Finite Group Databasebeta

← #16927103

Subgroups — #16927103

Order 4352.

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

normal subgroups only

This group has 540 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#124×Dic17C211yesmaximal, characteristic
2#14528638C212yesmaximal, complemented
3C2×#11762549C211yesmaximal, characteristic
4#14526666C212yesmaximal, complemented
5C2×C17×#329C211yesmaximal, characteristic
6C2×Q8×Dic17C411yescharacteristic
7C2×D4×Dic17C411yescharacteristic
8C2×#10587485—22no—
9C2×#10587474C2212yes—
10#11762716C411yescharacteristic
11#11762712C411yescharacteristic
12#11762702—22no—
13#11762675C2211yescharacteristic
14C22⋊C4×Dic17—22no—
15C42×Dic17C2211yescharacteristic
16#11762563—22no—
17#11762549—22no—
18C2×C17×#124C2211yescharacteristic
19C2×C17×#97C2212yes—
20C17×#329—22no—
21C22×Q8×C17C2×C411yescharacteristic
22C22×D4×C17C2×C411yescharacteristic
23C23×Dic17—21no—
24C22×C8×C17—22no—
25C17×#124—22no—
26C17×#124D412yes—
27C17×#124Q812yes—
28C2×C22⋊C4×C17—22no—
29C2×C42×C17C2311yescharacteristic
30C2×C22⋊Dic17—21no—
31C2×C22⋊Dic17—22no—
32Q8×Dic17—42no—
33D4×Dic17—42no—
34C2×C4⋊Dic17—21no—
35C2×C4⋊Dic17C2×C411yescharacteristic
36C2×C4⋊Dic17C2×C411yescharacteristic
37C2×C4×Dic17—21no—
38C2×C4×Dic17C2×C411yescharacteristic
39C2×C4×Dic17C2×C411yescharacteristic
40C2×C4×Dic17D412yes—
41C2×C4×Dic17D412yes—
42C22×#3622—22no—
43C17×#97—22no—
44C17×#97—22no—
45#10587502—22no—
46#10587502—22no—
47#10587485—28no—
48#10587474—22no—
49#10587474—22no—
50C24×C17—21no—
51C2×Q8×C17—21no—
52C2×Q8×C17—41no—
53C2×Q8×C17—41no—
54C2×Q8×C17C4⋊C412yes—
55C2×D4×C17—21no—
56C2×D4×C17—41no—
57C2×D4×C17—41no—
58C2×D4×C17C4⋊C412yes—
59C22×C4×C17—21no—
60C22×C4×C17C22×C411yescharacteristic
61C22×C4×C17C2×D412yes—
62C22×Dic17—21no—
63C22×Dic17—21no—
64C22×Dic17—44no—
65C22×Dic17—42no—
66C22×Dic17C22⋊C411yescharacteristic
67C22×Dic17C22⋊C411yescharacteristic
68C22×Dic17C22⋊C412yes—
69C2×C8×C17—24no—
70C2×C8×C17—42no—
71C2×C8×C17—42no—
72C22⋊C4×C17—24no—
73C22⋊C4×C17—24no—
74C42×C17C2×D411yescharacteristic
75C42×C17C2×Q811yescharacteristic
76C42×C17#4112yes—
77C22⋊Dic17—42no—
78C22⋊Dic17—44no—
79C4⋊Dic17—42no—
80C4⋊Dic17—41no—
81C4⋊Dic17—41no—
82C4×Dic17—22no—
83C4×Dic17—24no—
84C4×Dic17—41no—
85C4×Dic17—42no—
86C4×Dic17—42no—
87C2×#3622—24no—
88C2×#3622—42no—
89C2×#3622—42no—
90#8583—171nomaximal, Sylow
91C23×C17—22no—
92C23×C17—41no—
93C23×C17—41no—
94C23×C17—42no—
95C23×C17—42no—
96C23×C17C2×C22⋊C411yescharacteristic
97Q8×C17—42no—
98Q8×C17—42no—
99D4×C17—42no—
100D4×C17—42no—
101C2×C4×C17—21no—
102C2×C4×C17—21no—
103C2×C4×C17—22no—
104C2×C4×C17—22no—
105C2×C4×C17—22no—
106C2×C4×C17—41no—
107C2×C4×C17—41no—
108C2×C4×C17C2×C4⋊C411yescharacteristic
109C2×C4×C17#11711yescharacteristic
110C2×C4×C17#12112yes—
111C2×C4×C17C22⋊Q812yes—
112C2×Dic17—22no—
113C2×Dic17—22no—
114C2×Dic17—22no—
115C2×Dic17—21no—
116C2×Dic17—22no—
117C2×Dic17—41no—
118C2×Dic17—41no—
119C2×Dic17—42no—
120C2×Dic17—42no—
121C2×Dic17—84no—
122C2×Dic17#10012yes—
123C2×Dic17#10414yes—
124C8×C17—44no—
125#3622—44no—
126C4×#124—171no—
127#1767—172no—
128C2×#329—171no—
129C2×#329D1711yescharacteristic
130#1295—172no—
131C22×C17—42no—
132C22×C17—44no—
133C22×C17—81no—
134C22×C17—82no—
135C22×C17—81no—
136C22×C17#38011yescharacteristic
137C22×C17C2×#10011yescharacteristic
138C22×C17#41311yescharacteristic
139C22×C17C2×#10412yes—
140C22×C17#42112yes—
141C4×C17—21no—
142C4×C17—21no—
143C4×C17—22no—
144C4×C17—22no—
145C4×C17—42no—
146Dic17—24no—
147Dic17—22no—
148Dic17—41no—
149Dic17—81no—
150Dic17—81no—
151C2×#124C2×D1711yescharacteristic
152C2×C4×Q8—171no—
153C2×C4×D4—171no—
154C2×C4⋊C8—342no—
155C2×#97—172no—
156C2×#97C2×D1712yes—
157#391—171no—
158#390—171no—
159#388—342no—
160#382—171no—
161C4×C22⋊C4—342no—
162C43—171no—
163#336—342no—
164#329—22no—
165#329—342no—
166C2×C17—82no—
167C2×C17#152511yescharacteristic
168C2×C17#153212yes—
169C2×C17#153311yescharacteristic
170C2×C17#180011yescharacteristic
171C2×C17#180612yes—
172C22×Q8C4×D1711yescharacteristic
173C22×D4C4×D1711yescharacteristic
174C23×C4—341no—
175C22×C8—22no—
176C22×C8—342no—
177#124—22no—
178#124Dic3412yes—
179#124C22⋊D1712yes—
180C4×Q8—682no—
181C4×D4—682no—
182C2×C4⋊C4—171no—
183C2×C4⋊C4—171no—
184C2×C4⋊C4—341no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—341no—
187C2×C22⋊C4—342no—
188C2×C42—171no—
189C2×C42—172no—
190C2×C42—171no—
191C2×C42—172no—
192C2×C42—341no—
193C2×C42C22×D1711yescharacteristic
194C4⋊C8—348no—
195#97—22no—
196#97—22no—
197#97—342no—
198#97—342no—
199#95—342no—
200#95—342no—
201C17#858311yescharacteristic, complemented, Sylow
202C24—21no—
203C2×Q8—21no—
204C2×Q8—41no—
205C2×Q8—41no—
206C2×Q8#6322512yes—
207C2×D4—21no—
208C2×D4—41no—
209C2×D4—41no—
210C2×D4#6322512yes—
211C22×C4—21no—
212C22×C4—172no—
213C22×C4—171no—
214C22×C4—171no—
215C22×C4—341no—
216C22×C4—341no—
217C22×C4—682no—
218C22×C4—684no—
219C22×C4C2×C4×D1711yescharacteristic
220C22×C4D4×D1712yes—
221C2×C8—24no—
222C2×C8—42no—
223C2×C8—42no—
224C2×C8—344no—
225C2×C8—682no—
226C2×C8—682no—
227C4⋊C4—681no—
228C4⋊C4—681no—
229C4⋊C4—682no—
230C22⋊C4—24no—
231C22⋊C4—24no—
232C22⋊C4—684no—
233C22⋊C4—682no—
234C42—344no—
235C42—342no—
236C42—681no—
237C42—682no—
238C42—682no—
239C42C2×Dic3411yescharacteristic
240C42#6325412yes—
241C42C2×C22⋊D1711yescharacteristic
242C23—22no—
243C23—41no—
244C23—41no—
245C23—42no—
246C23—42no—
247C23C22⋊C4×D1711yescharacteristic
248Q8—42no—
249Q8—42no—
250D4—42no—
251D4—42no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—172no—
260C2×C4—174no—
261C2×C4—342no—
262C2×C4—342no—
263C2×C4—342no—
264C2×C4—341no—
265C2×C4—342no—
266C2×C4—681no—
267C2×C4—681no—
268C2×C4—682no—
269C2×C4—682no—
270C2×C4—1364no—
271C2×C4#1058756311yescharacteristic
272C2×C4#1058756412yes—
273C2×C4C2×#6322511yescharacteristic
274C2×C4#1058762512yes—
275C8—44no—
276C8—684no—
277C22—42no—
278C22—44no—
279C22—81no—
280C22—82no—
281C22—81no—
282C22#1176266711yescharacteristic
283C22#100×D1711yescharacteristic
284C22#1176279011yescharacteristic
285C22#104×D1712yes—
286C22#1176282412yes—
287C4—21no—
288C4—21no—
289C4—22no—
290C4—22no—
291C4—42no—
292C4—344no—
293C4—342no—
294C4—681no—
295C4—1361no—
296C4—1361no—
297C2—82no—
298C2#1452737411yescharacteristic
299C2#1452740112yes—
300C2#1452740611yescharacteristic
301C2#1452877411yescharacteristic
302C2#1452880212yes—
subgroupG/H
centerC23C22⋊C4×D17
commutatorC22×C4×C17C22×C4
FrattiniC2×C42C22×D17
FittingC2×C17×#329C2
socleC23×C17C2×C22⋊C4
prime radicalC22×D4×C17C2×C4
cosocleC2×C42×C17C23
hypercenterC2×#329D17
hypocenterC17#8583
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#329

Complements

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

NH ≅ G/Ntype
2. #14528638C2semidirect
4. #14526666C2semidirect
201. C17#8583semidirect

Maximal subgroups

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

subgroupindex
#124×Dic172
#145286382
C2×#117625492
#145266662
C2×C17×#3292
#858317