Finite Group Databasebeta

← #16931978

Subgroups — #16931978

Order 4352.

516 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 99 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#14533242C211yesmaximal, characteristic
2#14528861C211yesmaximal, characteristic, complemented
3#14527314C212yesmaximal
4#14526638C212yesmaximal, complemented
5C17×#2921C211yesmaximal, characteristic, complemented
6C2×#10587616C2211yescharacteristic
7#11762724—21no—
8#11762724—21no—
9#11762710C411yescharacteristic, complemented
10#11762710C411yescharacteristic, complemented
11#11762700—21no—
12#11762700—21no—
13#11762678C411yescharacteristic, complemented
14#11762674C411yescharacteristic, complemented
15#11762672C2212yes—
16C4×C22⋊Dic17C2211yescharacteristic
17#11762576—21no—
18#11762576—21no—
19#11762544—24no—
20C2×C17×#117C2211yescharacteristic
21C17×#433—24no—
22C17×#423—21no—
23C17×#407C2212yes—
24C4×M16×C17—21no—
25C23×C4×C17C2311yescharacteristic
26C22×C22⋊D17—21no—
27C22×D68C2×C411yescharacteristic
28C2×M16×C17—22no—
29C17×#117—22no—
30C17×#117—22no—
31C2×C4⋊C4×C17C2×C411yescharacteristic
32C2×C4⋊C4×C17C2×C411yescharacteristic
33C2×C22⋊C4×C17D411yescharacteristic
34C2×C22⋊C4×C17D411yescharacteristic
35C2×C42×C17D411yescharacteristic
36C2×C42×C17D411yescharacteristic
37C2×C22⋊Dic17—21no—
38C2×C22⋊Dic17—22no—
39#10587616—22no—
40#10587616—22no—
41C2×#63227—21no—
42C2×#63227—21no—
43C2×#63227—21no—
44C2×#63227—21no—
45C2×#63227—21no—
46C2×C4⋊Dic17C2×C411yescharacteristic
47C2×#63225—21no—
48C2×#63225—21no—
49C2×C4×Dic17C2×C411yescharacteristic
50C2×C4×Dic17C2×C411yescharacteristic
51C17×C4⋊C8—24no—
52C17×C22⋊C8—22no—
53C17×C22⋊C8D414yes—
54C17×#97—22no—
55C4×C8×C17—22no—
56#10587502—21no—
57#10587502—21no—
58#10587502—21no—
59#10587502—22no—
60#10587502—22no—
61C24×C17C2×D411yescharacteristic
62C23×D17—21no—
63C22×C4×C17—41no—
64C22×C4×C17—41no—
65C22×C4×C17C22⋊C411yescharacteristic
66C22×C4×C17C22⋊C411yescharacteristic
67C22×C4×C17C22⋊C411yescharacteristic
68C22×C4×C17C22⋊C411yescharacteristic
69C22×C4×C17C22×C411yescharacteristic
70C22×C4×C17C2×D411yescharacteristic
71C22×C4×C17C2×D412yes—
72C22×C4×C17#4112yes—
73C2×C22⋊D17—42no—
74C2×C22⋊D17—81no—
75C2×C22⋊D17—81no—
76C22×Dic17—21no—
77C22×Dic17—21no—
78C22×Dic17—21no—
79C22×Dic17—42no—
80C2×D68—22no—
81C2×D68—41no—
82C2×D68—41no—
83M16×C17—42no—
84M16×C17—42no—
85C2×C8×C17—24no—
86C4⋊C4×C17—22no—
87C4⋊C4×C17—22no—
88C22⋊C4×C17—22no—
89C22⋊C4×C17—22no—
90C42×C17—21no—
91C42×C17—21no—
92C42×C17—21no—
93C42×C17—21no—
94C22⋊Dic17—42no—
95C22⋊Dic17—42no—
96C22⋊Dic17—42no—
97#63227—42no—
98#63227—42no—
99#63227—41no—
100#63227—41no—
101#63227—81no—
102#63227—81no—
103C4⋊Dic17—22no—
104#63225—81no—
105#63225—81no—
106C4×Dic17—41no—
107C4×Dic17—41no—
108#9446—171nomaximal, Sylow
109C23×C17—21no—
110C23×C17—21no—
111C23×C17—41no—
112C23×C17—41no—
113C23×C17C2×C22⋊C411yescharacteristic
114C23×C17#12112yes—
115C22×D17—41no—
116C22×D17—41no—
117C22×D17—41no—
118C22×D17—41no—
119C22×D17—41no—
120C22×D17—41no—
121C22×D17—41no—
122C2×C4×C17—21no—
123C2×C4×C17—21no—
124C2×C4×C17—21no—
125C2×C4×C17—21no—
126C2×C4×C17—21no—
127C2×C4×C17—21no—
128C2×C4×C17—21no—
129C2×C4×C17—21no—
130C2×C4×C17—42no—
131C2×C4×C17—42no—
132C2×C4×C17—41no—
133C2×C4×C17—41no—
134C2×C4×C17—41no—
135C2×C4×C17—41no—
136C2×C4×C17C4×D412yes—
137C2×C4×C17#12011yescharacteristic
138C2×C4×C17#12311yescharacteristic
139C22⋊D17—84no—
140C2×Dic17—22no—
141C2×Dic17—41no—
142C2×Dic17—41no—
143C2×Dic17—41no—
144C2×Dic17—41no—
145C2×Dic17—41no—
146C2×Dic17—41no—
147C2×Dic17—41no—
148C2×Dic17—41no—
149C2×Dic17—82no—
150C2×Dic17—82no—
151C2×Dic17—82no—
152D68—82no—
153C8×C17—44no—
154#2921D1711yescharacteristic, complemented
155#2500—171no—
156#1820—171no—
157#1510—172no—
158#1284—172no—
159C22×C17—21no—
160C22×C17—21no—
161C22×C17—22no—
162C22×C17—22no—
163C22×C17—22no—
164C22×C17—81no—
165C22×C17—81no—
166C22×C17#38611yescharacteristic
167C22×C17#41011yescharacteristic
168C22×C17#41011yescharacteristic
169C2×D17—81no—
170C2×D17—81no—
171C2×D17—81no—
172C2×D17—81no—
173C2×D17—81no—
174C2×D17—81no—
175C2×D17—81no—
176C4×C17—21no—
177C4×C17—22no—
178C4×C17—21no—
179C4×C17—41no—
180C4×C17—41no—
181C4×C17—41no—
182C4×C17—41no—
183Dic17—42no—
184Dic17—81no—
185Dic17—81no—
186Dic17—162no—
187C2×#121—171no—
188C2×#117C2×D1711yescharacteristic
189#433—24no—
190#423—21no—
191#407C2×D1712yes—
192C4×M16—21no—
193#394—341no—
194#394—341no—
195#390—171no—
196#390—171no—
197#388—341no—
198#388—341no—
199#384—171no—
200#382—171no—
201#381—172no—
202C4×C22⋊C4—171no—
203#342—341no—
204#342—341no—
205#327—344no—
206C2×C17—21no—
207C2×C17—21no—
208C2×C17—42no—
209C2×C17#188411yescharacteristic
210C2×C17#189911yescharacteristic
211C2×C17#191011yescharacteristic
212D17—161no—
213C22×D4—171no—
214C22×D4—341no—
215C23×C4C22×D1711yescharacteristic
216C2×M16—22no—
217#121—342no—
218#121—342no—
219#117—22no—
220#117—22no—
221C2×C4⋊C4—171no—
222C2×C4⋊C4—341no—
223C2×C4⋊C4—341no—
224C2×C4⋊C4C4×D1711yescharacteristic
225C2×C4⋊C4C4×D1711yescharacteristic
226C2×C22⋊C4—341no—
227C2×C22⋊C4—341no—
228C2×C22⋊C4—341no—
229C2×C22⋊C4—341no—
230C2×C22⋊C4—342no—
231C2×C22⋊C4—341no—
232C2×C22⋊C4—341no—
233C2×C22⋊C4D6811yescharacteristic
234C2×C22⋊C4D6811yescharacteristic
235C2×C42—171no—
236C2×C42—171no—
237C2×C42C22⋊D1711yescharacteristic
238C2×C42C22⋊D1711yescharacteristic
239C4⋊C8—24no—
240C22⋊C8—22no—
241C22⋊C8C22⋊D1714yes—
242#97—22no—
243C4×C8—22no—
244#95—341no—
245#95—341no—
246#95—341no—
247#95—342no—
248#95—342no—
249C17#944611yescharacteristic, complemented, Sylow
250C24—341no—
251C24C2×D6811yescharacteristic
252C2×D4—342no—
253C2×D4—681no—
254C2×D4—681no—
255C2×D4—682no—
256C2×D4—1361no—
257C2×D4—1361no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—341no—
261C22×C4—341no—
262C22×C4—341no—
263C22×C4—682no—
264C22×C4#6322711yescharacteristic
265C22×C4#6322711yescharacteristic
266C22×C4#6322711yescharacteristic
267C22×C4#6322711yescharacteristic
268C22×C4C2×C4×D1711yescharacteristic
269C22×C4#6325212yes—
270C22×C4C2×C22⋊D1711yescharacteristic
271C22×C4C2×C22⋊D1712yes—
272M16—42no—
273M16—42no—
274C2×C8—24no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—342no—
278C4⋊C4—1361no—
279C4⋊C4—1361no—
280C22⋊C4—22no—
281C22⋊C4—22no—
282C22⋊C4—682no—
283C22⋊C4—682no—
284C22⋊C4—682no—
285C22⋊C4—681no—
286C22⋊C4—681no—
287C22⋊C4—682no—
288C22⋊C4—682no—
289C22⋊C4—1361no—
290C22⋊C4—1361no—
291C42—21no—
292C42—21no—
293C42—21no—
294C42—21no—
295C42—681no—
296C42—681no—
297C23—21no—
298C23—21no—
299C23—41no—
300C23—41no—
301C23—681no—
302C23—681no—
303C23—681no—
304C23—681no—
305C23—681no—
306C23—681no—
307C23—681no—
308C23C2×#6322711yescharacteristic
309C23#1058761612yes—
310D4—1362no—
311D4—1364no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—342no—
327C2×C4—681no—
328C2×C4—681no—
329C2×C4—681no—
330C2×C4—681no—
331C2×C4—681no—
332C2×C4—681no—
333C2×C4—681no—
334C2×C4—681no—
335C2×C4—1362no—
336C2×C4—1362no—
337C2×C4—1362no—
338C2×C4C4×C22⋊D1712yes—
339C2×C4#1058761511yescharacteristic
340C2×C4#1058763911yescharacteristic
341C8—44no—
342C22—21no—
343C22—21no—
344C22—22no—
345C22—22no—
346C22—22no—
347C22—81no—
348C22—81no—
349C22—1361no—
350C22—1361no—
351C22—1361no—
352C22—1361no—
353C22—1361no—
354C22—1361no—
355C22—1361no—
356C22#1176268811yescharacteristic
357C22#1176278211yescharacteristic
358C22#1176278211yescharacteristic
359C4—21no—
360C4—22no—
361C4—21no—
362C4—41no—
363C4—41no—
364C4—41no—
365C4—41no—
366C4—2722no—
367C4—682no—
368C4—1361no—
369C4—1361no—
370C2—21no—
371C2—21no—
372C2—42no—
373C2—2721no—
374C2#1452916711yescharacteristic
375C2#1452924211yescharacteristic
376C2#1452931111yescharacteristic
subgroupG/H
centerC22#11762688
commutatorC22×C4×C17C22×C4
FrattiniC23×C4C22×D17
FittingC17×#2921C2
socleC22×C17#386
prime radicalC2×#10587616C22
cosocleC23×C4×C17C23
hypercenter#2921D17
hypocenterC17#9446
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2921

Complements

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

NH ≅ G/Ntype
2. #14528861C2semidirect
4. #14526638C2semidirect
5. C17×#2921C2semidirect
9. #11762710C4semidirect
10. #11762710C4semidirect
13. #11762678C4semidirect
14. #11762674C4semidirect
154. #2921D17semidirect
249. C17#9446semidirect

Maximal subgroups

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

subgroupindex
#145332422
#145288612
#145273142
#145266382
C17×#29212
#944617