Finite Group Databasebeta

← #16932071

Subgroups — #16932071

Order 4352.

554 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 99 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#14533272C211yesmaximal, characteristic
2#14531387C211yesmaximal, characteristic, complemented
3#14527314C212yesmaximal
4#14526638C212yesmaximal, complemented
5C17×#1821C211yesmaximal, characteristic, complemented
6C2×#10587616C2211yescharacteristic
7#11762735—22no—
8#11762710C412yescomplemented
9#11762695—24no—
10#11762688—21no—
11#11762688—21no—
12#11762674C412yescomplemented
13#11762672C2212yes—
14#11762669—21no—
15C4×C22⋊Dic17—21no—
16C4×C22⋊Dic17C2211yescharacteristic
17#11762544—24no—
18C22×C4⋊C4×C17C2211yescharacteristic
19C17×#407C2212yes—
20C17×#343—22no—
21C23×C4×C17D411yescharacteristic
22C23×C4×C17D411yescharacteristic
23C23×C4×C17C2311yescharacteristic
24C22×C22⋊D17—21no—
25C22×D68C2×C411yescharacteristic
26C2×M16×C17—22no—
27C2×C4⋊C4×C17—22no—
28C2×C4⋊C4×C17—22no—
29C2×C4⋊C4×C17C2×C412yes—
30C2×C4⋊C4×C17D412yes—
31C2×C22⋊Dic17—21no—
32C2×C22⋊Dic17—22no—
33#10587616—22no—
34#10587616—22no—
35C2×#63227—22no—
36C2×#63227—22no—
37C2×#63227—21no—
38C2×C4⋊Dic17—22no—
39C2×C4⋊Dic17C2×C411yescharacteristic
40C2×C4×Dic17—22no—
41C2×C4×Dic17C2×C412yes—
42C17×C22⋊C8—22no—
43C17×C22⋊C8—22no—
44#10587502—22no—
45#10587502—22no—
46#10587502—22no—
47#10587502—22no—
48#10587502—22no—
49#10587502—21no—
50#10587502D414yes—
51C24×C17C2×D411yescharacteristic
52C23×D17—21no—
53C22×C4×C17—21no—
54C22×C4×C17—21no—
55C22×C4×C17—21no—
56C22×C4×C17—21no—
57C22×C4×C17—42no—
58C22×C4×C17—81no—
59C22×C4×C17—81no—
60C22×C4×C17C22⋊C412yes—
61C22×C4×C17C22⋊C412yes—
62C22×C4×C17C22×C411yescharacteristic
63C22×C4×C17C2×D412yes—
64C22×C4×C17C2×D411yescharacteristic
65C22×C4×C17#4112yes—
66C2×C22⋊D17—44no—
67C2×C22⋊D17—41no—
68C2×C22⋊D17—41no—
69C22×Dic17—21no—
70C22×Dic17—22no—
71C22×Dic17—24no—
72C2×D68—22no—
73C2×D68—42no—
74M16×C17—44no—
75C2×C8×C17—42no—
76C4⋊C4×C17—24no—
77C4⋊C4×C17—24no—
78C22⋊Dic17—44no—
79C22⋊Dic17—42no—
80#63227—42no—
81#63227—82no—
82#63227—82no—
83C4⋊Dic17—22no—
84C4⋊Dic17—82no—
85C4×Dic17—42no—
86C4×Dic17—82no—
87#9465—171nomaximal, Sylow
88C23×C17—22no—
89C23×C17—22no—
90C23×C17—41no—
91C23×C17C2×C22⋊C411yescharacteristic
92C23×C17#12011yescharacteristic
93C23×C17#12311yescharacteristic
94C22×D17—42no—
95C22×D17—42no—
96C22×D17—41no—
97C22×D17—42no—
98C2×C4×C17—22no—
99C2×C4×C17—22no—
100C2×C4×C17—22no—
101C2×C4×C17—22no—
102C2×C4×C17—42no—
103C2×C4×C17—42no—
104C2×C4×C17—42no—
105C2×C4×C17—42no—
106C2×C4×C17—42no—
107C2×C4×C17—42no—
108C2×C4×C17—81no—
109C2×C4×C17—81no—
110C2×C4×C17C4×D412yes—
111C2×C4×C17#12112yes—
112C22⋊D17—82no—
113C22⋊D17—82no—
114C2×Dic17—42no—
115C2×Dic17—44no—
116C2×Dic17—42no—
117C2×Dic17—44no—
118C2×Dic17—42no—
119C2×Dic17—41no—
120C2×Dic17—42no—
121C2×Dic17—44no—
122D68—82no—
123C8×C17—82no—
124#2504—171no—
125#2313—171no—
126#1821D1711yescharacteristic, complemented
127#1510—172no—
128#1284—172no—
129C22×C17—21no—
130C22×C17—21no—
131C22×C17—21no—
132C22×C17—21no—
133C22×C17—21no—
134C22×C17—21no—
135C22×C17—21no—
136C22×C17—44no—
137C22×C17#38611yescharacteristic
138C22×C17#41012yes—
139C22×C17#45412yes—
140C2×D17—84no—
141C2×D17—81no—
142C2×D17—81no—
143C2×D17—81no—
144C4×C17—22no—
145C4×C17—22no—
146C4×C17—42no—
147C4×C17—42no—
148Dic17—84no—
149Dic17—82no—
150Dic17—81no—
151C2×#121—171no—
152C22×C4⋊C4C2×D1711yescharacteristic
153#407C2×D1712yes—
154#397—342no—
155#390—172no—
156#387—344no—
157#386—341no—
158#386—341no—
159#382—172no—
160#381—172no—
161#380—341no—
162C4×C22⋊C4—171no—
163C4×C22⋊C4—341no—
164#343—22no—
165#327—344no—
166C2×C17—22no—
167C2×C17—22no—
168C2×C17—41no—
169C2×C17#188811yescharacteristic
170C2×C17#189911yescharacteristic
171C2×C17#191111yescharacteristic
172D17—161no—
173C22×D4—171no—
174C22×D4—341no—
175C23×C4D6811yescharacteristic
176C23×C4D6811yescharacteristic
177C23×C4C22×D1711yescharacteristic
178C2×M16—22no—
179#121—342no—
180#121—342no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—171no—
184C2×C4⋊C4—342no—
185C2×C4⋊C4C4×D1712yes—
186C2×C4⋊C4C22⋊D1712yes—
187C2×C22⋊C4—341no—
188C2×C22⋊C4—342no—
189C2×C22⋊C4—341no—
190C2×C22⋊C4—342no—
191C2×C22⋊C4—342no—
192C2×C42—172no—
193C2×C42—342no—
194C22⋊C8—22no—
195C22⋊C8—22no—
196#95—174no—
197#95—342no—
198#95—342no—
199#95—342no—
200#95—341no—
201#95—342no—
202#95—342no—
203C17#946511yescharacteristic, complemented, Sylow
204C24—341no—
205C24C2×D6811yescharacteristic
206C2×D4—342no—
207C2×D4—682no—
208C2×D4—684no—
209C2×D4—681no—
210C2×D4—681no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—42no—
216C22×C4—81no—
217C22×C4—81no—
218C22×C4—344no—
219C22×C4—341no—
220C22×C4—342no—
221C22×C4#6322712yes—
222C22×C4#6322712yes—
223C22×C4C2×C4×D1711yescharacteristic
224C22×C4D4×D1712yes—
225C22×C4#6325412yes—
226C22×C4C2×C22⋊D1711yescharacteristic
227M16—44no—
228C2×C8—42no—
229C4⋊C4—24no—
230C4⋊C4—24no—
231C4⋊C4—342no—
232C4⋊C4—1362no—
233C22⋊C4—682no—
234C22⋊C4—684no—
235C22⋊C4—682no—
236C22⋊C4—1362no—
237C22⋊C4—1362no—
238C42—682no—
239C42—1362no—
240C23—22no—
241C23—22no—
242C23—41no—
243C23—682no—
244C23—682no—
245C23—682no—
246C23—681no—
247C23#1058756811yescharacteristic
248C23#1058757211yescharacteristic
249C23C2×#6322711yescharacteristic
250D4—1362no—
251D4—1362no—
252D4—1362no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—81no—
264C2×C4—81no—
265C2×C4—684no—
266C2×C4—682no—
267C2×C4—684no—
268C2×C4—682no—
269C2×C4—682no—
270C2×C4—684no—
271C2×C4—682no—
272C2×C4—681no—
273C2×C4#1058756712yes—
274C2×C4#1058762512yes—
275C8—82no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—44no—
284C22—1364no—
285C22—1361no—
286C22—1361no—
287C22—1361no—
288C22#1176268611yescharacteristic
289C22#1176278212yes—
290C22#1176298412yes—
291C4—22no—
292C4—22no—
293C4—42no—
294C4—42no—
295C4—1364no—
296C4—1362no—
297C4—1361no—
298C2—22no—
299C2—22no—
300C2—41no—
301C2—2721no—
302C2#1452918511yescharacteristic
303C2#1452924011yescharacteristic
304C2#1452931411yescharacteristic
subgroupG/H
centerC22#11762686
commutatorC22×C4×C17C22×C4
FrattiniC23×C4C22×D17
FittingC17×#1821C2
socleC22×C17#386
prime radicalC2×#10587616C22
cosocleC23×C4×C17C23
hypercenter#1821D17
hypocenterC17#9465
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1821

Complements

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

NH ≅ G/Ntype
2. #14531387C2semidirect
4. #14526638C2semidirect
5. C17×#1821C2semidirect
8. #11762710C4semidirect
12. #11762674C4semidirect
126. #1821D17semidirect
203. C17#9465semidirect

Maximal subgroups

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

subgroupindex
#145332722
#145313872
#145273142
#145266382
C17×#18212
#946517