Finite Group Databasebeta

← #16936923

Subgroups — #16936923

Order 4352.

562 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763096C211yesmaximal, characteristic
2C2×#11763086C211yesmaximal, characteristic
3#14529578C211yesmaximal, characteristic
4C2×#11762558C211yesmaximal, characteristic, complemented
5C2×#11762555C211yesmaximal, characteristic, complemented
6#14526619C211yesmaximal, characteristic, complemented
7C2×C17×#487C211yesmaximal, characteristic, complemented
8C2×C4×D68C2211yescharacteristic
9#11763096—22no—
10#11763096—22no—
11#11763086—22no—
12#11763086—22no—
13C2×#10587478—21no—
14C2×#10587479—21no—
15C2×C4⋊#3622C2211yescharacteristic
16C2×C4⋊#3622C2211yescharacteristic
17C2×C4⋊#3622C2211yescharacteristic
18C2×#10587480—21no—
19C2×#10587480—21no—
20C2×#10587491—22no—
21#11762558—22no—
22#11762555—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×C17D412yes—
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⋊Q8D412yes—
41C17×C4⋊Q8D412yes—
42C2×C4⋊C4×C17—21no—
43C2×C4⋊C4×C17—21no—
44C2×C4⋊C4×C17—21no—
45C2×C4⋊C4×C17—21no—
46C2×C42×C17C2311yescharacteristic
47C2×#63227—21no—
48C2×C4⋊Dic17D411yescharacteristic
49C22×#3622—21no—
50C22×#3622—21no—
51C22×#3622—21no—
52C4×D68—21no—
53C4×D68—21no—
54C4×D68—21no—
55C4×D68—21no—
56C17×#103—28no—
57C4×C8×C17—21no—
58C4×C8×C17—21no—
59#10587491—44no—
60#10587480—24no—
61#10587480—24no—
62#10587479—24no—
63#10587478—24no—
64C4⋊#3622—21no—
65C4⋊#3622—21no—
66C4⋊#3622—21no—
67C4⋊#3622—21no—
68C4⋊#3622D412yes—
69C4⋊#3622D412yes—
70C23×D17—21no—
71C2×Q8×C17—22no—
72C2×Q8×C17—22no—
73C2×Q8×C17—41no—
74C2×Q8×C17—41no—
75C2×Q8×C17—42no—
76C2×Q8×C17—42no—
77C2×Q8×C17—82no—
78C22×C4×C17—41no—
79C22×C4×C17—41no—
80C22×C4×C17C2×D411yescharacteristic
81C22×C4×C17C2×D411yescharacteristic
82C22×C4×C17#4111yescharacteristic
83C22×Dic17—21no—
84C2×D68—21no—
85C2×D68—21no—
86C2×D68—21no—
87C2×D68—21no—
88C2×D68—21no—
89C2×D68Q1612yes—
90C2×C4×D17—41no—
91C2×C4×D17—41no—
92C2×C4×D17—42no—
93C2×C4×D17—42no—
94C2×C8×C17—22no—
95C2×C8×C17—22no—
96C2×C8×C17—22no—
97C4⋊C4×C17—22no—
98C4⋊C4×C17—22no—
99C4⋊C4×C17—22no—
100C4⋊C4×C17—22no—
101C4⋊C4×C17—41no—
102C4⋊C4×C17—41no—
103C4⋊C4×C17—41no—
104C4⋊C4×C17—41no—
105C42×C17C2×D411yescharacteristic
106C42×C17C2×D411yescharacteristic
107C42×C17C2×D411yescharacteristic
108C42×C17C2×D411yescharacteristic
109#63227—41no—
110#63227—41no—
111C4⋊Dic17—21no—
112C4⋊Dic17—21no—
113C2×#3622—22no—
114C2×#3622—22no—
115C2×#3622—41no—
116C2×#3622—41no—
117C2×#3622—41no—
118C2×#3622—41no—
119C2×#3622—41no—
120C2×#3622—42no—
121#10275—171nomaximal, Sylow
122C23×C17#12111yescharacteristic
123C22×D17—22no—
124C22×D17—41no—
125C22×D17—42no—
126C22×D17—41no—
127C22×D17—42no—
128Q8×C17—44no—
129Q8×C17—82no—
130Q8×C17—82no—
131Q8×C17—81no—
132Q8×C17—81no—
133C2×C4×C17—22no—
134C2×C4×C17—42no—
135C2×C4×C17—42no—
136C2×C4×C17—82no—
137C2×C4×C17—82no—
138C2×C4×C17#12011yescharacteristic
139C2×C4×C17#12011yescharacteristic
140C2×C4×C17#12011yescharacteristic
141C2×C4×C17#12111yescharacteristic
142C2×C4×C17#12111yescharacteristic
143C2×C4×C17#12711yescharacteristic
144C2×C4×C17C2×Q1611yescharacteristic
145C2×C4×C17#13511yescharacteristic
146C2×C4×C17#13512yes—
147C2×C4×C17AGL(1,ℤ/8)12yes—
148C2×C4×C17#13711yescharacteristic
149C2×C4×C17#13711yescharacteristic
150C2×Dic17—41no—
151C2×Dic17—42no—
152D68—22no—
153D68—22no—
154D68—41no—
155D68—41no—
156C4×D17—82no—
157C4×D17—81no—
158C4×D17—81no—
159C8×C17—42no—
160#3622—42no—
161#3622—42no—
162#3622—81no—
163C2×#487D1711yescharacteristic, complemented
164C2×#476—171no—
165C2×#474—171no—
166#1954—171no—
167C2×#333—171no—
168C2×#332—171no—
169#1277—171no—
170C22×C17#39211yescharacteristic
171C22×C17#46312yes—
172C22×C17#46711yescharacteristic
173C22×C17#47011yescharacteristic
174C22×C17#47112yes—
175C2×D17—44no—
176C2×D17—42no—
177C2×D17—82no—
178C2×D17—81no—
179C2×D17—81no—
180C4×C17—21no—
181C4×C17—21no—
182C4×C17—82no—
183C4×C17—82no—
184C4×C17#44912yes—
185C4×C17#45111yescharacteristic
186C4×C17#45111yescharacteristic
187C4×C17#45211yescharacteristic
188C4×C17#45211yescharacteristic
189C4×C17#49511yescharacteristic
190C4×C17#49611yescharacteristic
191Dic17—81no—
192C2×C4⋊Q8C2×D1711yescharacteristic
193C2×C4⋊Q8C2×D1711yescharacteristic
194C2×C4×D4—171no—
195#487—22no—
196#487—22no—
197#476—342no—
198#476—342no—
199#474—342no—
200#474—342no—
201C2×#107—341no—
202C2×#106—341no—
203C2×C4⋊C8—171no—
204C2×C4⋊C8—171no—
205C2×C4⋊C8—171no—
206C2×#103—22no—
207C2×#102—341no—
208C2×#102—341no—
209C2×C22⋊C8—342no—
210C2×C4×C8C2×D1711yescharacteristic
211#333—342no—
212#332—342no—
213C2×C17#168312yes—
214C2×C17#170312yes—
215C2×C17#200111yescharacteristic
216C2×C17#201211yescharacteristic
217C2×C17#201911yescharacteristic
218D17—82no—
219C22×Q8—21no—
220C22×Q8—21no—
221C22×D4—171no—
222C23×C4—341no—
223C22×C8—341no—
224C22×C8—341no—
225C22×C8—341no—
226C22×C8C22⋊D1712yes—
227C4⋊Q8—21no—
228C4⋊Q8—21no—
229C4⋊Q8—21no—
230C4⋊Q8—21no—
231C4⋊Q8—21no—
232C4⋊Q8—21no—
233C4⋊Q8C22⋊D1712yes—
234C4⋊Q8C22⋊D1712yes—
235C4×D4—341no—
236C4×D4—341no—
237C4×D4—341no—
238C4×D4—341no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—171no—
244C2×C22⋊C4—341no—
245C2×C42C22×D1711yescharacteristic
246#107—344no—
247#106—344no—
248C4⋊C8—172no—
249C4⋊C8—172no—
250C4⋊C8—341no—
251C4⋊C8—341no—
252C4⋊C8—341no—
253C4⋊C8—341no—
254#103—28no—
255#102—344no—
256#102—344no—
257C22⋊C8—684no—
258C4×C8—21no—
259C4×C8—21no—
260C17#1027511yescharacteristic, complemented, Sylow
261C24—341no—
262C2×Q8—22no—
263C2×Q8—22no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×Q8—42no—
267C2×Q8—42no—
268C2×Q8—82no—
269C2×D4—172no—
270C2×D4—341no—
271C2×D4—341no—
272C2×D4—341no—
273C2×D4—341no—
274C2×D4—341no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—341no—
278C22×C4—681no—
279C22×C4—682no—
280C22×C4—682no—
281C22×C4—681no—
282C22×C4D4×D1711yescharacteristic
283C22×C4#6325411yescharacteristic
284C22×C4C2×C22⋊D1711yescharacteristic
285C2×C8—22no—
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—342no—
289C2×C8—342no—
290C2×C8—681no—
291C2×C8—681no—
292C2×C8—681no—
293C2×C8—681no—
294C2×C8—681no—
295C2×C8—682no—
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—341no—
305C4⋊C4—341no—
306C22⋊C4—681no—
307C22⋊C4—681no—
308C42D4×D1711yescharacteristic
309C42D4×D1711yescharacteristic
310C42C2×C22⋊D1711yescharacteristic
311C42C2×C22⋊D1711yescharacteristic
312C23—342no—
313C23—681no—
314C23—682no—
315C23—681no—
316C23—682no—
317C23#1058762411yescharacteristic
318Q8—44no—
319Q8—82no—
320Q8—82no—
321Q8—81no—
322Q8—81no—
323D4—342no—
324D4—342no—
325D4—681no—
326D4—681no—
327C2×C4—22no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—82no—
331C2×C4—82no—
332C2×C4—681no—
333C2×C4—682no—
334C2×C4—1362no—
335C2×C4—1361no—
336C2×C4—1361no—
337C2×C4Q16×D1711yescharacteristic
338C2×C4#1058760411yescharacteristic
339C2×C4#1058760411yescharacteristic
340C2×C4#1058760511yescharacteristic
341C2×C4#1058762311yescharacteristic
342C2×C4#1058762311yescharacteristic
343C2×C4#1058762511yescharacteristic
344C2×C4#1058762511yescharacteristic
345C2×C4#1058762611yescharacteristic
346C2×C4#1058763512yes—
347C2×C4#1058763612yes—
348C2×C4#1058763911yescharacteristic
349C8—42no—
350C8—682no—
351C8—682no—
352C8—1361no—
353C22—684no—
354C22—682no—
355C22—1361no—
356C22—1362no—
357C22—1361no—
358C22#1176271711yescharacteristic
359C22#1176302112yes—
360C22#1176304711yescharacteristic
361C22#1176306411yescharacteristic
362C22#1176306912yes—
363C4—21no—
364C4—21no—
365C4—82no—
366C4—82no—
367C4—1361no—
368C4#1176295412yes—
369C4#1176296311yescharacteristic
370C4#1176296311yescharacteristic
371C4#1176297011yescharacteristic
372C4#1176297011yescharacteristic
373C4#1176319111yescharacteristic
374C4#1176319911yescharacteristic
375C2—1362no—
376C2#1452823212yes—
377C2#1452835612yes—
378C2#1452979111yescharacteristic
379C2#1452986111yescharacteristic
380C2#1452989611yescharacteristic
subgroupG/H
centerC23#10587624
commutator, cosocleC2×C42×C17C23
FrattiniC2×C42C22×D17
FittingC2×C17×#487C2
socleC23×C17#121
prime radicalC22×D68D4
hypercenterC2×#487D17
hypocenterC17#10275
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
4. C2×#11762558C2semidirect
5. C2×#11762555C2semidirect
6. #14526619C2semidirect
7. C2×C17×#487C2semidirect
163. C2×#487D17semidirect
260. C17#10275semidirect

Maximal subgroups

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

subgroupindex
C2×#117630962
C2×#117630862
#145295782
C2×#117625582
C2×#117625552
#145266192
C2×C17×#4872
#1027517