Finite Group Databasebeta

← #16929359

Subgroups — #16929359

Order 4352.

572 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 135 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#14531387C211yesmaximal, characteristic
2#14528874C212yesmaximal
3C2×#11762535C211yesmaximal, characteristic
4#14526663C212yesmaximal
5C2×C17×#400C211yesmaximal, characteristic
6C2×C22⋊#3622C2212yes—
7#11762695—24no—
8#11762672—21no—
9#11762669C2212yes—
10C4×C22⋊Dic17—21no—
11#11762571—24no—
12#11762535—22no—
13C22×C4⋊C4×C17C2211yescharacteristic
14C22×C4⋊C4×C17C2211yescharacteristic
15C17×#400—22no—
16C17×#400C414yescomplemented
17C2×C17×#95—21no—
18C2×C17×#95—21no—
19C2×C17×#95C2211yescharacteristic
20C23×C4×C17—21no—
21C23×C4×C17—21no—
22C23×C4×C17D411yescharacteristic
23C23×C4×C17D411yescharacteristic
24C23×C4×C17C2311yescharacteristic
25C2×C4⋊C4×C17—22no—
26C2×C4⋊C4×C17—22no—
27C2×C4⋊C4×C17—22no—
28C2×C4⋊C4×C17—22no—
29C2×C4⋊C4×C17—21no—
30C2×C4⋊C4×C17C2×C412yes—
31C2×C4⋊C4×C17C2×C412yes—
32C2×C4⋊C4×C17D412yes—
33C2×C22⋊Dic17—22no—
34C2×C4⋊Dic17D414yes—
35C2×C4×Dic17—22no—
36C22×#3622—22no—
37C17×#95—24no—
38C17×#95—24no—
39C17×#95—21no—
40C17×#95C2×C412yes—
41#10587502—22no—
42#10587502—24no—
43#10587502—22no—
44C22⋊#3622—22no—
45C22⋊#3622—22no—
46C24×C17C2×D411yescharacteristic
47C22×C4×C17—21no—
48C22×C4×C17—21no—
49C22×C4×C17—21no—
50C22×C4×C17—22no—
51C22×C4×C17—22no—
52C22×C4×C17—21no—
53C22×C4×C17—22no—
54C22×C4×C17—22no—
55C22×C4×C17—42no—
56C22×C4×C17—42no—
57C22×C4×C17—42no—
58C22×C4×C17—42no—
59C22×C4×C17—44no—
60C22×C4×C17—41no—
61C22×C4×C17—41no—
62C22×C4×C17—42no—
63C22×C4×C17C22⋊C412yes—
64C22×C4×C17C22⋊C412yes—
65C22×C4×C17C22×C411yescharacteristic
66C22×C4×C17C2×D412yes—
67C22×C4×C17C2×D411yescharacteristic
68C22×C4×C17#4112yes—
69C22×Dic17—24no—
70C4⋊C4×C17—24no—
71C4⋊C4×C17—24no—
72C4⋊C4×C17—24no—
73C4⋊C4×C17—42no—
74C22⋊Dic17—42no—
75C22⋊Dic17—42no—
76C4⋊Dic17—44no—
77C4×Dic17—82no—
78C2×#3622—44no—
79C2×#3622—42no—
80#9022—171nomaximal, Sylow
81C23×C17—22no—
82C23×C17—22no—
83C23×C17C2×C22⋊C411yescharacteristic
84C23×C17#12011yescharacteristic
85C23×C17#12311yescharacteristic
86C23×C17#13512yes—
87C23×C17#13712yes—
88C2×C4×C17—24no—
89C2×C4×C17—22no—
90C2×C4×C17—22no—
91C2×C4×C17—22no—
92C2×C4×C17—22no—
93C2×C4×C17—41no—
94C2×C4×C17—41no—
95C2×C4×C17—42no—
96C2×C4×C17—42no—
97C2×C4×C17—42no—
98C2×C4×C17—42no—
99C2×C4×C17—42no—
100C2×C4×C17—42no—
101C2×C4×C17—42no—
102C2×C4×C17—44no—
103C2×C4×C17—42no—
104C2×C4×C17—42no—
105C2×C4×C17—81no—
106C2×C4×C17—84no—
107C2×C4×C17—82no—
108C2×C4×C17—82no—
109C2×C4×C17C4×D412yes—
110C2×C4×C17#12112yes—
111C2×Dic17—44no—
112C2×Dic17—44no—
113C2×Dic17—44no—
114#3622—82no—
115C2×#400D1711yescharacteristic
116#2313—171no—
117#1824—172no—
118C2×#324—171no—
119#1294—172no—
120C22×C17—24no—
121C22×C17—24no—
122C22×C17—21no—
123C22×C17—21no—
124C22×C17—21no—
125C22×C17—21no—
126C22×C17#38611yescharacteristic
127C22×C17#41611yescharacteristic
128C22×C17#41911yescharacteristic
129C22×C17#45212yes—
130C22×C17#45412yes—
131C22×C17#45812yes—
132C22×C17#48512yes—
133C4×C17—22no—
134C4×C17—22no—
135C4×C17—42no—
136C4×C17—42no—
137C4×C17—82no—
138C4×C17—82no—
139Dic17—84no—
140C22×C4⋊C4C2×D1711yescharacteristic
141C22×C4⋊C4C2×D1711yescharacteristic
142C2×C22⋊C8—172no—
143#400—22no—
144#400Dic1714yescomplemented
145#387—344no—
146#381—341no—
147#380—172no—
148C4×C22⋊C4—341no—
149C2×#95—21no—
150C2×#95—21no—
151C2×#95C2×D1711yescharacteristic
152#340—344no—
153#324—342no—
154C2×C17—21no—
155C2×C17—21no—
156C2×C17—22no—
157C2×C17#161514yes—
158C2×C17#187311yescharacteristic
159C2×C17#188811yescharacteristic
160C2×C17#189011yescharacteristic
161C23×C4—21no—
162C23×C4—21no—
163C23×C4C22⋊D1711yescharacteristic
164C23×C4C22⋊D1711yescharacteristic
165C23×C4C22×D1711yescharacteristic
166C22×C8—342no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—174no—
173C2×C4⋊C4C2×Dic1712yes—
174C2×C4⋊C4C2×Dic1712yes—
175C2×C4⋊C4C22⋊D1712yes—
176C2×C22⋊C4—342no—
177C2×C42—342no—
178C22⋊C8—342no—
179C22⋊C8—342no—
180#95—24no—
181#95—24no—
182#95—21no—
183#95—342no—
184#95—344no—
185#95—342no—
186#95C2×Dic1712yes—
187C17#902211yescharacteristic, complemented, Sylow
188C24C2×C22⋊D1711yescharacteristic
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—21no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—44no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—42no—
205C22×C4—344no—
206C22×C4C22⋊Dic1712yes—
207C22×C4C22⋊Dic1712yes—
208C22×C4D4×D1712yes—
209C22×C4#6325412yes—
210C22×C4C22×Dic1711yescharacteristic
211C22×C4C2×C22⋊D1711yescharacteristic
212C2×C8—684no—
213C2×C8—682no—
214C4⋊C4—24no—
215C4⋊C4—24no—
216C4⋊C4—24no—
217C4⋊C4—42no—
218C4⋊C4—684no—
219C22⋊C4—682no—
220C22⋊C4—682no—
221C42—1362no—
222C23—22no—
223C23—22no—
224C23#1058762111yescharacteristic
225C23#1058762311yescharacteristic
226C23#1058762812yes—
227C23#1058763612yes—
228C23C2×C22⋊Dic1711yescharacteristic
229C2×C4—24no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—42no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—44no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—81no—
247C2×C4—84no—
248C2×C4—82no—
249C2×C4—82no—
250C2×C4—684no—
251C2×C4—684no—
252C2×C4—684no—
253C2×C4D4×Dic1712yes—
254C2×C4#1058762512yes—
255C8—1362no—
256C22—24no—
257C22—24no—
258C22—21no—
259C22—21no—
260C22—21no—
261C22—21no—
262C22#1176268911yescharacteristic
263C22#1176280511yescharacteristic
264C22#1176281811yescharacteristic
265C22#1176296912yes—
266C22#1176298112yes—
267C22#1176299812yes—
268C22#1176314712yes—
269C4—22no—
270C4—22no—
271C4—42no—
272C4—42no—
273C4—82no—
274C4—82no—
275C4—1364no—
276C2—21no—
277C2—21no—
278C2—22no—
279C2#1452781314yes—
280C2#1452909711yescharacteristic
281C2#1452918811yescharacteristic
282C2#1452919811yescharacteristic
subgroupG/H
centerC23#10587623
commutatorC22×C4×C17C22×C4
FrattiniC23×C4C22×D17
FittingC2×C17×#400C2
socleC23×C17#120
prime radicalC24×C17C2×D4
cosocleC23×C4×C17C23
hypercenterC2×#400D17
hypocenterC17#9022
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#400

Complements

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

NH ≅ G/Ntype
16. C17×#400C4semidirect
144. #400Dic17semidirect
187. C17#9022semidirect

Maximal subgroups

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

subgroupindex
#145313872
#145288742
C2×#117625352
#145266632
C2×C17×#4002
#902217