Finite Group Databasebeta

← #16933727

Subgroups — #16933727

Order 4352.

440 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14528470C211yesmaximal, characteristic, complemented
2#14527547C211yesmaximal, characteristic, complemented
3#14527399C211yesmaximal, characteristic, complemented
4#14527395C211yesmaximal, characteristic
5#14527296C211yesmaximal, characteristic, complemented
6#14527292C211yesmaximal, characteristic
7C17×#3167C211yesmaximal, characteristic
8#11762866C2211yescharacteristic
9#11762866C2211yescharacteristic
10C2×#10587474C2211yescharacteristic
11C2×#10587486—21no—
12C2×C8×Dic17—21no—
13#11762655C2211yescharacteristic
14#11762562—21no—
15#11762560—21no—
16#11762551—22no—
17#11762549—22no—
18#11762546—22no—
19#11762543—22no—
20C17×#533C2211yescharacteristic
21C17×#532C2211yescharacteristic
22C17×#498C411yescharacteristic, complemented
23C17×#492C411yescharacteristic, complemented
24C17×#487C411yescharacteristic, complemented
25C17×#486C411yescharacteristic, complemented
26C17×#465—21no—
27C17×#465—21no—
28C4×SD16×C17—21no—
29C4×SD16×C17—21no—
30C2×C4×C8×C17C2211yescharacteristic
31C2×#41×C17—21no—
32C2×SD16×C17—21no—
33C2×SD16×C17—21no—
34C22×C8×C17D411yescharacteristic
35C22×C8×C17D411yescharacteristic
36C17×C4⋊Q8C2×C411yescharacteristic
37C17×C4⋊Q8C2×C411yescharacteristic
38C17×C4⋊Q8C2×C411yescharacteristic
39C17×#127C2×C411yescharacteristic
40C17×#125—21no—
41C17×#124—21no—
42C17×C22⋊Q8—21no—
43C17×C22⋊Q8—21no—
44C17×#121—21no—
45C17×#121—21no—
46C4×Q8×C17—21no—
47C4×Q8×C17—21no—
48C4×D4×C17—21no—
49C4×D4×C17—21no—
50C17×#117—21no—
51C2×C42×C17C2311yescharacteristic
52C2×C4×Dic17—21no—
53C22×#3622—21no—
54C17×#106—21no—
55C17×#106—21no—
56C17×#103—21no—
57C17×#103—21no—
58C17×#102—21no—
59C17×#102—21no—
60C4×C8×C17C2×C411yescharacteristic
61C4×C8×C17C2×C411yescharacteristic
62C4×C8×C17D411yescharacteristic
63C4×C8×C17D411yescharacteristic
64#10587502—21no—
65#10587502D412yes—
66C22⋊#3622—21no—
67C22⋊#3622—21no—
68#10587486—24no—
69C8×Dic17—24no—
70C4⋊#3622—21no—
71C4⋊#3622—21no—
72#10587474—21no—
73#10587474—21no—
74#10587474—21no—
75#10587474D412yes—
76#41×C17—82no—
77C2×Q8×C17—21no—
78C2×Q8×C17—21no—
79C2×Q8×C17—21no—
80C2×D4×C17—21no—
81C2×D4×C17—21no—
82C2×D4×C17—21no—
83C2×D4×C17—41no—
84C22×C4×C17—41no—
85C22×C4×C17C2×D411yescharacteristic
86C22×C4×C17C2×D411yescharacteristic
87C22×C4×C17#4111yescharacteristic
88C22×Dic17—22no—
89SD16×C17—42no—
90SD16×C17—42no—
91C2×C8×C17—22no—
92C2×C8×C17—22no—
93C2×C8×C17C22⋊C411yescharacteristic
94C2×C8×C17C22⋊C411yescharacteristic
95C2×C8×C17C22⋊C411yescharacteristic
96C2×C8×C17C22⋊C411yescharacteristic
97C4⋊C4×C17—21no—
98C4⋊C4×C17—21no—
99C4⋊C4×C17—21no—
100C4⋊C4×C17—21no—
101C4⋊C4×C17—21no—
102C4⋊C4×C17—21no—
103C4⋊C4×C17—41no—
104C4⋊C4×C17—41no—
105C22⋊C4×C17—41no—
106C22⋊C4×C17—41no—
107C22⋊C4×C17—41no—
108C42×C17—41no—
109C42×C17C22×C411yescharacteristic
110C42×C17C2×D411yescharacteristic
111C42×C17C2×D411yescharacteristic
112C42×C17#4111yescharacteristic
113C4×Dic17—22no—
114C4×Dic17—22no—
115C2×#3622—22no—
116C2×#3622—22no—
117C2×#3622—22no—
118C2×#3622—41no—
119C2×#3622—41no—
120#9761—171nomaximal, Sylow
121C23×C17—41no—
122C23×C17#12111yescharacteristic
123Q8×C17—41no—
124Q8×C17—41no—
125Q8×C17—81no—
126Q8×C17—81no—
127D4×C17—41no—
128D4×C17—41no—
129D4×C17—81no—
130D4×C17—81no—
131D4×C17—82no—
132C2×C4×C17—22no—
133C2×C4×C17—21no—
134C2×C4×C17—21no—
135C2×C4×C17—41no—
136C2×C4×C17—41no—
137C2×C4×C17—41no—
138C2×C4×C17—41no—
139C2×C4×C17—82no—
140C2×C4×C17C2×C22⋊C411yescharacteristic
141C2×C4×C17C4×D411yescharacteristic
142C2×C4×C17C4×D411yescharacteristic
143C2×C4×C17#12111yescharacteristic
144C2×C4×C17#12411yescharacteristic
145C2×C4×C17#12711yescharacteristic
146C2×Dic17—24no—
147C2×Dic17—42no—
148C2×Dic17—42no—
149C8×C17—21no—
150C8×C17—21no—
151C8×C17#10412yes—
152C8×C17#10412yes—
153#3622—24no—
154#3622—81no—
155#3622—81no—
156#3167D1711yescharacteristic
157#1726—171no—
158#1568—171no—
159#1531—171no—
160#1530—171no—
161#1506—171no—
162#1505—171no—
163C22×C17—21no—
164C22×C17—21no—
165C22×C17—81no—
166C22×C17—81no—
167C22×C17—81no—
168C22×C17#39011yescharacteristic
169C22×C17#47112yes—
170C4×C17—21no—
171C4×C17—21no—
172C4×C17—21no—
173C4×C17—21no—
174C4×C17—21no—
175C4×C17—81no—
176C4×C17—81no—
177C4×C17—81no—
178C4×C17C2×#10411yescharacteristic
179C4×C17C2×#10411yescharacteristic
180Dic17—44no—
181#533C2×D1711yescharacteristic
182#532C2×D1711yescharacteristic
183#498Dic1711yescharacteristic, complemented
184#492Dic1711yescharacteristic, complemented
185#487Dic1711yescharacteristic, complemented
186#486Dic1711yescharacteristic, complemented
187#465—21no—
188#465—21no—
189C4×SD16—21no—
190C4×SD16—21no—
191#432—171no—
192#432—171no—
193C2×#97—171no—
194C2×#97—341no—
195C2×C4×C8—341no—
196C2×C4×C8C2×D1711yescharacteristic
197#376—171no—
198#335—341no—
199#334—341no—
200#330—342no—
201#329—342no—
202#328—342no—
203#327—342no—
204C2×C17—22no—
205C2×C17—161no—
206C2×C17#196211yescharacteristic
207C2×C17#197611yescharacteristic
208C2×C17#197711yescharacteristic
209C2×#41—21no—
210C2×SD16—21no—
211C2×SD16—21no—
212C22×C8—341no—
213C22×C8C22⋊D1711yescharacteristic
214C22×C8C22⋊D1711yescharacteristic
215C4⋊Q8C2×Dic1711yescharacteristic
216C4⋊Q8C2×Dic1711yescharacteristic
217C4⋊Q8C2×Dic1711yescharacteristic
218#127C2×Dic1711yescharacteristic
219#125—21no—
220#124—21no—
221C22⋊Q8—21no—
222C22⋊Q8—21no—
223#121—21no—
224#121—21no—
225C4×Q8—21no—
226C4×Q8—21no—
227C4×D4—21no—
228C4×D4—21no—
229#117—21no—
230C2×C42—341no—
231C2×C42C22×D1711yescharacteristic
232#106—21no—
233#106—21no—
234C4⋊C8—341no—
235C4⋊C8—341no—
236#103—21no—
237#103—21no—
238#102—21no—
239#102—21no—
240C22⋊C8—341no—
241C22⋊C8—341no—
242#97—172no—
243#97—344no—
244#97—341no—
245#97—341no—
246#97—341no—
247C4×C8—344no—
248C4×C8C2×Dic1711yescharacteristic
249C4×C8C2×Dic1711yescharacteristic
250C4×C8C22⋊D1711yescharacteristic
251C4×C8C22⋊D1711yescharacteristic
252#95—172no—
253#95—341no—
254C17#976111yescharacteristic, complemented, Sylow
255#41—82no—
256C2×Q8—21no—
257C2×Q8—21no—
258C2×Q8—21no—
259C2×D4—21no—
260C2×D4—21no—
261C2×D4—21no—
262C2×D4—41no—
263C22×C4—41no—
264C22×C4—342no—
265C22×C4D4×D1711yescharacteristic
266C22×C4#6325411yescharacteristic
267C22×C4C2×C22⋊D1711yescharacteristic
268SD16—42no—
269SD16—42no—
270C2×C8—22no—
271C2×C8—22no—
272C2×C8—342no—
273C2×C8—342no—
274C2×C8—342no—
275C2×C8—681no—
276C2×C8—681no—
277C2×C8C22⋊Dic1711yescharacteristic
278C2×C8C22⋊Dic1711yescharacteristic
279C2×C8C22⋊Dic1711yescharacteristic
280C2×C8C22⋊Dic1711yescharacteristic
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—21no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—41no—
291C22⋊C4—41no—
292C42—41no—
293C42—342no—
294C42—342no—
295C42D4×D1711yescharacteristic
296C42#6325411yescharacteristic
297C42C22×Dic1711yescharacteristic
298C42C2×C22⋊D1711yescharacteristic
299C23—41no—
300C23#1058762411yescharacteristic
301Q8—41no—
302Q8—41no—
303Q8—81no—
304Q8—81no—
305D4—41no—
306D4—41no—
307D4—81no—
308D4—81no—
309D4—82no—
310C2×C4—22no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—82no—
318C2×C4—344no—
319C2×C4—682no—
320C2×C4—682no—
321C2×C4D4×Dic1711yescharacteristic
322C2×C4D4×Dic1711yescharacteristic
323C2×C4#1058762211yescharacteristic
324C2×C4#1058762411yescharacteristic
325C2×C4#1058762611yescharacteristic
326C2×C4C2×C22⋊Dic1711yescharacteristic
327C8—21no—
328C8—21no—
329C8—344no—
330C8—1361no—
331C8—1361no—
332C8#1058750812yes—
333C8#1058750812yes—
334C22—21no—
335C22—21no—
336C22—81no—
337C22—81no—
338C22—81no—
339C22#1176271211yescharacteristic
340C22#1176306912yes—
341C4—21no—
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—81no—
347C4—81no—
348C4—81no—
349C4—684no—
350C4C2×#1058750811yescharacteristic
351C4C2×#1058750811yescharacteristic
352C2—22no—
353C2—161no—
354C2#1452961311yescharacteristic
355C2#1452968311yescharacteristic
356C2#1452968911yescharacteristic
subgroupG/H
centerC22#11762712
commutatorC42×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#3167C2
socleC22×C17#390
prime radicalC17×#532C22
cosocleC2×C42×C17C23
hypercenter#3167D17
hypocenterC17#9761
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3167

Complements

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

NH ≅ G/Ntype
1. #14528470C2semidirect
2. #14527547C2semidirect
3. #14527399C2semidirect
5. #14527296C2semidirect
22. C17×#498C4semidirect
23. C17×#492C4semidirect
24. C17×#487C4semidirect
25. C17×#486C4semidirect
183. #498Dic17semidirect
184. #492Dic17semidirect
185. #487Dic17semidirect
186. #486Dic17semidirect
254. C17#9761semidirect

Maximal subgroups

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

subgroupindex
#145284702
#145275472
#145273992
#145273952
#145272962
#145272922
C17×#31672
#976117