Finite Group Databasebeta

← #14533979

Subgroups — #14533979

Order 2176.

548 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10587575C211yesmaximal, characteristic, complemented
2C2×#10587567C211yesmaximal, characteristic
3#11762732C211yesmaximal, characteristic, complemented
4#11762731C211yesmaximal, characteristic, complemented
5#11762731C211yesmaximal, characteristic, complemented
6#11762709C211yesmaximal, characteristic
7#11762709C211yesmaximal, characteristic
8#11762684C211yesmaximal, characteristic, complemented
9#11762683C211yesmaximal, characteristic, complemented
10#11762681C211yesmaximal, characteristic, complemented
11#11762680C211yesmaximal, characteristic, complemented
12#11762678C211yesmaximal, characteristic, complemented
13#11762673C211yesmaximal, characteristic, complemented
14C4⋊C4×Dic17C211yesmaximal, characteristic, complemented
15C17×#396C211yesmaximal, characteristic, complemented
16C22×C22⋊D17C2211yescharacteristic
17C23×Dic17C2211yescharacteristic
18C22×C4×D17C2211yescharacteristic
19C2×C4⋊C4×C17C2211yescharacteristic
20C2×C4⋊C4×C17C2211yescharacteristic
21C2×C4⋊C4×C17C2211yescharacteristic
22C2×C22⋊C4×C17C2211yescharacteristic
23C2×C22⋊C4×C17C2211yescharacteristic
24C2×C22⋊C4×C17C2211yescharacteristic
25C2×C22⋊Dic17C2211yescharacteristic
26C2×C22⋊Dic17C2211yescharacteristic
27C2×#63227C2211yescharacteristic
28C2×#63227C2211yescharacteristic
29C2×#63227C2211yescharacteristic
30C2×#63227C2211yescharacteristic
31C2×#63227C2211yescharacteristic
32C2×C4⋊Dic17C2211yescharacteristic
33C2×C4⋊Dic17C2211yescharacteristic
34C2×C4⋊Dic17C2211yescharacteristic
35C2×#63225C2211yescharacteristic
36C2×#63225C2211yescharacteristic
37C2×#63225C2211yescharacteristic
38C2×#63225C2211yescharacteristic
39C2×#63225C2211yescharacteristic
40C2×#63225C2211yescharacteristic
41C2×#63225C2211yescharacteristic
42C2×C4×Dic17C2211yescharacteristic
43C2×C4×Dic17C2211yescharacteristic
44C2×C4×Dic17C2211yescharacteristic
45#10587575—24no—
46#10587567—24no—
47C17×#95C2211yescharacteristic
48#10587502C2211yescharacteristic
49#10587502C2211yescharacteristic
50#10587502C2211yescharacteristic
51#10587502C2211yescharacteristic
52#10587502C2211yescharacteristic
53C24×C17C2311yescharacteristic
54C23×D17C2311yescharacteristic
55C22×C4×C17C2311yescharacteristic
56C22×C4×C17C2311yescharacteristic
57C22×C4×C17C2311yescharacteristic
58C22×C4×C17C2311yescharacteristic
59C22×C4×C17C2311yescharacteristic
60C22×C4×C17C2311yescharacteristic
61C2×C22⋊D17—22no—
62C2×C22⋊D17—22no—
63C2×C22⋊D17Q814yes—
64C22×Dic17—22no—
65C22×Dic17—41no—
66C22×Dic17—41no—
67C22×Dic17C2311yescharacteristic
68C22×Dic17C2311yescharacteristic
69C22×Dic17C2311yescharacteristic
70C22×Dic17C2311yescharacteristic
71C22×Dic17C2311yescharacteristic
72C22×Dic17C2311yescharacteristic
73C22×Dic17C2311yescharacteristic
74C2×C4×D17—22no—
75C2×C4×D17—41no—
76C2×C4×D17—41no—
77C4⋊C4×C17—22no—
78C4⋊C4×C17—22no—
79C4⋊C4×C17—41no—
80C22⋊C4×C17—22no—
81C22⋊C4×C17—22no—
82C22⋊C4×C17—41no—
83C22⋊Dic17—22no—
84C22⋊Dic17—41no—
85#63227—22no—
86#63227—22no—
87#63227—22no—
88#63227—41no—
89#63227—41no—
90C4⋊Dic17—22no—
91C4⋊Dic17—22no—
92C4⋊Dic17—41no—
93#63225—22no—
94#63225—22no—
95#63225—22no—
96#63225—22no—
97#63225—22no—
98#63225—41no—
99#63225—41no—
100C4×Dic17—22no—
101C4×Dic17—22no—
102C4×Dic17D414yes—
103C23×C17—21no—
104C23×C17—21no—
105C23×C17—21no—
106C23×C17—21no—
107C23×C17—21no—
108C23×C17—21no—
109C23×C17C2×Q812yes—
110C23×C17C2411yescharacteristic
111C22×D17—21no—
112C22×D17—21no—
113C22×D17—21no—
114C22×D17—21no—
115C22×D17—21no—
116C22×D17—21no—
117C22×D17C2×Q812yes—
118C2×C4×C17—21no—
119C2×C4×C17—21no—
120C2×C4×C17—21no—
121C2×C4×C17—21no—
122C2×C4×C17—21no—
123C2×C4×C17—21no—
124C2×C4×C17—21no—
125C2×C4×C17—21no—
126C2×C4×C17—21no—
127C2×C4×C17—21no—
128C2×C4×C17—21no—
129C2×C4×C17—21no—
130C2×C4×C17—21no—
131C2×C4×C17—21no—
132C2×C4×C17—21no—
133C2×C4×C17—21no—
134C2×C4×C17C2×D412yes—
135C2×C4×C17#4112yes—
136C22⋊D17—28no—
137C2×Dic17—21no—
138C2×Dic17—21no—
139C2×Dic17—21no—
140C2×Dic17—21no—
141C2×Dic17—21no—
142C2×Dic17—21no—
143C2×Dic17—21no—
144C2×Dic17—21no—
145C2×Dic17—21no—
146C2×Dic17—21no—
147C2×Dic17—21no—
148C2×Dic17—21no—
149C2×Dic17—21no—
150C2×Dic17—21no—
151C2×Dic17—21no—
152C2×Dic17—21no—
153C2×Dic17—44no—
154C2×Dic17C2×D412yes—
155C2×Dic17C2×D412yes—
156C2×Dic17C2×Q812yes—
157C2×Dic17#4112yes—
158C2×Dic17#4112yes—
159C4×D17—44no—
160#2566—171nomaximal, Sylow
161C22×C17—22no—
162C22×C17—22no—
163C22×C17—22no—
164C22×C17—41no—
165C22×C17—41no—
166C22×C17—41no—
167C22×C17—41no—
168C22×C17C22×D411yescharacteristic
169C22×C17C22×Q811yescharacteristic
170C22×C17C2×#4111yescharacteristic
171C22×C17C2×#4111yescharacteristic
172C22×C17C2×#4111yescharacteristic
173C22×C17H(5,𝔽2)11yescharacteristic
174C22×C17H(5,𝔽2)11yescharacteristic
175C2×D17—22no—
176C2×D17—22no—
177C2×D17—22no—
178C2×D17—41no—
179C2×D17—41no—
180C2×D17—41no—
181C2×D17—41no—
182C4×C17—22no—
183C4×C17—22no—
184C4×C17—41no—
185C4×C17—41no—
186C4×C17—41no—
187C4×C17—41no—
188Dic17—22no—
189Dic17—22no—
190Dic17—41no—
191Dic17—41no—
192Dic17—41no—
193Dic17—41no—
194Dic17C22⋊Q814yes—
195C2×#125—171no—
196C2×C4×D4—171no—
197#396—171no—
198#396—171no—
199#396—171no—
200#396D1711yescharacteristic, complemented
201#390—171no—
202#390—171no—
203#385—171no—
204#385—171no—
205#385—171no—
206#385—171no—
207#384—171no—
208#382—171no—
209C4×C4⋊C4—171no—
210C2×C17—42no—
211C2×C17C2×C22⋊Q811yescharacteristic
212C2×C17#53211yescharacteristic
213C2×C17#53511yescharacteristic
214C2×C17#55311yescharacteristic
215C2×C17#55311yescharacteristic
216C2×C17#55411yescharacteristic
217C2×C17#55411yescharacteristic
218D17—42no—
219C22×D4—171no—
220C23×C4—171no—
221C23×C4—171no—
222#125—344no—
223C4×D4—344no—
224C2×C4⋊C4—171no—
225C2×C4⋊C4—171no—
226C2×C4⋊C4—171no—
227C2×C4⋊C4—171no—
228C2×C4⋊C4—171no—
229C2×C4⋊C4—171no—
230C2×C4⋊C4—171no—
231C2×C4⋊C4—171no—
232C2×C4⋊C4—171no—
233C2×C4⋊C4—171no—
234C2×C4⋊C4C2×D1711yescharacteristic
235C2×C4⋊C4C2×D1711yescharacteristic
236C2×C4⋊C4C2×D1711yescharacteristic
237C2×C22⋊C4—171no—
238C2×C22⋊C4—171no—
239C2×C22⋊C4—171no—
240C2×C22⋊C4—171no—
241C2×C22⋊C4—171no—
242C2×C22⋊C4—171no—
243C2×C22⋊C4—171no—
244C2×C22⋊C4C2×D1711yescharacteristic
245C2×C22⋊C4C2×D1711yescharacteristic
246C2×C22⋊C4C2×D1711yescharacteristic
247C2×C42—171no—
248C2×C42—171no—
249C2×C42—171no—
250#95—171no—
251#95—171no—
252#95—171no—
253#95—171no—
254#95—171no—
255#95C2×D1711yescharacteristic
256C17#256611yescharacteristic, complemented, Sylow
257C24—171no—
258C24C22×D1711yescharacteristic
259C2×D4—174no—
260C2×D4—342no—
261C2×D4—342no—
262C22×C4—171no—
263C22×C4—171no—
264C22×C4—171no—
265C22×C4—171no—
266C22×C4—171no—
267C22×C4—171no—
268C22×C4—171no—
269C22×C4—342no—
270C22×C4—342no—
271C22×C4—681no—
272C22×C4—681no—
273C22×C4—681no—
274C22×C4—681no—
275C22×C4C22×D1711yescharacteristic
276C22×C4C22×D1711yescharacteristic
277C22×C4C22×D1711yescharacteristic
278C22×C4C22×D1711yescharacteristic
279C22×C4C22×D1711yescharacteristic
280C22×C4C22×D1711yescharacteristic
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—41no—
284C4⋊C4—342no—
285C4⋊C4—342no—
286C4⋊C4—342no—
287C4⋊C4—342no—
288C4⋊C4—342no—
289C4⋊C4—342no—
290C4⋊C4—342no—
291C4⋊C4—681no—
292C4⋊C4—681no—
293C4⋊C4—681no—
294C22⋊C4—22no—
295C22⋊C4—22no—
296C22⋊C4—41no—
297C22⋊C4—342no—
298C22⋊C4—342no—
299C22⋊C4—342no—
300C22⋊C4—342no—
301C22⋊C4—681no—
302C22⋊C4—681no—
303C22⋊C4—681no—
304C42—174no—
305C42—342no—
306C42—342no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—172no—
314C23—341no—
315C23—341no—
316C23—341no—
317C23—341no—
318C23—341no—
319C23—341no—
320C23Q8×D1712yes—
321C23C23×D1711yescharacteristic
322D4—348no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—172no—
340C2×C4—172no—
341C2×C4—172no—
342C2×C4—172no—
343C2×C4—172no—
344C2×C4—341no—
345C2×C4—341no—
346C2×C4—341no—
347C2×C4—341no—
348C2×C4—341no—
349C2×C4—341no—
350C2×C4—341no—
351C2×C4—341no—
352C2×C4—341no—
353C2×C4—341no—
354C2×C4—341no—
355C2×C4—341no—
356C2×C4—341no—
357C2×C4—341no—
358C2×C4—341no—
359C2×C4—341no—
360C2×C4—684no—
361C2×C4—684no—
362C2×C4D4×D1712yes—
363C2×C4#6325412yes—
364C22—22no—
365C22—22no—
366C22—22no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—342no—
372C22—342no—
373C22—342no—
374C22—681no—
375C22—681no—
376C22—681no—
377C22—681no—
378C22C2×D4×D1711yescharacteristic
379C22C2×#6325411yescharacteristic
380C22#1058769111yescharacteristic
381C22C2×Q8×D1711yescharacteristic
382C22#41×D1711yescharacteristic
383C22#41×D1711yescharacteristic
384C22#1058769611yescharacteristic
385C4—22no—
386C4—22no—
387C4—41no—
388C4—41no—
389C4—41no—
390C4—41no—
391C4—174no—
392C4—342no—
393C4—342no—
394C4—681no—
395C4—681no—
396C4—681no—
397C4—681no—
398C2—42no—
399C2—682no—
400C2C22⋊Q8×D1711yescharacteristic
401C2#1176335411yescharacteristic
402C2#1176337511yescharacteristic
403C2#1176350211yescharacteristic
404C2#1176351011yescharacteristic
405C2#1176351311yescharacteristic
406C2#1176351711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#396C2
prime radicalC22×C22⋊D17C22
hypercenter#396D17
hypocenterC17#2566
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #396

Complements

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

NH ≅ G/Ntype
1. C2×#10587575C2semidirect
3. #11762732C2semidirect
4. #11762731C2semidirect
5. #11762731C2semidirect
8. #11762684C2semidirect
9. #11762683C2semidirect
10. #11762681C2semidirect
11. #11762680C2semidirect
12. #11762678C2semidirect
13. #11762673C2semidirect
14. C4⋊C4×Dic17C2semidirect
15. C17×#396C2semidirect
200. #396D17semidirect
256. C17#2566semidirect

Maximal subgroups

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

subgroupindex
C2×#105875752
C2×#105875672
#117627322
#117627312
#117627092
#117626842
#117626832
#117626812
#117626802
#117626782
#117626732
C4⋊C4×Dic172
C17×#3962
#256617