Finite Group Databasebeta

← #17039642

Subgroups — #17039642

Order 4352.

1214 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14542080C211yesmaximal, characteristic
2#14537417C211yesmaximal, characteristic, complemented
3#14529765C212yesmaximal, complemented
4#14529616C212yesmaximal, complemented
5#14529227C214yesmaximal, complemented
6#14528595C214yesmaximal, complemented
7C2×C17×#376C211yesmaximal, characteristic, complemented
8#11763366C412yescomplemented
9#11763366C2211yescharacteristic
10#11763359C412yescomplemented
11#11763359C2211yescharacteristic
12#11763343—22no—
13#11763343—22no—
14#11763343C414yescomplemented
15#11763320—22no—
16#11763320C2212yes—
17C4×#63252—22no—
18C4×#63252C2212yes—
19#11762866—22no—
20#11762862—22no—
21#11762833C2212yes—
22C2×#10587476C2218yes—
23#11762773C2212yes—
24C4×#63223C2212yes—
25#11762569C2218yes—
26C22×C42×C17C2212yes—
27C22×C42×C17C2211yescharacteristic
28C17×#376—22no—
29C17×#376C2212yes—
30C17×#376C2212yes—
31C2×C17×#95—22no—
32C23×C4×C17—22no—
33C23×C4×C17D412yes—
34C23×C4×C17C2311yescharacteristic
35C2×#63252—22no—
36C2×#63252C2312yes—
37C2×C42×C17—22no—
38C2×C42×C17—28no—
39C2×C42×C17C2×C412yes—
40C2×C42×C17C2×C412yes—
41C2×C42×C17D412yes—
42C2×C42×C17D412yes—
43C2×C42×C17D412yes—
44C2×C42×C17C2314yes—
45C2×C42×C17C2311yescharacteristic
46C2×C42×C17C2311yescharacteristic
47#10587639—22no—
48#10587616—22no—
49#10587616—22no—
50#10587616C2×C414yes—
51#10587615—24no—
52#10587615—22no—
53C4×C22⋊D17—24no—
54C4×C22⋊D17—24no—
55C4×C22⋊D17—24no—
56#10587612—22no—
57#10587612C2312yes—
58#10587610—22no—
59#10587610—22no—
60#10587610C2×C414yes—
61C2×#63223C2314yes—
62#10587562—24no—
63#10587561—22no—
64#10587561C2×C412yescomplemented
65#10587560C2×C411yescharacteristic, complemented
66C4×D68—22no—
67C4×D68—22no—
68C4×D68C2×C414yescomplemented
69#10587558—24no—
70#10587558—22no—
71C42×D17—22no—
72#10587556—22no—
73#10587556C2×C412yescomplemented
74#10587555C2×C412yescomplemented
75#10587555C2×C411yescharacteristic, complemented
76C4×Dic34—22no—
77C4×Dic34—22no—
78C4×Dic34C2×C414yescomplemented
79C17×#95—24no—
80C17×#95—24no—
81C22⋊#3622—24no—
82#10587476—216no—
83C4⋊#3622—24no—
84#10587474—22no—
85C4×#3622—22no—
86C24×C17C2×D411yescharacteristic
87C22×C4×C17—24no—
88C22×C4×C17—28no—
89C22×C4×C17—24no—
90C22×C4×C17—44no—
91C22×C4×C17—48no—
92C22×C4×C17C22⋊C412yes—
93C22×C4×C17C22⋊C412yes—
94C22×C4×C17C22⋊C412yes—
95C22×C4×C17C22⋊C412yes—
96C22×C4×C17C22×C412yes—
97C22×C4×C17C2×D412yes—
98C22×C4×C17C2×D411yescharacteristic
99C22×C4×C17C2×D412yes—
100C22×C4×C17C2×D412yes—
101C22×C4×C17C2411yescharacteristic
102C2×C22⋊D17—24no—
103C2×C22⋊D17—22no—
104#63252—28no—
105#63252—48no—
106C2×D68—22no—
107C2×D68C22×C412yes—
108C2×C4×D17—24no—
109C2×C4×D17—22no—
110C2×Dic34—22no—
111C2×Dic34C22×C412yes—
112C42×C17—28no—
113C42×C17—24no—
114C42×C17C22⋊C412yes—
115C42×C17C22⋊C412yes—
116C42×C17C22⋊C412yes—
117C42×C17C22⋊C412yes—
118C42×C17C22×C412yes—
119C42×C17C22×C411yescharacteristic
120C42×C17C22×C411yescharacteristic
121C42×C17C2×D411yescharacteristic
122C42×C17C2×D411yescharacteristic
123C42×C17C2×D412yes—
124C42×C17#4118yescomplemented
125C22⋊Dic17—24no—
126C22⋊Dic17—22no—
127#63227—24no—
128#63227—24no—
129#63227—24no—
130C4⋊Dic17—22no—
131C4⋊Dic17—22no—
132C4⋊Dic17C22×C414yes—
133#63225—24no—
134#63225—24no—
135#63225—24no—
136C4×Dic17—24no—
137C4×Dic17—22no—
138#63223—28no—
139C2×#3622—24no—
140#21417—171nomaximal, Sylow
141C23×C17—22no—
142C23×C17—24no—
143C23×C17C2×C22⋊C412yes—
144C23×C17C22×D411yescharacteristic
145C22×D17—22no—
146C2×C4×C17—22no—
147C2×C4×C17—24no—
148C2×C4×C17—24no—
149C2×C4×C17—24no—
150C2×C4×C17—24no—
151C2×C4×C17—28no—
152C2×C4×C17—42no—
153C2×C4×C17—42no—
154C2×C4×C17—42no—
155C2×C4×C17—42no—
156C2×C4×C17—416no—
157C2×C4×C17—48no—
158C2×C4×C17C2×C22⋊C412yes—
159C2×C4×C17C2×C22⋊C412yes—
160C2×C4×C17C2×C22⋊C412yes—
161C2×C4×C17C2×C22⋊C412yes—
162C2×C4×C17C2×C22⋊C412yes—
163C2×C4×C17C23×C411yescharacteristic
164C2×C4×C17C22×D411yescharacteristic
165C2×C4×C17C2×#4114yes—
166C22⋊D17—44no—
167C22⋊D17—48no—
168C2×Dic17—24no—
169C2×Dic17—22no—
170D68—24no—
171D68—44no—
172C4×D17—48no—
173C4×D17—44no—
174Dic34—24no—
175Dic34—44no—
176#3622—44no—
177#3444—171no—
178#2920—171no—
179C2×#376D1711yescharacteristic, complemented
180#1995—172no—
181#1963—172no—
182#1896—174no—
183#1759—174no—
184C22×C17—28no—
185C22×C17—22no—
186C22×C17—22no—
187C22×C17—22no—
188C22×C17#42114yes—
189C22×C17C22×C22⋊C411yescharacteristic
190C22×C17#52512yes—
191C2×D17—44no—
192C2×D17—42no—
193C4×C17—22no—
194C4×C17—22no—
195C4×C17—22no—
196C4×C17—22no—
197C4×C17—22no—
198C4×C17—48no—
199C4×C17C4×#4112yes—
200C4×C17#53212yes—
201Dic17—44no—
202Dic17—42no—
203#533—172no—
204#533—171no—
205#532—172no—
206#532—171no—
207#529—174no—
208#529—342no—
209#529—342no—
210#525—172no—
211#525—342no—
212C4×#41—172no—
213C4×#41—342no—
214C22×C42C2×D1712yes—
215C22×C42C2×D1711yescharacteristic
216#432—342no—
217#431—342no—
218#423—172no—
219C2×#104—178no—
220#407—172no—
221C4×M16—172no—
222#376—22no—
223#376C2×D1712yes—
224#376C2×D1712yes—
225C2×#95—22no—
226#339—178no—
227C2×C17—22no—
228C2×C17—24no—
229C2×C17#229711yescharacteristic
230C2×C17C2×#42112yes—
231D17—82no—
232C2×#41—172no—
233C2×#41—342no—
234C23×C4—22no—
235C23×C4C22⋊D1712yes—
236C23×C4C22×D1711yescharacteristic
237C2×M16—174no—
238C4⋊Q8—172no—
239C4⋊Q8—171no—
240#127—171no—
241#126—344no—
242#125—172no—
243#125—342no—
244#124—172no—
245#124—342no—
246#123—342no—
247#123—344no—
248C22⋊Q8—174no—
249C22⋊Q8—342no—
250C22⋊Q8—342no—
251#121—174no—
252#121—342no—
253#121—342no—
254#120—342no—
255C4×Q8—174no—
256C4×Q8—342no—
257C4×Q8—342no—
258C4×D4—174no—
259C4×D4—344no—
260C4×D4—344no—
261C4×D4—344no—
262C4×D4—342no—
263C4×D4—342no—
264#117—172no—
265#117—342no—
266#117—342no—
267#117—344no—
268C2×C42—22no—
269C2×C42—28no—
270C2×C42—342no—
271C2×C42C4×D1712yes—
272C2×C42C4×D1712yes—
273C2×C42D6812yes—
274C2×C42D6812yes—
275C2×C42C22⋊D1712yes—
276C2×C42C22×D1714yes—
277C2×C42C22×D1711yescharacteristic
278C2×C42C22×D1711yescharacteristic
279C4⋊C8—344no—
280#104—3416no—
281C22⋊C8—344no—
282#97—342no—
283C4×C8—342no—
284#95—24no—
285#95—24no—
286C17#2141711yescharacteristic, complemented, Sylow
287C24C2×C22⋊D1711yescharacteristic
288#41—348no—
289#41—688no—
290C2×Q8—172no—
291C2×Q8—342no—
292C2×D4—172no—
293C2×D4—342no—
294C2×D4—344no—
295C2×D4—342no—
296C22×C4—24no—
297C22×C4—28no—
298C22×C4—24no—
299C22×C4—44no—
300C22×C4—48no—
301C22×C4—342no—
302C22×C4—344no—
303C22×C4#6322712yes—
304C22×C4#6322712yes—
305C22×C4#6322712yes—
306C22×C4#6322712yes—
307C22×C4C2×C4×D1712yes—
308C22×C4C2×D6812yes—
309C22×C4C2×C22⋊D1711yescharacteristic
310C22×C4C2×C22⋊D1712yes—
311C22×C4C2×C22⋊D1712yes—
312C22×C4C23×D1711yescharacteristic
313M16—348no—
314C2×C8—344no—
315C4⋊C4—174no—
316C4⋊C4—344no—
317C4⋊C4—342no—
318C4⋊C4—342no—
319C4⋊C4—344no—
320C4⋊C4—344no—
321C22⋊C4—342no—
322C22⋊C4—344no—
323C22⋊C4—344no—
324C22⋊C4—344no—
325C22⋊C4—344no—
326C42—28no—
327C42—24no—
328C42—342no—
329C42—344no—
330C42#6322712yes—
331C42#6322712yes—
332C42#6322712yes—
333C42#6322712yes—
334C42C2×C4×D1712yes—
335C42C2×C4×D1711yescharacteristic
336C42C2×C4×D1711yescharacteristic
337C42C2×D6811yescharacteristic
338C42C2×D6811yescharacteristic
339C42C2×D6812yes—
340C42#6325218yescomplemented
341C23—22no—
342C23—24no—
343C23—342no—
344C23C2×#6322712yes—
345C23C22×C22⋊D1711yescharacteristic
346Q8—344no—
347Q8—684no—
348D4—344no—
349D4—684no—
350D4—684no—
351D4—688no—
352C2×C4—22no—
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—24no—
356C2×C4—24no—
357C2×C4—28no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—416no—
363C2×C4—48no—
364C2×C4—344no—
365C2×C4—342no—
366C2×C4—688no—
367C2×C4—684no—
368C2×C4C2×#6322712yes—
369C2×C4C2×#6322712yes—
370C2×C4C2×#6322712yes—
371C2×C4C2×#6322712yes—
372C2×C4C2×#6322712yes—
373C2×C4C22×C4×D1711yescharacteristic
374C2×C4C22×D6811yescharacteristic
375C2×C4C2×#6325214yes—
376C8—684no—
377C22—28no—
378C22—22no—
379C22—22no—
380C22—22no—
381C22—684no—
382C22—682no—
383C22#1176282714yes—
384C22C22×#6322711yescharacteristic
385C22#1176332012yes—
386C4—22no—
387C4—22no—
388C4—22no—
389C4—22no—
390C4—22no—
391C4—48no—
392C4—684no—
393C4—682no—
394C4C4×#6325212yes—
395C4#1176335912yes—
396C2—22no—
397C2—24no—
398C2—1362no—
399C2#1453127411yescharacteristic
400C2C2×#1176282712yes—
subgroupG/H
centerC2×C4C22×D68
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC2×C17×#376C2
socleC22×C17C22×C22⋊C4
prime radical#14542080C2
cosocleC22×C4×C17C24
hypercenterC2×#376D17
hypocenterC17#21417
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ C2×#376

Complements

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

NH ≅ G/Ntype
2. #14537417C2semidirect
3. #14529765C2semidirect
4. #14529616C2semidirect
5. #14529227C2semidirect
6. #14528595C2semidirect
7. C2×C17×#376C2semidirect
8. #11763366C4semidirect
10. #11763359C4semidirect
14. #11763343C4semidirect
64. #10587561C2×C4semidirect
65. #10587560C2×C4semidirect
68. C4×D68C2×C4semidirect
73. #10587556C2×C4semidirect
74. #10587555C2×C4semidirect
75. #10587555C2×C4semidirect
78. C4×Dic34C2×C4semidirect
124. C42×C17#41semidirect
179. C2×#376D17semidirect
286. C17#21417semidirect
340. C42#63252semidirect

Maximal subgroups

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

subgroupindex
#145420802
#145374172
#145297652
#145296162
#145292272
#145285952
C2×C17×#3762
#2141717