Finite Group Databasebeta

← #17950178

Subgroups — #17950178

Order 4352.

500 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 92 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#14544641C211yesmaximal, characteristic
2#14544448C211yesmaximal, characteristic
3#14543603C212yesmaximal, complemented
4#14543603C212yesmaximal, complemented
5#14537089C211yesmaximal, characteristic
6C22×#63243—21no—
7C2×#10587544—21no—
8C2×#10587543—21no—
9#11763675—21no—
10#11763675C2212yes—
11#11763675C2212yes—
12C2×C42×D17C2211yescharacteristic
13#11762705C2211yescharacteristic
14#11762705C2211yescharacteristic
15#11762690—21no—
16#11762690—21no—
17#11762680—21no—
18#11762680—21no—
19#11762675C411yescharacteristic
20#11762666C411yescharacteristic
21#11762666C411yescharacteristic
22C2×C17×#125C411yescharacteristic
23C22×C4×D17C2×C411yescharacteristic
24C22×C4×D17D411yescharacteristic
25C22×C4×D17D411yescharacteristic
26C22×#3624—22no—
27C22×#3624—22no—
28C2×#63243—24no—
29C2×#63243—42no—
30C2×#63243—42no—
31C17×#125—42no—
32C2×C4⋊C4×C17—21no—
33C2×C4⋊C4×C17—21no—
34C2×C4⋊C4×C17C2×C411yescharacteristic
35C2×C4⋊C4×C17C2×C411yescharacteristic
36C2×C42×C17C2×C411yescharacteristic
37C2×#63227—21no—
38C2×#63227—21no—
39C2×C4⋊Dic17C2×C411yescharacteristic
40C2×C4⋊Dic17C2×C411yescharacteristic
41C2×#63225—21no—
42C2×#63225—21no—
43C2×C4×Dic17D411yescharacteristic
44C2×C4×Dic17D411yescharacteristic
45C2×C4×Dic17C2311yescharacteristic
46C42×D17—42no—
47#10587544—42no—
48#10587543—42no—
49#10587502—21no—
50#10587502—21no—
51#10587502—21no—
52#10587502—21no—
53#10587502—21no—
54#10587502—21no—
55C23×D17C22⋊C411yescharacteristic
56C22×C4×C17—21no—
57C22×C4×C17—21no—
58C22×C4×C17C22⋊C411yescharacteristic
59C22×C4×C17C22⋊C411yescharacteristic
60C22×C4×C17C22×C411yescharacteristic
61C22×Dic17—21no—
62C22×Dic17—21no—
63C22×Dic17C22⋊C411yescharacteristic
64C22×Dic17C2×D411yescharacteristic
65C22×Dic17C2×D411yescharacteristic
66C2×C4×D17—22no—
67C2×C4×D17—22no—
68C2×C4×D17—21no—
69C2×C4×D17—21no—
70C2×C4×D17—22no—
71C2×C4×D17—22no—
72C2×C4×D17—21no—
73C2×C4×D17—21no—
74C2×C4×D17—41no—
75C2×C4×D17—41no—
76C2×C4×D17—41no—
77C2×#3624—24no—
78C2×#3624—42no—
79C2×#3624—42no—
80C2×#3624—42no—
81C2×#3624—42no—
82C2×#3624—42no—
83#63243—48no—
84C4⋊C4×C17—41no—
85C4⋊C4×C17—41no—
86C4⋊C4×C17—42no—
87C4⋊C4×C17—42no—
88C42×C17—22no—
89#63227—42no—
90#63227—42no—
91C4⋊Dic17—22no—
92C4⋊Dic17—22no—
93#63225—41no—
94#63225—41no—
95#63225—41no—
96#63225—41no—
97C4×Dic17—22no—
98C4×Dic17—41no—
99C4×Dic17—41no—
100#7711—171nomaximal, Sylow
101C23×C17C2×C22⋊C411yescharacteristic
102C22×D17—21no—
103C22×D17—21no—
104C22×D17—21no—
105C22×D17—21no—
106C22×D17—21no—
107C22×D17—21no—
108C22×D17#10412yes—
109C2×C4×C17—21no—
110C2×C4×C17—21no—
111C2×C4×C17—21no—
112C2×C4×C17—21no—
113C2×C4×C17—21no—
114C2×C4×C17—21no—
115C2×C4×C17—21no—
116C2×C4×C17—41no—
117C2×C4×C17—41no—
118C2×C4×C17—41no—
119C2×C4×C17—41no—
120C2×C4×C17—41no—
121C2×C4×C17—41no—
122C2×C4×C17#9912yes—
123C2×C4×C17#13112yes—
124C2×Dic17—21no—
125C2×Dic17—21no—
126C2×Dic17—21no—
127C2×Dic17—21no—
128C2×Dic17—21no—
129C2×Dic17—21no—
130C2×Dic17—41no—
131C2×Dic17—41no—
132C2×Dic17—41no—
133C2×Dic17—41no—
134C2×Dic17—41no—
135C2×Dic17—41no—
136C2×Dic17#10112yes—
137C2×Dic17AGL(1,ℤ/8)11yescharacteristic
138C2×Dic17AGL(1,ℤ/8)11yescharacteristic
139C2×Dic17#13711yescharacteristic
140C2×Dic17#13711yescharacteristic
141C4×D17—24no—
142C4×D17—24no—
143C4×D17—44no—
144C4×D17—42no—
145C4×D17—42no—
146#3624—44no—
147#3624—82no—
148#2851—171no—
149#1797—171no—
150#1728—171no—
151#1271—172no—
152#1271—172no—
153C22×C17#40811yescharacteristic
154C22×C17C2×#9911yescharacteristic
155C22×C17C2×#10111yescharacteristic
156C22×C17#41711yescharacteristic
157C22×C17#41711yescharacteristic
158C22×C17C2×#10411yescharacteristic
159C22×C17#42111yescharacteristic
160C2×D17—22no—
161C2×D17—22no—
162C2×D17—22no—
163C2×D17—41no—
164C2×D17—41no—
165C2×D17—41no—
166C2×D17—41no—
167C4×C17—22no—
168C4×C17—22no—
169C4×C17—41no—
170C4×C17—81no—
171C4×C17—81no—
172Dic17—22no—
173Dic17—21no—
174Dic17—21no—
175Dic17—21no—
176Dic17—21no—
177Dic17—81no—
178Dic17—81no—
179C22×M16—341no—
180C2×#125#59511yescharacteristic
181C22×C42—171no—
182C2×C4⋊C8—341no—
183C2×C22⋊C8—341no—
184#389—171no—
185#389—171no—
186#387—341no—
187#387—341no—
188#385—341no—
189#385—341no—
190#382—171no—
191#380—171no—
192#380—171no—
193#336—172no—
194#336—172no—
195#336—341no—
196C2×C17#147211yescharacteristic
197C2×C17#148311yescharacteristic
198C2×C17#149511yescharacteristic
199C2×C17#150411yescharacteristic
200C2×C17#151311yescharacteristic
201C2×C17#152811yescharacteristic
202C2×C17#156111yescharacteristic
203D17—42no—
204C23×C4—171no—
205C23×C4—171no—
206C23×C4—171no—
207C2×M16—344no—
208C2×M16—682no—
209C2×M16—682no—
210C22×C8—342no—
211C22×C8—342no—
212#125—42no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—171no—
216C2×C4⋊C4—171no—
217C2×C4⋊C4—341no—
218C2×C4⋊C4—341no—
219C2×C4⋊C4C2×#59511yescharacteristic
220C2×C4⋊C4C2×#59511yescharacteristic
221C2×C22⋊C4—341no—
222C2×C22⋊C4—341no—
223C2×C42—171no—
224C2×C42—171no—
225C2×C42—171no—
226C2×C42—682no—
227C2×C42C2×#59511yescharacteristic
228C4⋊C8—682no—
229C22⋊C8—682no—
230#95—341no—
231#95—341no—
232#95—341no—
233#95—341no—
234#95—341no—
235#95—341no—
236C17#771111yescharacteristic, complemented, Sylow
237C24—171no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—171no—
241C22×C4—171no—
242C22×C4—171no—
243C22×C4—341no—
244C22×C4—341no—
245C22×C4—341no—
246C22×C4—342no—
247C22×C4—341no—
248C22×C4—342no—
249C22×C4—342no—
250C22×C4—342no—
251C22×C4—341no—
252C22×C4—341no—
253C22×C4—681no—
254C22×C4—681no—
255C22×C4—681no—
256C22×C4C22⋊#59511yescharacteristic
257C22×C4C22⋊#59511yescharacteristic
258C22×C4C22×#59511yescharacteristic
259M16—688no—
260C2×C8—344no—
261C2×C8—682no—
262C2×C8—682no—
263C2×C8—682no—
264C2×C8—682no—
265C2×C8—682no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—42no—
269C4⋊C4—42no—
270C4⋊C4—342no—
271C4⋊C4—342no—
272C4⋊C4—681no—
273C4⋊C4—681no—
274C4⋊C4—681no—
275C4⋊C4—681no—
276C22⋊C4—682no—
277C22⋊C4—682no—
278C42—22no—
279C42—342no—
280C42—681no—
281C42—681no—
282C23—172no—
283C23—341no—
284C23—341no—
285C23—341no—
286C23—341no—
287C23—341no—
288C23—341no—
289C23C2×C22⋊#59511yescharacteristic
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—172no—
304C2×C4—171no—
305C2×C4—171no—
306C2×C4—171no—
307C2×C4—171no—
308C2×C4—344no—
309C2×C4—344no—
310C2×C4—341no—
311C2×C4—341no—
312C2×C4—341no—
313C2×C4—341no—
314C2×C4—341no—
315C2×C4—341no—
316C2×C4—684no—
317C2×C4—682no—
318C2×C4—682no—
319C2×C4—681no—
320C2×C4—681no—
321C2×C4—681no—
322C2×C4—681no—
323C2×C4—681no—
324C2×C4—681no—
325C2×C4#1058755112yes—
326C2×C4#1058768012yes—
327C8—684no—
328C8—1362no—
329C22—342no—
330C22—342no—
331C22—342no—
332C22—681no—
333C22—681no—
334C22—681no—
335C22—681no—
336C22#1176378411yescharacteristic
337C22C2×#1058755111yescharacteristic
338C22D17⋊#10111yescharacteristic
339C22#1176380011yescharacteristic
340C22#1176380011yescharacteristic
341C22D17⋊#10411yescharacteristic
342C22#1176380611yescharacteristic
343C4—22no—
344C4—22no—
345C4—41no—
346C4—81no—
347C4—81no—
348C4—341no—
349C4—341no—
350C4—341no—
351C4—341no—
352C4—342no—
353C4—1361no—
354C4—1361no—
355C2—682no—
356C2#1454406411yescharacteristic
357C2#1454410211yescharacteristic
358C2#1454414211yescharacteristic
359C2#1454417011yescharacteristic
360C2#1454419911yescharacteristic
361C2#1454424711yescharacteristic
362C2#1454432411yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC2×C17×#125C4
socleC23×C17C2×C22⋊C4
prime radicalC23×D17C22⋊C4
cosocleC2×C4×Dic17C23
hypercenterC2×#125#595
hypocenterC17#7711
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#125

Complements

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

NH ≅ G/Ntype
3. #14543603C2semidirect
4. #14543603C2semidirect
236. C17#7711semidirect

Maximal subgroups

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

subgroupindex
#145446412
#145444482
#145436032
#145370892
#771117