Finite Group Databasebeta

← #17019602

Subgroups — #17019602

Order 4352.

686 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14539722C211yesmaximal, characteristic
2C2×#11763009C211yesmaximal, characteristic
3#14537865C211yesmaximal, characteristic, complemented
4#14528202C214yesmaximal, complemented
5#14528134C214yesmaximal, complemented
6#14527556C211yesmaximal, characteristic, complemented
7#14527453C211yesmaximal, characteristic
8#14527441C211yesmaximal, characteristic, complemented
9C17×#3078C211yesmaximal, characteristic, complemented
10C22×#63230—21no—
11C2×#10587560C2211yescharacteristic
12#11763183C2212yes—
13#11763157C2212yes—
14#11763009—22no—
15#11763009—22no—
16#11763009C2214yes—
17#11762959—22no—
18Q8×#3622—22no—
19Q8×#3622C2214yescomplemented
20#11762862—22no—
21#11762862C2211yescharacteristic
22#11762862C2211yescharacteristic
23C2×C4⋊#3622C2211yescharacteristic
24C2×#10587480—21no—
25#11762559C2214yescomplemented
26#11762558C2212yescomplemented
27#11762555C2212yescomplemented
28#11762552C2214yes—
29C17×#533C2211yescharacteristic
30C2×C4×Q8×C17C2211yescharacteristic
31C17×#479—22no—
32C17×#477C2212yescomplemented
33C17×#475C2212yescomplemented
34C17×#428—21no—
35C17×#423C2211yescharacteristic
36C17×#419—21no—
37C22×Q8×C17D411yescharacteristic
38C22×D68—21no—
39C22×D68—21no—
40C22×D68—21no—
41C2×M16×C17—21no—
42C17×C4⋊Q8C2311yescharacteristic
43C17×C4⋊Q8C2311yescharacteristic
44C17×#125—21no—
45C17×C22⋊Q8—21no—
46C17×C22⋊Q8—21no—
47C4×Q8×C17—21no—
48C4×Q8×C17—21no—
49C4×Q8×C17—21no—
50C4×Q8×C17—21no—
51C4×Q8×C17D412yes—
52C4×Q8×C17C2312yes—
53C17×#117—21no—
54C2×C4⋊C4×C17—21no—
55C2×C4⋊C4×C17D411yescharacteristic
56C2×C42×C17—21no—
57C2×C42×C17C2311yescharacteristic
58C2×#63230—24no—
59C2×#63230—24no—
60C2×#63230—44no—
61C22×#3622—21no—
62#10587560—21no—
63#10587560—21no—
64#10587560—21no—
65#10587560C2312yes—
66C17×#107—22no—
67C17×#106—22no—
68C17×C4⋊C8—21no—
69C17×C4⋊C8C2312yes—
70C17×#103—22no—
71C17×#103—22no—
72C22⋊#3622—21no—
73C22⋊#3622D412yes—
74#10587480—24no—
75#10587480—24no—
76C4⋊#3622—24no—
77C4⋊#3622—21no—
78C4⋊#3622D412yes—
79C4⋊#3622C2312yes—
80C4⋊#3622C2312yes—
81C4×#3622—24no—
82C4×#3622—21no—
83C4×#3622C2312yes—
84C23×D17—41no—
85C2×Q8×C17—21no—
86C2×Q8×C17—22no—
87C2×Q8×C17—21no—
88C2×Q8×C17—21no—
89C2×Q8×C17SD1614yescomplemented
90C2×Q8×C17C2×D412yes—
91C22×C4×C17—21no—
92C22×C4×C17—21no—
93C22×C4×C17C2×D411yescharacteristic
94C22×C4×C17C2×D411yescharacteristic
95C22×C4×C17#4111yescharacteristic
96C2×D68—22no—
97C2×D68—22no—
98C2×D68—22no—
99C2×D68—42no—
100C2×D68—42no—
101C2×D68—41no—
102C2×D68—41no—
103C2×D68—41no—
104C2×D68—82no—
105C2×D68—82no—
106M16×C17—42no—
107C2×C8×C17—22no—
108C4⋊C4×C17—22no—
109C4⋊C4×C17—21no—
110C4⋊C4×C17—22no—
111C4⋊C4×C17—21no—
112C4⋊C4×C17—21no—
113C4⋊C4×C17—21no—
114C4⋊C4×C17—21no—
115C4⋊C4×C17—41no—
116C4⋊C4×C17—41no—
117C4⋊C4×C17C2×D412yes—
118C22⋊C4×C17—41no—
119C42×C17—22no—
120C42×C17—22no—
121C42×C17—41no—
122C42×C17C2×D411yescharacteristic
123C42×C17C2×D411yescharacteristic
124C42×C17#4111yescharacteristic
125C42×C17C2411yescharacteristic
126#63230—48no—
127#63230—84no—
128C2×#3622—22no—
129C2×#3622—22no—
130C2×#3622—42no—
131C2×#3622C2×D414yes—
132#19396—171nomaximal, Sylow
133C23×C17#12111yescharacteristic
134C22×D17—42no—
135C22×D17—82no—
136C22×D17—82no—
137C22×D17—82no—
138Q8×C17—22no—
139Q8×C17—24no—
140Q8×C17—42no—
141Q8×C17—81no—
142Q8×C17C2×SD1614yes—
143C2×C4×C17—22no—
144C2×C4×C17—24no—
145C2×C4×C17—22no—
146C2×C4×C17—22no—
147C2×C4×C17—41no—
148C2×C4×C17—41no—
149C2×C4×C17—41no—
150C2×C4×C17—81no—
151C2×C4×C17#12111yescharacteristic
152C2×C4×C17#12111yescharacteristic
153C2×C4×C17#12111yescharacteristic
154C2×C4×C17C2×SD1612yes—
155C2×C4×C17#13512yes—
156C2×C4×C17C22×D411yescharacteristic
157C2×C4×C17C22×D411yescharacteristic
158C2×C4×C17C2×#4111yescharacteristic
159D68—42no—
160D68—42no—
161D68—44no—
162D68—82no—
163D68—82no—
164D68—82no—
165D68—82no—
166C8×C17—42no—
167#3622—24no—
168#3622—42no—
169#3622—42no—
170#3208—171no—
171#3078D1711yescharacteristic, complemented
172C2×#460—171no—
173#2982—171no—
174#1678—174no—
175#1666—174no—
176#1570—171no—
177#1545—171no—
178#1542—171no—
179C22×C17—21no—
180C22×C17—21no—
181C22×C17#46512yes—
182C22×C17C2×#12111yescharacteristic
183C2×D17—82no—
184C2×D17—82no—
185C2×D17—82no—
186C2×D17—164no—
187C4×C17—21no—
188C4×C17—24no—
189C4×C17—21no—
190C4×C17—21no—
191C4×C17—42no—
192C4×C17—81no—
193C4×C17—81no—
194C4×C17#46512yes—
195C4×C17C22×SD1611yescharacteristic
196C4×C17C2×#13511yescharacteristic
197C4×C17#57712yes—
198C22×SD16—341no—
199#533C2×D1711yescharacteristic
200C2×#127—171no—
201C2×C4×Q8C2×D1711yescharacteristic
202#492—172no—
203#486—172no—
204#479—22no—
205#477C2×D1712yescomplemented
206#475C2×D1712yescomplemented
207#460—174no—
208#460—342no—
209#460—342no—
210#450—342no—
211C8×Q8—174no—
212C8×Q8—342no—
213#431—171no—
214#431—171no—
215#431—342no—
216#428—21no—
217#423C2×D1711yescharacteristic
218C2×C4⋊C8—171no—
219#419—21no—
220C2×#102—341no—
221#334—174no—
222#333—172no—
223#332—172no—
224#331—174no—
225C2×C17—22no—
226C2×C17#304311yescharacteristic
227C2×C17C2×#46511yescharacteristic
228C2×C17#306811yescharacteristic
229D17—162no—
230C22×Q8C22⋊D1711yescharacteristic
231C22×D4—341no—
232C22×D4—341no—
233C22×D4—341no—
234C2×SD16—344no—
235C2×SD16—344no—
236C2×SD16—684no—
237C2×M16—21no—
238C22×C8—341no—
239C4⋊Q8C22×D1711yescharacteristic
240C4⋊Q8C22×D1711yescharacteristic
241#127—172no—
242#127—341no—
243#127—341no—
244#127—341no—
245#125—21no—
246C22⋊Q8—21no—
247C22⋊Q8—21no—
248C4×Q8—21no—
249C4×Q8—21no—
250C4×Q8—21no—
251C4×Q8—21no—
252C4×Q8C22⋊D1712yes—
253C4×Q8C22×D1712yes—
254#117—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4C22⋊D1711yescharacteristic
257C2×C42—21no—
258C2×C42C22×D1711yescharacteristic
259#107—22no—
260#106—22no—
261C4⋊C8—21no—
262C4⋊C8—172no—
263C4⋊C8—172no—
264C4⋊C8—172no—
265C4⋊C8—344no—
266C4⋊C8—341no—
267C4⋊C8C22×D1712yes—
268#103—22no—
269#103—22no—
270#102—344no—
271#102—344no—
272C22⋊C8—172no—
273C22⋊C8—341no—
274C4×C8—172no—
275C4×C8—344no—
276C4×C8—341no—
277C17#1939611yescharacteristic, complemented, Sylow
278C24—681no—
279C2×Q8—21no—
280C2×Q8—22no—
281C2×Q8—21no—
282C2×Q8—21no—
283C2×Q8#6323014yescomplemented
284C2×Q8C2×C22⋊D1712yes—
285C2×D4—342no—
286C2×D4—342no—
287C2×D4—342no—
288C2×D4—681no—
289C2×D4—681no—
290C2×D4—681no—
291C2×D4—682no—
292C2×D4—682no—
293C2×D4—1362no—
294C2×D4—1362no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4D4×D1711yescharacteristic
298C22×C4#6325411yescharacteristic
299C22×C4C2×C22⋊D1711yescharacteristic
300SD16—688no—
301SD16—1364no—
302M16—42no—
303C2×C8—22no—
304C2×C8—174no—
305C2×C8—342no—
306C2×C8—342no—
307C2×C8—682no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4C2×C22⋊D1712yes—
318C22⋊C4—41no—
319C42—22no—
320C42—22no—
321C42—41no—
322C42D4×D1711yescharacteristic
323C42#6325411yescharacteristic
324C42C2×C22⋊D1711yescharacteristic
325C42C23×D1711yescharacteristic
326C23—682no—
327C23—1362no—
328C23—1362no—
329C23—1362no—
330C23#1058762511yescharacteristic
331Q8—24no—
332Q8—22no—
333Q8—42no—
334Q8—81no—
335Q8C2×#6323014yes—
336D4—682no—
337D4—682no—
338D4—684no—
339D4—1362no—
340D4—1362no—
341D4—1362no—
342D4—1362no—
343C2×C4—24no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—81no—
351C2×C4#1058762511yescharacteristic
352C2×C4#1058762511yescharacteristic
353C2×C4#1058762511yescharacteristic
354C2×C4C2×#6323012yes—
355C2×C4#1058763612yes—
356C2×C4C2×D4×D1711yescharacteristic
357C2×C4C2×#6325411yescharacteristic
358C2×C4C22×C22⋊D1711yescharacteristic
359C8—42no—
360C8—344no—
361C8—682no—
362C8—682no—
363C22—21no—
364C22—21no—
365C22—2724no—
366C22—1362no—
367C22—1362no—
368C22—1362no—
369C22#1176304012yes—
370C22C2×#1058762511yescharacteristic
371C4—21no—
372C4—21no—
373C4—21no—
374C4—24no—
375C4—42no—
376C4—81no—
377C4—81no—
378C4#1176304012yes—
379C4C22×#6323011yescharacteristic
380C4C2×#1058763611yescharacteristic
381C4#1176361812yes—
382C2—22no—
383C2—2722no—
384C2#1453830711yescharacteristic
385C2C2×#1176304011yescharacteristic
386C2#1453851411yescharacteristic
subgroupG/H
centerC22C2×#10587625
commutator, cosocleC42×C17C24
FrattiniC42C23×D17
FittingC17×#3078C2
socleC22×C17C2×#121
prime radicalC2×#10587560C22
hypercenter#3078D17
hypocenterC17#19396
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3078

Complements

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

NH ≅ G/Ntype
3. #14537865C2semidirect
4. #14528202C2semidirect
5. #14528134C2semidirect
6. #14527556C2semidirect
8. #14527441C2semidirect
9. C17×#3078C2semidirect
19. Q8×#3622C22semidirect
25. #11762559C22semidirect
26. #11762558C22semidirect
27. #11762555C22semidirect
32. C17×#477C22semidirect
33. C17×#475C22semidirect
89. C2×Q8×C17SD16semidirect
171. #3078D17semidirect
205. #477C2×D17semidirect
206. #475C2×D17semidirect
277. C17#19396semidirect
283. C2×Q8#63230semidirect

Maximal subgroups

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

subgroupindex
#145397222
C2×#117630092
#145378652
#145282022
#145281342
#145275562
#145274532
#145274412
C17×#30782
#1939617