Finite Group Databasebeta

← #16938721

Subgroups — #16938721

Order 4352.

490 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 99 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#14539556C211yesmaximal, characteristic
2#14529769C211yesmaximal, characteristic
3#14529208C211yesmaximal, characteristic
4#14529205C211yesmaximal, characteristic
5#14528964C211yesmaximal, characteristic
6#14526651C211yesmaximal, characteristic
7C17×#1996C211yesmaximal, characteristic
8C23×#3622C2211yescharacteristic
9C2×#10587610C2211yescharacteristic
10#11763144—21no—
11#11763142—21no—
12#11763048—21no—
13#11763048—21no—
14#11762965—21no—
15#11762964—21no—
16C2×C4⋊#3622—21no—
17C2×#10587506—21no—
18C2×#10587481—21no—
19C2×C22⋊#3622C2211yescharacteristic
20#11762669C2211yescharacteristic
21#11762575—21no—
22#11762575—21no—
23#11762571—21no—
24#11762571—21no—
25#11762564—21no—
26C2×C17×C22⋊Q8C2211yescharacteristic
27C17×#380C2211yescharacteristic
28C4×C22⋊C4×C17C2211yescharacteristic
29C17×#342—21no—
30C17×#342—21no—
31C22×Q8×C17D411yescharacteristic
32C23×C4×C17C2311yescharacteristic
33C22×Dic34D411yescharacteristic
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×C17D411yescharacteristic
41C2×C22⋊C4×C17—21no—
42C2×C22⋊C4×C17—21no—
43C2×C22⋊C4×C17D411yescharacteristic
44C2×C22⋊C4×C17D411yescharacteristic
45C2×C42×C17—21no—
46C2×C22⋊Dic17—21no—
47C2×C22⋊Dic17—21no—
48C2×#63231—21no—
49C2×#63231—21no—
50C2×C4⋊Dic17—21no—
51C2×C4⋊Dic17D411yescharacteristic
52#10587610—21no—
53#10587610—21no—
54#10587610—21no—
55#10587610—21no—
56C2×#63225—21no—
57C22×#3622—21no—
58C22×#3622—21no—
59C22×#3622—21no—
60C22×#3622—42no—
61C22×#3622D411yescharacteristic
62C22×#3622D411yescharacteristic
63C17×#95—21no—
64C17×#95—21no—
65#10587506—21no—
66#10587506—21no—
67#10587506—42no—
68#10587502—21no—
69C22⋊#3622—21no—
70C22⋊#3622—21no—
71#10587481—21no—
72#10587481—21no—
73#10587481—42no—
74#10587478—21no—
75#10587478—21no—
76C4⋊#3622—42no—
77C24×C17C2×D411yescharacteristic
78C2×Q8×C17—21no—
79C2×Q8×C17—21no—
80C2×Q8×C17—41no—
81C2×Q8×C17—41no—
82C22×C4×C17—21no—
83C22×C4×C17—21no—
84C22×C4×C17—21no—
85C22×C4×C17—21no—
86C22×C4×C17—41no—
87C22×C4×C17—41no—
88C22×C4×C17—41no—
89C22×C4×C17C2×D411yescharacteristic
90C22×C4×C17C2×D411yescharacteristic
91C22×C4×C17C2×D411yescharacteristic
92C22×C4×C17#4111yescharacteristic
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—21no—
109C22⋊C4×C17—21no—
110C22⋊C4×C17—21no—
111C22⋊C4×C17—41no—
112C22⋊C4×C17—41no—
113C22⋊C4×C17—41no—
114C22⋊C4×C17—41no—
115C22⋊C4×C17Q1612yescomplemented
116C42×C17—41no—
117C42×C17—41no—
118C22⋊Dic17—41no—
119C22⋊Dic17—41no—
120C22⋊Dic17—41no—
121C22⋊Dic17—41no—
122#63231—81no—
123#63231—81no—
124C4⋊Dic17—21no—
125C4⋊Dic17—21no—
126C4⋊Dic17—41no—
127C4⋊Dic17—41no—
128#63225—41no—
129#63225—41no—
130C2×#3622—22no—
131C2×#3622—22no—
132C2×#3622—22no—
133C2×#3622—21no—
134C2×#3622—21no—
135C2×#3622—41no—
136C2×#3622—41no—
137C2×#3622—41no—
138C2×#3622—41no—
139C2×#3622—41no—
140C2×#3622—41no—
141#10566—171nomaximal, Sylow
142C23×C17—41no—
143C23×C17—41no—
144C23×C17#12111yescharacteristic
145C23×C17#12111yescharacteristic
146C23×C17#12111yescharacteristic
147C23×C17C2×Q1611yescharacteristic
148C23×C17#13511yescharacteristic
149C23×C17#13711yescharacteristic
150C23×C17#13711yescharacteristic
151Q8×C17—42no—
152Q8×C17—41no—
153Q8×C17—41no—
154C2×C4×C17—21no—
155C2×C4×C17—21no—
156C2×C4×C17—21no—
157C2×C4×C17—21no—
158C2×C4×C17—21no—
159C2×C4×C17—21no—
160C2×C4×C17—22no—
161C2×C4×C17—41no—
162C2×C4×C17—41no—
163C2×C4×C17—41no—
164C2×C4×C17—41no—
165C2×C4×C17—41no—
166C2×C4×C17—41no—
167C2×C4×C17—41no—
168C2×C4×C17—41no—
169C2×C4×C17—41no—
170C2×C4×C17—41no—
171C2×C4×C17—81no—
172C2×C4×C17—81no—
173C2×C4×C17—81no—
174C2×C4×C17—81no—
175C2×C4×C17—81no—
176C2×C4×C17#12011yescharacteristic
177C2×C4×C17#12311yescharacteristic
178C2×C4×C17#12311yescharacteristic
179C2×C4×C17#12411yescharacteristic
180C2×Dic17—41no—
181C2×Dic17—41no—
182C2×Dic17—81no—
183C2×Dic17—81no—
184C2×Dic17—81no—
185C2×Dic17—81no—
186C2×Dic17—81no—
187Dic34—42no—
188Dic34—41no—
189Dic34—41no—
190#3622—42no—
191#3622—41no—
192#3622—41no—
193#3189—171no—
194#1996—171no—
195#1996D1711yescharacteristic
196#1892—171no—
197#1892—171no—
198#1850—171no—
199#1289—171no—
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#39411yescharacteristic
211C22×C17#46211yescharacteristic
212C22×C17#46211yescharacteristic
213C22×C17#46411yescharacteristic
214C22×C17#46411yescharacteristic
215C22×C17#46711yescharacteristic
216C22×C17#47011yescharacteristic
217C4×C17—21no—
218C4×C17—21no—
219C4×C17—21no—
220C4×C17—42no—
221C4×C17—81no—
222C4×C17—81no—
223C4×C17—81no—
224C4×C17—161no—
225C4×C17B(4,𝔽2)11yescharacteristic
226C4×C17#45811yescharacteristic
227Dic17—81no—
228Dic17—81no—
229Dic17—161no—
230C23×C8—171no—
231C2×C22⋊Q8—171no—
232C2×C22⋊Q8C2×D1711yescharacteristic
233#484—341no—
234#484—341no—
235#467—341no—
236#467—341no—
237#451—341no—
238#451—341no—
239C2×C4⋊C8—341no—
240C2×#103—341no—
241C2×#103—341no—
242C2×C22⋊C8—171no—
243#380—171no—
244#380C2×D1711yescharacteristic
245C4×C22⋊C4C2×D1711yescharacteristic
246#342—21no—
247#342—21no—
248#342—341no—
249#342—341no—
250#340—341no—
251#340—341no—
252#336—341no—
253C2×C17—21no—
254C2×C17—21no—
255C2×C17—21no—
256C2×C17—21no—
257C2×C17—21no—
258C2×C17—21no—
259C2×C17#204411yescharacteristic
260C2×C17#205311yescharacteristic
261C2×C17#205611yescharacteristic
262C22×Q8—171no—
263C22×Q8D6811yescharacteristic
264C23×C4C22×D1711yescharacteristic
265C2×Q16—341no—
266C2×Q16—341no—
267C22×C8—171no—
268C22×C8—171no—
269C22×C8—341no—
270C22×C8—341no—
271C22×C8—341no—
272C22×C8—682no—
273C22⋊Q8—21no—
274C22⋊Q8—21no—
275C22⋊Q8—21no—
276C22⋊Q8—21no—
277C22⋊Q8—341no—
278C22⋊Q8—341no—
279C22⋊Q8—341no—
280C22⋊Q8—341no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—171no—
284C2×C4⋊C4—341no—
285C2×C4⋊C4—341no—
286C2×C4⋊C4D6811yescharacteristic
287C2×C22⋊C4—21no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—341no—
290C2×C22⋊C4—341no—
291C2×C22⋊C4C22⋊D1711yescharacteristic
292C2×C22⋊C4C22⋊D1711yescharacteristic
293C2×C42—21no—
294#107—341no—
295#107—341no—
296C4⋊C8—682no—
297#103—341no—
298#103—341no—
299#103—341no—
300#103—341no—
301#103—682no—
302#103—682no—
303C22⋊C8—341no—
304C22⋊C8—341no—
305#95—21no—
306#95—21no—
307#95—341no—
308C17#1056611yescharacteristic, complemented, Sylow
309C24C2×C22⋊D1711yescharacteristic
310C2×Q8—21no—
311C2×Q8—21no—
312C2×Q8—41no—
313C2×Q8—41no—
314C2×Q8—341no—
315C2×Q8—341no—
316C2×Q8—681no—
317C2×Q8—681no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—341no—
326C22×C4—341no—
327C22×C4—681no—
328C22×C4C2×D6811yescharacteristic
329C22×C4#6325211yescharacteristic
330C22×C4D4×D1711yescharacteristic
331C22×C4D4×D1711yescharacteristic
332C22×C4#6325411yescharacteristic
333C22×C4#6325411yescharacteristic
334Q16—1361no—
335Q16—1361no—
336C2×C8—342no—
337C2×C8—342no—
338C2×C8—342no—
339C2×C8—341no—
340C2×C8—341no—
341C2×C8—681no—
342C2×C8—681no—
343C2×C8—681no—
344C2×C8—681no—
345C2×C8—681no—
346C2×C8—681no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—341no—
354C4⋊C4—341no—
355C4⋊C4—681no—
356C4⋊C4—681no—
357C4⋊C4—681no—
358C4⋊C4—681no—
359C22⋊C4—21no—
360C22⋊C4—21no—
361C22⋊C4—21no—
362C22⋊C4—41no—
363C22⋊C4—41no—
364C22⋊C4—41no—
365C22⋊C4—41no—
366C22⋊C4—681no—
367C22⋊C4—681no—
368C22⋊C4—681no—
369C22⋊C4—681no—
370C22⋊C4#6323112yescomplemented
371C42—41no—
372C42—41no—
373C23—41no—
374C23—41no—
375C23#1058761611yescharacteristic
376C23#1058762411yescharacteristic
377C23#1058762511yescharacteristic
378C23#1058762811yescharacteristic
379C23C2×#6323111yescharacteristic
380C23#1058763611yescharacteristic
381C23#1058763711yescharacteristic
382Q8—42no—
383Q8—41no—
384Q8—41no—
385Q8—682no—
386Q8—681no—
387Q8—681no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—22no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—681no—
411C2×C4—681no—
412C2×C4—1361no—
413C2×C4—1361no—
414C2×C4—1361no—
415C2×C4—1361no—
416C2×C4—1361no—
417C2×C4#1058756811yescharacteristic
418C2×C4#1058756911yescharacteristic
419C2×C4#1058757111yescharacteristic
420C2×C4#1058757211yescharacteristic
421C8—682no—
422C8—681no—
423C8—681no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—41no—
431C22—41no—
432C22—41no—
433C22—41no—
434C22#1176272511yescharacteristic
435C22#1176301611yescharacteristic
436C22#1176301911yescharacteristic
437C22#1176302811yescharacteristic
438C22#1176303111yescharacteristic
439C22#1176304811yescharacteristic
440C22#1176306511yescharacteristic
441C4—21no—
442C4—21no—
443C4—21no—
444C4—42no—
445C4—81no—
446C4—81no—
447C4—81no—
448C4—161no—
449C4—2721no—
450C4—1361no—
451C4—1361no—
452C4#1176299711yescharacteristic
453C4#1176300011yescharacteristic
454C2—21no—
455C2—21no—
456C2—21no—
457C2—21no—
458C2—21no—
459C2—21no—
460C2#1453003411yescharacteristic
461C2#1453008911yescharacteristic
462C2#1453010611yescharacteristic
subgroupG/H
centerC22#11762725
commutator, cosocleC23×C4×C17C23
FrattiniC23×C4C22×D17
FittingC17×#1996C2
socleC22×C17#394
prime radicalC24×C17C2×D4
hypercenter#1996D17
hypocenterC17#10566
LCSG ▷ C23×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1996

Complements

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

NH ≅ G/Ntype
115. C22⋊C4×C17Q16semidirect
308. C17#10566semidirect
370. C22⋊C4#63231semidirect

Maximal subgroups

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

subgroupindex
#145395562
#145297692
#145292082
#145292052
#145289642
#145266512
C17×#19962
#1056617