Finite Group Databasebeta

← #16943812

Subgroups — #16943812

Order 4352.

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#14527128C212yesmaximal, complemented
2#14526730C212yesmaximal, complemented
3#14526716C212yesmaximal
4C17×#2326C211yesmaximal, characteristic
5#11762866—22no—
6#11762535—22no—
7#11762535C2214yes—
8C2×C17×#124C411yescharacteristic
9C2×C4×Q8×C17C2211yescharacteristic
10C2×C4×D4×C17C411yescharacteristic
11C17×#391C411yescharacteristic
12C17×#390C2211yescharacteristic
13C17×#388—22no—
14C17×#385—22no—
15C17×#384C2211yescharacteristic
16C17×#376C411yescharacteristic
17C22×Q8×C17C2×C411yescharacteristic
18C22×D4×C17C2×C411yescharacteristic
19C23×C4×C17—21no—
20C17×#124—22no—
21C17×#124—22no—
22C4×Q8×C17—42no—
23C4×D4×C17—42no—
24C2×C4⋊C4×C17—22no—
25C2×C4⋊C4×C17C2×C411yescharacteristic
26C2×C4⋊C4×C17D412yes—
27C2×C4⋊C4×C17C2311yescharacteristic
28C2×C22⋊C4×C17—22no—
29C2×C22⋊C4×C17—21no—
30C2×C22⋊C4×C17—22no—
31C2×C42×C17C2×C411yescharacteristic
32C2×C42×C17C2×C411yescharacteristic
33C2×C42×C17C2×C411yescharacteristic
34C2×C42×C17D412yes—
35C17×#95—21no—
36C17×#95—21no—
37C17×#95—22no—
38C22⋊#3622—24no—
39C4⋊#3622—42no—
40#10587474—42no—
41C24×C17—21no—
42C2×Q8×C17—21no—
43C2×Q8×C17—21no—
44C2×Q8×C17—41no—
45C2×Q8×C17—41no—
46C2×D4×C17—21no—
47C2×D4×C17—41no—
48C2×D4×C17—41no—
49C2×D4×C17M1612yes—
50C22×C4×C17—21no—
51C22×C4×C17—21no—
52C22×C4×C17—21no—
53C22×C4×C17—42no—
54C22×C4×C17—82no—
55C22×C4×C17C22⋊C412yes—
56C22×C4×C17C22⋊C411yescharacteristic
57C22×C4×C17C22⋊C411yescharacteristic
58C22×C4×C17C22×C411yescharacteristic
59C22×C4×C17C2×D412yes—
60C4⋊C4×C17—42no—
61C4⋊C4×C17—41no—
62C4⋊C4×C17—41no—
63C4⋊C4×C17—82no—
64C22⋊C4×C17—28no—
65C22⋊C4×C17—42no—
66C22⋊C4×C17—82no—
67C42×C17—22no—
68C42×C17—22no—
69C42×C17—22no—
70C42×C17—21no—
71C42×C17—41no—
72C42×C17M1612yes—
73C2×#3622—44no—
74#11534—171nomaximal, Sylow
75C23×C17—22no—
76C23×C17—42no—
77C23×C17—42no—
78C23×C17—41no—
79C23×C17—41no—
80C23×C17C2×C22⋊C411yescharacteristic
81Q8×C17—42no—
82Q8×C17—42no—
83D4×C17—42no—
84D4×C17—42no—
85C2×C4×C17—21no—
86C2×C4×C17—21no—
87C2×C4×C17—22no—
88C2×C4×C17—22no—
89C2×C4×C17—21no—
90C2×C4×C17—22no—
91C2×C4×C17—22no—
92C2×C4×C17—22no—
93C2×C4×C17—41no—
94C2×C4×C17—41no—
95C2×C4×C17—41no—
96C2×C4×C17—42no—
97C2×C4×C17—41no—
98C2×C4×C17—41no—
99C2×C4×C17—42no—
100C2×C4×C17—41no—
101C2×C4×C17—41no—
102C2×C4×C17—41no—
103C2×C4×C17—84no—
104C2×C4×C17#9912yes—
105C2×C4×C17C2×M1611yescharacteristic
106C2×C4×C17C2×M1611yescharacteristic
107#3622—84no—
108#2326D1711yescharacteristic
109#1467—172no—
110#1325—172no—
111#1318—172no—
112C22×C17—21no—
113C22×C17—21no—
114C22×C17—44no—
115C22×C17—42no—
116C22×C17—82no—
117C22×C17—81no—
118C22×C17—81no—
119C22×C17#40711yescharacteristic
120C22×C17C2×#9911yescharacteristic
121C22×C17#41311yescharacteristic
122C4×C17—22no—
123C4×C17—21no—
124C4×C17—21no—
125C4×C17—42no—
126C4×C17—42no—
127C4×C17—41no—
128C4×C17—42no—
129C4×C17—81no—
130C4×C17—81no—
131C2×#124Dic1711yescharacteristic
132C2×C4×Q8C2×D1711yescharacteristic
133C2×C4×D4Dic1711yescharacteristic
134#432—342no—
135#391Dic1711yescharacteristic
136#390C2×D1711yescharacteristic
137#388—22no—
138#385—22no—
139#384C2×D1711yescharacteristic
140#376Dic1711yescharacteristic
141#324—174no—
142#324—342no—
143C2×C17—21no—
144C2×C17—21no—
145C2×C17—82no—
146C2×C17#146511yescharacteristic
147C2×C17#212811yescharacteristic
148C2×C17#213311yescharacteristic
149C22×Q8C2×Dic1711yescharacteristic
150C22×D4C2×Dic1711yescharacteristic
151C23×C4—21no—
152#124—22no—
153#124—22no—
154C4×Q8—42no—
155C4×D4—42no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4C2×Dic1711yescharacteristic
158C2×C4⋊C4C22⋊D1712yes—
159C2×C4⋊C4C22×D1711yescharacteristic
160C2×C22⋊C4—22no—
161C2×C22⋊C4—21no—
162C2×C22⋊C4—22no—
163C2×C42C2×Dic1711yescharacteristic
164C2×C42C2×Dic1711yescharacteristic
165C2×C42C2×Dic1711yescharacteristic
166C2×C42C22⋊D1712yes—
167C4⋊C8—682no—
168C22⋊C8—344no—
169#97—682no—
170#95—21no—
171#95—22no—
172#95—21no—
173C17#1153411yescharacteristic, complemented, Sylow
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#6322312yes—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—42no—
187C22×C4—82no—
188C22×C4C22⋊Dic1712yes—
189C22×C4C22⋊Dic1711yescharacteristic
190C22×C4C22⋊Dic1711yescharacteristic
191C22×C4C22×Dic1711yescharacteristic
192C22×C4C2×C22⋊D1712yes—
193C2×C8—684no—
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#6322312yes—
207C23—22no—
208C23—42no—
209C23—42no—
210C23—41no—
211C23—41no—
212C23C2×C22⋊Dic1711yescharacteristic
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—21no—
223C2×C4—22no—
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#1058750512yes—
237C2×C4C2×#6322311yescharacteristic
238C2×C4C2×#6322311yescharacteristic
239C8—1364no—
240C22—21no—
241C22—21no—
242C22—44no—
243C22—42no—
244C22—82no—
245C22—81no—
246C22—81no—
247C22#1176277311yescharacteristic
248C22C2×#1058750511yescharacteristic
249C22#1176279311yescharacteristic
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#1452712211yescharacteristic
263C2#1453045611yescharacteristic
264C2#1453047311yescharacteristic
subgroupG/H
centerC22C2×#10587505
commutatorC22×C4×C17C22×C4
FrattiniC2×C4⋊C4C22×D17
FittingC17×#2326C2
socleC22×C17C2×#99
prime radicalC22×D4×C17C2×C4
cosocleC2×C4⋊C4×C17C23
hypercenter#2326D17
hypocenterC17#11534
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2326

Complements

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

NH ≅ G/Ntype
1. #14527128C2semidirect
2. #14526730C2semidirect
173. C17#11534semidirect

Maximal subgroups

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

subgroupindex
#145271282
#145267302
#145267162
C17×#23262
#1153417