Finite Group Databasebeta

← #16927877

Subgroups — #16927877

Order 4352.

472 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 107 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#14531724C211yesmaximal, characteristic
2#14527102C211yesmaximal, characteristic
3#14526662C212yesmaximal, complemented
4#14526651C212yesmaximal, complemented
5C17×#1799C211yesmaximal, characteristic
6C2×C22⋊#3622C2212yes—
7#11762702—21no—
8#11762695—21no—
9#11762689—21no—
10#11762677C411yescharacteristic, complemented
11#11762672C411yescharacteristic, complemented
12#11762669C2211yescharacteristic
13C22⋊C4×Dic17—21no—
14C4×C22⋊Dic17C411yescharacteristic, complemented
15#11762655C411yescharacteristic, complemented
16C2×#10587502C2211yescharacteristic
17#11762535—21no—
18#11762533—21no—
19C2×C4×D4×C17C2211yescharacteristic
20C2×C17×#99—21no—
21C17×#380C2212yes—
22C17×#342—21no—
23C22×D4×C17D411yescharacteristic
24C23×C4×C17C2×C411yescharacteristic
25C23×C4×C17C2311yescharacteristic
26C23×Dic17—21no—
27C4×D4×C17—22no—
28C4×D4×C17—22no—
29C2×C4⋊C4×C17—22no—
30C2×C4⋊C4×C17D411yescharacteristic
31C2×C22⋊C4×C17—22no—
32C2×C22⋊C4×C17D411yescharacteristic
33C2×C22⋊C4×C17D411yescharacteristic
34C2×C42×C17C2×C411yescharacteristic
35C2×C22⋊Dic17—21no—
36C2×C22⋊Dic17—21no—
37C2×C22⋊Dic17—21no—
38C2×C4⋊Dic17—21no—
39C2×C4×Dic17—21no—
40C2×C4×Dic17—21no—
41C22×#3622—22no—
42C17×#99—42no—
43C17×#95—22no—
44#10587502—21no—
45#10587502—21no—
46#10587502—21no—
47#10587502—21no—
48#10587502—21no—
49#10587502C2×C411yescharacteristic
50#10587502C2×C411yescharacteristic
51#10587502C2×C411yescharacteristic
52#10587502C2×C411yescharacteristic
53C22⋊#3622—22no—
54C22⋊#3622—22no—
55C24×C17C22⋊C411yescharacteristic
56C24×C17C2×D411yescharacteristic
57C2×D4×C17—22no—
58C2×D4×C17—22no—
59C2×D4×C17—21no—
60C2×D4×C17—21no—
61C22×C4×C17—21no—
62C22×C4×C17—21no—
63C22×C4×C17—42no—
64C22×C4×C17—81no—
65C22×C4×C17—81no—
66C22×C4×C17C22⋊C411yescharacteristic
67C22×C4×C17C22⋊C411yescharacteristic
68C22×C4×C17C22⋊C411yescharacteristic
69C22×C4×C17C22×C411yescharacteristic
70C22×C4×C17C2×D411yescharacteristic
71C22×C4×C17#4112yes—
72C22×C4×C17#4112yes—
73C22×Dic17—21no—
74C22×Dic17—21no—
75C22×Dic17—21no—
76C22×Dic17—21no—
77C22×Dic17—42no—
78C22×Dic17—41no—
79C22×Dic17—41no—
80C22×Dic17—81no—
81C4⋊C4×C17—22no—
82C4⋊C4×C17—82no—
83C22⋊C4×C17—22no—
84C22⋊C4×C17—22no—
85C22⋊C4×C17—42no—
86C22⋊C4×C17—42no—
87C42×C17—21no—
88C42×C17—21no—
89C22⋊Dic17—42no—
90C22⋊Dic17—42no—
91C22⋊Dic17—81no—
92C4⋊Dic17—81no—
93C4×Dic17—42no—
94C4×Dic17—81no—
95C2×#3622—42no—
96C2×#3622—82no—
97#8762—171nomaximal, Sylow
98C23×C17—21no—
99C23×C17—21no—
100C23×C17—21no—
101C23×C17—41no—
102C23×C17—41no—
103C23×C17—41no—
104C23×C17—41no—
105C23×C17C2×C22⋊C411yescharacteristic
106C23×C17#12312yes—
107C23×C17#13512yes—
108C23×C17AGL(1,ℤ/8)11yescharacteristic
109C23×C17#13711yescharacteristic
110D4×C17—42no—
111D4×C17—42no—
112C2×C4×C17—21no—
113C2×C4×C17—21no—
114C2×C4×C17—21no—
115C2×C4×C17—21no—
116C2×C4×C17—21no—
117C2×C4×C17—42no—
118C2×C4×C17—41no—
119C2×C4×C17—41no—
120C2×C4×C17—42no—
121C2×C4×C17—41no—
122C2×C4×C17—41no—
123C2×C4×C17—81no—
124C2×C4×C17—81no—
125C2×C4×C17—82no—
126C2×C4×C17—82no—
127C2×C4×C17—82no—
128C2×C4×C17#11711yescharacteristic
129C2×C4×C17#11711yescharacteristic
130C2×C4×C17#12312yes—
131C2×Dic17—22no—
132C2×Dic17—41no—
133C2×Dic17—41no—
134C2×Dic17—41no—
135C2×Dic17—41no—
136C2×Dic17—41no—
137C2×Dic17—41no—
138C2×Dic17—41no—
139C2×Dic17—41no—
140C2×Dic17—41no—
141C2×Dic17—41no—
142C2×Dic17—41no—
143C2×Dic17—82no—
144C2×Dic17—82no—
145#3622—82no—
146#2353—171no—
147#1799D1711yescharacteristic
148#1459—171no—
149#1294—172no—
150#1289—172no—
151C22×C17—21no—
152C22×C17—22no—
153C22×C17—21no—
154C22×C17—22no—
155C22×C17—41no—
156C22×C17—41no—
157C22×C17—42no—
158C22×C17—41no—
159C22×C17—41no—
160C22×C17—41no—
161C22×C17—81no—
162C22×C17—81no—
163C22×C17#38111yescharacteristic
164C22×C17#41611yescharacteristic
165C22×C17#41711yescharacteristic
166C22×C17#48212yes—
167C22×C17#48512yes—
168C4×C17—22no—
169C4×C17—21no—
170C4×C17—21no—
171C4×C17—41no—
172C4×C17—41no—
173C4×C17—41no—
174C4×C17—162no—
175Dic17—42no—
176Dic17—81no—
177Dic17—81no—
178Dic17—81no—
179C2×C4×D4C2×D1711yescharacteristic
180C2×#99—21no—
181C2×C22⋊C8—172no—
182#388—341no—
183#387—341no—
184#386—341no—
185#383—171no—
186#381—171no—
187#380—171no—
188#380C2×D1712yes—
189C4×C22⋊C4—171no—
190C4×C22⋊C4—341no—
191#376—171no—
192C2×#95—171no—
193#342—21no—
194#324—341no—
195#323—341no—
196C2×C17—21no—
197C2×C17—21no—
198C2×C17—22no—
199C2×C17—22no—
200C2×C17—81no—
201C2×C17#182511yescharacteristic
202C2×C17#182711yescharacteristic
203C2×C17#182811yescharacteristic
204C22×D4C22⋊D1711yescharacteristic
205C23×C4—341no—
206C23×C4C4×D1711yescharacteristic
207C23×C4C22×D1711yescharacteristic
208C22×C8—342no—
209C4×D4—22no—
210C4×D4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—341no—
213C2×C4⋊C4D6811yescharacteristic
214C2×C22⋊C4—22no—
215C2×C22⋊C4—341no—
216C2×C22⋊C4—341no—
217C2×C22⋊C4—341no—
218C2×C22⋊C4D6811yescharacteristic
219C2×C22⋊C4C22⋊D1711yescharacteristic
220C2×C42—341no—
221C2×C42—341no—
222C2×C42C4×D1711yescharacteristic
223#99—42no—
224C22⋊C8—342no—
225C22⋊C8—342no—
226#95—22no—
227#95—171no—
228#95—171no—
229#95—171no—
230#95—171no—
231#95—341no—
232#95—341no—
233#95—341no—
234#95—341no—
235#95—341no—
236C17#876211yescharacteristic, complemented, Sylow
237C24#6322711yescharacteristic
238C24C2×C22⋊D1711yescharacteristic
239C2×D4—21no—
240C2×D4—22no—
241C2×D4—21no—
242C2×D4—22no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—42no—
246C22×C4—81no—
247C22×C4—81no—
248C22×C4—341no—
249C22×C4—341no—
250C22×C4—341no—
251C22×C4—341no—
252C22×C4—682no—
253C22×C4—681no—
254C22×C4—681no—
255C22×C4—1361no—
256C22×C4#6322711yescharacteristic
257C22×C4#6322711yescharacteristic
258C22×C4#6322711yescharacteristic
259C22×C4C2×C4×D1711yescharacteristic
260C22×C4C2×D6811yescharacteristic
261C22×C4#6325412yes—
262C22×C4#6325412yes—
263C2×C8—682no—
264C2×C8—1362no—
265C4⋊C4—22no—
266C4⋊C4—82no—
267C4⋊C4—1361no—
268C22⋊C4—22no—
269C22⋊C4—22no—
270C22⋊C4—42no—
271C22⋊C4—42no—
272C22⋊C4—682no—
273C22⋊C4—682no—
274C22⋊C4—1361no—
275C42—21no—
276C42—21no—
277C42—682no—
278C42—1361no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23C2×#6322711yescharacteristic
287C23#1058761811yescharacteristic
288C23#1058762112yes—
289C23#1058762811yescharacteristic
290C23#1058763612yes—
291D4—42no—
292D4—42no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—42no—
299C2×C4—41no—
300C2×C4—42no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4—342no—
310C2×C4—681no—
311C2×C4—681no—
312C2×C4—681no—
313C2×C4—681no—
314C2×C4—681no—
315C2×C4—681no—
316C2×C4—681no—
317C2×C4—681no—
318C2×C4—681no—
319C2×C4—681no—
320C2×C4—681no—
321C2×C4—1362no—
322C2×C4—1362no—
323C2×C4#1058756311yescharacteristic
324C2×C4#1058756311yescharacteristic
325C2×C4#1058757212yes—
326C8—1362no—
327C22—21no—
328C22—22no—
329C22—21no—
330C22—22no—
331C22—41no—
332C22—41no—
333C22—42no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—81no—
338C22—81no—
339C22#1176267011yescharacteristic
340C22#1176280311yescharacteristic
341C22#1176280811yescharacteristic
342C22#1176313012yes—
343C22#1176314712yes—
344C4—22no—
345C4—21no—
346C4—21no—
347C4—41no—
348C4—41no—
349C4—41no—
350C4—162no—
351C4—682no—
352C4—1361no—
353C4—1361no—
354C4—1361no—
355C2—22no—
356C2—21no—
357C2—21no—
358C2—22no—
359C2—81no—
360C2#1452887611yescharacteristic
361C2#1452888511yescharacteristic
362C2#1452889011yescharacteristic
subgroupG/H
centerC22#11762670
commutatorC22×C4×C17C22×C4
FrattiniC23×C4C22×D17
FittingC17×#1799C2
socleC22×C17#381
prime radicalC22×D4×C17D4
cosocleC23×C4×C17C23
hypercenter#1799D17
hypocenterC17#8762
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1799

Complements

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

NH ≅ G/Ntype
3. #14526662C2semidirect
4. #14526651C2semidirect
10. #11762677C4semidirect
11. #11762672C4semidirect
14. C4×C22⋊Dic17C4semidirect
15. #11762655C4semidirect
236. C17#8762semidirect

Maximal subgroups

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

subgroupindex
#145317242
#145271022
#145266622
#145266512
C17×#17992
#876217