Finite Group Databasebeta

← #14535132

Subgroups — #14535132

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×#10587615C211yesmaximal, characteristic, complemented
2C2×#10587632C211yesmaximal, characteristic, complemented
3C2×#10587570C211yesmaximal, characteristic
4#11762740C211yesmaximal, characteristic, complemented
5#11762731C211yesmaximal, characteristic, complemented
6#11762726C211yesmaximal, characteristic, complemented
7#11762724C211yesmaximal, characteristic, complemented
8#11762724C211yesmaximal, characteristic, complemented
9#11762720C211yesmaximal, characteristic, complemented
10#11762710C211yesmaximal, characteristic, complemented
11#11762704C211yesmaximal, characteristic, complemented
12#11762702C211yesmaximal, characteristic, complemented
13#11762688C211yesmaximal, characteristic, complemented
14#11762680C211yesmaximal, characteristic, complemented
15C2×C17×C22⋊Q8C211yesmaximal, characteristic, complemented
16C22×Q8×C17C2211yescharacteristic
17C23×C4×C17C2211yescharacteristic
18C22×C22⋊D17C2211yescharacteristic
19C22×C22⋊D17C2211yescharacteristic
20C22×D68C2211yescharacteristic
21C22×C4×D17C2211yescharacteristic
22C17×C22⋊Q8—24no—
23C2×C4⋊C4×C17C2211yescharacteristic
24C2×C4⋊C4×C17C2211yescharacteristic
25C2×C4⋊C4×C17C2211yescharacteristic
26C2×C22⋊C4×C17C2211yescharacteristic
27C2×C22⋊C4×C17C2211yescharacteristic
28C2×C22⋊Dic17C2211yescharacteristic
29C2×C22⋊Dic17C2211yescharacteristic
30#10587632—24no—
31#10587615—24no—
32C2×#63227C2211yescharacteristic
33C2×#63227C2211yescharacteristic
34C2×#63227C2211yescharacteristic
35C2×#63227C2211yescharacteristic
36C2×#63227C2211yescharacteristic
37C2×#63227C2211yescharacteristic
38C2×#63227C2211yescharacteristic
39C2×#63227C2211yescharacteristic
40C2×#63227C2211yescharacteristic
41C2×C4⋊Dic17C2211yescharacteristic
42C2×#63225C2211yescharacteristic
43C2×#63225C2211yescharacteristic
44C2×#63225C2211yescharacteristic
45C2×#63225C2211yescharacteristic
46C2×#63225C2211yescharacteristic
47C2×C4×Dic17C2211yescharacteristic
48#10587570—24no—
49#10587502C2211yescharacteristic
50#10587502C2211yescharacteristic
51#10587502C2211yescharacteristic
52#10587502C2211yescharacteristic
53#10587502C2211yescharacteristic
54#10587502C2211yescharacteristic
55C24×C17C2311yescharacteristic
56C23×D17C2311yescharacteristic
57C23×D17C2311yescharacteristic
58C2×Q8×C17—22no—
59C2×Q8×C17—22no—
60C2×Q8×C17—41no—
61C22×C4×C17—22no—
62C22×C4×C17—22no—
63C22×C4×C17—41no—
64C22×C4×C17C2311yescharacteristic
65C22×C4×C17C2311yescharacteristic
66C22×C4×C17C2311yescharacteristic
67C22×C4×C17C2311yescharacteristic
68C22×C4×C17C2311yescharacteristic
69C22×C4×C17C2311yescharacteristic
70C2×C22⋊D17—22no—
71C2×C22⋊D17—22no—
72C2×C22⋊D17—22no—
73C2×C22⋊D17—22no—
74C2×C22⋊D17—41no—
75C2×C22⋊D17—41no—
76C22×Dic17C2311yescharacteristic
77C22×Dic17C2311yescharacteristic
78C22×Dic17C2311yescharacteristic
79C22×Dic17C2311yescharacteristic
80C22×Dic17C2311yescharacteristic
81C22×Dic17C2311yescharacteristic
82C2×D68—22no—
83C2×D68—22no—
84C2×D68—41no—
85C2×C4×D17—22no—
86C2×C4×D17—22no—
87C2×C4×D17—41no—
88C4⋊C4×C17—22no—
89C4⋊C4×C17—22no—
90C4⋊C4×C17—22no—
91C22⋊C4×C17—22no—
92C22⋊C4×C17D414yes—
93C22⋊Dic17—22no—
94C22⋊Dic17—41no—
95#63227—22no—
96#63227—22no—
97#63227—22no—
98#63227—22no—
99#63227—22no—
100#63227—22no—
101#63227—41no—
102#63227—41no—
103#63227—41no—
104C4⋊Dic17—22no—
105#63225—22no—
106#63225—22no—
107#63225—22no—
108#63225—41no—
109#63225D414yescomplemented
110C4×Dic17—41no—
111C23×C17—21no—
112C23×C17—21no—
113C23×C17—21no—
114C23×C17—21no—
115C23×C17—21no—
116C23×C17—21no—
117C23×C17C2×D412yes—
118C23×C17C2411yescharacteristic
119C22×D17—21no—
120C22×D17—21no—
121C22×D17—21no—
122C22×D17—21no—
123C22×D17—21no—
124C22×D17—21no—
125C22×D17—21no—
126C22×D17—21no—
127C22×D17—21no—
128C22×D17—21no—
129C22×D17—21no—
130C22×D17—21no—
131C22×D17—21no—
132C22×D17#4112yes—
133Q8×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×C17—21no—
149C2×C4×C17—44no—
150C2×C4×C17C2×D412yes—
151C2×C4×C17C2×D412yes—
152C2×C4×C17C2×D412yes—
153C22⋊D17—44no—
154C22⋊D17—44no—
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—21no—
169C2×Dic17—21no—
170C2×Dic17—21no—
171C2×Dic17C2×D412yes—
172C2×Dic17C2×D412yes—
173D68—44no—
174C4×D17—44no—
175#2681—171nomaximal, Sylow
176C22×C17—22no—
177C22×C17—22no—
178C22×C17—22no—
179C22×C17—41no—
180C22×C17—41no—
181C22×C17—41no—
182C22×C17—41no—
183C22×C17C22×D411yescharacteristic
184C22×C17C22×D411yescharacteristic
185C22×C17C2×#4111yescharacteristic
186C22×C17H(5,𝔽2)11yescharacteristic
187C22×C17H(5,𝔽2)11yescharacteristic
188C22×C172−1+411yescharacteristic
189C22×C172−1+411yescharacteristic
190C2×D17—22no—
191C2×D17—22no—
192C2×D17—22no—
193C2×D17—41no—
194C2×D17—41no—
195C2×D17—41no—
196C2×D17—41no—
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—22no—
206C4×C17—22no—
207C4×C17—41no—
208C4×C17—41no—
209C4×C17—41no—
210Dic17—22no—
211Dic17—22no—
212Dic17—41no—
213Dic17—41no—
214Dic17—41no—
215Dic17—41no—
216C2×#123—171no—
217C2×C22⋊Q8—171no—
218C2×C22⋊Q8D1711yescharacteristic, complemented
219C2×#121—171no—
220#398—171no—
221#396—171no—
222#394—171no—
223#394—171no—
224#394—171no—
225#393—171no—
226#390—171no—
227#389—171no—
228#388—171no—
229#386—171no—
230#385—171no—
231C2×C17—42no—
232C2×C17#53611yescharacteristic
233C2×C17#53711yescharacteristic
234C2×C17#53911yescharacteristic
235C2×C17D4211yescharacteristic
236C2×C17#54611yescharacteristic
237C2×C17#54811yescharacteristic
238C2×C17#56211yescharacteristic
239D17—42no—
240D17—81no—
241C22×Q8C2×D1711yescharacteristic
242C22×D4—171no—
243C22×D4—171no—
244C22×D4—171no—
245C23×C4—171no—
246C23×C4C2×D1711yescharacteristic
247#123—344no—
248C22⋊Q8—24no—
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⋊C4C2×D1711yescharacteristic
258C2×C4⋊C4C2×D1711yescharacteristic
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⋊C4—171no—
270C2×C22⋊C4—171no—
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#95—171no—
280C17#268111yescharacteristic, complemented, Sylow
281C24—171no—
282C24—171no—
283C24C22×D1711yescharacteristic
284C2×Q8—22no—
285C2×Q8—22no—
286C2×Q8—41no—
287C2×D4—342no—
288C2×D4—342no—
289C2×D4—342no—
290C2×D4—342no—
291C2×D4—342no—
292C2×D4—342no—
293C2×D4—681no—
294C2×D4—681no—
295C2×D4—681no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—41no—
299C22×C4—171no—
300C22×C4—171no—
301C22×C4—171no—
302C22×C4—171no—
303C22×C4—171no—
304C22×C4—171no—
305C22×C4—342no—
306C22×C4—342no—
307C22×C4—681no—
308C22×C4C22×D1711yescharacteristic
309C22×C4C22×D1711yescharacteristic
310C22×C4C22×D1711yescharacteristic
311C22×C4C22×D1711yescharacteristic
312C22×C4C22×D1711yescharacteristic
313C22×C4C22×D1711yescharacteristic
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—174no—
318C4⋊C4—342no—
319C4⋊C4—342no—
320C4⋊C4—342no—
321C4⋊C4—342no—
322C4⋊C4—681no—
323C22⋊C4—22no—
324C22⋊C4—342no—
325C22⋊C4—342no—
326C22⋊C4—342no—
327C22⋊C4—342no—
328C22⋊C4—342no—
329C22⋊C4—342no—
330C22⋊C4—342no—
331C22⋊C4—681no—
332C22⋊C4—681no—
333C22⋊C4—681no—
334C22⋊C4—681no—
335C22⋊C4C22⋊D1714yes—
336C42—681no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—172no—
344C23—341no—
345C23—341no—
346C23—341no—
347C23—341no—
348C23—341no—
349C23—341no—
350C23—341no—
351C23—341no—
352C23—341no—
353C23—341no—
354C23—341no—
355C23—341no—
356C23—341no—
357C23C2×C22⋊D1712yes—
358C23C23×D1711yescharacteristic
359Q8—44no—
360D4—684no—
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—21no—
378C2×C4—44no—
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—684no—
398C2×C4D4×D1712yes—
399C2×C4C2×C22⋊D1712yes—
400C2×C4C2×C22⋊D1712yes—
401C22—22no—
402C22—22no—
403C22—22no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—342no—
409C22—342no—
410C22—342no—
411C22—681no—
412C22—681no—
413C22—681no—
414C22—681no—
415C22—681no—
416C22—681no—
417C22—681no—
418C22—681no—
419C22—681no—
420C22—681no—
421C22—681no—
422C22C2×D4×D1711yescharacteristic
423C22#1058769111yescharacteristic
424C22#1058769411yescharacteristic
425C22#1058769411yescharacteristic
426C22#41×D1711yescharacteristic
427C22#1058769611yescharacteristic
428C22C22×C22⋊D1711yescharacteristic
429C4—22no—
430C4—22no—
431C4—22no—
432C4—41no—
433C4—41no—
434C4—41no—
435C4—342no—
436C4—342no—
437C4—681no—
438C4—681no—
439C4—681no—
440C4—681no—
441C2—42no—
442C2—682no—
443C2—1361no—
444C2#1176338311yescharacteristic
445C2#1176338911yescharacteristic
446C2#1176340511yescharacteristic
447C2D4×C22⋊D1711yescharacteristic
448C2#1176344711yescharacteristic
449C2#1176346911yescharacteristic
450C2#1176355511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC2×C17×C22⋊Q8C2
prime radicalC2×#10587570C2
hypercenterC2×C22⋊Q8D17
hypocenterC17#2681
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#10587615C2semidirect
2. C2×#10587632C2semidirect
4. #11762740C2semidirect
5. #11762731C2semidirect
6. #11762726C2semidirect
7. #11762724C2semidirect
8. #11762724C2semidirect
9. #11762720C2semidirect
10. #11762710C2semidirect
11. #11762704C2semidirect
12. #11762702C2semidirect
13. #11762688C2semidirect
14. #11762680C2semidirect
15. C2×C17×C22⋊Q8C2semidirect
109. #63225D4semidirect
218. C2×C22⋊Q8D17semidirect
280. C17#2681semidirect

Maximal subgroups

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

subgroupindex
C2×#105876152
C2×#105876322
C2×#105875702
#117627402
#117627312
#117627262
#117627242
#117627202
#117627102
#117627042
#117627022
#117626882
#117626802
C2×C17×C22⋊Q82
#268117