Finite Group Databasebeta

← #16933849

Subgroups — #16933849

Order 4352.

510 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 111 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
1C2×#11763048C211yesmaximal, characteristic
2#14529769C211yesmaximal, characteristic
3#14528807C211yesmaximal, characteristic
4#14528562C211yesmaximal, characteristic
5#14527356C211yesmaximal, characteristic
6#14527355C211yesmaximal, characteristic
7C17×#1996C211yesmaximal, characteristic
8C22×#63231C411yescharacteristic, complemented
9C23×#3622C2211yescharacteristic
10C2×#10587610C2211yescharacteristic
11#11763048—21no—
12#11763048—21no—
13#11763048—21no—
14#11763048—21no—
15C2×#10587478C411yescharacteristic, complemented
16C2×#10587479—21no—
17C2×#10587506C411yescharacteristic, complemented
18C2×#10587481C411yescharacteristic, complemented
19C2×C22⋊#3622C2211yescharacteristic
20C2×#10587474—21no—
21#11762669C2211yescharacteristic
22#11762575—22no—
23#11762571—21no—
24#11762564—21no—
25#11762546—22no—
26#11762545—22no—
27C2×C17×C22⋊Q8C2211yescharacteristic
28C17×#380C2211yescharacteristic
29C4×C22⋊C4×C17C2211yescharacteristic
30C17×#342—22no—
31C22×Q8×C17C2×C411yescharacteristic
32C23×C4×C17C2311yescharacteristic
33C22×Dic34C2×C411yescharacteristic
34C17×C22⋊Q8—21no—
35C17×C22⋊Q8—21no—
36C17×C22⋊Q8—21no—
37C17×C22⋊Q8—21no—
38C2×C4⋊C4×C17—21no—
39C2×C4⋊C4×C17—21no—
40C2×C4⋊C4×C17C2×C411yescharacteristic
41C2×C22⋊C4×C17—21no—
42C2×C22⋊C4×C17—21no—
43C2×C22⋊C4×C17D412yes—
44C2×C42×C17—21no—
45C2×C22⋊Dic17—21no—
46C2×C22⋊Dic17—21no—
47C2×#63231—21no—
48C2×#63231—21no—
49C2×#63231—42no—
50C2×C4⋊Dic17—21no—
51C2×C4⋊Dic17C2×C411yescharacteristic
52#10587610—21no—
53#10587610—21no—
54#10587610—21no—
55#10587610—21no—
56C2×#63225—21no—
57C22×#3622—21no—
58C22×#3622—42no—
59C22×#3622C2×C411yescharacteristic
60C22×#3622C2×C411yescharacteristic
61C22×#3622D411yescharacteristic
62C22×#3622D411yescharacteristic
63C22×#3622D411yescharacteristic
64C22×#3622D411yescharacteristic
65C17×#95—21no—
66C17×#95—21no—
67#10587506—21no—
68#10587506—21no—
69#10587502—21no—
70C22⋊#3622—21no—
71C22⋊#3622D412yes—
72#10587481—21no—
73#10587481—21no—
74#10587479—42no—
75#10587478—21no—
76#10587478—21no—
77#10587474—42no—
78C24×C17C2×D411yescharacteristic
79C2×Q8×C17—21no—
80C2×Q8×C17—21no—
81C2×Q8×C17—41no—
82C2×Q8×C17—41no—
83C22×C4×C17—22no—
84C22×C4×C17—21no—
85C22×C4×C17—21no—
86C22×C4×C17—41no—
87C22×C4×C17—41no—
88C22×C4×C17—41no—
89C22×C4×C17C22×C411yescharacteristic
90C22×C4×C17C2×D411yescharacteristic
91C22×C4×C17C2×D411yescharacteristic
92C22×C4×C17C2×D411yescharacteristic
93C22×C4×C17#4111yescharacteristic
94C22×C4×C17#4111yescharacteristic
95C22×Dic17—21no—
96C22×Dic17—21no—
97C22×Dic17—41no—
98C2×Dic34—21no—
99C2×Dic34—21no—
100C2×Dic34—41no—
101C2×Dic34—41no—
102C4⋊C4×C17—21no—
103C4⋊C4×C17—21no—
104C4⋊C4×C17—41no—
105C4⋊C4×C17—41no—
106C4⋊C4×C17—41no—
107C4⋊C4×C17—41no—
108C22⋊C4×C17—22no—
109C22⋊C4×C17—22no—
110C22⋊C4×C17—41no—
111C22⋊C4×C17—41no—
112C22⋊C4×C17—42no—
113C42×C17—41no—
114C42×C17—41no—
115C22⋊Dic17—41no—
116C22⋊Dic17—41no—
117C22⋊Dic17—42no—
118#63231—42no—
119#63231—42no—
120C4⋊Dic17—21no—
121C4⋊Dic17—21no—
122C4⋊Dic17—41no—
123C4⋊Dic17—41no—
124#63225—41no—
125#63225—41no—
126C2×#3622—22no—
127C2×#3622—22no—
128C2×#3622—22no—
129C2×#3622—22no—
130C2×#3622—41no—
131C2×#3622—41no—
132C2×#3622—41no—
133C2×#3622—41no—
134C2×#3622—41no—
135C2×#3622C22⋊C411yescharacteristic
136C2×#3622C22⋊C411yescharacteristic
137C2×#3622C22⋊C411yescharacteristic
138C2×#3622C22⋊C411yescharacteristic
139#9784—171nomaximal, Sylow
140C23×C17—41no—
141C23×C17—41no—
142C23×C17C4×D411yescharacteristic
143C23×C17#12111yescharacteristic
144C23×C17#12711yescharacteristic
145C23×C17#13712yes—
146C23×C17#13712yes—
147Q8×C17—41no—
148Q8×C17—41no—
149Q8×C17—81no—
150C2×C4×C17—21no—
151C2×C4×C17—21no—
152C2×C4×C17—21no—
153C2×C4×C17—21no—
154C2×C4×C17—21no—
155C2×C4×C17—21no—
156C2×C4×C17—41no—
157C2×C4×C17—41no—
158C2×C4×C17—41no—
159C2×C4×C17—42no—
160C2×C4×C17—42no—
161C2×C4×C17—41no—
162C2×C4×C17—41no—
163C2×C4×C17—42no—
164C2×C4×C17—81no—
165C2×C4×C17—81no—
166C2×C4×C17—81no—
167C2×C4×C17—81no—
168C2×C4×C17—81no—
169C2×C4×C17C2×C22⋊C411yescharacteristic
170C2×C4×C17C4×D411yescharacteristic
171C2×C4×C17#12111yescharacteristic
172C2×C4×C17#12411yescharacteristic
173C2×Dic17—41no—
174C2×Dic17—41no—
175C2×Dic17—81no—
176C2×Dic17—81no—
177C2×Dic17—81no—
178C2×Dic17—81no—
179C2×Dic17—81no—
180Dic34—41no—
181Dic34—41no—
182Dic34—81no—
183#3622—21no—
184#3622—21no—
185#3622—21no—
186#3622—42no—
187#3622#9912yes—
188C2×#467—171no—
189#1996—171no—
190#1996D1711yescharacteristic
191#1807—171no—
192#1752—171no—
193#1520—171no—
194#1520—171no—
195C22×C17—21no—
196C22×C17—21no—
197C22×C17—21no—
198C22×C17—21no—
199C22×C17—21no—
200C22×C17—21no—
201C22×C17—42no—
202C22×C17—42no—
203C22×C17#39011yescharacteristic
204C22×C17#44112yes—
205C22×C17#47012yes—
206C22×C17#49711yescharacteristic
207C22×C17#49711yescharacteristic
208C4×C17—21no—
209C4×C17—21no—
210C4×C17—21no—
211C4×C17—82no—
212C4×C17—81no—
213C4×C17—81no—
214C4×C17—161no—
215C4×C17C2×#9911yescharacteristic
216C4×C17#41011yescharacteristic
217Dic17—81no—
218Dic17—81no—
219Dic17—161no—
220C22×Q16—171no—
221C23×C8—171no—
222C2×C22⋊Q8—171no—
223C2×C22⋊Q8C2×D1711yescharacteristic
224#467—341no—
225#467—341no—
226#467—341no—
227#467—341no—
228C2×#107—171no—
229C2×#106—341no—
230C2×#103—171no—
231C2×#103—171no—
232C2×C22⋊C8—171no—
233C2×#97—341no—
234#380—171no—
235#380C2×D1711yescharacteristic
236C4×C22⋊C4C2×D1711yescharacteristic
237#342—22no—
238#342—342no—
239#340—341no—
240#336—341no—
241#328—342no—
242#328—342no—
243C2×C17—21no—
244C2×C17—22no—
245C2×C17—21no—
246C2×C17—22no—
247C2×C17#196411yescharacteristic
248C2×C17#197111yescharacteristic
249C2×C17#197611yescharacteristic
250C22×Q8—171no—
251C22×Q8C4×D1711yescharacteristic
252C23×C4C22×D1711yescharacteristic
253C2×Q16—341no—
254C2×Q16—341no—
255C2×Q16—682no—
256C22×C8—171no—
257C22×C8—171no—
258C22×C8—171no—
259C22×C8—171no—
260C22×C8—171no—
261C22×C8—171no—
262C22×C8—341no—
263C22×C8—682no—
264C22⋊Q8—21no—
265C22⋊Q8—21no—
266C22⋊Q8—21no—
267C22⋊Q8—21no—
268C22⋊Q8—341no—
269C22⋊Q8—341no—
270C22⋊Q8—341no—
271C22⋊Q8—341no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—171no—
275C2×C4⋊C4—341no—
276C2×C4⋊C4—341no—
277C2×C4⋊C4C4×D1711yescharacteristic
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—341no—
281C2×C22⋊C4—341no—
282C2×C22⋊C4C22⋊D1712yes—
283C2×C42—21no—
284#107—341no—
285#107—341no—
286#106—682no—
287#103—341no—
288#103—341no—
289#103—341no—
290#103—341no—
291C22⋊C8—172no—
292C22⋊C8—341no—
293#97—682no—
294#95—21no—
295#95—21no—
296#95—341no—
297C17#978411yescharacteristic, complemented, Sylow
298C24C2×C22⋊D1711yescharacteristic
299C2×Q8—21no—
300C2×Q8—21no—
301C2×Q8—41no—
302C2×Q8—41no—
303C2×Q8—341no—
304C2×Q8—341no—
305C2×Q8—681no—
306C2×Q8—681no—
307C22×C4—22no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—341no—
314C22×C4—341no—
315C22×C4—681no—
316C22×C4C2×C4×D1711yescharacteristic
317C22×C4#6325211yescharacteristic
318C22×C4D4×D1711yescharacteristic
319C22×C4D4×D1711yescharacteristic
320C22×C4D4×D1711yescharacteristic
321C22×C4#6325411yescharacteristic
322Q16—682no—
323Q16—682no—
324C2×C8—171no—
325C2×C8—171no—
326C2×C8—171no—
327C2×C8—171no—
328C2×C8—342no—
329C2×C8—342no—
330C2×C8—342no—
331C2×C8—342no—
332C2×C8—681no—
333C2×C8—681no—
334C2×C8—681no—
335C2×C8—681no—
336C2×C8—681no—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—341no—
344C4⋊C4—341no—
345C4⋊C4—681no—
346C4⋊C4—681no—
347C4⋊C4—681no—
348C4⋊C4—681no—
349C22⋊C4—22no—
350C22⋊C4—22no—
351C22⋊C4—41no—
352C22⋊C4—41no—
353C22⋊C4—42no—
354C22⋊C4—681no—
355C22⋊C4—681no—
356C22⋊C4—682no—
357C42—41no—
358C42—41no—
359C23—41no—
360C23—41no—
361C23C4×C22⋊D1711yescharacteristic
362C23#1058762511yescharacteristic
363C23#1058762611yescharacteristic
364C23#1058762812yes—
365C23#1058763712yes—
366Q8—41no—
367Q8—41no—
368Q8—81no—
369Q8—681no—
370Q8—681no—
371Q8—1361no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—42no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—681no—
392C2×C4—681no—
393C2×C4—1361no—
394C2×C4—1361no—
395C2×C4—1361no—
396C2×C4—1361no—
397C2×C4—1361no—
398C2×C4C22⋊C4×D1711yescharacteristic
399C2×C4#1058756711yescharacteristic
400C2×C4#1058757011yescharacteristic
401C2×C4#1058757111yescharacteristic
402C8—172no—
403C8—341no—
404C8—341no—
405C8—341no—
406C8—682no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—42no—
414C22—42no—
415C22#1176270911yescharacteristic
416C22#1176291412yes—
417C22#1176306512yes—
418C22#1176320111yescharacteristic
419C22#1176320411yescharacteristic
420C4—21no—
421C4—21no—
422C4—21no—
423C4—82no—
424C4—81no—
425C4—81no—
426C4—161no—
427C4—2721no—
428C4—1361no—
429C4—1361no—
430C4#99×D1711yescharacteristic
431C4#1176278111yescharacteristic
432C2—22no—
433C2—21no—
434C2—21no—
435C2—22no—
436C2#1452961811yescharacteristic
437C2#1452965411yescharacteristic
438C2#1452967911yescharacteristic
subgroupG/H
centerC22#11762709
commutatorC22×C4×C17C22×C4
FrattiniC23×C4C22×D17
FittingC17×#1996C2
socleC22×C17#390
prime radicalC24×C17C2×D4
cosocleC23×C4×C17C23
hypercenter#1996D17
hypocenterC17#9784
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1996

Complements

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

NH ≅ G/Ntype
8. C22×#63231C4semidirect
15. C2×#10587478C4semidirect
17. C2×#10587506C4semidirect
18. C2×#10587481C4semidirect
297. C17#9784semidirect

Maximal subgroups

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

subgroupindex
C2×#117630482
#145297692
#145288072
#145285622
#145273562
#145273552
C17×#19962
#978417