Finite Group Databasebeta

← #16930199

Subgroups — #16930199

Order 4352.

432 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14538593C211yesmaximal, characteristic
2#14537449C211yesmaximal, characteristic
3#14527469C211yesmaximal, characteristic, complemented
4#14527169C211yesmaximal, characteristic
5#14526681C211yesmaximal, characteristic, complemented
6#14526659C211yesmaximal, characteristic, complemented
7C2×C17×#333C211yesmaximal, characteristic, complemented
8C4×#63252C2211yescharacteristic
9#11763101—21no—
10#11763096—21no—
11#11763091C412yes—
12#11763086C412yes—
13#11762869—22no—
14#11762865—21no—
15#11762862C2211yescharacteristic
16#11762861—21no—
17#11762849—21no—
18#11762833C2211yescharacteristic
19#11762833C2211yescharacteristic
20#11762808—21no—
21#11762775—21no—
22#11762764—21no—
23#11762555—22no—
24#11762539—21no—
25#11762536—21no—
26C2×C17×C4⋊Q8C2211yescharacteristic
27C2×C17×C4⋊C8C2211yescharacteristic
28C2×C17×C4⋊C8C2211yescharacteristic
29C17×#333—21no—
30C17×#333—21no—
31C22×Q8×C17—21no—
32C2×#63252D411yescharacteristic
33C22×C8×C17—21no—
34C22×C8×C17—21no—
35C17×C4⋊Q8—21no—
36C17×C4⋊Q8—21no—
37C17×C4⋊Q8—21no—
38C17×C4⋊Q8C2×C412yes—
39C2×C4⋊C4×C17—21no—
40C2×C4⋊C4×C17—21no—
41C2×C42×C17C2311yescharacteristic
42C4×C22⋊D17—21no—
43C4×C22⋊D17—21no—
44#10587612D411yescharacteristic
45C2×#63223—21no—
46C2×#63223—21no—
47C4×D68C2×C411yescharacteristic
48C4×D68D411yescharacteristic
49#10587558—21no—
50C42×D17—21no—
51C4×Dic34C2×C411yescharacteristic
52C4×Dic34D411yescharacteristic
53C17×C4⋊C8—21no—
54C17×C4⋊C8—21no—
55C17×C4⋊C8—21no—
56C17×C4⋊C8D411yescharacteristic
57C17×C4⋊C8D411yescharacteristic
58C22⋊#3622—21no—
59#10587491—22no—
60#10587491—22no—
61#10587486—22no—
62C8×Dic17—22no—
63#10587481—22no—
64#10587480—22no—
65#10587479—22no—
66#10587478—22no—
67C4⋊#3622—21no—
68C4⋊#3622—21no—
69C4⋊#3622—21no—
70C4⋊#3622C2×C412yes—
71C4⋊#3622D412yes—
72C4×#3622—21no—
73C2×Q8×C17—22no—
74C2×Q8×C17—41no—
75C2×Q8×C17—82no—
76C22×C4×C17—41no—
77C22×C4×C17C2×D411yescharacteristic
78C22×C4×C17#4111yescharacteristic
79C22×C4×C17#4111yescharacteristic
80C2×C22⋊D17—21no—
81#63252—21no—
82#63252—21no—
83#63252—21no—
84#63252—21no—
85C2×D68C22⋊C411yescharacteristic
86C2×C4×D17—21no—
87C2×C4×D17—21no—
88C2×C4×D17—21no—
89C2×Dic34C22⋊C411yescharacteristic
90C2×C8×C17—21no—
91C2×C8×C17—21no—
92C2×C8×C17—41no—
93C2×C8×C17—41no—
94C2×C8×C17—41no—
95C2×C8×C17—81no—
96C4⋊C4×C17—22no—
97C4⋊C4×C17—22no—
98C4⋊C4×C17—41no—
99C4⋊C4×C17—41no—
100C42×C17C22×C411yescharacteristic
101C42×C17C2×D411yescharacteristic
102C42×C17C2×D411yescharacteristic
103C42×C17C2×D411yescharacteristic
104C22⋊Dic17—21no—
105#63227—21no—
106#63227—21no—
107C4⋊Dic17C22⋊C411yescharacteristic
108C4⋊Dic17C22⋊C411yescharacteristic
109#63225—21no—
110#63225—21no—
111C4×Dic17—21no—
112C4×Dic17—21no—
113C4×Dic17—21no—
114#63223—41no—
115#63223—41no—
116#63223—41no—
117#63223—41no—
118C2×#3622—22no—
119C2×#3622—22no—
120C2×#3622—41no—
121#9170—171nomaximal, Sylow
122C23×C17#12311yescharacteristic
123C22×D17—21no—
124Q8×C17—82no—
125Q8×C17—82no—
126C2×C4×C17—21no—
127C2×C4×C17—21no—
128C2×C4×C17—41no—
129C2×C4×C17—42no—
130C2×C4×C17—82no—
131C2×C4×C17C2×C22⋊C411yescharacteristic
132C2×C4×C17C4×D411yescharacteristic
133C2×C4×C17C4×D411yescharacteristic
134C2×C4×C17#12011yescharacteristic
135C2×C4×C17#12111yescharacteristic
136C2×C4×C17#12111yescharacteristic
137C2×C4×C17#13511yescharacteristic
138C2×C4×C17#13511yescharacteristic
139C2×C4×C17#13711yescharacteristic
140C2×C4×C17#13711yescharacteristic
141C22⋊D17—41no—
142C22⋊D17—41no—
143C2×Dic17—21no—
144C2×Dic17—21no—
145C2×Dic17—21no—
146D68—21no—
147D68—21no—
148C4×D17—41no—
149C4×D17—41no—
150C4×D17—41no—
151C4×D17—41no—
152C4×D17—81no—
153Dic34—21no—
154Dic34—21no—
155C8×C17—41no—
156C8×C17—41no—
157C8×C17—81no—
158#3622—42no—
159#3622—42no—
160#3622—81no—
161#3079—171no—
162#2925—171no—
163#1549—171no—
164C2×#333D1711yescharacteristic, complemented
165#1477—171no—
166#1301—171no—
167#1292—171no—
168C22×C17—21no—
169C22×C17—21no—
170C22×C17#38611yescharacteristic
171C22×C17#48511yescharacteristic
172C22×C17#48511yescharacteristic
173C2×D17—41no—
174C2×D17—41no—
175C2×D17—41no—
176C4×C17—21no—
177C4×C17—21no—
178C4×C17—21no—
179C4×C17—82no—
180C4×C17#41611yescharacteristic
181C4×C17#41911yescharacteristic
182C4×C17#45211yescharacteristic
183C4×C17#45211yescharacteristic
184C4×C17#47111yescharacteristic
185C4×C17#47311yescharacteristic
186Dic17—41no—
187Dic17—41no—
188Dic17—41no—
189C2×C4⋊Q8C2×D1711yescharacteristic
190C4×#41—171no—
191#477—341no—
192#476—341no—
193#475—172no—
194#474—172no—
195#433—342no—
196#432—341no—
197#431—171no—
198#431—341no—
199#428—341no—
200#423—171no—
201#423—171no—
202C2×C4⋊C8C2×D1711yescharacteristic
203C2×C4⋊C8C2×D1711yescharacteristic
204#417—341no—
205#408—341no—
206#405—341no—
207#333—21no—
208#333—21no—
209#332—342no—
210Q8⋊C8—341no—
211D4⋊C8—341no—
212C2×C17—21no—
213C2×C17—21no—
214C2×C17#187311yescharacteristic
215C2×C17#190311yescharacteristic
216C2×C17#190811yescharacteristic
217D17—81no—
218C2×#41—171no—
219C22×Q8—21no—
220C2×M16—341no—
221C2×M16—341no—
222C22×C8—21no—
223C22×C8—21no—
224C4⋊Q8—21no—
225C4⋊Q8—21no—
226C4⋊Q8—21no—
227C4⋊Q8C4×D1712yes—
228C4×Q8—171no—
229C4×Q8—171no—
230C4×D4—171no—
231C4×D4—171no—
232C4×D4—341no—
233C4×D4—341no—
234#117—171no—
235#117—341no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C42—341no—
239C2×C42C22×D1711yescharacteristic
240#107—342no—
241#106—342no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—172no—
246C4⋊C8—172no—
247C4⋊C8—341no—
248C4⋊C8—341no—
249C4⋊C8—341no—
250C4⋊C8C22⋊D1711yescharacteristic
251C4⋊C8C22⋊D1711yescharacteristic
252#103—342no—
253#102—342no—
254C22⋊C8—342no—
255C22⋊C8—342no—
256C22⋊C8—341no—
257#97—342no—
258C4×C8—342no—
259C4×C8—341no—
260C17#917011yescharacteristic, complemented, Sylow
261#41—341no—
262#41—341no—
263#41—341no—
264#41—341no—
265C2×Q8—22no—
266C2×Q8—41no—
267C2×Q8—82no—
268C2×Q8—171no—
269C2×D4—171no—
270C2×D4—341no—
271C22×C4—41no—
272C22×C4—341no—
273C22×C4—341no—
274C22×C4—341no—
275C22×C4#6325211yescharacteristic
276C22×C4D4×D1711yescharacteristic
277C22×C4#6325411yescharacteristic
278M16—681no—
279M16—681no—
280M16—681no—
281M16—681no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—81no—
288C2×C8—342no—
289C2×C8—342no—
290C2×C8—681no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—171no—
296C4⋊C4—171no—
297C4⋊C4—341no—
298C4⋊C4—341no—
299C22⋊C4—341no—
300C22⋊C4—341no—
301C22⋊C4—341no—
302C42—341no—
303C42—341no—
304C42—341no—
305C42C2×C4×D1711yescharacteristic
306C42D4×D1711yescharacteristic
307C42D4×D1711yescharacteristic
308C42C2×C22⋊D1711yescharacteristic
309C23—341no—
310C23#1058756911yescharacteristic
311Q8—82no—
312Q8—82no—
313Q8—341no—
314Q8—341no—
315D4—341no—
316D4—341no—
317D4—681no—
318D4—681no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—41no—
322C2×C4—42no—
323C2×C4—82no—
324C2×C4—341no—
325C2×C4—341no—
326C2×C4—341no—
327C2×C4—681no—
328C2×C4—681no—
329C2×C4—681no—
330C2×C4—681no—
331C2×C4—1361no—
332C2×C4C22⋊C4×D1711yescharacteristic
333C2×C4#1058756711yescharacteristic
334C2×C4#1058757011yescharacteristic
335C2×C4#1058760111yescharacteristic
336C2×C4#1058760211yescharacteristic
337C2×C4#1058760411yescharacteristic
338C2×C4#1058760511yescharacteristic
339C2×C4C4×C22⋊D1711yescharacteristic
340C2×C4#1058762311yescharacteristic
341C2×C4#1058762411yescharacteristic
342C8—41no—
343C8—41no—
344C8—81no—
345C8—682no—
346C8—682no—
347C8—1361no—
348C22—21no—
349C22—21no—
350C22—681no—
351C22—681no—
352C22—681no—
353C22#1176268511yescharacteristic
354C22#1176314811yescharacteristic
355C22#1176314911yescharacteristic
356C4—21no—
357C4—21no—
358C4—21no—
359C4—82no—
360C4—681no—
361C4—681no—
362C4—681no—
363C4#1176280411yescharacteristic
364C4#1176281611yescharacteristic
365C4#1176296811yescharacteristic
366C4#1176297011yescharacteristic
367C4#1176306911yescharacteristic
368C4#1176308111yescharacteristic
369C2—21no—
370C2—21no—
371C2—1361no—
372C2#1452909311yescharacteristic
373C2#1452926711yescharacteristic
374C2#1452930211yescharacteristic
subgroupG/H
centerC22#11762685
commutatorC42×C17C22×C4
FrattiniC2×C42C22×D17
FittingC2×C17×#333C2
socleC22×C17#386
prime radicalC2×#63252D4
cosocleC2×C42×C17C23
hypercenterC2×#333D17
hypocenterC17#9170
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#333

Complements

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

NH ≅ G/Ntype
3. #14527469C2semidirect
5. #14526681C2semidirect
6. #14526659C2semidirect
7. C2×C17×#333C2semidirect
164. C2×#333D17semidirect
260. C17#9170semidirect

Maximal subgroups

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

subgroupindex
#145385932
#145374492
#145274692
#145271692
#145266812
#145266592
C2×C17×#3332
#917017