Finite Group Databasebeta

← #17959480

Subgroups — #17959480

Order 4352.

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

normal subgroups only

This group has 586 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
1#14544683C211yesmaximal, characteristic
2#14544034C211yesmaximal, characteristic
3#14543785C214yesmaximal, complemented
4C2×#11762728C211yesmaximal, characteristic
5#11763779—21no—
6#11763779—21no—
7#11763689—21no—
8#11763689C2212yes—
9#11763655—21no—
10#11763655C2212yes—
11C22×#63225—21no—
12C22×#63225C2211yescharacteristic
13#11762728—21no—
14#11762728—21no—
15#11762728C412yes—
16#11762728C2212yes—
17C2×#10587502—21no—
18C2×#10587502C411yescharacteristic
19C23×Q8×C17C411yescharacteristic
20C22×Q8×C17—21no—
21C22×Q8×C17—44no—
22C22×Q8×C17—41no—
23C22×Q8×C17—41no—
24C22×Q8×C17C2×C412yes—
25C23×C4×C17—21no—
26C23×C4×C17C2×C411yescharacteristic
27C23×Dic17—21no—
28C23×Dic17D411yescharacteristic
29C23×Dic17D411yescharacteristic
30C2×#63247—22no—
31C2×#63247—41no—
32C2×#63225—22no—
33C2×#63225—22no—
34C2×#63225—41no—
35C2×#63225—41no—
36C2×#63225—44no—
37C2×#63225C2×C411yescharacteristic
38C2×#63225D411yescharacteristic
39C2×#63225D411yescharacteristic
40C2×#63225C2311yescharacteristic
41C22⋊#3624—22no—
42C22⋊#3624—41no—
43C22⋊#3624—41no—
44C22⋊#3624—41no—
45#10587502—21no—
46#10587502—22no—
47#10587502—22no—
48#10587502C2×C412yes—
49C24×C17C22⋊C411yescharacteristic
50C2×Q8×C17—44no—
51C2×Q8×C17—44no—
52C2×Q8×C17—42no—
53C2×Q8×C17—42no—
54C2×Q8×C17—48no—
55C2×Q8×C17—44no—
56C2×Q8×C17—44no—
57C22×C4×C17—24no—
58C22×C4×C17—21no—
59C22×C4×C17—21no—
60C22×C4×C17—22no—
61C22×C4×C17—42no—
62C22×C4×C17—41no—
63C22×C4×C17—41no—
64C22×C4×C17—42no—
65C22×C4×C17C22⋊C411yescharacteristic
66C22×C4×C17C22×C411yescharacteristic
67C22×Dic17—24no—
68C22×Dic17—21no—
69C22×Dic17—21no—
70C22×Dic17—21no—
71C22×Dic17—22no—
72C22×Dic17—21no—
73C22×Dic17—22no—
74C22×Dic17—41no—
75C22×Dic17—42no—
76C22×Dic17—42no—
77C22×Dic17—42no—
78C22×Dic17C22⋊C411yescharacteristic
79C22×Dic17C22⋊C411yescharacteristic
80C22×Dic17C2×D411yescharacteristic
81C22×Dic17C2×D411yescharacteristic
82#63247—42no—
83#63247—42no—
84#63247—82no—
85C2×#3624—42no—
86C2×#3624—42no—
87#63225—41no—
88#63225—41no—
89#63225—41no—
90#63225—41no—
91#63225—44no—
92#63225—44no—
93#11815—171nomaximal, Sylow
94C23×C17—22no—
95C23×C17—22no—
96C23×C17—21no—
97C23×C17—21no—
98C23×C17#9911yescharacteristic
99C23×C17#9911yescharacteristic
100C23×C17C2×C22⋊C411yescharacteristic
101Q8×C17—44no—
102Q8×C17—44no—
103Q8×C17—48no—
104C2×C4×C17—24no—
105C2×C4×C17—21no—
106C2×C4×C17—21no—
107C2×C4×C17—42no—
108C2×C4×C17—44no—
109C2×C4×C17—42no—
110C2×C4×C17—44no—
111C2×C4×C17—42no—
112C2×C4×C17—42no—
113C2×C4×C17—41no—
114C2×C4×C17—41no—
115C2×Dic17—21no—
116C2×Dic17—21no—
117C2×Dic17—24no—
118C2×Dic17—21no—
119C2×Dic17—21no—
120C2×Dic17—21no—
121C2×Dic17—21no—
122C2×Dic17—21no—
123C2×Dic17—42no—
124C2×Dic17—42no—
125C2×Dic17—48no—
126C2×Dic17—42no—
127C2×Dic17—42no—
128C2×Dic17—42no—
129C2×Dic17—41no—
130C2×Dic17—41no—
131C2×Dic17—81no—
132C2×Dic17#13712yes—
133#3624—82no—
134#3624—82no—
135C2×#395—171no—
136#1821—171no—
137#1463—171no—
138#1354—174no—
139C22×C17—24no—
140C22×C17—24no—
141C22×C17—21no—
142C22×C17—21no—
143C22×C17—21no—
144C22×C17—21no—
145C22×C17—21no—
146C22×C17—21no—
147C22×C17AΣL(1,𝔽16)12yes—
148C22×C17#35612yes—
149C22×C17C2×#9911yescharacteristic
150C22×C17#41911yescharacteristic
151C22×C17#42111yescharacteristic
152C4×C17—44no—
153C4×C17—41no—
154C4×C17—41no—
155Dic17—22no—
156Dic17—44no—
157Dic17—41no—
158Dic17—41no—
159Dic17—41no—
160C23×Q8#59511yescharacteristic
161C22×C4⋊C4—171no—
162C22×C4⋊C4—341no—
163#407—341no—
164#407—341no—
165#395—172no—
166#395—172no—
167#395—341no—
168#395—341no—
169C2×#95—171no—
170C2×#95—341no—
171#343—172no—
172#343—341no—
173#324—172no—
174#324—341no—
175C2×C17—21no—
176C2×C17—22no—
177C2×C17—22no—
178C2×C17—21no—
179C2×C17#151111yescharacteristic
180C2×C17C2×AΣL(1,𝔽16)11yescharacteristic
181C2×C17C2×#35611yescharacteristic
182C22×Q8—21no—
183C22×Q8—44no—
184C22×Q8—41no—
185C22×Q8—41no—
186C22×Q8C2×#59512yes—
187C23×C4—21no—
188C23×C4—171no—
189C23×C4—171no—
190C23×C4—341no—
191C23×C4C2×#59511yescharacteristic
192C2×M16—342no—
193C2×M16—681no—
194C2×C4⋊C4—171no—
195C2×C4⋊C4—171no—
196C2×C4⋊C4—171no—
197C2×C4⋊C4—171no—
198C2×C4⋊C4—342no—
199C2×C4⋊C4—342no—
200C2×C4⋊C4—681no—
201C2×C4⋊C4—681no—
202C2×C4⋊C4—684no—
203C22⋊C8—342no—
204C22⋊C8—681no—
205C22⋊C8—681no—
206C22⋊C8—681no—
207#95—172no—
208#95—342no—
209#95—342no—
210#95—341no—
211C17#1181511yescharacteristic, complemented, Sylow
212C24C22⋊#59511yescharacteristic
213C2×Q8—44no—
214C2×Q8—44no—
215C2×Q8—42no—
216C2×Q8—42no—
217C2×Q8—48no—
218C2×Q8—44no—
219C2×Q8—44no—
220C22×C4—24no—
221C22×C4—22no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—42no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—42no—
228C22×C4—171no—
229C22×C4—171no—
230C22×C4—171no—
231C22×C4—171no—
232C22×C4—342no—
233C22×C4—344no—
234C22×C4—342no—
235C22×C4—341no—
236C22×C4—341no—
237C22×C4—341no—
238C22×C4—341no—
239C22×C4—681no—
240C22×C4—682no—
241C22×C4—682no—
242C22×C4—682no—
243C22×C4C22⋊#59511yescharacteristic
244C22×C4C22×#59511yescharacteristic
245M16—682no—
246M16—682no—
247M16—1362no—
248C2×C8—682no—
249C2×C8—682no—
250C4⋊C4—681no—
251C4⋊C4—681no—
252C4⋊C4—681no—
253C4⋊C4—681no—
254C4⋊C4—684no—
255C4⋊C4—684no—
256C23—22no—
257C23—21no—
258C23—21no—
259C23—22no—
260C23#1058755111yescharacteristic
261C23#1058755111yescharacteristic
262C23C2×C22⋊#59511yescharacteristic
263Q8—44no—
264Q8—44no—
265Q8—48no—
266C2×C4—24no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—42no—
270C2×C4—44no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—44no—
274C2×C4—42no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—172no—
278C2×C4—341no—
279C2×C4—341no—
280C2×C4—344no—
281C2×C4—341no—
282C2×C4—341no—
283C2×C4—341no—
284C2×C4—341no—
285C2×C4—341no—
286C2×C4—682no—
287C2×C4—682no—
288C2×C4—682no—
289C2×C4—688no—
290C2×C4—682no—
291C2×C4—682no—
292C2×C4—681no—
293C2×C4—681no—
294C2×C4—1361no—
295C8—1362no—
296C8—1362no—
297C22—24no—
298C22—24no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22#1176370912yes—
306C22#1176371912yes—
307C22C2×#1058755111yescharacteristic
308C22#1176380211yescharacteristic
309C22#1176380811yescharacteristic
310C4—44no—
311C4—41no—
312C4—41no—
313C4—342no—
314C4—684no—
315C4—681no—
316C4—681no—
317C4—681no—
318C2—21no—
319C2—22no—
320C2—21no—
321C2—22no—
322C2#1454419411yescharacteristic
323C2C2×#1176370911yescharacteristic
324C2C2×#1176371911yescharacteristic
subgroupG/H
centerC22C2×#10587551
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC23×Q8×C17C4
socleC22×C17C2×#99
prime radicalC24×C17C22⋊C4
cosocleC2×#63225C23
hypercenterC23×Q8#595
hypocenterC17#11815
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C23×Q8

Complements

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

NH ≅ G/Ntype
3. #14543785C2semidirect
211. C17#11815semidirect

Maximal subgroups

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

subgroupindex
#145446832
#145440342
#145437852
C2×#117627282
#1181517