Finite Group Databasebeta

← C2×#14534614 #17201391

Subgroups — C2×#14534614 #17201391

Order 4352.

2614 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#10587556C211yesmaximal, characteristic, complemented
2#14534614C2116yesmaximal, direct product factor
3C2×#11762742C214yesmaximal, complemented
4C2×#11762732C214yesmaximal
5C2×#11762707C212yesmaximal, complemented
6C2×#11762679C211yesmaximal, characteristic, complemented
7C2×#11762669C212yesmaximal
8C22×C17×#117C211yesmaximal, characteristic
9C2×#10587556—28no—
10C2×#10587556C2218yescomplemented
11C22×C4⋊Dic17C2212yes—
12C22×C4⋊Dic17C2212yes—
13C22×C4⋊Dic17C2214yes—
14C22×#63225C2214yes—
15C22×#63225C2214yes—
16C22×C22⋊Dic17C2214yes—
17#11762742C22132yescomplemented
18#11762732C22132yes—
19#11762707C22116yescomplemented
20#11762679C2218yescomplemented
21#11762669C22116yes—
22C2×#10587502C2214yes—
23C2×#10587502C2214yes—
24C24×C4×C17C2211yescharacteristic
25C2×C17×#117—28no—
26C2×C17×#117C2218yes—
27C22×C4⋊C4×C17C2212yes—
28C22×C22⋊C4×C17C2212yes—
29C22×C42×C17C2211yescharacteristic
30C22×C42×C17C2211yescharacteristic
31C25×C17C2311yescharacteristic
32C23×C4×C17—24no—
33C23×C4×C17—24no—
34C23×C4×C17Q818yes—
35C23×C4×C17C2312yes—
36C23×C4×C17C2312yes—
37C23×C4×C17C2311yescharacteristic
38C23×C4×C17C2314yes—
39C23×C4×C17C2311yescharacteristic
40C23×Dic17C2314yes—
41C23×Dic17C2314yes—
42C17×#117—232no—
43C2×C4⋊C4×C17—28no—
44C2×C4⋊C4×C17C2318yes—
45C2×C22⋊C4×C17—28no—
46C2×C22⋊C4×C17C2318yes—
47C2×C42×C17—24no—
48C2×C42×C17D418yes—
49C2×C42×C17C2314yes—
50C2×C42×C17C2314yes—
51C2×C22⋊Dic17—48no—
52C2×C22⋊Dic17C23116yes—
53C2×C4⋊Dic17—28no—
54C2×C4⋊Dic17—28no—
55C2×C4⋊Dic17—48no—
56C2×C4⋊Dic17C2318yes—
57C2×C4⋊Dic17C2318yes—
58C2×C4⋊Dic17C23116yes—
59C2×#63225—216no—
60C2×#63225—48no—
61C2×#63225C23116yes—
62C2×#63225C23116yes—
63#10587556—232no—
64#10587502C23116yes—
65#10587502C23116yes—
66C24×C17—22no—
67C24×C17—22no—
68C24×C17—24no—
69C24×C17—24no—
70C24×C17C2×Q814yes—
71C24×C17C2412yes—
72C24×C17C2411yescharacteristic
73C22×C4×C17—22no—
74C22×C4×C17—22no—
75C22×C4×C17—24no—
76C22×C4×C17—24no—
77C22×C4×C17—232no—
78C22×C4×C17—28no—
79C22×C4×C17—28no—
80C22×C4×C17—24no—
81C22×C4×C17—24no—
82C22×C4×C17—24no—
83C22×C4×C17C2×D418yes—
84C22×C4×C17C2×D414yes—
85C22×C4×C17C2×Q814yes—
86C22×C4×C17C2×Q8116yes—
87C22×C4×C17C2×Q814yes—
88C22×C4×C17#4114yes—
89C22×C4×C17C2414yes—
90C22×C4×C17C2414yes—
91C22×C4×C17C2412yes—
92C22×C4×C17C2412yes—
93C22×Dic17—28no—
94C22×Dic17—28no—
95C22×Dic17—28no—
96C22×Dic17—28no—
97C22×Dic17—28no—
98C22×Dic17—28no—
99C22×Dic17C2418yes—
100C22×Dic17C2418yes—
101C4⋊C4×C17—216no—
102C22⋊C4×C17—216no—
103C42×C17—28no—
104C42×C17C2×D4116yes—
105C22⋊Dic17—416no—
106C4⋊Dic17—216no—
107C4⋊Dic17—216no—
108C4⋊Dic17—416no—
109#63225—232no—
110#63225—416no—
111C2×#2629—171nomaximal, Sylow
112C23×C17—28no—
113C23×C17—216no—
114C23×C17—22no—
115C23×C17—22no—
116C23×C17—24no—
117C23×C17—24no—
118C23×C17—48no—
119C23×C17—44no—
120C23×C17—44no—
121C23×C17C22×D412yes—
122C23×C17C22×Q814yes—
123C23×C17C22×Q812yes—
124C23×C17C2×#4112yes—
125C23×C17H(5,𝔽2)14yes—
126C23×C172−1+414yes—
127C23×C17C2511yescharacteristic
128C2×C4×C17—28no—
129C2×C4×C17—216no—
130C2×C4×C17—22no—
131C2×C4×C17—22no—
132C2×C4×C17—24no—
133C2×C4×C17—24no—
134C2×C4×C17—232no—
135C2×C4×C17—24no—
136C2×C4×C17—24no—
137C2×C4×C17—24no—
138C2×C4×C17—48no—
139C2×C4×C17C22⋊Q8116yes—
140C2×C4×C17C22×D418yes—
141C2×C4×C17C22×D414yes—
142C2×C4×C17C22×Q814yes—
143C2×C4×C17C22×Q814yes—
144C2×C4×C17C2×#4114yes—
145C2×Dic17—28no—
146C2×Dic17—28no—
147C2×Dic17—28no—
148C2×Dic17—28no—
149C2×Dic17—28no—
150C2×Dic17—28no—
151C2×Dic17—416no—
152C2×Dic17—416no—
153C22×#125—171no—
154C22×#117D1711yescharacteristic
155#2629—1716no—
156C2×#398—174no—
157C2×#396—174no—
158C2×#389—172no—
159C2×#384—171no—
160C2×#380—172no—
161C22×C17—24no—
162C22×C17—28no—
163C22×C17—416no—
164C22×C17—44no—
165C22×C17—42no—
166C22×C17—42no—
167C22×C17C2×C22⋊Q814yes—
168C22×C17#53718yes—
169C22×C17#54014yes—
170C22×C17#54114yes—
171C22×C17#54418yes—
172C22×C17C23×D411yescharacteristic
173C22×C17C23×Q811yescharacteristic
174C22×C17C22×#4111yescharacteristic
175C22×C17C2×H(5,𝔽2)12yes—
176C22×C17C2×2−1+412yes—
177C4×C17—24no—
178C4×C17—28no—
179C4×C17—44no—
180C4×C17C2×C22⋊Q818yes—
181Dic17—48no—
182Dic17—48no—
183C24×C4C2×D1711yescharacteristic
184C2×#125—178no—
185C2×#125—348no—
186C2×#117—28no—
187C2×#117C2×D1718yes—
188C22×C4⋊C4—174no—
189C22×C4⋊C4—172no—
190C22×C4⋊C4—172no—
191C22×C4⋊C4—174no—
192C22×C4⋊C4—174no—
193C22×C4⋊C4C2×D1712yes—
194C22×C22⋊C4—174no—
195C22×C22⋊C4C2×D1712yes—
196C22×C42C2×D1711yescharacteristic
197C22×C42C2×D1711yescharacteristic
198#398—1732no—
199#396—1732no—
200#389—1716no—
201#384—178no—
202#380—1716no—
203C2×#95—174no—
204C2×#95—174no—
205C2×C17—44no—
206C2×C17#262918yescomplemented
207C2×C17C22×C22⋊Q811yescharacteristic
208C2×C17C2×#53712yes—
209C2×C17C2×#54011yescharacteristic
210C2×C17C2×#54111yescharacteristic
211C2×C17C2×#54412yes—
212C25C22×D1711yescharacteristic
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—174no—
216C23×C4—174no—
217C23×C4Dic3418yes—
218C23×C4C22×D1712yes—
219C23×C4C22×D1714yes—
220C23×C4C22×D1711yescharacteristic
221C23×C4C22×D1712yes—
222C23×C4C22×D1711yescharacteristic
223#125—3432no—
224#117—232no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—178no—
227C2×C4⋊C4—1716no—
228C2×C4⋊C4—178no—
229C2×C4⋊C4—1716no—
230C2×C4⋊C4—1716no—
231C2×C4⋊C4—348no—
232C2×C4⋊C4—3416no—
233C2×C4⋊C4—348no—
234C2×C4⋊C4—688no—
235C2×C4⋊C4—688no—
236C2×C4⋊C4C22×D1718yes—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—1716no—
239C2×C22⋊C4—688no—
240C2×C22⋊C4C22×D1718yes—
241C2×C42—24no—
242C2×C42C22⋊D1718yes—
243C2×C42C22×D1714yes—
244C2×C42C22×D1714yes—
245#95—1716no—
246#95—1716no—
247C17C2×#262911yescharacteristic, complemented, Sylow
248C24—22no—
249C24—24no—
250C24—24no—
251C24—22no—
252C24C2×Dic3414yes—
253C24C23×D1712yes—
254C24C23×D1711yescharacteristic
255C22×C4—22no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—22no—
259C22×C4—28no—
260C22×C4—28no—
261C22×C4—232no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—178no—
266C22×C4—178no—
267C22×C4—348no—
268C22×C4—348no—
269C22×C4—348no—
270C22×C4—348no—
271C22×C4—348no—
272C22×C4—348no—
273C22×C4C2×Dic3414yes—
274C22×C4C2×Dic34116yes—
275C22×C4C2×Dic3414yes—
276C22×C4#6325214yes—
277C22×C4C2×C22⋊D1718yes—
278C22×C4C2×C22⋊D1714yes—
279C22×C4C23×D1714yes—
280C22×C4C23×D1712yes—
281C22×C4C23×D1714yes—
282C22×C4C23×D1712yes—
283C4⋊C4—216no—
284C4⋊C4—3416no—
285C4⋊C4—3432no—
286C4⋊C4—3416no—
287C4⋊C4—6816no—
288C4⋊C4—6816no—
289C22⋊C4—216no—
290C22⋊C4—6816no—
291C42—28no—
292C42C2×C22⋊D17116yes—
293C23—28no—
294C23—22no—
295C23—22no—
296C23—24no—
297C23—24no—
298C23—216no—
299C23—48no—
300C23—44no—
301C23—44no—
302C23C22×Dic3414yes—
303C23C22×Dic3412yes—
304C23C2×#6325212yes—
305C23#1058769614yes—
306C23#1058769714yes—
307C23C22×C22⋊D1712yes—
308C23C24×D1711yescharacteristic
309C2×C4—28no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—24no—
313C2×C4—24no—
314C2×C4—216no—
315C2×C4—232no—
316C2×C4—24no—
317C2×C4—24no—
318C2×C4—24no—
319C2×C4—48no—
320C2×C4—348no—
321C2×C4—348no—
322C2×C4—348no—
323C2×C4—348no—
324C2×C4—348no—
325C2×C4—348no—
326C2×C4—6816no—
327C2×C4—6816no—
328C2×C4#10587610116yes—
329C2×C4C22×Dic3414yes—
330C2×C4C22×Dic3414yes—
331C2×C4C2×#6325214yes—
332C2×C4C22×C22⋊D1718yes—
333C2×C4C22×C22⋊D1714yes—
334C22—28no—
335C22—24no—
336C22—44no—
337C22—42no—
338C22—42no—
339C22—416no—
340C22C2×#1058761014yes—
341C22#1176338718yes—
342C22#1176341414yes—
343C22#1176342114yes—
344C22#1176343718yes—
345C22C23×C22⋊D1711yescharacteristic
346C22C23×Dic3411yescharacteristic
347C22C22×#6325211yescharacteristic
348C22C2×#1058769612yes—
349C22C2×#1058769712yes—
350C4—28no—
351C4—24no—
352C4—44no—
353C4—688no—
354C4—688no—
355C4C2×#1058761018yes—
356C2—44no—
357C2#1453461418yesdirect product factor
358C2C22×#1058761011yescharacteristic
359C2C2×#1176338712yes—
360C2C2×#1176341411yescharacteristic
361C2C2×#1176342111yescharacteristic
362C2C2×#1176343712yes—
subgroupG/H
centerC24C23×D17
commutator, cosocleC23×C17C25
FrattiniC23C24×D17
FittingC22×C17×#117C2
socleC24×C17C24
prime radicalC25×C17C23
hypercenterC22×#117D17
hypocenterC17C2×#2629
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ C22×#117

Complements

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

NH ≅ G/Ntype
1. C22×#10587556C2semidirect
2. #14534614C2direct (N × H)
3. C2×#11762742C2semidirect
5. C2×#11762707C2semidirect
6. C2×#11762679C2semidirect
10. C2×#10587556C22semidirect
17. #11762742C22semidirect
19. #11762707C22semidirect
20. #11762679C22semidirect
206. C2×C17#2629semidirect
247. C17C2×#2629semidirect
357. C2#14534614direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#105875562
#145346142
C2×#117627422
C2×#117627322
C2×#117627072
C2×#117626792
C2×#117626692
C22×C17×#1172
C2×#262917