Finite Group Databasebeta

← #16943950

Subgroups — #16943950

Order 4352.

444 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 87 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#14533776C211yesmaximal, characteristic
2#14527102C211yesmaximal, characteristic
3#14526717C214yesmaximal, complemented
4C17×#1465C211yesmaximal, characteristic
5C2×#10587555C411yescharacteristic
6C2×#10587622C412yes—
7C2×C22⋊#3622—21no—
8#11762734—21no—
9#11762734—21no—
10#11762714C2212yes—
11#11762669—21no—
12#11762669—21no—
13C4×C4⋊Dic17C411yescharacteristic
14#11762535C2212yes—
15#11762533—21no—
16C2×C4×D4×C17C2211yescharacteristic
17C17×#407—21no—
18C17×#324C2212yes—
19C17×#323—21no—
20C22×D4×C17C2×C411yescharacteristic
21C23×C4×C17D411yescharacteristic
22C23×C4×C17D411yescharacteristic
23C22×Dic34C2×C412yes—
24C2×M16×C17—41no—
25C4×D4×C17—41no—
26C4×D4×C17—41no—
27C2×C4⋊C4×C17C2311yescharacteristic
28C2×C22⋊C4×C17D411yescharacteristic
29C2×C22⋊C4×C17D411yescharacteristic
30C2×C42×C17C2×C411yescharacteristic
31C2×C22⋊Dic17—22no—
32C2×C22⋊Dic17—22no—
33#10587622—44no—
34C2×C4⋊Dic17—21no—
35C2×C4⋊Dic17—21no—
36C2×C4⋊Dic17—21no—
37C2×C4×Dic17C2×C412yes—
38C22×#3622—41no—
39#10587555—41no—
40#10587555—41no—
41C17×C22⋊C8—22no—
42C17×C22⋊C8—41no—
43#10587502—21no—
44#10587502—22no—
45#10587502—22no—
46C22⋊#3622—22no—
47C22⋊#3622—41no—
48C24×C17C22⋊C411yescharacteristic
49C24×C17C22⋊C411yescharacteristic
50C2×D4×C17—22no—
51C2×D4×C17—22no—
52C2×D4×C17—41no—
53C22×C4×C17—21no—
54C22×C4×C17—21no—
55C22×C4×C17—21no—
56C22×C4×C17—21no—
57C22×C4×C17—42no—
58C22×C4×C17—42no—
59C22×C4×C17C22⋊C411yescharacteristic
60C22×C4×C17C22⋊C411yescharacteristic
61C22×C4×C17C22×C411yescharacteristic
62C22×C4×C17C2×D411yescharacteristic
63C22×C4×C17C2×D411yescharacteristic
64C22×Dic17—22no—
65C22×Dic17—22no—
66C2×Dic34—24no—
67C2×Dic34—42no—
68C2×Dic34—42no—
69M16×C17—82no—
70C2×C8×C17—42no—
71C4⋊C4×C17—41no—
72C22⋊C4×C17—41no—
73C22⋊C4×C17—41no—
74C42×C17—21no—
75C42×C17—21no—
76C22⋊Dic17—44no—
77C22⋊Dic17—44no—
78C4⋊Dic17—42no—
79C4⋊Dic17—42no—
80C4⋊Dic17—41no—
81C4⋊Dic17—41no—
82C4×Dic17—42no—
83C2×#3622—42no—
84C2×#3622—82no—
85#11564—171nomaximal, Sylow
86C23×C17—21no—
87C23×C17—21no—
88C23×C17—21no—
89C23×C17—21no—
90C23×C17—21no—
91C23×C17—41no—
92C23×C17—41no—
93C23×C17—41no—
94C23×C17—41no—
95C23×C17#10012yes—
96C23×C17C2×C22⋊C411yescharacteristic
97D4×C17—44no—
98C2×C4×C17—21no—
99C2×C4×C17—21no—
100C2×C4×C17—22no—
101C2×C4×C17—21no—
102C2×C4×C17—21no—
103C2×C4×C17—21no—
104C2×C4×C17—42no—
105C2×C4×C17—42no—
106C2×C4×C17—41no—
107C2×C4×C17—41no—
108C2×C4×C17—41no—
109C2×C4×C17—81no—
110C2×C4×C17—81no—
111C2×C4×C17#9911yescharacteristic
112C2×C4×C17#9911yescharacteristic
113C2×C4×C17#13112yes—
114C2×Dic17—42no—
115C2×Dic17—42no—
116C2×Dic17—42no—
117C2×Dic17—42no—
118C2×Dic17—42no—
119C2×Dic17—42no—
120Dic34—84no—
121C8×C17—82no—
122#3622—82no—
123#2547—171no—
124#1465D1711yescharacteristic
125#1459—171no—
126#1319—174no—
127C22×C17—21no—
128C22×C17—21no—
129C22×C17—22no—
130C22×C17—42no—
131C22×C17—42no—
132C22×C17—41no—
133C22×C17—41no—
134C22×C17—41no—
135C22×C17—41no—
136C22×C17—41no—
137C22×C17—81no—
138C22×C17—81no—
139C22×C17#40811yescharacteristic
140C22×C17C2×#9911yescharacteristic
141C22×C17C2×#10011yescharacteristic
142C4×C17—21no—
143C4×C17—22no—
144C4×C17—41no—
145C4×C17—41no—
146C4×C17—41no—
147C4×C17#35412yes—
148Dic17—82no—
149Dic17—82no—
150C2×C4⋊Q8—171no—
151C2×#124—172no—
152C2×C4×D4C2×D1711yescharacteristic
153#407—21no—
154C2×C22⋊C8—341no—
155#397—341no—
156#397—341no—
157#391—172no—
158#380—341no—
159#380—341no—
160C4×C4⋊C4—171no—
161#324—172no—
162#324C2×D1712yes—
163#323—21no—
164#323—341no—
165C2×C17—21no—
166C2×C17—21no—
167C2×C17—42no—
168C2×C17—81no—
169C2×C17#147111yescharacteristic
170C2×C17C2×#35411yescharacteristic
171C2×C17#213511yescharacteristic
172C22×Q8—172no—
173C22×D4C4×D1711yescharacteristic
174C23×C4D6811yescharacteristic
175C23×C4C22⋊D1711yescharacteristic
176C2×M16—41no—
177C22×C8—681no—
178C4⋊Q8—681no—
179C4⋊Q8—681no—
180#124—684no—
181C4×D4—41no—
182C4×D4—41no—
183C2×C4⋊C4—341no—
184C2×C4⋊C4—341no—
185C2×C4⋊C4—341no—
186C2×C4⋊C4C22×D1711yescharacteristic
187C2×C22⋊C4—342no—
188C2×C22⋊C4—342no—
189C2×C22⋊C4D6811yescharacteristic
190C2×C22⋊C4C22⋊D1711yescharacteristic
191C2×C42—172no—
192C2×C42C4×D1711yescharacteristic
193C22⋊C8—22no—
194C22⋊C8—41no—
195C22⋊C8—342no—
196C22⋊C8—681no—
197#95—342no—
198#95—342no—
199#95—341no—
200C17#1156411yescharacteristic, complemented, Sylow
201C24#6322711yescharacteristic
202C24#6322711yescharacteristic
203C2×Q8—344no—
204C2×Q8—682no—
205C2×Q8—682no—
206C2×D4—22no—
207C2×D4—22no—
208C2×D4—41no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—342no—
216C22×C4—342no—
217C22×C4#6322711yescharacteristic
218C22×C4#6322711yescharacteristic
219C22×C4C2×C4×D1711yescharacteristic
220C22×C4C2×D6811yescharacteristic
221C22×C4C2×C22⋊D1711yescharacteristic
222M16—82no—
223C2×C8—42no—
224C2×C8—682no—
225C2×C8—1362no—
226C4⋊C4—41no—
227C4⋊C4—682no—
228C4⋊C4—682no—
229C4⋊C4—681no—
230C4⋊C4—681no—
231C22⋊C4—41no—
232C22⋊C4—41no—
233C22⋊C4—684no—
234C22⋊C4—684no—
235C42—21no—
236C42—21no—
237C42—682no—
238C23—21no—
239C23—21no—
240C23—21no—
241C23—21no—
242C23—21no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23#1058749412yes—
248C23C2×#6322711yescharacteristic
249Q8—1364no—
250D4—44no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—81no—
263C2×C4—81no—
264C2×C4—682no—
265C2×C4—682no—
266C2×C4—682no—
267C2×C4—682no—
268C2×C4—682no—
269C2×C4—682no—
270C2×C4#1058747711yescharacteristic
271C2×C4#1058747711yescharacteristic
272C2×C4#1058759312yes—
273C8—82no—
274C8—1362no—
275C22—21no—
276C22—21no—
277C22—22no—
278C22—42no—
279C22—42no—
280C22—41no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—41no—
285C22—81no—
286C22—81no—
287C22#1176277711yescharacteristic
288C22C2×#1058747711yescharacteristic
289C22C2×#1058749411yescharacteristic
290C4—21no—
291C4—22no—
292C4—41no—
293C4—41no—
294C4—41no—
295C4—1362no—
296C4—1362no—
297C4#1176260312yes—
298C2—21no—
299C2—21no—
300C2—42no—
301C2—81no—
302C2#1452714511yescharacteristic
303C2C2×#1176260311yescharacteristic
304C2#1453047811yescharacteristic
subgroupG/H
centerC22C2×#10587477
commutatorC22×C4×C17C22×C4
FrattiniC2×C4⋊C4C22×D17
FittingC17×#1465C2
socleC22×C17C2×#99
prime radicalC22×D4×C17C2×C4
cosocleC2×C4⋊C4×C17C23
hypercenter#1465D17
hypocenterC17#11564
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1465

Complements

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

NH ≅ G/Ntype
3. #14526717C2semidirect
200. C17#11564semidirect

Maximal subgroups

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

subgroupindex
#145337762
#145271022
#145267172
C17×#14652
#1156417