Finite Group Databasebeta

← #16935537

Subgroups — #16935537

Order 4352.

670 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763207C211yesmaximal, characteristic, complemented
2C2×#11763015C211yesmaximal, characteristic
3#14533733C211yesmaximal, characteristic
4#14529736C211yesmaximal, characteristic, complemented
5C2×#11762554C211yesmaximal, characteristic, complemented
6C2×#11762538C211yesmaximal, characteristic
7C17×#1279C211yesmaximal, characteristic, complemented
8C22×#63219—21no—
9C2×#10587555C2211yescharacteristic
10C2×#10587555C2211yescharacteristic
11C2×#10587583—22no—
12C2×C4×D68C2211yescharacteristic
13#11763207—22no—
14#11763207—22no—
15#11763015—22no—
16#11763015—22no—
17C2×#10587488—22no—
18C2×C4⋊#3622C2211yescharacteristic
19C2×#10587487—21no—
20#11762746—22no—
21#11762711—21no—
22#11762679—21no—
23#11762571—22no—
24#11762554—22no—
25#11762538—22no—
26C2×C17×C4⋊C8C2211yescharacteristic
27C2×C4×C8×C17C2211yescharacteristic
28C4×C4⋊C4×C17C2211yescharacteristic
29C22×D68D411yescharacteristic
30C22×C4×D17—21no—
31C22×Dic34—21no—
32C22×Dic34—21no—
33C22×C8×C17—21no—
34C22×C8×C17—21no—
35C2×C4⋊C4×C17D412yes—
36C2×C42×C17—21no—
37C2×C42×C17C2311yescharacteristic
38C2×#63227—24no—
39C2×#63227—21no—
40C2×C4⋊Dic17—24no—
41C2×C4⋊Dic17—21no—
42C2×C4⋊Dic17—21no—
43C2×C4⋊Dic17—21no—
44C2×C4⋊Dic17D412yes—
45C2×C4⋊Dic17D411yescharacteristic
46C22×#3622—21no—
47C2×#63219—24no—
48C2×#63219—42no—
49C2×#63219—42no—
50#10587583—44no—
51#10587583—44no—
52C4×D68—21no—
53C4×D68—21no—
54C4×D68—21no—
55C4×D68—21no—
56#10587555—21no—
57#10587555—21no—
58#10587555—21no—
59#10587555—21no—
60#10587555—21no—
61#10587555—21no—
62#10587555—21no—
63#10587555D412yes—
64C17×C4⋊C8—21no—
65C17×C4⋊C8D412yes—
66C4×C8×C17—21no—
67C4×C8×C17D412yes—
68C17×#95—21no—
69#10587488—28no—
70#10587487—24no—
71C4⋊#3622—21no—
72C4⋊#3622—21no—
73C23×D17—21no—
74C22×C4×C17—22no—
75C22×C4×C17C2×D411yescharacteristic
76C22×C4×C17C2×D411yescharacteristic
77C22×C4×C17C2×D411yescharacteristic
78C22×Dic17—22no—
79C22×Dic17—21no—
80C22×Dic17—41no—
81C2×D68—21no—
82C2×D68—21no—
83C2×D68—21no—
84C2×D68—21no—
85C2×D68—21no—
86C2×D68SD1612yes—
87C2×C4×D17—41no—
88C2×C4×D17—41no—
89C2×C4×D17—42no—
90C2×C4×D17—42no—
91C2×Dic34—22no—
92C2×Dic34—41no—
93C2×Dic34—41no—
94C2×Dic34—41no—
95C2×Dic34—41no—
96C2×Dic34—41no—
97C2×Dic34—41no—
98C2×Dic34—41no—
99C2×Dic34—42no—
100C2×Dic34—42no—
101C2×C8×C17—24no—
102C2×C8×C17—22no—
103C2×C8×C17—22no—
104C2×C8×C17—41no—
105C2×C8×C17—41no—
106C4⋊C4×C17—22no—
107C4⋊C4×C17SD1614yescomplemented
108C42×C17—41no—
109C42×C17—41no—
110C42×C17C2×D411yescharacteristic
111C42×C17C2×D411yescharacteristic
112C42×C17C2×D411yescharacteristic
113C42×C17#4111yescharacteristic
114#63227—48no—
115#63227—41no—
116#63227—41no—
117C4⋊Dic17—21no—
118C4⋊Dic17—22no—
119C4⋊Dic17—22no—
120C4⋊Dic17—22no—
121C4⋊Dic17—22no—
122C4⋊Dic17—21no—
123C4⋊Dic17—48no—
124C4⋊Dic17—41no—
125C4⋊Dic17—41no—
126C4⋊Dic17—41no—
127C4⋊Dic17—41no—
128C2×#3622—41no—
129C2×#3622—42no—
130#63219—48no—
131#10068—171nomaximal, Sylow
132C23×C17#12011yescharacteristic
133C22×D17—22no—
134C22×D17—41no—
135C22×D17—41no—
136C22×D17—42no—
137C22×D17—42no—
138C2×C4×C17—22no—
139C2×C4×C17—24no—
140C2×C4×C17—42no—
141C2×C4×C17—42no—
142C2×C4×C17#12011yescharacteristic
143C2×C4×C17#12011yescharacteristic
144C2×C4×C17#12111yescharacteristic
145C2×C4×C17#12111yescharacteristic
146C2×C4×C17#12111yescharacteristic
147C2×C4×C17#12711yescharacteristic
148C2×C4×C17C2×SD1612yes—
149C2×C4×C17C2×SD1611yescharacteristic
150C2×C4×C17#13511yescharacteristic
151C2×C4×C17AGL(1,ℤ/8)12yes—
152C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
153C2×C4×C17#13711yescharacteristic
154C2×Dic17—42no—
155C2×Dic17—42no—
156C2×Dic17—41no—
157C2×Dic17—44no—
158C2×Dic17—42no—
159C2×Dic17—81no—
160C2×Dic17—81no—
161D68—22no—
162D68—22no—
163D68—41no—
164D68—41no—
165C4×D17—82no—
166C4×D17—81no—
167C4×D17—81no—
168Dic34—42no—
169Dic34—42no—
170Dic34—82no—
171Dic34—81no—
172Dic34—81no—
173Dic34—81no—
174Dic34—81no—
175C8×C17—24no—
176C8×C17—42no—
177#3622—81no—
178C2×#498—171no—
179C2×#461—171no—
180#2542—171no—
181#1988—171no—
182C2×#332—171no—
183C2×D4⋊C8—171no—
184#1279D1711yescharacteristic, complemented
185C22×C17#39211yescharacteristic
186C22×C17#44811yescharacteristic
187C22×C17#44911yescharacteristic
188C22×C17#45012yes—
189C22×C17#45612yes—
190C2×D17—44no—
191C2×D17—42no—
192C2×D17—82no—
193C2×D17—81no—
194C2×D17—81no—
195C4×C17—21no—
196C4×C17—21no—
197C4×C17—44no—
198C4×C17#45011yescharacteristic
199C4×C17#45211yescharacteristic
200C4×C17#46012yes—
201C4×C17#46511yescharacteristic
202C4×C17#46811yescharacteristic
203C4×C17#49211yescharacteristic
204C4×C17#49611yescharacteristic
205Dic17—82no—
206Dic17—82no—
207Dic17—81no—
208C22×SD16—341no—
209C2×C4⋊Q8—171no—
210C2×C4⋊Q8—171no—
211C2×C22⋊Q8—342no—
212C2×C4×D4—171no—
213#498—342no—
214#498—342no—
215#461—342no—
216#461—342no—
217C2×#106—342no—
218C2×C4⋊C8—171no—
219C2×C4⋊C8C2×D1711yescharacteristic
220C2×#103—341no—
221C2×C4×C8C2×D1711yescharacteristic
222#399—342no—
223#390—341no—
224#384—341no—
225C4×C4⋊C4C2×D1711yescharacteristic
226#340—342no—
227#332—342no—
228D4⋊C8—342no—
229C2×C17#165112yes—
230C2×C17#165412yes—
231C2×C17#200011yescharacteristic
232C2×C17#200511yescharacteristic
233C2×C17#201411yescharacteristic
234D17—82no—
235C22×Q8—341no—
236C22×Q8—341no—
237C22×D4—171no—
238C23×C4—341no—
239C2×SD16—344no—
240C2×SD16—682no—
241C2×SD16—682no—
242C22×C8—21no—
243C22×C8—21no—
244C22×C8—341no—
245C4⋊Q8—172no—
246C4⋊Q8—341no—
247C4⋊Q8—341no—
248C4⋊Q8—341no—
249C4⋊Q8—341no—
250C4⋊Q8—341no—
251C4⋊Q8—341no—
252C4⋊Q8—341no—
253C22⋊Q8—684no—
254C22⋊Q8—684no—
255C4×D4—341no—
256C4×D4—341no—
257C4×D4—341no—
258C4×D4—341no—
259C2×C4⋊C4—171no—
260C2×C4⋊C4—172no—
261C2×C4⋊C4—341no—
262C2×C4⋊C4—341no—
263C2×C4⋊C4—344no—
264C2×C4⋊C4—341no—
265C2×C4⋊C4D6812yes—
266C2×C22⋊C4—344no—
267C2×C22⋊C4—341no—
268C2×C42—21no—
269C2×C42C22×D1711yescharacteristic
270#106—348no—
271C4⋊C8—21no—
272C4⋊C8—341no—
273C4⋊C8—341no—
274C4⋊C8D6812yes—
275#103—344no—
276C4×C8—21no—
277C4×C8D6812yes—
278#95—21no—
279C17#1006811yescharacteristic, complemented, Sylow
280C24—341no—
281C2×Q8—342no—
282C2×Q8—681no—
283C2×Q8—681no—
284C2×Q8—681no—
285C2×Q8—682no—
286C2×Q8—682no—
287C2×Q8—681no—
288C2×Q8—681no—
289C2×Q8—681no—
290C2×Q8—681no—
291C2×D4—172no—
292C2×D4—341no—
293C2×D4—341no—
294C2×D4—341no—
295C2×D4—341no—
296C2×D4—341no—
297C22×C4—22no—
298C22×C4—342no—
299C22×C4—341no—
300C22×C4—681no—
301C22×C4—681no—
302C22×C4—681no—
303C22×C4—682no—
304C22×C4—682no—
305C22×C4C2×D6811yescharacteristic
306C22×C4D4×D1711yescharacteristic
307C22×C4D4×D1711yescharacteristic
308SD16—688no—
309C2×C8—24no—
310C2×C8—22no—
311C2×C8—22no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—681no—
315C2×C8—682no—
316C4⋊C4—22no—
317C4⋊C4—341no—
318C4⋊C4—342no—
319C4⋊C4—342no—
320C4⋊C4—341no—
321C4⋊C4—342no—
322C4⋊C4—342no—
323C4⋊C4—688no—
324C4⋊C4—681no—
325C4⋊C4—681no—
326C4⋊C4—681no—
327C4⋊C4—681no—
328C4⋊C4#6321914yescomplemented
329C22⋊C4—688no—
330C22⋊C4—681no—
331C22⋊C4—681no—
332C42—41no—
333C42—41no—
334C42C2×D6811yescharacteristic
335C42C2×D6811yescharacteristic
336C42D4×D1711yescharacteristic
337C42#6325611yescharacteristic
338C23—342no—
339C23—681no—
340C23—682no—
341C23—681no—
342C23—682no—
343C23#1058756811yescharacteristic
344Q8—682no—
345Q8—682no—
346Q8—1362no—
347Q8—1361no—
348Q8—1361no—
349Q8—1361no—
350Q8—1361no—
351D4—342no—
352D4—342no—
353D4—681no—
354D4—681no—
355C2×C4—22no—
356C2×C4—24no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—684no—
360C2×C4—682no—
361C2×C4—681no—
362C2×C4—682no—
363C2×C4—682no—
364C2×C4—1361no—
365C2×C4—1361no—
366C2×C4—1362no—
367C2×C4—1361no—
368C2×C4—1361no—
369C2×C4#1058756011yescharacteristic
370C2×C4#1058756811yescharacteristic
371C2×C4#1058756811yescharacteristic
372C2×C4#1058758111yescharacteristic
373C2×C4#1058758111yescharacteristic
374C2×C4#1058758111yescharacteristic
375C2×C4C2×#6321912yes—
376C2×C4#1058759412yes—
377C2×C4#1058759711yescharacteristic
378C2×C4SD16×D1711yescharacteristic
379C2×C4#1058760211yescharacteristic
380C2×C4#1058760411yescharacteristic
381C8—24no—
382C8—42no—
383C8—1361no—
384C22—684no—
385C22—682no—
386C22—1361no—
387C22—1362no—
388C22—1361no—
389C22#1176271911yescharacteristic
390C22#1176294911yescharacteristic
391C22#1176295511yescharacteristic
392C22#1176296212yes—
393C22#1176299312yes—
394C4—21no—
395C4—21no—
396C4—44no—
397C4—1361no—
398C4—1362no—
399C4—1362no—
400C4#1176296111yescharacteristic
401C4#1176297411yescharacteristic
402C4#1176301012yes—
403C4#1176303911yescharacteristic
404C4#1176305611yescharacteristic
405C4#1176318211yescharacteristic
406C4#1176319811yescharacteristic
407C2—1362no—
408C2#1452805012yes—
409C2#1452807112yes—
410C2#1452979011yescharacteristic
411C2#1452982311yescharacteristic
412C2#1452987411yescharacteristic
subgroupG/H
centerC23#10587568
commutator, cosocleC2×C42×C17C23
FrattiniC2×C42C22×D17
FittingC17×#1279C2
socleC23×C17#120
prime radicalC22×D68D4
hypercenter#1279D17
hypocenterC17#10068
LCSG ▷ C2×C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1279

Complements

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

NH ≅ G/Ntype
1. C2×#11763207C2semidirect
4. #14529736C2semidirect
5. C2×#11762554C2semidirect
7. C17×#1279C2semidirect
107. C4⋊C4×C17SD16semidirect
184. #1279D17semidirect
279. C17#10068semidirect
328. C4⋊C4#63219semidirect

Maximal subgroups

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

subgroupindex
C2×#117632072
C2×#117630152
#145337332
#145297362
C2×#117625542
C2×#117625382
C17×#12792
#1006817