Finite Group Databasebeta

← #14540904

Subgroups — #14540904

Order 2176.

392 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11763480C211yesmaximal, characteristic
2#11763175C211yesmaximal, characteristic, complemented
3#11763133C211yesmaximal, characteristic, complemented
4#11763127C211yesmaximal, characteristic, complemented
5#11763113C211yesmaximal, characteristic, complemented
6#11763057C211yesmaximal, characteristic, complemented
7#11763040C211yesmaximal, characteristic, complemented
8#11763009C211yesmaximal, characteristic, complemented
9#11762969C211yesmaximal, characteristic, complemented
10#11762969C211yesmaximal, characteristic, complemented
11#11762914C211yesmaximal, characteristic, complemented
12#11762881C211yesmaximal, characteristic, complemented
13#11762808C211yesmaximal, characteristic, complemented
14#11762803C211yesmaximal, characteristic, complemented
15C17×#552C211yesmaximal, characteristic, complemented
16C2×#63252C2211yescharacteristic
17C2×#63252C2211yescharacteristic
18C17×#126—21no—
19C17×#125—21no—
20C17×#125C2211yescharacteristic
21C17×#123—21no—
22C17×C22⋊Q8C2211yescharacteristic
23C17×C22⋊Q8C2211yescharacteristic
24C4×Q8×C17C2211yescharacteristic
25C4×D4×C17C2211yescharacteristic
26C17×#117—21no—
27C17×#117C2211yescharacteristic
28C2×C4⋊C4×C17C2211yescharacteristic
29C2×#63231C2211yescharacteristic
30C2×#63230C2211yescharacteristic
31C2×#63230C2211yescharacteristic
32#10587626—21no—
33#10587616C2211yescharacteristic
34#10587616C2211yescharacteristic
35C2×#63223C2211yescharacteristic
36C22×#3622C2211yescharacteristic
37#10587579—21no—
38#10587570—21no—
39#10587570—21no—
40#10587560C2211yescharacteristic
41C4×Dic34C2211yescharacteristic
42#10587506C2211yescharacteristic
43C22⋊#3622C2211yescharacteristic
44C22⋊#3622C2211yescharacteristic
45#10587481C2211yescharacteristic
46#10587481C2211yescharacteristic
47#10587481C2211yescharacteristic
48#10587480C2211yescharacteristic
49#10587480C2211yescharacteristic
50#10587480C2211yescharacteristic
51#10587480C2211yescharacteristic
52#10587480C2211yescharacteristic
53#10587480C2211yescharacteristic
54#10587479C2211yescharacteristic
55#10587479C2211yescharacteristic
56#10587478C2211yescharacteristic
57C4⋊#3622C2211yescharacteristic
58#10587474C2211yescharacteristic
59C2×Q8×C17C2311yescharacteristic
60C2×D4×C17D411yescharacteristic
61C22×C4×C17—21no—
62C22×C4×C17C2311yescharacteristic
63C22×C4×C17C2311yescharacteristic
64C2×C22⋊D17—21no—
65C2×C22⋊D17—21no—
66C2×C22⋊D17—21no—
67C2×C22⋊D17—21no—
68#63252—22no—
69#63252—22no—
70#63252—41no—
71#63252—41no—
72C2×D68—21no—
73C2×D68—21no—
74C2×D68C2311yescharacteristic
75C2×D68C2311yescharacteristic
76C2×C4×D17—21no—
77C2×C4×D17—21no—
78C2×Dic34C2311yescharacteristic
79C4⋊C4×C17—21no—
80C4⋊C4×C17—21no—
81C4⋊C4×C17—21no—
82C4⋊C4×C17—21no—
83C4⋊C4×C17—21no—
84C4⋊C4×C17—21no—
85C4⋊C4×C17D411yescharacteristic
86C4⋊C4×C17C2311yescharacteristic
87C4⋊C4×C17C2311yescharacteristic
88C4⋊C4×C17C2311yescharacteristic
89C22⋊C4×C17—21no—
90C22⋊C4×C17—21no—
91C22⋊C4×C17—21no—
92C22⋊C4×C17D411yescharacteristic
93C22⋊C4×C17D411yescharacteristic
94C42×C17—21no—
95C42×C17—21no—
96C42×C17C2311yescharacteristic
97#63231—22no—
98#63230—41no—
99#63230—41no—
100#63227—21no—
101#63227—21no—
102C4⋊Dic17C2311yescharacteristic
103#63225—21no—
104C4×Dic17—21no—
105#63223—41no—
106C2×#3622—41no—
107C2×#3622C2311yescharacteristic
108C2×#3622C2311yescharacteristic
109C2×#3622C2311yescharacteristic
110C2×#3622C2311yescharacteristic
111C23×C17C2×D411yescharacteristic
112C23×C17C2×D411yescharacteristic
113C22×D17—21no—
114C22×D17—21no—
115Q8×C17—21no—
116Q8×C17—21no—
117D4×C17—41no—
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—42no—
127C2×C4×C17C2×D411yescharacteristic
128C2×C4×C17C2×D411yescharacteristic
129C2×C4×C17C2×D411yescharacteristic
130C2×C4×C17C2×D411yescharacteristic
131C2×C4×C17C2411yescharacteristic
132C22⋊D17—42no—
133C22⋊D17—42no—
134C22⋊D17—81no—
135C22⋊D17—81no—
136C2×Dic17—21no—
137C2×Dic17—21no—
138D68—21no—
139D68—21no—
140D68—21no—
141D68—21no—
142D68—42no—
143D68—81no—
144C4×D17—42no—
145C4×D17—42no—
146Dic34—21no—
147Dic34#4112yes—
148#3622—21no—
149#3622—21no—
150#3622—21no—
151#3622—21no—
152#3317—171nomaximal, Sylow
153C22×C17—21no—
154C22×C17—21no—
155C22×C17—21no—
156C22×C17—21no—
157C22×C17—21no—
158C22×C17—21no—
159C22×C17C22×D411yescharacteristic
160C2×D17—41no—
161C2×D17—41no—
162C2×D17—41no—
163C2×D17—41no—
164C2×D17—41no—
165C2×D17—41no—
166C4×C17—21no—
167C4×C17—21no—
168C4×C17—21no—
169C4×C17—21no—
170C4×C17—41no—
171C4×C17—41no—
172C4×C17—41no—
173C4×C17—41no—
174C4×C17C2×#4111yescharacteristic
175C4×C17H(5,𝔽2)11yescharacteristic
176Dic17—22no—
177Dic17—41no—
178#552D1711yescharacteristic, complemented
179#550—171no—
180#490—171no—
181#482—171no—
182#481—171no—
183#479—171no—
184#468—171no—
185#465—171no—
186#460—171no—
187#452—171no—
188#452—171no—
189#441—171no—
190#435—171no—
191#417—171no—
192#416—171no—
193C2×C17—41no—
194C2×C17—41no—
195C2×C17#54611yescharacteristic
196C2×C17#57511yescharacteristic
197C2×C17#57711yescharacteristic
198D17—81no—
199D17—81no—
200C2×#41—171no—
201C2×#41—171no—
202C2×Q16—171no—
203C2×SD16—171no—
204C2×SD16—171no—
205C2×M16—171no—
206C22×C8—171no—
207#127—171no—
208#127—341no—
209#126—21no—
210#125—21no—
211#125C2×D1711yescharacteristic
212#123—21no—
213C22⋊Q8C2×D1711yescharacteristic
214C22⋊Q8C2×D1711yescharacteristic
215#121—171no—
216#121—171no—
217#121—341no—
218#121—341no—
219C4×Q8—171no—
220C4×Q8C2×D1711yescharacteristic
221C4×D4—341no—
222C4×D4C2×D1711yescharacteristic
223#117—21no—
224#117C2×D1711yescharacteristic
225C2×C4⋊C4C2×D1711yescharacteristic
226#107—171no—
227#106—171no—
228#106—171no—
229C4⋊C8—171no—
230#103—171no—
231#103—171no—
232#103—171no—
233#103—171no—
234#102—171no—
235#102—171no—
236#102—171no—
237#102—171no—
238#102—171no—
239#102—171no—
240C22⋊C8—171no—
241C22⋊C8—171no—
242#97—171no—
243C17#331711yescharacteristic, complemented, Sylow
244#41—342no—
245#41—342no—
246#41—681no—
247#41—681no—
248C2×Q8—171no—
249C2×Q8C22×D1711yescharacteristic
250C2×D4—171no—
251C2×D4—171no—
252C2×D4—341no—
253C2×D4—341no—
254C2×D4—341no—
255C2×D4—341no—
256C2×D4—341no—
257C2×D4—341no—
258C2×D4C22⋊D1711yescharacteristic
259C22×C4—21no—
260C22×C4—341no—
261C22×C4—341no—
262C22×C4C22×D1711yescharacteristic
263C22×C4C22×D1711yescharacteristic
264Q16—342no—
265SD16—681no—
266SD16—681no—
267M16—681no—
268C2×C8—171no—
269C2×C8—171no—
270C2×C8—171no—
271C2×C8—171no—
272C2×C8—681no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—171no—
280C4⋊C4—341no—
281C4⋊C4C22⋊D1711yescharacteristic
282C4⋊C4C22×D1711yescharacteristic
283C4⋊C4C22×D1711yescharacteristic
284C4⋊C4C22×D1711yescharacteristic
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—21no—
288C22⋊C4—341no—
289C22⋊C4—341no—
290C22⋊C4C22⋊D1711yescharacteristic
291C22⋊C4C22⋊D1711yescharacteristic
292C42—21no—
293C42—21no—
294C42—341no—
295C42C22×D1711yescharacteristic
296C23—341no—
297C23—341no—
298C23C2×C22⋊D1711yescharacteristic
299C23C2×C22⋊D1711yescharacteristic
300Q8—21no—
301Q8—21no—
302Q8—172no—
303Q8—341no—
304D4—41no—
305D4—341no—
306D4—341no—
307D4—341no—
308D4—341no—
309D4—682no—
310D4—682no—
311D4—682no—
312D4—1361no—
313D4—1361no—
314D4—1361no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—42no—
324C2×C4—341no—
325C2×C4—341no—
326C2×C4—682no—
327C2×C4—682no—
328C2×C4C2×C22⋊D1711yescharacteristic
329C2×C4C2×C22⋊D1711yescharacteristic
330C2×C4C2×C22⋊D1711yescharacteristic
331C2×C4C2×C22⋊D1711yescharacteristic
332C2×C4C23×D1711yescharacteristic
333C8—341no—
334C8—341no—
335C8—341no—
336C8—341no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—681no—
344C22—681no—
345C22—681no—
346C22—681no—
347C22—681no—
348C22—681no—
349C22C22×C22⋊D1711yescharacteristic
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—41no—
358C4—342no—
359C4—681no—
360C4#41×D1711yescharacteristic
361C4#1058769611yescharacteristic
362C2—41no—
363C2—41no—
364C2—1361no—
365C2—1361no—
366C2#1176344711yescharacteristic
367C2#1176360711yescharacteristic
368C2#1176361711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#552C2
socleC22×C17C22×D4
prime radical#11763480C2
hypercenter#552D17
hypocenterC17#3317
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C4×D4 ◁ #552

Complements

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

NH ≅ G/Ntype
2. #11763175C2semidirect
3. #11763133C2semidirect
4. #11763127C2semidirect
5. #11763113C2semidirect
6. #11763057C2semidirect
7. #11763040C2semidirect
8. #11763009C2semidirect
9. #11762969C2semidirect
10. #11762969C2semidirect
11. #11762914C2semidirect
12. #11762881C2semidirect
13. #11762808C2semidirect
14. #11762803C2semidirect
15. C17×#552C2semidirect
178. #552D17semidirect
243. C17#3317semidirect

Maximal subgroups

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

subgroupindex
#117634802
#117631752
#117631332
#117631272
#117631132
#117630572
#117630402
#117630092
#117629692
#117629142
#117628812
#117628082
#117628032
C17×#5522
#331717