Finite Group Databasebeta

← #14537101

Subgroups — #14537101

Order 2176.

582 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10587584C211yesmaximal, characteristic, complemented
2C22×C4×Dic17C211yesmaximal, characteristic, complemented
3#11762701C211yesmaximal, characteristic, complemented
4#11762699C211yesmaximal, characteristic, complemented
5#11762698C211yesmaximal, characteristic, complemented
6#11762695C211yesmaximal, characteristic, complemented
7#11762692C211yesmaximal, characteristic, complemented
8#11762690C211yesmaximal, characteristic, complemented
9#11762688C211yesmaximal, characteristic
10#11762686C211yesmaximal, characteristic
11#11762686C211yesmaximal, characteristic
12#11762684C211yesmaximal, characteristic, complemented
13#11762681C211yesmaximal, characteristic, complemented
14#11762676C211yesmaximal, characteristic, complemented
15C17×#385C211yesmaximal, characteristic, complemented
16C23×C4×C17C2211yescharacteristic
17C22×C22⋊D17C2211yescharacteristic
18C23×Dic17C2211yescharacteristic
19C23×Dic17C2211yescharacteristic
20C2×C4⋊C4×C17C2211yescharacteristic
21C2×C4⋊C4×C17C2211yescharacteristic
22C2×C22⋊C4×C17C2211yescharacteristic
23C2×C22⋊C4×C17C2211yescharacteristic
24C2×C22⋊Dic17C2211yescharacteristic
25C2×#63227C2211yescharacteristic
26C2×#63227C2211yescharacteristic
27C2×#63227C2211yescharacteristic
28C2×#63227C2211yescharacteristic
29C2×#63227C2211yescharacteristic
30C2×#63227C2211yescharacteristic
31C2×C4⋊Dic17C2211yescharacteristic
32C2×C4⋊Dic17C2211yescharacteristic
33C2×#63225C2211yescharacteristic
34C2×#63225C2211yescharacteristic
35C2×C4×Dic17—24no—
36C2×C4×Dic17C2211yescharacteristic
37C2×C4×Dic17C2211yescharacteristic
38C2×C4×Dic17C2211yescharacteristic
39C2×C4×Dic17C2211yescharacteristic
40#10587584—24no—
41C17×#95C2211yescharacteristic
42C17×#95C2211yescharacteristic
43#10587502C2211yescharacteristic
44#10587502C2211yescharacteristic
45#10587502C2211yescharacteristic
46#10587502C2211yescharacteristic
47#10587502C2211yescharacteristic
48#10587502C2211yescharacteristic
49#10587502C2211yescharacteristic
50#10587502C2211yescharacteristic
51#10587502C2211yescharacteristic
52#10587502C2211yescharacteristic
53C24×C17C2311yescharacteristic
54C23×D17C2311yescharacteristic
55C22×C4×C17—22no—
56C22×C4×C17—22no—
57C22×C4×C17—22no—
58C22×C4×C17C2311yescharacteristic
59C22×C4×C17C2311yescharacteristic
60C22×C4×C17C2311yescharacteristic
61C22×C4×C17C2311yescharacteristic
62C22×C4×C17C2311yescharacteristic
63C22×C4×C17C2311yescharacteristic
64C2×C22⋊D17—22no—
65C2×C22⋊D17—22no—
66C2×C22⋊D17—22no—
67C22×Dic17—22no—
68C22×Dic17—22no—
69C22×Dic17—22no—
70C22×Dic17—22no—
71C22×Dic17—22no—
72C22×Dic17—22no—
73C22×Dic17C2311yescharacteristic
74C22×Dic17C2311yescharacteristic
75C22×Dic17C2311yescharacteristic
76C22×Dic17C2311yescharacteristic
77C22×Dic17C2311yescharacteristic
78C22×Dic17C2311yescharacteristic
79C22×Dic17C2311yescharacteristic
80C4⋊C4×C17—22no—
81C4⋊C4×C17—22no—
82C22⋊C4×C17—22no—
83C22⋊C4×C17—22no—
84C22⋊Dic17—22no—
85#63227—22no—
86#63227—22no—
87#63227—22no—
88#63227—22no—
89#63227—22no—
90#63227—22no—
91C4⋊Dic17—22no—
92C4⋊Dic17—22no—
93#63225—22no—
94#63225—22no—
95C4×Dic17—22no—
96C4×Dic17—22no—
97C4×Dic17—22no—
98C4×Dic17D414yescomplemented
99C23×C17—21no—
100C23×C17—21no—
101C23×C17—21no—
102C23×C17—21no—
103C23×C17#4112yes—
104C23×C17#4112yes—
105C23×C17#4112yes—
106C23×C17C2411yescharacteristic
107C22×D17—21no—
108C22×D17—21no—
109C22×D17—21no—
110C22×D17—21no—
111C22×D17—21no—
112C22×D17—21no—
113C22×D17—21no—
114C2×C4×C17—28no—
115C2×C4×C17—21no—
116C2×C4×C17—21no—
117C2×C4×C17—21no—
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×C17C2×D412yes—
132C2×C4×C17#4112yes—
133C22⋊D17—44no—
134C2×Dic17—28no—
135C2×Dic17—28no—
136C2×Dic17—21no—
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×Dic17C2×D412yes—
154C2×Dic17C2×D412yes—
155C2×Dic17#4112yes—
156C2×Dic17#4112yes—
157#2852—171nomaximal, Sylow
158C22×C17—22no—
159C22×C17—22no—
160C22×C17—22no—
161C22×C17—22no—
162C22×C17—22no—
163C22×C17—22no—
164C22×C17#12614yescomplemented
165C22×C17C22×D411yescharacteristic
166C22×C17C2×#4111yescharacteristic
167C22×C17C2×#4111yescharacteristic
168C22×C17C2×#4111yescharacteristic
169C22×C17C2×#4111yescharacteristic
170C22×C17C2×#4111yescharacteristic
171C22×C17C2×#4111yescharacteristic
172C2×D17—41no—
173C2×D17—41no—
174C2×D17—41no—
175C2×D17—41no—
176C2×D17—41no—
177C2×D17—41no—
178C2×D17—41no—
179C4×C17—22no—
180C4×C17—22no—
181C4×C17—41no—
182C4×C17—41no—
183C4×C17—41no—
184C4×C17—41no—
185Dic17—22no—
186Dic17—22no—
187Dic17—22no—
188Dic17—22no—
189Dic17—41no—
190Dic17—41no—
191Dic17—41no—
192C2×#126—171no—
193C22×C42—171no—
194#388—171no—
195#388—171no—
196#388—171no—
197#387—171no—
198#387—171no—
199#387—171no—
200#386—171no—
201#386—171no—
202#386—171no—
203#385—171no—
204#385—171no—
205#385D1711yescharacteristic, complemented
206#383—171no—
207C2×C17—24no—
208C2×C17#52511yescharacteristic
209C2×C17#52511yescharacteristic
210C2×C17#52511yescharacteristic
211C2×C17C2×#12611yescharacteristic
212C2×C17#52911yescharacteristic
213C2×C17#52911yescharacteristic
214C2×C17#52911yescharacteristic
215D17—81no—
216C22×D4—171no—
217C23×C4—171no—
218C23×C4—171no—
219C23×C4C2×D1711yescharacteristic
220#126—344no—
221C2×C4⋊C4—171no—
222C2×C4⋊C4—171no—
223C2×C4⋊C4—171no—
224C2×C4⋊C4—171no—
225C2×C4⋊C4C2×D1711yescharacteristic
226C2×C4⋊C4C2×D1711yescharacteristic
227C2×C22⋊C4—171no—
228C2×C22⋊C4—171no—
229C2×C22⋊C4—171no—
230C2×C22⋊C4—171no—
231C2×C22⋊C4—171no—
232C2×C22⋊C4—171no—
233C2×C22⋊C4—171no—
234C2×C22⋊C4C2×D1711yescharacteristic
235C2×C22⋊C4C2×D1711yescharacteristic
236C2×C42—171no—
237C2×C42—171no—
238C2×C42—171no—
239C2×C42—171no—
240C2×C42—344no—
241#95—171no—
242#95—171no—
243#95—171no—
244#95—171no—
245#95—171no—
246#95—171no—
247#95—171no—
248#95—171no—
249#95—171no—
250#95—171no—
251#95C2×D1711yescharacteristic
252#95C2×D1711yescharacteristic
253C17#285211yescharacteristic, complemented, Sylow
254C24—171no—
255C24C22×D1711yescharacteristic
256C2×D4—342no—
257C2×D4—342no—
258C2×D4—342no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
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—342no—
272C22×C4—342no—
273C22×C4—342no—
274C22×C4—342no—
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—342no—
284C4⋊C4—342no—
285C4⋊C4—342no—
286C4⋊C4—342no—
287C22⋊C4—22no—
288C22⋊C4—22no—
289C22⋊C4—342no—
290C22⋊C4—342no—
291C22⋊C4—342no—
292C22⋊C4—342no—
293C22⋊C4—342no—
294C22⋊C4—342no—
295C22⋊C4—342no—
296C42—174no—
297C42—342no—
298C42—342no—
299C42—342no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—341no—
305C23—341no—
306C23—341no—
307C23—341no—
308C23—341no—
309C23—341no—
310C23—341no—
311C23#6325212yes—
312C23#6325412yes—
313C23#6325612yes—
314C23C23×D1711yescharacteristic
315D4—684no—
316C2×C4—28no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
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—172no—
334C2×C4—172no—
335C2×C4—172no—
336C2×C4—172no—
337C2×C4—348no—
338C2×C4—348no—
339C2×C4—341no—
340C2×C4—341no—
341C2×C4—341no—
342C2×C4—341no—
343C2×C4—341no—
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×C4D4×D1712yes—
357C2×C4#6325412yes—
358C22—22no—
359C22—22no—
360C22—22no—
361C22—22no—
362C22—22no—
363C22—22no—
364C22—681no—
365C22—681no—
366C22—681no—
367C22—681no—
368C22—681no—
369C22—681no—
370C22—681no—
371C22#1058758414yescomplemented
372C22C2×#6325211yescharacteristic
373C22C2×D4×D1711yescharacteristic
374C22C2×#6325411yescharacteristic
375C22C2×#6325411yescharacteristic
376C22C2×#6325611yescharacteristic
377C22#41×D1711yescharacteristic
378C22#41×D1711yescharacteristic
379C4—22no—
380C4—22no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—41no—
385C4—342no—
386C4—342no—
387C4—342no—
388C4—342no—
389C4—681no—
390C4—681no—
391C4—681no—
392C2—24no—
393C2—1361no—
394C2#1176331511yescharacteristic
395C2#1176331811yescharacteristic
396C2#1176332211yescharacteristic
397C2C2×#1058758411yescharacteristic
398C2#1176333611yescharacteristic
399C2#1176333911yescharacteristic
400C2#1176334511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#385C2
prime radicalC22×C22⋊D17C22
hypercenter#385D17
hypocenterC17#2852
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
1. C2×#10587584C2semidirect
2. C22×C4×Dic17C2semidirect
3. #11762701C2semidirect
4. #11762699C2semidirect
5. #11762698C2semidirect
6. #11762695C2semidirect
7. #11762692C2semidirect
8. #11762690C2semidirect
12. #11762684C2semidirect
13. #11762681C2semidirect
14. #11762676C2semidirect
15. C17×#385C2semidirect
98. C4×Dic17D4semidirect
164. C22×C17#126semidirect
205. #385D17semidirect
253. C17#2852semidirect
371. C22#10587584semidirect

Maximal subgroups

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

subgroupindex
C2×#105875842
C22×C4×Dic172
#117627012
#117626992
#117626982
#117626952
#117626922
#117626902
#117626882
#117626862
#117626842
#117626812
#117626762
C17×#3852
#285217