Finite Group Databasebeta

← #16932001

Subgroups — #16932001

Order 4352.

518 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 518 conjugacy classes of subgroups — too many to draw. Only its 103 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#14537813C211yesmaximal, characteristic
2#14533776C211yesmaximal, characteristic
3#14527509C212yesmaximal, complemented
4#14526639C212yesmaximal, complemented
5C17×#1466C211yesmaximal, characteristic
6C2×#10587555C2211yescharacteristic
7C2×#10587622C412yes—
8#11762881—24no—
9#11762866C2212yes—
10#11762773—21no—
11C2×C22⋊#3622—21no—
12#11762734—21no—
13#11762734—21no—
14#11762714C412yes—
15#11762669—21no—
16#11762669—21no—
17C4×C4⋊Dic17C2211yescharacteristic
18#11762557—24no—
19C2×C4×D4×C17C2211yescharacteristic
20C17×#432C2212yes—
21C17×#323—21no—
22C17×#323—21no—
23C22×D4×C17C2×C411yescharacteristic
24C23×C4×C17D411yescharacteristic
25C23×C4×C17D411yescharacteristic
26C22×Dic34C2×C412yes—
27C4×D4×C17—22no—
28C4×D4×C17—22no—
29C2×C4⋊C4×C17C2×C411yescharacteristic
30C2×C22⋊C4×C17D411yescharacteristic
31C2×C22⋊C4×C17D411yescharacteristic
32C2×C42×C17C2311yescharacteristic
33C2×C22⋊Dic17—22no—
34C2×C22⋊Dic17—22no—
35#10587622—44no—
36C2×C4⋊Dic17—21no—
37C2×C4⋊Dic17—21no—
38C2×C4⋊Dic17—21no—
39C2×C4×Dic17C2×C412yes—
40C2×#63223—22no—
41C22×#3622—22no—
42#10587555—22no—
43#10587555—22no—
44C17×C4⋊C8—22no—
45C17×C22⋊C8—22no—
46C17×#97—22no—
47#10587502—22no—
48#10587502—22no—
49#10587502—21no—
50C22⋊#3622—22no—
51C22⋊#3622—22no—
52C22⋊#3622—22no—
53C4⋊#3622D414yes—
54#10587474—22no—
55C24×C17C22⋊C411yescharacteristic
56C24×C17C22⋊C411yescharacteristic
57C2×D4×C17—22no—
58C2×D4×C17—22no—
59C2×D4×C17—41no—
60C22×C4×C17—21no—
61C22×C4×C17—21no—
62C22×C4×C17—21no—
63C22×C4×C17—21no—
64C22×C4×C17—42no—
65C22×C4×C17—42no—
66C22×C4×C17C22⋊C411yescharacteristic
67C22×C4×C17C22⋊C411yescharacteristic
68C22×C4×C17C22×C411yescharacteristic
69C22×C4×C17C2×D411yescharacteristic
70C22×C4×C17C2×D411yescharacteristic
71C22×Dic17—22no—
72C22×Dic17—22no—
73C2×Dic34—24no—
74C2×Dic34—42no—
75C2×Dic34—42no—
76C2×C8×C17—42no—
77C4⋊C4×C17—22no—
78C22⋊C4×C17—22no—
79C22⋊C4×C17—22no—
80C42×C17C2×D412yes—
81C42×C17#4112yes—
82C22⋊Dic17—44no—
83C22⋊Dic17—44no—
84C4⋊Dic17—42no—
85C4⋊Dic17—41no—
86C4⋊Dic17—42no—
87C4⋊Dic17—41no—
88C4×Dic17—42no—
89#63223—82no—
90C2×#3622—24no—
91C2×#3622—82no—
92#9451—171nomaximal, Sylow
93C23×C17—21no—
94C23×C17—21no—
95C23×C17—21no—
96C23×C17—21no—
97C23×C17—21no—
98C23×C17—41no—
99C23×C17—41no—
100C23×C17—41no—
101C23×C17—41no—
102C23×C17#10012yes—
103C23×C17C2×C22⋊C411yescharacteristic
104D4×C17—44no—
105C2×C4×C17—21no—
106C2×C4×C17—21no—
107C2×C4×C17—22no—
108C2×C4×C17—21no—
109C2×C4×C17—21no—
110C2×C4×C17—41no—
111C2×C4×C17—42no—
112C2×C4×C17—42no—
113C2×C4×C17—41no—
114C2×C4×C17—41no—
115C2×C4×C17—81no—
116C2×C4×C17—81no—
117C2×C4×C17C4×D412yes—
118C2×C4×C17#12011yescharacteristic
119C2×C4×C17#12112yes—
120C2×C4×C17#12311yescharacteristic
121C2×Dic17—42no—
122C2×Dic17—42no—
123C2×Dic17—42no—
124C2×Dic17—42no—
125C2×Dic17—42no—
126C2×Dic17—42no—
127Dic34—84no—
128C8×C17—82no—
129#3622—44no—
130#2973—171no—
131#2547—171no—
132#1558—172no—
133#1466D1711yescharacteristic
134#1285—172no—
135C22×C17—21no—
136C22×C17—21no—
137C22×C17—22no—
138C22×C17—42no—
139C22×C17—42no—
140C22×C17—41no—
141C22×C17—41no—
142C22×C17—41no—
143C22×C17—41no—
144C22×C17—41no—
145C22×C17—81no—
146C22×C17—81no—
147C22×C17#38611yescharacteristic
148C22×C17C2×#10011yescharacteristic
149C22×C17#41311yescharacteristic
150C4×C17—21no—
151C4×C17—22no—
152C4×C17—22no—
153C4×C17—41no—
154C4×C17—41no—
155C4×C17#45512yes—
156Dic17—82no—
157Dic17—82no—
158C2×C4⋊Q8—171no—
159C2×#124—172no—
160C2×C4×D4C2×D1711yescharacteristic
161#435—344no—
162#432—172no—
163#432C2×D1712yes—
164#407—341no—
165C2×C22⋊C8—341no—
166#397—341no—
167#397—341no—
168#391—172no—
169#380—341no—
170#380—341no—
171C4×C4⋊C4—171no—
172#333—344no—
173#323—21no—
174#323—21no—
175C2×C17—21no—
176C2×C17—21no—
177C2×C17—42no—
178C2×C17—81no—
179C2×C17#188511yescharacteristic
180C2×C17#190011yescharacteristic
181C2×C17#190911yescharacteristic
182C22×Q8—172no—
183C22×D4C4×D1711yescharacteristic
184C23×C4D6811yescharacteristic
185C23×C4D6811yescharacteristic
186C2×M16—342no—
187C22×C8—342no—
188C4⋊Q8—342no—
189C4⋊Q8—342no—
190#124—684no—
191C4×D4—22no—
192C4×D4—22no—
193C2×C4⋊C4—341no—
194C2×C4⋊C4—341no—
195C2×C4⋊C4—341no—
196C2×C4⋊C4C4×D1711yescharacteristic
197C2×C22⋊C4—342no—
198C2×C22⋊C4—342no—
199C2×C22⋊C4C22⋊D1711yescharacteristic
200C2×C22⋊C4C22⋊D1711yescharacteristic
201C2×C42—172no—
202C2×C42C22×D1711yescharacteristic
203C4⋊C8—22no—
204C4⋊C8—174no—
205C22⋊C8—22no—
206C22⋊C8—342no—
207C22⋊C8—342no—
208C22⋊C8—342no—
209#97—22no—
210#97—342no—
211#95—341no—
212#95—342no—
213#95—342no—
214C17#945111yescharacteristic, complemented, Sylow
215C24#6322711yescharacteristic
216C24#6322711yescharacteristic
217C2×Q8—344no—
218C2×Q8—682no—
219C2×Q8—682no—
220C2×D4—22no—
221C2×D4—22no—
222C2×D4—41no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—342no—
230C22×C4—342no—
231C22×C4#6322711yescharacteristic
232C22×C4#6322711yescharacteristic
233C22×C4C2×C4×D1711yescharacteristic
234C22×C4C2×D6811yescharacteristic
235C22×C4C2×C22⋊D1711yescharacteristic
236M16—1362no—
237C2×C8—42no—
238C2×C8—344no—
239C2×C8—1362no—
240C4⋊C4—22no—
241C4⋊C4—682no—
242C4⋊C4—681no—
243C4⋊C4—681no—
244C4⋊C4—682no—
245C22⋊C4—22no—
246C22⋊C4—22no—
247C22⋊C4—684no—
248C22⋊C4—684no—
249C42—682no—
250C42D4×D1712yes—
251C42#6325412yes—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23#1058749412yes—
262C23C2×#6322711yescharacteristic
263Q8—1364no—
264D4—44no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—81no—
276C2×C4—81no—
277C2×C4—682no—
278C2×C4—682no—
279C2×C4—682no—
280C2×C4—682no—
281C2×C4—682no—
282C2×C4—682no—
283C2×C4#1058756712yes—
284C2×C4#1058756811yescharacteristic
285C2×C4#1058757211yescharacteristic
286C2×C4#1058762512yes—
287C8—82no—
288C8—684no—
289C22—21no—
290C22—21no—
291C22—22no—
292C22—42no—
293C22—42no—
294C22—41no—
295C22—41no—
296C22—41no—
297C22—41no—
298C22—41no—
299C22—81no—
300C22—81no—
301C22#1176268611yescharacteristic
302C22C2×#1058749411yescharacteristic
303C22#1176279111yescharacteristic
304C4—22no—
305C4—21no—
306C4—22no—
307C4—41no—
308C4—41no—
309C4—1362no—
310C4—1362no—
311C4#1176298912yes—
312C2—21no—
313C2—21no—
314C2—42no—
315C2—81no—
316C2#1452917011yescharacteristic
317C2#1452924511yescharacteristic
318C2#1452930411yescharacteristic
subgroupG/H
centerC22#11762686
commutatorC22×C4×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#1466C2
socleC22×C17#386
prime radicalC22×D4×C17C2×C4
cosocleC2×C42×C17C23
hypercenter#1466D17
hypocenterC17#9451
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1466

Complements

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

NH ≅ G/Ntype
3. #14527509C2semidirect
4. #14526639C2semidirect
214. C17#9451semidirect

Maximal subgroups

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

subgroupindex
#145378132
#145337762
#145275092
#145266392
C17×#14662
#945117