Finite Group Databasebeta

← #16935089

Subgroups — #16935089

Order 4352.

568 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 101 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#14538792C211yesmaximal, characteristic
2#14538589C211yesmaximal, characteristic, complemented
3#14527347C211yesmaximal, characteristic, complemented
4#14527209C211yesmaximal, characteristic
5#14527179C211yesmaximal, characteristic, complemented
6#14526642C211yesmaximal, characteristic, complemented
7C17×#2536C211yesmaximal, characteristic
8#11763366C2211yescharacteristic
9#11763139—21no—
10#11763116—22no—
11#11763104—21no—
12#11763099—21no—
13#11763094—21no—
14#11763014—21no—
15#11762960—21no—
16C4×#63229—22no—
17#11762862C2211yescharacteristic
18#11762862C2211yescharacteristic
19#11762849—21no—
20#11762833C2211yescharacteristic
21#11762818—21no—
22#11762545—22no—
23Q8⋊#3622—22no—
24D4⋊#3622—22no—
25C2×C17×C22⋊Q8—21no—
26C2×C4×Q8×C17C2211yescharacteristic
27C2×C4×D4×C17C2211yescharacteristic
28C17×#398—21no—
29C17×#394—21no—
30C17×#388—22no—
31C17×#382C2211yescharacteristic
32C17×#380—21no—
33C22×Q8×C17D411yescharacteristic
34C22×D4×C17D411yescharacteristic
35C23×C4×C17—21no—
36C17×C22⋊Q8—42no—
37C17×C22⋊Q8—42no—
38C4×Q8×C17—21no—
39C4×Q8×C17—21no—
40C4×Q8×C17—21no—
41C4×Q8×C17—21no—
42C4×D4×C17—21no—
43C4×D4×C17—21no—
44C4×D4×C17—21no—
45C4×D4×C17—21no—
46C2×C4⋊C4×C17—22no—
47C2×C4⋊C4×C17—21no—
48C2×C4⋊C4×C17—21no—
49C2×C4⋊C4×C17D411yescharacteristic
50C2×C4⋊C4×C17D411yescharacteristic
51C2×C22⋊C4×C17—22no—
52C2×C22⋊C4×C17—22no—
53C2×C22⋊C4×C17—21no—
54C2×C42×C17—21no—
55C2×C42×C17C2311yescharacteristic
56C2×#63229—22no—
57#10587612—21no—
58#10587610—21no—
59#10587610—21no—
60C2×#63223—21no—
61#10587556—21no—
62#10587555D411yescharacteristic
63#10587555D411yescharacteristic
64C4×Dic34—21no—
65C4×Dic34—21no—
66C17×#95—22no—
67C17×#95—21no—
68C17×#95D412yes—
69#10587506—22no—
70#10587506—22no—
71#10587503—22no—
72C22⋊#3622—21no—
73C22⋊#3622—21no—
74#10587481—22no—
75#10587479—22no—
76#10587479—22no—
77#10587478—22no—
78C4⋊#3622—21no—
79C4⋊#3622—21no—
80C4⋊#3622—21no—
81C4⋊#3622D412yes—
82C4×#3622—21no—
83C4×#3622D412yes—
84C24×C17—21no—
85C2×Q8×C17—21no—
86C2×Q8×C17—21no—
87C2×Q8×C17—21no—
88C2×Q8×C17—21no—
89C2×Q8×C17—21no—
90C2×Q8×C17SD1612yescomplemented
91C2×D4×C17—21no—
92C2×D4×C17—21no—
93C2×D4×C17—42no—
94C22×C4×C17—21no—
95C22×C4×C17—21no—
96C22×C4×C17—22no—
97C22×C4×C17—21no—
98C22×C4×C17—41no—
99C22×C4×C17—41no—
100C22×C4×C17—41no—
101C22×C4×C17—41no—
102C22×C4×C17—41no—
103C22×C4×C17—41no—
104C22×C4×C17C2×D411yescharacteristic
105C22×C4×C17C2×D411yescharacteristic
106C22×C4×C17C2×D411yescharacteristic
107C2×Dic34—21no—
108C2×Dic34—21no—
109C4⋊C4×C17—21no—
110C4⋊C4×C17—21no—
111C4⋊C4×C17—21no—
112C4⋊C4×C17—21no—
113C4⋊C4×C17—44no—
114C4⋊C4×C17—41no—
115C4⋊C4×C17—41no—
116C4⋊C4×C17—41no—
117C4⋊C4×C17—41no—
118C22⋊C4×C17—44no—
119C22⋊C4×C17—44no—
120C22⋊C4×C17—41no—
121C22⋊C4×C17—41no—
122C42×C17—41no—
123C42×C17—41no—
124C42×C17C2×D411yescharacteristic
125C42×C17C2×D411yescharacteristic
126C42×C17C2×D411yescharacteristic
127C42×C17#4111yescharacteristic
128C22⋊Dic17—41no—
129#63229—44no—
130C4⋊Dic17—21no—
131C4⋊Dic17—21no—
132C4⋊Dic17—21no—
133C4⋊Dic17—21no—
134#63225—41no—
135#63225—41no—
136C4×Dic17—41no—
137#63223—41no—
138#63223—41no—
139C2×#3622—22no—
140C2×#3622—22no—
141C2×#3622—41no—
142#10001—171nomaximal, Sylow
143C23×C17—22no—
144C23×C17—41no—
145C23×C17—41no—
146C23×C17—41no—
147C23×C17—41no—
148C23×C17—41no—
149C23×C17—41no—
150C23×C17#12011yescharacteristic
151Q8×C17—22no—
152Q8×C17—22no—
153Q8×C17—41no—
154Q8×C17—41no—
155D4×C17—22no—
156D4×C17—22no—
157D4×C17—41no—
158D4×C17—41no—
159C2×C4×C17—22no—
160C2×C4×C17—41no—
161C2×C4×C17—41no—
162C2×C4×C17—41no—
163C2×C4×C17—41no—
164C2×C4×C17—42no—
165C2×C4×C17—42no—
166C2×C4×C17—42no—
167C2×C4×C17—41no—
168C2×C4×C17—41no—
169C2×C4×C17—42no—
170C2×C4×C17—41no—
171C2×C4×C17—41no—
172C2×C4×C17—81no—
173C2×C4×C17—81no—
174C2×C4×C17—81no—
175C2×C4×C17—81no—
176C2×C4×C17#12011yescharacteristic
177C2×C4×C17#12011yescharacteristic
178C2×C4×C17#12111yescharacteristic
179C2×C4×C17#12111yescharacteristic
180C2×C4×C17#12111yescharacteristic
181C2×C4×C17#12711yescharacteristic
182C2×C4×C17C2×SD1611yescharacteristic
183C2×C4×C17#13511yescharacteristic
184C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
185C2×C4×C17#13711yescharacteristic
186C2×Dic17—41no—
187C2×Dic17—41no—
188Dic34—42no—
189Dic34—81no—
190#3622—42no—
191#3622—42no—
192#3622—81no—
193#3101—171no—
194#3078—171no—
195#2536D1711yescharacteristic
196#1518—171no—
197#1486—171no—
198#1479—171no—
199#1286—171no—
200C22×C17—21no—
201C22×C17—21no—
202C22×C17—42no—
203C22×C17—42no—
204C22×C17—42no—
205C22×C17—81no—
206C22×C17—81no—
207C22×C17—81no—
208C22×C17—81no—
209C22×C17#39211yescharacteristic
210C22×C17#45011yescharacteristic
211C22×C17#45211yescharacteristic
212C4×C17—21no—
213C4×C17—21no—
214C4×C17—21no—
215C4×C17—21no—
216C4×C17—42no—
217C4×C17—82no—
218C4×C17—81no—
219C4×C17—81no—
220C4×C17#44811yescharacteristic
221C4×C17#44911yescharacteristic
222C4×C17#46511yescharacteristic
223C4×C17#46811yescharacteristic
224Dic17—81no—
225Dic17—81no—
226#533—171no—
227C2×C22⋊Q8—21no—
228C2×C4×Q8C2×D1711yescharacteristic
229C2×C4×D4C2×D1711yescharacteristic
230#483—341no—
231#479—342no—
232#477—341no—
233#476—341no—
234#475—341no—
235#461—341no—
236#450—341no—
237C4×SD16—342no—
238#431—171no—
239#431—171no—
240#428—341no—
241#423—171no—
242#419—341no—
243#398—21no—
244#394—21no—
245#388—22no—
246#382C2×D1711yescharacteristic
247#380—21no—
248#328—342no—
249Q8⋊C8—342no—
250D4⋊C8—342no—
251C2×C17—21no—
252C2×C17—21no—
253C2×C17—82no—
254C2×C17#200011yescharacteristic
255C2×C17#200911yescharacteristic
256C2×C17#201011yescharacteristic
257C22×Q8C22⋊D1711yescharacteristic
258C22×D4C22⋊D1711yescharacteristic
259C23×C4—21no—
260C2×SD16—342no—
261C2×M16—341no—
262C4⋊Q8—171no—
263C4⋊Q8—171no—
264#125—341no—
265C22⋊Q8—42no—
266C22⋊Q8—42no—
267C22⋊Q8—341no—
268C22⋊Q8—341no—
269C4×Q8—21no—
270C4×Q8—21no—
271C4×Q8—21no—
272C4×Q8—21no—
273C4×Q8—341no—
274C4×Q8—341no—
275C4×D4—21no—
276C4×D4—21no—
277C4×D4—21no—
278C4×D4—21no—
279#117—341no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4C22⋊D1711yescharacteristic
284C2×C4⋊C4C22⋊D1711yescharacteristic
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—21no—
288C2×C42—21no—
289C2×C42C22×D1711yescharacteristic
290#107—342no—
291#106—342no—
292#106—342no—
293C4⋊C8—172no—
294C4⋊C8—341no—
295C4⋊C8—341no—
296C4⋊C8—341no—
297#103—342no—
298#103—342no—
299#103—342no—
300#102—342no—
301C22⋊C8—341no—
302C22⋊C8—341no—
303C4×C8—172no—
304C4×C8—341no—
305#95—22no—
306#95—21no—
307#95C22⋊D1712yes—
308C17#1000111yescharacteristic, complemented, Sylow
309C24—21no—
310C2×Q8—21no—
311C2×Q8—21no—
312C2×Q8—21no—
313C2×Q8—21no—
314C2×Q8—21no—
315C2×Q8—341no—
316C2×Q8—341no—
317C2×Q8#6322912yescomplemented
318C2×D4—21no—
319C2×D4—21no—
320C2×D4—42no—
321C22×C4—21no—
322C22×C4—22no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4C2×C22⋊D1711yescharacteristic
332C22×C4C2×C22⋊D1711yescharacteristic
333C22×C4C2×C22⋊D1711yescharacteristic
334SD16—684no—
335M16—681no—
336M16—681no—
337C2×C8—342no—
338C2×C8—342no—
339C2×C8—681no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—44no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—341no—
350C4⋊C4—341no—
351C4⋊C4—341no—
352C4⋊C4—341no—
353C4⋊C4—681no—
354C4⋊C4—681no—
355C22⋊C4—44no—
356C22⋊C4—44no—
357C22⋊C4—41no—
358C22⋊C4—41no—
359C22⋊C4—681no—
360C42—41no—
361C42—41no—
362C42—681no—
363C42D4×D1711yescharacteristic
364C42D4×D1711yescharacteristic
365C42D4×D1711yescharacteristic
366C42#6325411yescharacteristic
367C23—22no—
368C23—41no—
369C23—41no—
370C23—41no—
371C23—41no—
372C23—41no—
373C23—41no—
374C23#1058763911yescharacteristic
375Q8—22no—
376Q8—22no—
377Q8—41no—
378Q8—41no—
379Q8—682no—
380Q8—1361no—
381D4—22no—
382D4—22no—
383D4—41no—
384D4—41no—
385C2×C4—22no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—681no—
403C2×C4—681no—
404C2×C4#1058761811yescharacteristic
405C2×C4C2×#6322911yescharacteristic
406C2×C4#1058762311yescharacteristic
407C2×C4#1058762311yescharacteristic
408C2×C4#1058762411yescharacteristic
409C2×C4#1058762511yescharacteristic
410C2×C4#1058762511yescharacteristic
411C2×C4#1058762611yescharacteristic
412C2×C4#1058763611yescharacteristic
413C2×C4#1058763711yescharacteristic
414C8—682no—
415C8—682no—
416C8—1361no—
417C22—21no—
418C22—21no—
419C22—42no—
420C22—42no—
421C22—42no—
422C22—81no—
423C22—81no—
424C22—81no—
425C22—81no—
426C22#1176271711yescharacteristic
427C22#1176296011yescharacteristic
428C22#1176297111yescharacteristic
429C4—21no—
430C4—21no—
431C4—21no—
432C4—21no—
433C4—42no—
434C4—82no—
435C4—81no—
436C4—81no—
437C4—1361no—
438C4—1361no—
439C4#1176294711yescharacteristic
440C4#1176295211yescharacteristic
441C4#1176303611yescharacteristic
442C4#1176305311yescharacteristic
443C2—21no—
444C2—21no—
445C2—82no—
446C2#1452978711yescharacteristic
447C2#1452984411yescharacteristic
448C2#1452984911yescharacteristic
subgroupG/H
centerC22#11762717
commutator, cosocleC2×C42×C17C23
FrattiniC2×C42C22×D17
FittingC17×#2536C2
socleC22×C17#392
prime radicalC22×D4×C17D4
hypercenter#2536D17
hypocenterC17#10001
LCSG ▷ C2×C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2536

Complements

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

NH ≅ G/Ntype
2. #14538589C2semidirect
3. #14527347C2semidirect
5. #14527179C2semidirect
6. #14526642C2semidirect
90. C2×Q8×C17SD16semidirect
308. C17#10001semidirect
317. C2×Q8#63229semidirect

Maximal subgroups

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

subgroupindex
#145387922
#145385892
#145273472
#145272092
#145271792
#145266422
C17×#25362
#1000117