Finite Group Databasebeta

← #17953564

Subgroups — #17953564

Order 4352.

544 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 100 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#14544676C211yesmaximal, characteristic
2#14543960C211yesmaximal, characteristic
3#14543657C212yesmaximal
4#14543657C212yesmaximal
5#14531180C211yesmaximal, characteristic
6C2×#10587543C2212yes—
7C2×(C2×#595)⋊C4—21no—
8C2×(C2×#595)⋊C4—21no—
9C2×(C2×#595)⋊C4C2212yes—
10#11763675—21no—
11#11763675—21no—
12C2×C42×D17C2211yescharacteristic
13#11762676—21no—
14#11762674C411yescharacteristic, complemented
15#11762674C411yescharacteristic, complemented
16#11762671—21no—
17#11762666C2211yescharacteristic
18#11762666C2211yescharacteristic
19C4×C4⋊Dic17C411yescharacteristic, complemented
20C4×#63227—21no—
21#11762655—21no—
22C4×C4⋊C4×C17C411yescharacteristic, complemented
23C23×#595—22no—
24C22×C4×D17D411yescharacteristic
25C22×C4×D17D411yescharacteristic
26C22×C4×D17C2311yescharacteristic
27C22×#3624—22no—
28C2×C4⋊C4×C17C2×C411yescharacteristic
29C2×C4⋊C4×C17C2×C411yescharacteristic
30C2×C42×C17—21no—
31C2×C42×C17C2×C411yescharacteristic
32C2×#63227—21no—
33C2×#63227—21no—
34C2×C4⋊Dic17C2×C411yescharacteristic
35C2×C4⋊Dic17C2×C411yescharacteristic
36C2×C4×Dic17—21no—
37C2×C4×Dic17C2×C411yescharacteristic
38C2×C4×Dic17D411yescharacteristic
39C2×C4×Dic17D411yescharacteristic
40C42×D17—24no—
41(C2×#595)⋊C4—24no—
42(C2×#595)⋊C4—24no—
43(C2×#595)⋊C4—42no—
44#10587543—24no—
45C17×#95—21no—
46#10587502—21no—
47#10587502—21no—
48#10587502—21no—
49#10587502—21no—
50#10587502—21no—
51#10587502—21no—
52#10587502—21no—
53C23×D17C2×D411yescharacteristic
54C22×#595—24no—
55C22×#595—42no—
56C22×#595—42no—
57C22×#595—42no—
58C22×#595—42no—
59C22×#595—42no—
60C22×#595—42no—
61C22×C4×C17—21no—
62C22×C4×C17—21no—
63C22×C4×C17C22⋊C411yescharacteristic
64C22×C4×C17C22⋊C411yescharacteristic
65C22×C4×C17C22×C411yescharacteristic
66C22×Dic17—21no—
67C22×Dic17—21no—
68C22×Dic17C22⋊C411yescharacteristic
69C22×Dic17C22⋊C411yescharacteristic
70C22×Dic17C2×D411yescharacteristic
71C2×C4×D17—22no—
72C2×C4×D17—22no—
73C2×C4×D17—22no—
74C2×C4×D17—21no—
75C2×C4×D17—21no—
76C2×C4×D17—22no—
77C2×C4×D17—41no—
78C2×C4×D17—41no—
79C2×C4×D17—41no—
80C2×C4×D17#4114yes—
81C2×#3624—42no—
82C2×#3624—42no—
83C2×#3624—42no—
84C4⋊C4×C17—41no—
85C4⋊C4×C17—41no—
86C42×C17—22no—
87C42×C17—42no—
88#63227—42no—
89#63227—41no—
90#63227—41no—
91C4⋊Dic17—41no—
92C4⋊Dic17—41no—
93C4×Dic17—22no—
94C4×Dic17—22no—
95C4×Dic17—22no—
96C4×Dic17—42no—
97#8714—171nomaximal, Sylow
98C23×C17C2×C22⋊C411yescharacteristic
99C22×D17—21no—
100C22×D17—21no—
101C22×D17—21no—
102C22×D17—21no—
103C22×D17#12312yes—
104C22×D17#13512yes—
105C22×D17AGL(1,ℤ/8)11yescharacteristic
106C22×D17#13711yescharacteristic
107C2×#595—44no—
108C2×#595—44no—
109C2×#595—44no—
110C2×#595—82no—
111C2×#595—82no—
112C2×#595—82no—
113C2×#595—82no—
114C2×C4×C17—21no—
115C2×C4×C17—21no—
116C2×C4×C17—21no—
117C2×C4×C17—21no—
118C2×C4×C17—21no—
119C2×C4×C17—21no—
120C2×C4×C17—21no—
121C2×C4×C17—41no—
122C2×C4×C17—41no—
123C2×C4×C17—41no—
124C2×C4×C17—41no—
125C2×C4×C17—41no—
126C2×C4×C17—41no—
127C2×C4×C17#10412yes—
128C2×C4×C17#11712yes—
129C2×Dic17—21no—
130C2×Dic17—21no—
131C2×Dic17—21no—
132C2×Dic17—21no—
133C2×Dic17—21no—
134C2×Dic17—21no—
135C2×Dic17—21no—
136C2×Dic17—21no—
137C2×Dic17—41no—
138C2×Dic17—41no—
139C2×Dic17—41no—
140C2×Dic17—41no—
141C2×Dic17—41no—
142C2×Dic17—41no—
143C2×Dic17#12312yes—
144C4×D17—44no—
145C4×D17—44no—
146C4×D17—42no—
147C4×D17—42no—
148#3624—82no—
149#2283—171no—
150#1817—171no—
151#1434—171no—
152#1294—172no—
153#1294—172no—
154C22×C17#38111yescharacteristic
155C22×C17#41011yescharacteristic
156C22×C17#41011yescharacteristic
157C22×C17#41611yescharacteristic
158C22×C17#41711yescharacteristic
159C22×C17C2×#10411yescharacteristic
160C22×C17#42111yescharacteristic
161C2×D17—22no—
162C2×D17—22no—
163C2×D17—22no—
164C2×D17—21no—
165C2×D17—21no—
166C2×D17—21no—
167C2×D17—21no—
168C2×D17—22no—
169C2×D17#48212yes—
170C2×D17#48512yes—
171#595—84no—
172C4×C17—22no—
173C4×C17—22no—
174C4×C17—41no—
175C4×C17—81no—
176C4×C17—81no—
177Dic17—22no—
178Dic17—41no—
179Dic17—41no—
180Dic17—81no—
181Dic17—81no—
182C22×C42—171no—
183C2×C22⋊C8—172no—
184#383—341no—
185#382—171no—
186#382—171no—
187#381—341no—
188#380—171no—
189#380—171no—
190C4×C4⋊C4—171no—
191C4×C4⋊C4#59511yescharacteristic, complemented
192C4×C22⋊C4—341no—
193#376—341no—
194C2×#95—172no—
195C2×#95—341no—
196C2×#95—341no—
197#336—341no—
198#336—341no—
199C2×C17#150011yescharacteristic
200C2×C17#150911yescharacteristic
201C2×C17#151211yescharacteristic
202C2×C17#152111yescharacteristic
203C2×C17#182011yescharacteristic
204C2×C17#182511yescharacteristic
205C2×C17#182611yescharacteristic
206D17—22no—
207D17—22no—
208C23×C4—171no—
209C23×C4—171no—
210C23×C4—171no—
211C23×C4—342no—
212C22×C8—342no—
213C2×C4⋊C4—171no—
214C2×C4⋊C4—171no—
215C2×C4⋊C4C2×#59511yescharacteristic
216C2×C4⋊C4C2×#59511yescharacteristic
217C2×C22⋊C4—341no—
218C2×C22⋊C4—341no—
219C2×C42—21no—
220C2×C42—171no—
221C2×C42—171no—
222C2×C42—171no—
223C2×C42—344no—
224C2×C42—341no—
225C2×C42C2×#59511yescharacteristic
226C22⋊C8—344no—
227#95—21no—
228#95—341no—
229#95—341no—
230#95—341no—
231#95—341no—
232#95—341no—
233#95—341no—
234#95—344no—
235#95—344no—
236#95—341no—
237#95—682no—
238C17#871411yescharacteristic, complemented, Sylow
239C24—171no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—171no—
243C22×C4—174no—
244C22×C4—171no—
245C22×C4—171no—
246C22×C4—341no—
247C22×C4—342no—
248C22×C4—342no—
249C22×C4—342no—
250C22×C4—341no—
251C22×C4—342no—
252C22×C4—344no—
253C22×C4—341no—
254C22×C4—341no—
255C22×C4—681no—
256C22×C4—681no—
257C22×C4—681no—
258C22×C4—682no—
259C22×C4—682no—
260C22×C4—682no—
261C22×C4—682no—
262C22×C4—682no—
263C22×C4—682no—
264C22×C4C22⋊#59511yescharacteristic
265C22×C4C22⋊#59511yescharacteristic
266C22×C4C22×#59511yescharacteristic
267C2×C8—682no—
268C2×C8—682no—
269C2×C8—682no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—681no—
273C4⋊C4—681no—
274C22⋊C4—682no—
275C22⋊C4—681no—
276C22⋊C4—681no—
277C42—22no—
278C42—42no—
279C42—342no—
280C42—342no—
281C42—342no—
282C42—682no—
283C23—172no—
284C23—172no—
285C23—171no—
286C23—171no—
287C23—341no—
288C23—341no—
289C23—341no—
290C23—341no—
291C23C2×C22⋊#59511yescharacteristic
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—172no—
306C2×C4—341no—
307C2×C4—341no—
308C2×C4—341no—
309C2×C4—341no—
310C2×C4—341no—
311C2×C4—341no—
312C2×C4—341no—
313C2×C4—341no—
314C2×C4—684no—
315C2×C4—684no—
316C2×C4—682no—
317C2×C4—682no—
318C2×C4—684no—
319C2×C4—684no—
320C2×C4—684no—
321C2×C4—681no—
322C2×C4—681no—
323C2×C4—681no—
324C2×C4—681no—
325C2×C4—681no—
326C2×C4—681no—
327C2×C4—1362no—
328C2×C4—1362no—
329C2×C4—1362no—
330C2×C4—1362no—
331C2×C4#1058754812yes—
332C2×C4#1058767712yes—
333C8—1362no—
334C22—172no—
335C22—172no—
336C22—342no—
337C22—342no—
338C22—342no—
339C22—341no—
340C22—341no—
341C22—341no—
342C22—341no—
343C22—342no—
344C22#1176375211yescharacteristic
345C22#1176378811yescharacteristic
346C22#1176378811yescharacteristic
347C22#1176379911yescharacteristic
348C22#1176380011yescharacteristic
349C22C2×#1058754811yescharacteristic
350C22#1176380811yescharacteristic
351C4—22no—
352C4—22no—
353C4—41no—
354C4—81no—
355C4—81no—
356C4—342no—
357C4—681no—
358C4—681no—
359C4—1364no—
360C4—1361no—
361C4—1361no—
362C2—342no—
363C2—342no—
364C2#1454415511yescharacteristic
365C2#1454418611yescharacteristic
366C2#1454419711yescharacteristic
367C2#1454422811yescharacteristic
368C2#1454468111yescharacteristic
369C2#1454468911yescharacteristic
370C2#1454469011yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC4×C4⋊C4×C17C4
socleC23×C17C2×C22⋊C4
prime radicalC23×D17C2×D4
cosocleC22×C4×D17C23
hypercenterC4×C4⋊C4#595
hypocenterC17#8714
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
14. #11762674C4semidirect
15. #11762674C4semidirect
19. C4×C4⋊Dic17C4semidirect
22. C4×C4⋊C4×C17C4semidirect
191. C4×C4⋊C4#595semidirect
238. C17#8714semidirect

Maximal subgroups

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

subgroupindex
#145446762
#145439602
#145436572
#145311802
#871417