Finite Group Databasebeta

← C2×#14527826 #17006559

Subgroups — C2×#14527826 #17006559

Order 4352.

984 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763142C212yesmaximal
2C2×#11762748C211yesmaximal, characteristic
3#14527826C218yesmaximal, direct product factor
4C2×#11762546C212yesmaximal
5C2×#11762535C211yesmaximal, characteristic
6C2×C17×#544C211yesmaximal, characteristic
7C22×C4⋊Dic17C2211yescharacteristic
8#11763142—24no—
9#11763142C2218yes—
10C2×#10587478—22no—
11C2×#10587506—22no—
12C2×C22⋊#3622C2212yes—
13#11762748—22no—
14#11762748C2214yes—
15C2×#10587502—21no—
16C2×#10587502—21no—
17C2×#10587502C2211yescharacteristic
18#11762546C2218yes—
19#11762535C2214yes—
20C17×#544—22no—
21C17×#544—24no—
22C17×#544C2214yes—
23C2×C17×C4⋊Q8—22no—
24C2×C17×#125—22no—
25C2×C17×C22⋊Q8—21no—
26C2×C17×C22⋊Q8C2212yes—
27C2×C17×#117—21no—
28C22×C4⋊C4×C17C2211yescharacteristic
29C22×Q8×C17—21no—
30C22×Q8×C17D412yes—
31C23×C4×C17D411yescharacteristic
32C23×C4×C17D411yescharacteristic
33C23×C4×C17C2311yescharacteristic
34C23×Dic17—21no—
35C23×Dic17—21no—
36C17×C4⋊Q8—28no—
37C17×C4⋊Q8—28no—
38C17×#125—28no—
39C17×#125—28no—
40C17×C22⋊Q8—24no—
41C17×C22⋊Q8—24no—
42C17×C22⋊Q8—22no—
43C17×C22⋊Q8—22no—
44C17×C22⋊Q8D414yes—
45C17×C22⋊Q8C2314yes—
46C17×#117—24no—
47C17×#117—24no—
48C2×C4⋊C4×C17—21no—
49C2×C4⋊C4×C17—21no—
50C2×C4⋊C4×C17—21no—
51C2×C4⋊C4×C17—22no—
52C2×C4⋊C4×C17—22no—
53C2×C4⋊C4×C17—22no—
54C2×C4⋊C4×C17—22no—
55C2×C4⋊C4×C17D412yes—
56C2×C4⋊C4×C17D412yes—
57C2×C4⋊C4×C17C2312yes—
58C2×C22⋊C4×C17—22no—
59C2×C42×C17—22no—
60C2×C4⋊Dic17—21no—
61C2×C4⋊Dic17—22no—
62C2×C4⋊Dic17—22no—
63C2×C4⋊Dic17C2312yes—
64C22×#3622—22no—
65#10587506—28no—
66#10587502—21no—
67#10587502—24no—
68#10587502—24no—
69#10587502C2312yes—
70C22⋊#3622—22no—
71C22⋊#3622C2314yes—
72#10587478—28no—
73C24×C17C2×D411yescharacteristic
74C2×Q8×C17—24no—
75C2×Q8×C17—22no—
76C2×Q8×C17—44no—
77C2×Q8×C17—82no—
78C2×Q8×C17C2×D414yes—
79C22×C4×C17—21no—
80C22×C4×C17—21no—
81C22×C4×C17—22no—
82C22×C4×C17—21no—
83C22×C4×C17—22no—
84C22×C4×C17—22no—
85C22×C4×C17—21no—
86C22×C4×C17—42no—
87C22×C4×C17—42no—
88C22×C4×C17Q1614yes—
89C22×C4×C17C2×D412yes—
90C22×C4×C17C2×D412yes—
91C22×C4×C17C2×D412yes—
92C22×C4×C17C2×D411yescharacteristic
93C22×C4×C17C2×D412yes—
94C22×C4×C17C2411yescharacteristic
95C22×Dic17—22no—
96C22×Dic17—22no—
97C22×Dic17—42no—
98C22×Dic17—42no—
99C22×Dic17—44no—
100C22×Dic17—41no—
101C22×Dic17—41no—
102C22×Dic17—42no—
103C4⋊C4×C17—24no—
104C4⋊C4×C17—28no—
105C4⋊C4×C17—24no—
106C4⋊C4×C17—24no—
107C4⋊C4×C17—22no—
108C4⋊C4×C17—24no—
109C4⋊C4×C17—24no—
110C4⋊C4×C17—24no—
111C4⋊C4×C17—24no—
112C4⋊C4×C17C2×D414yes—
113C22⋊C4×C17—24no—
114C22⋊C4×C17—24no—
115C42×C17—28no—
116C4⋊Dic17—24no—
117C4⋊Dic17—42no—
118C2×#3622—24no—
119C2×#3622—44no—
120C2×#1617—171nomaximal, Sylow
121C23×C17—22no—
122C23×C17—22no—
123C23×C17#12011yescharacteristic
124C23×C17#12011yescharacteristic
125C23×C17C2×Q1612yes—
126C23×C17#13712yes—
127C23×C17C22×D411yescharacteristic
128Q8×C17—44no—
129Q8×C17—82no—
130Q8×C17—84no—
131C2×C4×C17—22no—
132C2×C4×C17—22no—
133C2×C4×C17—24no—
134C2×C4×C17—24no—
135C2×C4×C17—24no—
136C2×C4×C17—24no—
137C2×C4×C17—24no—
138C2×C4×C17—22no—
139C2×C4×C17—42no—
140C2×C4×C17—41no—
141C2×C4×C17—41no—
142C2×C4×C17—44no—
143C2×C4×C17—42no—
144C2×C4×C17—44no—
145C2×C4×C17—42no—
146C2×C4×C17—81no—
147C2×C4×C17#12012yes—
148C2×C4×C17C2×Q1614yes—
149C2×C4×C17C22×D412yes—
150C2×Dic17—42no—
151C2×Dic17—42no—
152C2×Dic17—44no—
153C2×Dic17—42no—
154C2×Dic17—42no—
155C2×Dic17—84no—
156C2×Dic17—82no—
157C2×Dic17—82no—
158#3622—44no—
159C2×#544D1711yescharacteristic
160C2×#484—172no—
161C2×#400—171no—
162#1617—178no—
163C2×#328—172no—
164C2×#324—171no—
165C22×C17—24no—
166C22×C17—24no—
167C22×C17—21no—
168C22×C17—21no—
169C22×C17—21no—
170C22×C17—21no—
171C22×C17#45112yes—
172C22×C17#45112yes—
173C22×C17#45412yes—
174C22×C17#45812yes—
175C22×C17C2×#12011yescharacteristic
176C22×C17C22×Q1611yescharacteristic
177C22×C17C2×#13711yescharacteristic
178C4×C17—24no—
179C4×C17—22no—
180C4×C17—22no—
181C4×C17—44no—
182C4×C17—42no—
183C4×C17—44no—
184Dic17—82no—
185Dic17—82no—
186#544—24no—
187#544—22no—
188#544C2×D1714yes—
189C2×C4⋊Q8—22no—
190C2×#125—22no—
191C2×C22⋊Q8—21no—
192C2×C22⋊Q8C2×D1712yes—
193C2×#117—21no—
194C22×C4⋊C4—171no—
195C22×C4⋊C4C2×D1711yescharacteristic
196#484—178no—
197#484—344no—
198C2×#107—342no—
199C2×#103—342no—
200C2×C22⋊C8—172no—
201#400—174no—
202#400—342no—
203C2×#95—171no—
204C2×#95—341no—
205C2×#95—341no—
206#328—178no—
207#324—174no—
208C2×C17—21no—
209C2×C17—21no—
210C2×C17—22no—
211C2×C17#161714yescomplemented
212C2×C17C2×#45111yescharacteristic
213C2×C17C2×#45411yescharacteristic
214C2×C17C2×#45811yescharacteristic
215C22×Q8—21no—
216C22×Q8C22⋊D1712yes—
217C23×C4—341no—
218C23×C4—341no—
219C23×C4C22⋊D1711yescharacteristic
220C23×C4C22⋊D1711yescharacteristic
221C23×C4C22×D1711yescharacteristic
222C22×C8—342no—
223C4⋊Q8—28no—
224C4⋊Q8—28no—
225#125—28no—
226#125—28no—
227C22⋊Q8—24no—
228C22⋊Q8—24no—
229C22⋊Q8—22no—
230C22⋊Q8—22no—
231C22⋊Q8C22⋊D1714yes—
232C22⋊Q8C22×D1714yes—
233#117—24no—
234#117—24no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—172no—
243C2×C4⋊C4—342no—
244C2×C4⋊C4—342no—
245C2×C4⋊C4—341no—
246C2×C4⋊C4C22⋊D1712yes—
247C2×C4⋊C4C22⋊D1712yes—
248C2×C4⋊C4C22×D1712yes—
249C2×C22⋊C4—22no—
250C2×C42—22no—
251#107—348no—
252#103—348no—
253C22⋊C8—174no—
254C22⋊C8—342no—
255#95—172no—
256#95—341no—
257#95—344no—
258#95—344no—
259C17C2×#161711yescharacteristic, complemented, Sylow
260C24C2×C22⋊D1711yescharacteristic
261C2×Q8—24no—
262C2×Q8—22no—
263C2×Q8—44no—
264C2×Q8—82no—
265C2×Q8C2×C22⋊D1714yes—
266C22×C4—22no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—342no—
276C22×C4—342no—
277C22×C4—682no—
278C22×C4—682no—
279C22×C4—684no—
280C22×C4—682no—
281C22×C4—681no—
282C22×C4—681no—
283C22×C4#6323114yes—
284C22×C4C2×C22⋊D1712yes—
285C22×C4C2×C22⋊D1712yes—
286C22×C4C2×C22⋊D1712yes—
287C22×C4C2×C22⋊D1712yes—
288C22×C4C2×C22⋊D1711yescharacteristic
289C22×C4C23×D1711yescharacteristic
290C2×C8—344no—
291C2×C8—684no—
292C4⋊C4—24no—
293C4⋊C4—24no—
294C4⋊C4—24no—
295C4⋊C4—28no—
296C4⋊C4—24no—
297C4⋊C4—24no—
298C4⋊C4—24no—
299C4⋊C4—24no—
300C4⋊C4—22no—
301C4⋊C4—344no—
302C4⋊C4—682no—
303C4⋊C4C2×C22⋊D1714yes—
304C22⋊C4—24no—
305C22⋊C4—24no—
306C42—28no—
307C23—22no—
308C23—22no—
309C23#1058762812yes—
310C23C2×#6323112yes—
311C23#1058763911yescharacteristic
312C23#1058763911yescharacteristic
313C23C22×C22⋊D1711yescharacteristic
314Q8—44no—
315Q8—82no—
316Q8—84no—
317C2×C4—24no—
318C2×C4—24no—
319C2×C4—24no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—24no—
323C2×C4—24no—
324C2×C4—22no—
325C2×C4—42no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—44no—
329C2×C4—42no—
330C2×C4—44no—
331C2×C4—42no—
332C2×C4—81no—
333C2×C4—682no—
334C2×C4—682no—
335C2×C4—684no—
336C2×C4—682no—
337C2×C4—682no—
338C2×C4—1364no—
339C2×C4—1362no—
340C2×C4—1362no—
341C2×C4C2×#6323114yes—
342C2×C4#1058763912yes—
343C2×C4C22×C22⋊D1712yes—
344C8—684no—
345C22—24no—
346C22—24no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22#1176296512yes—
352C22#1176296512yes—
353C22#1176298212yes—
354C22#1176299912yes—
355C22C2×#1058763911yescharacteristic
356C22C22×#6323111yescharacteristic
357C22C2×#1058762811yescharacteristic
358C4—22no—
359C4—22no—
360C4—24no—
361C4—44no—
362C4—42no—
363C4—44no—
364C4—1362no—
365C4—1362no—
366C2—21no—
367C2—22no—
368C2—21no—
369C2#1452782614yesdirect product factor
370C2C2×#1176296511yescharacteristic
371C2C2×#1176298211yescharacteristic
372C2C2×#1176299911yescharacteristic
subgroupG/H
centerC23#10587639
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC2×C17×#544C2
socleC23×C17#120
prime radicalC24×C17C2×D4
hypercenterC2×#544D17
hypocenterC17C2×#1617
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#544

Complements

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

NH ≅ G/Ntype
3. #14527826C2direct (N × H)
211. C2×C17#1617semidirect
259. C17C2×#1617semidirect
369. C2#14527826direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#117631422
C2×#117627482
#145278262
C2×#117625462
C2×#117625352
C2×C17×#5442
C2×#161717