Finite Group Databasebeta

← #16944944

Subgroups — #16944944

Order 4352.

516 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 87 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#14533737C211yesmaximal, characteristic
2#14527122C211yesmaximal, characteristic
3#14526807C214yesmaximal, complemented
4C4×C17×#99C211yesmaximal, characteristic
5C2×#10587555C411yescharacteristic, complemented
6C2×#10587610—21no—
7C2×#10587610—21no—
8#11762773—21no—
9#11762745C412yescomplemented
10#11762745C2212yes—
11#11762712—21no—
12#11762678—21no—
13C4×C4⋊Dic17C411yescharacteristic, complemented
14#11762535—21no—
15#11762533C2212yes—
16C2×C4×D4×C17C2211yescharacteristic
17C2×C17×#99—21no—
18C4×C22⋊C4×C17—21no—
19C17×#342C2212yes—
20C22×D4×C17D411yescharacteristic
21C23×C4×C17D411yescharacteristic
22C23×C4×C17D411yescharacteristic
23C22×Dic34—21no—
24C4×D4×C17—41no—
25C4×D4×C17—41no—
26C2×C4⋊C4×C17D411yescharacteristic
27C2×C22⋊C4×C17—22no—
28C2×C22⋊C4×C17C2×C411yescharacteristic
29C2×C22⋊C4×C17C2311yescharacteristic
30C2×C42×C17—41no—
31C2×C42×C17C2×C411yescharacteristic
32C2×C22⋊Dic17—22no—
33C2×C22⋊Dic17—22no—
34C2×C4⋊Dic17—21no—
35C2×C4⋊Dic17—21no—
36C2×C4⋊Dic17C2×C412yes—
37#10587610—44no—
38#10587610—44no—
39C2×#63225—22no—
40C2×#63225—22no—
41C2×C4×Dic17—21no—
42C2×#63223—41no—
43#10587555—41no—
44#10587555—41no—
45C17×#99—24no—
46C17×#95—41no—
47#10587502C2×C412yes—
48C22⋊#3622—22no—
49C22⋊#3622—41no—
50C24×C17C22⋊C411yescharacteristic
51C24×C17C2×D411yescharacteristic
52C2×D4×C17—22no—
53C2×D4×C17—22no—
54C2×D4×C17—41no—
55C22×C4×C17—21no—
56C22×C4×C17—21no—
57C22×C4×C17—21no—
58C22×C4×C17—21no—
59C22×C4×C17—42no—
60C22×C4×C17—42no—
61C22×C4×C17—42no—
62C22×C4×C17C22⋊C411yescharacteristic
63C22×C4×C17C22⋊C411yescharacteristic
64C22×C4×C17C22⋊C411yescharacteristic
65C22×C4×C17C22×C411yescharacteristic
66C22×C4×C17C2×D411yescharacteristic
67C22×Dic17—22no—
68C22×Dic17—22no—
69C2×Dic34—42no—
70C2×Dic34—42no—
71C2×Dic34—41no—
72C2×Dic34—41no—
73C4⋊C4×C17—41no—
74C22⋊C4×C17—24no—
75C22⋊C4×C17—41no—
76C22⋊C4×C17—41no—
77C22⋊C4×C17—42no—
78C42×C17—21no—
79C42×C17—21no—
80C42×C17—82no—
81C22⋊Dic17—44no—
82C22⋊Dic17—44no—
83C4⋊Dic17—24no—
84C4⋊Dic17—42no—
85C4⋊Dic17—42no—
86#63225—44no—
87#63225—44no—
88C4×Dic17—42no—
89#63223—82no—
90C2×#3622—42no—
91#11788—171nomaximal, Sylow
92C23×C17—21no—
93C23×C17—21no—
94C23×C17—21no—
95C23×C17—21no—
96C23×C17—21no—
97C23×C17—41no—
98C23×C17—41no—
99C23×C17—41no—
100C23×C17—41no—
101C23×C17C2×C22⋊C411yescharacteristic
102C23×C17#13712yes—
103D4×C17—44no—
104C2×C4×C17—21no—
105C2×C4×C17—21no—
106C2×C4×C17—22no—
107C2×C4×C17—21no—
108C2×C4×C17—21no—
109C2×C4×C17—21no—
110C2×C4×C17—42no—
111C2×C4×C17—42no—
112C2×C4×C17—41no—
113C2×C4×C17—41no—
114C2×C4×C17—41no—
115C2×C4×C17—44no—
116C2×C4×C17—81no—
117C2×C4×C17—81no—
118C2×C4×C17—82no—
119C2×C4×C17—82no—
120C2×C4×C17#9911yescharacteristic
121C2×C4×C17#9911yescharacteristic
122C2×C4×C17#10412yes—
123C2×Dic17—24no—
124C2×Dic17—42no—
125C2×Dic17—42no—
126C2×Dic17—42no—
127C2×Dic17—42no—
128C2×Dic17—42no—
129Dic34—82no—
130Dic34—82no—
131#3622—82no—
132#2542—171no—
133C4×#99D1711yescharacteristic
134#1465—171no—
135#1352—174no—
136C22×C17—21no—
137C22×C17—21no—
138C22×C17—22no—
139C22×C17—42no—
140C22×C17—42no—
141C22×C17—41no—
142C22×C17—41no—
143C22×C17—41no—
144C22×C17—41no—
145C22×C17—41no—
146C22×C17—81no—
147C22×C17—81no—
148C22×C17C2×#9911yescharacteristic
149C22×C17#41911yescharacteristic
150C22×C17C2×#10411yescharacteristic
151C4×C17—21no—
152C4×C17—22no—
153C4×C17—41no—
154C4×C17—41no—
155C4×C17—41no—
156C4×C17—84no—
157C4×C17#35312yes—
158Dic17—44no—
159Dic17—82no—
160C2×C4⋊Q8—171no—
161C2×C22⋊Q8—341no—
162C2×C22⋊Q8—341no—
163C2×C4×D4C2×D1711yescharacteristic
164C2×#99—21no—
165#407—341no—
166#399—172no—
167#399—172no—
168#390—341no—
169#384—341no—
170C4×C4⋊C4—171no—
171C4×C22⋊C4—21no—
172#342C2×D1712yes—
173#324—341no—
174#323—172no—
175C2×C17—21no—
176C2×C17—21no—
177C2×C17—42no—
178C2×C17—81no—
179C2×C17#150811yescharacteristic
180C2×C17#212311yescharacteristic
181C2×C17C2×#35311yescharacteristic
182C22×Q8—341no—
183C22×D4C22⋊D1711yescharacteristic
184C23×C4D6811yescharacteristic
185C23×C4C22⋊D1711yescharacteristic
186C2×M16—681no—
187C4⋊Q8—681no—
188C4⋊Q8—681no—
189C22⋊Q8—684no—
190C22⋊Q8—684no—
191C4×D4—41no—
192C4×D4—41no—
193C2×C4⋊C4—172no—
194C2×C4⋊C4—341no—
195C2×C4⋊C4—342no—
196C2×C4⋊C4—342no—
197C2×C4⋊C4—341no—
198C2×C4⋊C4D6811yescharacteristic
199C2×C22⋊C4—22no—
200C2×C22⋊C4—342no—
201C2×C22⋊C4—342no—
202C2×C22⋊C4C4×D1711yescharacteristic
203C2×C22⋊C4C22×D1711yescharacteristic
204C2×C42—41no—
205C2×C42—341no—
206C2×C42C4×D1711yescharacteristic
207#99—24no—
208C22⋊C8—342no—
209C22⋊C8—681no—
210#95—41no—
211#95—172no—
212C17#1178811yescharacteristic, complemented, Sylow
213C24#6322711yescharacteristic
214C24C2×C22⋊D1711yescharacteristic
215C2×Q8—682no—
216C2×Q8—682no—
217C2×Q8—681no—
218C2×Q8—681no—
219C2×D4—22no—
220C2×D4—22no—
221C2×D4—41no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—342no—
230C22×C4—342no—
231C22×C4#6322711yescharacteristic
232C22×C4#6322711yescharacteristic
233C22×C4#6322711yescharacteristic
234C22×C4C2×C4×D1711yescharacteristic
235C22×C4C2×D6811yescharacteristic
236M16—1362no—
237C2×C8—682no—
238C4⋊C4—41no—
239C4⋊C4—344no—
240C4⋊C4—682no—
241C4⋊C4—682no—
242C4⋊C4—684no—
243C4⋊C4—684no—
244C22⋊C4—24no—
245C22⋊C4—41no—
246C22⋊C4—41no—
247C22⋊C4—42no—
248C22⋊C4—684no—
249C22⋊C4—684no—
250C42—21no—
251C42—21no—
252C42—82no—
253C42—682no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23C2×#6322711yescharacteristic
264C23#1058763712yes—
265Q8—1362no—
266Q8—1362no—
267D4—44no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—22no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—44no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4—82no—
283C2×C4—82no—
284C2×C4—344no—
285C2×C4—682no—
286C2×C4—682no—
287C2×C4—682no—
288C2×C4—682no—
289C2×C4—682no—
290C2×C4#1058747612yes—
291C2×C4#1058747711yescharacteristic
292C2×C4#1058747711yescharacteristic
293C8—1362no—
294C22—21no—
295C22—21no—
296C22—22no—
297C22—42no—
298C22—42no—
299C22—41no—
300C22—41no—
301C22—41no—
302C22—41no—
303C22—41no—
304C22—81no—
305C22—81no—
306C22C2×#1058747711yescharacteristic
307C22#1176281711yescharacteristic
308C22C2×#1058747611yescharacteristic
309C4—22no—
310C4—21no—
311C4—41no—
312C4—41no—
313C4—41no—
314C4—84no—
315C4—684no—
316C4—1362no—
317C4#1176260112yes—
318C2—21no—
319C2—21no—
320C2—42no—
321C2—81no—
322C2#1452730311yescharacteristic
323C2#1453043711yescharacteristic
324C2C2×#1176260111yescharacteristic
subgroupG/H
centerC22C2×#10587477
commutatorC22×C4×C17C22×C4
FrattiniC2×C22⋊C4C22×D17
FittingC4×C17×#99C2
socleC22×C17C2×#99
prime radicalC22×D4×C17D4
cosocleC2×C22⋊C4×C17C23
hypercenterC4×#99D17
hypocenterC17#11788
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#99

Complements

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

NH ≅ G/Ntype
3. #14526807C2semidirect
5. C2×#10587555C4semidirect
9. #11762745C4semidirect
13. C4×C4⋊Dic17C4semidirect
212. C17#11788semidirect

Maximal subgroups

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

subgroupindex
#145337372
#145271222
#145268072
C4×C17×#992
#1178817