Finite Group Databasebeta

← C2×#14528205 #17016607

Subgroups — C2×#14528205 #17016607

Order 4352.

758 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763159C211yesmaximal, characteristic, complemented
2C2×#11763014C211yesmaximal, characteristic
3C2×D4×#3622C211yesmaximal, characteristic
4#14528205C218yesmaximal, direct product factor
5C2×#11762559C211yesmaximal, characteristic, complemented
6C2×#11762556C211yesmaximal, characteristic, complemented
7C2×#11762555C211yesmaximal, characteristic, complemented
8C2×C17×#474C211yesmaximal, characteristic
9C22×#63229—21no—
10C23×#3622—21no—
11C2×#10587555C2211yescharacteristic
12#11763159—22no—
13#11763159C2214yescomplemented
14#11763014—22no—
15#11763014C2214yes—
16D4×#3622—22no—
17D4×#3622C2214yes—
18C2×C4⋊#3622C2211yescharacteristic
19C2×C4⋊#3622C2211yescharacteristic
20C2×#10587506—21no—
21C2×#10587506—21no—
22C2×#10587481—21no—
23C2×C22⋊#3622—21no—
24C2×C4×#3622C2211yescharacteristic
25#11762559C2214yescomplemented
26#11762556C2214yescomplemented
27#11762555C2214yescomplemented
28C2×C17×C4⋊Q8C2211yescharacteristic
29C2×C4×D4×C17C2211yescharacteristic
30C17×#474—22no—
31C17×#474C2214yes—
32C2×C17×#107—21no—
33C2×C17×C4⋊C8C2211yescharacteristic
34C2×C17×#102—21no—
35C22×Q8×C17—21no—
36C22×D4×C17D411yescharacteristic
37C23×C4×C17—21no—
38C22×Dic34—21no—
39C22×C8×C17—21no—
40C17×C4⋊Q8—21no—
41C17×C4⋊Q8—21no—
42C17×C4⋊Q8—21no—
43C17×C4⋊Q8C2312yes—
44C4×D4×C17—21no—
45C4×D4×C17—21no—
46C4×D4×C17D412yes—
47C4×D4×C17C2312yes—
48C2×C4⋊C4×C17—21no—
49C2×C4⋊C4×C17—21no—
50C2×C4⋊C4×C17D411yescharacteristic
51C2×C22⋊C4×C17—21no—
52C2×C42×C17C2311yescharacteristic
53C2×#63229—24no—
54C2×#63229—44no—
55C2×C4⋊Dic17—21no—
56C2×C4⋊Dic17—21no—
57C22×#3622—24no—
58C22×#3622—21no—
59C22×#3622—21no—
60C22×#3622—44no—
61C22×#3622D411yescharacteristic
62C22×#3622D411yescharacteristic
63#10587555—21no—
64#10587555—21no—
65#10587555—21no—
66#10587555C2312yes—
67C17×#107—24no—
68C17×C4⋊C8—21no—
69C17×C4⋊C8C2312yes—
70C17×#102—24no—
71#10587506—24no—
72#10587506—24no—
73C22⋊#3622—24no—
74#10587481—24no—
75C4⋊#3622—21no—
76C4⋊#3622—21no—
77C4⋊#3622C2312yes—
78C4⋊#3622C2312yes—
79C4×#3622D412yes—
80C4×#3622C2312yes—
81C24×C17—21no—
82C2×Q8×C17—22no—
83C2×Q8×C17—41no—
84C2×Q8×C17—42no—
85C2×Q8×C17—42no—
86C2×D4×C17—21no—
87C2×D4×C17—22no—
88C2×D4×C17SD1614yes—
89C2×D4×C17C2×D412yes—
90C22×C4×C17—21no—
91C22×C4×C17—22no—
92C22×C4×C17—22no—
93C22×C4×C17—42no—
94C22×C4×C17—42no—
95C22×C4×C17—41no—
96C22×C4×C17C2×D411yescharacteristic
97C22×C4×C17C2×D411yescharacteristic
98C22×C4×C17#4111yescharacteristic
99C22×Dic17—41no—
100C2×Dic34—22no—
101C2×Dic34—42no—
102C2×Dic34—42no—
103C2×Dic34—41no—
104C2×C8×C17—22no—
105C2×C8×C17—42no—
106C4⋊C4×C17—21no—
107C4⋊C4×C17—22no—
108C4⋊C4×C17—22no—
109C4⋊C4×C17—41no—
110C4⋊C4×C17—41no—
111C4⋊C4×C17C2×D412yes—
112C22⋊C4×C17—22no—
113C22⋊C4×C17—22no—
114C42×C17C2×D411yescharacteristic
115C42×C17C2×D411yescharacteristic
116C42×C17#4111yescharacteristic
117C42×C17C2411yescharacteristic
118#63229—48no—
119C4⋊Dic17—22no—
120C4⋊Dic17—22no—
121C4⋊Dic17—41no—
122C4⋊Dic17—41no—
123C2×#3622—22no—
124C2×#3622—22no—
125C2×#3622—22no—
126C2×#3622—22no—
127C2×#3622—48no—
128C2×#3622—42no—
129C2×#3622—42no—
130C2×#3622C2×D412yes—
131C2×#3622C2×D412yes—
132C2×#1679—171nomaximal, Sylow
133C23×C17—22no—
134C23×C17—24no—
135C23×C17—42no—
136C23×C17—42no—
137C23×C17#12111yescharacteristic
138Q8×C17—42no—
139Q8×C17—42no—
140Q8×C17—82no—
141D4×C17—22no—
142D4×C17—24no—
143D4×C17C2×SD1614yes—
144C2×C4×C17—22no—
145C2×C4×C17—22no—
146C2×C4×C17—42no—
147C2×C4×C17—42no—
148C2×C4×C17—42no—
149C2×C4×C17—44no—
150C2×C4×C17—42no—
151C2×C4×C17—82no—
152C2×C4×C17#12111yescharacteristic
153C2×C4×C17#12111yescharacteristic
154C2×C4×C17#12111yescharacteristic
155C2×C4×C17C2×SD1612yes—
156C2×C4×C17#13512yes—
157C2×C4×C17#13512yes—
158C2×C4×C17AGL(1,ℤ/8)12yes—
159C2×C4×C17C22×D411yescharacteristic
160C2×C4×C17C22×D411yescharacteristic
161C2×C4×C17C2×#4111yescharacteristic
162C2×Dic17—42no—
163C2×Dic17—82no—
164Dic34—42no—
165Dic34—42no—
166Dic34—82no—
167C8×C17—42no—
168#3622—22no—
169#3622—22no—
170#3622—42no—
171#3622—42no—
172C2×#487—171no—
173C2×#474D1711yescharacteristic
174C2×#461—171no—
175C2×C8×D4—171no—
176#1679—178no—
177C2×#334—171no—
178C2×#332—171no—
179C2×#332—171no—
180C22×C17—24no—
181C22×C17—48no—
182C22×C17—42no—
183C22×C17—42no—
184C22×C17#46312yes—
185C22×C17#46512yes—
186C22×C17#47312yes—
187C22×C17C2×#12111yescharacteristic
188C4×C17—21no—
189C4×C17—21no—
190C4×C17—42no—
191C4×C17—82no—
192C4×C17#46312yes—
193C4×C17#46512yes—
194C4×C17C22×SD1611yescharacteristic
195C4×C17C2×#13511yescharacteristic
196C4×C17C2×#13511yescharacteristic
197C4×C17C2×AGL(1,ℤ/8)11yescharacteristic
198Dic17—82no—
199C22×SD16—341no—
200C23×C8—341no—
201C2×C4⋊Q8—171no—
202C2×C4⋊Q8C2×D1711yescharacteristic
203C2×C4×D4C2×D1711yescharacteristic
204#487—174no—
205#487—342no—
206#474—22no—
207#474C2×D1714yes—
208#461—174no—
209#461—342no—
210C8×D4—174no—
211C8×D4—342no—
212C2×#107—21no—
213C2×C4⋊C8—171no—
214C2×C4⋊C8—171no—
215C2×C4⋊C8C2×D1711yescharacteristic
216C2×#103—341no—
217C2×#103—341no—
218C2×#103—341no—
219C2×#102—21no—
220C2×C22⋊C8—341no—
221C2×C4×C8—171no—
222#334—174no—
223#332—174no—
224#332—174no—
225C2×C17—44no—
226C2×C17#167914yescomplemented
227C2×C17C2×#46311yescharacteristic
228C2×C17C2×#46511yescharacteristic
229C2×C17C2×#47311yescharacteristic
230C22×Q8—21no—
231C22×Q8—341no—
232C22×D4C22⋊D1711yescharacteristic
233C23×C4—21no—
234C2×SD16—344no—
235C2×SD16—684no—
236C22×C8—21no—
237C22×C8—171no—
238C22×C8—171no—
239C22×C8—344no—
240C22×C8—341no—
241C22×C8—341no—
242C22×C8—684no—
243C4⋊Q8—21no—
244C4⋊Q8—21no—
245C4⋊Q8—21no—
246C4⋊Q8—172no—
247C4⋊Q8—341no—
248C4⋊Q8—341no—
249C4⋊Q8—341no—
250C4⋊Q8C22×D1712yes—
251C4×D4—21no—
252C4×D4—21no—
253C4×D4C22⋊D1712yes—
254C4×D4C22×D1712yes—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—341no—
258C2×C4⋊C4—341no—
259C2×C4⋊C4C22⋊D1711yescharacteristic
260C2×C22⋊C4—21no—
261C2×C42C22×D1711yescharacteristic
262#107—24no—
263C4⋊C8—21no—
264C4⋊C8—172no—
265C4⋊C8—172no—
266C4⋊C8—341no—
267C4⋊C8—341no—
268C4⋊C8C22×D1712yes—
269#103—344no—
270#103—344no—
271#103—344no—
272#102—24no—
273C22⋊C8—344no—
274C4×C8—172no—
275C4×C8—172no—
276C17C2×#167911yescharacteristic, complemented, Sylow
277C24—21no—
278C2×Q8—22no—
279C2×Q8—41no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×Q8—342no—
283C2×Q8—681no—
284C2×Q8—682no—
285C2×Q8—682no—
286C2×D4—21no—
287C2×D4—22no—
288C2×D4#6323014yes—
289C2×D4C2×C22⋊D1712yes—
290C22×C4—21no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—41no—
296C22×C4—681no—
297C22×C4D4×D1711yescharacteristic
298C22×C4#6325411yescharacteristic
299C22×C4C2×C22⋊D1711yescharacteristic
300SD16—688no—
301C2×C8—22no—
302C2×C8—42no—
303C2×C8—172no—
304C2×C8—172no—
305C2×C8—342no—
306C2×C8—342no—
307C2×C8—342no—
308C2×C8—342no—
309C2×C8—688no—
310C2×C8—682no—
311C2×C8—682no—
312C4⋊C4—21no—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—342no—
318C4⋊C4—342no—
319C4⋊C4—681no—
320C4⋊C4—681no—
321C4⋊C4C2×C22⋊D1712yes—
322C22⋊C4—22no—
323C22⋊C4—22no—
324C42D4×D1711yescharacteristic
325C42#6325411yescharacteristic
326C42C2×C22⋊D1711yescharacteristic
327C42C23×D1711yescharacteristic
328C23—24no—
329C23—22no—
330C23—42no—
331C23—42no—
332C23#1058762511yescharacteristic
333Q8—42no—
334Q8—42no—
335Q8—82no—
336Q8—682no—
337Q8—682no—
338Q8—1362no—
339D4—24no—
340D4—22no—
341D4C2×#6323014yes—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—42no—
345C2×C4—44no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—82no—
350C2×C4—682no—
351C2×C4—1362no—
352C2×C4#1058759712yes—
353C2×C4#1058759812yes—
354C2×C4#1058762511yescharacteristic
355C2×C4#1058762511yescharacteristic
356C2×C4#1058762511yescharacteristic
357C2×C4C2×#6323012yes—
358C2×C4#1058763612yes—
359C2×C4C2×D4×D1711yescharacteristic
360C2×C4C2×#6325411yescharacteristic
361C2×C4C22×C22⋊D1711yescharacteristic
362C8—42no—
363C8—342no—
364C8—342no—
365C8—682no—
366C8—682no—
367C22—24no—
368C22—48no—
369C22—42no—
370C22—42no—
371C22#1176302212yes—
372C22#1176304012yes—
373C22#1176308212yes—
374C22C2×#1058762511yescharacteristic
375C4—21no—
376C4—21no—
377C4—42no—
378C4—82no—
379C4—1362no—
380C4#1176302212yes—
381C4#1176304012yes—
382C4C22×#6323011yescharacteristic
383C4C2×#1058759811yescharacteristic
384C4C2×#1058763611yescharacteristic
385C4C2×#1058759711yescharacteristic
386C2—44no—
387C2#1452820514yesdirect product factor
388C2C2×#1176302211yescharacteristic
389C2C2×#1176304011yescharacteristic
390C2C2×#1176308211yescharacteristic
subgroupG/H
centerC23#10587625
commutator, cosocleC42×C17C24
FrattiniC42C23×D17
FittingC2×C17×#474C2
socleC23×C17#121
prime radicalC22×D4×C17D4
hypercenterC2×#474D17
hypocenterC17C2×#1679
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#474

Complements

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

NH ≅ G/Ntype
1. C2×#11763159C2semidirect
4. #14528205C2direct (N × H)
5. C2×#11762559C2semidirect
6. C2×#11762556C2semidirect
7. C2×#11762555C2semidirect
13. #11763159C22semidirect
25. #11762559C22semidirect
26. #11762556C22semidirect
27. #11762555C22semidirect
226. C2×C17#1679semidirect
276. C17C2×#1679semidirect
387. C2#14528205direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#117631592
C2×#117630142
C2×D4×#36222
#145282052
C2×#117625592
C2×#117625562
C2×#117625552
C2×C17×#4742
C2×#167917