Finite Group Databasebeta

← #16952992

Subgroups — #16952992

Order 4352.

556 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 93 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
1#14533735C211yesmaximal, characteristic
2#14528770C211yesmaximal, characteristic
3#99×Dic17C211yesmaximal, characteristic
4#14526909C211yesmaximal, characteristic
5#14526908C211yesmaximal, characteristic
6#14526908C211yesmaximal, characteristic
7C17×#1393C211yesmaximal, characteristic
8C2×#10587574C411yescharacteristic, complemented
9C2×#10587564—21no—
10C2×#10587564—21no—
11C2×D4×Dic17C2211yescharacteristic
12C2×#10587505C2211yescharacteristic
13C2×#10587505C2211yescharacteristic
14#11762745—21no—
15#11762745—21no—
16#11762712C2211yescharacteristic
17#11762679C411yescharacteristic, complemented
18#11762678C411yescharacteristic, complemented
19#11762667—21no—
20C4×#63225C411yescharacteristic, complemented
21C22⋊C4×Dic17—21no—
22#11762575—21no—
23#11762575—21no—
24C2×C17×#99C2211yescharacteristic
25C2×C17×#99C2211yescharacteristic
26C17×#390C2211yescharacteristic
27C22×D4×C17C2311yescharacteristic
28C23×Dic17D411yescharacteristic
29C23×Dic17D411yescharacteristic
30C22×Dic34—21no—
31C2×C4⋊C4×C17C2×C411yescharacteristic
32C2×C22⋊C4×C17—21no—
33C2×C22⋊C4×C17—21no—
34C2×C22⋊C4×C17—21no—
35C2×C22⋊C4×C17—21no—
36C2×C42×C17C2×C411yescharacteristic
37C2×C22⋊Dic17—21no—
38C2×C22⋊Dic17—21no—
39C2×C22⋊Dic17—21no—
40C2×C22⋊Dic17—21no—
41C2×C22⋊Dic17D411yescharacteristic
42C2×C22⋊Dic17D411yescharacteristic
43D4×Dic17—41no—
44D4×Dic17—41no—
45C2×C4⋊Dic17—21no—
46C2×C4⋊Dic17—21no—
47C2×C4⋊Dic17C2×C411yescharacteristic
48C2×C4⋊Dic17C2×C411yescharacteristic
49C2×#63225—21no—
50C2×#63225—21no—
51C2×#63225—21no—
52C2×#63225—21no—
53C2×#63225—41no—
54C2×C4×Dic17—41no—
55C2×C4×Dic17C2×C411yescharacteristic
56C2×C4×Dic17C2×C411yescharacteristic
57#10587574—41no—
58#10587574—41no—
59#10587564—44no—
60#10587564—44no—
61C17×#99—21no—
62C17×#99—21no—
63C17×#99—21no—
64C17×#99Q812yescomplemented
65#10587505—21no—
66#10587505—21no—
67#10587505—21no—
68#10587505Q812yescomplemented
69#10587502—21no—
70#10587502—41no—
71#10587502—41no—
72C24×C17C2×D411yescharacteristic
73C24×C17C2×D411yescharacteristic
74C2×D4×C17—41no—
75C2×D4×C17—41no—
76C2×D4×C17C2×Q811yescharacteristic
77C2×D4×C17C2×Q811yescharacteristic
78C2×D4×C17#4111yescharacteristic
79C2×D4×C17#4111yescharacteristic
80C22×C4×C17—21no—
81C22×C4×C17—21no—
82C22×C4×C17—41no—
83C22×C4×C17—41no—
84C22×C4×C17C22×C411yescharacteristic
85C22×Dic17—21no—
86C22×Dic17—21no—
87C22×Dic17—21no—
88C22×Dic17—21no—
89C22×Dic17—21no—
90C22×Dic17—21no—
91C22×Dic17—42no—
92C22×Dic17—42no—
93C22×Dic17—41no—
94C22×Dic17—41no—
95C22×Dic17C22⋊C411yescharacteristic
96C22×Dic17C22⋊C411yescharacteristic
97C22×Dic17C22⋊C411yescharacteristic
98C22×Dic17C22⋊C411yescharacteristic
99C2×Dic34—42no—
100C2×Dic34—42no—
101C2×Dic34—41no—
102C2×Dic34—41no—
103C4⋊C4×C17—41no—
104C22⋊C4×C17—22no—
105C22⋊C4×C17—42no—
106C22⋊C4×C17—42no—
107C22⋊C4×C17—41no—
108C22⋊C4×C17—41no—
109C22⋊C4×C17—41no—
110C42×C17—41no—
111C22⋊Dic17—22no—
112C22⋊Dic17—41no—
113C22⋊Dic17—41no—
114C22⋊Dic17—42no—
115C22⋊Dic17—42no—
116C22⋊Dic17—41no—
117C22⋊Dic17—41no—
118C22⋊Dic17—41no—
119C4⋊Dic17—41no—
120C4⋊Dic17—42no—
121C4⋊Dic17—42no—
122C4⋊Dic17—41no—
123#63225—24no—
124#63225—42no—
125#63225—42no—
126#63225—42no—
127#63225—82no—
128C4×Dic17—21no—
129C4×Dic17—21no—
130C4×Dic17—41no—
131C4×Dic17—82no—
132#13298—171nomaximal, Sylow
133C23×C17—21no—
134C23×C17—21no—
135C23×C17—21no—
136C23×C17—21no—
137C23×C17—41no—
138C23×C17—41no—
139C23×C17—41no—
140C23×C17—41no—
141C23×C17C2×C22⋊C411yescharacteristic
142C23×C17C22⋊Q811yescharacteristic
143C23×C17C22⋊Q811yescharacteristic
144C23×C17#12411yescharacteristic
145C23×C17C4⋊Q811yescharacteristic
146D4×C17—41no—
147D4×C17—41no—
148D4×C17—41no—
149D4×C17—41no—
150C2×C4×C17—21no—
151C2×C4×C17—21no—
152C2×C4×C17—22no—
153C2×C4×C17—41no—
154C2×C4×C17—41no—
155C2×C4×C17—42no—
156C2×C4×C17—41no—
157C2×C4×C17—41no—
158C2×C4×C17—41no—
159C2×C4×C17—81no—
160C2×C4×C17—81no—
161C2×C4×C17—81no—
162C2×C4×C17—81no—
163C2×C4×C17—81no—
164C2×C4×C17C4×Q811yescharacteristic
165C2×C4×C17C4×Q811yescharacteristic
166C2×Dic17—22no—
167C2×Dic17—22no—
168C2×Dic17—21no—
169C2×Dic17—21no—
170C2×Dic17—21no—
171C2×Dic17—41no—
172C2×Dic17—41no—
173C2×Dic17—42no—
174C2×Dic17—42no—
175C2×Dic17—41no—
176C2×Dic17—41no—
177C2×Dic17—41no—
178C2×Dic17—42no—
179C2×Dic17—41no—
180C2×Dic17—41no—
181C2×Dic17—41no—
182C2×Dic17—81no—
183C2×Dic17—81no—
184C2×Dic17—81no—
185C2×Dic17—81no—
186C2×Dic17—81no—
187C2×Dic17—81no—
188C2×Dic17—81no—
189C2×Dic17#9911yescharacteristic
190C2×Dic17#9911yescharacteristic
191Dic34—82no—
192Dic34—82no—
193#2542—171no—
194#1799—171no—
195C4×#99—171no—
196#1393—171no—
197#1393—171no—
198#1393—171no—
199#1393D1711yescharacteristic
200C22×C17—21no—
201C22×C17—21no—
202C22×C17—21no—
203C22×C17—21no—
204C22×C17—21no—
205C22×C17—21no—
206C22×C17—41no—
207C22×C17—41no—
208C22×C17—41no—
209C22×C17—41no—
210C22×C17—41no—
211C22×C17—41no—
212C22×C17—41no—
213C22×C17—41no—
214C22×C17—81no—
215C22×C17—81no—
216C22×C17#39111yescharacteristic
217C22×C17C2×#9911yescharacteristic
218C22×C17#41011yescharacteristic
219C4×C17—21no—
220C4×C17—21no—
221C4×C17—42no—
222C4×C17—82no—
223C4×C17—81no—
224C4×C17—161no—
225Dic17—21no—
226Dic17—42no—
227Dic17—41no—
228Dic17—41no—
229Dic17—41no—
230Dic17—82no—
231Dic17—81no—
232Dic17—161no—
233Dic17#35312yes—
234C2×C4⋊Q8—171no—
235C2×C22⋊Q8—341no—
236C2×C22⋊Q8—341no—
237C2×C4×D4—171no—
238C2×#99—171no—
239C2×#99—171no—
240C2×#99C2×D1711yescharacteristic
241C2×#99C2×D1711yescharacteristic
242#399—341no—
243#399—341no—
244#390—171no—
245#390C2×D1711yescharacteristic
246#384—171no—
247#384—171no—
248#380—341no—
249C4×C4⋊C4—171no—
250C4×C22⋊C4—341no—
251#342—341no—
252#342—341no—
253C2×C17—21no—
254C2×C17—21no—
255C2×C17—41no—
256C2×C17—41no—
257C2×C17—41no—
258C2×C17—41no—
259C2×C17#199611yescharacteristic
260C2×C17C2×#35311yescharacteristic
261C2×C17#212911yescharacteristic
262C22×Q8—341no—
263C22×D4C22×D1711yescharacteristic
264C23×C4—171no—
265C23×C4—171no—
266C4⋊Q8—681no—
267C4⋊Q8—681no—
268C22⋊Q8—684no—
269C22⋊Q8—684no—
270C4×D4—681no—
271C4×D4—681no—
272C2×C4⋊C4—171no—
273C2×C4⋊C4—171no—
274C2×C4⋊C4—341no—
275C2×C4⋊C4—341no—
276C2×C4⋊C4—341no—
277C2×C4⋊C4—341no—
278C2×C4⋊C4—341no—
279C2×C4⋊C4—341no—
280C2×C4⋊C4—681no—
281C2×C4⋊C4C4×D1711yescharacteristic
282C2×C22⋊C4—21no—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—21no—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—171no—
287C2×C22⋊C4—171no—
288C2×C22⋊C4—341no—
289C2×C22⋊C4—341no—
290C2×C22⋊C4—341no—
291C2×C22⋊C4—341no—
292C2×C42—171no—
293C2×C42—171no—
294C2×C42—681no—
295C2×C42C4×D1711yescharacteristic
296#99—21no—
297#99—21no—
298#99—21no—
299#99—172no—
300#99—341no—
301#99—341no—
302#99—341no—
303#99Dic3412yescomplemented
304#95—341no—
305#95—681no—
306#95—681no—
307C17#1329811yescharacteristic, complemented, Sylow
308C24D4×D1711yescharacteristic
309C24D4×D1711yescharacteristic
310C2×Q8—681no—
311C2×Q8—681no—
312C2×Q8—682no—
313C2×Q8—682no—
314C2×D4—41no—
315C2×D4—41no—
316C2×D4C2×Dic3411yescharacteristic
317C2×D4#6325211yescharacteristic
318C2×D4#6325411yescharacteristic
319C2×D4Q8×D1711yescharacteristic
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—171no—
325C22×C4—171no—
326C22×C4—171no—
327C22×C4—171no—
328C22×C4—341no—
329C22×C4—341no—
330C22×C4—341no—
331C22×C4—341no—
332C22×C4—341no—
333C22×C4—341no—
334C22×C4—682no—
335C22×C4—682no—
336C22×C4—681no—
337C22×C4—681no—
338C22×C4C2×C4×D1711yescharacteristic
339C4⋊C4—41no—
340C4⋊C4—344no—
341C4⋊C4—681no—
342C4⋊C4—681no—
343C4⋊C4—682no—
344C4⋊C4—682no—
345C4⋊C4—682no—
346C4⋊C4—682no—
347C4⋊C4—682no—
348C4⋊C4—1362no—
349C22⋊C4—22no—
350C22⋊C4—42no—
351C22⋊C4—42no—
352C22⋊C4—41no—
353C22⋊C4—41no—
354C22⋊C4—41no—
355C22⋊C4—342no—
356C22⋊C4—681no—
357C22⋊C4—681no—
358C22⋊C4—682no—
359C22⋊C4—682no—
360C22⋊C4—681no—
361C22⋊C4—681no—
362C22⋊C4—681no—
363C42—41no—
364C42—341no—
365C42—341no—
366C42—681no—
367C42—1362no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—41no—
373C23—41no—
374C23—41no—
375C23—41no—
376C23#1058756411yescharacteristic
377C23C22⋊C4×D1711yescharacteristic
378C23#1058757111yescharacteristic
379C23#1058757411yescharacteristic
380C23#1058758211yescharacteristic
381Q8—1362no—
382Q8—1362no—
383D4—41no—
384D4—41no—
385D4—41no—
386D4—41no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—22no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—42no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—171no—
402C2×C4—171no—
403C2×C4—342no—
404C2×C4—342no—
405C2×C4—341no—
406C2×C4—341no—
407C2×C4—341no—
408C2×C4—682no—
409C2×C4—682no—
410C2×C4—681no—
411C2×C4—681no—
412C2×C4—681no—
413C2×C4—681no—
414C2×C4—681no—
415C2×C4—682no—
416C2×C4—681no—
417C2×C4—681no—
418C2×C4—681no—
419C2×C4—1361no—
420C2×C4—1361no—
421C2×C4—1361no—
422C2×C4—1361no—
423C2×C4—1361no—
424C2×C4—1361no—
425C2×C4—1361no—
426C2×C4C4×Dic3411yescharacteristic
427C2×C4#1058757311yescharacteristic
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—41no—
435C22—41no—
436C22—41no—
437C22—41no—
438C22—41no—
439C22—41no—
440C22—41no—
441C22—41no—
442C22—81no—
443C22—81no—
444C22#1176271311yescharacteristic
445C22#99×D1711yescharacteristic
446C22#1176278111yescharacteristic
447C4—21no—
448C4—21no—
449C4—42no—
450C4—82no—
451C4—81no—
452C4—161no—
453C4—2721no—
454C4—172no—
455C4—341no—
456C4—682no—
457C4—681no—
458C4—681no—
459C4—681no—
460C4—1362no—
461C4—1361no—
462C2—21no—
463C2—21no—
464C2—41no—
465C2—41no—
466C2—41no—
467C2—41no—
468C2#1452976711yescharacteristic
469C2D17×#35311yescharacteristic
470C2#1453045711yescharacteristic
subgroupG/H
centerC22#99×D17
commutatorC22×C4×C17C22×C4
FrattiniC22×D4C22×D17
FittingC17×#1393C2
socleC22×C17C2×#99
prime radical, cosocleC22×D4×C17C23
hypercenter#1393D17
hypocenterC17#13298
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ #1393

Complements

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

NH ≅ G/Ntype
8. C2×#10587574C4semidirect
17. #11762679C4semidirect
18. #11762678C4semidirect
20. C4×#63225C4semidirect
64. C17×#99Q8semidirect
68. #10587505Q8semidirect
303. #99Dic34semidirect
307. C17#13298semidirect

Maximal subgroups

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

subgroupindex
#145337352
#145287702
#99×Dic172
#145269092
#145269082
C17×#13932
#1329817