Finite Group Databasebeta

← #17959394

Subgroups — #17959394

Order 4352.

462 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 70 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×#10587551C211yesmaximal, characteristic
2#14544043C211yesmaximal, characteristic
3#14543781C212yesmaximal, complemented
4#14543779C212yesmaximal, complemented
5#14533966C211yesmaximal, characteristic
6#11763830—21no—
7C2×#10587551—21no—
8C4×C22⋊#595—21no—
9#11763687C2212yes—
10#11763653C2212yes—
11#11763652—21no—
12C2×C4×C22⋊D17C2211yescharacteristic
13#11762747C411yescharacteristic, complemented
14#11762744C2211yescharacteristic
15#11762728C411yescharacteristic, complemented
16#11762720C2211yescharacteristic
17#11762690—21no—
18#11762686—21no—
19#11762680—21no—
20#11762667—21no—
21C2×#10587502C411yescharacteristic, complemented
22C2×C17×C22⋊Q8C411yescharacteristic, complemented
23C22×Q8×C17C2×C411yescharacteristic
24C23×C4×C17C2×C411yescharacteristic
25C22×C22⋊D17D411yescharacteristic
26C23×Dic17—21no—
27C22×C4×D17D411yescharacteristic
28C2×C22⋊#595—22no—
29C2×C4×#595—41no—
30C17×C22⋊Q8—41no—
31C17×C22⋊Q8—41no—
32C2×C4⋊C4×C17—21no—
33C2×C4⋊C4×C17C2×C411yescharacteristic
34C2×C22⋊C4×C17—21no—
35C2×C22⋊Dic17D411yescharacteristic
36C4×C22⋊D17—41no—
37C4×C22⋊D17—41no—
38C2×#63227—21no—
39C2×#63227—21no—
40C2×#63227C2311yescharacteristic
41C2×#63225—21no—
42C2×#63225—21no—
43C2×#63225C2×C411yescharacteristic
44C2×C4×Dic17D411yescharacteristic
45C22⋊#3624—22no—
46#10587551—24no—
47(C2×#595)⋊C4—41no—
48#10587542—41no—
49C4⋊#3624—41no—
50#10587502—21no—
51#10587502—21no—
52#10587502—21no—
53#10587502—21no—
54#10587502—21no—
55#10587502C2×C411yescharacteristic
56#10587502C2×C411yescharacteristic
57C24×C17C22⋊C411yescharacteristic
58C23×D17C2×D411yescharacteristic
59C22×#595—42no—
60C2×Q8×C17—41no—
61C2×Q8×C17—81no—
62C22×C4×C17—21no—
63C22×C4×C17—21no—
64C22×C4×C17—21no—
65C22×C4×C17—21no—
66C22×C4×C17—81no—
67C22×C4×C17C22⋊C411yescharacteristic
68C22×C4×C17C22×C411yescharacteristic
69C2×C22⋊D17—22no—
70C2×C22⋊D17—22no—
71C2×C22⋊D17—41no—
72C22×Dic17—21no—
73C22×Dic17—21no—
74C22×Dic17—42no—
75C22×Dic17—42no—
76C22×Dic17—81no—
77C22×Dic17C22⋊C411yescharacteristic
78C22×Dic17C22⋊C411yescharacteristic
79C22×Dic17C2×D411yescharacteristic
80C2×C4×D17—21no—
81C2×C4×D17—21no—
82C2×C4×D17—81no—
83C22⋊#595—24no—
84C22⋊#595—42no—
85C2×#3624—42no—
86C4×#595—82no—
87C4⋊C4×C17—41no—
88C4⋊C4×C17—41no—
89C4⋊C4×C17—41no—
90C22⋊C4×C17—42no—
91C22⋊Dic17—41no—
92#63227—41no—
93#63227—42no—
94#63227—81no—
95#63225—41no—
96#63225—42no—
97#63225—81no—
98C4×Dic17—21no—
99C4×Dic17—21no—
100#11789—171nomaximal, Sylow
101C23×C17—21no—
102C23×C17—21no—
103C23×C17—41no—
104C23×C17—41no—
105C23×C17#10412yes—
106C23×C17C2×C22⋊C411yescharacteristic
107C22×D17—21no—
108C22×D17—21no—
109C22×D17—41no—
110C22×D17—41no—
111C22×D17#13712yes—
112C2×#595—44no—
113C2×#595—82no—
114C2×#595—82no—
115Q8×C17—82no—
116C2×C4×C17—21no—
117C2×C4×C17—41no—
118C2×C4×C17—41no—
119C2×C4×C17—41no—
120C2×C4×C17—41no—
121C2×C4×C17—41no—
122C2×C4×C17—41no—
123C2×C4×C17—41no—
124C2×C4×C17—41no—
125C2×C4×C17—81no—
126C2×C4×C17—81no—
127C2×C4×C17#9911yescharacteristic
128C2×C4×C17#9911yescharacteristic
129C22⋊D17—44no—
130C2×Dic17—22no—
131C2×Dic17—21no—
132C2×Dic17—21no—
133C2×Dic17—21no—
134C2×Dic17—21no—
135C2×Dic17—21no—
136C2×Dic17—41no—
137C2×Dic17—41no—
138C2×Dic17—41no—
139C2×Dic17—41no—
140C2×Dic17—41no—
141C2×Dic17—41no—
142C2×Dic17—41no—
143C2×Dic17—84no—
144C4×D17—42no—
145C4×D17—81no—
146#3624—82no—
147#2564—171no—
148C4×#99—171no—
149#1466—171no—
150#1352—172no—
151#1351—172no—
152C22×C17—21no—
153C22×C17—21no—
154C22×C17—22no—
155C22×C17—42no—
156C22×C17—42no—
157C22×C17—41no—
158C22×C17—41no—
159C22×C17C2×#9911yescharacteristic
160C22×C17#41911yescharacteristic
161C22×C17C2×#10411yescharacteristic
162C2×D17—22no—
163C2×D17—41no—
164C2×D17—41no—
165C2×D17—42no—
166C2×D17—81no—
167#595—84no—
168C4×C17—21no—
169C4×C17—21no—
170C4×C17—41no—
171C4×C17—81no—
172C4×C17—81no—
173Dic17—42no—
174Dic17—41no—
175Dic17—41no—
176Dic17—41no—
177Dic17—81no—
178C2×C22⋊Q8#59511yescharacteristic, complemented
179C2×C4×D4—171no—
180#432—341no—
181C2×#99—341no—
182#400—171no—
183#399—171no—
184#395—171no—
185#393—171no—
186#387—341no—
187#386—341no—
188#385—341no—
189#380—341no—
190C4×C22⋊C4—341no—
191C2×#95—171no—
192#342—172no—
193#323—172no—
194#323—341no—
195C2×C17—21no—
196C2×C17—21no—
197C2×C17—42no—
198C2×C17#150811yescharacteristic
199C2×C17#212111yescharacteristic
200C2×C17#212611yescharacteristic
201D17—42no—
202C22×Q8C2×#59511yescharacteristic
203C22×D4—171no—
204C23×C4—171no—
205C23×C4—341no—
206C23×C4C2×#59511yescharacteristic
207C22⋊Q8—41no—
208C22⋊Q8—41no—
209C4×D4—681no—
210C4×D4—681no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—171no—
213C2×C4⋊C4—341no—
214C2×C4⋊C4—341no—
215C2×C4⋊C4C2×#59511yescharacteristic
216C2×C22⋊C4—21no—
217C2×C22⋊C4—171no—
218C2×C22⋊C4—171no—
219C2×C22⋊C4—342no—
220C2×C22⋊C4—341no—
221C2×C22⋊C4—341no—
222C2×C42—171no—
223C2×C42—681no—
224C4⋊C8—681no—
225#99—344no—
226C22⋊C8—342no—
227#97—681no—
228#95—171no—
229#95—171no—
230#95—341no—
231#95—341no—
232#95—341no—
233#95—341no—
234#95—341no—
235#95—681no—
236C17#1178911yescharacteristic, complemented, Sylow
237C24—171no—
238C24C22⋊#59511yescharacteristic
239C2×Q8—41no—
240C2×Q8—81no—
241C2×D4—342no—
242C2×D4—342no—
243C2×D4—681no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—81no—
249C22×C4—171no—
250C22×C4—171no—
251C22×C4—171no—
252C22×C4—341no—
253C22×C4—341no—
254C22×C4—341no—
255C22×C4—341no—
256C22×C4—682no—
257C22×C4—682no—
258C22×C4—682no—
259C22×C4—1361no—
260C22×C4—1361no—
261C22×C4C22⋊#59511yescharacteristic
262C22×C4C22×#59511yescharacteristic
263C2×C8—682no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—681no—
268C4⋊C4—682no—
269C4⋊C4—1361no—
270C22⋊C4—42no—
271C22⋊C4—344no—
272C22⋊C4—681no—
273C22⋊C4—682no—
274C22⋊C4—681no—
275C22⋊C4—682no—
276C22⋊C4—1361no—
277C42—341no—
278C42—341no—
279C42—1362no—
280C23—21no—
281C23—21no—
282C23—41no—
283C23—41no—
284C23—172no—
285C23—341no—
286C23—341no—
287C23—681no—
288C23—681no—
289C23#1058755012yes—
290C23C2×C22⋊#59511yescharacteristic
291Q8—82no—
292D4—684no—
293C2×C4—21no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—81no—
303C2×C4—81no—
304C2×C4—342no—
305C2×C4—341no—
306C2×C4—341no—
307C2×C4—341no—
308C2×C4—341no—
309C2×C4—341no—
310C2×C4—682no—
311C2×C4—681no—
312C2×C4—681no—
313C2×C4—681no—
314C2×C4—681no—
315C2×C4—681no—
316C2×C4—681no—
317C2×C4—684no—
318C2×C4—681no—
319C2×C4—1361no—
320C2×C4—1364no—
321C2×C4—1362no—
322C2×C4—1362no—
323C2×C4#1058755111yescharacteristic
324C2×C4#1058755111yescharacteristic
325C8—1362no—
326C22—21no—
327C22—21no—
328C22—22no—
329C22—42no—
330C22—42no—
331C22—41no—
332C22—41no—
333C22—342no—
334C22—682no—
335C22—681no—
336C22—681no—
337C22—1361no—
338C22C2×#1058755111yescharacteristic
339C22#1176380211yescharacteristic
340C22C2×#1058755011yescharacteristic
341C4—21no—
342C4—21no—
343C4—41no—
344C4—81no—
345C4—81no—
346C4—682no—
347C4—681no—
348C4—681no—
349C4—681no—
350C4—1361no—
351C4—1364no—
352C2—21no—
353C2—21no—
354C2—42no—
355C2—682no—
356C2#1454418411yescharacteristic
357C2#1454503911yescharacteristic
358C2#1454505511yescharacteristic
subgroupG/H
centerC22C2×#10587551
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC2×C17×C22⋊Q8C4
socleC22×C17C2×#99
prime radicalC22×C22⋊D17D4
cosocleC2×#63227C23
hypercenterC2×C22⋊Q8#595
hypocenterC17#11789
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #14543781C2semidirect
4. #14543779C2semidirect
13. #11762747C4semidirect
15. #11762728C4semidirect
21. C2×#10587502C4semidirect
22. C2×C17×C22⋊Q8C4semidirect
178. C2×C22⋊Q8#595semidirect
236. C17#11789semidirect

Maximal subgroups

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

subgroupindex
C4×#105875512
#145440432
#145437812
#145437792
#145339662
#1178917