Finite Group Databasebeta

← #16932213

Subgroups — #16932213

Order 4352.

458 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 99 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#14534361C211yesmaximal, characteristic
2#14527232C212yesmaximal, complemented
3#14527118C211yesmaximal, characteristic
4#14526649C212yesmaximal, complemented
5C17×#2921C211yesmaximal, characteristic
6C2×C4×Dic34C2211yescharacteristic
7#11762866C2212yes—
8#11762739C411yescharacteristic
9#11762734C411yescharacteristic
10#11762729C411yescharacteristic
11#11762721C411yescharacteristic
12#11762697—21no—
13#11762693—21no—
14#11762690—21no—
15#11762682—21no—
16#11762677C2211yescharacteristic
17#11762541—24no—
18#11762535—21no—
19#11762533—21no—
20C2×C17×#117C2211yescharacteristic
21C17×#433—24no—
22C17×#423—21no—
23C17×#407—21no—
24C4×M16×C17C2212yes—
25C23×C4×C17D411yescharacteristic
26C22×Dic34C2×C411yescharacteristic
27C2×M16×C17—22no—
28C17×#117—22no—
29C17×#117—22no—
30C2×C4⋊C4×C17C2×C411yescharacteristic
31C2×C4⋊C4×C17D411yescharacteristic
32C2×C22⋊C4×C17C2×C411yescharacteristic
33C2×C22⋊C4×C17D411yescharacteristic
34C2×C42×C17D411yescharacteristic
35C2×C42×C17C2311yescharacteristic
36C2×C22⋊Dic17—21no—
37C2×C22⋊Dic17—21no—
38C2×C4⋊Dic17C2×C411yescharacteristic
39C2×#63225—21no—
40C2×#63225—21no—
41C2×#63225—21no—
42C2×C4×Dic17—21no—
43C4×Dic34—22no—
44C4×Dic34—22no—
45C17×C4⋊C8—24no—
46C17×C22⋊C8—24no—
47C17×#97—22no—
48C4×C8×C17D414yes—
49#10587502—21no—
50#10587502—21no—
51#10587502—21no—
52#10587502—21no—
53#10587502—21no—
54#10587502—21no—
55#10587502C2×C411yescharacteristic
56#10587502C2×C411yescharacteristic
57C22⋊#3622—22no—
58C4⋊#3622—22no—
59#10587474—22no—
60C24×C17C22⋊C411yescharacteristic
61C22×C4×C17—21no—
62C22×C4×C17—21no—
63C22×C4×C17—41no—
64C22×C4×C17—41no—
65C22×C4×C17C22⋊C411yescharacteristic
66C22×C4×C17C22⋊C411yescharacteristic
67C22×C4×C17C22⋊C411yescharacteristic
68C22×C4×C17C22×C411yescharacteristic
69C22×C4×C17C2×D411yescharacteristic
70C22×C4×C17C2×D411yescharacteristic
71C22×Dic17—21no—
72C22×Dic17—21no—
73C22×Dic17—21no—
74C22×Dic17—21no—
75C2×Dic34—22no—
76C2×Dic34—42no—
77M16×C17—42no—
78M16×C17—42no—
79C2×C8×C17—24no—
80C4⋊C4×C17—22no—
81C4⋊C4×C17—22no—
82C22⋊C4×C17—22no—
83C22⋊C4×C17—22no—
84C42×C17—21no—
85C42×C17—21no—
86C42×C17C2×D412yes—
87C42×C17#4112yes—
88C22⋊Dic17—42no—
89C22⋊Dic17—42no—
90C4⋊Dic17—22no—
91#63225—42no—
92#63225—42no—
93#63225—41no—
94#63225—41no—
95C4×Dic17—42no—
96C2×#3622—42no—
97#9493—171nomaximal, Sylow
98C23×C17—21no—
99C23×C17—21no—
100C23×C17—21no—
101C23×C17—41no—
102C23×C17—41no—
103C23×C17C2×C22⋊C411yescharacteristic
104C2×C4×C17—21no—
105C2×C4×C17—21no—
106C2×C4×C17—21no—
107C2×C4×C17—21no—
108C2×C4×C17—21no—
109C2×C4×C17—21no—
110C2×C4×C17—21no—
111C2×C4×C17—41no—
112C2×C4×C17—41no—
113C2×C4×C17—42no—
114C2×C4×C17—42no—
115C2×C4×C17—41no—
116C2×C4×C17—41no—
117C2×C4×C17C4×D412yes—
118C2×C4×C17#12011yescharacteristic
119C2×C4×C17#12112yes—
120C2×C4×C17#12311yescharacteristic
121C2×Dic17—22no—
122C2×Dic17—41no—
123C2×Dic17—41no—
124C2×Dic17—41no—
125C2×Dic17—41no—
126C2×Dic17—41no—
127C2×Dic17—41no—
128C2×Dic17—41no—
129C2×Dic17—41no—
130C2×Dic17—41no—
131C2×Dic17—41no—
132C2×Dic17—41no—
133Dic34—24no—
134Dic34—42no—
135C8×C17—44no—
136#3622—82no—
137#2921D1711yescharacteristic
138#2599—171no—
139#1491—172no—
140#1464—171no—
141#1288—172no—
142C22×C17—21no—
143C22×C17—21no—
144C22×C17—42no—
145C22×C17—41no—
146C22×C17—41no—
147C22×C17—41no—
148C22×C17—81no—
149C22×C17#38611yescharacteristic
150C22×C17#42111yescharacteristic
151C22×C17#42111yescharacteristic
152C4×C17—22no—
153C4×C17—21no—
154C4×C17—21no—
155C4×C17—22no—
156C4×C17—41no—
157C4×C17—41no—
158C4×C17—41no—
159Dic17—42no—
160Dic17—81no—
161Dic17—81no—
162Dic17—81no—
163C2×C4×Q8—171no—
164C2×#117C2×D1711yescharacteristic
165#433—24no—
166#432—172no—
167#423—21no—
168#407—21no—
169C4×M16C2×D1712yes—
170#398—171no—
171#397—171no—
172#395—171no—
173#393—171no—
174#388—341no—
175#387—341no—
176#387—341no—
177#385—341no—
178#383—171no—
179Q8⋊C8—344no—
180#324—341no—
181#323—341no—
182C2×C17—21no—
183C2×C17—21no—
184C2×C17—81no—
185C2×C17#189111yescharacteristic
186C2×C17#189711yescharacteristic
187C2×C17#190911yescharacteristic
188C22×Q8—171no—
189C23×C4C22⋊D1711yescharacteristic
190C2×M16—22no—
191C4×Q8—342no—
192C4×Q8—342no—
193#117—22no—
194#117—22no—
195C2×C4⋊C4—171no—
196C2×C4⋊C4—341no—
197C2×C4⋊C4—341no—
198C2×C4⋊C4—341no—
199C2×C4⋊C4C4×D1711yescharacteristic
200C2×C4⋊C4D6811yescharacteristic
201C2×C22⋊C4—341no—
202C2×C22⋊C4—341no—
203C2×C22⋊C4C4×D1711yescharacteristic
204C2×C22⋊C4D6811yescharacteristic
205C2×C42—341no—
206C2×C42C22⋊D1711yescharacteristic
207C2×C42C22×D1711yescharacteristic
208C4⋊C8—24no—
209C4⋊C8—342no—
210C22⋊C8—24no—
211C22⋊C8—342no—
212#97—22no—
213#97—342no—
214C4×C8C22⋊D1714yes—
215#95—171no—
216#95—171no—
217#95—341no—
218#95—341no—
219#95—341no—
220#95—341no—
221#95—341no—
222#95—341no—
223C17#949311yescharacteristic, complemented, Sylow
224C24#6322711yescharacteristic
225C2×Q8—342no—
226C2×Q8—682no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—341no—
232C22×C4—341no—
233C22×C4—341no—
234C22×C4—341no—
235C22×C4#6322711yescharacteristic
236C22×C4#6322711yescharacteristic
237C22×C4#6322711yescharacteristic
238C22×C4C2×C4×D1711yescharacteristic
239C22×C4C2×D6811yescharacteristic
240C22×C4C2×C22⋊D1711yescharacteristic
241M16—42no—
242M16—42no—
243C2×C8—24no—
244C2×C8—682no—
245C4⋊C4—22no—
246C4⋊C4—22no—
247C4⋊C4—342no—
248C4⋊C4—682no—
249C4⋊C4—682no—
250C4⋊C4—681no—
251C4⋊C4—681no—
252C22⋊C4—22no—
253C22⋊C4—22no—
254C22⋊C4—682no—
255C22⋊C4—682no—
256C42—21no—
257C42—21no—
258C42—682no—
259C42#6325212yes—
260C42C2×C22⋊D1712yes—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—41no—
265C23—41no—
266C23C2×#6322711yescharacteristic
267Q8—344no—
268Q8—682no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—342no—
283C2×C4—681no—
284C2×C4—681no—
285C2×C4—681no—
286C2×C4—681no—
287C2×C4—681no—
288C2×C4—681no—
289C2×C4—681no—
290C2×C4—681no—
291C2×C4—681no—
292C2×C4—681no—
293C2×C4—681no—
294C2×C4C4×C22⋊D1712yes—
295C2×C4#1058761511yescharacteristic
296C2×C4#1058761612yes—
297C2×C4#1058763911yescharacteristic
298C8—44no—
299C8—1362no—
300C22—21no—
301C22—21no—
302C22—42no—
303C22—41no—
304C22—41no—
305C22—41no—
306C22—81no—
307C22#1176268811yescharacteristic
308C22#1176282711yescharacteristic
309C22#1176282711yescharacteristic
310C4—22no—
311C4—22no—
312C4—21no—
313C4—21no—
314C4—41no—
315C4—41no—
316C4—41no—
317C4—682no—
318C4—1361no—
319C4—1361no—
320C4—1361no—
321C2—21no—
322C2—21no—
323C2—81no—
324C2#1452920211yescharacteristic
325C2#1452923211yescharacteristic
326C2#1452930611yescharacteristic
subgroupG/H
centerC22#11762688
commutatorC22×C4×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#2921C2
socleC22×C17#386
prime radicalC24×C17C22⋊C4
cosocleC2×C42×C17C23
hypercenter#2921D17
hypocenterC17#9493
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2921

Complements

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

NH ≅ G/Ntype
2. #14527232C2semidirect
4. #14526649C2semidirect
223. C17#9493semidirect

Maximal subgroups

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

subgroupindex
#145343612
#145272322
#145271182
#145266492
C17×#29212
#949317