Finite Group Databasebeta

← #17044816

Subgroups — #17044816

Order 4352.

770 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762864C211yesmaximal, characteristic
2C2×#11762831C211yesmaximal, characteristic
3#14531180C211yesmaximal, characteristic
4#14529508C211yesmaximal, characteristic, complemented
5#14529507C211yesmaximal, characteristic, complemented
6#14529505C211yesmaximal, characteristic, complemented
7#14528968C212yesmaximal
8#14528965C212yesmaximal
9#14528937C212yesmaximal, complemented
10#14528902C212yesmaximal, complemented
11C17×#1940C211yesmaximal, characteristic, complemented
12C22×#63218C2212yes—
13C2×C42×D17C2211yescharacteristic
14#11762864—24no—
15#11762831—24no—
16C2×C4⋊#3622C2211yescharacteristic
17C2×C4⋊#3622C2212yes—
18C2×#10587485C2212yes—
19C2×#10587485C2211yescharacteristic
20C2×#10587485C2211yescharacteristic
21C2×#10587485C2212yes—
22C2×#10587485C2212yes—
23C2×#10587491C2211yescharacteristic
24C2×#10587491C2211yescharacteristic
25C2×#10587474C2211yescharacteristic
26#11762676C412yes—
27#11762674C412yes—
28#11762671C412yes—
29#11762666C412yes—
30C4×C4⋊Dic17C2211yescharacteristic
31C4×#63227C2212yes—
32#11762655C2212yes—
33#11762564C2212yes—
34#11762563C2212yes—
35#11762563C2212yes—
36C2×C17×C4⋊C8C2211yescharacteristic
37C2×C17×C4⋊C8C2212yes—
38C2×C17×#97C2211yescharacteristic
39C4×C4⋊C4×C17C2211yescharacteristic
40C17×#336C2212yes—
41C22×C4×D17C2×C411yescharacteristic
42C22×C4×D17C2×C411yescharacteristic
43C22×C4×D17C2311yescharacteristic
44C22×C8×C17C2311yescharacteristic
45C22×C8×C17C2311yescharacteristic
46C22×C8×C17C2312yes—
47C2×C4⋊C4×C17C2×C412yes—
48C2×C42×C17C2312yes—
49C2×C42×C17C2311yescharacteristic
50C2×#63227C2×C414yes—
51C2×C4⋊Dic17C2×C412yes—
52C2×C4×Dic17C2×C411yescharacteristic
53C2×C4×Dic17C2×C411yescharacteristic
54C2×C4×Dic17C2312yes—
55C2×C4×Dic17C2311yescharacteristic
56C22×#3622C2311yescharacteristic
57C22×#3622C2311yescharacteristic
58C22×#3622C2312yes—
59C2×#63218—24no—
60C2×#63218—42no—
61C2×#63218—42no—
62C42×D17D414yes—
63C42×D17Q814yes—
64C17×C4⋊C8—22no—
65C17×C4⋊C8—42no—
66C17×#97—22no—
67C17×#95C2×C412yes—
68#10587502C2×C414yes—
69#10587502C2×C414yes—
70#10587502C2×C414yes—
71#10587502C2×C412yes—
72#10587491—22no—
73#10587491—22no—
74#10587485—24no—
75#10587485—22no—
76#10587485—22no—
77#10587485—42no—
78#10587485—42no—
79C4⋊#3622—22no—
80C4⋊#3622—42no—
81#10587474—22no—
82C23×D17C22×C411yescharacteristic
83C22×C4×C17C22×C414yes—
84C22×C4×C17C22×C411yescharacteristic
85C22×C4×C17C22×C411yescharacteristic
86C22×C4×C17C2411yescharacteristic
87C22×Dic17C22×C414yes—
88C22×Dic17C22×C411yescharacteristic
89C22×Dic17C22×C411yescharacteristic
90C22×Dic17C22×C411yescharacteristic
91C2×C4×D17—21no—
92C2×C4×D17—21no—
93C2×C4×D17—21no—
94C2×C4×D17—21no—
95C2×C4×D17—42no—
96C2×C4×D17—42no—
97C2×C4×D17C4⋊C414yes—
98C2×C4×D17C4⋊C414yes—
99C2×C4×D17C2×D412yes—
100C2×C4×D17C2×Q812yes—
101C2×C8×C17—21no—
102C2×C8×C17—21no—
103C2×C8×C17—22no—
104C2×C8×C17—21no—
105C2×C8×C17—22no—
106C2×C8×C17—21no—
107C2×C8×C17—22no—
108C2×C8×C17—21no—
109C2×C8×C17—21no—
110C4⋊C4×C17—42no—
111C42×C17—22no—
112C42×C17—22no—
113C42×C17C2×D412yes—
114C42×C17C2×Q812yes—
115#63227—44no—
116C4⋊Dic17—42no—
117C4×Dic17—22no—
118C4×Dic17—22no—
119C4×Dic17C4⋊C414yes—
120C4×Dic17C4⋊C414yes—
121C4×Dic17C2×D412yes—
122C4×Dic17C2×Q812yes—
123C2×#3622—21no—
124C2×#3622—21no—
125C2×#3622—22no—
126C2×#3622—21no—
127C2×#3622—22no—
128C2×#3622—21no—
129C2×#3622—22no—
130C2×#3622—21no—
131C2×#3622—21no—
132#63218—48no—
133#21979—171nomaximal, Sylow
134C23×C17C23×C411yescharacteristic
135C22×D17—21no—
136C22×D17—21no—
137C22×D17—22no—
138C22×D17—22no—
139C22×D17C2×C4⋊C412yes—
140C2×C4×C17—22no—
141C2×C4×C17—22no—
142C2×C4×C17—24no—
143C2×C4×C17—24no—
144C2×C4×C17—24no—
145C2×C4×C17C2×C4⋊C412yes—
146C2×C4×C17C2×C4⋊C412yes—
147C2×C4×C17C22×D411yescharacteristic
148C2×C4×C17C22×Q811yescharacteristic
149C2×C4×C17H(5,𝔽2)11yescharacteristic
150C2×C4×C17H(5,𝔽2)11yescharacteristic
151C2×C4×C172−1+411yescharacteristic
152C2×C4×C172−1+411yescharacteristic
153C2×Dic17—21no—
154C2×Dic17—21no—
155C2×Dic17—22no—
156C2×Dic17—22no—
157C2×Dic17—24no—
158C2×Dic17—24no—
159C2×Dic17—24no—
160C2×Dic17C2×C4⋊C412yes—
161C2×Dic17C2×C4⋊C412yes—
162C2×Dic17C2×C4⋊C412yes—
163C4×D17—22no—
164C4×D17—22no—
165C4×D17—22no—
166C4×D17—22no—
167C4×D17—44no—
168C4×D17—44no—
169C8×C17—41no—
170C8×C17—42no—
171C8×C17—41no—
172#3622—42no—
173#3622—41no—
174#3622—41no—
175C2×#432—171no—
176C2×#423—171no—
177#2283—171no—
178#1940—171no—
179#1940—171no—
180#1940—171no—
181#1940D1711yescharacteristic, complemented
182#1851—172no—
183#1851—172no—
184#1843—172no—
185#1832—172no—
186C22×C17C22×C4⋊C411yescharacteristic
187C22×C17#52011yescharacteristic
188C22×C17#52011yescharacteristic
189C22×C17#56812yes—
190C22×C17#56812yes—
191C2×D17—22no—
192C2×D17—22no—
193C2×D17—22no—
194C2×D17—42no—
195C2×D17—42no—
196C4×C17—22no—
197C4×C17—22no—
198C4×C17—44no—
199C4×C17#53511yescharacteristic
200C4×C17#53611yescharacteristic
201C4×C17#54411yescharacteristic
202C4×C17#54411yescharacteristic
203Dic17—22no—
204Dic17—22no—
205Dic17—22no—
206Dic17—44no—
207C22×M16—172no—
208C22×C42—171no—
209#432—344no—
210#423—344no—
211C2×C4⋊C8—171no—
212C2×C4⋊C8—172no—
213C2×C4⋊C8—171no—
214C2×C4⋊C8—171no—
215C2×C4⋊C8—172no—
216C2×C4⋊C8—172no—
217C2×C4⋊C8—172no—
218C2×C4⋊C8C2×D1711yescharacteristic
219C2×C4⋊C8C2×D1712yes—
220C2×C22⋊C8—171no—
221C2×C22⋊C8—171no—
222C2×#97—171no—
223C2×#97C2×D1711yescharacteristic
224#383—172no—
225#382—172no—
226#381—172no—
227#380—172no—
228C4×C4⋊C4—171no—
229C4×C4⋊C4C2×D1711yescharacteristic
230C4×C22⋊C4—172no—
231#376—172no—
232#336—172no—
233#336—172no—
234#336—172no—
235#336C2×D1712yes—
236C2×C17#231711yescharacteristic
237C2×C17#290511yescharacteristic
238C2×C17#290511yescharacteristic
239C2×C17#299512yes—
240C2×C17#299512yes—
241D17—42no—
242C23×C4—171no—
243C23×C4—171no—
244C23×C4—171no—
245C2×M16—344no—
246C2×M16—682no—
247C2×M16—682no—
248C22×C8—172no—
249C22×C8—171no—
250C22×C8—171no—
251C22×C8C22×D1712yes—
252C22×C8C22×D1711yescharacteristic
253C22×C8C22×D1711yescharacteristic
254C2×C4⋊C4—172no—
255C2×C4⋊C4C4×D1712yes—
256C2×C22⋊C4—174no—
257C2×C42—171no—
258C2×C42—171no—
259C2×C42—171no—
260C2×C42—174no—
261C2×C42—174no—
262C2×C42—172no—
263C2×C42C22×D1712yes—
264C2×C42C22×D1711yescharacteristic
265C4⋊C8—22no—
266C4⋊C8—42no—
267C4⋊C8—344no—
268C4⋊C8—342no—
269C4⋊C8—342no—
270C4⋊C8—342no—
271C4⋊C8—682no—
272C4⋊C8—682no—
273C4⋊C8—682no—
274C22⋊C8—342no—
275C22⋊C8—342no—
276#97—22no—
277#97—342no—
278#95—174no—
279#95—174no—
280#95—174no—
281#95—172no—
282#95C4×D1712yes—
283C17#2197911yescharacteristic, complemented, Sylow
284C24—171no—
285C22×C4—171no—
286C22×C4—171no—
287C22×C4—171no—
288C22×C4—172no—
289C22×C4—172no—
290C22×C4—174no—
291C22×C4—174no—
292C22×C4—174no—
293C22×C4—341no—
294C22×C4—341no—
295C22×C4—341no—
296C22×C4—341no—
297C22×C4—682no—
298C22×C4—682no—
299C22×C4C2×C4×D1714yes—
300C22×C4C2×C4×D1711yescharacteristic
301C22×C4C2×C4×D1711yescharacteristic
302C22×C4C23×D1711yescharacteristic
303M16—688no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—22no—
307C2×C8—21no—
308C2×C8—22no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—22no—
312C2×C8—21no—
313C2×C8—341no—
314C2×C8—341no—
315C2×C8—342no—
316C2×C8—341no—
317C2×C8—342no—
318C2×C8—341no—
319C2×C8—341no—
320C2×C8—342no—
321C2×C8—341no—
322C4⋊C4—42no—
323C4⋊C4—682no—
324C22⋊C4—684no—
325C42—22no—
326C42—22no—
327C42—172no—
328C42—172no—
329C42—174no—
330C42—174no—
331C42—342no—
332C42—342no—
333C42D4×D1712yes—
334C42Q8×D1712yes—
335C23—172no—
336C23—341no—
337C23—341no—
338C23—342no—
339C23—342no—
340C23C22×C4×D1711yescharacteristic
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—172no—
347C2×C4—172no—
348C2×C4—172no—
349C2×C4—341no—
350C2×C4—341no—
351C2×C4—342no—
352C2×C4—342no—
353C2×C4—342no—
354C2×C4—342no—
355C2×C4—342no—
356C2×C4—342no—
357C2×C4—344no—
358C2×C4—344no—
359C2×C4—344no—
360C2×C4—684no—
361C2×C4—684no—
362C2×C4C4⋊C4×D1712yes—
363C2×C4C4⋊C4×D1712yes—
364C2×C4C2×D4×D1711yescharacteristic
365C2×C4#1058769111yescharacteristic
366C2×C4C2×Q8×D1711yescharacteristic
367C2×C4#1058769411yescharacteristic
368C2×C4#1058769611yescharacteristic
369C2×C4#1058769711yescharacteristic
370C8—42no—
371C8—41no—
372C8—41no—
373C8—682no—
374C8—681no—
375C8—681no—
376C22—342no—
377C22—342no—
378C22—342no—
379C22—682no—
380C22—682no—
381C22C2×C4⋊C4×D1711yescharacteristic
382C22#1176328811yescharacteristic
383C22#1176329111yescharacteristic
384C22#1176357412yes—
385C22#1176357512yes—
386C4—22no—
387C4—22no—
388C4—44no—
389C4—342no—
390C4—342no—
391C4—342no—
392C4—684no—
393C4#1176337511yescharacteristic
394C4#1176338211yescharacteristic
395C4#1176343611yescharacteristic
396C4#1176343911yescharacteristic
397C2—682no—
398C2#1453140611yescharacteristic
399C2#1453734511yescharacteristic
400C2#1453734611yescharacteristic
401C2#1453795012yes—
402C2#1453795312yes—
subgroupG/H
center, FrattiniC22×C4C23×D17
commutator, socleC23×C17C23×C4
FittingC17×#1940C2
prime radicalC23×D17C22×C4
cosocleC22×C4×C17C24
hypercenter#1940D17
hypocenterC17#21979
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C22×C4 ◁ #1940

Complements

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

NH ≅ G/Ntype
4. #14529508C2semidirect
5. #14529507C2semidirect
6. #14529505C2semidirect
9. #14528937C2semidirect
10. #14528902C2semidirect
11. C17×#1940C2semidirect
181. #1940D17semidirect
283. C17#21979semidirect

Maximal subgroups

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

subgroupindex
C2×#117628642
C2×#117628312
#145311802
#145295082
#145295072
#145295052
#145289682
#145289652
#145289372
#145289022
C17×#19402
#2197917