Finite Group Databasebeta

← #16930991

Subgroups — #16930991

Order 4352.

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

normal subgroups only

This group has 422 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#14539732C211yesmaximal, characteristic
2#14537724C211yesmaximal, characteristic
3#14527469C211yesmaximal, characteristic
4#14527347C211yesmaximal, characteristic
5#14526642C211yesmaximal, characteristic
6#14526615C211yesmaximal, characteristic
7C17×#1993C211yesmaximal, characteristic
8#11763366C2211yescharacteristic
9#11763188C412yescomplemented
10#11763159C412yescomplemented
11#11763016—22no—
12#11762965—22no—
13#11762937—22no—
14#11762937—22no—
15#11762862C2211yescharacteristic
16#11762862C2211yescharacteristic
17#11762835—21no—
18#11762833C2211yescharacteristic
19#11762764—21no—
20#11762555—22no—
21#11762545—22no—
22C2×C17×C4⋊Q8C2211yescharacteristic
23C2×C17×#97C2211yescharacteristic
24C17×#382C2211yescharacteristic
25C17×#340—21no—
26C17×#340—21no—
27C22×Q8×C17—21no—
28C22×C8×C17D411yescharacteristic
29C22×C8×C17D411yescharacteristic
30C17×C4⋊Q8—21no—
31C17×C4⋊Q8—21no—
32C17×C4⋊Q8—21no—
33C17×C4⋊Q8C2×C412yes—
34C2×C4⋊C4×C17—21no—
35C2×C4⋊C4×C17—21no—
36C2×C4⋊C4×C17—21no—
37C2×C42×C17C2311yescharacteristic
38C2×#63231—24no—
39#10587612—21no—
40#10587610—21no—
41#10587610D412yes—
42C2×#63223—21no—
43#10587556—21no—
44#10587555C2×C411yescharacteristic
45#10587555C2×C411yescharacteristic
46C4×Dic34—21no—
47C4×Dic34D412yes—
48C17×#97—21no—
49C17×#97—21no—
50C17×#95—21no—
51C17×#95—21no—
52#10587506—24no—
53C22⋊#3622—21no—
54C22⋊#3622—21no—
55#10587486—22no—
56#10587485—24no—
57C8×Dic17—22no—
58C4⋊#3622—21no—
59C4⋊#3622—21no—
60C4⋊#3622—21no—
61C4⋊#3622D412yes—
62C4×#3622—21no—
63C4×#3622C2×C412yes—
64C2×Q8×C17—22no—
65C2×Q8×C17—41no—
66C2×Q8×C17—82no—
67C22×C4×C17—41no—
68C22×C4×C17—41no—
69C22×C4×C17C2×D411yescharacteristic
70C22×C4×C17C2×D411yescharacteristic
71C22×C4×C17#4111yescharacteristic
72C2×Dic34—21no—
73C2×Dic34C22⋊C412yes—
74C2×C8×C17—21no—
75C2×C8×C17—21no—
76C2×C8×C17—21no—
77C2×C8×C17—21no—
78C2×C8×C17—21no—
79C2×C8×C17Q1612yescomplemented
80C4⋊C4×C17—22no—
81C4⋊C4×C17—22no—
82C4⋊C4×C17—41no—
83C4⋊C4×C17—41no—
84C4⋊C4×C17—41no—
85C4⋊C4×C17—41no—
86C42×C17C22×C411yescharacteristic
87C42×C17C2×D411yescharacteristic
88C42×C17C2×D411yescharacteristic
89C42×C17#4111yescharacteristic
90C22⋊Dic17—22no—
91#63231—84no—
92C4⋊Dic17—21no—
93C4⋊Dic17—21no—
94C4⋊Dic17—21no—
95C4⋊Dic17C22⋊C412yes—
96#63225—22no—
97#63225—22no—
98C4×Dic17—22no—
99#63223—41no—
100#63223—41no—
101C2×#3622—22no—
102C2×#3622—22no—
103C2×#3622—41no—
104#9290—171nomaximal, Sylow
105C23×C17#12111yescharacteristic
106Q8×C17—44no—
107Q8×C17—82no—
108C2×C4×C17—41no—
109C2×C4×C17—41no—
110C2×C4×C17—42no—
111C2×C4×C17—82no—
112C2×C4×C17—81no—
113C2×C4×C17—81no—
114C2×C4×C17—81no—
115C2×C4×C17C2×C22⋊C411yescharacteristic
116C2×C4×C17C4×D411yescharacteristic
117C2×C4×C17C4×D411yescharacteristic
118C2×C4×C17#12011yescharacteristic
119C2×C4×C17#12111yescharacteristic
120C2×C4×C17#12311yescharacteristic
121C2×C4×C17C2×Q1611yescharacteristic
122C2×C4×C17#13511yescharacteristic
123C2×C4×C17#13711yescharacteristic
124C2×C4×C17#13711yescharacteristic
125C2×Dic17—22no—
126C2×Dic17—22no—
127Dic34—42no—
128Dic34—81no—
129C8×C17—22no—
130C8×C17—41no—
131#3622—42no—
132#3622—42no—
133#3622—81no—
134#3209—171no—
135#2961—171no—
136#1993D1711yescharacteristic
137#1549—171no—
138#1518—171no—
139#1286—171no—
140#1275—171no—
141C22×C17—21no—
142C22×C17—21no—
143C22×C17#38611yescharacteristic
144C22×C17#46211yescharacteristic
145C22×C17#46411yescharacteristic
146C4×C17—21no—
147C4×C17—21no—
148C4×C17—21no—
149C4×C17—21no—
150C4×C17—82no—
151C4×C17—161no—
152C4×C17C4×Q1611yescharacteristic
153C4×C17#44111yescharacteristic
154C4×C17#45111yescharacteristic
155C4×C17#45211yescharacteristic
156Dic17—42no—
157Dic17—42no—
158#533—171no—
159C2×C4⋊Q8C2×D1711yescharacteristic
160#494—172no—
161#487—172no—
162#462—342no—
163#451—342no—
164#446—342no—
165#446—342no—
166#431—171no—
167#431—171no—
168#424—341no—
169#423—171no—
170#405—341no—
171C2×#97C2×D1711yescharacteristic
172#382C2×D1711yescharacteristic
173#340—21no—
174#340—21no—
175#332—342no—
176#328—342no—
177C2×C17—21no—
178C2×C17—21no—
179C2×C17#187811yescharacteristic
180C2×C17#190211yescharacteristic
181C2×C17#191011yescharacteristic
182C22×Q8—21no—
183C2×Q16—344no—
184C2×M16—341no—
185C22×C8C22⋊D1711yescharacteristic
186C22×C8C22⋊D1711yescharacteristic
187C4⋊Q8—21no—
188C4⋊Q8—21no—
189C4⋊Q8—21no—
190C4⋊Q8—171no—
191C4⋊Q8—171no—
192C4⋊Q8C4×D1712yes—
193#125—341no—
194C22⋊Q8—172no—
195C22⋊Q8—341no—
196C4×Q8—172no—
197C4×Q8—341no—
198#117—341no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C42C22×D1711yescharacteristic
203C4⋊C8—172no—
204C4⋊C8—344no—
205C4⋊C8—341no—
206C4⋊C8—341no—
207C4⋊C8—341no—
208#103—344no—
209C22⋊C8—341no—
210C22⋊C8—341no—
211#97—21no—
212#97—21no—
213#97—342no—
214C4×C8—172no—
215C4×C8—342no—
216C4×C8—341no—
217#95—21no—
218#95—21no—
219C17#929011yescharacteristic, complemented, Sylow
220C2×Q8—22no—
221C2×Q8—41no—
222C2×Q8—82no—
223C2×Q8—172no—
224C2×Q8—341no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4D4×D1711yescharacteristic
228C22×C4#6325411yescharacteristic
229C22×C4C2×C22⋊D1711yescharacteristic
230Q16—1364no—
231M16—681no—
232M16—681no—
233C2×C8—21no—
234C2×C8—21no—
235C2×C8—21no—
236C2×C8—21no—
237C2×C8—21no—
238C2×C8—342no—
239C2×C8—342no—
240C2×C8—681no—
241C2×C8#6323112yescomplemented
242C4⋊C4—22no—
243C4⋊C4—22no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—172no—
249C4⋊C4—341no—
250C4⋊C4—342no—
251C4⋊C4—342no—
252C4⋊C4—341no—
253C4⋊C4—341no—
254C22⋊C4—342no—
255C42—342no—
256C42C2×C4×D1711yescharacteristic
257C42#6325211yescharacteristic
258C42D4×D1711yescharacteristic
259C42D4×D1711yescharacteristic
260C23#1058762411yescharacteristic
261Q8—44no—
262Q8—82no—
263Q8—682no—
264Q8—1361no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—42no—
268C2×C4—82no—
269C2×C4—81no—
270C2×C4—81no—
271C2×C4—81no—
272C2×C4—342no—
273C2×C4—342no—
274C2×C4C22⋊C4×D1711yescharacteristic
275C2×C4#1058756711yescharacteristic
276C2×C4#1058756911yescharacteristic
277C2×C4#1058757011yescharacteristic
278C2×C4C4×C22⋊D1711yescharacteristic
279C2×C4#1058762311yescharacteristic
280C2×C4#1058762811yescharacteristic
281C2×C4C2×#6323111yescharacteristic
282C2×C4#1058763611yescharacteristic
283C2×C4#1058763711yescharacteristic
284C8—22no—
285C8—41no—
286C8—682no—
287C8—682no—
288C8—1361no—
289C22—21no—
290C22—21no—
291C22#1176268511yescharacteristic
292C22#1176301611yescharacteristic
293C22#1176302811yescharacteristic
294C4—21no—
295C4—21no—
296C4—21no—
297C4—21no—
298C4—82no—
299C4—161no—
300C4—682no—
301C4—682no—
302C4C4×#6323111yescharacteristic
303C4#1176291411yescharacteristic
304C4#1176296411yescharacteristic
305C4#1176296911yescharacteristic
306C2—21no—
307C2—21no—
308C2#1452912811yescharacteristic
309C2#1452925411yescharacteristic
310C2#1452930811yescharacteristic
subgroupG/H
centerC22#11762685
commutatorC42×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#1993C2
socleC22×C17#386
prime radicalC23×C17#121
cosocleC2×C42×C17C23
hypercenter#1993D17
hypocenterC17#9290
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1993

Complements

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

NH ≅ G/Ntype
9. #11763188C4semidirect
10. #11763159C4semidirect
79. C2×C8×C17Q16semidirect
219. C17#9290semidirect
241. C2×C8#63231semidirect

Maximal subgroups

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

subgroupindex
#145397322
#145377242
#145274692
#145273472
#145266422
#145266152
C17×#19932
#929017