Finite Group Databasebeta

← #16952996

Subgroups — #16952996

Order 4352.

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

normal subgroups only

This group has 500 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#14533955C211yesmaximal, characteristic
2#14528770C211yesmaximal, characteristic
3#99×Dic17C211yesmaximal, characteristic
4#14526909C211yesmaximal, characteristic
5#14526908C211yesmaximal, characteristic
6#14526908C211yesmaximal, characteristic
7C17×#1393C211yesmaximal, characteristic
8C2×#10587574C411yescharacteristic, complemented
9C2×D4×Dic17C2211yescharacteristic
10C2×#10587505C2211yescharacteristic
11C2×#10587505C2211yescharacteristic
12#11762721—21no—
13#11762721—21no—
14#11762712C2211yescharacteristic
15#11762675C411yescharacteristic, complemented
16#11762674C411yescharacteristic, complemented
17#11762667—21no—
18#11762667—21no—
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⋊Dic17C2×C411yescharacteristic
46C2×C4⋊Dic17C2×C411yescharacteristic
47C2×#63225—21no—
48C2×#63225—21no—
49C2×#63225—41no—
50C2×C4×Dic17—41no—
51C2×C4×Dic17C2×C411yescharacteristic
52C2×C4×Dic17C2×C411yescharacteristic
53#10587574—41no—
54#10587574—41no—
55C17×#99—21no—
56C17×#99—21no—
57C17×#99—21no—
58C17×#99Q812yescomplemented
59#10587505—21no—
60#10587505—21no—
61#10587505—21no—
62#10587505Q812yescomplemented
63#10587502—21no—
64#10587502—21no—
65#10587502—21no—
66#10587502—21no—
67#10587502—21no—
68#10587502—41no—
69#10587502—41no—
70C24×C17C2×D411yescharacteristic
71C24×C17C2×D411yescharacteristic
72C2×D4×C17—41no—
73C2×D4×C17—41no—
74C2×D4×C17C2×Q811yescharacteristic
75C2×D4×C17C2×Q811yescharacteristic
76C2×D4×C17#4111yescharacteristic
77C2×D4×C17#4111yescharacteristic
78C22×C4×C17—21no—
79C22×C4×C17—21no—
80C22×C4×C17—41no—
81C22×C4×C17—41no—
82C22×C4×C17C22×C411yescharacteristic
83C22×Dic17—21no—
84C22×Dic17—21no—
85C22×Dic17—21no—
86C22×Dic17—21no—
87C22×Dic17—21no—
88C22×Dic17—21no—
89C22×Dic17—42no—
90C22×Dic17—42no—
91C22×Dic17—41no—
92C22×Dic17—41no—
93C22×Dic17C22⋊C411yescharacteristic
94C22×Dic17C22⋊C411yescharacteristic
95C22×Dic17C22⋊C411yescharacteristic
96C22×Dic17C22⋊C411yescharacteristic
97C2×Dic34—42no—
98C2×Dic34—81no—
99C2×Dic34—81no—
100C4⋊C4×C17—41no—
101C22⋊C4×C17—22no—
102C22⋊C4×C17—41no—
103C22⋊C4×C17—41no—
104C22⋊C4×C17—41no—
105C22⋊C4×C17—81no—
106C22⋊C4×C17—81no—
107C42×C17—41no—
108C22⋊Dic17—22no—
109C22⋊Dic17—41no—
110C22⋊Dic17—41no—
111C22⋊Dic17—41no—
112C22⋊Dic17—41no—
113C22⋊Dic17—41no—
114C22⋊Dic17—81no—
115C22⋊Dic17—81no—
116C4⋊Dic17—41no—
117C4⋊Dic17—41no—
118#63225—42no—
119#63225—41no—
120#63225—41no—
121#63225—82no—
122C4×Dic17—21no—
123C4×Dic17—21no—
124C4×Dic17—41no—
125C4×Dic17—82no—
126#13299—171nomaximal, Sylow
127C23×C17—21no—
128C23×C17—21no—
129C23×C17—21no—
130C23×C17—21no—
131C23×C17—41no—
132C23×C17—41no—
133C23×C17—41no—
134C23×C17—41no—
135C23×C17C2×C22⋊C411yescharacteristic
136C23×C17C22⋊Q811yescharacteristic
137C23×C17C22⋊Q811yescharacteristic
138C23×C17#12411yescharacteristic
139C23×C17C4⋊Q811yescharacteristic
140D4×C17—41no—
141D4×C17—41no—
142D4×C17—41no—
143D4×C17—41no—
144C2×C4×C17—21no—
145C2×C4×C17—21no—
146C2×C4×C17—42no—
147C2×C4×C17—41no—
148C2×C4×C17—41no—
149C2×C4×C17—41no—
150C2×C4×C17—41no—
151C2×C4×C17—41no—
152C2×C4×C17—41no—
153C2×C4×C17—81no—
154C2×C4×C17—81no—
155C2×C4×C17—81no—
156C2×C4×C17—81no—
157C2×C4×C17—81no—
158C2×C4×C17C4×Q811yescharacteristic
159C2×C4×C17C4×Q811yescharacteristic
160C2×Dic17—22no—
161C2×Dic17—21no—
162C2×Dic17—21no—
163C2×Dic17—21no—
164C2×Dic17—42no—
165C2×Dic17—42no—
166C2×Dic17—41no—
167C2×Dic17—41no—
168C2×Dic17—41no—
169C2×Dic17—42no—
170C2×Dic17—41no—
171C2×Dic17—41no—
172C2×Dic17—41no—
173C2×Dic17—41no—
174C2×Dic17—41no—
175C2×Dic17—41no—
176C2×Dic17—81no—
177C2×Dic17—81no—
178C2×Dic17—81no—
179C2×Dic17—81no—
180C2×Dic17—81no—
181C2×Dic17—81no—
182C2×Dic17—81no—
183C2×Dic17#9911yescharacteristic
184C2×Dic17#9911yescharacteristic
185Dic34—84no—
186#2563—171no—
187#1799—171no—
188C4×#99—171no—
189#1393—171no—
190#1393—171no—
191#1393—171no—
192#1393D1711yescharacteristic
193C22×C17—21no—
194C22×C17—21no—
195C22×C17—21no—
196C22×C17—21no—
197C22×C17—21no—
198C22×C17—21no—
199C22×C17—41no—
200C22×C17—41no—
201C22×C17—41no—
202C22×C17—41no—
203C22×C17—41no—
204C22×C17—41no—
205C22×C17—41no—
206C22×C17—41no—
207C22×C17—81no—
208C22×C17—81no—
209C22×C17#39111yescharacteristic
210C22×C17C2×#9911yescharacteristic
211C22×C17#41011yescharacteristic
212C4×C17—21no—
213C4×C17—21no—
214C4×C17—81no—
215C4×C17—82no—
216C4×C17—81no—
217C4×C17—161no—
218Dic17—21no—
219Dic17—41no—
220Dic17—41no—
221Dic17—41no—
222Dic17—81no—
223Dic17—82no—
224Dic17—81no—
225Dic17—161no—
226Dic17#35412yes—
227C2×C4⋊Q8—171no—
228C2×C4×D4—171no—
229C2×#99—171no—
230C2×#99—171no—
231C2×#99C2×D1711yescharacteristic
232C2×#99C2×D1711yescharacteristic
233#393—341no—
234#393—341no—
235#390—171no—
236#390C2×D1711yescharacteristic
237#382—171no—
238#382—171no—
239#380—341no—
240#380—341no—
241#380—341no—
242C4×C4⋊C4—171no—
243C4×C22⋊C4—341no—
244#342—341no—
245#342—341no—
246C2×C17—21no—
247C2×C17—21no—
248C2×C17—41no—
249C2×C17—41no—
250C2×C17—41no—
251C2×C17—41no—
252C2×C17#199611yescharacteristic
253C2×C17C2×#35411yescharacteristic
254C2×C17#212811yescharacteristic
255C22×Q8—341no—
256C22×D4C22×D1711yescharacteristic
257C23×C4—171no—
258C23×C4—171no—
259C4⋊Q8—681no—
260C4⋊Q8—681no—
261C4×D4—681no—
262C4×D4—681no—
263C2×C4⋊C4—171no—
264C2×C4⋊C4—171no—
265C2×C4⋊C4—341no—
266C2×C4⋊C4—341no—
267C2×C4⋊C4—681no—
268C2×C4⋊C4C4×D1711yescharacteristic
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—171no—
274C2×C22⋊C4—171no—
275C2×C22⋊C4—341no—
276C2×C22⋊C4—341no—
277C2×C22⋊C4—341no—
278C2×C22⋊C4—341no—
279C2×C42—171no—
280C2×C42—171no—
281C2×C42—681no—
282C2×C42C4×D1711yescharacteristic
283#99—21no—
284#99—21no—
285#99—21no—
286#99—172no—
287#99—341no—
288#99—341no—
289#99—341no—
290#99Dic3412yescomplemented
291#95—341no—
292#95—341no—
293#95—341no—
294#95—341no—
295#95—341no—
296#95—681no—
297#95—681no—
298C17#1329911yescharacteristic, complemented, Sylow
299C24D4×D1711yescharacteristic
300C24D4×D1711yescharacteristic
301C2×Q8—682no—
302C2×Q8—1361no—
303C2×Q8—1361no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4C2×Dic3411yescharacteristic
307C2×D4#6325211yescharacteristic
308C2×D4#6325411yescharacteristic
309C2×D4Q8×D1711yescharacteristic
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—171no—
315C22×C4—171no—
316C22×C4—171no—
317C22×C4—171no—
318C22×C4—341no—
319C22×C4—341no—
320C22×C4—341no—
321C22×C4—341no—
322C22×C4—341no—
323C22×C4—341no—
324C22×C4—682no—
325C22×C4—682no—
326C22×C4—681no—
327C22×C4—681no—
328C22×C4C2×C4×D1711yescharacteristic
329C4⋊C4—41no—
330C4⋊C4—681no—
331C4⋊C4—681no—
332C4⋊C4—681no—
333C4⋊C4—681no—
334C4⋊C4—682no—
335C4⋊C4—1362no—
336C22⋊C4—22no—
337C22⋊C4—41no—
338C22⋊C4—41no—
339C22⋊C4—41no—
340C22⋊C4—81no—
341C22⋊C4—81no—
342C22⋊C4—342no—
343C22⋊C4—681no—
344C22⋊C4—681no—
345C22⋊C4—681no—
346C22⋊C4—681no—
347C22⋊C4—681no—
348C22⋊C4—1361no—
349C22⋊C4—1361no—
350C42—41no—
351C42—341no—
352C42—341no—
353C42—681no—
354C42—1362no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23#1058756411yescharacteristic
364C23C22⋊C4×D1711yescharacteristic
365C23#1058757111yescharacteristic
366C23#1058757411yescharacteristic
367C23#1058758211yescharacteristic
368Q8—1364no—
369D4—41no—
370D4—41no—
371D4—41no—
372D4—41no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—42no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—171no—
388C2×C4—171no—
389C2×C4—342no—
390C2×C4—341no—
391C2×C4—341no—
392C2×C4—341no—
393C2×C4—682no—
394C2×C4—682no—
395C2×C4—681no—
396C2×C4—681no—
397C2×C4—681no—
398C2×C4—682no—
399C2×C4—681no—
400C2×C4—681no—
401C2×C4—681no—
402C2×C4—681no—
403C2×C4—681no—
404C2×C4—681no—
405C2×C4—1361no—
406C2×C4—1361no—
407C2×C4—1361no—
408C2×C4—1361no—
409C2×C4—1361no—
410C2×C4—1361no—
411C2×C4—1361no—
412C2×C4C4×Dic3411yescharacteristic
413C2×C4#1058757311yescharacteristic
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—41no—
421C22—41no—
422C22—41no—
423C22—41no—
424C22—41no—
425C22—41no—
426C22—41no—
427C22—41no—
428C22—81no—
429C22—81no—
430C22#1176271311yescharacteristic
431C22#99×D1711yescharacteristic
432C22#1176278111yescharacteristic
433C4—21no—
434C4—21no—
435C4—81no—
436C4—82no—
437C4—81no—
438C4—161no—
439C4—2721no—
440C4—172no—
441C4—341no—
442C4—681no—
443C4—681no—
444C4—681no—
445C4—1361no—
446C4—1362no—
447C4—1361no—
448C2—21no—
449C2—21no—
450C2—41no—
451C2—41no—
452C2—41no—
453C2—41no—
454C2#1452976711yescharacteristic
455C2D17×#35411yescharacteristic
456C2#1453045411yescharacteristic
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#13299
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
15. #11762675C4semidirect
16. #11762674C4semidirect
20. C4×#63225C4semidirect
58. C17×#99Q8semidirect
62. #10587505Q8semidirect
290. #99Dic34semidirect
298. C17#13299semidirect

Maximal subgroups

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

subgroupindex
#145339552
#145287702
#99×Dic172
#145269092
#145269082
C17×#13932
#1329917