Finite Group Databasebeta

← #16950308

Subgroups — #16950308

Order 4352.

544 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 107 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#10587622C211yesmaximal, characteristic
2C2×#11762603C212yesmaximal
3#14528772C211yesmaximal, characteristic
4#14526908C212yesmaximal
5C17×#1799C211yesmaximal, characteristic
6C2×#10587622C2211yescharacteristic
7C2×C4×Dic34C411yescharacteristic, complemented
8C2×#10587505C2212yes—
9#11762714C411yescharacteristic, complemented
10#11762712C2211yescharacteristic
11#11762703—21no—
12#11762703—21no—
13#11762674C411yescharacteristic, complemented
14#11762669—21no—
15C4×C22⋊Dic17—21no—
16C4×C22⋊Dic17—21no—
17C42×Dic17C411yescharacteristic, complemented
18#11762603—24no—
19#11762576—21no—
20C2×C4×D4×C17C2211yescharacteristic
21C2×C17×#99C2212yes—
22C17×#380—21no—
23C17×#342—21no—
24C22×D4×C17C2311yescharacteristic
25C23×C4×C17D411yescharacteristic
26C23×C4×C17D411yescharacteristic
27C22×Dic34C2×C411yescharacteristic
28C4×D4×C17—41no—
29C4×D4×C17—41no—
30C2×C4⋊C4×C17—41no—
31C2×C4⋊C4×C17C2×C411yescharacteristic
32C2×C22⋊C4×C17—22no—
33C2×C22⋊C4×C17D411yescharacteristic
34C2×C22⋊C4×C17D411yescharacteristic
35C2×C42×C17C2×C411yescharacteristic
36C2×C22⋊Dic17—21no—
37C2×C22⋊Dic17—21no—
38C2×C22⋊Dic17—21no—
39C2×C22⋊Dic17—21no—
40C2×C22⋊Dic17—22no—
41#10587622—21no—
42#10587622—21no—
43#10587622D412yes—
44#10587622Q812yes—
45C2×C4⋊Dic17—41no—
46C2×C4⋊Dic17C2×C411yescharacteristic
47C2×#63225—21no—
48C2×C4×Dic17—21no—
49C2×C4×Dic17—21no—
50C2×C4×Dic17—21no—
51C2×C4×Dic17C2×C411yescharacteristic
52C2×C4×Dic17C2×C411yescharacteristic
53C4×Dic34—41no—
54C4×Dic34—41no—
55C17×#99—22no—
56C17×#99—22no—
57C17×#95—41no—
58#10587505—22no—
59#10587505—22no—
60#10587502—21no—
61#10587502—21no—
62#10587502—21no—
63#10587502—21no—
64#10587502—41no—
65C24×C17C2×D411yescharacteristic
66C24×C17C2×D411yescharacteristic
67C2×D4×C17—41no—
68C2×D4×C17—41no—
69C2×D4×C17C2×D411yescharacteristic
70C2×D4×C17C2×Q811yescharacteristic
71C2×D4×C17#4111yescharacteristic
72C2×D4×C17#4111yescharacteristic
73C22×C4×C17—21no—
74C22×C4×C17—21no—
75C22×C4×C17—21no—
76C22×C4×C17—21no—
77C22×C4×C17—42no—
78C22×C4×C17—42no—
79C22×C4×C17—42no—
80C22×C4×C17C22⋊C411yescharacteristic
81C22×C4×C17C22⋊C411yescharacteristic
82C22×C4×C17C22⋊C411yescharacteristic
83C22×C4×C17C22⋊C411yescharacteristic
84C22×C4×C17C22×C411yescharacteristic
85C22×Dic17—21no—
86C22×Dic17—21no—
87C22×Dic17—21no—
88C22×Dic17—21no—
89C22×Dic17—42no—
90C2×Dic34—21no—
91C2×Dic34—41no—
92C2×Dic34—41no—
93C2×Dic34C4⋊C412yescomplemented
94C4⋊C4×C17—41no—
95C4⋊C4×C17—82no—
96C22⋊C4×C17—41no—
97C22⋊C4×C17—41no—
98C22⋊C4×C17—42no—
99C22⋊C4×C17—42no—
100C42×C17—21no—
101C42×C17—21no—
102C22⋊Dic17—22no—
103C22⋊Dic17—22no—
104C22⋊Dic17—22no—
105C22⋊Dic17—22no—
106C22⋊Dic17—42no—
107C22⋊Dic17—42no—
108C22⋊Dic17—42no—
109C22⋊Dic17—42no—
110C4⋊Dic17—41no—
111C4⋊Dic17—82no—
112#63225—42no—
113C4×Dic17—21no—
114C4×Dic17—42no—
115C4×Dic17—42no—
116C4×Dic17—41no—
117C4×Dic17—41no—
118C4×Dic17—41no—
119C4×Dic17C4⋊C412yescomplemented
120#12798—171nomaximal, Sylow
121C23×C17—21no—
122C23×C17—21no—
123C23×C17—21no—
124C23×C17—21no—
125C23×C17—41no—
126C23×C17—41no—
127C23×C17—41no—
128C23×C17—41no—
129C23×C17C2×C22⋊C411yescharacteristic
130C23×C17#12111yescharacteristic
131C23×C17#12111yescharacteristic
132C23×C17C22⋊Q811yescharacteristic
133C23×C17C22⋊Q811yescharacteristic
134D4×C17—41no—
135D4×C17—41no—
136D4×C17—41no—
137D4×C17—41no—
138C2×C4×C17—21no—
139C2×C4×C17—21no—
140C2×C4×C17—22no—
141C2×C4×C17—21no—
142C2×C4×C17—21no—
143C2×C4×C17—21no—
144C2×C4×C17—42no—
145C2×C4×C17—42no—
146C2×C4×C17—41no—
147C2×C4×C17—41no—
148C2×C4×C17—41no—
149C2×C4×C17—81no—
150C2×C4×C17—81no—
151C2×C4×C17—82no—
152C2×C4×C17—82no—
153C2×C4×C17—82no—
154C2×C4×C17#9911yescharacteristic
155C2×C4×C17#9911yescharacteristic
156C2×C4×C17C2×C4⋊C411yescharacteristic
157C2×C4×C17#11711yescharacteristic
158C2×Dic17—22no—
159C2×Dic17—22no—
160C2×Dic17—22no—
161C2×Dic17—41no—
162C2×Dic17—41no—
163C2×Dic17—41no—
164C2×Dic17—41no—
165C2×Dic17—41no—
166C2×Dic17—41no—
167C2×Dic17—41no—
168C2×Dic17—41no—
169C2×Dic17—41no—
170C2×Dic17—82no—
171C2×Dic17—82no—
172C2×Dic17—82no—
173Dic34—42no—
174Dic34—42no—
175C4×#124—171no—
176C2×#354—172no—
177#1799—171no—
178#1799D1711yescharacteristic
179#1393—172no—
180C22×C17—21no—
181C22×C17—21no—
182C22×C17—21no—
183C22×C17—21no—
184C22×C17—21no—
185C22×C17—21no—
186C22×C17—41no—
187C22×C17—41no—
188C22×C17—41no—
189C22×C17—41no—
190C22×C17—41no—
191C22×C17—41no—
192C22×C17—41no—
193C22×C17—41no—
194C22×C17—81no—
195C22×C17—81no—
196C22×C17#38011yescharacteristic
197C22×C17C2×#9911yescharacteristic
198C22×C17#41011yescharacteristic
199C4×C17—21no—
200C4×C17—21no—
201C4×C17—21no—
202C4×C17—41no—
203C4×C17—41no—
204C4×C17—41no—
205C4×C17—162no—
206C4×C17#35412yescomplemented
207Dic17—42no—
208Dic17—42no—
209Dic17—42no—
210Dic17—81no—
211Dic17—162no—
212C2×#124—171no—
213C2×C4×Q8—171no—
214C2×C4×D4C2×D1711yescharacteristic
215C2×#99—172no—
216C2×#99C2×D1712yes—
217#391—171no—
218#390—171no—
219#388—341no—
220#388—341no—
221#382—171no—
222#380—21no—
223#380—341no—
224C4×C22⋊C4—341no—
225C4×C22⋊C4—341no—
226C43—171no—
227#354—344no—
228#342—21no—
229#342—341no—
230C2×C17—21no—
231C2×C17—21no—
232C2×C17—41no—
233C2×C17—41no—
234C2×C17—41no—
235C2×C17—41no—
236C2×C17#179911yescharacteristic
237C2×C17C2×#35411yescharacteristic
238C2×C17#212611yescharacteristic
239C22×Q8—171no—
240C22×D4C22×D1711yescharacteristic
241C23×C4D6811yescharacteristic
242C23×C4C22⋊D1711yescharacteristic
243#124—172no—
244#124—172no—
245#124—341no—
246#124—341no—
247C4×Q8—681no—
248C4×Q8—681no—
249C4×D4—41no—
250C4×D4—41no—
251C2×C4⋊C4—41no—
252C2×C4⋊C4—171no—
253C2×C4⋊C4—341no—
254C2×C4⋊C4—681no—
255C2×C4⋊C4C4×D1711yescharacteristic
256C2×C22⋊C4—22no—
257C2×C22⋊C4—341no—
258C2×C22⋊C4—341no—
259C2×C22⋊C4—341no—
260C2×C22⋊C4—341no—
261C2×C22⋊C4—342no—
262C2×C22⋊C4D6811yescharacteristic
263C2×C22⋊C4C22⋊D1711yescharacteristic
264C2×C42—171no—
265C2×C42—171no—
266C2×C42—341no—
267C2×C42—341no—
268C2×C42—341no—
269C2×C42C4×D1711yescharacteristic
270#99—22no—
271#99—22no—
272#99—342no—
273#99—342no—
274#95—41no—
275#95—341no—
276#95—341no—
277#95—341no—
278#95—341no—
279#95—681no—
280C17#1279811yescharacteristic, complemented, Sylow
281C24C2×D6811yescharacteristic
282C24C2×C22⋊D1711yescharacteristic
283C2×Q8—172no—
284C2×Q8—341no—
285C2×Q8—681no—
286C2×Q8—681no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4D4×D1711yescharacteristic
290C2×D4#6325411yescharacteristic
291C2×D4Q8×D1711yescharacteristic
292C2×D4#6325611yescharacteristic
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—341no—
301C22×C4—341no—
302C22×C4—341no—
303C22×C4—341no—
304C22×C4—682no—
305C22×C4#6322711yescharacteristic
306C22×C4#6322711yescharacteristic
307C22×C4#6322711yescharacteristic
308C22×C4#6322711yescharacteristic
309C22×C4C2×C4×D1711yescharacteristic
310C4⋊C4—41no—
311C4⋊C4—82no—
312C4⋊C4—681no—
313C4⋊C4—682no—
314C4⋊C4—1362no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C22⋊C4—42no—
318C22⋊C4—42no—
319C22⋊C4—342no—
320C22⋊C4—342no—
321C22⋊C4—342no—
322C22⋊C4—342no—
323C22⋊C4—682no—
324C22⋊C4—682no—
325C22⋊C4—682no—
326C22⋊C4—682no—
327C42—21no—
328C42—21no—
329C42—172no—
330C42—341no—
331C42—681no—
332C42—682no—
333C42—682no—
334C42—681no—
335C42—681no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23#1058758111yescharacteristic
345C23#1058758311yescharacteristic
346C23C2×#6322711yescharacteristic
347C23#1058762411yescharacteristic
348C23#1058763211yescharacteristic
349Q8—682no—
350Q8—682no—
351D4—41no—
352D4—41no—
353D4—41no—
354D4—41no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—22no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—342no—
372C2×C4—342no—
373C2×C4—342no—
374C2×C4—681no—
375C2×C4—681no—
376C2×C4—681no—
377C2×C4—681no—
378C2×C4—681no—
379C2×C4—681no—
380C2×C4—681no—
381C2×C4—681no—
382C2×C4—681no—
383C2×C4—1362no—
384C2×C4—1362no—
385C2×C4—1362no—
386C2×C4#1058747711yescharacteristic
387C2×C4#1058747711yescharacteristic
388C2×C4C4⋊C4×D1711yescharacteristic
389C2×C4#1058757811yescharacteristic
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—81no—
405C22—81no—
406C22#1176266611yescharacteristic
407C22C2×#1058747711yescharacteristic
408C22#1176278211yescharacteristic
409C4—21no—
410C4—21no—
411C4—21no—
412C4—41no—
413C4—41no—
414C4—41no—
415C4—162no—
416C4—2722no—
417C4—682no—
418C4—682no—
419C4—682no—
420C4—1361no—
421C4#1176260312yescomplemented
422C2—21no—
423C2—21no—
424C2—41no—
425C2—41no—
426C2—41no—
427C2—41no—
428C2#1452876911yescharacteristic
429C2C2×#1176260311yescharacteristic
430C2#1453044711yescharacteristic
subgroupG/H
centerC22C2×#10587477
commutatorC22×C4×C17C22×C4
FrattiniC22×D4C22×D17
FittingC17×#1799C2
socleC22×C17C2×#99
prime radical, cosocleC22×D4×C17C23
hypercenter#1799D17
hypocenterC17#12798
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1799

Complements

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

NH ≅ G/Ntype
7. C2×C4×Dic34C4semidirect
9. #11762714C4semidirect
13. #11762674C4semidirect
17. C42×Dic17C4semidirect
93. C2×Dic34C4⋊C4semidirect
119. C4×Dic17C4⋊C4semidirect
206. C4×C17#354semidirect
280. C17#12798semidirect
421. C4#11762603semidirect

Maximal subgroups

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

subgroupindex
C4×#105876222
C2×#117626032
#145287722
#145269082
C17×#17992
#1279817