Finite Group Databasebeta

← #16934214

Subgroups — #16934214

Order 4352.

460 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14529762C211yesmaximal, characteristic, complemented
2#14529613C211yesmaximal, characteristic
3#14528529C211yesmaximal, characteristic, complemented
4#14527540C211yesmaximal, characteristic, complemented
5#14527239C211yesmaximal, characteristic, complemented
6#14527235C211yesmaximal, characteristic
7C17×#3167C211yesmaximal, characteristic
8#11762833C2211yescharacteristic
9#11762833C2211yescharacteristic
10C2×#10587508—22no—
11C2×C4×#3622C2211yescharacteristic
12C42×Dic17C2211yescharacteristic
13#11762568—22no—
14#11762563—22no—
15#11762560—22no—
16Q8⋊#3622—22no—
17D4⋊#3622—22no—
18C17×#533C2211yescharacteristic
19C17×#532C2211yescharacteristic
20C17×#498C411yescharacteristic, complemented
21C17×#492C411yescharacteristic, complemented
22C17×#487C411yescharacteristic, complemented
23C17×#486C411yescharacteristic, complemented
24C17×#465—22no—
25C4×SD16×C17—22no—
26C2×C4×C8×C17C2211yescharacteristic
27C2×#41×C17—21no—
28C2×SD16×C17—22no—
29C22×C8×C17D412yes—
30C17×C4⋊Q8C2×C411yescharacteristic
31C17×C4⋊Q8C2×C411yescharacteristic
32C17×C4⋊Q8C2×C411yescharacteristic
33C17×#127C2×C411yescharacteristic
34C17×#125—21no—
35C17×#124—21no—
36C17×C22⋊Q8—21no—
37C17×C22⋊Q8—21no—
38C17×#121—21no—
39C17×#121—21no—
40C4×Q8×C17—21no—
41C4×Q8×C17—21no—
42C4×D4×C17—21no—
43C4×D4×C17—21no—
44C17×#117—21no—
45C2×C42×C17C2311yescharacteristic
46C2×C4×Dic17—21no—
47C2×C4×Dic17—21no—
48C2×C4×Dic17D412yes—
49C2×#63223—21no—
50C2×#63223—21no—
51C22×#3622—21no—
52C17×#106—22no—
53C17×#103—22no—
54C17×#102—22no—
55C4×C8×C17C2×C411yescharacteristic
56C4×C8×C17C2×C411yescharacteristic
57C4×C8×C17D412yes—
58#10587508—44no—
59C4⋊#3622—21no—
60C4⋊#3622—21no—
61C4⋊#3622—21no—
62C4⋊#3622—21no—
63C4×#3622—21no—
64C4×#3622D412yes—
65#41×C17—42no—
66#41×C17—42no—
67C2×Q8×C17—21no—
68C2×Q8×C17—21no—
69C2×Q8×C17—21no—
70C2×D4×C17—21no—
71C2×D4×C17—21no—
72C2×D4×C17—21no—
73C2×D4×C17—41no—
74C22×C4×C17—41no—
75C22×C4×C17C2×D411yescharacteristic
76C22×C4×C17C2×D411yescharacteristic
77C22×C4×C17#4111yescharacteristic
78C22×Dic17—22no—
79SD16×C17—44no—
80C2×C8×C17—42no—
81C2×C8×C17C22⋊C412yes—
82C2×C8×C17C22⋊C412yes—
83C4⋊C4×C17—21no—
84C4⋊C4×C17—21no—
85C4⋊C4×C17—21no—
86C4⋊C4×C17—21no—
87C4⋊C4×C17—21no—
88C4⋊C4×C17—21no—
89C4⋊C4×C17—41no—
90C4⋊C4×C17—41no—
91C22⋊C4×C17—41no—
92C22⋊C4×C17—41no—
93C22⋊C4×C17—41no—
94C42×C17—41no—
95C42×C17C22×C411yescharacteristic
96C42×C17C2×D411yescharacteristic
97C42×C17C2×D411yescharacteristic
98C42×C17#4111yescharacteristic
99C4×Dic17—22no—
100C4×Dic17—42no—
101C4×Dic17—42no—
102C4×Dic17SD1614yescomplemented
103#63223—42no—
104#63223—42no—
105C2×#3622—22no—
106C2×#3622—42no—
107C2×#3622—41no—
108C2×#3622—41no—
109#9846—171nomaximal, Sylow
110C23×C17—41no—
111C23×C17#12111yescharacteristic
112Q8×C17—42no—
113Q8×C17—42no—
114Q8×C17—42no—
115D4×C17—42no—
116D4×C17—42no—
117D4×C17—42no—
118D4×C17—82no—
119C2×C4×C17—22no—
120C2×C4×C17—22no—
121C2×C4×C17—41no—
122C2×C4×C17—41no—
123C2×C4×C17—41no—
124C2×C4×C17—81no—
125C2×C4×C17—81no—
126C2×C4×C17C2×C22⋊C411yescharacteristic
127C2×C4×C17C4×D411yescharacteristic
128C2×C4×C17C4×D411yescharacteristic
129C2×C4×C17#12111yescharacteristic
130C2×C4×C17#12411yescharacteristic
131C2×C4×C17#12711yescharacteristic
132C2×C4×C17C2×SD1612yes—
133C2×C4×C17#13512yes—
134C2×Dic17—24no—
135C2×Dic17—42no—
136C2×Dic17—42no—
137C8×C17—22no—
138C8×C17—22no—
139#3622—42no—
140#3622—81no—
141#3622—81no—
142#3167D1711yescharacteristic
143#1994—171no—
144#1962—171no—
145#1744—171no—
146#1566—171no—
147#1493—171no—
148#1492—171no—
149C22×C17—21no—
150C22×C17—21no—
151C22×C17—81no—
152C22×C17—81no—
153C22×C17—81no—
154C22×C17#39011yescharacteristic
155C22×C17#46512yes—
156C4×C17—21no—
157C4×C17—21no—
158C4×C17—81no—
159C4×C17—81no—
160C4×C17—81no—
161C4×C17#42112yes—
162C4×C17C4×SD1612yes—
163C4×C17#44412yes—
164C4×C17#49211yescharacteristic
165C4×C17#49511yescharacteristic
166Dic17—44no—
167#533C2×D1711yescharacteristic
168#532C2×D1711yescharacteristic
169#498Dic1711yescharacteristic, complemented
170#492Dic1711yescharacteristic, complemented
171#487Dic1711yescharacteristic, complemented
172#486Dic1711yescharacteristic, complemented
173#465—22no—
174C4×SD16—22no—
175#423—171no—
176#423—171no—
177C2×#104—342no—
178C2×C4×C8—171no—
179C2×C4×C8C2×D1711yescharacteristic
180C43—171no—
181#339—342no—
182#336—342no—
183#334—342no—
184Q8⋊C8—342no—
185D4⋊C8—342no—
186C2×C17—22no—
187C2×C17—161no—
188C2×C17#196311yescharacteristic
189C2×C17#196811yescharacteristic
190C2×C17#197511yescharacteristic
191C2×#41—21no—
192C2×SD16—22no—
193C2×M16—341no—
194C2×M16—341no—
195C22×C8—341no—
196C22×C8C22⋊D1712yes—
197C4⋊Q8C2×Dic1711yescharacteristic
198C4⋊Q8C2×Dic1711yescharacteristic
199C4⋊Q8C2×Dic1711yescharacteristic
200#127C2×Dic1711yescharacteristic
201#125—21no—
202#124—21no—
203C22⋊Q8—21no—
204C22⋊Q8—21no—
205#121—21no—
206#121—21no—
207C4×Q8—21no—
208C4×Q8—21no—
209C4×D4—21no—
210C4×D4—21no—
211#117—21no—
212C2×C42—172no—
213C2×C42—341no—
214C2×C42—341no—
215C2×C42C22×D1711yescharacteristic
216#106—22no—
217C4⋊C8—341no—
218C4⋊C8—341no—
219C4⋊C8—341no—
220C4⋊C8—341no—
221#104—684no—
222#103—22no—
223#102—22no—
224C4×C8—172no—
225C4×C8—341no—
226C4×C8C2×Dic1711yescharacteristic
227C4×C8C2×Dic1711yescharacteristic
228C4×C8C22⋊D1712yes—
229C17#984611yescharacteristic, complemented, Sylow
230#41—42no—
231#41—42no—
232C2×Q8—21no—
233C2×Q8—21no—
234C2×Q8—21no—
235C2×D4—21no—
236C2×D4—21no—
237C2×D4—21no—
238C2×D4—41no—
239C22×C4—41no—
240C22×C4—342no—
241C22×C4D4×D1711yescharacteristic
242C22×C4#6325411yescharacteristic
243C22×C4C2×C22⋊D1711yescharacteristic
244SD16—44no—
245M16—682no—
246M16—682no—
247C2×C8—42no—
248C2×C8—342no—
249C2×C8—682no—
250C2×C8—681no—
251C2×C8—681no—
252C2×C8C22⋊Dic1712yes—
253C2×C8C22⋊Dic1712yes—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C22⋊C4—41no—
263C22⋊C4—41no—
264C22⋊C4—41no—
265C42—41no—
266C42—174no—
267C42—342no—
268C42—682no—
269C42—682no—
270C42D4×D1711yescharacteristic
271C42#6325411yescharacteristic
272C42C22×Dic1711yescharacteristic
273C42C2×C22⋊D1711yescharacteristic
274C23—41no—
275C23#1058762411yescharacteristic
276Q8—42no—
277Q8—42no—
278Q8—42no—
279D4—42no—
280D4—42no—
281D4—42no—
282D4—82no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4—344no—
291C2×C4—682no—
292C2×C4—682no—
293C2×C4SD16×D1712yes—
294C2×C4#1058760212yes—
295C2×C4D4×Dic1711yescharacteristic
296C2×C4D4×Dic1711yescharacteristic
297C2×C4#1058762211yescharacteristic
298C2×C4#1058762411yescharacteristic
299C2×C4#1058762611yescharacteristic
300C2×C4C2×C22⋊Dic1711yescharacteristic
301C8—22no—
302C8—22no—
303C8—682no—
304C8—1361no—
305C8—1361no—
306C22—21no—
307C22—21no—
308C22—81no—
309C22—81no—
310C22—81no—
311C22#1176271211yescharacteristic
312C22#1176303512yes—
313C4—21no—
314C4—21no—
315C4—81no—
316C4—81no—
317C4—81no—
318C4—684no—
319C4#1176282612yes—
320C4SD16×Dic1712yes—
321C4#1176293112yes—
322C4#1176318011yescharacteristic
323C4#1176319011yescharacteristic
324C2—22no—
325C2—161no—
326C2#1452961711yescharacteristic
327C2#1452964011yescharacteristic
328C2#1452967811yescharacteristic
subgroupG/H
centerC2×C4#10587626
commutatorC42×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#3167C2
socleC22×C17#390
prime radicalC17×#532C22
cosocleC2×C42×C17C23
hypercenter#3167D17
hypocenterC17#9846
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3167

Complements

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

NH ≅ G/Ntype
1. #14529762C2semidirect
3. #14528529C2semidirect
4. #14527540C2semidirect
5. #14527239C2semidirect
20. C17×#498C4semidirect
21. C17×#492C4semidirect
22. C17×#487C4semidirect
23. C17×#486C4semidirect
102. C4×Dic17SD16semidirect
169. #498Dic17semidirect
170. #492Dic17semidirect
171. #487Dic17semidirect
172. #486Dic17semidirect
229. C17#9846semidirect

Maximal subgroups

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

subgroupindex
#145297622
#145296132
#145285292
#145275402
#145272392
#145272352
C17×#31672
#984617