Finite Group Databasebeta

← #16935415

Subgroups — #16935415

Order 4352.

670 conjugacy classes of subgroups.

The subgroup lattice has 428 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×#2546C211yesmaximal, 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⋊Q8—21no—
23C2×C17×C4⋊Q8C2211yescharacteristic
24C2×C4×Q8×C17C2211yescharacteristic
25C17×#398—22no—
26C17×#391—21no—
27C17×#389—22no—
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—
38C17×C4⋊Q8—42no—
39C17×C4⋊Q8—42no—
40C4×Q8×C17—21no—
41C4×Q8×C17—21no—
42C4×Q8×C17—21no—
43C4×Q8×C17—21no—
44C2×C4⋊C4×C17—24no—
45C2×C4⋊C4×C17—21no—
46C2×C4⋊C4×C17—21no—
47C2×C4⋊C4×C17D411yescharacteristic
48C2×C4⋊C4×C17D412yes—
49C2×C4⋊C4×C17D412yes—
50C2×C42×C17—21no—
51C2×C42×C17—21no—
52C2×C42×C17C2311yescharacteristic
53C2×#63230—24no—
54C2×#63230—42no—
55C2×#63230—42no—
56C22×#3622—21no—
57C22×#3622—21no—
58C22×#3622—21no—
59#10587560—21no—
60#10587560—21no—
61#10587560—21no—
62#10587560D412yes—
63C17×#95—24no—
64C17×#95—21no—
65#10587480—24no—
66#10587480—28no—
67#10587480—44no—
68C4⋊#3622—21no—
69C4⋊#3622—21no—
70C4⋊#3622—21no—
71C4⋊#3622D412yes—
72C4×#3622—21no—
73C4×#3622D412yes—
74C23×D17—41no—
75C2×Q8×C17—21no—
76C2×Q8×C17—21no—
77C2×Q8×C17—21no—
78C2×Q8×C17—21no—
79C2×Q8×C17—21no—
80C2×Q8×C17—41no—
81C2×Q8×C17—41no—
82C2×Q8×C17—41no—
83C2×Q8×C17—41no—
84C2×Q8×C17—41no—
85C2×Q8×C17—41no—
86C2×Q8×C17D812yes—
87C22×C4×C17—21no—
88C22×C4×C17—22no—
89C22×C4×C17—22no—
90C22×C4×C17—21no—
91C22×C4×C17C2×D411yescharacteristic
92C22×C4×C17C2×D411yescharacteristic
93C22×C4×C17C2×D411yescharacteristic
94C2×D68—22no—
95C2×D68—22no—
96C2×D68—22no—
97C2×D68—44no—
98C2×D68—41no—
99C2×D68—42no—
100C2×D68—41no—
101C2×D68—41no—
102C2×D68—41no—
103C2×D68—42no—
104C2×D68—41no—
105C2×D68—41no—
106C2×D68—41no—
107C4⋊C4×C17—22no—
108C4⋊C4×C17—22no—
109C4⋊C4×C17—21no—
110C4⋊C4×C17—22no—
111C4⋊C4×C17—21no—
112C4⋊C4×C17—48no—
113C4⋊C4×C17—41no—
114C4⋊C4×C17—41no—
115C4⋊C4×C17—41no—
116C4⋊C4×C17—41no—
117C4⋊C4×C17SD1614yescomplemented
118C42×C17—41no—
119C42×C17—41no—
120C42×C17—41no—
121C42×C17—41no—
122C42×C17C2×D411yescharacteristic
123C42×C17C2×D411yescharacteristic
124C42×C17C2×D411yescharacteristic
125C42×C17#4111yescharacteristic
126#63230—48no—
127C2×#3622—22no—
128C2×#3622—22no—
129C2×#3622—42no—
130C2×#3622—41no—
131C2×#3622—41no—
132C2×#3622—41no—
133C2×#3622—42no—
134#10051—171nomaximal, Sylow
135C23×C17#12011yescharacteristic
136C22×D17—42no—
137C22×D17—81no—
138C22×D17—82no—
139C22×D17—81no—
140C22×D17—82no—
141Q8×C17—22no—
142Q8×C17—22no—
143Q8×C17—41no—
144Q8×C17—41no—
145Q8×C17—81no—
146Q8×C17—81no—
147Q8×C17—81no—
148Q8×C17—81no—
149C2×C4×C17—22no—
150C2×C4×C17—22no—
151C2×C4×C17—24no—
152C2×C4×C17—41no—
153C2×C4×C17—42no—
154C2×C4×C17—41no—
155C2×C4×C17—42no—
156C2×C4×C17—41no—
157C2×C4×C17—42no—
158C2×C4×C17—42no—
159C2×C4×C17—42no—
160C2×C4×C17—42no—
161C2×C4×C17#12011yescharacteristic
162C2×C4×C17#12011yescharacteristic
163C2×C4×C17#12111yescharacteristic
164C2×C4×C17#12111yescharacteristic
165C2×C4×C17#12111yescharacteristic
166C2×C4×C17#12711yescharacteristic
167C2×C4×C17C2×D811yescharacteristic
168C2×C4×C17C2×SD1612yes—
169C2×C4×C17#13511yescharacteristic
170C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
171C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
172C2×C4×C17#13712yes—
173D68—44no—
174D68—42no—
175D68—42no—
176D68—44no—
177D68—82no—
178D68—81no—
179D68—81no—
180D68—81no—
181D68—81no—
182#3622—42no—
183#3622—42no—
184#3622—81no—
185C2×#486—171no—
186C2×#460—171no—
187#2546D1711yescharacteristic, complemented
188#1982—171no—
189C2×#332—171no—
190C2×Q8⋊C8—171no—
191#1279—171no—
192C22×C17#39211yescharacteristic
193C22×C17#44711yescharacteristic
194C22×C17#44812yes—
195C22×C17#44911yescharacteristic
196C22×C17#45412yes—
197C2×D17—82no—
198C2×D17—82no—
199C2×D17—82no—
200C2×D17—162no—
201C2×D17—162no—
202C4×C17—21no—
203C4×C17—21no—
204C4×C17—44no—
205C4×C17—42no—
206C4×C17—82no—
207C4×C17—81no—
208C4×C17#44711yescharacteristic
209C4×C17#44911yescharacteristic
210C4×C17#46112yes—
211C4×C17#46611yescharacteristic
212C4×C17#46911yescharacteristic
213C4×C17#49211yescharacteristic
214C4×C17#49711yescharacteristic
215C22×SD16—341no—
216C2×C4⋊Q8—21no—
217C2×C4⋊Q8C2×D1711yescharacteristic
218C2×#127—171no—
219C2×C4×Q8C2×D1711yescharacteristic
220#486—342no—
221#486—342no—
222#460—342no—
223#460—342no—
224C2×C4⋊C8—171no—
225C2×C4⋊C8—171no—
226C2×#102—342no—
227C2×#102—341no—
228C2×#102—342no—
229C2×C4×C8—171no—
230#398—22no—
231#391—21no—
232#389—22no—
233C4×C4⋊C4C2×D1711yescharacteristic
234#332—342no—
235Q8⋊C8—342no—
236C2×C17#162612yes—
237C2×C17#162712yes—
238C2×C17#199911yescharacteristic
239C2×C17#200311yescharacteristic
240C2×C17#201311yescharacteristic
241D17—162no—
242C22×Q8—21no—
243C22×Q8C22⋊D1711yescharacteristic
244C22×D4—341no—
245C22×D4—341no—
246C22×D4—341no—
247C2×SD16—344no—
248C2×SD16—682no—
249C2×SD16—682no—
250C22×C8—341no—
251C22×C8—341no—
252C22×C8—341no—
253C4⋊Q8—21no—
254C4⋊Q8—21no—
255C4⋊Q8—21no—
256C4⋊Q8—21no—
257C4⋊Q8—42no—
258C4⋊Q8—42no—
259#127—172no—
260#127—341no—
261#127—341no—
262#127—341no—
263C4×Q8—21no—
264C4×Q8—21no—
265C4×Q8—21no—
266C4×Q8—21no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4C22⋊D1711yescharacteristic
271C2×C4⋊C4C22⋊D1712yes—
272C2×C4⋊C4C22⋊D1712yes—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42C22×D1711yescharacteristic
276C4⋊C8—172no—
277C4⋊C8—341no—
278C4⋊C8—341no—
279C4⋊C8—341no—
280#102—344no—
281#102—348no—
282#102—684no—
283C4×C8—172no—
284C4×C8—341no—
285#95—24no—
286#95—21no—
287C17#1005111yescharacteristic, complemented, Sylow
288C24—681no—
289C2×Q8—21no—
290C2×Q8—21no—
291C2×Q8—21no—
292C2×Q8—21no—
293C2×Q8—21no—
294C2×Q8—41no—
295C2×Q8—41no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×Q8#6322812yes—
301C2×D4—342no—
302C2×D4—342no—
303C2×D4—342no—
304C2×D4—684no—
305C2×D4—681no—
306C2×D4—681no—
307C2×D4—681no—
308C2×D4—681no—
309C2×D4—681no—
310C2×D4—682no—
311C2×D4—682no—
312C2×D4—681no—
313C2×D4—681no—
314C22×C4—22no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4C2×C22⋊D1711yescharacteristic
319C22×C4C2×C22⋊D1711yescharacteristic
320C22×C4C2×C22⋊D1711yescharacteristic
321SD16—688no—
322C2×C8—342no—
323C2×C8—342no—
324C2×C8—682no—
325C2×C8—681no—
326C2×C8—681no—
327C2×C8—681no—
328C2×C8—682no—
329C4⋊C4—22no—
330C4⋊C4—21no—
331C4⋊C4—22no—
332C4⋊C4—21no—
333C4⋊C4—22no—
334C4⋊C4—48no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4#6323014yescomplemented
340C42—41no—
341C42—41no—
342C42—41no—
343C42—41no—
344C42D4×D1711yescharacteristic
345C42D4×D1711yescharacteristic
346C42D4×D1711yescharacteristic
347C42#6325411yescharacteristic
348C23—682no—
349C23—1362no—
350C23—1361no—
351C23—1362no—
352C23—1361no—
353C23#1058763911yescharacteristic
354Q8—22no—
355Q8—22no—
356Q8—41no—
357Q8—41no—
358Q8—81no—
359Q8—81no—
360Q8—81no—
361Q8—81no—
362D4—684no—
363D4—682no—
364D4—684no—
365D4—682no—
366D4—1362no—
367D4—1361no—
368D4—1361no—
369D4—1361no—
370D4—1361no—
371C2×C4—22no—
372C2×C4—24no—
373C2×C4—22no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—41no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4C2×#6322811yescharacteristic
384C2×C4#1058761811yescharacteristic
385C2×C4#1058762311yescharacteristic
386C2×C4#1058762311yescharacteristic
387C2×C4#1058762411yescharacteristic
388C2×C4#1058762511yescharacteristic
389C2×C4#1058762511yescharacteristic
390C2×C4#1058762611yescharacteristic
391C2×C4C2×#6323012yes—
392C2×C4#1058762812yes—
393C2×C4#1058763511yescharacteristic
394C2×C4#1058763611yescharacteristic
395C8—682no—
396C8—682no—
397C8—1361no—
398C22—2722no—
399C22—2722no—
400C22—1362no—
401C22—1362no—
402C22—1362no—
403C22#1176271711yescharacteristic
404C22#1176294311yescharacteristic
405C22#1176294812yes—
406C22#1176295411yescharacteristic
407C22#1176298212yes—
408C4—21no—
409C4—21no—
410C4—44no—
411C4—42no—
412C4—81no—
413C4—82no—
414C4#1176294211yescharacteristic
415C4#1176295211yescharacteristic
416C4#1176301112yes—
417C4#1176304311yescharacteristic
418C4#1176306011yescharacteristic
419C4#1176318311yescharacteristic
420C4#1176320111yescharacteristic
421C2—2722no—
422C2#1452788012yes—
423C2#1452788712yes—
424C2#1452978011yescharacteristic
425C2#1452980411yescharacteristic
426C2#1452986711yescharacteristic
subgroupG/H
centerC23#10587639
commutator, cosocleC2×C42×C17C23
FrattiniC2×C42C22×D17
FittingC17×#2546C2
socleC23×C17#120
prime radicalC2×#10587560C22
hypercenter#2546D17
hypocenterC17#10051
LCSG ▷ C2×C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2546

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×#2546C2semidirect
117. C4⋊C4×C17SD16semidirect
187. #2546D17semidirect
287. C17#10051semidirect
339. C4⋊C4#63230semidirect

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×#25462
#1005117