Finite Group Databasebeta

← #17054147

Subgroups — #17054147

Order 4352.

850 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
1C2×D8×Dic17C211yesmaximal, characteristic
2#14531756C212yesmaximal, complemented
3#14529661C211yesmaximal, characteristic
4#14529407C212yesmaximal, complemented
5#14529401C212yesmaximal, complemented
6#14528945C211yesmaximal, characteristic, complemented
7#14528792C212yesmaximal, complemented
8#14528684C211yesmaximal, characteristic, complemented
9#14528681C211yesmaximal, characteristic, complemented
10#14528652C211yesmaximal, characteristic, complemented
11C17×#1972C211yesmaximal, characteristic
12C2×D4×Dic17C2212yes—
13D8×Dic17C418yescomplemented
14C2×#10587489C2211yescharacteristic
15C2×#10587478C2212yes—
16C2×#10587488C2211yescharacteristic
17C2×#10587485C2211yescharacteristic
18C2×#10587503C2212yes—
19C2×#10587503C2212yes—
20C2×#10587486C2211yescharacteristic
21C2×C8×Dic17C2211yescharacteristic
22#11762712C2212yes—
23#11762703—22no—
24#11762697—22no—
25#11762674C2212yes—
26#11762670—22no—
27C4⋊C4×Dic17C2212yes—
28C4×#63225C2211yescharacteristic
29C4×C4⋊Dic17C2212yes—
30C22⋊C4×Dic17—22no—
31#11762572C2212yes—
32#11762571C2212yes—
33#11762564C2211yescharacteristic
34#11762563C2211yescharacteristic
35C22×D8×C17C2211yescharacteristic
36C2×C17×#106C2211yescharacteristic
37C2×C17×#102C2212yes—
38C2×C17×#97C2211yescharacteristic
39C17×#390C2212yes—
40C22×D4×C17C2312yes—
41C23×Dic17—22no—
42C2×D8×C17—22no—
43C2×D8×C17—22no—
44C2×D8×C17C2×C414yes—
45C22×C8×C17C2311yescharacteristic
46C22×C8×C17C2311yescharacteristic
47C2×C4⋊C4×C17C2312yes—
48C2×C22⋊C4×C17—22no—
49C2×C22⋊C4×C17—22no—
50C2×C42×C17C2311yescharacteristic
51C2×C22⋊Dic17—22no—
52C2×C22⋊Dic17—22no—
53C2×C22⋊Dic17—22no—
54D4×Dic17—24no—
55D4×Dic17C2×C418yes—
56C2×C4⋊Dic17C2312yes—
57C2×C4⋊Dic17C2312yes—
58C2×#63225D412yes—
59C2×C4×Dic17—22no—
60C2×C4×Dic17—22no—
61C2×C4×Dic17C2311yescharacteristic
62C2×C4×Dic17C2311yescharacteristic
63C22×#3622C2311yescharacteristic
64C22×#3622C2311yescharacteristic
65C17×#106—22no—
66C17×#102—42no—
67C17×#97—22no—
68#10587503—42no—
69#10587503C2×C418yes—
70#10587502—22no—
71#10587502—22no—
72#10587502—22no—
73#10587502—22no—
74#10587502D412yes—
75#10587489C2×C414yes—
76#10587488—22no—
77#10587486—22no—
78#10587485—22no—
79C8×Dic17C2×C414yes—
80#10587478—42no—
81C24×C17—22no—
82C2×D4×C17—22no—
83C2×D4×C17—22no—
84C2×D4×C17—22no—
85C2×D4×C17—22no—
86C2×D4×C17—22no—
87C2×D4×C17C22×C414yes—
88C22×C4×C17—22no—
89C22×C4×C17C2×D412yes—
90C22×C4×C17C2411yescharacteristic
91C22×Dic17—22no—
92C22×Dic17—28no—
93C22×Dic17—22no—
94C22×Dic17—82no—
95C22×Dic17—82no—
96C22×Dic17C2×D412yes—
97C22×Dic17C2×D411yescharacteristic
98C22×Dic17C2×D411yescharacteristic
99D8×C17—28no—
100C2×C8×C17—21no—
101C2×C8×C17—21no—
102C2×C8×C17—21no—
103C2×C8×C17C22×C412yes—
104C2×C8×C17#4112yes—
105C2×C8×C17#4112yes—
106C4⋊C4×C17—22no—
107C4⋊C4×C17—22no—
108C22⋊C4×C17—44no—
109C22⋊C4×C17—82no—
110C42×C17—21no—
111C42×C17—21no—
112C22⋊Dic17—28no—
113C22⋊Dic17—44no—
114C22⋊Dic17—82no—
115C4⋊Dic17—22no—
116C4⋊Dic17—22no—
117C4⋊Dic17—22no—
118C4⋊Dic17C22×C414yes—
119#63225—42no—
120C4×Dic17—21no—
121C4×Dic17—21no—
122C4×Dic17—82no—
123C4×Dic17—82no—
124C4×Dic17C22×C412yes—
125C4×Dic17#4112yes—
126C2×#3622—21no—
127C2×#3622—21no—
128C2×#3622—21no—
129C2×#3622—21no—
130C2×#3622—21no—
131C2×#3622C22×C412yes—
132#22816—171nomaximal, Sylow
133C23×C17—24no—
134C23×C17—42no—
135C23×C17—42no—
136C23×C17—42no—
137C23×C17—42no—
138C23×C17—42no—
139C23×C17—42no—
140C23×C17C22×D411yescharacteristic
141D4×C17—24no—
142D4×C17—24no—
143D4×C17—42no—
144D4×C17—42no—
145C2×C4×C17—22no—
146C2×C4×C17—22no—
147C2×C4×C17—22no—
148C2×C4×C17—42no—
149C2×C4×C17—42no—
150C2×C4×C17—42no—
151C2×C4×C17C23×C411yescharacteristic
152C2×C4×C17C2×#4111yescharacteristic
153C2×C4×C17C2×#4111yescharacteristic
154C2×C4×C17C2×#4111yescharacteristic
155C2×C4×C17H(5,𝔽2)11yescharacteristic
156C2×C4×C172−1+411yescharacteristic
157C2×Dic17—22no—
158C2×Dic17—22no—
159C2×Dic17—24no—
160C2×Dic17—22no—
161C2×Dic17—22no—
162C2×Dic17—22no—
163C2×Dic17—42no—
164C2×Dic17—42no—
165C2×Dic17—42no—
166C2×Dic17—42no—
167C2×Dic17—42no—
168C2×Dic17—88no—
169C2×Dic17C4×D412yes—
170C2×Dic17C4×D412yes—
171C8×C17—41no—
172C8×C17#11714yes—
173#3622—22no—
174#3622—41no—
175C2×C4×D8—171no—
176#2357—172no—
177#1972—171no—
178#1972D1711yescharacteristic
179#1925—172no—
180#1925—172no—
181#1846—171no—
182#1804—172no—
183#1776—171no—
184#1776—171no—
185#1770—171no—
186C22×C17—44no—
187C22×C17—44no—
188C22×C17—44no—
189C22×C17—82no—
190C22×C17—82no—
191C22×C17—82no—
192C22×C17—82no—
193C22×C17C2×C4×D411yescharacteristic
194C22×C17#54711yescharacteristic
195C22×C17#55011yescharacteristic
196C22×C17#57512yes—
197C22×C17#57612yes—
198C4×C17—42no—
199C4×C17—82no—
200C4×C17C2×#11711yescharacteristic
201C4×C17#52011yescharacteristic
202C4×C17#55511yescharacteristic
203C4×C17#55611yescharacteristic
204Dic17—22no—
205Dic17—22no—
206Dic17—42no—
207Dic17—44no—
208Dic17—82no—
209C22×D8C2×D1711yescharacteristic
210C2×C4×D4—172no—
211C4×D8—178no—
212C2×#107—171no—
213C2×#107—172no—
214C2×#106—171no—
215C2×#106C2×D1711yescharacteristic
216C2×C4⋊C8—171no—
217C2×#102—172no—
218C2×#102—172no—
219C2×#102C2×D1712yes—
220C2×#97—171no—
221C2×#97C2×D1711yescharacteristic
222C2×C4×C8—171no—
223#390—172no—
224#390C2×D1712yes—
225#388—342no—
226#388—342no—
227#382—172no—
228#381—342no—
229C4×C4⋊C4—171no—
230C4×C4⋊C4—172no—
231C4×C4⋊C4—172no—
232C4×C22⋊C4—342no—
233#340—172no—
234#340—172no—
235#336—171no—
236#336—171no—
237C2×C17—84no—
238C2×C17#235411yescharacteristic
239C2×C17#294311yescharacteristic
240C2×C17#294811yescharacteristic
241C2×C17#335512yes—
242C2×C17#340312yes—
243C22×D4C22×D1712yes—
244C23×C4—342no—
245C2×D8—22no—
246C2×D8—22no—
247C2×D8C4×D1714yes—
248C22×C8—171no—
249C22×C8—171no—
250C22×C8C22×D1711yescharacteristic
251C22×C8C22×D1711yescharacteristic
252C4×D4—178no—
253C4×D4—344no—
254C2×C4⋊C4—172no—
255C2×C4⋊C4—172no—
256C2×C4⋊C4—172no—
257C2×C4⋊C4C22×D1712yes—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—342no—
261C2×C22⋊C4—342no—
262C2×C22⋊C4—342no—
263C2×C42—171no—
264C2×C42—171no—
265C2×C42—342no—
266C2×C42—342no—
267C2×C42C22×D1711yescharacteristic
268#107—174no—
269#107—682no—
270#106—22no—
271#106—342no—
272C4⋊C8—342no—
273#102—42no—
274#102—178no—
275#102—682no—
276#97—22no—
277#97—342no—
278C4×C8—174no—
279#95—172no—
280#95—342no—
281#95—342no—
282#95—342no—
283#95—342no—
284C17#2281611yescharacteristic, complemented, Sylow
285C24—22no—
286C2×D4—22no—
287C2×D4—22no—
288C2×D4—22no—
289C2×D4—22no—
290C2×D4—22no—
291C2×D4C2×C4×D1714yes—
292C22×C4—22no—
293C22×C4—171no—
294C22×C4—171no—
295C22×C4—172no—
296C22×C4—348no—
297C22×C4—342no—
298C22×C4—342no—
299C22×C4—1362no—
300C22×C4—1362no—
301C22×C4D4×D1712yes—
302C22×C4C23×D1711yescharacteristic
303D8—28no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—172no—
308C2×C8—341no—
309C2×C8—341no—
310C2×C8—341no—
311C2×C8—341no—
312C2×C8—341no—
313C2×C8C2×C4×D1712yes—
314C2×C8#6325412yes—
315C2×C8#6325412yes—
316C4⋊C4—22no—
317C4⋊C4—22no—
318C4⋊C4—174no—
319C4⋊C4—342no—
320C4⋊C4—342no—
321C4⋊C4—342no—
322C4⋊C4—682no—
323C22⋊C4—44no—
324C22⋊C4—82no—
325C22⋊C4—348no—
326C22⋊C4—684no—
327C22⋊C4—1362no—
328C42—21no—
329C42—21no—
330C42—172no—
331C42—172no—
332C42—341no—
333C42—341no—
334C42—1362no—
335C42—1362no—
336C23—24no—
337C23—42no—
338C23—42no—
339C23—42no—
340C23—42no—
341C23—42no—
342C23—42no—
343C23C2×D4×D1711yescharacteristic
344D4—24no—
345D4—24no—
346D4—42no—
347D4—42no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—172no—
355C2×C4—172no—
356C2×C4—342no—
357C2×C4—342no—
358C2×C4—342no—
359C2×C4—344no—
360C2×C4—342no—
361C2×C4—342no—
362C2×C4—682no—
363C2×C4—682no—
364C2×C4—682no—
365C2×C4—682no—
366C2×C4—682no—
367C2×C4—1368no—
368C2×C4C22×C4×D1711yescharacteristic
369C2×C4C2×#6325411yescharacteristic
370C2×C4C2×#6325411yescharacteristic
371C2×C4#1058769111yescharacteristic
372C2×C4#41×D1711yescharacteristic
373C2×C4#1058769711yescharacteristic
374C8—41no—
375C8—342no—
376C8—681no—
377C8#1058756314yes—
378C22—44no—
379C22—44no—
380C22—44no—
381C22—82no—
382C22—82no—
383C22—82no—
384C22—82no—
385C22C4×D4×D1711yescharacteristic
386C22#1176345911yescharacteristic
387C22#1176347911yescharacteristic
388C22#1176360512yes—
389C22#1176361312yes—
390C4—42no—
391C4—82no—
392C4—342no—
393C4—342no—
394C4—682no—
395C4—684no—
396C4—1362no—
397C4C2×#1058756311yescharacteristic
398C4#1176328811yescharacteristic
399C4#1176352311yescharacteristic
400C4#1176352811yescharacteristic
401C2—84no—
402C2#1453173211yescharacteristic
403C2#1453758111yescharacteristic
404C2#1453763311yescharacteristic
405C2#1454138512yes—
406C2#1454186912yes—
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1972C2
socleC23×C17C22×D4
prime radicalC22×D8×C17C22
cosocleC22×C4×C17C24
hypercenter#1972D17
hypocenterC17#22816
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1972

Complements

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

NH ≅ G/Ntype
2. #14531756C2semidirect
4. #14529407C2semidirect
5. #14529401C2semidirect
6. #14528945C2semidirect
7. #14528792C2semidirect
8. #14528684C2semidirect
9. #14528681C2semidirect
10. #14528652C2semidirect
13. D8×Dic17C4semidirect
284. C17#22816semidirect

Maximal subgroups

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

subgroupindex
C2×D8×Dic172
#145317562
#145296612
#145294072
#145294012
#145289452
#145287922
#145286842
#145286812
#145286522
C17×#19722
#2281617