Finite Group Databasebeta

← #28498177

Subgroups — #28498177

Order 15104.

424 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 95 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#21422440C212yesmaximal, complemented
2#21422042C212yesmaximal, complemented
3#21422028C212yesmaximal
4C59×#2326C211yesmaximal, characteristic
5#16837459—22no—
6#16837128—22no—
7#16837128C2214yes—
8C2×#124×C59C411yescharacteristic
9C2×C4×Q8×C59C2211yescharacteristic
10C2×C4×D4×C59C411yescharacteristic
11C59×#391C411yescharacteristic
12C59×#390C2211yescharacteristic
13C59×#388—22no—
14C59×#385—22no—
15C59×#384C2211yescharacteristic
16C59×#376C411yescharacteristic
17C4⋊(C59⋊C8)—42no—
18C22⋊(C59⋊C8)—24no—
19#14273638—42no—
20C22×Q8×C59C2×C411yescharacteristic
21C22×D4×C59C2×C411yescharacteristic
22C23×C4×C59—21no—
23#124×C59—22no—
24#124×C59—22no—
25C4×Q8×C59—42no—
26C4×D4×C59—42no—
27C2×C4⋊C4×C59—22no—
28C2×C4⋊C4×C59C2×C411yescharacteristic
29C2×C4⋊C4×C59D412yes—
30C2×C4⋊C4×C59C2311yescharacteristic
31C2×C22⋊C4×C59—22no—
32C2×C22⋊C4×C59—21no—
33C2×C22⋊C4×C59—22no—
34C2×C42×C59C2×C411yescharacteristic
35C2×C42×C59C2×C411yescharacteristic
36C2×C42×C59C2×C411yescharacteristic
37C2×C42×C59D412yes—
38#95×C59—21no—
39#95×C59—21no—
40#95×C59—22no—
41C24×C59—21no—
42C2×Q8×C59—21no—
43C2×Q8×C59—21no—
44C2×Q8×C59—41no—
45C2×Q8×C59—41no—
46C2×D4×C59—21no—
47C2×D4×C59—41no—
48C2×D4×C59—41no—
49C2×D4×C59M1612yes—
50C22×C4×C59—21no—
51C22×C4×C59—21no—
52C22×C4×C59—21no—
53C22×C4×C59—42no—
54C22×C4×C59—82no—
55C22×C4×C59C22⋊C412yes—
56C22×C4×C59C22⋊C411yescharacteristic
57C22×C4×C59C22⋊C411yescharacteristic
58C22×C4×C59C22×C411yescharacteristic
59C22×C4×C59C2×D412yes—
60C4⋊C4×C59—42no—
61C4⋊C4×C59—41no—
62C4⋊C4×C59—41no—
63C4⋊C4×C59—82no—
64C22⋊C4×C59—28no—
65C22⋊C4×C59—42no—
66C22⋊C4×C59—82no—
67C42×C59—22no—
68C42×C59—22no—
69C42×C59—22no—
70C42×C59—21no—
71C42×C59—41no—
72C42×C59M1612yes—
73C2×C59⋊C8—44no—
74C23×C59—22no—
75C23×C59—42no—
76C23×C59—42no—
77C23×C59—41no—
78C23×C59—41no—
79C23×C59C2×C22⋊C411yescharacteristic
80Q8×C59—42no—
81Q8×C59—42no—
82D4×C59—42no—
83D4×C59—42no—
84C2×C4×C59—21no—
85C2×C4×C59—21no—
86C2×C4×C59—22no—
87C2×C4×C59—22no—
88C2×C4×C59—22no—
89C2×C4×C59—21no—
90C2×C4×C59—22no—
91C2×C4×C59—22no—
92C2×C4×C59—41no—
93C2×C4×C59—41no—
94C2×C4×C59—41no—
95C2×C4×C59—42no—
96C2×C4×C59—41no—
97C2×C4×C59—41no—
98C2×C4×C59—42no—
99C2×C4×C59—41no—
100C2×C4×C59—41no—
101C2×C4×C59—41no—
102C2×C4×C59—84no—
103C2×C4×C59#9912yes—
104C2×C4×C59C2×M1611yescharacteristic
105C2×C4×C59C2×M1611yescharacteristic
106C59⋊C8—84no—
107#11534—591nomaximal, Sylow
108C22×C59—21no—
109C22×C59—21no—
110C22×C59—44no—
111C22×C59—42no—
112C22×C59—82no—
113C22×C59—81no—
114C22×C59—81no—
115C22×C59#40711yescharacteristic
116C22×C59C2×#9911yescharacteristic
117C22×C59#41311yescharacteristic
118C4×C59—22no—
119C4×C59—21no—
120C4×C59—21no—
121C4×C59—42no—
122C4×C59—42no—
123C4×C59—41no—
124C4×C59—42no—
125C4×C59—81no—
126C4×C59—81no—
127#2326D5911yescharacteristic
128#1467—592no—
129#1325—592no—
130#1318—592no—
131C2×C59—21no—
132C2×C59—21no—
133C2×C59—82no—
134C2×C59#146511yescharacteristic
135C2×C59#212811yescharacteristic
136C2×C59#213311yescharacteristic
137C2×#124Dic5911yescharacteristic
138C2×C4×Q8C2×D5911yescharacteristic
139C2×C4×D4Dic5911yescharacteristic
140#432—1182no—
141#391Dic5911yescharacteristic
142#390C2×D5911yescharacteristic
143#388—22no—
144#385—22no—
145#384C2×D5911yescharacteristic
146#376Dic5911yescharacteristic
147#324—594no—
148#324—1182no—
149C59#1153411yescharacteristic, complemented, Sylow
150C22×Q8C2×Dic5911yescharacteristic
151C22×D4C2×Dic5911yescharacteristic
152C23×C4—21no—
153#124—22no—
154#124—22no—
155C4×Q8—42no—
156C4×D4—42no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4C2×Dic5911yescharacteristic
159C2×C4⋊C4C22⋊D5912yes—
160C2×C4⋊C4C22×D5911yescharacteristic
161C2×C22⋊C4—22no—
162C2×C22⋊C4—21no—
163C2×C22⋊C4—22no—
164C2×C42C2×Dic5911yescharacteristic
165C2×C42C2×Dic5911yescharacteristic
166C2×C42C2×Dic5911yescharacteristic
167C2×C42C22⋊D5912yes—
168C4⋊C8—2362no—
169C22⋊C8—1184no—
170#97—2362no—
171#95—21no—
172#95—22no—
173#95—21no—
174C24—21no—
175C2×Q8—21no—
176C2×Q8—21no—
177C2×Q8—41no—
178C2×Q8—41no—
179C2×D4—21no—
180C2×D4—41no—
181C2×D4—41no—
182C2×D4#1174566412yes—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—42no—
187C22×C4—82no—
188C22×C4C22⋊Dic5912yes—
189C22×C4C22⋊Dic5911yescharacteristic
190C22×C4C22⋊Dic5911yescharacteristic
191C22×C4C22×Dic5911yescharacteristic
192C22×C4C2×C22⋊D5912yes—
193C2×C8—2364no—
194C4⋊C4—42no—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4—82no—
198C22⋊C4—28no—
199C22⋊C4—42no—
200C22⋊C4—82no—
201C42—22no—
202C42—22no—
203C42—22no—
204C42—21no—
205C42—41no—
206C42#1174566412yes—
207C23—22no—
208C23—42no—
209C23—42no—
210C23—41no—
211C23—41no—
212C23C2×C22⋊Dic5911yescharacteristic
213Q8—42no—
214Q8—42no—
215D4—42no—
216D4—42no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—22no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—42no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—42no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—84no—
236C2×C4#1427364212yes—
237C2×C4C2×#1174566411yescharacteristic
238C2×C4C2×#1174566411yescharacteristic
239C8—4724no—
240C22—21no—
241C22—21no—
242C22—44no—
243C22—42no—
244C22—82no—
245C22—81no—
246C22—81no—
247C22#1683736611yescharacteristic
248C22C2×#1427364211yescharacteristic
249C22#1683738611yescharacteristic
250C4—21no—
251C4—21no—
252C4—22no—
253C4—42no—
254C4—42no—
255C4—42no—
256C4—41no—
257C4—81no—
258C4—81no—
259C2—21no—
260C2—21no—
261C2—82no—
262C2#2142243411yescharacteristic
263C2#2142576811yescharacteristic
264C2#2142578511yescharacteristic
subgroupG/H
centerC22C2×#14273642
commutatorC22×C4×C59C22×C4
FrattiniC2×C4⋊C4C22×D59
FittingC59×#2326C2
socleC22×C59C2×#99
prime radicalC22×D4×C59C2×C4
cosocleC2×C4⋊C4×C59C23
hypercenter#2326D59
hypocenterC59#11534
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2326

Complements

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

NH ≅ G/Ntype
1. #21422440C2semidirect
2. #21422042C2semidirect
149. C59#11534semidirect

Maximal subgroups

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

subgroupindex
#214224402
#214220422
#214220282
C59×#23262
#1153459