Finite Group Databasebeta

← #16951874

Subgroups — #16951874

Order 4352.

676 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 676 conjugacy classes of subgroups — too many to draw. Only its 115 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
1C22×#10587507C211yesmaximal, characteristic
2C2×#11762608C212yesmaximal, complemented
3#14528708C211yesmaximal, characteristic
4#14526911C212yesmaximal, complemented
5C17×#2434C211yesmaximal, characteristic
6C22×#63223—21no—
7C2×#10587507—24no—
8C2×#10587507—21no—
9C2×#10587507—21no—
10C2×#10587507C2212yes—
11C2×#10587507C2212yes—
12#11762773—21no—
13#11762608—24no—
14C23×Q8×C17C2211yescharacteristic
15C2×C17×C4⋊Q8C411yescharacteristic
16C2×C17×C4⋊Q8C2211yescharacteristic
17C2×C17×C22⋊Q8C411yescharacteristic
18C17×#395—21no—
19C17×#395—21no—
20C17×#391C411yescharacteristic
21C17×#391C2211yescharacteristic
22C17×#385—21no—
23C17×#385—21no—
24C4×C22⋊C4×C17C411yescharacteristic
25C22×Q8×C17—22no—
26C22×Q8×C17—22no—
27C22×Q8×C17—41no—
28C22×Q8×C17—41no—
29C22×Q8×C17—42no—
30C22×Q8×C17C2×C411yescharacteristic
31C22×Q8×C17C2×C411yescharacteristic
32C22×Q8×C17D411yescharacteristic
33C22×Q8×C17D411yescharacteristic
34C22×Q8×C17C2311yescharacteristic
35C23×C4×C17C2×C411yescharacteristic
36C23×C4×C17D411yescharacteristic
37C23×C4×C17D411yescharacteristic
38C17×C4⋊Q8—22no—
39C17×C4⋊Q8—22no—
40C17×C4⋊Q8—42no—
41C17×C22⋊Q8—41no—
42C17×C22⋊Q8—41no—
43C2×C4⋊C4×C17—21no—
44C2×C4⋊C4×C17—21no—
45C2×C4⋊C4×C17—21no—
46C2×C4⋊C4×C17—21no—
47C2×C4⋊C4×C17—21no—
48C2×C4⋊C4×C17C2×C411yescharacteristic
49C2×C22⋊C4×C17—21no—
50C2×C22⋊C4×C17—21no—
51C2×C42×C17C2×C411yescharacteristic
52C2×C42×C17C2×C411yescharacteristic
53C2×#63223—24no—
54C2×#63223—22no—
55C2×#63223—22no—
56C2×#63223—41no—
57C2×#63223—42no—
58C22×#3622—41no—
59C17×#95—21no—
60C17×#95—21no—
61C17×#95—21no—
62C17×#95—21no—
63C17×#95—21no—
64#10587507—24no—
65#10587507—22no—
66#10587507—22no—
67#10587507—22no—
68#10587507—42no—
69#10587507D414yes—
70C22⋊#3622—41no—
71C22⋊#3622—41no—
72C24×C17C22⋊C411yescharacteristic
73C2×Q8×C17—22no—
74C2×Q8×C17—22no—
75C2×Q8×C17—22no—
76C2×Q8×C17—21no—
77C2×Q8×C17—22no—
78C2×Q8×C17—21no—
79C2×Q8×C17—22no—
80C2×Q8×C17—41no—
81C2×Q8×C17—41no—
82C2×Q8×C17—41no—
83C2×Q8×C17—41no—
84C2×Q8×C17—41no—
85C2×Q8×C17—41no—
86C2×Q8×C17—44no—
87C2×Q8×C17—44no—
88C2×Q8×C17—44no—
89C2×Q8×C17—44no—
90C2×Q8×C17—41no—
91C2×Q8×C17—41no—
92C2×Q8×C17C2×D412yes—
93C2×Q8×C17#4112yes—
94C22×C4×C17—21no—
95C22×C4×C17—21no—
96C22×C4×C17—22no—
97C22×C4×C17—21no—
98C22×C4×C17—21no—
99C22×C4×C17—21no—
100C22×C4×C17—21no—
101C22×C4×C17—21no—
102C22×C4×C17—21no—
103C22×C4×C17—21no—
104C22×C4×C17—21no—
105C22×C4×C17—22no—
106C22×C4×C17—22no—
107C22×C4×C17—21no—
108C22×C4×C17—21no—
109C22×C4×C17—41no—
110C22×C4×C17—41no—
111C22×C4×C17C22⋊C411yescharacteristic
112C22×C4×C17C22⋊C411yescharacteristic
113C22×C4×C17C22⋊C411yescharacteristic
114C22×C4×C17C22×C411yescharacteristic
115C22×C4×C17C2×D411yescharacteristic
116C22×C4×C17C2×D411yescharacteristic
117C4⋊C4×C17—41no—
118C4⋊C4×C17—42no—
119C4⋊C4×C17—42no—
120C4⋊C4×C17—42no—
121C4⋊C4×C17—41no—
122C4⋊C4×C17—42no—
123C4⋊C4×C17—41no—
124C22⋊C4×C17—24no—
125C22⋊C4×C17—41no—
126C22⋊C4×C17—41no—
127C42×C17—22no—
128C42×C17—41no—
129#63223—24no—
130#63223—44no—
131#63223—42no—
132#63223—42no—
133#63223—42no—
134C2×#3622—44no—
135C2×#3622—42no—
136C2×#3622—42no—
137#13094—171nomaximal, Sylow
138C23×C17—22no—
139C23×C17—21no—
140C23×C17—21no—
141C23×C17—41no—
142C23×C17#9911yescharacteristic
143C23×C17#9911yescharacteristic
144C23×C17C2×C22⋊C411yescharacteristic
145Q8×C17—42no—
146Q8×C17—42no—
147Q8×C17—44no—
148Q8×C17—44no—
149Q8×C17—42no—
150Q8×C17—42no—
151Q8×C17—82no—
152C2×C4×C17—22no—
153C2×C4×C17—22no—
154C2×C4×C17—22no—
155C2×C4×C17—22no—
156C2×C4×C17—21no—
157C2×C4×C17—21no—
158C2×C4×C17—21no—
159C2×C4×C17—21no—
160C2×C4×C17—21no—
161C2×C4×C17—21no—
162C2×C4×C17—21no—
163C2×C4×C17—21no—
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—42no—
173C2×C4×C17—42no—
174C2×C4×C17—42no—
175C2×C4×C17—42no—
176C2×C4×C17—41no—
177C2×C4×C17—41no—
178C2×C4×C17—41no—
179C2×C4×C17—41no—
180C2×C4×C17—42no—
181C2×C4×C17—41no—
182C2×C4×C17—41no—
183C2×C4×C17—41no—
184C2×C4×C17—41no—
185C2×C4×C17#9912yes—
186C2×C4×C17C4×D412yes—
187C2×C4×C17#12011yescharacteristic
188C2×C4×C17#12112yes—
189C2×C4×C17#12311yescharacteristic
190#3622—44no—
191#3622—82no—
192C22×#101—171no—
193#2434D1711yescharacteristic
194C2×#356—172no—
195#1784—171no—
196#1394—172no—
197C22×C17—21no—
198C22×C17—21no—
199C22×C17—21no—
200C22×C17—21no—
201C22×C17—21no—
202C22×C17—21no—
203C22×C17—21no—
204C22×C17—42no—
205C22×C17—42no—
206C22×C17#35612yescomplemented
207C22×C17#38611yescharacteristic
208C22×C17C2×#9911yescharacteristic
209C22×C17C2×#9911yescharacteristic
210C4×C17—21no—
211C4×C17—22no—
212C4×C17—22no—
213C4×C17—21no—
214C4×C17—22no—
215C4×C17—42no—
216C4×C17—42no—
217C4×C17—41no—
218C4×C17—41no—
219C4×C17—81no—
220C4×C17—81no—
221C23×Q8C2×D1711yescharacteristic
222C22×M16—341no—
223C2×C4⋊Q8Dic1711yescharacteristic
224C2×C4⋊Q8C2×D1711yescharacteristic
225C2×C22⋊Q8Dic1711yescharacteristic
226C2×#101—172no—
227C2×#101—172no—
228C2×#101—344no—
229C2×#101—341no—
230C2×#101—341no—
231#407—341no—
232#395—21no—
233#395—21no—
234#391Dic1711yescharacteristic
235#391C2×D1711yescharacteristic
236#385—21no—
237#385—21no—
238C4×C22⋊C4Dic1711yescharacteristic
239#356—344no—
240C2×C17—21no—
241C2×C17—21no—
242C2×C17—22no—
243C2×C17—41no—
244C2×C17#189811yescharacteristic
245C2×C17C2×#35611yescharacteristic
246C2×C17#213311yescharacteristic
247C22×Q8—22no—
248C22×Q8—22no—
249C22×Q8—41no—
250C22×Q8—41no—
251C22×Q8—42no—
252C22×Q8C2×Dic1711yescharacteristic
253C22×Q8C2×Dic1711yescharacteristic
254C22×Q8C22⋊D1711yescharacteristic
255C22×Q8C22⋊D1711yescharacteristic
256C22×Q8C22×D1711yescharacteristic
257C23×C4C2×Dic1711yescharacteristic
258C23×C4C22⋊D1711yescharacteristic
259C23×C4C22⋊D1711yescharacteristic
260C2×M16—344no—
261C2×M16—342no—
262C2×M16—342no—
263C2×M16—681no—
264C2×M16—682no—
265C22×C8—681no—
266C4⋊Q8—22no—
267C4⋊Q8—22no—
268C4⋊Q8—42no—
269C22⋊Q8—41no—
270C22⋊Q8—41no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4C2×Dic1711yescharacteristic
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C42C2×Dic1711yescharacteristic
280C2×C42C2×Dic1711yescharacteristic
281#101—174no—
282#101—344no—
283#101—342no—
284#101—342no—
285#101—342no—
286#101—682no—
287C22⋊C8—681no—
288C22⋊C8—681no—
289#95—21no—
290#95—21no—
291#95—21no—
292#95—21no—
293#95—21no—
294C17#1309411yescharacteristic, complemented, Sylow
295C24C22⋊Dic1711yescharacteristic
296C2×Q8—22no—
297C2×Q8—22no—
298C2×Q8—22no—
299C2×Q8—21no—
300C2×Q8—21no—
301C2×Q8—22no—
302C2×Q8—22no—
303C2×Q8—41no—
304C2×Q8—41no—
305C2×Q8—41no—
306C2×Q8—41no—
307C2×Q8—41no—
308C2×Q8—41no—
309C2×Q8—44no—
310C2×Q8—44no—
311C2×Q8—44no—
312C2×Q8—44no—
313C2×Q8—41no—
314C2×Q8—41no—
315C2×Q8D4×D1712yes—
316C2×Q8#6325412yes—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—22no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4C22⋊Dic1711yescharacteristic
335C22×C4C22⋊Dic1711yescharacteristic
336C22×C4C22⋊Dic1711yescharacteristic
337C22×C4C22×Dic1711yescharacteristic
338C22×C4C2×C22⋊D1711yescharacteristic
339C22×C4C2×C22⋊D1711yescharacteristic
340M16—344no—
341M16—684no—
342M16—682no—
343M16—682no—
344M16—682no—
345C2×C8—684no—
346C2×C8—682no—
347C2×C8—682no—
348C4⋊C4—41no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—42no—
355C22⋊C4—24no—
356C22⋊C4—41no—
357C22⋊C4—41no—
358C42—22no—
359C42—41no—
360C23—21no—
361C23—21no—
362C23—22no—
363C23—41no—
364C23#1058750511yescharacteristic
365C23#1058750511yescharacteristic
366C23C2×C22⋊Dic1711yescharacteristic
367Q8—42no—
368Q8—42no—
369Q8—44no—
370Q8—44no—
371Q8—42no—
372Q8—42no—
373Q8—82no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—42no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4#1058750512yes—
408C2×C4D4×Dic1712yes—
409C2×C4#1058762111yescharacteristic
410C2×C4#1058762311yescharacteristic
411C2×C4#1058762512yes—
412C8—684no—
413C8—1362no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—42no—
422C22—42no—
423C22#1176260812yescomplemented
424C22#1176268911yescharacteristic
425C22C2×#1058750511yescharacteristic
426C22C2×#1058750511yescharacteristic
427C4—21no—
428C4—22no—
429C4—22no—
430C4—21no—
431C4—22no—
432C4—42no—
433C4—42no—
434C4—41no—
435C4—41no—
436C4—81no—
437C4—81no—
438C2—21no—
439C2—21no—
440C2—22no—
441C2—41no—
442C2#1452923811yescharacteristic
443C2C2×#1176260811yescharacteristic
444C2#1453047311yescharacteristic
subgroupG/H
centerC22C2×#10587505
commutatorC22×C4×C17C22×C4
FrattiniC22×Q8C22×D17
FittingC17×#2434C2
socleC22×C17C2×#99
prime radicalC24×C17C22⋊C4
cosocleC22×Q8×C17C23
hypercenter#2434D17
hypocenterC17#13094
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C24 ◁ C23×Q8 ◁ #2434

Complements

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

NH ≅ G/Ntype
2. C2×#11762608C2semidirect
4. #14526911C2semidirect
206. C22×C17#356semidirect
294. C17#13094semidirect
423. C22#11762608semidirect

Maximal subgroups

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

subgroupindex
C22×#105875072
C2×#117626082
#145287082
#145269112
C17×#24342
#1309417