Finite Group Databasebeta

← #16943946

Subgroups — #16943946

Order 4352.

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

normal subgroups only

This group has 616 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#14533730C211yesmaximal, characteristic
2#14527102C211yesmaximal, characteristic, complemented
3#14526715C214yesmaximal, complemented
4C17×#1465C211yesmaximal, characteristic, complemented
5C2×#10587560C411yescharacteristic
6C2×#10587626C412yes—
7C2×#10587616—21no—
8C2×#10587616—21no—
9C2×C22⋊#3622—21no—
10#11762718—21no—
11#11762718—21no—
12#11762710C2212yes—
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×C22⋊D17—22no—
24C22×C22⋊D17—22no—
25C22×D68—21no—
26C22×D68—21no—
27C22×D68C2×C412yes—
28C2×M16×C17—41no—
29C4×D4×C17—41no—
30C4×D4×C17—41no—
31C2×C4⋊C4×C17C2311yescharacteristic
32C2×C22⋊C4×C17D411yescharacteristic
33C2×C22⋊C4×C17D411yescharacteristic
34C2×C42×C17C2×C411yescharacteristic
35#10587626—44no—
36#10587616—44no—
37#10587616—44no—
38C2×#63227—22no—
39C2×#63227—22no—
40C2×C4⋊Dic17—21no—
41C2×C4×Dic17C2×C412yes—
42C22×#3622—41no—
43#10587560—41no—
44#10587560—41no—
45C17×C22⋊C8—22no—
46C17×C22⋊C8—41no—
47#10587502—21no—
48C22⋊#3622—22no—
49C22⋊#3622—41no—
50C24×C17C22⋊C411yescharacteristic
51C24×C17C22⋊C411yescharacteristic
52C23×D17—22no—
53C2×D4×C17—22no—
54C2×D4×C17—22no—
55C2×D4×C17—41no—
56C22×C4×C17—21no—
57C22×C4×C17—21no—
58C22×C4×C17—21no—
59C22×C4×C17—21no—
60C22×C4×C17—42no—
61C22×C4×C17—42no—
62C22×C4×C17C22⋊C411yescharacteristic
63C22×C4×C17C22⋊C411yescharacteristic
64C22×C4×C17C22×C411yescharacteristic
65C22×C4×C17C2×D411yescharacteristic
66C22×C4×C17C2×D411yescharacteristic
67C2×C22⋊D17—44no—
68C2×C22⋊D17—44no—
69C2×C22⋊D17—44no—
70C2×C22⋊D17—44no—
71C2×C22⋊D17—82no—
72C2×C22⋊D17—82no—
73C22×Dic17—22no—
74C2×D68—24no—
75C2×D68—42no—
76C2×D68—42no—
77C2×D68—42no—
78C2×D68—42no—
79C2×D68—42no—
80C2×D68—41no—
81C2×D68—41no—
82C2×D68—82no—
83M16×C17—82no—
84C2×C8×C17—42no—
85C4⋊C4×C17—41no—
86C22⋊C4×C17—41no—
87C22⋊C4×C17—41no—
88C42×C17—21no—
89C42×C17—21no—
90#63227—44no—
91#63227—44no—
92C4⋊Dic17—24no—
93C4×Dic17—42no—
94C2×#3622—42no—
95C2×#3622—82no—
96#11563—171nomaximal, Sylow
97C23×C17—21no—
98C23×C17—21no—
99C23×C17—21no—
100C23×C17—21no—
101C23×C17—21no—
102C23×C17—41no—
103C23×C17—41no—
104C23×C17—41no—
105C23×C17—41no—
106C23×C17#10012yes—
107C23×C17C2×C22⋊C411yescharacteristic
108C22×D17—42no—
109C22×D17—42no—
110C22×D17—42no—
111C22×D17—42no—
112C22×D17—42no—
113C22×D17—42no—
114C22×D17—42no—
115D4×C17—44no—
116C2×C4×C17—21no—
117C2×C4×C17—21no—
118C2×C4×C17—22no—
119C2×C4×C17—21no—
120C2×C4×C17—21no—
121C2×C4×C17—21no—
122C2×C4×C17—42no—
123C2×C4×C17—42no—
124C2×C4×C17—41no—
125C2×C4×C17—41no—
126C2×C4×C17—41no—
127C2×C4×C17—81no—
128C2×C4×C17—81no—
129C2×C4×C17#9911yescharacteristic
130C2×C4×C17#9911yescharacteristic
131C2×C4×C17#13112yes—
132C22⋊D17—88no—
133C22⋊D17—88no—
134C2×Dic17—24no—
135C2×Dic17—42no—
136C2×Dic17—42no—
137D68—84no—
138D68—82no—
139D68—84no—
140D68—82no—
141C8×C17—82no—
142#3622—82no—
143#2541—171no—
144#1465D1711yescharacteristic, complemented
145#1459—171no—
146#1318—174no—
147C22×C17—21no—
148C22×C17—21no—
149C22×C17—22no—
150C22×C17—42no—
151C22×C17—42no—
152C22×C17—41no—
153C22×C17—41no—
154C22×C17—41no—
155C22×C17—41no—
156C22×C17—41no—
157C22×C17—81no—
158C22×C17—81no—
159C22×C17#40811yescharacteristic
160C22×C17C2×#9911yescharacteristic
161C22×C17C2×#10011yescharacteristic
162C2×D17—82no—
163C2×D17—82no—
164C2×D17—82no—
165C2×D17—82no—
166C2×D17—82no—
167C2×D17—82no—
168C2×D17—82no—
169C4×C17—21no—
170C4×C17—22no—
171C4×C17—41no—
172C4×C17—41no—
173C4×C17—41no—
174C4×C17#35312yes—
175Dic17—44no—
176C2×#127—172no—
177C2×#127—171no—
178C2×#121—341no—
179C2×#121—341no—
180C2×C4×D4C2×D1711yescharacteristic
181#407—21no—
182C2×C22⋊C8—341no—
183#392—341no—
184#392—341no—
185#390—172no—
186C4×C4⋊C4—171no—
187#324—172no—
188#324C2×D1712yes—
189#323—21no—
190#323—341no—
191C2×C17—21no—
192C2×C17—21no—
193C2×C17—42no—
194C2×C17—81no—
195C2×C17#147111yescharacteristic
196C2×C17C2×#35311yescharacteristic
197C2×C17#213411yescharacteristic
198D17—162no—
199C22×D4—172no—
200C22×D4—342no—
201C22×D4—342no—
202C22×D4—341no—
203C22×D4—341no—
204C22×D4C4×D1711yescharacteristic
205C23×C4D6811yescharacteristic
206C23×C4C22⋊D1711yescharacteristic
207C2×M16—41no—
208C22×C8—681no—
209#127—681no—
210#127—681no—
211#127—684no—
212#121—684no—
213#121—684no—
214C4×D4—41no—
215C4×D4—41no—
216C2×C4⋊C4—341no—
217C2×C4⋊C4C22×D1711yescharacteristic
218C2×C22⋊C4—342no—
219C2×C22⋊C4—342no—
220C2×C22⋊C4D6811yescharacteristic
221C2×C22⋊C4C22⋊D1711yescharacteristic
222C2×C42—172no—
223C2×C42C4×D1711yescharacteristic
224C22⋊C8—22no—
225C22⋊C8—41no—
226C22⋊C8—342no—
227C22⋊C8—681no—
228#95—341no—
229C17#1156311yescharacteristic, complemented, Sylow
230C24—342no—
231C24#6322711yescharacteristic
232C24#6322711yescharacteristic
233C2×D4—22no—
234C2×D4—22no—
235C2×D4—41no—
236C2×D4—344no—
237C2×D4—682no—
238C2×D4—682no—
239C2×D4—682no—
240C2×D4—682no—
241C2×D4—682no—
242C2×D4—684no—
243C2×D4—684no—
244C2×D4—684no—
245C2×D4—684no—
246C2×D4—681no—
247C2×D4—681no—
248C2×D4—1362no—
249C2×D4—1362no—
250C2×D4—1362no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—342no—
258C22×C4#6322711yescharacteristic
259C22×C4#6322711yescharacteristic
260C22×C4C2×C4×D1711yescharacteristic
261C22×C4C2×D6811yescharacteristic
262C22×C4C2×C22⋊D1711yescharacteristic
263M16—82no—
264C2×C8—42no—
265C2×C8—682no—
266C2×C8—1362no—
267C4⋊C4—41no—
268C4⋊C4—344no—
269C22⋊C4—41no—
270C22⋊C4—41no—
271C22⋊C4—684no—
272C22⋊C4—684no—
273C42—21no—
274C42—21no—
275C42—682no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—682no—
286C23—682no—
287C23—682no—
288C23—682no—
289C23—682no—
290C23—682no—
291C23—682no—
292C23#1058749412yes—
293C23C2×#6322711yescharacteristic
294D4—44no—
295D4—1364no—
296D4—1362no—
297D4—1364no—
298D4—1368no—
299D4—1368no—
300D4—1362no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—344no—
315C2×C4—682no—
316C2×C4—682no—
317C2×C4#1058747711yescharacteristic
318C2×C4#1058747711yescharacteristic
319C2×C4#1058759312yes—
320C8—82no—
321C8—1362no—
322C22—21no—
323C22—21no—
324C22—22no—
325C22—42no—
326C22—42no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—81no—
333C22—81no—
334C22—1362no—
335C22—1362no—
336C22—1362no—
337C22—1362no—
338C22—1362no—
339C22—1362no—
340C22—1362no—
341C22#1176277711yescharacteristic
342C22C2×#1058747711yescharacteristic
343C22C2×#1058749411yescharacteristic
344C4—21no—
345C4—22no—
346C4—41no—
347C4—41no—
348C4—41no—
349C4—684no—
350C4#1176260112yes—
351C2—21no—
352C2—21no—
353C2—42no—
354C2—81no—
355C2—2722no—
356C2#1452714511yescharacteristic
357C2C2×#1176260111yescharacteristic
358C2#1453047511yescharacteristic
subgroupG/H
centerC22C2×#10587477
commutatorC22×C4×C17C22×C4
FrattiniC2×C4⋊C4C22×D17
FittingC17×#1465C2
socleC22×C17C2×#99
prime radical#14533730C2
cosocleC2×C4⋊C4×C17C23
hypercenter#1465D17
hypocenterC17#11563
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
2. #14527102C2semidirect
3. #14526715C2semidirect
4. C17×#1465C2semidirect
144. #1465D17semidirect
229. C17#11563semidirect

Maximal subgroups

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

subgroupindex
#145337302
#145271022
#145267152
C17×#14652
#1156317