C2×#41—42no—
143C22×Q8C2×Dic5911yescharacteristic
144C22×D4C22⋊D5912yes—
145C22×D4C22×D5911yescharacteristic
146C23×C4—21no—
147C23×C4C2×Dic5911yescharacteristic
148C23×C4C22⋊D5912yes—
149#117—42no—
150C2×C4⋊C4C2×Dic5912yes—
151C2×C22⋊C4—21no—
152C2×C22⋊C4—24no—
153C2×C22⋊C4—1184no—
154C2×C42—21no—
155C2×C42—21no—
156C2×C42—42no—
157C2×C42C2×Dic5912yes—
158#99—1188no—
159#99—2364no—
160#95—24no—
161#95—2364no—
162C24C22⋊Dic5911yescharacteristic
163C24C2×C22⋊D5912yes—
164#41—416no—
165C2×Q8—22no—
166C2×Q8—41no—
167C2×Q8—41no—
168C2×D4—24no—
169C2×D4—24no—
170C2×D4—42no—
171C2×D4—41no—
172C2×D4—41no—
173C2×D4#1174568514yes—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—24no—
179C22×C4—24no—
180C22×C4—22no—
181C22×C4—42no—
182C22×C4—44no—
183C22×C4—42no—
184C22×C4—2364no—
185C22×C4C22⋊Dic5912yes—
186C22×C4C22⋊Dic5911yescharacteristic
187C22×C4C22×Dic5911yescharacteristic
188C4⋊C4—42no—
189C22⋊C4—48no—
190C22⋊C4—42no—
191C22⋊C4—2364no—
192C22⋊C4—2364no—
193C42—42no—
194C42—42no—
195C42—42no—
196C23—21no—
197C23—21no—
198C23—22no—
199C23—22no—
200C23—42no—
201C23—44no—
202C23#1427364212yes—
203C23C2×C22⋊Dic5911yescharacteristic
204C23#1427371014yes—
205Q8—44no—
206D4—44no—
207D4—48no—
208C2×C4—28no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—24no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—48no—
218C2×C4—42no—
219C2×C4—48no—
220C2×C4—42no—
221C2×C4—44no—
222C2×C4—42no—
223C2×C4—4724no—
224C2×C4—4728no—
225C2×C4#1427364212yes—
226C2×C4#1427368612yes—
227C22—22no—
228C22—22no—
229C22—24no—
230C22—42no—
231C22—44no—
232C22—42no—
233C22—44no—
234C22—42no—
235C22—82no—
236C22#1683726511yescharacteristic
237C22C2×#1427364211yescharacteristic
238C22C2×#1427364211yescharacteristic
239C4—22no—
240C4—22no—
241C4—44no—
242C4—42no—
243C4—82no—
244C4—9444no—
245C2—21no—
246C2—21no—
247C2—44no—
248C2—42no—
249C2#2142417011yescharacteristic
250C2#2142576112yes—
subgroupG/H
centerC22C2×#14273642
commutatorC22×C4×C59C22×C4
FrattiniC22×D4C22×D59
FittingC59×#2297C2
socleC22×C59C2×#99
prime radicalC22×#41×C59C22
cosocleC22×D4×C59C23
hypercenter#2297D59
hypocenterC59#12857
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ #2297

Complements

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

NH ≅ G/Ntype
2. #21422221C2semidirect
8. C2×#117×C59C4semidirect
9. C22×C42×C59C4semidirect
10. C59×#391C4semidirect
130. C2×#117Dic59semidirect
131. C22×C42Dic59semidirect
134. #391Dic59semidirect
139. C59#12857semidirect

Maximal subgroups

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

subgroupindex
#214240142
#214222212
C59×#22972
#1285759