Finite Group Databasebeta

← #17019486

Subgroups — #17019486

Order 4352.

738 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14539722C211yesmaximal, characteristic, complemented
2C2×#11763004C211yesmaximal, characteristic
3#14537808C211yesmaximal, characteristic, complemented
4#14528195C214yesmaximal, complemented
5#14528127C214yesmaximal, complemented
6#14527556C211yesmaximal, characteristic, complemented
7#14527453C211yesmaximal, characteristic, complemented
8#14527441C211yesmaximal, characteristic, complemented
9C17×#3077C211yesmaximal, characteristic, complemented
10C22×#63228—21no—
11C2×#10587560C2211yescharacteristic
12#11763183C2212yescomplemented
13#11763157C2212yescomplemented
14#11763009—22no—
15#11763004—22no—
16#11763004C2214yes—
17#11762959—22no—
18D4×#3622—22no—
19D4×#3622C2214yescomplemented
20#11762862C2211yescharacteristic
21#11762862C2211yescharacteristic
22C2×C4⋊#3622C2211yescharacteristic
23C2×#10587480—21no—
24C2×C22⋊#3622—22no—
25#11762559C2214yescomplemented
26#11762558C2212yescomplemented
27#11762555C2212yescomplemented
28#11762552C2214yescomplemented
29C17×#533C2211yescharacteristic
30C2×C4×D4×C17C2211yescharacteristic
31C17×#478—22no—
32C17×#476C2212yescomplemented
33C17×#474C2212yescomplemented
34C17×#428—21no—
35C17×#423C2211yescharacteristic
36C17×#418—21no—
37C22×D4×C17D411yescharacteristic
38C23×C4×C17—21no—
39C22×D68—21no—
40C22×D68—21no—
41C22×D68—21no—
42C2×M16×C17—21no—
43C17×C4⋊Q8C2311yescharacteristic
44C17×C4⋊Q8C2311yescharacteristic
45C17×#125—21no—
46C17×C22⋊Q8—21no—
47C17×C22⋊Q8—21no—
48C4×Q8×C17—21no—
49C4×Q8×C17—21no—
50C4×D4×C17—21no—
51C4×D4×C17—21no—
52C4×D4×C17D412yes—
53C4×D4×C17C2312yes—
54C17×#117—21no—
55C2×C4⋊C4×C17D411yescharacteristic
56C2×C22⋊C4×C17—21no—
57C2×C42×C17C2311yescharacteristic
58C2×#63230—24no—
59C2×#63228—24no—
60C2×#63228—44no—
61C22×#3622—24no—
62C22×#3622—21no—
63#10587560—21no—
64#10587560—21no—
65#10587560—21no—
66#10587560C2312yes—
67C17×#107—22no—
68C17×#106—22no—
69C17×C4⋊C8—21no—
70C17×C4⋊C8C2312yes—
71C17×#102—22no—
72C17×#102—22no—
73C22⋊#3622—24no—
74C22⋊#3622—21no—
75C22⋊#3622D412yes—
76#10587480—24no—
77#10587480—24no—
78C4⋊#3622—21no—
79C4⋊#3622D412yes—
80C4⋊#3622C2312yes—
81C4⋊#3622C2312yes—
82C4×#3622—21no—
83C4×#3622C2312yes—
84C24×C17—21no—
85C23×D17—41no—
86C2×Q8×C17—21no—
87C2×Q8×C17—21no—
88C2×D4×C17—21no—
89C2×D4×C17—22no—
90C2×D4×C17SD1614yescomplemented
91C2×D4×C17C2×D412yes—
92C22×C4×C17—21no—
93C22×C4×C17—22no—
94C22×C4×C17—22no—
95C22×C4×C17—82no—
96C22×C4×C17C2×D411yescharacteristic
97C22×C4×C17C2×D411yescharacteristic
98C22×C4×C17#4111yescharacteristic
99C2×D68—22no—
100C2×D68—22no—
101C2×D68—22no—
102C2×D68—42no—
103C2×D68—42no—
104C2×D68—41no—
105C2×D68—41no—
106C2×D68—41no—
107C2×D68—82no—
108C2×D68—82no—
109M16×C17—42no—
110C2×C8×C17—22no—
111C4⋊C4×C17—21no—
112C4⋊C4×C17—21no—
113C4⋊C4×C17—21no—
114C4⋊C4×C17—21no—
115C4⋊C4×C17—21no—
116C4⋊C4×C17—41no—
117C4⋊C4×C17—41no—
118C4⋊C4×C17C2×D412yes—
119C22⋊C4×C17—22no—
120C22⋊C4×C17—22no—
121C22⋊C4×C17—41no—
122C42×C17—41no—
123C42×C17C2×D411yescharacteristic
124C42×C17C2×D411yescharacteristic
125C42×C17#4111yescharacteristic
126C42×C17C2411yescharacteristic
127#63230—84no—
128#63228—48no—
129C2×#3622—22no—
130C2×#3622—22no—
131C2×#3622—42no—
132C2×#3622—84no—
133C2×#3622C2×D414yes—
134#19384—171nomaximal, Sylow
135C23×C17—22no—
136C23×C17—24no—
137C23×C17—81no—
138C23×C17—81no—
139C23×C17#12111yescharacteristic
140C22×D17—42no—
141C22×D17—82no—
142C22×D17—82no—
143C22×D17—82no—
144Q8×C17—42no—
145Q8×C17—81no—
146D4×C17—22no—
147D4×C17—24no—
148D4×C17C2×SD1614yes—
149C2×C4×C17—22no—
150C2×C4×C17—22no—
151C2×C4×C17—41no—
152C2×C4×C17—44no—
153C2×C4×C17—41no—
154C2×C4×C17—41no—
155C2×C4×C17—82no—
156C2×C4×C17—81no—
157C2×C4×C17#12111yescharacteristic
158C2×C4×C17#12111yescharacteristic
159C2×C4×C17#12111yescharacteristic
160C2×C4×C17C2×SD1612yes—
161C2×C4×C17#13512yes—
162C2×C4×C17C22×D411yescharacteristic
163C2×C4×C17C22×D411yescharacteristic
164C2×C4×C17C2×#4111yescharacteristic
165D68—42no—
166D68—42no—
167D68—44no—
168D68—82no—
169D68—82no—
170D68—82no—
171D68—82no—
172C8×C17—42no—
173#3622—24no—
174#3622—42no—
175#3622—42no—
176#3208—171no—
177#3077D1711yescharacteristic, complemented
178C2×#459—171no—
179#2972—171no—
180#1677—174no—
181#1665—174no—
182#1570—171no—
183#1545—171no—
184#1542—171no—
185C22×C17—21no—
186C22×C17—21no—
187C22×C17—24no—
188C22×C17—42no—
189C22×C17—42no—
190C22×C17—84no—
191C22×C17#46512yes—
192C22×C17C2×#12111yescharacteristic
193C2×D17—82no—
194C2×D17—82no—
195C2×D17—82no—
196C2×D17—164no—
197C4×C17—21no—
198C4×C17—21no—
199C4×C17—21no—
200C4×C17—42no—
201C4×C17—81no—
202C4×C17—81no—
203C4×C17#46512yes—
204C4×C17C22×SD1611yescharacteristic
205C4×C17C2×#13511yescharacteristic
206C4×C17#57612yes—
207C22×D8—341no—
208#533C2×D1711yescharacteristic
209C2×#127—171no—
210C2×C4×D4C2×D1711yescharacteristic
211#492—172no—
212#486—172no—
213#478—22no—
214#476C2×D1712yescomplemented
215#474C2×D1712yescomplemented
216#460—342no—
217#459—174no—
218#459—342no—
219#450—342no—
220C8×D4—174no—
221C8×D4—342no—
222#431—171no—
223#431—171no—
224#428—21no—
225#423C2×D1711yescharacteristic
226C2×C4⋊C8—171no—
227#418—21no—
228C2×#102—341no—
229C2×C22⋊C8—342no—
230#334—174no—
231#333—172no—
232#332—172no—
233#331—174no—
234C2×C17—22no—
235C2×C17—44no—
236C2×C17#304111yescharacteristic
237C2×C17C2×#46511yescharacteristic
238C2×C17#306811yescharacteristic
239D17—162no—
240C22×D4—341no—
241C22×D4—341no—
242C22×D4—341no—
243C22×D4C22⋊D1711yescharacteristic
244C23×C4—21no—
245C2×SD16—344no—
246C2×D8—344no—
247C2×D8—684no—
248C2×M16—21no—
249C22×C8—344no—
250C22×C8—341no—
251C4⋊Q8C22×D1711yescharacteristic
252C4⋊Q8C22×D1711yescharacteristic
253#127—172no—
254#127—341no—
255#127—341no—
256#127—341no—
257#125—21no—
258C22⋊Q8—21no—
259C22⋊Q8—21no—
260C4×Q8—21no—
261C4×Q8—21no—
262C4×D4—21no—
263C4×D4—21no—
264C4×D4C22⋊D1712yes—
265C4×D4C22×D1712yes—
266#117—21no—
267C2×C4⋊C4C22⋊D1711yescharacteristic
268C2×C22⋊C4—21no—
269C2×C42C22×D1711yescharacteristic
270#107—22no—
271#106—22no—
272C4⋊C8—21no—
273C4⋊C8—172no—
274C4⋊C8—172no—
275C4⋊C8—172no—
276C4⋊C8—341no—
277C4⋊C8C22×D1712yes—
278#102—22no—
279#102—22no—
280#102—344no—
281#102—344no—
282C22⋊C8—172no—
283C22⋊C8—344no—
284C22⋊C8—341no—
285C4×C8—172no—
286C4×C8—341no—
287C17#1938411yescharacteristic, complemented, Sylow
288C24—21no—
289C24—681no—
290C2×Q8—21no—
291C2×Q8—21no—
292C2×D4—21no—
293C2×D4—22no—
294C2×D4—342no—
295C2×D4—342no—
296C2×D4—342no—
297C2×D4—681no—
298C2×D4—681no—
299C2×D4—681no—
300C2×D4—682no—
301C2×D4—682no—
302C2×D4—1362no—
303C2×D4—1362no—
304C2×D4#6323014yescomplemented
305C2×D4C2×C22⋊D1712yes—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—82no—
310C22×C4D4×D1711yescharacteristic
311C22×C4#6325411yescharacteristic
312C22×C4C2×C22⋊D1711yescharacteristic
313SD16—1364no—
314D8—688no—
315M16—42no—
316C2×C8—22no—
317C2×C8—174no—
318C2×C8—342no—
319C2×C8—342no—
320C2×C8—682no—
321C2×C8—1364no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4C2×C22⋊D1712yes—
330C22⋊C4—22no—
331C22⋊C4—22no—
332C22⋊C4—41no—
333C42—41no—
334C42D4×D1711yescharacteristic
335C42#6325411yescharacteristic
336C42C2×C22⋊D1711yescharacteristic
337C42C23×D1711yescharacteristic
338C23—22no—
339C23—24no—
340C23—81no—
341C23—81no—
342C23—682no—
343C23—1362no—
344C23—1362no—
345C23—1362no—
346C23#1058762511yescharacteristic
347Q8—42no—
348Q8—81no—
349D4—24no—
350D4—22no—
351D4—682no—
352D4—682no—
353D4—684no—
354D4—1362no—
355D4—1362no—
356D4—1362no—
357D4—1362no—
358D4C2×#6323014yes—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—41no—
362C2×C4—44no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—82no—
366C2×C4—81no—
367C2×C4#1058762511yescharacteristic
368C2×C4#1058762511yescharacteristic
369C2×C4#1058762511yescharacteristic
370C2×C4C2×#6323012yes—
371C2×C4#1058763612yes—
372C2×C4C2×D4×D1711yescharacteristic
373C2×C4C2×#6325411yescharacteristic
374C2×C4C22×C22⋊D1711yescharacteristic
375C8—42no—
376C8—344no—
377C8—682no—
378C8—682no—
379C22—21no—
380C22—21no—
381C22—24no—
382C22—42no—
383C22—42no—
384C22—84no—
385C22—2724no—
386C22—1362no—
387C22—1362no—
388C22—1362no—
389C22#1176304012yes—
390C22C2×#1058762511yescharacteristic
391C4—21no—
392C4—21no—
393C4—21no—
394C4—42no—
395C4—81no—
396C4—81no—
397C4#1176304012yes—
398C4C22×#6323011yescharacteristic
399C4C2×#1058763611yescharacteristic
400C4#1176361312yes—
401C2—22no—
402C2—44no—
403C2—2722no—
404C2#1453828911yescharacteristic
405C2C2×#1176304011yescharacteristic
406C2#1453851411yescharacteristic
subgroupG/H
centerC22C2×#10587625
commutator, cosocleC42×C17C24
FrattiniC42C23×D17
FittingC17×#3077C2
socleC22×C17C2×#121
prime radicalC2×#11763004C2
hypercenter#3077D17
hypocenterC17#19384
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3077

Complements

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

NH ≅ G/Ntype
1. #14539722C2semidirect
3. #14537808C2semidirect
4. #14528195C2semidirect
5. #14528127C2semidirect
6. #14527556C2semidirect
7. #14527453C2semidirect
8. #14527441C2semidirect
9. C17×#3077C2semidirect
12. #11763183C22semidirect
13. #11763157C22semidirect
19. D4×#3622C22semidirect
25. #11762559C22semidirect
26. #11762558C22semidirect
27. #11762555C22semidirect
28. #11762552C22semidirect
32. C17×#476C22semidirect
33. C17×#474C22semidirect
90. C2×D4×C17SD16semidirect
177. #3077D17semidirect
214. #476C2×D17semidirect
215. #474C2×D17semidirect
287. C17#19384semidirect
304. C2×D4#63230semidirect

Maximal subgroups

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

subgroupindex
#145397222
C2×#117630042
#145378082
#145281952
#145281272
#145275562
#145274532
#145274412
C17×#30772
#1938417