Finite Group Databasebeta

← #16931001

Subgroups — #16931001

Order 4352.

426 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 109 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#14539930C211yesmaximal, characteristic
2#14537724C211yesmaximal, characteristic
3#14529604C211yesmaximal, characteristic
4#14527455C211yesmaximal, characteristic
5#14527349C211yesmaximal, characteristic
6#14526643C211yesmaximal, characteristic
7C17×#1275C211yesmaximal, characteristic
8#11763366C2211yescharacteristic
9C2×#10587555C2211yescharacteristic
10#11763207C412yescomplemented
11#11763160C412yescomplemented
12#11763140—22no—
13#11763095—22no—
14#11762937—22no—
15#11762937—22no—
16#11762835—21no—
17#11762833C2211yescharacteristic
18C2×#10587487—21no—
19C2×#10587487—21no—
20#11762764—21no—
21#11762675C2211yescharacteristic
22#11762554—22no—
23#11762547—22no—
24C17×#431C2211yescharacteristic
25C17×#431C2211yescharacteristic
26C2×C17×#97C2211yescharacteristic
27C22×Dic34—21no—
28C22×C8×C17D411yescharacteristic
29C22×C8×C17D411yescharacteristic
30C2×C42×C17C2311yescharacteristic
31#10587612—21no—
32C2×C4⋊Dic17—21no—
33C2×C4⋊Dic17—21no—
34C2×C4⋊Dic17—21no—
35#10587610—21no—
36#10587610D412yes—
37C2×#63223—21no—
38#10587556—21no—
39#10587555—21no—
40#10587555—21no—
41#10587555—21no—
42#10587555C2×C412yes—
43#10587555C2×C411yescharacteristic
44#10587555C2×C411yescharacteristic
45C4×Dic34—21no—
46C4×Dic34D412yes—
47C17×C4⋊C8—21no—
48C17×C4⋊C8—21no—
49C17×C22⋊C8—21no—
50C17×C22⋊C8—21no—
51C17×#97—21no—
52C17×#97—21no—
53C4×C8×C17—21no—
54C4×C8×C17C2×C412yes—
55#10587502—21no—
56#10587502—21no—
57#10587488—24no—
58#10587487—24no—
59#10587487—42no—
60#10587487—42no—
61#10587486—22no—
62#10587485—24no—
63C8×Dic17—22no—
64C4⋊#3622—21no—
65C4⋊#3622D412yes—
66C22×C4×C17C2×D411yescharacteristic
67C22×C4×C17C2×D411yescharacteristic
68C22×C4×C17#4111yescharacteristic
69C22×Dic17—41no—
70C22×Dic17—41no—
71C2×Dic34—22no—
72C2×Dic34—21no—
73C2×Dic34—41no—
74C2×Dic34—41no—
75C2×Dic34—41no—
76C2×Dic34—41no—
77C2×Dic34—41no—
78C2×Dic34C22⋊C412yes—
79C2×C8×C17—21no—
80C2×C8×C17—21no—
81C2×C8×C17—22no—
82C2×C8×C17—21no—
83C2×C8×C17—21no—
84C2×C8×C17—21no—
85C2×C8×C17—41no—
86C2×C8×C17SD1612yes—
87C42×C17C22×C411yescharacteristic
88C42×C17C2×D411yescharacteristic
89C42×C17C2×D411yescharacteristic
90C42×C17#4111yescharacteristic
91C22⋊Dic17—22no—
92C4⋊Dic17—21no—
93C4⋊Dic17—21no—
94C4⋊Dic17—21no—
95C4⋊Dic17—22no—
96C4⋊Dic17—22no—
97C4⋊Dic17—41no—
98C4⋊Dic17—41no—
99C4⋊Dic17—41no—
100C4⋊Dic17—41no—
101C4⋊Dic17C22⋊C412yes—
102#63225—22no—
103#63225—22no—
104C4×Dic17—22no—
105#63223—41no—
106#63223—41no—
107C2×#3622—22no—
108#9292—171nomaximal, Sylow
109C23×C17#12111yescharacteristic
110C2×C4×C17—41no—
111C2×C4×C17—41no—
112C2×C4×C17C2×C22⋊C411yescharacteristic
113C2×C4×C17C4×D411yescharacteristic
114C2×C4×C17C4×D411yescharacteristic
115C2×C4×C17#12011yescharacteristic
116C2×C4×C17#12111yescharacteristic
117C2×C4×C17#12311yescharacteristic
118C2×C4×C17C2×SD1611yescharacteristic
119C2×C4×C17#13511yescharacteristic
120C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
121C2×C4×C17#13711yescharacteristic
122C2×Dic17—22no—
123C2×Dic17—22no—
124C2×Dic17—42no—
125C2×Dic17—81no—
126C2×Dic17—81no—
127C2×Dic17—81no—
128C2×Dic17—81no—
129C2×Dic17—81no—
130Dic34—42no—
131Dic34—42no—
132Dic34—42no—
133Dic34—81no—
134Dic34—81no—
135Dic34—81no—
136C8×C17—22no—
137C8×C17—42no—
138C8×C17—41no—
139C8×C17—81no—
140#3622—42no—
141#3229—171no—
142#2961—171no—
143#1960—171no—
144#1546—171no—
145#1518—171no—
146#1286—171no—
147#1275D1711yescharacteristic
148C22×C17—21no—
149C22×C17—21no—
150C22×C17#38611yescharacteristic
151C22×C17#46011yescharacteristic
152C22×C17#46411yescharacteristic
153C4×C17—21no—
154C4×C17—21no—
155C4×C17—21no—
156C4×C17—21no—
157C4×C17C4×SD1611yescharacteristic
158C4×C17#44011yescharacteristic
159C4×C17#44811yescharacteristic
160C4×C17#44911yescharacteristic
161Dic17—42no—
162Dic17—42no—
163Dic17—82no—
164Dic17—161no—
165#533—171no—
166C2×C4⋊Q8—171no—
167#498—172no—
168#487—172no—
169#483—342no—
170#475—342no—
171#446—342no—
172#446—342no—
173#431C2×D1711yescharacteristic
174#431C2×D1711yescharacteristic
175#424—341no—
176#423—171no—
177C2×#103—341no—
178C2×#103—341no—
179#405—341no—
180C2×#97C2×D1711yescharacteristic
181#382—171no—
182#332—342no—
183#328—342no—
184C2×C17—21no—
185C2×C17—21no—
186C2×C17#188011yescharacteristic
187C2×C17#190211yescharacteristic
188C2×C17#190911yescharacteristic
189C22×Q8—341no—
190C2×M16—341no—
191C22×C8D6811yescharacteristic
192C22×C8D6811yescharacteristic
193C4⋊Q8—171no—
194C4⋊Q8—171no—
195C4⋊Q8—172no—
196C4⋊Q8—341no—
197C4⋊Q8—341no—
198C4⋊Q8—341no—
199#125—341no—
200C22⋊Q8—172no—
201C22⋊Q8—341no—
202C4×Q8—172no—
203C4×Q8—341no—
204#117—341no—
205C2×C4⋊C4—341no—
206C2×C4⋊C4—341no—
207C2×C4⋊C4—341no—
208C2×C42C22×D1711yescharacteristic
209#106—344no—
210C4⋊C8—21no—
211C4⋊C8—21no—
212C4⋊C8—172no—
213C4⋊C8—344no—
214C4⋊C8—341no—
215#103—344no—
216#103—682no—
217#103—682no—
218C22⋊C8—21no—
219C22⋊C8—21no—
220#97—21no—
221#97—21no—
222#97—342no—
223C4×C8—21no—
224C4×C8—342no—
225C4×C8C4×D1712yes—
226#95—341no—
227#95—341no—
228C17#929211yescharacteristic, complemented, Sylow
229C2×Q8—172no—
230C2×Q8—341no—
231C2×Q8—342no—
232C2×Q8—681no—
233C2×Q8—681no—
234C2×Q8—681no—
235C2×Q8—681no—
236C2×Q8—681no—
237C22×C4—681no—
238C22×C4—681no—
239C22×C4C2×D6811yescharacteristic
240C22×C4D4×D1711yescharacteristic
241C22×C4#6325611yescharacteristic
242M16—681no—
243M16—681no—
244C2×C8—21no—
245C2×C8—21no—
246C2×C8—22no—
247C2×C8—21no—
248C2×C8—21no—
249C2×C8—21no—
250C2×C8—41no—
251C2×C8—342no—
252C2×C8#6321912yes—
253C4⋊C4—172no—
254C4⋊C4—341no—
255C4⋊C4—341no—
256C4⋊C4—342no—
257C4⋊C4—341no—
258C4⋊C4—342no—
259C4⋊C4—342no—
260C4⋊C4—342no—
261C4⋊C4—681no—
262C4⋊C4—681no—
263C4⋊C4—681no—
264C4⋊C4—681no—
265C22⋊C4—342no—
266C42—342no—
267C42C2×C4×D1711yescharacteristic
268C42#6325211yescharacteristic
269C42D4×D1711yescharacteristic
270C42D4×D1711yescharacteristic
271C23#1058758111yescharacteristic
272Q8—682no—
273Q8—682no—
274Q8—682no—
275Q8—1361no—
276Q8—1361no—
277Q8—1361no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—342no—
281C2×C4—342no—
282C2×C4—682no—
283C2×C4—1361no—
284C2×C4—1361no—
285C2×C4—1361no—
286C2×C4—1361no—
287C2×C4—1361no—
288C2×C4C4×D6811yescharacteristic
289C2×C4C22⋊C4×D1711yescharacteristic
290C2×C4#1058756811yescharacteristic
291C2×C4#1058757011yescharacteristic
292C2×C4#1058757911yescharacteristic
293C2×C4#1058758011yescharacteristic
294C2×C4C2×#6321911yescharacteristic
295C2×C4#1058759011yescharacteristic
296C2×C4#1058759411yescharacteristic
297C2×C4#1058759511yescharacteristic
298C8—22no—
299C8—42no—
300C8—41no—
301C8—81no—
302C8—682no—
303C22—21no—
304C22—21no—
305C22#1176268711yescharacteristic
306C22#1176301011yescharacteristic
307C22#1176303211yescharacteristic
308C4—21no—
309C4—21no—
310C4—21no—
311C4—21no—
312C4—2721no—
313C4—682no—
314C4—682no—
315C4—1362no—
316C4C4×#6321911yescharacteristic
317C4#1176291111yescharacteristic
318C4#1176295011yescharacteristic
319C4#1176295611yescharacteristic
320C2—21no—
321C2—21no—
322C2#1452913811yescharacteristic
323C2#1452925811yescharacteristic
324C2#1452930511yescharacteristic
subgroupG/H
centerC22#11762687
commutatorC42×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#1275C2
socleC22×C17#386
prime radicalC23×C17#121
cosocleC2×C42×C17C23
hypercenter#1275D17
hypocenterC17#9292
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1275

Complements

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

NH ≅ G/Ntype
10. #11763207C4semidirect
11. #11763160C4semidirect
228. C17#9292semidirect

Maximal subgroups

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

subgroupindex
#145399302
#145377242
#145296042
#145274552
#145273492
#145266432
C17×#12752
#929217