Finite Group Databasebeta

← #14535142

Subgroups — #14535142

Order 2176.

618 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10587572C211yesmaximal, characteristic, complemented
2C2×#10587564C211yesmaximal, characteristic, complemented
3C2×#10587564C211yesmaximal, characteristic, complemented
4C2×#10587625C211yesmaximal, characteristic
5#11762743C211yesmaximal, characteristic, complemented
6#11762731C211yesmaximal, characteristic, complemented
7#11762724C211yesmaximal, characteristic, complemented
8#11762724C211yesmaximal, characteristic, complemented
9#11762722C211yesmaximal, characteristic, complemented
10#11762709C211yesmaximal, characteristic, complemented
11#11762705C211yesmaximal, characteristic, complemented
12#11762703C211yesmaximal, characteristic, complemented
13#11762689C211yesmaximal, characteristic, complemented
14#11762684C211yesmaximal, characteristic, complemented
15C17×#394C211yesmaximal, characteristic, complemented
16C22×D4×C17C2211yescharacteristic
17C22×C22⋊D17C2211yescharacteristic
18C22×C22⋊D17C2211yescharacteristic
19C23×Dic17C2211yescharacteristic
20C23×Dic17C2211yescharacteristic
21C22×Dic34C2211yescharacteristic
22C2×C4⋊C4×C17C2211yescharacteristic
23C2×C22⋊C4×C17C2211yescharacteristic
24C2×C22⋊C4×C17C2211yescharacteristic
25C2×C22⋊C4×C17C2211yescharacteristic
26C2×C22⋊C4×C17C2211yescharacteristic
27C2×C22⋊Dic17C2211yescharacteristic
28C2×C22⋊Dic17C2211yescharacteristic
29C2×C22⋊Dic17C2211yescharacteristic
30C2×C22⋊Dic17C2211yescharacteristic
31#10587625—24no—
32C2×#63227C2211yescharacteristic
33C2×#63227C2211yescharacteristic
34C2×#63227C2211yescharacteristic
35C2×#63227C2211yescharacteristic
36C2×#63227C2211yescharacteristic
37C2×C4⋊Dic17C2211yescharacteristic
38C2×C4⋊Dic17C2211yescharacteristic
39C2×C4⋊Dic17C2211yescharacteristic
40C2×#63225C2211yescharacteristic
41C2×#63225C2211yescharacteristic
42C2×#63225C2211yescharacteristic
43C2×#63225C2211yescharacteristic
44C2×#63225C2211yescharacteristic
45C2×C4×Dic17C2211yescharacteristic
46#10587572—24no—
47#10587564—24no—
48#10587564—24no—
49C17×#95C2211yescharacteristic
50#10587502C2211yescharacteristic
51#10587502C2211yescharacteristic
52#10587502C2211yescharacteristic
53#10587502C2211yescharacteristic
54#10587502C2211yescharacteristic
55C24×C17C2311yescharacteristic
56C24×C17C2311yescharacteristic
57C23×D17C2311yescharacteristic
58C2×D4×C17—22no—
59C2×D4×C17—22no—
60C2×D4×C17—41no—
61C22×C4×C17C2311yescharacteristic
62C22×C4×C17C2311yescharacteristic
63C22×C4×C17C2311yescharacteristic
64C22×C4×C17C2311yescharacteristic
65C22×C4×C17C2311yescharacteristic
66C2×C22⋊D17—22no—
67C2×C22⋊D17—22no—
68C2×C22⋊D17—22no—
69C2×C22⋊D17—22no—
70C2×C22⋊D17—41no—
71C2×C22⋊D17—41no—
72C22×Dic17—22no—
73C22×Dic17—22no—
74C22×Dic17—22no—
75C22×Dic17—22no—
76C22×Dic17—41no—
77C22×Dic17—41no—
78C22×Dic17C2311yescharacteristic
79C22×Dic17C2311yescharacteristic
80C22×Dic17C2311yescharacteristic
81C22×Dic17C2311yescharacteristic
82C22×Dic17C2311yescharacteristic
83C22×Dic17C2311yescharacteristic
84C22×Dic17C2311yescharacteristic
85C2×Dic34—22no—
86C2×Dic34—22no—
87C2×Dic34—41no—
88C4⋊C4×C17—41no—
89C22⋊C4×C17—22no—
90C22⋊C4×C17—22no—
91C22⋊C4×C17—22no—
92C22⋊C4×C17—22no—
93C22⋊Dic17—22no—
94C22⋊Dic17—41no—
95C22⋊Dic17—41no—
96C22⋊Dic17D414yes—
97#63227—22no—
98#63227—22no—
99#63227—22no—
100#63227—41no—
101#63227—41no—
102C4⋊Dic17—22no—
103C4⋊Dic17—22no—
104C4⋊Dic17D414yescomplemented
105#63225—22no—
106#63225—22no—
107#63225—22no—
108#63225—22no—
109#63225—22no—
110C4×Dic17—41no—
111C23×C17—21no—
112C23×C17—21no—
113C23×C17—21no—
114C23×C17—21no—
115C23×C17—21no—
116C23×C17—21no—
117C23×C17—21no—
118C23×C17—21no—
119C23×C17—21no—
120C23×C17—21no—
121C23×C17—21no—
122C23×C17—21no—
123C23×C17C2×D412yes—
124C23×C17#4112yes—
125C23×C17C2411yescharacteristic
126C22×D17—21no—
127C22×D17—21no—
128C22×D17—21no—
129C22×D17—21no—
130C22×D17—21no—
131C22×D17—21no—
132C22×D17—21no—
133D4×C17—44no—
134C2×C4×C17—21no—
135C2×C4×C17—21no—
136C2×C4×C17—21no—
137C2×C4×C17—21no—
138C2×C4×C17—21no—
139C2×C4×C17—21no—
140C2×C4×C17—21no—
141C2×C4×C17—21no—
142C2×C4×C17—21no—
143C2×C4×C17—21no—
144C2×C4×C17—21no—
145C2×C4×C17—21no—
146C2×C4×C17—21no—
147C2×C4×C17—21no—
148C2×C4×C17C2×D412yes—
149C22⋊D17—44no—
150C22⋊D17—44no—
151C2×Dic17—21no—
152C2×Dic17—21no—
153C2×Dic17—21no—
154C2×Dic17—21no—
155C2×Dic17—21no—
156C2×Dic17—21no—
157C2×Dic17—21no—
158C2×Dic17—21no—
159C2×Dic17—21no—
160C2×Dic17—21no—
161C2×Dic17—21no—
162C2×Dic17—21no—
163C2×Dic17—21no—
164C2×Dic17—21no—
165C2×Dic17—21no—
166C2×Dic17—21no—
167C2×Dic17—21no—
168C2×Dic17—44no—
169C2×Dic17—44no—
170C2×Dic17C2×D412yes—
171C2×Dic17C2×D412yes—
172C2×Dic17C2×D412yes—
173C2×Dic17C2×D412yes—
174Dic34—44no—
175#2681—171nomaximal, Sylow
176C22×C17—22no—
177C22×C17—22no—
178C22×C17—22no—
179C22×C17—22no—
180C22×C17—22no—
181C22×C17—22no—
182C22×C17—41no—
183C22×C17—41no—
184C22×C17—41no—
185C22×C17—41no—
186C22×C17—41no—
187C22×C17—41no—
188C22×C17—41no—
189C22×C17—41no—
190C22×C17C22×D411yescharacteristic
191C22×C17C22×D411yescharacteristic
192C22×C17C2×#4111yescharacteristic
193C22×C17H(5,𝔽2)11yescharacteristic
194C22×C17H(5,𝔽2)11yescharacteristic
195C22×C172−1+411yescharacteristic
196C22×C172−1+411yescharacteristic
197C2×D17—41no—
198C2×D17—41no—
199C2×D17—41no—
200C2×D17—41no—
201C2×D17—41no—
202C2×D17—41no—
203C2×D17—41no—
204C4×C17—22no—
205C4×C17—41no—
206C4×C17—41no—
207C4×C17—41no—
208C4×C17—41no—
209Dic17—22no—
210Dic17—22no—
211Dic17—22no—
212Dic17—22no—
213Dic17—41no—
214Dic17—41no—
215Dic17—41no—
216C2×#123—171no—
217C2×C22⋊Q8—171no—
218C2×C22⋊Q8—171no—
219C2×#121—171no—
220#398—171no—
221#396—171no—
222#394—171no—
223#394—171no—
224#394D1711yescharacteristic, complemented
225#393—171no—
226#390—171no—
227#389—171no—
228#388—171no—
229#386—171no—
230#385—171no—
231C2×C17—42no—
232C2×C17—42no—
233C2×C17#53611yescharacteristic
234C2×C17#53711yescharacteristic
235C2×C17#53911yescharacteristic
236C2×C17D4211yescharacteristic
237C2×C17#54611yescharacteristic
238C2×C17#54811yescharacteristic
239C2×C17#56211yescharacteristic
240D17—81no—
241C22×Q8—171no—
242C22×D4—171no—
243C22×D4—171no—
244C22×D4C2×D1711yescharacteristic
245C23×C4—171no—
246C23×C4—171no—
247#123—344no—
248C22⋊Q8—344no—
249C22⋊Q8—344no—
250#121—344no—
251C2×C4⋊C4—171no—
252C2×C4⋊C4—171no—
253C2×C4⋊C4—171no—
254C2×C4⋊C4—171no—
255C2×C4⋊C4—171no—
256C2×C4⋊C4—171no—
257C2×C4⋊C4—171no—
258C2×C4⋊C4—171no—
259C2×C4⋊C4C2×D1711yescharacteristic
260C2×C22⋊C4—171no—
261C2×C22⋊C4—171no—
262C2×C22⋊C4—171no—
263C2×C22⋊C4—171no—
264C2×C22⋊C4—171no—
265C2×C22⋊C4—171no—
266C2×C22⋊C4—171no—
267C2×C22⋊C4—171no—
268C2×C22⋊C4—171no—
269C2×C22⋊C4C2×D1711yescharacteristic
270C2×C22⋊C4C2×D1711yescharacteristic
271C2×C22⋊C4C2×D1711yescharacteristic
272C2×C22⋊C4C2×D1711yescharacteristic
273C2×C42—171no—
274#95—171no—
275#95—171no—
276#95—171no—
277#95—171no—
278#95—171no—
279#95C2×D1711yescharacteristic
280C17#268111yescharacteristic, complemented, Sylow
281C24—171no—
282C24C22×D1711yescharacteristic
283C24C22×D1711yescharacteristic
284C2×Q8—342no—
285C2×Q8—342no—
286C2×Q8—681no—
287C2×D4—22no—
288C2×D4—22no—
289C2×D4—41no—
290C2×D4—342no—
291C2×D4—342no—
292C2×D4—342no—
293C2×D4—342no—
294C2×D4—681no—
295C2×D4—681no—
296C22×C4—171no—
297C22×C4—171no—
298C22×C4—171no—
299C22×C4—171no—
300C22×C4—171no—
301C22×C4—171no—
302C22×C4—171no—
303C22×C4—342no—
304C22×C4—342no—
305C22×C4—342no—
306C22×C4—342no—
307C22×C4—681no—
308C22×C4—681no—
309C22×C4C22×D1711yescharacteristic
310C22×C4C22×D1711yescharacteristic
311C22×C4C22×D1711yescharacteristic
312C22×C4C22×D1711yescharacteristic
313C22×C4C22×D1711yescharacteristic
314C4⋊C4—41no—
315C4⋊C4—174no—
316C4⋊C4—342no—
317C4⋊C4—342no—
318C4⋊C4—342no—
319C4⋊C4—342no—
320C4⋊C4—342no—
321C4⋊C4—342no—
322C4⋊C4—342no—
323C22⋊C4—22no—
324C22⋊C4—22no—
325C22⋊C4—22no—
326C22⋊C4—22no—
327C22⋊C4—174no—
328C22⋊C4—342no—
329C22⋊C4—342no—
330C22⋊C4—342no—
331C22⋊C4—342no—
332C22⋊C4—681no—
333C22⋊C4—681no—
334C22⋊C4—681no—
335C22⋊C4—681no—
336C42—681no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—341no—
350C23—341no—
351C23—341no—
352C23—341no—
353C23—341no—
354C23—341no—
355C23—341no—
356C23D4×D1712yes—
357C23#6325412yes—
358C23C23×D1711yescharacteristic
359Q8—684no—
360D4—44no—
361D4—684no—
362D4—684no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—172no—
378C2×C4—172no—
379C2×C4—172no—
380C2×C4—172no—
381C2×C4—341no—
382C2×C4—341no—
383C2×C4—341no—
384C2×C4—341no—
385C2×C4—341no—
386C2×C4—341no—
387C2×C4—341no—
388C2×C4—341no—
389C2×C4—341no—
390C2×C4—341no—
391C2×C4—341no—
392C2×C4—341no—
393C2×C4—341no—
394C2×C4—341no—
395C2×C4—341no—
396C2×C4—341no—
397C2×C4—341no—
398C2×C4—684no—
399C2×C4—684no—
400C2×C4D4×D1712yes—
401C22—22no—
402C22—22no—
403C22—22no—
404C22—22no—
405C22—22no—
406C22—22no—
407C22—41no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—681no—
416C22—681no—
417C22—681no—
418C22—681no—
419C22—681no—
420C22—681no—
421C22—681no—
422C22C2×D4×D1711yescharacteristic
423C22C2×D4×D1711yescharacteristic
424C22C2×#6325411yescharacteristic
425C22#1058769111yescharacteristic
426C22#1058769611yescharacteristic
427C22#1058769711yescharacteristic
428C22#1058769711yescharacteristic
429C4—22no—
430C4—41no—
431C4—41no—
432C4—41no—
433C4—41no—
434C4—342no—
435C4—342no—
436C4—342no—
437C4—342no—
438C4—681no—
439C4—681no—
440C4—681no—
441C2—42no—
442C2—42no—
443C2—1361no—
444C2#1176338111yescharacteristic
445C2#1176338611yescharacteristic
446C2#1176340311yescharacteristic
447C2#1176344311yescharacteristic
448C2#1176344611yescharacteristic
449C2#1176346511yescharacteristic
450C2#1176355311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#394C2
prime radicalC2×#10587625C2
hypercenter#394D17
hypocenterC17#2681
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#10587572C2semidirect
2. C2×#10587564C2semidirect
3. C2×#10587564C2semidirect
5. #11762743C2semidirect
6. #11762731C2semidirect
7. #11762724C2semidirect
8. #11762724C2semidirect
9. #11762722C2semidirect
10. #11762709C2semidirect
11. #11762705C2semidirect
12. #11762703C2semidirect
13. #11762689C2semidirect
14. #11762684C2semidirect
15. C17×#394C2semidirect
104. C4⋊Dic17D4semidirect
224. #394D17semidirect
280. C17#2681semidirect

Maximal subgroups

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

subgroupindex
C2×#105875722
C2×#105875642
C2×#105876252
#117627432
#117627312
#117627242
#117627222
#117627092
#117627052
#117627032
#117626892
#117626842
C17×#3942
#268117