Finite Group Databasebeta

← #16929002

Subgroups — #16929002

Order 4352.

542 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 119 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#14537869C211yesmaximal, characteristic, complemented
2C4×#10587633C211yesmaximal, characteristic
3#14527176C212yesmaximal
4#14526616C212yesmaximal, complemented
5C17×#1461C211yesmaximal, characteristic, complemented
6C2×#10587633C411yescharacteristic
7C2×C4×D68C2211yescharacteristic
8#11762940—24no—
9#11762866—22no—
10#11762862C2212yes—
11#11762716C411yescharacteristic
12#11762710C411yescharacteristic
13#11762699—22no—
14#11762674C411yescharacteristic
15C4×#63227—22no—
16C42×Dic17C2211yescharacteristic
17#11762538—24no—
18C2×C4×Q8×C17C2211yescharacteristic
19C17×#431C2212yes—
20C17×#324—22no—
21C22×Q8×C17C2×C411yescharacteristic
22C22×D68C2×C411yescharacteristic
23C22×C4×D17—21no—
24C4×Q8×C17—22no—
25C4×Q8×C17—22no—
26C2×C4⋊C4×C17C2×C411yescharacteristic
27C2×C4⋊C4×C17D412yes—
28C2×C42×C17D412yes—
29C2×C42×C17C2311yescharacteristic
30#10587633—22no—
31#10587633—22no—
32C2×#63227—22no—
33C2×#63227—22no—
34C2×#63227—21no—
35C2×C4⋊Dic17C2×C411yescharacteristic
36C2×C4×Dic17—21no—
37C2×C4×Dic17—21no—
38C2×C4×Dic17C2×C411yescharacteristic
39C2×C4×Dic17C2×C411yescharacteristic
40C4×D68—22no—
41C4×D68—22no—
42C17×C4⋊C8—22no—
43C17×C22⋊C8—22no—
44C4×C8×C17—22no—
45#10587502—22no—
46#10587502—22no—
47C22⋊#3622—22no—
48C4⋊#3622—24no—
49C4⋊#3622D414yes—
50#10587474—24no—
51C4×#3622—22no—
52C23×D17—21no—
53C2×Q8×C17—22no—
54C2×Q8×C17—21no—
55C2×Q8×C17—21no—
56C2×Q8×C17—22no—
57C22×C4×C17C22⋊C412yes—
58C22×C4×C17C22⋊C412yes—
59C22×C4×C17C22×C411yescharacteristic
60C22×C4×C17C2×D411yescharacteristic
61C22×C4×C17C2×D411yescharacteristic
62C22×Dic17—21no—
63C22×Dic17—21no—
64C22×Dic17—21no—
65C2×D68—22no—
66C2×D68—22no—
67C2×D68—21no—
68C2×D68M1612yes—
69C2×C4×D17—44no—
70C2×C4×D17—41no—
71C2×C4×D17—41no—
72C2×C8×C17—42no—
73C4⋊C4×C17—22no—
74C4⋊C4×C17—24no—
75C42×C17—22no—
76C42×C17—22no—
77C42×C17C2×D412yes—
78C42×C17#4112yes—
79#63227—28no—
80#63227—44no—
81#63227—42no—
82C4⋊Dic17—22no—
83C4×Dic17—21no—
84C4×Dic17—22no—
85C4×Dic17—42no—
86C4×Dic17—42no—
87C4×Dic17M1612yes—
88C2×#3622—24no—
89#8964—171nomaximal, Sylow
90C23×C17C2×C22⋊C411yescharacteristic
91C22×D17—22no—
92C22×D17—42no—
93C22×D17—42no—
94C22×D17—41no—
95C22×D17—41no—
96Q8×C17—42no—
97Q8×C17—42no—
98C2×C4×C17—22no—
99C2×C4×C17—22no—
100C2×C4×C17—22no—
101C2×C4×C17—22no—
102C2×C4×C17—22no—
103C2×C4×C17—41no—
104C2×C4×C17—42no—
105C2×C4×C17C4×D412yes—
106C2×C4×C17#12011yescharacteristic
107C2×C4×C17#12112yes—
108C2×C4×C17#12311yescharacteristic
109C2×C4×C17C2×M1611yescharacteristic
110C2×C4×C17C2×M1611yescharacteristic
111C2×C4×C17#13112yes—
112C2×Dic17—22no—
113C2×Dic17—22no—
114C2×Dic17—41no—
115C2×Dic17—41no—
116C2×Dic17—41no—
117C2×Dic17—41no—
118C2×Dic17—42no—
119C2×Dic17—41no—
120D68—24no—
121D68—42no—
122C4×D17—82no—
123C4×D17—82no—
124C8×C17—82no—
125#3622—44no—
126#2983—171no—
127C4×#124—171no—
128#1478—172no—
129#1461D1711yescharacteristic, complemented
130#1276—172no—
131C22×C17—21no—
132C22×C17—21no—
133C22×C17#38611yescharacteristic
134C22×C17#40711yescharacteristic
135C22×C17#40811yescharacteristic
136C2×D17—44no—
137C2×D17—42no—
138C2×D17—82no—
139C2×D17—81no—
140C2×D17—81no—
141C4×C17—22no—
142C4×C17—21no—
143C4×C17—42no—
144C4×C17—42no—
145C4×C17#43512yes—
146C4×C17#43612yes—
147C4×C17#45412yes—
148Dic17—42no—
149Dic17—42no—
150Dic17—81no—
151C2×#124—171no—
152C2×C4×Q8C2×D1711yescharacteristic
153C2×C4×D4—171no—
154#446—344no—
155#432—342no—
156#431—172no—
157#431C2×D1712yes—
158#391—171no—
159#390—171no—
160#388—342no—
161#382—171no—
162C4×C22⋊C4—342no—
163C43—171no—
164D4⋊C8—344no—
165#324—22no—
166C2×C17—21no—
167C2×C17—21no—
168C2×C17#187011yescharacteristic
169C2×C17#188411yescharacteristic
170C2×C17#188811yescharacteristic
171D17—82no—
172C22×Q8C4×D1711yescharacteristic
173C22×D4—171no—
174C23×C4—341no—
175#124—342no—
176#124—342no—
177C4×Q8—22no—
178C4×Q8—22no—
179C4×D4—342no—
180C4×D4—342no—
181C2×C4⋊C4—171no—
182C2×C4⋊C4C4×D1711yescharacteristic
183C2×C4⋊C4C22⋊D1712yes—
184C2×C22⋊C4—342no—
185C2×C22⋊C4—341no—
186C2×C22⋊C4—342no—
187C2×C42—171no—
188C2×C42—171no—
189C2×C42—341no—
190C2×C42—341no—
191C2×C42D6812yes—
192C2×C42C22×D1711yescharacteristic
193C4⋊C8—22no—
194C4⋊C8—174no—
195C4⋊C8—344no—
196C22⋊C8—22no—
197C22⋊C8—342no—
198#97—344no—
199C4×C8—22no—
200C4×C8—342no—
201#95—342no—
202#95—342no—
203C17#896411yescharacteristic, complemented, Sylow
204C24—341no—
205C2×Q8—22no—
206C2×Q8—22no—
207C2×Q8—21no—
208C2×Q8—21no—
209C2×D4—172no—
210C2×D4—342no—
211C2×D4—342no—
212C2×D4—341no—
213C22×C4—341no—
214C22×C4—341no—
215C22×C4—341no—
216C22×C4—684no—
217C22×C4—681no—
218C22×C4—681no—
219C22×C4#6322712yes—
220C22×C4#6322712yes—
221C22×C4C2×C4×D1711yescharacteristic
222C22×C4C2×D6811yescharacteristic
223C22×C4C2×C22⋊D1711yescharacteristic
224C2×C8—42no—
225C2×C8—344no—
226C4⋊C4—22no—
227C4⋊C4—24no—
228C4⋊C4—342no—
229C22⋊C4—348no—
230C22⋊C4—684no—
231C22⋊C4—682no—
232C42—22no—
233C42—22no—
234C42—172no—
235C42—341no—
236C42—342no—
237C42—682no—
238C42—682no—
239C42D4×D1712yes—
240C42#6325412yes—
241C23—342no—
242C23—682no—
243C23—682no—
244C23—681no—
245C23—681no—
246C23C2×#6322711yescharacteristic
247Q8—42no—
248Q8—42no—
249D4—344no—
250D4—682no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—342no—
259C2×C4—342no—
260C2×C4—681no—
261C2×C4—681no—
262C2×C4—681no—
263C2×C4—681no—
264C2×C4—682no—
265C2×C4—681no—
266C2×C4—1362no—
267C2×C4—1362no—
268C2×C4#1058756712yes—
269C2×C4#1058756811yescharacteristic
270C2×C4#1058757211yescharacteristic
271C2×C4#1058758712yes—
272C2×C4M16×D1711yescharacteristic
273C2×C4M16×D1711yescharacteristic
274C2×C4#1058762512yes—
275C8—82no—
276C8—684no—
277C22—21no—
278C22—21no—
279C22—684no—
280C22—682no—
281C22—1362no—
282C22—1361no—
283C22—1361no—
284C22#1176268611yescharacteristic
285C22#1176277211yescharacteristic
286C22#1176277511yescharacteristic
287C4—22no—
288C4—21no—
289C4—42no—
290C4—42no—
291C4—682no—
292C4—682no—
293C4—1361no—
294C4#1176288312yes—
295C4#1176288412yes—
296C4#1176298412yes—
297C2—21no—
298C2—21no—
299C2—1362no—
300C2#1452907911yescharacteristic
301C2#1452916511yescharacteristic
302C2#1452918511yescharacteristic
subgroupG/H
centerC2×C4#10587568
commutatorC22×C4×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#1461C2
socleC22×C17#386
prime radicalC22×D68C2×C4
cosocleC2×C42×C17C23
hypercenter#1461D17
hypocenterC17#8964
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1461

Complements

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

NH ≅ G/Ntype
1. #14537869C2semidirect
4. #14526616C2semidirect
5. C17×#1461C2semidirect
129. #1461D17semidirect
203. C17#8964semidirect

Maximal subgroups

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

subgroupindex
#145378692
C4×#105876332
#145271762
#145266162
C17×#14612
#896417