Finite Group Databasebeta

← #16935403

Subgroups — #16935403

Order 4352.

714 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763182C211yesmaximal, characteristic
2C2×#11763009C211yesmaximal, characteristic
3#14533989C211yesmaximal, characteristic, complemented
4#14529710C211yesmaximal, characteristic
5C2×#11762557C211yesmaximal, characteristic, complemented
6#14526624C211yesmaximal, characteristic, complemented
7C2×C17×Q8⋊C8C211yesmaximal, characteristic, complemented
8C22×#63219—22no—
9C22×#63230—21no—
10C2×#10587555C2211yescharacteristic
11C2×#10587560C2211yescharacteristic
12#11763182—22no—
13#11763182—22no—
14#11763009—22no—
15#11763009—22no—
16C2×C4⋊#3622C2211yescharacteristic
17C2×#10587492—22no—
18C2×#10587480—21no—
19#11762729—22no—
20#11762716—21no—
21#11762707—22no—
22#11762678—21no—
23C4×C4⋊Dic17C2211yescharacteristic
24#11762557—22no—
25C2×C4×Q8×C17C2211yescharacteristic
26C2×C17×C4⋊C8C2211yescharacteristic
27C2×C4×C8×C17C2211yescharacteristic
28C17×Q8⋊C8—22no—
29C22×Q8×C17D411yescharacteristic
30C22×D68—21no—
31C22×D68—21no—
32C22×D68—21no—
33C22×Dic34D412yes—
34C22×C8×C17—21no—
35C22×C8×C17—21no—
36C4×Q8×C17—21no—
37C4×Q8×C17—21no—
38C4×Q8×C17—21no—
39C4×Q8×C17—21no—
40C2×C4⋊C4×C17—21no—
41C2×C4⋊C4×C17D411yescharacteristic
42C2×C42×C17—21no—
43C2×C42×C17C2311yescharacteristic
44C2×#63230—24no—
45C2×#63230—42no—
46C2×#63230—42no—
47C2×C4⋊Dic17—21no—
48C2×C4⋊Dic17—21no—
49C2×C4⋊Dic17D412yes—
50C2×#63225—24no—
51C2×C4×Dic17—21no—
52C22×#3622—21no—
53C2×#63219—28no—
54C2×#63219—48no—
55#10587560—21no—
56#10587560—21no—
57#10587560—21no—
58#10587560D412yes—
59#10587555—21no—
60#10587555—21no—
61#10587555—21no—
62#10587555—21no—
63C17×C4⋊C8—21no—
64C17×C4⋊C8—21no—
65C4×C8×C17—21no—
66C4×C8×C17D412yes—
67#10587502—24no—
68#10587502—21no—
69#10587492—44no—
70#10587480—24no—
71C4⋊#3622—21no—
72C4⋊#3622D412yes—
73C23×D17—41no—
74C2×Q8×C17—21no—
75C2×Q8×C17—21no—
76C2×Q8×C17—21no—
77C2×Q8×C17—21no—
78C2×Q8×C17—21no—
79C2×Q8×C17SD1612yescomplemented
80C22×C4×C17—21no—
81C22×C4×C17—21no—
82C22×C4×C17C2×D411yescharacteristic
83C22×C4×C17C2×D411yescharacteristic
84C22×C4×C17C2×D411yescharacteristic
85C22×Dic17—22no—
86C22×Dic17—22no—
87C2×D68—22no—
88C2×D68—22no—
89C2×D68—22no—
90C2×D68—44no—
91C2×D68—42no—
92C2×D68—41no—
93C2×D68—41no—
94C2×D68—41no—
95C2×D68—41no—
96C2×D68—42no—
97C2×D68—41no—
98C2×D68—41no—
99C2×D68—41no—
100C2×Dic34—22no—
101C2×Dic34—22no—
102C2×Dic34—44no—
103C2×C8×C17—24no—
104C2×C8×C17—22no—
105C2×C8×C17—41no—
106C2×C8×C17—42no—
107C4⋊C4×C17—21no—
108C4⋊C4×C17—21no—
109C4⋊C4×C17—41no—
110C4⋊C4×C17—41no—
111C42×C17—41no—
112C42×C17—41no—
113C42×C17C2×D411yescharacteristic
114C42×C17C2×D411yescharacteristic
115C42×C17C2×D411yescharacteristic
116C42×C17#4111yescharacteristic
117#63230—48no—
118C4⋊Dic17—22no—
119C4⋊Dic17—41no—
120C4⋊Dic17—41no—
121C4⋊Dic17—41no—
122C4⋊Dic17—41no—
123C4⋊Dic17SD1614yescomplemented
124#63225—84no—
125C4×Dic17—41no—
126C4×Dic17—41no—
127C2×#3622—22no—
128C2×#3622—41no—
129C2×#3622—41no—
130#63219—416no—
131#10049—171nomaximal, Sylow
132C23×C17#12011yescharacteristic
133C22×D17—42no—
134C22×D17—82no—
135C22×D17—81no—
136C22×D17—81no—
137C22×D17—82no—
138Q8×C17—22no—
139Q8×C17—22no—
140Q8×C17—41no—
141Q8×C17—41no—
142C2×C4×C17—22no—
143C2×C4×C17—22no—
144C2×C4×C17—41no—
145C2×C4×C17—41no—
146C2×C4×C17—41no—
147C2×C4×C17—42no—
148C2×C4×C17#12011yescharacteristic
149C2×C4×C17#12011yescharacteristic
150C2×C4×C17#12111yescharacteristic
151C2×C4×C17#12111yescharacteristic
152C2×C4×C17#12111yescharacteristic
153C2×C4×C17#12711yescharacteristic
154C2×C4×C17C2×SD1611yescharacteristic
155C2×C4×C17C2×SD1612yes—
156C2×C4×C17#13511yescharacteristic
157C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
158C2×C4×C17#13711yescharacteristic
159C2×C4×C17#13712yes—
160C2×Dic17—24no—
161C2×Dic17—42no—
162C2×Dic17—42no—
163C2×Dic17—42no—
164C2×Dic17—44no—
165D68—44no—
166D68—42no—
167D68—42no—
168D68—44no—
169D68—82no—
170D68—81no—
171D68—81no—
172D68—81no—
173D68—81no—
174Dic34—44no—
175Dic34—82no—
176C8×C17—24no—
177C8×C17—81no—
178#3622—42no—
179C2×#492—171no—
180C2×#460—171no—
181#2567—171no—
182#1982—171no—
183C2×#333—171no—
184C2×Q8⋊C8D1711yescharacteristic, complemented
185#1279—171no—
186C22×C17#39211yescharacteristic
187C22×C17#44812yes—
188C22×C17#44811yescharacteristic
189C22×C17#44911yescharacteristic
190C22×C17#45312yes—
191C2×D17—82no—
192C2×D17—82no—
193C2×D17—82no—
194C2×D17—162no—
195C2×D17—162no—
196C4×C17—21no—
197C4×C17—21no—
198C4×C17—42no—
199C4×C17—81no—
200C4×C17#45011yescharacteristic
201C4×C17#45211yescharacteristic
202C4×C17#46112yes—
203C4×C17#46511yescharacteristic
204C4×C17#46811yescharacteristic
205C4×C17#49211yescharacteristic
206C4×C17#49711yescharacteristic
207Dic17—44no—
208Dic17—82no—
209C22×SD16—342no—
210C22×SD16—341no—
211C2×C4⋊Q8—171no—
212C2×#127—171no—
213C2×C4×Q8C2×D1711yescharacteristic
214#492—342no—
215#492—342no—
216#460—342no—
217#460—342no—
218C2×C4⋊C8—171no—
219C2×C4⋊C8C2×D1711yescharacteristic
220C2×#102—341no—
221C2×#102—342no—
222C2×C4×C8C2×D1711yescharacteristic
223#395—342no—
224#391—341no—
225#389—342no—
226#384—341no—
227C4×C4⋊C4—171no—
228#333—342no—
229Q8⋊C8—22no—
230C2×C17#162212yes—
231C2×C17#162312yes—
232C2×C17#200011yescharacteristic
233C2×C17#200211yescharacteristic
234C2×C17#201311yescharacteristic
235D17—162no—
236C22×Q8—172no—
237C22×Q8D6811yescharacteristic
238C22×D4—341no—
239C22×D4—341no—
240C22×D4—341no—
241C2×SD16—348no—
242C2×SD16—344no—
243C2×SD16—688no—
244C2×SD16—682no—
245C2×SD16—682no—
246C22×C8—21no—
247C22×C8—21no—
248C22×C8—341no—
249C4⋊Q8—341no—
250C4⋊Q8—341no—
251C4⋊Q8—341no—
252C4⋊Q8—341no—
253#127—172no—
254#127—341no—
255#127—341no—
256#127—341no—
257C4×Q8—21no—
258C4×Q8—21no—
259C4×Q8—21no—
260C4×Q8—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—172no—
263C2×C4⋊C4—341no—
264C2×C4⋊C4—341no—
265C2×C4⋊C4—344no—
266C2×C4⋊C4D6811yescharacteristic
267C2×C42—21no—
268C2×C42—341no—
269C2×C42C22×D1711yescharacteristic
270C4⋊C8—21no—
271C4⋊C8—21no—
272C4⋊C8—172no—
273C4⋊C8—341no—
274#102—344no—
275#102—684no—
276C4×C8—21no—
277C4×C8C22⋊D1712yes—
278#95—344no—
279#95—341no—
280C17#1004911yescharacteristic, complemented, Sylow
281C24—681no—
282C2×Q8—21no—
283C2×Q8—21no—
284C2×Q8—21no—
285C2×Q8—21no—
286C2×Q8—21no—
287C2×Q8—342no—
288C2×Q8—342no—
289C2×Q8—684no—
290C2×Q8#6321912yescomplemented
291C2×D4—342no—
292C2×D4—342no—
293C2×D4—342no—
294C2×D4—684no—
295C2×D4—681no—
296C2×D4—681no—
297C2×D4—681no—
298C2×D4—681no—
299C2×D4—681no—
300C2×D4—682no—
301C2×D4—682no—
302C2×D4—681no—
303C2×D4—681no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—342no—
307C22×C4—342no—
308C22×C4C2×D6811yescharacteristic
309C22×C4D4×D1711yescharacteristic
310C22×C4D4×D1711yescharacteristic
311SD16—6816no—
312SD16—688no—
313C2×C8—24no—
314C2×C8—22no—
315C2×C8—42no—
316C2×C8—41no—
317C2×C8—342no—
318C2×C8—681no—
319C2×C8—681no—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—174no—
325C4⋊C4—342no—
326C4⋊C4—681no—
327C4⋊C4—681no—
328C4⋊C4—681no—
329C4⋊C4—681no—
330C4⋊C4—1364no—
331C42—41no—
332C42—41no—
333C42—681no—
334C42—681no—
335C42#6325211yescharacteristic
336C42D4×D1711yescharacteristic
337C42D4×D1711yescharacteristic
338C42C2×C22⋊D1711yescharacteristic
339C23—682no—
340C23—1362no—
341C23—1361no—
342C23—1361no—
343C23—1362no—
344C23#1058756811yescharacteristic
345Q8—22no—
346Q8—22no—
347Q8—41no—
348Q8—41no—
349Q8—684no—
350Q8—1362no—
351D4—684no—
352D4—682no—
353D4—682no—
354D4—684no—
355D4—1362no—
356D4—1361no—
357D4—1361no—
358D4—1361no—
359D4—1361no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—344no—
367C2×C4—682no—
368C2×C4—682no—
369C2×C4—682no—
370C2×C4—684no—
371C2×C4#1058756811yescharacteristic
372C2×C4#1058757011yescharacteristic
373C2×C4#1058757011yescharacteristic
374C2×C4C2×#6321911yescharacteristic
375C2×C4#1058759011yescharacteristic
376C2×C4#1058759411yescharacteristic
377C2×C4#1058759511yescharacteristic
378C2×C4SD16×D1712yes—
379C2×C4#1058760412yes—
380C2×C4#1058761611yescharacteristic
381C2×C4#1058762311yescharacteristic
382C2×C4#1058762611yescharacteristic
383C8—24no—
384C8—81no—
385C8—682no—
386C22—2722no—
387C22—2722no—
388C22—1362no—
389C22—1362no—
390C22—1362no—
391C22#1176271811yescharacteristic
392C22#1176294912yes—
393C22#1176295011yescharacteristic
394C22#1176295611yescharacteristic
395C22#1176297812yes—
396C4—21no—
397C4—21no—
398C4—42no—
399C4—81no—
400C4—684no—
401C4—1362no—
402C4#1176296211yescharacteristic
403C4#1176297311yescharacteristic
404C4#1176301312yes—
405C4#1176303811yescharacteristic
406C4#1176305511yescharacteristic
407C4#1176318011yescharacteristic
408C4#1176320211yescharacteristic
409C2—2722no—
410C2#1452785912yes—
411C2#1452786412yes—
412C2#1452978911yescharacteristic
413C2#1452979811yescharacteristic
414C2#1452986811yescharacteristic
subgroupG/H
centerC23#10587568
commutator, cosocleC2×C42×C17C23
FrattiniC2×C42C22×D17
FittingC2×C17×Q8⋊C8C2
socleC23×C17#120
prime radicalC2×#10587560C22
hypercenterC2×Q8⋊C8D17
hypocenterC17#10049
LCSG ▷ C2×C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×Q8⋊C8

Complements

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

NH ≅ G/Ntype
3. #14533989C2semidirect
5. C2×#11762557C2semidirect
6. #14526624C2semidirect
7. C2×C17×Q8⋊C8C2semidirect
79. C2×Q8×C17SD16semidirect
123. C4⋊Dic17SD16semidirect
184. C2×Q8⋊C8D17semidirect
280. C17#10049semidirect
290. C2×Q8#63219semidirect

Maximal subgroups

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

subgroupindex
C2×#117631822
C2×#117630092
#145339892
#145297102
C2×#117625572
#145266242
C2×C17×Q8⋊C82
#1004917