Finite Group Databasebeta

← #16928586

Subgroups — #16928586

Order 4352.

530 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14531735C211yesmaximal, characteristic
2#14529540C211yesmaximal, characteristic, complemented
3#14528781C211yesmaximal, characteristic
4C2×#11762538C211yesmaximal, characteristic
5#14526667C211yesmaximal, characteristic, complemented
6#14526624C211yesmaximal, characteristic, complemented
7C17×#1834C211yesmaximal, characteristic, complemented
8C2×C4×D68C2211yescharacteristic
9C2×#10587578—21no—
10C2×#10587478—21no—
11C2×C4⋊#3622C2211yescharacteristic
12C2×C4⋊#3622C2211yescharacteristic
13C2×#10587480—21no—
14#11762710—21no—
15#11762699—22no—
16#11762666—21no—
17C4⋊C4×Dic17—21no—
18C4×C4⋊Dic17C2211yescharacteristic
19#11762571—21no—
20#11762571—21no—
21#11762538C414yescomplemented
22C2×C4×C8×C17C2211yescharacteristic
23C17×#384C2211yescharacteristic
24C17×#384C2211yescharacteristic
25C17×#340—22no—
26C22×D68D411yescharacteristic
27C22×C4×D17—21no—
28C22×C8×C17—21no—
29C2×C4⋊C4×C17—21no—
30C2×C4⋊C4×C17—21no—
31C2×C4⋊C4×C17—21no—
32C2×C4⋊C4×C17—21no—
33C2×C4⋊C4×C17—21no—
34C2×C4⋊C4×C17D412yes—
35C2×C42×C17C2311yescharacteristic
36C2×#63227—22no—
37C2×#63227—22no—
38C2×#63227—21no—
39C2×C4⋊Dic17D411yescharacteristic
40C2×C4⋊Dic17D411yescharacteristic
41C2×C4⋊Dic17D411yescharacteristic
42C2×C4×Dic17—22no—
43C2×C4×Dic17—21no—
44C22×#3622—21no—
45C22×#3622—21no—
46#10587578—42no—
47#10587578—42no—
48C4×D68—21no—
49C4×D68—21no—
50C4×D68C2×C412yes—
51C4×D68Q812yes—
52C4×C8×C17—21no—
53C4×C8×C17C2×C412yes—
54#10587502—22no—
55#10587502—21no—
56#10587480—42no—
57#10587478—42no—
58C4⋊#3622—21no—
59C4⋊#3622—21no—
60C4⋊#3622—21no—
61C4⋊#3622C2×C412yes—
62C23×D17—21no—
63C22×C4×C17—22no—
64C22×C4×C17—41no—
65C22×C4×C17C2×D411yescharacteristic
66C22×C4×C17C2×D411yescharacteristic
67C22×C4×C17C2×D411yescharacteristic
68C22×Dic17—21no—
69C22×Dic17—21no—
70C22×Dic17—21no—
71C2×D68—21no—
72C2×D68—21no—
73C2×D68—21no—
74C2×D68—21no—
75C2×D68—21no—
76C2×D68C4⋊C412yes—
77C2×C4×D17—22no—
78C2×C4×D17—22no—
79C2×C4×D17—41no—
80C2×C4×D17—41no—
81C2×C4×D17—41no—
82C2×C4×D17—41no—
83C2×C8×C17—22no—
84C2×C8×C17—41no—
85C2×C8×C17—41no—
86C4⋊C4×C17—42no—
87C4⋊C4×C17—42no—
88C4⋊C4×C17—41no—
89C4⋊C4×C17—41no—
90C4⋊C4×C17—41no—
91C4⋊C4×C17—41no—
92C4⋊C4×C17—41no—
93C4⋊C4×C17—41no—
94C4⋊C4×C17—41no—
95C4⋊C4×C17—41no—
96C42×C17C22×C411yescharacteristic
97C42×C17C2×Q811yescharacteristic
98C42×C17#4111yescharacteristic
99C42×C17#4111yescharacteristic
100#63227—22no—
101#63227—22no—
102#63227—44no—
103#63227—82no—
104C4⋊Dic17—21no—
105C4⋊Dic17—21no—
106C4⋊Dic17—21no—
107C4⋊Dic17—21no—
108C4⋊Dic17C4⋊C412yes—
109C4⋊Dic17D812yes—
110C4×Dic17—44no—
111C4×Dic17—41no—
112C4×Dic17—41no—
113C2×#3622—22no—
114C2×#3622—41no—
115C2×#3622—41no—
116C2×#3622—41no—
117C2×#3622—41no—
118C2×#3622—41no—
119#8897—171nomaximal, Sylow
120C23×C17#12011yescharacteristic
121C22×D17—22no—
122C22×D17—41no—
123C22×D17—41no—
124C22×D17—41no—
125C22×D17—41no—
126C22×D17—41no—
127C22×D17—41no—
128C2×C4×C17—21no—
129C2×C4×C17—21no—
130C2×C4×C17—42no—
131C2×C4×C17—42no—
132C2×C4×C17—42no—
133C2×C4×C17—81no—
134C2×C4×C17—81no—
135C2×C4×C17—81no—
136C2×C4×C17C2×C4⋊C411yescharacteristic
137C2×C4×C17C4×D411yescharacteristic
138C2×C4×C17C4×D411yescharacteristic
139C2×C4×C17C22⋊Q811yescharacteristic
140C2×C4×C17C22⋊Q811yescharacteristic
141C2×C4×C17#12311yescharacteristic
142C2×C4×C17C2×D811yescharacteristic
143C2×C4×C17#13511yescharacteristic
144C2×C4×C17#13511yescharacteristic
145C2×C4×C17#13511yescharacteristic
146C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
147C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
148C2×C4×C17#13711yescharacteristic
149C2×C4×C17#13711yescharacteristic
150C2×Dic17—22no—
151C2×Dic17—22no—
152C2×Dic17—41no—
153C2×Dic17—41no—
154C2×Dic17—41no—
155C2×Dic17—41no—
156C2×Dic17—41no—
157C2×Dic17—41no—
158C2×Dic17—41no—
159D68—22no—
160D68—22no—
161D68—41no—
162D68—41no—
163C4×D17—42no—
164C4×D17—42no—
165C4×D17—42no—
166C4×D17—42no—
167C8×C17—42no—
168#3622—42no—
169#3622—81no—
170#2354—171no—
171#1947—171no—
172#1834D1711yescharacteristic, complemented
173#1802—171no—
174C2×D4⋊C8—171no—
175#1296—171no—
176#1279—171no—
177C22×C17#38511yescharacteristic
178C22×C17#44711yescharacteristic
179C22×C17#44911yescharacteristic
180C22×C17#45211yescharacteristic
181C22×C17#45211yescharacteristic
182C22×C17#45411yescharacteristic
183C22×C17#45511yescharacteristic
184C2×D17—42no—
185C2×D17—42no—
186C2×D17—42no—
187C2×D17—81no—
188C2×D17—81no—
189C2×D17—81no—
190C2×D17—81no—
191C4×C17—21no—
192C4×C17—21no—
193C4×C17—82no—
194C4×C17—161no—
195C4×C17#42711yescharacteristic
196C4×C17#42811yescharacteristic
197C4×C17C4×D811yescharacteristic
198C4×C17#44011yescharacteristic
199C4×C17#47411yescharacteristic
200C4×C17#47811yescharacteristic
201C4×C17#48211yescharacteristic
202C4×C17#48511yescharacteristic
203Dic17—42no—
204Dic17—42no—
205Dic17—81no—
206C2×C4×D4—171no—
207C2×#117—341no—
208C2×#107—341no—
209C2×C4⋊C8—171no—
210C2×C4⋊C8—171no—
211C2×#102—341no—
212C2×C4×C8C2×D1711yescharacteristic
213#390—341no—
214#388—342no—
215#384C2×D1711yescharacteristic
216#384C2×D1711yescharacteristic
217#380—341no—
218C4×C4⋊C4—171no—
219C4×C4⋊C4—341no—
220#340—22no—
221#340—341no—
222#340—341no—
223D4⋊C8—174no—
224C2×C17#162511yescharacteristic
225C2×C17#163811yescharacteristic
226C2×C17#164111yescharacteristic
227C2×C17#164611yescharacteristic
228C2×C17#185411yescharacteristic
229C2×C17#186011yescharacteristic
230C2×C17#186211yescharacteristic
231D17—82no—
232C22×D4—171no—
233C23×C4—341no—
234C22×C8—21no—
235C22×C8—341no—
236C22×C8—341no—
237C4×D4—172no—
238C4×D4—172no—
239C4×D4—341no—
240C4×D4—341no—
241#117—682no—
242#117—682no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—171no—
249C2×C4⋊C4—171no—
250C2×C4⋊C4—171no—
251C2×C4⋊C4C22⋊D1712yes—
252C2×C22⋊C4—341no—
253C2×C22⋊C4—342no—
254C2×C22⋊C4—342no—
255C2×C42—342no—
256C2×C42—341no—
257C2×C42C22×D1711yescharacteristic
258#107—682no—
259C4⋊C8—172no—
260C4⋊C8—341no—
261C4⋊C8—341no—
262C4⋊C8—341no—
263#102—682no—
264C4×C8—21no—
265C4×C8C4×D1712yes—
266#95—342no—
267#95—341no—
268C17#889711yescharacteristic, complemented, Sylow
269C24—341no—
270C2×D4—172no—
271C2×D4—341no—
272C2×D4—341no—
273C2×D4—341no—
274C2×D4—341no—
275C2×D4—341no—
276C22×C4—22no—
277C22×C4—41no—
278C22×C4—341no—
279C22×C4—341no—
280C22×C4—342no—
281C22×C4—342no—
282C22×C4—341no—
283C22×C4—681no—
284C22×C4—681no—
285C22×C4—681no—
286C22×C4—681no—
287C22×C4D4×D1711yescharacteristic
288C22×C4D4×D1711yescharacteristic
289C22×C4C2×C22⋊D1711yescharacteristic
290C2×C8—22no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—342no—
294C2×C8—681no—
295C2×C8—681no—
296C2×C8—681no—
297C2×C8—681no—
298C2×C8—681no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—172no—
310C4⋊C4—172no—
311C4⋊C4—341no—
312C4⋊C4—341no—
313C4⋊C4—341no—
314C4⋊C4—341no—
315C22⋊C4—342no—
316C22⋊C4—342no—
317C22⋊C4—684no—
318C22⋊C4—1362no—
319C42—684no—
320C42—681no—
321C42—681no—
322C42C2×C4×D1711yescharacteristic
323C42#6325211yescharacteristic
324C42Q8×D1711yescharacteristic
325C42#6325611yescharacteristic
326C23—342no—
327C23—681no—
328C23—681no—
329C23—681no—
330C23—681no—
331C23—681no—
332C23—681no—
333C23#1058762311yescharacteristic
334D4—342no—
335D4—342no—
336D4—681no—
337D4—681no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—342no—
347C2×C4—342no—
348C2×C4—681no—
349C2×C4—681no—
350C2×C4—681no—
351C2×C4—681no—
352C2×C4—682no—
353C2×C4—682no—
354C2×C4—682no—
355C2×C4—682no—
356C2×C4—681no—
357C2×C4—681no—
358C2×C4—681no—
359C2×C4C4⋊C4×D1711yescharacteristic
360C2×C4#1058757911yescharacteristic
361C2×C4#1058758011yescharacteristic
362C2×C4#1058758211yescharacteristic
363C2×C4D8×D1711yescharacteristic
364C2×C4#1058759711yescharacteristic
365C2×C4#1058760011yescharacteristic
366C2×C4#1058760111yescharacteristic
367C2×C4#1058760211yescharacteristic
368C2×C4#1058760211yescharacteristic
369C2×C4#1058760411yescharacteristic
370C2×C4#1058760511yescharacteristic
371C2×C4C4×C22⋊D1711yescharacteristic
372C2×C4#1058763211yescharacteristic
373C8—42no—
374C8—682no—
375C8—1361no—
376C22—682no—
377C22—682no—
378C22—682no—
379C22—1361no—
380C22—1361no—
381C22—1361no—
382C22—1361no—
383C22#1176268011yescharacteristic
384C22#1176294111yescharacteristic
385C22#1176295311yescharacteristic
386C22#1176296811yescharacteristic
387C22#1176297011yescharacteristic
388C22#1176298111yescharacteristic
389C22#1176298611yescharacteristic
390C4—21no—
391C4—21no—
392C4—82no—
393C4—161no—
394C4—682no—
395C4—682no—
396C4—1361no—
397C4#1176284411yescharacteristic
398C4#1176284811yescharacteristic
399C4#1176289111yescharacteristic
400C4#1176290811yescharacteristic
401C4#1176308811yescharacteristic
402C4#1176311211yescharacteristic
403C4#1176313511yescharacteristic
404C4#1176315311yescharacteristic
405C2—1362no—
406C2#1452787011yescharacteristic
407C2#1452795311yescharacteristic
408C2#1452797411yescharacteristic
409C2#1452800911yescharacteristic
410C2#1452898211yescharacteristic
411C2#1452901611yescharacteristic
412C2#1452902811yescharacteristic
subgroupG/H
centerC23#10587623
commutatorC42×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#1834C2
socleC23×C17#120
prime radicalC22×D68D4
cosocleC2×C42×C17C23
hypercenter#1834D17
hypocenterC17#8897
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1834

Complements

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

NH ≅ G/Ntype
2. #14529540C2semidirect
5. #14526667C2semidirect
6. #14526624C2semidirect
7. C17×#1834C2semidirect
21. #11762538C4semidirect
172. #1834D17semidirect
268. C17#8897semidirect

Maximal subgroups

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

subgroupindex
#145317352
#145295402
#145287812
C2×#117625382
#145266672
#145266242
C17×#18342
#889717