Finite Group Databasebeta

← #16950330

Subgroups — #16950330

Order 4352.

480 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 93 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#14531498C211yesmaximal, characteristic
2C2×#11762605C212yesmaximal
3#14528778C212yesmaximal, complemented
4#14526911C211yesmaximal, characteristic, complemented
5C17×#1394C211yesmaximal, characteristic, complemented
6C2×#10587633C2211yescharacteristic
7C2×Q8×Dic17C2211yescharacteristic
8C2×#10587579C411yescharacteristic
9#11762832—22no—
10C2×#10587507C2212yes—
11#11762711C411yescharacteristic
12#11762698—22no—
13#11762683—22no—
14#11762678C411yescharacteristic
15#11762655C411yescharacteristic
16#11762605—24no—
17#11762577—22no—
18C2×C17×#101C2212yes—
19C17×#391C2211yescharacteristic
20C22×Q8×C17C2311yescharacteristic
21C22×D68C2×C411yescharacteristic
22C22×C4×D17—21no—
23C2×M16×C17—22no—
24C2×C4⋊C4×C17C2×C411yescharacteristic
25C2×C42×C17C2×C411yescharacteristic
26#10587633—22no—
27#10587633D412yes—
28#10587633Q812yes—
29Q8×Dic17—42no—
30C2×#63227—22no—
31C2×#63227—21no—
32C2×#63227—22no—
33C2×C4⋊Dic17C2×C411yescharacteristic
34C2×C4⋊Dic17D412yes—
35C2×#63225—22no—
36C2×C4×Dic17C2×C411yescharacteristic
37C2×C4×Dic17C2×C411yescharacteristic
38C2×C4×Dic17D412yes—
39C2×#63223—22no—
40#10587579—42no—
41C17×#101—22no—
42C17×#101—22no—
43C17×#95—22no—
44#10587507—22no—
45#10587507—22no—
46#10587502—21no—
47#10587502—21no—
48#10587485—44no—
49C23×D17—21no—
50C2×Q8×C17—41no—
51C2×Q8×C17—41no—
52C2×Q8×C17C2×D411yescharacteristic
53C2×Q8×C17C2×Q811yescharacteristic
54C2×Q8×C17#4112yes—
55C22×C4×C17—21no—
56C22×C4×C17—21no—
57C22×C4×C17C22×C411yescharacteristic
58C22×C4×C17C2×D412yes—
59C22×Dic17—21no—
60C22×Dic17C22⋊C411yescharacteristic
61C22×Dic17C22⋊C412yes—
62C22×Dic17C22⋊C411yescharacteristic
63C2×D68—21no—
64C2×D68—41no—
65C2×D68—41no—
66C2×D68C4⋊C412yes—
67C2×C4×D17—42no—
68C2×C4×D17—82no—
69M16×C17—42no—
70M16×C17—42no—
71C2×C8×C17—42no—
72C4⋊C4×C17—41no—
73C42×C17—41no—
74#63227—24no—
75#63227—24no—
76#63227—42no—
77#63227—82no—
78C4⋊Dic17—41no—
79C4⋊Dic17—42no—
80#63225—82no—
81C4×Dic17—21no—
82C4×Dic17—24no—
83C4×Dic17—22no—
84C4×Dic17C4⋊C412yes—
85#63223—42no—
86#63223—42no—
87C2×#3622—42no—
88#12803—171nomaximal, Sylow
89C23×C17C2×C22⋊C411yescharacteristic
90C22×D17—22no—
91C22×D17—42no—
92C22×D17—42no—
93C22×D17—41no—
94C22×D17—41no—
95Q8×C17—41no—
96Q8×C17—41no—
97Q8×C17—42no—
98C2×C4×C17—21no—
99C2×C4×C17—22no—
100C2×C4×C17—21no—
101C2×C4×C17—22no—
102C2×C4×C17—42no—
103C2×C4×C17—41no—
104C2×C4×C17—41no—
105C2×C4×C17—41no—
106C2×C4×C17—41no—
107C2×C4×C17C2×C4⋊C411yescharacteristic
108C2×C4×C17#11711yescharacteristic
109C2×C4×C17#12112yes—
110C2×C4×C17C22⋊Q812yes—
111C2×Dic17—22no—
112C2×Dic17—21no—
113C2×Dic17—22no—
114C2×Dic17—41no—
115C2×Dic17—41no—
116C2×Dic17—41no—
117C2×Dic17—41no—
118C2×Dic17—42no—
119C2×Dic17#9912yes—
120D68—42no—
121D68—42no—
122C4×D17—84no—
123C8×C17—82no—
124#3622—82no—
125#2326—171no—
126C2×#355—172no—
127#1801—172no—
128#1394—171no—
129#1394D1711yescharacteristic, complemented
130C22×C17—21no—
131C22×C17—21no—
132C22×C17#38011yescharacteristic
133C22×C17C2×#9911yescharacteristic
134C22×C17#41011yescharacteristic
135C2×D17—44no—
136C2×D17—42no—
137C2×D17—82no—
138C2×D17—81no—
139C2×D17—81no—
140C4×C17—22no—
141C4×C17—21no—
142C4×C17—21no—
143C4×C17—22no—
144C4×C17—81no—
145C4×C17—81no—
146Dic17—22no—
147Dic17—42no—
148Dic17—41no—
149Dic17—42no—
150C2×#124—171no—
151C2×C4×Q8—171no—
152C2×C4×D4—171no—
153#423—342no—
154C2×#101—172no—
155C2×#101C2×D1712yes—
156#391C2×D1711yescharacteristic
157#390—171no—
158#388—342no—
159#385—342no—
160#384—171no—
161#376—171no—
162#355—344no—
163#343—342no—
164C2×C17—21no—
165C2×C17—21no—
166C2×C17#179911yescharacteristic
167C2×C17#213311yescharacteristic
168C2×C17#213411yescharacteristic
169D17—82no—
170C22×Q8C22×D1711yescharacteristic
171C22×D4—171no—
172C23×C4—341no—
173C2×M16—22no—
174C2×M16—342no—
175#124—172no—
176#124—172no—
177#124—342no—
178C4×Q8—682no—
179C4×D4—682no—
180C2×C4⋊C4—171no—
181C2×C4⋊C4—172no—
182C2×C4⋊C4—342no—
183C2×C4⋊C4C4×D1711yescharacteristic
184C2×C22⋊C4—342no—
185C2×C22⋊C4—341no—
186C2×C22⋊C4—342no—
187C2×C42—171no—
188C2×C42—171no—
189C2×C42—172no—
190C2×C42C4×D1711yescharacteristic
191C4⋊C8—684no—
192#101—22no—
193#101—22no—
194#101—342no—
195#101—342no—
196#95—22no—
197#95—341no—
198#95—341no—
199C17#1280311yescharacteristic, complemented, Sylow
200C24—341no—
201C2×Q8—41no—
202C2×Q8—41no—
203C2×Q8#6325212yes—
204C2×Q8D4×D1711yescharacteristic
205C2×Q8Q8×D1711yescharacteristic
206C2×D4—172no—
207C2×D4—341no—
208C2×D4—681no—
209C2×D4—681no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—171no—
213C22×C4—171no—
214C22×C4—172no—
215C22×C4—341no—
216C22×C4—682no—
217C22×C4—1362no—
218C22×C4C2×C4×D1711yescharacteristic
219C22×C4D4×D1712yes—
220M16—42no—
221M16—42no—
222M16—682no—
223M16—682no—
224C2×C8—42no—
225C2×C8—682no—
226C4⋊C4—41no—
227C4⋊C4—681no—
228C4⋊C4—682no—
229C4⋊C4—1362no—
230C22⋊C4—344no—
231C22⋊C4—344no—
232C22⋊C4—682no—
233C22⋊C4—1362no—
234C42—41no—
235C42—172no—
236C42—344no—
237C42—342no—
238C42—341no—
239C23—342no—
240C23—682no—
241C23—682no—
242C23—681no—
243C23—681no—
244C23C22⋊C4×D1711yescharacteristic
245Q8—42no—
246Q8—41no—
247Q8—41no—
248D4—682no—
249D4—682no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—41no—
255C2×C4—42no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—172no—
260C2×C4—341no—
261C2×C4—342no—
262C2×C4—342no—
263C2×C4—681no—
264C2×C4—681no—
265C2×C4—681no—
266C2×C4—682no—
267C2×C4—681no—
268C2×C4—1364no—
269C2×C4#1058755811yescharacteristic
270C2×C4#1058757012yes—
271C2×C4C4⋊C4×D1711yescharacteristic
272C2×C4#1058758212yes—
273C8—82no—
274C8—1362no—
275C22—21no—
276C22—21no—
277C22—684no—
278C22—682no—
279C22—1362no—
280C22—1361no—
281C22—1361no—
282C22#1176266811yescharacteristic
283C22#99×D1711yescharacteristic
284C22#1176278111yescharacteristic
285C4—22no—
286C4—21no—
287C4—21no—
288C4—22no—
289C4—81no—
290C4—81no—
291C4—342no—
292C4—682no—
293C4—681no—
294C4—682no—
295C2—21no—
296C2—21no—
297C2—1362no—
298C2#1452877111yescharacteristic
299C2#1453047211yescharacteristic
300C2#1453047411yescharacteristic
subgroupG/H
centerC22#99×D17
commutatorC22×C4×C17C22×C4
FrattiniC22×Q8C22×D17
FittingC17×#1394C2
socleC22×C17C2×#99
prime radicalC22×D68C2×C4
cosocleC22×Q8×C17C23
hypercenter#1394D17
hypocenterC17#12803
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #1394

Complements

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

NH ≅ G/Ntype
3. #14528778C2semidirect
4. #14526911C2semidirect
5. C17×#1394C2semidirect
129. #1394D17semidirect
199. C17#12803semidirect

Maximal subgroups

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

subgroupindex
#145314982
C2×#117626052
#145287782
#145269112
C17×#13942
#1280317