Finite Group Databasebeta

← #16951878

Subgroups — #16951878

Order 4352.

694 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 694 conjugacy classes of subgroups — too many to draw. Only its 107 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#14532408C211yesmaximal, characteristic
2C2×#11762607C212yesmaximal, complemented
3#14528712C211yesmaximal, characteristic
4#14526910C212yesmaximal, complemented
5C2×C17×#413C211yesmaximal, characteristic
6C2×#10587630C2211yescharacteristic
7C2×#10587622C411yescharacteristic
8C2×#10587610C411yescharacteristic
9C2×#10587507C2212yes—
10C2×#10587504—21no—
11#11762773—21no—
12#11762723—21no—
13#11762723—21no—
14#11762716C2211yescharacteristic
15#11762712C411yescharacteristic
16#11762682—21no—
17#11762682—21no—
18C4×C22⋊Dic17C411yescharacteristic
19#11762607—24no—
20C22×#41×C17C2211yescharacteristic
21C22×M16×C17—21no—
22C17×#413—24no—
23C2×C17×#101C2212yes—
24C2×C17×#100—21no—
25C2×#41×C17—22no—
26C2×#41×C17—22no—
27C2×#41×C17—41no—
28C2×#41×C17—41no—
29C2×#41×C17—42no—
30C22×Q8×C17C2311yescharacteristic
31C22×D4×C17C2×C411yescharacteristic
32C22×D4×C17D411yescharacteristic
33C22×D4×C17D411yescharacteristic
34C23×C4×C17C2×C411yescharacteristic
35C23×C4×C17D411yescharacteristic
36C23×C4×C17D411yescharacteristic
37C22×Dic34C2×C411yescharacteristic
38C2×M16×C17—24no—
39C2×M16×C17—22no—
40C2×M16×C17—42no—
41C2×M16×C17—41no—
42C22×C8×C17—41no—
43C2×C22⋊Dic17—21no—
44C2×C22⋊Dic17—21no—
45C2×C22⋊Dic17—21no—
46C2×C22⋊Dic17—21no—
47C2×C22⋊Dic17—21no—
48C2×C22⋊Dic17—21no—
49#10587630—22no—
50#10587630—22no—
51#10587622—42no—
52C2×C4⋊Dic17C2×C411yescharacteristic
53#10587610—41no—
54#10587610—41no—
55C2×#63225—21no—
56C2×#63225—21no—
57C2×#63225—21no—
58C2×C4×Dic17C2×C411yescharacteristic
59C2×C4×Dic17C2×C411yescharacteristic
60C2×#63223—22no—
61C17×#101—22no—
62C17×#101D414yes—
63C17×#100—24no—
64#10587507—22no—
65#10587507—22no—
66#10587504—42no—
67#10587502—21no—
68#10587502—21no—
69#10587502—21no—
70C22⋊#3622—41no—
71C22⋊#3622—41no—
72C24×C17C22⋊C411yescharacteristic
73C24×C17C22⋊C411yescharacteristic
74C24×C17C22⋊C411yescharacteristic
75#41×C17—44no—
76#41×C17—44no—
77#41×C17—44no—
78#41×C17—44no—
79C2×Q8×C17—41no—
80C2×Q8×C17—41no—
81C2×Q8×C17C2×D412yes—
82C2×Q8×C17#4112yes—
83C2×D4×C17—22no—
84C2×D4×C17—22no—
85C2×D4×C17—22no—
86C2×D4×C17—21no—
87C2×D4×C17—22no—
88C2×D4×C17—21no—
89C2×D4×C17—41no—
90C2×D4×C17—41no—
91C2×D4×C17—41no—
92C2×D4×C17—41no—
93C22×C4×C17—22no—
94C22×C4×C17—21no—
95C22×C4×C17—21no—
96C22×C4×C17—21no—
97C22×C4×C17—21no—
98C22×C4×C17—21no—
99C22×C4×C17—21no—
100C22×C4×C17—21no—
101C22×C4×C17—21no—
102C22×C4×C17—22no—
103C22×C4×C17—22no—
104C22×C4×C17—41no—
105C22×C4×C17—41no—
106C22×C4×C17C22⋊C411yescharacteristic
107C22×C4×C17C22×C411yescharacteristic
108C22×C4×C17C2×D411yescharacteristic
109C22×C4×C17C2×D411yescharacteristic
110C22×Dic17—21no—
111C22×Dic17—21no—
112C22×Dic17—21no—
113C22×Dic17—21no—
114C2×Dic34—22no—
115C2×Dic34—41no—
116C2×Dic34—41no—
117M16×C17—24no—
118M16×C17—44no—
119M16×C17—42no—
120C2×C8×C17—44no—
121C2×C8×C17—42no—
122C22⋊Dic17—42no—
123C22⋊Dic17—42no—
124C22⋊Dic17—42no—
125C22⋊Dic17—42no—
126C22⋊Dic17—42no—
127C22⋊Dic17—42no—
128C4⋊Dic17—41no—
129#63225—41no—
130#63225—41no—
131#63225—41no—
132#63225—42no—
133#63225—41no—
134C4×Dic17—22no—
135C4×Dic17—41no—
136#63223—42no—
137#63223—42no—
138C2×#3622—42no—
139#13095—171nomaximal, Sylow
140C23×C17—21no—
141C23×C17—21no—
142C23×C17—21no—
143C23×C17—21no—
144C23×C17—21no—
145C23×C17—21no—
146C23×C17—21no—
147C23×C17—21no—
148C23×C17—41no—
149C23×C17—41no—
150C23×C17—41no—
151C23×C17—41no—
152C23×C17—41no—
153C23×C17—41no—
154C23×C17#9912yes—
155C23×C17C2×C22⋊C411yescharacteristic
156Q8×C17—42no—
157Q8×C17—42no—
158D4×C17—44no—
159D4×C17—44no—
160D4×C17—42no—
161D4×C17—42no—
162C2×C4×C17—22no—
163C2×C4×C17—22no—
164C2×C4×C17—22no—
165C2×C4×C17—22no—
166C2×C4×C17—21no—
167C2×C4×C17—21no—
168C2×C4×C17—21no—
169C2×C4×C17—21no—
170C2×C4×C17—21no—
171C2×C4×C17—21no—
172C2×C4×C17—21no—
173C2×C4×C17—21no—
174C2×C4×C17—42no—
175C2×C4×C17—42no—
176C2×C4×C17—42no—
177C2×C4×C17—42no—
178C2×C4×C17#9911yescharacteristic
179C2×C4×C17#9911yescharacteristic
180C2×C4×C17C4×D412yes—
181C2×C4×C17#12011yescharacteristic
182C2×C4×C17#12112yes—
183C2×C4×C17#12311yescharacteristic
184C2×Dic17—22no—
185C2×Dic17—41no—
186C2×Dic17—41no—
187C2×Dic17—41no—
188C2×Dic17—41no—
189C2×Dic17—41no—
190C2×Dic17—41no—
191C2×Dic17—41no—
192C2×Dic17—41no—
193C2×Dic17—41no—
194C2×Dic17—41no—
195C2×Dic17—41no—
196Dic34—82no—
197C8×C17—44no—
198#3622—82no—
199C2×#413D1711yescharacteristic
200#2432—171no—
201C2×#356—172no—
202#1785—171no—
203#1394—172no—
204C22×C17—21no—
205C22×C17—21no—
206C22×C17—22no—
207C22×C17—42no—
208C22×C17—42no—
209C22×C17—41no—
210C22×C17—41no—
211C22×C17—41no—
212C22×C17—41no—
213C22×C17—41no—
214C22×C17—41no—
215C22×C17—41no—
216C22×C17—41no—
217C22×C17—81no—
218C22×C17—81no—
219C22×C17—81no—
220C22×C17—81no—
221C22×C17#38611yescharacteristic
222C22×C17C2×#9911yescharacteristic
223C22×C17C2×#9911yescharacteristic
224C4×C17—22no—
225C4×C17—21no—
226C4×C17—22no—
227C4×C17—21no—
228C4×C17—21no—
229C4×C17—21no—
230Dic17—42no—
231Dic17—81no—
232Dic17—81no—
233Dic17—81no—
234C22×#41C2×D1711yescharacteristic
235C22×M16—21no—
236C2×C4⋊Q8—171no—
237C2×#124—171no—
238C2×C22⋊Q8—171no—
239#413—24no—
240C2×#101—172no—
241C2×#101C2×D1712yes—
242C2×#100—21no—
243C2×#100—341no—
244#407—341no—
245#394—341no—
246#394—341no—
247#391—171no—
248#390—171no—
249#385—341no—
250#385—341no—
251C4×C22⋊C4—171no—
252#356—344no—
253C2×C17—21no—
254C2×C17—21no—
255C2×C17—42no—
256C2×C17—81no—
257C2×C17—81no—
258C2×C17#189811yescharacteristic
259C2×C17#213211yescharacteristic
260C2×C17#213511yescharacteristic
261C2×#41—22no—
262C2×#41—22no—
263C2×#41—41no—
264C2×#41—41no—
265C2×#41—42no—
266C22×Q8—171no—
267C22×Q8C22×D1711yescharacteristic
268C22×D4C4×D1711yescharacteristic
269C22×D4D6811yescharacteristic
270C22×D4C22⋊D1711yescharacteristic
271C23×C4C4×D1711yescharacteristic
272C23×C4D6811yescharacteristic
273C23×C4C22⋊D1711yescharacteristic
274C2×M16—24no—
275C2×M16—22no—
276C2×M16—42no—
277C2×M16—41no—
278C2×M16—342no—
279C22×C8—41no—
280C4⋊Q8—342no—
281C4⋊Q8—342no—
282#124—682no—
283C22⋊Q8—681no—
284C22⋊Q8—681no—
285C2×C4⋊C4—171no—
286C2×C4⋊C4—341no—
287C2×C4⋊C4—341no—
288C2×C4⋊C4—341no—
289C2×C22⋊C4—341no—
290C2×C22⋊C4—341no—
291C2×C22⋊C4—341no—
292C2×C22⋊C4—341no—
293C2×C22⋊C4—341no—
294C2×C22⋊C4—341no—
295C2×C42—171no—
296C2×C42—171no—
297#101—22no—
298#101—342no—
299#101—342no—
300#101C22⋊D1714yes—
301#100—24no—
302#100—682no—
303C22⋊C8—681no—
304C22⋊C8—681no—
305#95—341no—
306#95—341no—
307#95—341no—
308C17#1309511yescharacteristic, complemented, Sylow
309C24#6322711yescharacteristic
310C24#6322711yescharacteristic
311C24#6322711yescharacteristic
312#41—44no—
313#41—44no—
314#41—44no—
315#41—44no—
316C2×Q8—41no—
317C2×Q8—41no—
318C2×Q8—342no—
319C2×Q8—681no—
320C2×Q8—681no—
321C2×Q8#6325212yes—
322C2×Q8C2×C22⋊D1712yes—
323C2×D4—22no—
324C2×D4—22no—
325C2×D4—22no—
326C2×D4—21no—
327C2×D4—22no—
328C2×D4—21no—
329C2×D4—41no—
330C2×D4—41no—
331C2×D4—41no—
332C2×D4—41no—
333C22×C4—22no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—341no—
347C22×C4—341no—
348C22×C4—341no—
349C22×C4—341no—
350C22×C4#6322711yescharacteristic
351C22×C4C2×C4×D1711yescharacteristic
352C22×C4C2×D6811yescharacteristic
353C22×C4C2×C22⋊D1711yescharacteristic
354M16—24no—
355M16—44no—
356M16—42no—
357M16—682no—
358M16—682no—
359C2×C8—44no—
360C2×C8—42no—
361C2×C8—682no—
362C4⋊C4—681no—
363C4⋊C4—681no—
364C4⋊C4—681no—
365C4⋊C4—681no—
366C4⋊C4—682no—
367C4⋊C4—681no—
368C22⋊C4—682no—
369C22⋊C4—682no—
370C22⋊C4—682no—
371C22⋊C4—682no—
372C22⋊C4—682no—
373C22⋊C4—682no—
374C42—342no—
375C42—681no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—41no—
385C23—41no—
386C23—41no—
387C23—41no—
388C23—41no—
389C23—41no—
390C23#1058747712yes—
391C23C2×#6322711yescharacteristic
392Q8—42no—
393Q8—42no—
394Q8—1362no—
395D4—44no—
396D4—44no—
397D4—42no—
398D4—42no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—342no—
416C2×C4—681no—
417C2×C4—681no—
418C2×C4—681no—
419C2×C4—681no—
420C2×C4—681no—
421C2×C4—681no—
422C2×C4—681no—
423C2×C4—681no—
424C2×C4—681no—
425C2×C4—681no—
426C2×C4—681no—
427C2×C4#1058747711yescharacteristic
428C2×C4#1058747711yescharacteristic
429C2×C4C4×C22⋊D1712yes—
430C2×C4#1058761511yescharacteristic
431C2×C4#1058761612yes—
432C2×C4#1058763911yescharacteristic
433C8—44no—
434C8—1362no—
435C22—21no—
436C22—21no—
437C22—22no—
438C22—42no—
439C22—42no—
440C22—41no—
441C22—41no—
442C22—41no—
443C22—41no—
444C22—41no—
445C22—41no—
446C22—41no—
447C22—41no—
448C22—81no—
449C22—81no—
450C22—81no—
451C22—81no—
452C22#1176268811yescharacteristic
453C22C2×#1058747711yescharacteristic
454C22C2×#1058747711yescharacteristic
455C4—21no—
456C4—22no—
457C4—21no—
458C4—21no—
459C4—21no—
460C4—22no—
461C4—682no—
462C4—1361no—
463C4—1361no—
464C4—1361no—
465C2—21no—
466C2—21no—
467C2—42no—
468C2—81no—
469C2—81no—
470C2#1452923711yescharacteristic
471C2#1453046711yescharacteristic
472C2#1453047811yescharacteristic
subgroupG/H
centerC22C2×#10587477
commutatorC22×C4×C17C22×C4
FrattiniC22×Q8C22×D17
FittingC2×C17×#413C2
socleC22×C17C2×#99
prime radicalC22×#41×C17C22
cosocleC22×Q8×C17C23
hypercenterC2×#413D17
hypocenterC17#13095
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ C2×#413

Complements

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

NH ≅ G/Ntype
2. C2×#11762607C2semidirect
4. #14526910C2semidirect
308. C17#13095semidirect

Maximal subgroups

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

subgroupindex
#145324082
C2×#117626072
#145287122
#145269102
C2×C17×#4132
#1309517