Finite Group Databasebeta

← #14532258

Subgroups — #14532258

Order 2176.

560 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#10587569C211yesmaximal, characteristic
2C2×C4×C22⋊D17C211yesmaximal, characteristic
3#11762751C211yesmaximal, characteristic, complemented
4#11762743C211yesmaximal, characteristic, complemented
5#11762737C211yesmaximal, characteristic, complemented
6#11762734C211yesmaximal, characteristic, complemented
7#11762696C211yesmaximal, characteristic, complemented
8#11762691C211yesmaximal, characteristic, complemented
9#11762685C211yesmaximal, characteristic
10#11762683C211yesmaximal, characteristic, complemented
11#11762682C211yesmaximal, characteristic, complemented
12#11762671C211yesmaximal, characteristic, complemented
13#11762668C211yesmaximal, characteristic, complemented
14#95×D17C211yesmaximal, characteristic, complemented
15C17×#381C211yesmaximal, characteristic, complemented
16C23×C4×C17C2211yescharacteristic
17C22×C22⋊D17C2211yescharacteristic
18C22×C4×D17C2211yescharacteristic
19C22×C4×D17C2211yescharacteristic
20C22×C4×D17C2211yescharacteristic
21C2×C22⋊C4×C17C2211yescharacteristic
22C2×C22⋊C4×C17C2211yescharacteristic
23C2×C22⋊Dic17C2211yescharacteristic
24C2×C22⋊Dic17C2211yescharacteristic
25C2×C22⋊Dic17C2211yescharacteristic
26C4×C22⋊D17—24no—
27C2×#63227C2211yescharacteristic
28C2×#63227C2211yescharacteristic
29C2×#63227C2211yescharacteristic
30C2×C4⋊Dic17C2211yescharacteristic
31C2×C4⋊Dic17C2211yescharacteristic
32C2×#63225C2211yescharacteristic
33C2×#63225C2211yescharacteristic
34C2×#63225C2211yescharacteristic
35C2×#63225C2211yescharacteristic
36C2×#63225C2211yescharacteristic
37C2×C4×Dic17C2211yescharacteristic
38#10587569—24no—
39C17×#95C2211yescharacteristic
40C17×#95C2211yescharacteristic
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—41no—
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⋊D17D414yes—
67C22×Dic17C2311yescharacteristic
68C22×Dic17C2311yescharacteristic
69C22×Dic17C2311yescharacteristic
70C22×Dic17C2311yescharacteristic
71C22×Dic17C2311yescharacteristic
72C22×Dic17C2311yescharacteristic
73C22×Dic17C2311yescharacteristic
74C2×C4×D17—22no—
75C2×C4×D17—22no—
76C2×C4×D17—22no—
77C2×C4×D17—22no—
78C2×C4×D17—22no—
79C2×C4×D17—22no—
80C2×C4×D17—22no—
81C2×C4×D17—41no—
82C2×C4×D17—41no—
83C22⋊C4×C17—22no—
84C22⋊C4×C17—41no—
85C22⋊Dic17—22no—
86C22⋊Dic17—22no—
87C22⋊Dic17—22no—
88#63227—22no—
89#63227—22no—
90#63227—22no—
91C4⋊Dic17—22no—
92C4⋊Dic17—41no—
93#63225—22no—
94#63225—22no—
95#63225—22no—
96#63225—22no—
97#63225—41no—
98C4×Dic17—22no—
99C23×C17—21no—
100C23×C17—21no—
101C23×C17—21no—
102C23×C17—21no—
103C23×C17—21no—
104C23×C17—21no—
105C23×C17C2×D412yes—
106C23×C17C2411yescharacteristic
107C22×D17—21no—
108C22×D17—21no—
109C22×D17—21no—
110C22×D17—21no—
111C22×D17C2×D412yes—
112C22×D17#4112yes—
113C22×D17#4112yes—
114C2×C4×C17—21no—
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×C17—44no—
132C2×C4×C17#4112yes—
133C22⋊D17—28no—
134C2×Dic17—21no—
135C2×Dic17—21no—
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×Dic17#4112yes—
155C4×D17—44no—
156C4×D17—44no—
157C4×D17—44no—
158#2419—171nomaximal, Sylow
159C22×C17—22no—
160C22×C17—22no—
161C22×C17—22no—
162C22×C17—41no—
163C22×C17—41no—
164C22×C17—41no—
165C22×C17—41no—
166C22×C17C22×D411yescharacteristic
167C22×C17C2×#4111yescharacteristic
168C22×C17C2×#4111yescharacteristic
169C22×C17C2×#4111yescharacteristic
170C22×C17C2×#4111yescharacteristic
171C22×C17H(5,𝔽2)11yescharacteristic
172C22×C172−1+411yescharacteristic
173C2×D17—22no—
174C2×D17—22no—
175C2×D17—22no—
176C2×D17—22no—
177C2×D17—22no—
178C2×D17—22no—
179C2×D17#12314yes—
180C4×C17—22no—
181C4×C17—41no—
182C4×C17—41no—
183C4×C17—41no—
184C4×C17—41no—
185C4×C17—41no—
186Dic17—22no—
187Dic17—22no—
188Dic17—41no—
189Dic17—41no—
190Dic17—41no—
191Dic17—41no—
192Dic17—41no—
193C2×#123—171no—
194C2×C4×D4—171no—
195#400—171no—
196#398—171no—
197#397—171no—
198#397—171no—
199#387—171no—
200#387—171no—
201#386—171no—
202#385—171no—
203#385—171no—
204#381—171no—
205#381D1711yescharacteristic, complemented
206#380—171no—
207C2×#95—171no—
208C2×C17—42no—
209C2×C17C2×#12311yescharacteristic
210C2×C17#52911yescharacteristic
211C2×C17#53911yescharacteristic
212C2×C17#54611yescharacteristic
213C2×C17#54711yescharacteristic
214C2×C17#55111yescharacteristic
215C2×C17#55211yescharacteristic
216D17—24no—
217C22×D4—171no—
218C23×C4—171no—
219C23×C4—171no—
220C23×C4—171no—
221C23×C4C2×D1711yescharacteristic
222#123—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×C22⋊C4—171no—
232C2×C22⋊C4—171no—
233C2×C22⋊C4—171no—
234C2×C22⋊C4—171no—
235C2×C22⋊C4—171no—
236C2×C22⋊C4—171no—
237C2×C22⋊C4C2×D1711yescharacteristic
238C2×C22⋊C4C2×D1711yescharacteristic
239C2×C42—171no—
240#95—171no—
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#95C2×D1711yescharacteristic
251#95C2×D1711yescharacteristic
252#95C2×D1711yescharacteristic
253#95C2×D1711yescharacteristic
254C17#241911yescharacteristic, complemented, Sylow
255C24—171no—
256C24C22×D1711yescharacteristic
257C2×D4—174no—
258C2×D4—342no—
259C2×D4—342no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—41no—
263C22×C4—171no—
264C22×C4—171no—
265C22×C4—171no—
266C22×C4—171no—
267C22×C4—171no—
268C22×C4—171no—
269C22×C4—171no—
270C22×C4—342no—
271C22×C4—342no—
272C22×C4—342no—
273C22×C4—342no—
274C22×C4—342no—
275C22×C4—342no—
276C22×C4—342no—
277C22×C4—681no—
278C22×C4—681no—
279C22×C4C22×D1711yescharacteristic
280C22×C4C22×D1711yescharacteristic
281C22×C4C22×D1711yescharacteristic
282C22×C4C22×D1711yescharacteristic
283C22×C4C22×D1711yescharacteristic
284C22×C4C22×D1711yescharacteristic
285C4⋊C4—342no—
286C4⋊C4—342no—
287C4⋊C4—342no—
288C4⋊C4—342no—
289C4⋊C4—342no—
290C4⋊C4—681no—
291C4⋊C4—681no—
292C22⋊C4—22no—
293C22⋊C4—41no—
294C22⋊C4—342no—
295C22⋊C4—342no—
296C22⋊C4—342no—
297C22⋊C4—342no—
298C22⋊C4—342no—
299C22⋊C4—342no—
300C42—342no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—172no—
308C23—172no—
309C23—172no—
310C23—341no—
311C23—341no—
312C23—341no—
313C23—341no—
314C23D4×D1712yes—
315C23C23×D1711yescharacteristic
316D4—348no—
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—21no—
334C2×C4—44no—
335C2×C4—172no—
336C2×C4—172no—
337C2×C4—341no—
338C2×C4—341no—
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×C4—684no—
357C2×C4—684no—
358C2×C4—684no—
359C2×C4#6325212yes—
360C22—22no—
361C22—22no—
362C22—22no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—174no—
368C22—342no—
369C22—342no—
370C22—342no—
371C22—342no—
372C22—342no—
373C22—342no—
374C22C2×#6325211yescharacteristic
375C22C2×D4×D1711yescharacteristic
376C22#1058769111yescharacteristic
377C22#41×D1711yescharacteristic
378C22#41×D1711yescharacteristic
379C22#41×D1711yescharacteristic
380C22#1058769711yescharacteristic
381C4—22no—
382C4—41no—
383C4—41no—
384C4—41no—
385C4—41no—
386C4—41no—
387C4—342no—
388C4—342no—
389C4—681no—
390C4—681no—
391C4—681no—
392C4—681no—
393C4—681no—
394C2—42no—
395C2—344no—
396C2#123×D1711yescharacteristic
397C2#1176333711yescharacteristic
398C2#1176340211yescharacteristic
399C2#1176344911yescharacteristic
400C2#1176345711yescharacteristic
401C2#1176348711yescharacteristic
402C2#1176349811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#381C2
prime radicalC22×C22⋊D17C22
hypercenter#381D17
hypocenterC17#2419
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
3. #11762751C2semidirect
4. #11762743C2semidirect
5. #11762737C2semidirect
6. #11762734C2semidirect
7. #11762696C2semidirect
8. #11762691C2semidirect
10. #11762683C2semidirect
11. #11762682C2semidirect
12. #11762671C2semidirect
13. #11762668C2semidirect
14. #95×D17C2semidirect
15. C17×#381C2semidirect
205. #381D17semidirect
254. C17#2419semidirect

Maximal subgroups

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

subgroupindex
C2×#105875692
C2×C4×C22⋊D172
#117627512
#117627432
#117627372
#117627342
#117626962
#117626912
#117626852
#117626832
#117626822
#117626712
#117626682
#95×D172
C17×#3812
#241917