Finite Group Databasebeta

← #16944945

Subgroups — #16944945

Order 4352.

516 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 95 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
1C4×#10587505C211yesmaximal, characteristic
2#14527122C211yesmaximal, characteristic
3#14526808C214yesmaximal, complemented
4C17×#2542C211yesmaximal, characteristic
5C2×#10587505—21no—
6#11762773—21no—
7C4×C22⋊Dic17—21no—
8#11762576C2212yes—
9#11762535—21no—
10#11762533C2212yes—
11C2×C17×C4⋊Q8C411yescharacteristic, complemented
12C2×C17×C22⋊Q8—21no—
13C2×C17×C22⋊Q8—21no—
14C2×C4×D4×C17C2211yescharacteristic
15C17×#399C412yescomplemented
16C17×#399C2212yes—
17C17×#390—21no—
18C17×#384—21no—
19C4×C4⋊C4×C17C411yescharacteristic, complemented
20C22×Q8×C17—21no—
21C22×D4×C17D411yescharacteristic
22C23×C4×C17D411yescharacteristic
23C23×C4×C17D411yescharacteristic
24C17×C4⋊Q8—41no—
25C17×C4⋊Q8—41no—
26C17×C22⋊Q8—44no—
27C17×C22⋊Q8—44no—
28C4×D4×C17—41no—
29C4×D4×C17—41no—
30C2×C4⋊C4×C17—21no—
31C2×C4⋊C4×C17—22no—
32C2×C4⋊C4×C17—21no—
33C2×C4⋊C4×C17—22no—
34C2×C4⋊C4×C17C2×C412yes—
35C2×C4⋊C4×C17D411yescharacteristic
36C2×C22⋊C4×C17—22no—
37C2×C22⋊C4×C17—22no—
38C2×C22⋊C4×C17C2×C411yescharacteristic
39C2×C22⋊C4×C17C2311yescharacteristic
40C2×C42×C17—21no—
41C2×C42×C17C2×C411yescharacteristic
42C2×C22⋊Dic17—22no—
43C2×C4×Dic17—41no—
44C2×#63223—41no—
45C17×#95C2×C412yes—
46#10587505—24no—
47#10587502—41no—
48C22⋊#3622—22no—
49C22⋊#3622—41no—
50C24×C17C22⋊C411yescharacteristic
51C24×C17C2×D411yescharacteristic
52C2×Q8×C17—42no—
53C2×Q8×C17—42no—
54C2×Q8×C17—41no—
55C2×Q8×C17—41no—
56C2×D4×C17—22no—
57C2×D4×C17—22no—
58C2×D4×C17—41no—
59C22×C4×C17—22no—
60C22×C4×C17—21no—
61C22×C4×C17—21no—
62C22×C4×C17—21no—
63C22×C4×C17—21no—
64C22×C4×C17—22no—
65C22×C4×C17—42no—
66C22×C4×C17—42no—
67C22×C4×C17C22⋊C411yescharacteristic
68C22×C4×C17C22⋊C411yescharacteristic
69C22×C4×C17C22⋊C411yescharacteristic
70C22×C4×C17C22×C411yescharacteristic
71C22×C4×C17C2×D411yescharacteristic
72C22×Dic17—42no—
73C4⋊C4×C17—24no—
74C4⋊C4×C17—41no—
75C4⋊C4×C17—42no—
76C4⋊C4×C17—44no—
77C4⋊C4×C17—44no—
78C4⋊C4×C17—42no—
79C22⋊C4×C17—41no—
80C22⋊C4×C17—41no—
81C22⋊C4×C17—44no—
82C22⋊C4×C17—44no—
83C42×C17—21no—
84C42×C17—21no—
85C42×C17—42no—
86C22⋊Dic17—24no—
87C22⋊Dic17—42no—
88C4×Dic17—82no—
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—
103Q8×C17—82no—
104Q8×C17—82no—
105D4×C17—44no—
106C2×C4×C17—21no—
107C2×C4×C17—21no—
108C2×C4×C17—22no—
109C2×C4×C17—24no—
110C2×C4×C17—21no—
111C2×C4×C17—21no—
112C2×C4×C17—21no—
113C2×C4×C17—42no—
114C2×C4×C17—42no—
115C2×C4×C17—42no—
116C2×C4×C17—42no—
117C2×C4×C17—41no—
118C2×C4×C17—41no—
119C2×C4×C17—41no—
120C2×C4×C17—42no—
121C2×C4×C17—42no—
122C2×C4×C17—42no—
123C2×C4×C17—81no—
124C2×C4×C17—81no—
125C2×C4×C17#9911yescharacteristic
126C2×C4×C17#9911yescharacteristic
127C2×C4×C17#10412yes—
128C2×Dic17—44no—
129C2×Dic17—82no—
130C2×Dic17—82no—
131#3622—82no—
132#2542D1711yescharacteristic
133C4×#99—171no—
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—44no—
154C4×C17—41no—
155C4×C17—41no—
156C4×C17—41no—
157C4×C17—82no—
158C4×C17#35312yes—
159Dic17—84no—
160C2×C4⋊Q8Dic1711yescharacteristic, complemented
161C2×C22⋊Q8—21no—
162C2×C22⋊Q8—21no—
163C2×C4×D4C2×D1711yescharacteristic
164C2×#99—341no—
165#407—341no—
166#399Dic1712yescomplemented
167#399C2×D1712yes—
168#390—21no—
169#384—21no—
170C4×C4⋊C4Dic1711yescharacteristic, complemented
171C4×C22⋊C4—341no—
172#342—172no—
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—21no—
183C22×D4C22⋊D1711yescharacteristic
184C23×C4C22⋊D1711yescharacteristic
185C23×C4C22⋊D1711yescharacteristic
186C2×M16—681no—
187C4⋊Q8—41no—
188C4⋊Q8—41no—
189C22⋊Q8—44no—
190C22⋊Q8—44no—
191C4×D4—41no—
192C4×D4—41no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4C2×Dic1712yes—
198C2×C4⋊C4C22⋊D1711yescharacteristic
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—342no—
202C2×C22⋊C4C2×Dic1711yescharacteristic
203C2×C22⋊C4C22×D1711yescharacteristic
204C2×C42—21no—
205C2×C42—681no—
206C2×C42C2×Dic1711yescharacteristic
207#99—344no—
208C22⋊C8—342no—
209C22⋊C8—681no—
210#95—681no—
211#95C2×Dic1712yes—
212C17#1178811yescharacteristic, complemented, Sylow
213C24C22⋊Dic1711yescharacteristic
214C24C2×C22⋊D1711yescharacteristic
215C2×Q8—42no—
216C2×Q8—42no—
217C2×Q8—41no—
218C2×Q8—41no—
219C2×D4—22no—
220C2×D4—22no—
221C2×D4—41no—
222C22×C4—22no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—22no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—682no—
231C22×C4C22⋊Dic1711yescharacteristic
232C22×C4C22⋊Dic1711yescharacteristic
233C22×C4C22⋊Dic1711yescharacteristic
234C22×C4C22×Dic1711yescharacteristic
235C22×C4C2×C22⋊D1711yescharacteristic
236M16—1362no—
237C2×C8—682no—
238C4⋊C4—24no—
239C4⋊C4—41no—
240C4⋊C4—42no—
241C4⋊C4—44no—
242C4⋊C4—44no—
243C4⋊C4—42no—
244C22⋊C4—41no—
245C22⋊C4—41no—
246C22⋊C4—44no—
247C22⋊C4—44no—
248C22⋊C4—344no—
249C22⋊C4—682no—
250C42—21no—
251C42—21no—
252C42—42no—
253C42—1362no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23#1058762812yes—
264C23C2×C22⋊Dic1711yescharacteristic
265Q8—82no—
266Q8—82no—
267D4—44no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—22no—
271C2×C4—24no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—81no—
286C2×C4—81no—
287C2×C4—684no—
288C2×C4—1362no—
289C2×C4—1362no—
290C2×C4#1058750511yescharacteristic
291C2×C4#1058750511yescharacteristic
292C2×C4#1058750812yes—
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×#1058750511yescharacteristic
307C22#1176281811yescharacteristic
308C22C2×#1058750811yescharacteristic
309C4—22no—
310C4—21no—
311C4—44no—
312C4—41no—
313C4—41no—
314C4—41no—
315C4—82no—
316C4—1364no—
317C4#1176260212yes—
318C2—21no—
319C2—21no—
320C2—42no—
321C2—81no—
322C2#1452730411yescharacteristic
323C2#1453043811yescharacteristic
324C2C2×#1176260211yescharacteristic
subgroupG/H
centerC22C2×#10587505
commutatorC22×C4×C17C22×C4
FrattiniC2×C22⋊C4C22×D17
FittingC17×#2542C2
socleC22×C17C2×#99
prime radicalC22×D4×C17D4
cosocleC2×C22⋊C4×C17C23
hypercenter#2542D17
hypocenterC17#11788
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2542

Complements

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

NH ≅ G/Ntype
3. #14526808C2semidirect
11. C2×C17×C4⋊Q8C4semidirect
15. C17×#399C4semidirect
19. C4×C4⋊C4×C17C4semidirect
160. C2×C4⋊Q8Dic17semidirect
166. #399Dic17semidirect
170. C4×C4⋊C4Dic17semidirect
212. C17#11788semidirect

Maximal subgroups

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

subgroupindex
C4×#105875052
#145271222
#145268082
C17×#25422
#1178817