Finite Group Databasebeta

← #16938148

Subgroups — #16938148

Order 4352.

506 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763011C211yesmaximal, characteristic
2#14529578C211yesmaximal, characteristic
3#14528781C211yesmaximal, characteristic
4C2×#11762560C211yesmaximal, characteristic, complemented
5C2×#11762555C211yesmaximal, characteristic, complemented
6#14526667C211yesmaximal, characteristic, complemented
7C17×#1990C211yesmaximal, characteristic, complemented
8C22×#63230—21no—
9C2×C4×D68C2211yescharacteristic
10#11763011—22no—
11#11763011—22no—
12C2×C4⋊#3622C2211yescharacteristic
13C2×C4⋊#3622C2211yescharacteristic
14C2×C4⋊#3622C2211yescharacteristic
15C2×#10587506—21no—
16C2×#10587480—21no—
17C2×#10587491—21no—
18C2×#10587491—21no—
19#11762571—21no—
20#11762560—22no—
21#11762555—22no—
22C2×C17×C4⋊Q8C2211yescharacteristic
23C2×C4×C8×C17C2211yescharacteristic
24C17×#384C2211yescharacteristic
25C17×#340—21no—
26C17×#340—21no—
27C22×Q8×C17—21no—
28C22×D68D411yescharacteristic
29C22×C4×D17—21no—
30C22×C8×C17D411yescharacteristic
31C22×C8×C17D411yescharacteristic
32C17×C4⋊Q8—21no—
33C17×C4⋊Q8—21no—
34C17×C4⋊Q8—21no—
35C17×C4⋊Q8D412yes—
36C2×C4⋊C4×C17—21no—
37C2×C4⋊C4×C17—21no—
38C2×C4⋊C4×C17—21no—
39C2×C4⋊C4×C17—21no—
40C2×C4⋊C4×C17—21no—
41C2×C42×C17C2311yescharacteristic
42C2×#63230—24no—
43C2×#63230—42no—
44C2×#63230—42no—
45C2×#63227—21no—
46C2×C4⋊Dic17D411yescharacteristic
47C22×#3622—21no—
48C22×#3622—21no—
49C22×#3622—21no—
50C4×D68—21no—
51C4×D68—21no—
52C4×D68—21no—
53C4×D68—21no—
54C4×C8×C17—21no—
55C4×C8×C17—21no—
56#10587506—24no—
57#10587491—42no—
58#10587491—42no—
59#10587480—42no—
60C4⋊#3622—21no—
61C4⋊#3622—21no—
62C4⋊#3622—21no—
63C4⋊#3622—21no—
64C4⋊#3622—21no—
65C4⋊#3622D412yes—
66C23×D17—21no—
67C2×Q8×C17—22no—
68C2×Q8×C17—41no—
69C2×Q8×C17—41no—
70C2×Q8×C17—41no—
71C2×Q8×C17—41no—
72C2×Q8×C17—41no—
73C22×C4×C17—41no—
74C22×C4×C17—41no—
75C22×C4×C17C2×D411yescharacteristic
76C22×C4×C17C2×D411yescharacteristic
77C22×C4×C17#4111yescharacteristic
78C22×Dic17—21no—
79C2×D68—21no—
80C2×D68—21no—
81C2×D68—21no—
82C2×D68—21no—
83C2×D68—21no—
84C2×D68SD1612yes—
85C2×C4×D17—41no—
86C2×C4×D17—41no—
87C2×C4×D17—41no—
88C2×C4×D17—41no—
89C2×C4×D17—41no—
90C2×C4×D17—41no—
91C2×C8×C17—21no—
92C2×C8×C17—21no—
93C2×C8×C17—21no—
94C2×C8×C17—21no—
95C2×C8×C17—21no—
96C2×C8×C17SD1612yescomplemented
97C4⋊C4×C17—22no—
98C4⋊C4×C17—22no—
99C4⋊C4×C17—41no—
100C4⋊C4×C17—41no—
101C4⋊C4×C17—41no—
102C4⋊C4×C17—41no—
103C4⋊C4×C17—41no—
104C4⋊C4×C17—41no—
105C4⋊C4×C17—81no—
106C42×C17C2×D411yescharacteristic
107C42×C17C2×D411yescharacteristic
108C42×C17#4111yescharacteristic
109C42×C17#4111yescharacteristic
110#63230—48no—
111#63227—41no—
112#63227—41no—
113C4⋊Dic17—21no—
114C4⋊Dic17—21no—
115C2×#3622—22no—
116C2×#3622—41no—
117C2×#3622—41no—
118C2×#3622—41no—
119C2×#3622—41no—
120C2×#3622—41no—
121C2×#3622—41no—
122C2×#3622—41no—
123C2×#3622—41no—
124#10477—171nomaximal, Sylow
125C23×C17#12111yescharacteristic
126C22×D17—22no—
127C22×D17—41no—
128C22×D17—41no—
129C22×D17—41no—
130C22×D17—41no—
131C22×D17—41no—
132C22×D17—41no—
133Q8×C17—42no—
134Q8×C17—42no—
135Q8×C17—81no—
136Q8×C17—81no—
137C2×C4×C17—21no—
138C2×C4×C17—21no—
139C2×C4×C17—42no—
140C2×C4×C17—81no—
141C2×C4×C17—81no—
142C2×C4×C17—81no—
143C2×C4×C17—81no—
144C2×C4×C17—81no—
145C2×C4×C17#12011yescharacteristic
146C2×C4×C17#12111yescharacteristic
147C2×C4×C17#12111yescharacteristic
148C2×C4×C17#12311yescharacteristic
149C2×C4×C17#12311yescharacteristic
150C2×C4×C17#12411yescharacteristic
151C2×C4×C17C2×SD1611yescharacteristic
152C2×C4×C17C2×SD1611yescharacteristic
153C2×C4×C17#13511yescharacteristic
154C2×C4×C17#13511yescharacteristic
155C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
156C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
157C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
158C2×C4×C17#13711yescharacteristic
159C2×Dic17—41no—
160C2×Dic17—41no—
161C2×Dic17—41no—
162D68—22no—
163D68—22no—
164D68—41no—
165D68—41no—
166C4×D17—81no—
167C4×D17—81no—
168C4×D17—81no—
169C4×D17—81no—
170C8×C17—22no—
171C8×C17—41no—
172#3622—42no—
173#3622—81no—
174#3622—81no—
175C2×#461—171no—
176#1990D1711yescharacteristic, complemented
177#1954—171no—
178#1802—171no—
179C2×#334—171no—
180C2×#332—171no—
181#1296—171no—
182C22×C17#39411yescharacteristic
183C22×C17#46111yescharacteristic
184C22×C17#46311yescharacteristic
185C22×C17#46511yescharacteristic
186C22×C17#46911yescharacteristic
187C22×C17#47111yescharacteristic
188C22×C17#47311yescharacteristic
189C2×D17—42no—
190C2×D17—42no—
191C2×D17—42no—
192C2×D17—81no—
193C2×D17—81no—
194C2×D17—81no—
195C2×D17—81no—
196C4×C17—21no—
197C4×C17—21no—
198C4×C17—82no—
199C4×C17—161no—
200C4×C17#44911yescharacteristic
201C4×C17#45011yescharacteristic
202C4×C17#46011yescharacteristic
203C4×C17#46411yescharacteristic
204C4×C17#46511yescharacteristic
205C4×C17#46811yescharacteristic
206C4×C17#48111yescharacteristic
207C4×C17#48211yescharacteristic
208Dic17—81no—
209C22×SD16—341no—
210C2×C4⋊Q8C2×D1711yescharacteristic
211C2×C4×D4—171no—
212#461—342no—
213#461—342no—
214C2×C4⋊C8—171no—
215C2×C4⋊C8—171no—
216C2×C4⋊C8—171no—
217C2×#103—341no—
218C2×#102—341no—
219C2×C22⋊C8—341no—
220C2×C22⋊C8—341no—
221C2×C4×C8C2×D1711yescharacteristic
222#384C2×D1711yescharacteristic
223#340—21no—
224#340—21no—
225#340—341no—
226#334—342no—
227#332—342no—
228C2×C17#167111yescharacteristic
229C2×C17#167911yescharacteristic
230C2×C17#169311yescharacteristic
231C2×C17#170111yescharacteristic
232C2×C17#203411yescharacteristic
233C2×C17#205211yescharacteristic
234C2×C17#205511yescharacteristic
235D17—82no—
236C22×Q8—21no—
237C22×D4—171no—
238C23×C4—341no—
239C2×SD16—344no—
240C2×SD16—682no—
241C2×SD16—682no—
242C22×C8—341no—
243C22×C8—341no—
244C22×C8—341no—
245C22×C8C22⋊D1711yescharacteristic
246C22×C8C22⋊D1711yescharacteristic
247C4⋊Q8—21no—
248C4⋊Q8—21no—
249C4⋊Q8—21no—
250C4⋊Q8D6812yes—
251C4×D4—341no—
252C4×D4—341no—
253C4×D4—341no—
254C4×D4—341no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—171no—
261C2×C22⋊C4—341no—
262C2×C42C22×D1711yescharacteristic
263C4⋊C8—172no—
264C4⋊C8—341no—
265C4⋊C8—341no—
266C4⋊C8—341no—
267C4⋊C8—341no—
268C4⋊C8—341no—
269#103—344no—
270#102—682no—
271C22⋊C8—682no—
272C22⋊C8—682no—
273C4×C8—21no—
274C4×C8—21no—
275C17#1047711yescharacteristic, complemented, Sylow
276C24—341no—
277C2×Q8—22no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×Q8—41no—
283C2×D4—172no—
284C2×D4—341no—
285C2×D4—341no—
286C2×D4—341no—
287C2×D4—341no—
288C2×D4—341no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—341no—
292C22×C4—681no—
293C22×C4—681no—
294C22×C4—681no—
295C22×C4—681no—
296C22×C4—681no—
297C22×C4—681no—
298C22×C4D4×D1711yescharacteristic
299C22×C4#6325411yescharacteristic
300C22×C4C2×C22⋊D1711yescharacteristic
301SD16—688no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—342no—
308C2×C8—681no—
309C2×C8—681no—
310C2×C8—681no—
311C2×C8—681no—
312C2×C8—681no—
313C2×C8—681no—
314C2×C8—681no—
315C2×C8—681no—
316C2×C8#6323012yescomplemented
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—81no—
326C4⋊C4—341no—
327C4⋊C4—341no—
328C22⋊C4—681no—
329C22⋊C4—681no—
330C42C2×D6811yescharacteristic
331C42#6325211yescharacteristic
332C42D4×D1711yescharacteristic
333C42#6325411yescharacteristic
334C23—342no—
335C23—681no—
336C23—681no—
337C23—681no—
338C23—681no—
339C23—681no—
340C23—681no—
341C23#1058762411yescharacteristic
342Q8—42no—
343Q8—42no—
344Q8—81no—
345Q8—81no—
346D4—342no—
347D4—342no—
348D4—681no—
349D4—681no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—42no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—681no—
359C2×C4—681no—
360C2×C4—681no—
361C2×C4—1361no—
362C2×C4—1361no—
363C2×C4—1361no—
364C2×C4—1361no—
365C2×C4#1058756811yescharacteristic
366C2×C4#1058756911yescharacteristic
367C2×C4#1058757111yescharacteristic
368C2×C4#1058757211yescharacteristic
369C2×C4#1058759711yescharacteristic
370C2×C4#1058759711yescharacteristic
371C2×C4SD16×D1711yescharacteristic
372C2×C4#1058760211yescharacteristic
373C2×C4#1058761611yescharacteristic
374C2×C4#1058762511yescharacteristic
375C2×C4C2×#6323011yescharacteristic
376C2×C4#1058762811yescharacteristic
377C2×C4#1058763511yescharacteristic
378C2×C4#1058763611yescharacteristic
379C8—22no—
380C8—41no—
381C8—682no—
382C8—1361no—
383C8—1361no—
384C22—682no—
385C22—682no—
386C22—682no—
387C22—1361no—
388C22—1361no—
389C22—1361no—
390C22—1361no—
391C22#1176272511yescharacteristic
392C22#1176301111yescharacteristic
393C22#1176302111yescharacteristic
394C22#1176303511yescharacteristic
395C22#1176305911yescharacteristic
396C22#1176306911yescharacteristic
397C22#1176308111yescharacteristic
398C4—21no—
399C4—21no—
400C4—82no—
401C4—161no—
402C4—1361no—
403C4#1176295511yescharacteristic
404C4#1176296111yescharacteristic
405C4#1176300911yescharacteristic
406C4#1176303111yescharacteristic
407C4#1176304011yescharacteristic
408C4#1176305711yescharacteristic
409C4#1176312611yescharacteristic
410C4#1176313211yescharacteristic
411C2—1362no—
412C2#1452815611yescharacteristic
413C2#1452820411yescharacteristic
414C2#1452829411yescharacteristic
415C2#1452834211yescharacteristic
416C2#1452997211yescharacteristic
417C2#1453008411yescharacteristic
418C2#1453009911yescharacteristic
subgroupG/H
centerC23#10587624
commutator, cosocleC2×C42×C17C23
FrattiniC2×C42C22×D17
FittingC17×#1990C2
socleC23×C17#121
prime radicalC22×D68D4
hypercenter#1990D17
hypocenterC17#10477
LCSG ▷ C2×C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1990

Complements

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

NH ≅ G/Ntype
4. C2×#11762560C2semidirect
5. C2×#11762555C2semidirect
6. #14526667C2semidirect
7. C17×#1990C2semidirect
96. C2×C8×C17SD16semidirect
176. #1990D17semidirect
275. C17#10477semidirect
316. C2×C8#63230semidirect

Maximal subgroups

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

subgroupindex
C2×#117630112
#145295782
#145287812
C2×#117625602
C2×#117625552
#145266672
C17×#19902
#1047717