Finite Group Databasebeta

← #16944895

Subgroups — #16944895

Order 4352.

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

normal subgroups only

This group has 652 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
2C4×#10587505C211yesmaximal, characteristic, complemented
3#14526782C214yesmaximal, complemented
4C17×#1465C211yesmaximal, characteristic, complemented
5C2×#10587560C411yescharacteristic, complemented
6C2×#10587626—21no—
7C2×#10587616—21no—
8C2×#10587616—21no—
9C2×#10587505—21no—
10#11762718C412yescomplemented
11#11762718C2212yes—
12#11762710—21no—
13C4×C4⋊Dic17C411yescharacteristic, complemented
14C4×C22⋊Dic17—21no—
15#11762576C2212yes—
16C2×C4×D4×C17C2211yescharacteristic
17C17×#407—21no—
18C17×#324—21no—
19C17×#323C2212yes—
20C22×D4×C17D411yescharacteristic
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×C17D411yescharacteristic
32C2×C22⋊C4×C17C2×C411yescharacteristic
33C2×C22⋊C4×C17C2311yescharacteristic
34C2×C42×C17C2×C411yescharacteristic
35C2×C22⋊Dic17—22no—
36#10587626—44no—
37#10587616—44no—
38#10587616—44no—
39C2×#63227—22no—
40C2×#63227—22no—
41C2×C4⋊Dic17—21no—
42C2×C4×Dic17—21no—
43C2×C4×Dic17—41no—
44#10587560—41no—
45#10587560—41no—
46C17×C22⋊C8—22no—
47C17×C22⋊C8—41no—
48#10587505—24no—
49#10587502—41no—
50#10587502C2×C412yes—
51C24×C17C22⋊C411yescharacteristic
52C24×C17C2×D411yescharacteristic
53C23×D17—22no—
54C2×D4×C17—22no—
55C2×D4×C17—22no—
56C2×D4×C17—41no—
57C22×C4×C17—21no—
58C22×C4×C17—21no—
59C22×C4×C17—21no—
60C22×C4×C17—21no—
61C22×C4×C17—42no—
62C22×C4×C17—42no—
63C22×C4×C17C22⋊C411yescharacteristic
64C22×C4×C17C22⋊C411yescharacteristic
65C22×C4×C17C22⋊C411yescharacteristic
66C22×C4×C17C22×C411yescharacteristic
67C22×C4×C17C2×D411yescharacteristic
68C2×C22⋊D17—44no—
69C2×C22⋊D17—44no—
70C2×C22⋊D17—44no—
71C2×C22⋊D17—44no—
72C2×C22⋊D17—82no—
73C2×C22⋊D17—82no—
74C22×Dic17—22no—
75C22×Dic17—42no—
76C2×D68—42no—
77C2×D68—42no—
78C2×D68—42no—
79C2×D68—42no—
80C2×D68—42no—
81C2×D68—41no—
82C2×D68—41no—
83C2×D68—42no—
84C2×D68—82no—
85M16×C17—82no—
86C2×C8×C17—42no—
87C4⋊C4×C17—41no—
88C22⋊C4×C17—41no—
89C22⋊C4×C17—41no—
90C42×C17—21no—
91C42×C17—21no—
92C22⋊Dic17—24no—
93C22⋊Dic17—42no—
94#63227—44no—
95#63227—44no—
96C4⋊Dic17—24no—
97C4×Dic17—42no—
98C4×Dic17—82no—
99#11776—171nomaximal, Sylow
100C23×C17—21no—
101C23×C17—21no—
102C23×C17—21no—
103C23×C17—21no—
104C23×C17—21no—
105C23×C17—41no—
106C23×C17—41no—
107C23×C17—41no—
108C23×C17—41no—
109C23×C17C2×C22⋊C411yescharacteristic
110C23×C17AGL(1,ℤ/8)12yes—
111C22×D17—42no—
112C22×D17—42no—
113C22×D17—42no—
114C22×D17—42no—
115C22×D17—42no—
116C22×D17—42no—
117C22×D17—42no—
118D4×C17—44no—
119C2×C4×C17—21no—
120C2×C4×C17—21no—
121C2×C4×C17—22no—
122C2×C4×C17—21no—
123C2×C4×C17—21no—
124C2×C4×C17—21no—
125C2×C4×C17—42no—
126C2×C4×C17—42no—
127C2×C4×C17—41no—
128C2×C4×C17—41no—
129C2×C4×C17—41no—
130C2×C4×C17—81no—
131C2×C4×C17—81no—
132C2×C4×C17#9911yescharacteristic
133C2×C4×C17#9911yescharacteristic
134C2×C4×C17#10412yes—
135C22⋊D17—88no—
136C22⋊D17—88no—
137C2×Dic17—24no—
138C2×Dic17—42no—
139C2×Dic17—42no—
140C2×Dic17—44no—
141C2×Dic17—82no—
142C2×Dic17—82no—
143D68—82no—
144D68—84no—
145D68—84no—
146D68—82no—
147C8×C17—82no—
148#2541—171no—
149C4×#99—171no—
150#1465D1711yescharacteristic, complemented
151#1343—174no—
152C22×C17—21no—
153C22×C17—21no—
154C22×C17—22no—
155C22×C17—42no—
156C22×C17—42no—
157C22×C17—41no—
158C22×C17—41no—
159C22×C17—41no—
160C22×C17—41no—
161C22×C17—41no—
162C22×C17—81no—
163C22×C17—81no—
164C22×C17C2×#9911yescharacteristic
165C22×C17#41811yescharacteristic
166C22×C17C2×#10411yescharacteristic
167C2×D17—82no—
168C2×D17—82no—
169C2×D17—82no—
170C2×D17—82no—
171C2×D17—82no—
172C2×D17—82no—
173C2×D17—82no—
174C4×C17—21no—
175C4×C17—22no—
176C4×C17—41no—
177C4×C17—41no—
178C4×C17—41no—
179C4×C17#35312yes—
180Dic17—44no—
181Dic17—84no—
182C2×#127—171no—
183C2×#127—341no—
184C2×#121—341no—
185C2×#121—341no—
186C2×C4×D4C2×D1711yescharacteristic
187C2×#99—341no—
188#407—21no—
189#392—172no—
190#392—172no—
191#390—341no—
192C4×C4⋊C4—171no—
193C4×C22⋊C4—341no—
194#342—172no—
195#324—21no—
196#323C2×D1712yes—
197C2×C17—21no—
198C2×C17—21no—
199C2×C17—42no—
200C2×C17—81no—
201C2×C17#150711yescharacteristic
202C2×C17#212211yescharacteristic
203C2×C17C2×#35311yescharacteristic
204D17—162no—
205C22×D4—172no—
206C22×D4—342no—
207C22×D4—342no—
208C22×D4—341no—
209C22×D4—341no—
210C22×D4D6811yescharacteristic
211C23×C4D6811yescharacteristic
212C23×C4C22⋊D1711yescharacteristic
213C2×M16—41no—
214#127—681no—
215#127—681no—
216#127—684no—
217#121—684no—
218#121—684no—
219C4×D4—41no—
220C4×D4—41no—
221C2×C4⋊C4—341no—
222C2×C4⋊C4C22⋊D1711yescharacteristic
223C2×C22⋊C4—342no—
224C2×C22⋊C4—342no—
225C2×C22⋊C4—342no—
226C2×C22⋊C4C4×D1711yescharacteristic
227C2×C22⋊C4C22×D1711yescharacteristic
228C2×C42—341no—
229C2×C42—681no—
230C2×C42C4×D1711yescharacteristic
231#99—344no—
232C22⋊C8—22no—
233C22⋊C8—41no—
234#95—172no—
235#95—681no—
236C17#1177611yescharacteristic, complemented, Sylow
237C24—342no—
238C24#6322711yescharacteristic
239C24C2×D6811yescharacteristic
240C2×D4—22no—
241C2×D4—22no—
242C2×D4—41no—
243C2×D4—682no—
244C2×D4—682no—
245C2×D4—682no—
246C2×D4—682no—
247C2×D4—682no—
248C2×D4—684no—
249C2×D4—684no—
250C2×D4—684no—
251C2×D4—684no—
252C2×D4—681no—
253C2×D4—681no—
254C2×D4—682no—
255C2×D4—1362no—
256C2×D4—1362no—
257C2×D4—1362no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—342no—
265C22×C4—682no—
266C22×C4#6322711yescharacteristic
267C22×C4#6322711yescharacteristic
268C22×C4#6322711yescharacteristic
269C22×C4C2×C4×D1711yescharacteristic
270C22×C4C2×C22⋊D1711yescharacteristic
271M16—82no—
272C2×C8—42no—
273C4⋊C4—41no—
274C4⋊C4—344no—
275C22⋊C4—41no—
276C22⋊C4—41no—
277C22⋊C4—344no—
278C22⋊C4—684no—
279C22⋊C4—684no—
280C22⋊C4—682no—
281C42—21no—
282C42—21no—
283C42—682no—
284C42—1362no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23—682no—
295C23—682no—
296C23—682no—
297C23—682no—
298C23—682no—
299C23—682no—
300C23—682no—
301C23#1058759412yes—
302C23C2×#6322711yescharacteristic
303D4—44no—
304D4—1364no—
305D4—1362no—
306D4—1364no—
307D4—1368no—
308D4—1368no—
309D4—1362no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—22no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—344no—
324C2×C4—682no—
325C2×C4—682no—
326C2×C4—684no—
327C2×C4—1362no—
328C2×C4—1362no—
329C2×C4#1058747711yescharacteristic
330C2×C4#1058747711yescharacteristic
331C2×C4#1058749612yes—
332C8—82no—
333C22—21no—
334C22—21no—
335C22—22no—
336C22—42no—
337C22—42no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—81no—
344C22—81no—
345C22—1362no—
346C22—1362no—
347C22—1362no—
348C22—1362no—
349C22—1362no—
350C22—1362no—
351C22—1362no—
352C22C2×#1058747711yescharacteristic
353C22#1176281511yescharacteristic
354C22C2×#1058749611yescharacteristic
355C4—22no—
356C4—21no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—684no—
361C4—1364no—
362C4#1176260112yes—
363C2—21no—
364C2—21no—
365C2—42no—
366C2—81no—
367C2—2722no—
368C2#1452730111yescharacteristic
369C2#1453043511yescharacteristic
370C2C2×#1176260111yescharacteristic
subgroupG/H
centerC22C2×#10587477
commutatorC22×C4×C17C22×C4
FrattiniC2×C22⋊C4C22×D17
FittingC17×#1465C2
socleC22×C17C2×#99
prime radical#14533730C2
cosocleC2×C22⋊C4×C17C23
hypercenter#1465D17
hypocenterC17#11776
LCSG ▷ C22×C4×C17 ▷ C23×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. C4×#10587505C2semidirect
3. #14526782C2semidirect
4. C17×#1465C2semidirect
5. C2×#10587560C4semidirect
10. #11762718C4semidirect
13. C4×C4⋊Dic17C4semidirect
150. #1465D17semidirect
236. C17#11776semidirect

Maximal subgroups

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

subgroupindex
#145337302
C4×#105875052
#145267822
C17×#14652
#1177617