Finite Group Databasebeta

← #17959390

Subgroups — #17959390

Order 4352.

516 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 74 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×#10587539C211yesmaximal, characteristic
2#14544041C211yesmaximal, characteristic
3#14543781C214yesmaximal, complemented
4#14533738C211yesmaximal, characteristic
5C2×#10587539—21no—
6#11763780—21no—
7C4×C22⋊#595—21no—
8#11763687C2212yes—
9#11763654—21no—
10#11763653C2212yes—
11C2×#10587632—21no—
12C2×#10587632—21no—
13C2×C4×D68C2211yescharacteristic
14#11762744C412yescomplemented
15#11762744C2212yes—
16#11762710—21no—
17#11762678—21no—
18C4×C4⋊Dic17C411yescharacteristic, complemented
19C2×C17×C4⋊Q8C411yescharacteristic, complemented
20C22×Q8×C17—21no—
21C22×D68D411yescharacteristic
22C22×C4×D17D411yescharacteristic
23C22×C4×D17D411yescharacteristic
24C2×C22⋊#595—22no—
25C2×C4×#595—41no—
26C2×#63243—41no—
27C17×C4⋊Q8—41no—
28C17×C4⋊Q8—41no—
29C2×C4⋊C4×C17—21no—
30C2×C4⋊C4×C17C2×C412yes—
31C2×C42×C17C2×C411yescharacteristic
32#10587632—44no—
33#10587632—44no—
34C2×#63227—22no—
35C2×#63227—22no—
36C2×#63227C2×C411yescharacteristic
37C2×#63227C2311yescharacteristic
38C2×C4⋊Dic17—21no—
39C2×C4⋊Dic17D411yescharacteristic
40C2×#63225—22no—
41C2×#63225—22no—
42C2×C4×Dic17—21no—
43C4×D68—41no—
44C4×D68—41no—
45C22⋊#3624—22no—
46(C2×#595)⋊C4—41no—
47#10587543—41no—
48#10587539—24no—
49#10587502C2×C412yes—
50C23×D17C22⋊C411yescharacteristic
51C23×D17C2×D411yescharacteristic
52C22×#595—42no—
53C2×Q8×C17—42no—
54C2×Q8×C17—42no—
55C2×Q8×C17—41no—
56C2×Q8×C17—41no—
57C22×C4×C17—22no—
58C22×C4×C17C22⋊C411yescharacteristic
59C22×C4×C17C22⋊C411yescharacteristic
60C22×C4×C17C22×C411yescharacteristic
61C22×Dic17—22no—
62C22×Dic17C22⋊C411yescharacteristic
63C22×Dic17C2×D411yescharacteristic
64C2×D68—22no—
65C2×D68—22no—
66C2×D68—41no—
67C2×C4×D17—21no—
68C2×C4×D17—21no—
69C2×C4×D17—21no—
70C2×C4×D17—21no—
71C2×C4×D17—42no—
72C2×C4×D17—42no—
73C22⋊#595—24no—
74C22⋊#595—42no—
75C2×#3624—42no—
76C4×#595—82no—
77#63243—82no—
78C4⋊C4×C17—42no—
79C4⋊C4×C17—42no—
80C42×C17—21no—
81C42×C17—21no—
82#63227—41no—
83#63227—41no—
84#63227—44no—
85#63227—44no—
86C4⋊Dic17—24no—
87C4⋊Dic17—41no—
88#63225—44no—
89#63225—44no—
90C4×Dic17—42no—
91#11788—171nomaximal, Sylow
92C23×C17C2×C22⋊C411yescharacteristic
93C22×D17—21no—
94C22×D17—21no—
95C22×D17—21no—
96C22×D17—21no—
97C22×D17—21no—
98C22×D17—41no—
99C22×D17—41no—
100C22×D17—41no—
101C22×D17—41no—
102C22×D17#13712yes—
103C2×#595—44no—
104C2×#595—82no—
105C2×#595—82no—
106Q8×C17—82no—
107Q8×C17—82no—
108C2×C4×C17—21no—
109C2×C4×C17—21no—
110C2×C4×C17—22no—
111C2×C4×C17—21no—
112C2×C4×C17—41no—
113C2×C4×C17—42no—
114C2×C4×C17—42no—
115C2×C4×C17—42no—
116C2×C4×C17#9911yescharacteristic
117C2×C4×C17#9911yescharacteristic
118C2×C4×C17#10412yes—
119C2×Dic17—24no—
120C2×Dic17—21no—
121C2×Dic17—21no—
122C2×Dic17—42no—
123C2×Dic17—42no—
124C2×Dic17—41no—
125C2×Dic17—41no—
126D68—44no—
127C4×D17—42no—
128C4×D17—42no—
129C4×D17—81no—
130C4×D17—81no—
131#3624—82no—
132#2542—171no—
133C4×#99—171no—
134#1465—171no—
135#1352—174no—
136C22×C17—21no—
137C22×C17—21no—
138C22×C17C2×#9911yescharacteristic
139C22×C17#41911yescharacteristic
140C22×C17C2×#10411yescharacteristic
141C2×D17—22no—
142C2×D17—41no—
143C2×D17—41no—
144C2×D17—42no—
145C2×D17—42no—
146C2×D17—41no—
147C2×D17—41no—
148C2×D17—41no—
149C2×D17—81no—
150C2×D17—81no—
151#595—84no—
152C4×C17—21no—
153C4×C17—22no—
154C4×C17—41no—
155C4×C17—82no—
156C4×C17#35312yes—
157Dic17—44no—
158Dic17—41no—
159Dic17—41no—
160C2×C4⋊Q8#59511yescharacteristic, complemented
161C2×C22⋊Q8—341no—
162C2×C22⋊Q8—341no—
163C2×C4×D4—171no—
164C2×#99—341no—
165#407—341no—
166#399—172no—
167#399—172no—
168#390—341no—
169#384—341no—
170C4×C4⋊C4—171no—
171C4×C22⋊C4—341no—
172#342—172no—
173#324—341no—
174#323—172no—
175C2×C17—21no—
176C2×C17—21no—
177C2×C17#150811yescharacteristic
178C2×C17#212311yescharacteristic
179C2×C17C2×#35311yescharacteristic
180D17—42no—
181D17—81no—
182C22×Q8—21no—
183C22×D4—171no—
184C23×C4—171no—
185C23×C4—171no—
186C2×M16—681no—
187C4⋊Q8—41no—
188C4⋊Q8—41no—
189C22⋊Q8—684no—
190C22⋊Q8—684no—
191C4×D4—681no—
192C4×D4—681no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—171no—
195C2×C4⋊C4—341no—
196C2×C4⋊C4—342no—
197C2×C4⋊C4—342no—
198C2×C4⋊C4C2×#59512yes—
199C2×C22⋊C4—171no—
200C2×C22⋊C4—171no—
201C2×C22⋊C4—342no—
202C2×C22⋊C4—342no—
203C2×C22⋊C4—342no—
204C2×C42—341no—
205C2×C42—681no—
206C2×C42C2×#59511yescharacteristic
207#99—344no—
208C22⋊C8—342no—
209C22⋊C8—681no—
210#95—172no—
211#95—681no—
212C17#1178811yescharacteristic, complemented, Sylow
213C24—171no—
214C24—171no—
215C2×Q8—42no—
216C2×Q8—42no—
217C2×Q8—41no—
218C2×Q8—41no—
219C2×D4—342no—
220C2×D4—342no—
221C2×D4—681no—
222C22×C4—22no—
223C22×C4—171no—
224C22×C4—171no—
225C22×C4—342no—
226C22×C4—341no—
227C22×C4—341no—
228C22×C4—341no—
229C22×C4—341no—
230C22×C4—682no—
231C22×C4—682no—
232C22×C4—682no—
233C22×C4C22⋊#59511yescharacteristic
234C22×C4C22⋊#59511yescharacteristic
235C22×C4C22×#59511yescharacteristic
236M16—1362no—
237C2×C8—682no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—344no—
241C4⋊C4—681no—
242C4⋊C4—684no—
243C4⋊C4—684no—
244C22⋊C4—344no—
245C22⋊C4—681no—
246C22⋊C4—681no—
247C22⋊C4—684no—
248C22⋊C4—684no—
249C22⋊C4—682no—
250C42—21no—
251C42—21no—
252C42—682no—
253C42—1362no—
254C23—172no—
255C23—341no—
256C23—341no—
257C23—341no—
258C23—341no—
259C23—341no—
260C23—681no—
261C23—681no—
262C23—681no—
263C23—681no—
264C23C2×C22⋊#59511yescharacteristic
265Q8—82no—
266Q8—82no—
267D4—684no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—22no—
271C2×C4—21no—
272C2×C4—41no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—344no—
277C2×C4—341no—
278C2×C4—341no—
279C2×C4—682no—
280C2×C4—682no—
281C2×C4—682no—
282C2×C4—682no—
283C2×C4—681no—
284C2×C4—681no—
285C2×C4—684no—
286C2×C4—1361no—
287C2×C4—1361no—
288C2×C4—1362no—
289C2×C4—1362no—
290C2×C4#1058755012yes—
291C2×C4#1058755111yescharacteristic
292C2×C4#1058755111yescharacteristic
293C8—1362no—
294C22—21no—
295C22—21no—
296C22—342no—
297C22—682no—
298C22—682no—
299C22—681no—
300C22—681no—
301C22—681no—
302C22—681no—
303C22—681no—
304C22—1361no—
305C22—1361no—
306C22C2×#1058755111yescharacteristic
307C22#1176380211yescharacteristic
308C22C2×#1058755011yescharacteristic
309C4—22no—
310C4—21no—
311C4—41no—
312C4—82no—
313C4—684no—
314C4—681no—
315C4—681no—
316C4—1364no—
317C4#1176371312yes—
318C2—21no—
319C2—21no—
320C2—682no—
321C2—1361no—
322C2#1454418411yescharacteristic
323C2#1454504611yescharacteristic
324C2C2×#1176371311yescharacteristic
subgroupG/H
centerC22C2×#10587551
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC2×C17×C4⋊Q8C4
socleC22×C17C2×#99
prime radicalC22×D68D4
cosocleC2×#63227C23
hypercenterC2×C4⋊Q8#595
hypocenterC17#11788
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
3. #14543781C2semidirect
14. #11762744C4semidirect
18. C4×C4⋊Dic17C4semidirect
19. C2×C17×C4⋊Q8C4semidirect
160. C2×C4⋊Q8#595semidirect
212. C17#11788semidirect

Maximal subgroups

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

subgroupindex
C4×#105875392
#145440412
#145437812
#145337382
#1178817