Finite Group Databasebeta

← #16935429

Subgroups — #16935429

Order 4352.

634 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×#11763157C211yesmaximal, characteristic
2C2×#11763009C211yesmaximal, characteristic
3#14529711C211yesmaximal, characteristic
4C2×#11762555C211yesmaximal, characteristic, complemented
5C2×Q8⋊#3622C211yesmaximal, characteristic, complemented
6#14526625C211yesmaximal, characteristic, complemented
7C17×#2568C211yesmaximal, characteristic, complemented
8C22×#63230—21no—
9C2×#10587560C2211yescharacteristic
10#11763157—22no—
11#11763157—22no—
12#11763009—22no—
13#11763009—22no—
14C2×C4⋊#3622C2211yescharacteristic
15C2×C4⋊#3622C2211yescharacteristic
16C2×#10587480—22no—
17C2×#10587480—21no—
18C2×#10587480—22no—
19C2×C4×#3622C2211yescharacteristic
20#11762555—22no—
21Q8⋊#3622—22no—
22C2×C17×C4⋊Q8C2211yescharacteristic
23C2×C4×Q8×C17C2211yescharacteristic
24C17×#400—22no—
25C17×#391—22no—
26C17×#391—21no—
27C17×#382—21no—
28C4×C4⋊C4×C17C2211yescharacteristic
29C22×Q8×C17—21no—
30C22×Q8×C17D411yescharacteristic
31C22×D68—21no—
32C22×D68—21no—
33C22×D68—21no—
34C17×C4⋊Q8—21no—
35C17×C4⋊Q8—21no—
36C17×C4⋊Q8—21no—
37C17×C4⋊Q8—21no—
38C4×Q8×C17—21no—
39C4×Q8×C17—21no—
40C4×Q8×C17—21no—
41C4×Q8×C17—21no—
42C2×C4⋊C4×C17—21no—
43C2×C4⋊C4×C17—21no—
44C2×C4⋊C4×C17D411yescharacteristic
45C2×C4⋊C4×C17D412yes—
46C2×C4⋊C4×C17D412yes—
47C2×C42×C17—21no—
48C2×C42×C17—21no—
49C2×C42×C17C2311yescharacteristic
50C2×#63230—24no—
51C2×#63230—42no—
52C2×#63230—42no—
53C22×#3622—21no—
54C22×#3622—21no—
55C22×#3622—21no—
56#10587560—21no—
57#10587560—21no—
58#10587560—21no—
59#10587560D412yes—
60C17×#95—24no—
61C17×#95—24no—
62C17×#95—21no—
63#10587480—24no—
64#10587480—28no—
65#10587480—44no—
66C4⋊#3622—21no—
67C4⋊#3622—21no—
68C4⋊#3622—21no—
69C4⋊#3622D412yes—
70C4×#3622—21no—
71C4×#3622D412yes—
72C23×D17—41no—
73C2×Q8×C17—21no—
74C2×Q8×C17—21no—
75C2×Q8×C17—21no—
76C2×Q8×C17—21no—
77C2×Q8×C17—21no—
78C2×Q8×C17—41no—
79C2×Q8×C17—41no—
80C2×Q8×C17—42no—
81C2×Q8×C17—42no—
82C2×Q8×C17SD1612yes—
83C22×C4×C17—21no—
84C22×C4×C17—22no—
85C22×C4×C17—22no—
86C22×C4×C17—21no—
87C22×C4×C17C2×D411yescharacteristic
88C22×C4×C17C2×D411yescharacteristic
89C22×C4×C17C2×D411yescharacteristic
90C2×D68—22no—
91C2×D68—22no—
92C2×D68—22no—
93C2×D68—44no—
94C2×D68—41no—
95C2×D68—41no—
96C2×D68—42no—
97C2×D68—42no—
98C2×D68—41no—
99C2×D68—41no—
100C2×D68—41no—
101C2×D68—41no—
102C2×D68—41no—
103C4⋊C4×C17—22no—
104C4⋊C4×C17—21no—
105C4⋊C4×C17—22no—
106C4⋊C4×C17—21no—
107C4⋊C4×C17—22no—
108C4⋊C4×C17—22no—
109C4⋊C4×C17—41no—
110C4⋊C4×C17—41no—
111C4⋊C4×C17—41no—
112C4⋊C4×C17—41no—
113C42×C17—41no—
114C42×C17—41no—
115C42×C17—41no—
116C42×C17—41no—
117C42×C17C2×D411yescharacteristic
118C42×C17C2×D411yescharacteristic
119C42×C17C2×D411yescharacteristic
120C42×C17#4111yescharacteristic
121#63230—48no—
122C2×#3622—22no—
123C2×#3622—22no—
124C2×#3622—42no—
125C2×#3622—41no—
126C2×#3622—41no—
127C2×#3622—41no—
128C2×#3622—42no—
129#10053—171nomaximal, Sylow
130C23×C17#12011yescharacteristic
131C22×D17—42no—
132C22×D17—81no—
133C22×D17—81no—
134C22×D17—82no—
135C22×D17—82no—
136Q8×C17—22no—
137Q8×C17—22no—
138Q8×C17—41no—
139Q8×C17—41no—
140Q8×C17—82no—
141Q8×C17—81no—
142Q8×C17—81no—
143C2×C4×C17—22no—
144C2×C4×C17—22no—
145C2×C4×C17—41no—
146C2×C4×C17—41no—
147C2×C4×C17—41no—
148C2×C4×C17—42no—
149C2×C4×C17—42no—
150C2×C4×C17—42no—
151C2×C4×C17—44no—
152C2×C4×C17—42no—
153C2×C4×C17—42no—
154C2×C4×C17#12011yescharacteristic
155C2×C4×C17#12011yescharacteristic
156C2×C4×C17#12111yescharacteristic
157C2×C4×C17#12111yescharacteristic
158C2×C4×C17#12111yescharacteristic
159C2×C4×C17#12711yescharacteristic
160C2×C4×C17C2×SD1611yescharacteristic
161C2×C4×C17#13511yescharacteristic
162C2×C4×C17#13512yes—
163C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
164C2×C4×C17AGL(1,ℤ/8)12yes—
165C2×C4×C17#13711yescharacteristic
166D68—44no—
167D68—42no—
168D68—42no—
169D68—44no—
170D68—82no—
171D68—81no—
172D68—81no—
173D68—81no—
174D68—81no—
175#3622—42no—
176#3622—42no—
177#3622—81no—
178C2×#486—171no—
179C2×#460—171no—
180#2568D1711yescharacteristic, complemented
181#1982—171no—
182C2×#332—171no—
183C2×Q8⋊C8—171no—
184#1279—171no—
185C22×C17#39211yescharacteristic
186C22×C17#44811yescharacteristic
187C22×C17#44912yes—
188C22×C17#44911yescharacteristic
189C22×C17#45412yes—
190C2×D17—82no—
191C2×D17—82no—
192C2×D17—82no—
193C2×D17—162no—
194C2×D17—162no—
195C4×C17—21no—
196C4×C17—21no—
197C4×C17—42no—
198C4×C17—82no—
199C4×C17—81no—
200C4×C17—82no—
201C4×C17#45011yescharacteristic
202C4×C17#45211yescharacteristic
203C4×C17#46312yes—
204C4×C17#46511yescharacteristic
205C4×C17#46811yescharacteristic
206C4×C17#49511yescharacteristic
207C4×C17#49611yescharacteristic
208C22×SD16—341no—
209C2×C4⋊Q8C2×D1711yescharacteristic
210C2×#127—171no—
211C2×C4×Q8C2×D1711yescharacteristic
212#486—342no—
213#486—342no—
214#460—342no—
215#460—342no—
216C2×C4⋊C8—171no—
217C2×C4⋊C8—171no—
218C2×#102—342no—
219C2×#102—341no—
220C2×#102—342no—
221C2×C4×C8—171no—
222#400—22no—
223#391—21no—
224#391—22no—
225#382—21no—
226C4×C4⋊C4C2×D1711yescharacteristic
227#332—342no—
228Q8⋊C8—342no—
229C2×C17#162612yes—
230C2×C17#162812yes—
231C2×C17#200011yescharacteristic
232C2×C17#200311yescharacteristic
233C2×C17#201211yescharacteristic
234D17—162no—
235C22×Q8—21no—
236C22×Q8C22⋊D1711yescharacteristic
237C22×D4—341no—
238C22×D4—341no—
239C22×D4—341no—
240C2×SD16—344no—
241C2×SD16—682no—
242C2×SD16—682no—
243C22×C8—341no—
244C22×C8—341no—
245C22×C8—341no—
246C4⋊Q8—21no—
247C4⋊Q8—21no—
248C4⋊Q8—21no—
249C4⋊Q8—21no—
250#127—172no—
251#127—341no—
252#127—341no—
253#127—341no—
254C4×Q8—21no—
255C4×Q8—21no—
256C4×Q8—21no—
257C4×Q8—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4C22⋊D1711yescharacteristic
261C2×C4⋊C4C22⋊D1712yes—
262C2×C4⋊C4C22⋊D1712yes—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42C22×D1711yescharacteristic
266C4⋊C8—172no—
267C4⋊C8—341no—
268C4⋊C8—341no—
269C4⋊C8—341no—
270#102—344no—
271#102—348no—
272#102—684no—
273C4×C8—172no—
274C4×C8—341no—
275#95—24no—
276#95—24no—
277#95—21no—
278C17#1005311yescharacteristic, complemented, Sylow
279C24—681no—
280C2×Q8—21no—
281C2×Q8—21no—
282C2×Q8—21no—
283C2×Q8—21no—
284C2×Q8—21no—
285C2×Q8—41no—
286C2×Q8—41no—
287C2×Q8—42no—
288C2×Q8—42no—
289C2×Q8#6323012yes—
290C2×D4—342no—
291C2×D4—342no—
292C2×D4—342no—
293C2×D4—684no—
294C2×D4—681no—
295C2×D4—681no—
296C2×D4—681no—
297C2×D4—681no—
298C2×D4—681no—
299C2×D4—682no—
300C2×D4—681no—
301C2×D4—681no—
302C2×D4—682no—
303C22×C4—21no—
304C22×C4—22no—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4C2×C22⋊D1711yescharacteristic
308C22×C4C2×C22⋊D1711yescharacteristic
309C22×C4C2×C22⋊D1711yescharacteristic
310SD16—688no—
311C2×C8—342no—
312C2×C8—342no—
313C2×C8—682no—
314C2×C8—681no—
315C2×C8—681no—
316C2×C8—681no—
317C2×C8—682no—
318C4⋊C4—21no—
319C4⋊C4—22no—
320C4⋊C4—21no—
321C4⋊C4—22no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C42—41no—
329C42—41no—
330C42—41no—
331C42—41no—
332C42D4×D1711yescharacteristic
333C42D4×D1711yescharacteristic
334C42D4×D1711yescharacteristic
335C42#6325411yescharacteristic
336C23—682no—
337C23—1361no—
338C23—1362no—
339C23—1361no—
340C23—1362no—
341C23#1058763911yescharacteristic
342Q8—22no—
343Q8—22no—
344Q8—41no—
345Q8—41no—
346Q8—82no—
347Q8—81no—
348Q8—81no—
349D4—684no—
350D4—682no—
351D4—682no—
352D4—684no—
353D4—1362no—
354D4—1361no—
355D4—1361no—
356D4—1361no—
357D4—1361no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—44no—
365C2×C4—41no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4#1058762311yescharacteristic
370C2×C4#1058762311yescharacteristic
371C2×C4#1058762411yescharacteristic
372C2×C4#1058762511yescharacteristic
373C2×C4#1058762511yescharacteristic
374C2×C4#1058762611yescharacteristic
375C2×C4C2×#6323011yescharacteristic
376C2×C4#1058762811yescharacteristic
377C2×C4#1058763511yescharacteristic
378C2×C4#1058763512yes—
379C2×C4#1058763611yescharacteristic
380C2×C4#1058763612yes—
381C8—682no—
382C8—682no—
383C8—1361no—
384C22—2722no—
385C22—2722no—
386C22—1362no—
387C22—1362no—
388C22—1362no—
389C22#1176271711yescharacteristic
390C22#1176294811yescharacteristic
391C22#1176295412yes—
392C22#1176295411yescharacteristic
393C22#1176298212yes—
394C4—21no—
395C4—21no—
396C4—42no—
397C4—81no—
398C4—82no—
399C4—82no—
400C4#1176295911yescharacteristic
401C4#1176296911yescharacteristic
402C4#1176302112yes—
403C4#1176304011yescharacteristic
404C4#1176305711yescharacteristic
405C4#1176319111yescharacteristic
406C4#1176319911yescharacteristic
407C2—2722no—
408C2#1452788012yes—
409C2#1452789412yes—
410C2#1452978511yescharacteristic
411C2#1452980411yescharacteristic
412C2#1452986111yescharacteristic
subgroupG/H
centerC23#10587639
commutator, cosocleC2×C42×C17C23
FrattiniC2×C42C22×D17
FittingC17×#2568C2
socleC23×C17#120
prime radicalC2×#10587560C22
hypercenter#2568D17
hypocenterC17#10053
LCSG ▷ C2×C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2568

Complements

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

NH ≅ G/Ntype
4. C2×#11762555C2semidirect
5. C2×Q8⋊#3622C2semidirect
6. #14526625C2semidirect
7. C17×#2568C2semidirect
180. #2568D17semidirect
278. C17#10053semidirect

Maximal subgroups

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

subgroupindex
C2×#117631572
C2×#117630092
#145297112
C2×#117625552
C2×Q8⋊#36222
#145266252
C17×#25682
#1005317