Finite Group Databasebeta

← #16934206

Subgroups — #16934206

Order 4352.

478 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 121 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14539366C211yesmaximal, characteristic
2#14529762C211yesmaximal, characteristic
3#14529760C211yesmaximal, characteristic
4C4×C8×Dic17C211yesmaximal, characteristic
5#14527542C211yesmaximal, characteristic
6#14527240C211yesmaximal, characteristic
7C17×#1493C211yesmaximal, characteristic
8#11763366C2211yescharacteristic
9#11763212C411yescharacteristic, complemented
10#11763188C411yescharacteristic, complemented
11#11763161C411yescharacteristic, complemented
12#11763159C411yescharacteristic, complemented
13#11763048—22no—
14C4×#63231—22no—
15#11762833C2211yescharacteristic
16C2×C4×#3622C2211yescharacteristic
17C2×C8×Dic17—22no—
18C42×Dic17C2211yescharacteristic
19#11762570—22no—
20#11762568—22no—
21#11762561—22no—
22#11762541—22no—
23C17×#533C2211yescharacteristic
24C17×#423C2211yescharacteristic
25C2×C4×C8×C17C2211yescharacteristic
26C17×Q8⋊C8—22no—
27C2×M16×C17—21no—
28C22×C8×C17—21no—
29C17×C4⋊Q8C2×C411yescharacteristic
30C17×C4⋊Q8C2×C411yescharacteristic
31C17×#125—21no—
32C17×C22⋊Q8—21no—
33C17×C22⋊Q8—21no—
34C4×Q8×C17—21no—
35C4×Q8×C17—21no—
36C17×#117—21no—
37C2×C42×C17C2311yescharacteristic
38C2×#63231—22no—
39#10587612—21no—
40#10587610—21no—
41#10587610—21no—
42C2×C4×Dic17—21no—
43C2×C4×Dic17—21no—
44C2×C4×Dic17D412yes—
45C2×#63223—21no—
46C22×#3622D412yes—
47#10587556—21no—
48#10587555C2×C411yescharacteristic
49#10587555C2×C411yescharacteristic
50C4×Dic34—21no—
51C4×Dic34—21no—
52C17×C4⋊C8—21no—
53C17×C4⋊C8—21no—
54C4×C8×C17—21no—
55C4×C8×C17D412yes—
56#10587506—22no—
57C8×Dic17—28no—
58#10587481—22no—
59#10587478—22no—
60C4⋊#3622—21no—
61C4⋊#3622—21no—
62C4×#3622C2×C411yescharacteristic
63C4×#3622C2×C411yescharacteristic
64C4×#3622D412yes—
65C2×Q8×C17—21no—
66C2×Q8×C17—21no—
67C22×C4×C17C2×D411yescharacteristic
68C22×C4×C17C2×D411yescharacteristic
69C22×C4×C17#4111yescharacteristic
70C22×Dic17—22no—
71C2×Dic34—21no—
72C2×Dic34—21no—
73M16×C17—42no—
74C2×C8×C17—24no—
75C2×C8×C17—22no—
76C2×C8×C17—41no—
77C4⋊C4×C17—21no—
78C4⋊C4×C17—21no—
79C4⋊C4×C17—21no—
80C4⋊C4×C17—21no—
81C4⋊C4×C17—41no—
82C4⋊C4×C17—41no—
83C22⋊C4×C17—41no—
84C42×C17—41no—
85C42×C17C22×C411yescharacteristic
86C42×C17C2×D411yescharacteristic
87C42×C17C2×D411yescharacteristic
88C42×C17#4111yescharacteristic
89C22⋊Dic17—41no—
90#63231—44no—
91C4⋊Dic17—21no—
92C4⋊Dic17—21no—
93C4⋊Dic17—21no—
94C4⋊Dic17—21no—
95#63225—41no—
96#63225—41no—
97C4×Dic17—24no—
98C4×Dic17—24no—
99C4×Dic17—22no—
100C4×Dic17—41no—
101C4×Dic17Q1614yescomplemented
102#63223—42no—
103C2×#3622—24no—
104C2×#3622—41no—
105C2×#3622C22⋊C412yes—
106C2×#3622C22⋊C412yes—
107#9845—171nomaximal, Sylow
108C23×C17#12111yescharacteristic
109Q8×C17—42no—
110Q8×C17—42no—
111C2×C4×C17—22no—
112C2×C4×C17—22no—
113C2×C4×C17—41no—
114C2×C4×C17—41no—
115C2×C4×C17C2×C22⋊C411yescharacteristic
116C2×C4×C17C4×D411yescharacteristic
117C2×C4×C17C4×D411yescharacteristic
118C2×C4×C17#12111yescharacteristic
119C2×C4×C17#12411yescharacteristic
120C2×C4×C17#12711yescharacteristic
121C2×C4×C17C2×Q1612yes—
122C2×C4×C17#13512yes—
123C2×Dic17—24no—
124C2×Dic17—24no—
125C2×Dic17—24no—
126C2×Dic17—41no—
127C2×Dic17—41no—
128Dic34—42no—
129Dic34—42no—
130C8×C17—24no—
131C8×C17—81no—
132#3622—22no—
133#3622—81no—
134#3622#10414yes—
135#3170—171no—
136#1994—171no—
137#1994—171no—
138C42×C8—171no—
139#1567—171no—
140#1493—171no—
141#1493D1711yescharacteristic
142C22×C17—21no—
143C22×C17—21no—
144C22×C17#39011yescharacteristic
145C22×C17#46712yes—
146C4×C17—21no—
147C4×C17—21no—
148C4×C17—81no—
149C4×C17—81no—
150C4×C17C2×#10412yes—
151C4×C17C4×Q1612yes—
152C4×C17#44312yes—
153C4×C17#49411yescharacteristic
154C4×C17#49511yescharacteristic
155Dic17—28no—
156Dic17—81no—
157Dic17—81no—
158#533—171no—
159#533C2×D1711yescharacteristic
160#500—171no—
161#494—171no—
162#487—171no—
163#487—171no—
164#467—342no—
165C4×Q16—342no—
166#423—171no—
167#423C2×D1711yescharacteristic
168C2×C4×C8—171no—
169C2×C4×C8—342no—
170C2×C4×C8C2×D1711yescharacteristic
171C43—171no—
172#339—342no—
173#339—342no—
174#335—342no—
175Q8⋊C8—22no—
176Q8⋊C8—342no—
177C2×C17—22no—
178C2×C17#196211yescharacteristic
179C2×C17#196911yescharacteristic
180C2×C17#197411yescharacteristic
181C2×Q16—342no—
182C2×M16—21no—
183C2×M16—341no—
184C22×C8—21no—
185C22×C8—172no—
186C4⋊Q8—171no—
187C4⋊Q8—171no—
188C4⋊Q8C4×D1711yescharacteristic
189C4⋊Q8C4×D1711yescharacteristic
190#125—21no—
191#125—341no—
192C22⋊Q8—21no—
193C22⋊Q8—21no—
194C22⋊Q8—341no—
195C22⋊Q8—341no—
196C4×Q8—21no—
197C4×Q8—21no—
198C4×Q8—341no—
199C4×Q8—341no—
200#117—21no—
201#117—341no—
202C2×C42—172no—
203C2×C42—341no—
204C2×C42—341no—
205C2×C42C22×D1711yescharacteristic
206#107—342no—
207C4⋊C8—21no—
208C4⋊C8—21no—
209C4⋊C8—341no—
210C4⋊C8—341no—
211#103—342no—
212#103—342no—
213C4×C8—21no—
214C4×C8—171no—
215C4×C8—171no—
216C4×C8—172no—
217C4×C8—348no—
218C4×C8C22⋊D1712yes—
219C17#984511yescharacteristic, complemented, Sylow
220C2×Q8—21no—
221C2×Q8—21no—
222C2×Q8—341no—
223C2×Q8—341no—
224C22×C4—342no—
225C22×C4#6325211yescharacteristic
226C22×C4D4×D1711yescharacteristic
227C22×C4D4×D1711yescharacteristic
228Q16—684no—
229M16—42no—
230M16—682no—
231C2×C8—24no—
232C2×C8—22no—
233C2×C8—41no—
234C2×C8—172no—
235C2×C8—172no—
236C2×C8—344no—
237C2×C8—681no—
238C4⋊C4—21no—
239C4⋊C4—21no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—341no—
245C4⋊C4—341no—
246C4⋊C4—341no—
247C4⋊C4—341no—
248C4⋊C4—681no—
249C4⋊C4—681no—
250C22⋊C4—41no—
251C22⋊C4—681no—
252C42—41no—
253C42—174no—
254C42—344no—
255C42—344no—
256C42—342no—
257C42—681no—
258C42C2×C4×D1711yescharacteristic
259C42D4×D1711yescharacteristic
260C42#6325411yescharacteristic
261C42C2×C22⋊D1711yescharacteristic
262C23#1058757011yescharacteristic
263Q8—42no—
264Q8—42no—
265Q8—682no—
266Q8—682no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—344no—
272C2×C4—344no—
273C2×C4—344no—
274C2×C4—681no—
275C2×C4—681no—
276C2×C4C22⋊C4×D1711yescharacteristic
277C2×C4#1058756711yescharacteristic
278C2×C4#1058757111yescharacteristic
279C2×C4#1058760212yes—
280C2×C4Q16×D1712yes—
281C2×C4C4×C22⋊D1711yescharacteristic
282C2×C4#1058762511yescharacteristic
283C2×C4#1058762611yescharacteristic
284C8—24no—
285C8—81no—
286C8—174no—
287C8—342no—
288C8—1361no—
289C22—21no—
290C22—21no—
291C22#1176270911yescharacteristic
292C22#1176304912yes—
293C4—21no—
294C4—21no—
295C4—81no—
296C4—81no—
297C4—348no—
298C4—1361no—
299C4—1361no—
300C4#104×D1712yes—
301C4#1176290112yes—
302C4#1176292412yes—
303C4#1176318711yescharacteristic
304C4#1176319011yescharacteristic
305C2—22no—
306C2#1452961011yescharacteristic
307C2#1452964211yescharacteristic
308C2#1452967111yescharacteristic
subgroupG/H
centerC2×C4#10587626
commutatorC42×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#1493C2
socleC22×C17#390
prime radicalC23×C17#121
cosocleC2×C42×C17C23
hypercenter#1493D17
hypocenterC17#9845
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1493

Complements

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

NH ≅ G/Ntype
9. #11763212C4semidirect
10. #11763188C4semidirect
11. #11763161C4semidirect
12. #11763159C4semidirect
101. C4×Dic17Q16semidirect
219. C17#9845semidirect

Maximal subgroups

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

subgroupindex
#145393662
#145297622
#145297602
C4×C8×Dic172
#145275422
#145272402
C17×#14932
#984517