Finite Group Databasebeta

← #16933271

Subgroups — #16933271

Order 4352.

466 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#63230C211yesmaximal, characteristic
2#14527530C211yesmaximal, characteristic, complemented
3#14527179C211yesmaximal, characteristic, complemented
4#14527176C211yesmaximal, characteristic
5#14527104C211yesmaximal, characteristic
6#14527077C211yesmaximal, characteristic, complemented
7C17×#1461C211yesmaximal, characteristic, complemented
8C22×#63230C411yescharacteristic
9C2×C4×D68C2211yescharacteristic
10C4×#63230—21no—
11C4×#63230—21no—
12C4×#63230—21no—
13C4×#63230—21no—
14#11762862C2211yescharacteristic
15#11762862C2211yescharacteristic
16C2×#10587479C411yescharacteristic
17C2×#10587506C411yescharacteristic
18C2×#10587480C411yescharacteristic
19C2×C4×#3622C2211yescharacteristic
20#11762561—21no—
21#11762559—21no—
22Q8⋊#3622—22no—
23#11762538—22no—
24#11762532—22no—
25#11762531—21no—
26#11762530—21no—
27C2×C4×Q8×C17C2211yescharacteristic
28C17×#431C2211yescharacteristic
29C17×#431C2211yescharacteristic
30C17×#324—22no—
31C22×Q8×C17C2×C411yescharacteristic
32C22×D68C2×C411yescharacteristic
33C22×C4×D17—21no—
34C4×Q8×C17—21no—
35C4×Q8×C17—21no—
36C4×Q8×C17—21no—
37C4×Q8×C17—21no—
38C2×C4⋊C4×C17—21no—
39C2×C4⋊C4×C17C2×C411yescharacteristic
40C2×C42×C17—21no—
41C2×C42×C17C2311yescharacteristic
42C2×#63230—21no—
43C2×#63230—21no—
44C2×#63230—22no—
45C2×#63230—22no—
46C2×#63227—21no—
47C2×C4⋊Dic17C2×C411yescharacteristic
48C22×#3622C2×C411yescharacteristic
49C22×#3622C2×C411yescharacteristic
50C4×D68—21no—
51C4×D68—21no—
52C4×D68—21no—
53C4×D68—21no—
54C17×C4⋊C8—21no—
55C17×C4⋊C8—21no—
56C17×C22⋊C8—21no—
57C17×C22⋊C8D412yes—
58C4×C8×C17—21no—
59C4×C8×C17—21no—
60#10587506—21no—
61#10587506—21no—
62C22⋊#3622—21no—
63C22⋊#3622—21no—
64#10587480—21no—
65#10587480—21no—
66#10587479—21no—
67#10587479—21no—
68C4⋊#3622—21no—
69C4⋊#3622D412yes—
70C4×#3622—21no—
71C4×#3622—21no—
72C4×#3622D411yescharacteristic
73C4×#3622D411yescharacteristic
74C4×#3622D411yescharacteristic
75C4×#3622D411yescharacteristic
76C23×D17—21no—
77C2×Q8×C17—21no—
78C2×Q8×C17—21no—
79C2×Q8×C17—21no—
80C2×Q8×C17—21no—
81C2×Q8×C17—21no—
82C2×Q8×C17M1612yes—
83C22×C4×C17—21no—
84C22×C4×C17—21no—
85C22×C4×C17C22×C411yescharacteristic
86C22×C4×C17C2×D411yescharacteristic
87C22×C4×C17#4111yescharacteristic
88C22×Dic17—21no—
89C2×D68—21no—
90C2×D68—21no—
91C2×D68—21no—
92C2×D68—21no—
93C2×D68—21no—
94C2×D68M1612yes—
95C2×C4×D17—41no—
96C2×C4×D17—41no—
97C2×C4×D17—82no—
98C2×C8×C17—22no—
99C2×C8×C17—41no—
100C4⋊C4×C17—21no—
101C4⋊C4×C17—21no—
102C4⋊C4×C17—41no—
103C4⋊C4×C17—41no—
104C42×C17—41no—
105C42×C17—41no—
106C42×C17C2×D411yescharacteristic
107C42×C17C2×D411yescharacteristic
108C42×C17C2×D411yescharacteristic
109C42×C17#4111yescharacteristic
110#63230—24no—
111#63230—24no—
112#63227—41no—
113#63227—41no—
114C4⋊Dic17—21no—
115C4⋊Dic17—21no—
116C2×#3622—21no—
117C2×#3622—21no—
118C2×#3622—21no—
119C2×#3622—21no—
120C2×#3622—22no—
121C2×#3622—41no—
122C2×#3622C22⋊C411yescharacteristic
123C2×#3622C22⋊C411yescharacteristic
124C2×#3622C22⋊C411yescharacteristic
125C2×#3622C22⋊C411yescharacteristic
126#9672—171nomaximal, Sylow
127C23×C17C4×D411yescharacteristic
128C22×D17—22no—
129C22×D17—41no—
130C22×D17—41no—
131C22×D17—81no—
132C22×D17—81no—
133Q8×C17—22no—
134Q8×C17—22no—
135Q8×C17—41no—
136Q8×C17—41no—
137C2×C4×C17—22no—
138C2×C4×C17—22no—
139C2×C4×C17—41no—
140C2×C4×C17—41no—
141C2×C4×C17—41no—
142C2×C4×C17—41no—
143C2×C4×C17—81no—
144C2×C4×C17C2×C22⋊C411yescharacteristic
145C2×C4×C17C4×D411yescharacteristic
146C2×C4×C17#12111yescharacteristic
147C2×C4×C17#12111yescharacteristic
148C2×C4×C17#12411yescharacteristic
149C2×C4×C17#12711yescharacteristic
150C2×C4×C17C2×M1611yescharacteristic
151C2×C4×C17C2×M1611yescharacteristic
152C2×C4×C17#13111yescharacteristic
153C2×C4×C17#13111yescharacteristic
154C2×Dic17—41no—
155C2×Dic17—81no—
156D68—22no—
157D68—22no—
158D68—41no—
159D68—41no—
160C4×D17—82no—
161C4×D17—82no—
162C8×C17—42no—
163C8×C17—81no—
164#3622—21no—
165#3622—21no—
166#3622—21no—
167#3622—21no—
168#3622—42no—
169#3622—81no—
170C2×C4×SD16—171no—
171#1564—171no—
172#1479—171no—
173#1478—171no—
174#1461D1711yescharacteristic, complemented
175#1460—171no—
176#1452—171no—
177C22×C17—21no—
178C22×C17—21no—
179C22×C17#39011yescharacteristic
180C22×C17#43511yescharacteristic
181C22×C17#43511yescharacteristic
182C2×D17—42no—
183C2×D17—82no—
184C2×D17—81no—
185C2×D17—81no—
186C2×D17—82no—
187C4×C17—21no—
188C4×C17—21no—
189C4×C17—21no—
190C4×C17—42no—
191C4×C17—81no—
192C4×C17#40711yescharacteristic
193C4×C17#40811yescharacteristic
194C4×C17#43511yescharacteristic
195C4×C17#43511yescharacteristic
196C4×C17#47112yes—
197Dic17—81no—
198C22×SD16—171no—
199C2×C4×Q8C2×D1711yescharacteristic
200C2×C4×D4—171no—
201C4×SD16—341no—
202C4×SD16—341no—
203C4×SD16—341no—
204C4×SD16—341no—
205#431—171no—
206#431—171no—
207#431C2×D1711yescharacteristic
208#431C2×D1711yescharacteristic
209C2×#106—171no—
210C2×#103—171no—
211C2×#102—171no—
212C2×C4×C8—171no—
213#335—341no—
214#334—341no—
215Q8⋊C8—342no—
216D4⋊C8—342no—
217#324—22no—
218#323—342no—
219#322—341no—
220#322—341no—
221C2×C17—21no—
222C2×C17—21no—
223C2×C17#195511yescharacteristic
224C2×C17#197611yescharacteristic
225C2×C17#197711yescharacteristic
226D17—82no—
227C22×Q8C4×D1711yescharacteristic
228C22×D4—171no—
229C23×C4—341no—
230C2×SD16—341no—
231C2×SD16—341no—
232C2×SD16—342no—
233C2×SD16—342no—
234C22×C8—171no—
235C22×C8—171no—
236C4×Q8—21no—
237C4×Q8—21no—
238C4×Q8—21no—
239C4×Q8—21no—
240C4×D4—341no—
241C4×D4—341no—
242C4×D4—341no—
243C4×D4—341no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—171no—
246C2×C4⋊C4C4×D1711yescharacteristic
247C2×C22⋊C4—341no—
248C2×C42—21no—
249C2×C42C22×D1711yescharacteristic
250#106—341no—
251#106—341no—
252C4⋊C8—21no—
253C4⋊C8—21no—
254C4⋊C8—172no—
255C4⋊C8—341no—
256#103—341no—
257#103—341no—
258#102—341no—
259#102—341no—
260C22⋊C8—21no—
261C22⋊C8—341no—
262C22⋊C8—341no—
263C22⋊C8C22⋊D1712yes—
264C4×C8—21no—
265C4×C8—21no—
266C4×C8—171no—
267C4×C8—171no—
268C4×C8—171no—
269C4×C8—171no—
270C4×C8—341no—
271C4×C8—341no—
272C17#967211yescharacteristic, complemented, Sylow
273C24—341no—
274C2×Q8—21no—
275C2×Q8—21no—
276C2×Q8—21no—
277C2×Q8—21no—
278C2×Q8—21no—
279C2×Q8#6321812yes—
280C2×D4—172no—
281C2×D4—341no—
282C2×D4—341no—
283C2×D4—341no—
284C2×D4—341no—
285C2×D4—341no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—341no—
289C22×C4—681no—
290C22×C4—681no—
291C22×C4—1362no—
292C22×C4C2×C4×D1711yescharacteristic
293C22×C4#6325211yescharacteristic
294C22×C4C2×C22⋊D1711yescharacteristic
295SD16—344no—
296SD16—344no—
297C2×C8—22no—
298C2×C8—41no—
299C2×C8—171no—
300C2×C8—171no—
301C2×C8—171no—
302C2×C8—171no—
303C2×C8—341no—
304C2×C8—341no—
305C2×C8—341no—
306C2×C8—341no—
307C2×C8—342no—
308C2×C8—681no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—341no—
314C4⋊C4—341no—
315C22⋊C4—681no—
316C22⋊C4—681no—
317C42—41no—
318C42—41no—
319C42D4×D1711yescharacteristic
320C42D4×D1711yescharacteristic
321C42D4×D1711yescharacteristic
322C42#6325411yescharacteristic
323C23—342no—
324C23—681no—
325C23—681no—
326C23—1361no—
327C23—1361no—
328C23C4×C22⋊D1711yescharacteristic
329Q8—22no—
330Q8—22no—
331Q8—41no—
332Q8—41no—
333D4—342no—
334D4—342no—
335D4—681no—
336D4—681no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—81no—
344C2×C4—681no—
345C2×C4—1362no—
346C2×C4—1362no—
347C2×C4—1361no—
348C2×C4C22⋊C4×D1711yescharacteristic
349C2×C4#1058756711yescharacteristic
350C2×C4#1058757011yescharacteristic
351C2×C4#1058757111yescharacteristic
352C2×C4C2×#6321811yescharacteristic
353C2×C4#1058758711yescharacteristic
354C2×C4M16×D1711yescharacteristic
355C2×C4#1058759311yescharacteristic
356C2×C4#1058762511yescharacteristic
357C2×C4#1058762611yescharacteristic
358C8—42no—
359C8—81no—
360C8—341no—
361C8—341no—
362C8—341no—
363C8—341no—
364C8—682no—
365C8—1361no—
366C22—21no—
367C22—21no—
368C22—682no—
369C22—1362no—
370C22—1362no—
371C22—1361no—
372C22—1361no—
373C22#1176270911yescharacteristic
374C22#1176287811yescharacteristic
375C22#1176288011yescharacteristic
376C4—21no—
377C4—21no—
378C4—21no—
379C4—42no—
380C4—81no—
381C4—1361no—
382C4#1176277111yescharacteristic
383C4#1176277411yescharacteristic
384C4#1176287711yescharacteristic
385C4#1176288311yescharacteristic
386C4#1176307512yes—
387C2—21no—
388C2—21no—
389C2—1362no—
390C2#1452958011yescharacteristic
391C2#1452967911yescharacteristic
392C2#1452968811yescharacteristic
subgroupG/H
centerC2×C4#10587570
commutatorC22×C4×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#1461C2
socleC22×C17#390
prime radicalC22×D68C2×C4
cosocleC2×C42×C17C23
hypercenter#1461D17
hypocenterC17#9672
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1461

Complements

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

NH ≅ G/Ntype
2. #14527530C2semidirect
3. #14527179C2semidirect
6. #14527077C2semidirect
7. C17×#1461C2semidirect
174. #1461D17semidirect
272. C17#9672semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#632302
#145275302
#145271792
#145271762
#145271042
#145270772
C17×#14612
#967217