Finite Group Databasebeta

← #16935461

Subgroups — #16935461

Order 4352.

602 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763096C211yesmaximal, characteristic
2#14533780C211yesmaximal, characteristic
3#14529733C211yesmaximal, characteristic, complemented
4C2×#11762555C211yesmaximal, characteristic, complemented
5C2×#11762538C211yesmaximal, characteristic
6#14526624C211yesmaximal, characteristic, complemented
7C2×C17×#487C211yesmaximal, characteristic, complemented
8C2×#10587633—21no—
9C2×C4×D68C2211yescharacteristic
10#11763096—22no—
11#11763096—22no—
12C2×#10587479—21no—
13C2×C4⋊#3622C2211yescharacteristic
14C2×C4⋊#3622C2211yescharacteristic
15C2×#10587480—21no—
16#11762735—22no—
17#11762716—21no—
18#11762666—22no—
19C4×C4⋊Dic17C2211yescharacteristic
20#11762571—22no—
21#11762555—22no—
22#11762538—22no—
23C2×C17×C4⋊Q8C2211yescharacteristic
24C2×C17×C4⋊Q8C2211yescharacteristic
25C17×#487—22no—
26C17×#487—22no—
27C2×C17×#103—22no—
28C2×C4×C8×C17C2211yescharacteristic
29C22×Q8×C17—21no—
30C22×Q8×C17—21no—
31C22×D68D411yescharacteristic
32C22×C4×D17—21no—
33C22×C8×C17—21no—
34C17×C4⋊Q8—21no—
35C17×C4⋊Q8—21no—
36C17×C4⋊Q8—21no—
37C17×C4⋊Q8—21no—
38C17×C4⋊Q8—21no—
39C17×C4⋊Q8—21no—
40C17×C4⋊Q8—21no—
41C17×C4⋊Q8D412yes—
42C2×C4⋊C4×C17—21no—
43C2×C4⋊C4×C17—21no—
44C2×C4⋊C4×C17—21no—
45C2×C4⋊C4×C17D412yes—
46C2×C42×C17C2311yescharacteristic
47#10587633—42no—
48#10587633—42no—
49C2×#63227—24no—
50C2×#63227—21no—
51C2×C4⋊Dic17D411yescharacteristic
52C2×C4⋊Dic17D412yes—
53C2×C4×Dic17—21no—
54C22×#3622—21no—
55C22×#3622—21no—
56C4×D68—21no—
57C4×D68—21no—
58C4×D68—21no—
59C4×D68—21no—
60C17×#103—28no—
61C4×C8×C17—21no—
62C4×C8×C17D412yes—
63#10587502—24no—
64#10587502—21no—
65#10587480—24no—
66#10587479—24no—
67C4⋊#3622—21no—
68C4⋊#3622—21no—
69C4⋊#3622—21no—
70C4⋊#3622D412yes—
71C23×D17—21no—
72C2×Q8×C17—22no—
73C2×Q8×C17—44no—
74C2×Q8×C17—41no—
75C2×Q8×C17—41no—
76C2×Q8×C17—41no—
77C2×Q8×C17—41no—
78C2×Q8×C17—41no—
79C2×Q8×C17—41no—
80C2×Q8×C17—41no—
81C22×C4×C17—22no—
82C22×C4×C17—41no—
83C22×C4×C17C2×D411yescharacteristic
84C22×C4×C17C2×D411yescharacteristic
85C22×C4×C17C2×D411yescharacteristic
86C22×Dic17—21no—
87C22×Dic17—22no—
88C2×D68—21no—
89C2×D68—21no—
90C2×D68—21no—
91C2×D68—21no—
92C2×D68—21no—
93C2×D68SD1612yes—
94C2×C4×D17—41no—
95C2×C4×D17—41no—
96C2×C4×D17—42no—
97C2×C4×D17—42no—
98C2×C8×C17—22no—
99C2×C8×C17—42no—
100C4⋊C4×C17—22no—
101C4⋊C4×C17—22no—
102C4⋊C4×C17—22no—
103C4⋊C4×C17—22no—
104C4⋊C4×C17—41no—
105C4⋊C4×C17—41no—
106C4⋊C4×C17—41no—
107C4⋊C4×C17—41no—
108C42×C17C2×D411yescharacteristic
109C42×C17C2×D411yescharacteristic
110C42×C17C2×D411yescharacteristic
111C42×C17#4111yescharacteristic
112#63227—48no—
113#63227—41no—
114#63227—41no—
115C4⋊Dic17—21no—
116C4⋊Dic17—21no—
117C4⋊Dic17—22no—
118C4⋊Dic17—22no—
119C4×Dic17—41no—
120C4×Dic17—41no—
121C2×#3622—22no—
122C2×#3622—41no—
123C2×#3622—41no—
124C2×#3622—41no—
125C2×#3622—42no—
126#10058—171nomaximal, Sylow
127C23×C17#12011yescharacteristic
128C22×D17—22no—
129C22×D17—41no—
130C22×D17—42no—
131C22×D17—41no—
132C22×D17—42no—
133Q8×C17—44no—
134Q8×C17—82no—
135Q8×C17—81no—
136Q8×C17—81no—
137Q8×C17—81no—
138Q8×C17—81no—
139C2×C4×C17—22no—
140C2×C4×C17—42no—
141C2×C4×C17—42no—
142C2×C4×C17—42no—
143C2×C4×C17—42no—
144C2×C4×C17—82no—
145C2×C4×C17#12011yescharacteristic
146C2×C4×C17#12011yescharacteristic
147C2×C4×C17#12111yescharacteristic
148C2×C4×C17#12111yescharacteristic
149C2×C4×C17#12111yescharacteristic
150C2×C4×C17#12711yescharacteristic
151C2×C4×C17C2×SD1611yescharacteristic
152C2×C4×C17#13511yescharacteristic
153C2×C4×C17#13512yes—
154C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
155C2×C4×C17#13711yescharacteristic
156C2×C4×C17#13712yes—
157C2×Dic17—41no—
158C2×Dic17—42no—
159C2×Dic17—42no—
160C2×Dic17—42no—
161C2×Dic17—42no—
162D68—22no—
163D68—22no—
164D68—41no—
165D68—41no—
166C4×D17—82no—
167C4×D17—81no—
168C4×D17—81no—
169C8×C17—42no—
170#3622—42no—
171#3622—81no—
172C2×#487D1711yescharacteristic, complemented
173C2×#476—171no—
174#2547—171no—
175#1987—171no—
176C2×#332—171no—
177C2×D4⋊C8—171no—
178#1279—171no—
179C22×C17#39211yescharacteristic
180C22×C17#45011yescharacteristic
181C22×C17#45211yescharacteristic
182C22×C17#45212yes—
183C22×C17#45412yes—
184C2×D17—44no—
185C2×D17—42no—
186C2×D17—82no—
187C2×D17—81no—
188C2×D17—81no—
189C4×C17—21no—
190C4×C17—21no—
191C4×C17—82no—
192C4×C17—82no—
193C4×C17#44811yescharacteristic
194C4×C17#44911yescharacteristic
195C4×C17#46412yes—
196C4×C17#46511yescharacteristic
197C4×C17#46811yescharacteristic
198C4×C17#49511yescharacteristic
199C4×C17#49711yescharacteristic
200Dic17—82no—
201Dic17—81no—
202C2×C4⋊Q8C2×D1711yescharacteristic
203C2×C4⋊Q8C2×D1711yescharacteristic
204C2×#124—341no—
205C2×C4×D4—171no—
206#487—22no—
207#487—22no—
208#476—342no—
209#476—342no—
210C2×#106—341no—
211C2×C4⋊C8—171no—
212C2×C4⋊C8—171no—
213C2×#103—22no—
214C2×#102—341no—
215C2×C4×C8C2×D1711yescharacteristic
216#397—342no—
217#391—341no—
218#380—342no—
219C4×C4⋊C4—171no—
220#340—342no—
221#332—342no—
222D4⋊C8—342no—
223C2×C17#163512yes—
224C2×C17#163812yes—
225C2×C17#200011yescharacteristic
226C2×C17#200311yescharacteristic
227C2×C17#201511yescharacteristic
228D17—82no—
229C22×Q8—21no—
230C22×Q8—21no—
231C22×D4—171no—
232C23×C4—341no—
233C22×C8—21no—
234C22×C8—341no—
235C22×C8—341no—
236C4⋊Q8—21no—
237C4⋊Q8—21no—
238C4⋊Q8—21no—
239C4⋊Q8—21no—
240C4⋊Q8—21no—
241C4⋊Q8—21no—
242C4⋊Q8—21no—
243C4⋊Q8C22⋊D1712yes—
244#124—682no—
245#124—682no—
246C4×D4—341no—
247C4×D4—341no—
248C4×D4—341no—
249C4×D4—341no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—171no—
254C2×C4⋊C4—172no—
255C2×C4⋊C4C22⋊D1712yes—
256C2×C22⋊C4—344no—
257C2×C22⋊C4—341no—
258C2×C42—341no—
259C2×C42C22×D1711yescharacteristic
260#106—344no—
261C4⋊C8—172no—
262C4⋊C8—341no—
263C4⋊C8—341no—
264C4⋊C8—341no—
265#103—28no—
266#102—344no—
267C4×C8—21no—
268C4×C8C22⋊D1712yes—
269#95—344no—
270#95—341no—
271C17#1005811yescharacteristic, complemented, Sylow
272C24—341no—
273C2×Q8—22no—
274C2×Q8—44no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×D4—172no—
283C2×D4—341no—
284C2×D4—341no—
285C2×D4—341no—
286C2×D4—341no—
287C2×D4—341no—
288C22×C4—22no—
289C22×C4—41no—
290C22×C4—341no—
291C22×C4—342no—
292C22×C4—682no—
293C22×C4—682no—
294C22×C4—681no—
295C22×C4—681no—
296C22×C4D4×D1711yescharacteristic
297C22×C4D4×D1711yescharacteristic
298C22×C4C2×C22⋊D1711yescharacteristic
299C2×C8—22no—
300C2×C8—42no—
301C2×C8—342no—
302C2×C8—681no—
303C2×C8—681no—
304C2×C8—682no—
305C2×C8—681no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—342no—
315C4⋊C4—342no—
316C4⋊C4—341no—
317C4⋊C4—341no—
318C22⋊C4—688no—
319C22⋊C4—681no—
320C22⋊C4—681no—
321C42—681no—
322C42—681no—
323C42D4×D1711yescharacteristic
324C42#6325411yescharacteristic
325C42C2×C22⋊D1711yescharacteristic
326C42C2×C22⋊D1711yescharacteristic
327C23—342no—
328C23—681no—
329C23—682no—
330C23—681no—
331C23—682no—
332C23#1058762311yescharacteristic
333Q8—44no—
334Q8—82no—
335Q8—81no—
336Q8—81no—
337Q8—81no—
338Q8—81no—
339D4—342no—
340D4—342no—
341D4—681no—
342D4—681no—
343C2×C4—22no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—82no—
349C2×C4—682no—
350C2×C4—682no—
351C2×C4—682no—
352C2×C4—681no—
353C2×C4—682no—
354C2×C4—1361no—
355C2×C4—1362no—
356C2×C4—1361no—
357C2×C4SD16×D1711yescharacteristic
358C2×C4#1058760011yescharacteristic
359C2×C4#1058760111yescharacteristic
360C2×C4#1058760211yescharacteristic
361C2×C4#1058760412yes—
362C2×C4#1058760512yes—
363C2×C4#1058762311yescharacteristic
364C2×C4#1058762411yescharacteristic
365C2×C4#1058762511yescharacteristic
366C2×C4#1058762511yescharacteristic
367C2×C4#1058762611yescharacteristic
368C2×C4#1058763911yescharacteristic
369C8—42no—
370C8—682no—
371C8—1361no—
372C22—684no—
373C22—682no—
374C22—1361no—
375C22—1361no—
376C22—1362no—
377C22#1176271711yescharacteristic
378C22#1176295811yescharacteristic
379C22#1176296811yescharacteristic
380C22#1176297012yes—
381C22#1176298112yes—
382C4—21no—
383C4—21no—
384C4—82no—
385C4—82no—
386C4—1361no—
387C4—1362no—
388C4#1176294611yescharacteristic
389C4#1176295311yescharacteristic
390C4#1176303312yes—
391C4#1176303511yescharacteristic
392C4#1176305211yescharacteristic
393C4#1176319411yescharacteristic
394C4#1176320211yescharacteristic
395C2—1362no—
396C2#1452793212yes—
397C2#1452795312yes—
398C2#1452978411yescharacteristic
399C2#1452980611yescharacteristic
400C2#1452987511yescharacteristic
subgroupG/H
centerC23#10587623
commutator, cosocleC2×C42×C17C23
FrattiniC2×C42C22×D17
FittingC2×C17×#487C2
socleC23×C17#120
prime radicalC22×D68D4
hypercenterC2×#487D17
hypocenterC17#10058
LCSG ▷ C2×C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#487

Complements

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

NH ≅ G/Ntype
3. #14529733C2semidirect
4. C2×#11762555C2semidirect
6. #14526624C2semidirect
7. C2×C17×#487C2semidirect
172. C2×#487D17semidirect
271. C17#10058semidirect

Maximal subgroups

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

subgroupindex
C2×#117630962
#145337802
#145297332
C2×#117625552
C2×#117625382
#145266242
C2×C17×#4872
#1005817