Finite Group Databasebeta

← #16932116

Subgroups — #16932116

Order 4352.

576 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 115 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#14537813C211yesmaximal, characteristic
2#14533092C211yesmaximal, characteristic
3#14528868C212yesmaximal, complemented
4#14526651C212yesmaximal, complemented
5C17×#1799C211yesmaximal, characteristic
6C2×#10587621—21no—
7C22×C4⋊Dic17C2211yescharacteristic
8#11762881—24no—
9#11762866—21no—
10#11762773—21no—
11C2×C22⋊#3622C2212yes—
12#11762739C412yescomplemented
13#11762734C412yescomplemented
14#11762707—21no—
15#11762670—21no—
16#11762669—21no—
17#11762669C2211yescharacteristic
18#11762571—24no—
19C2×C4×D4×C17C2211yescharacteristic
20C2×C17×#99—21no—
21C17×#380C2212yes—
22C17×#342—21no—
23C22×D4×C17D411yescharacteristic
24C23×C4×C17D411yescharacteristic
25C23×C4×C17C2311yescharacteristic
26C23×Dic17—21no—
27C4×D4×C17—22no—
28C4×D4×C17—22no—
29C2×C4⋊C4×C17—22no—
30C2×C4⋊C4×C17C2×C411yescharacteristic
31C2×C22⋊C4×C17—22no—
32C2×C22⋊C4×C17C2×C411yescharacteristic
33C2×C22⋊C4×C17D411yescharacteristic
34C2×C42×C17D411yescharacteristic
35C2×C22⋊Dic17—22no—
36C2×C22⋊Dic17—21no—
37#10587621—44no—
38C2×C4⋊Dic17—21no—
39C2×C4⋊Dic17—21no—
40C2×C4⋊Dic17—22no—
41C2×C4⋊Dic17—22no—
42C2×C4⋊Dic17C2×C412yes—
43C2×#63225—22no—
44C2×#63223—22no—
45C22×#3622D414yes—
46C17×#99—42no—
47C17×#95—22no—
48#10587502—22no—
49#10587502—22no—
50#10587502C2×C412yes—
51C22⋊#3622—22no—
52C22⋊#3622—22no—
53C22⋊#3622—22no—
54C4⋊#3622—22no—
55#10587474—22no—
56C24×C17C22⋊C411yescharacteristic
57C24×C17C2×D411yescharacteristic
58C2×D4×C17—22no—
59C2×D4×C17—21no—
60C2×D4×C17—22no—
61C2×D4×C17—21no—
62C22×C4×C17—21no—
63C22×C4×C17—21no—
64C22×C4×C17—42no—
65C22×C4×C17—42no—
66C22×C4×C17—42no—
67C22×C4×C17C22⋊C411yescharacteristic
68C22×C4×C17C22⋊C411yescharacteristic
69C22×C4×C17C22⋊C411yescharacteristic
70C22×C4×C17C22×C411yescharacteristic
71C22×C4×C17C2×D411yescharacteristic
72C22×C4×C17C2×D412yes—
73C22×C4×C17#4112yes—
74C22×Dic17—22no—
75C22×Dic17—22no—
76C22×Dic17—42no—
77C22×Dic17—41no—
78C22×Dic17—42no—
79C22×Dic17—41no—
80C4⋊C4×C17—22no—
81C4⋊C4×C17—44no—
82C22⋊C4×C17—22no—
83C22⋊C4×C17—22no—
84C22⋊C4×C17—42no—
85C22⋊C4×C17—42no—
86C42×C17—21no—
87C42×C17—21no—
88C22⋊Dic17—44no—
89C22⋊Dic17—42no—
90C4⋊Dic17—24no—
91C4⋊Dic17—42no—
92C4⋊Dic17—41no—
93C4⋊Dic17—41no—
94#63225—44no—
95#63223—82no—
96C2×#3622—24no—
97C2×#3622—44no—
98#9474—171nomaximal, Sylow
99C23×C17—21no—
100C23×C17—21no—
101C23×C17—21no—
102C23×C17—41no—
103C23×C17—41no—
104C23×C17—41no—
105C23×C17—41no—
106C23×C17C2×C22⋊C411yescharacteristic
107C23×C17#12112yes—
108C23×C17AGL(1,ℤ/8)11yescharacteristic
109C23×C17AGL(1,ℤ/8)12yes—
110C23×C17#13711yescharacteristic
111D4×C17—42no—
112D4×C17—42no—
113C2×C4×C17—24no—
114C2×C4×C17—22no—
115C2×C4×C17—21no—
116C2×C4×C17—21no—
117C2×C4×C17—21no—
118C2×C4×C17—21no—
119C2×C4×C17—21no—
120C2×C4×C17—41no—
121C2×C4×C17—42no—
122C2×C4×C17—42no—
123C2×C4×C17—41no—
124C2×C4×C17—41no—
125C2×C4×C17—81no—
126C2×C4×C17—82no—
127C2×C4×C17—82no—
128C2×C4×C17—82no—
129C2×C4×C17C4×D412yes—
130C2×C4×C17#12011yescharacteristic
131C2×C4×C17#12311yescharacteristic
132C2×Dic17—42no—
133C2×Dic17—42no—
134C2×Dic17—42no—
135C2×Dic17—42no—
136C2×Dic17—42no—
137C2×Dic17—42no—
138C2×Dic17—82no—
139C2×Dic17—82no—
140#3622—44no—
141#2973—171no—
142#2486—171no—
143#1823—172no—
144#1799D1711yescharacteristic
145#1289—172no—
146C22×C17—21no—
147C22×C17—22no—
148C22×C17—21no—
149C22×C17—22no—
150C22×C17—41no—
151C22×C17—42no—
152C22×C17—41no—
153C22×C17—42no—
154C22×C17—41no—
155C22×C17—41no—
156C22×C17—41no—
157C22×C17—81no—
158C22×C17#38611yescharacteristic
159C22×C17#41711yescharacteristic
160C22×C17#41811yescharacteristic
161C22×C17#46812yes—
162C22×C17#46912yes—
163C4×C17—21no—
164C4×C17—22no—
165C4×C17—41no—
166C4×C17—41no—
167C4×C17—41no—
168C4×C17—162no—
169C4×C17#45812yes—
170Dic17—82no—
171Dic17—82no—
172C2×#123—341no—
173C2×C4×D4C2×D1711yescharacteristic
174C22×C4⋊C4—171no—
175#435—344no—
176#432—341no—
177C2×#99—21no—
178#407—341no—
179C2×C22⋊C8—172no—
180#398—172no—
181#397—172no—
182#389—341no—
183#381—341no—
184#380—171no—
185#380—341no—
186#380C2×D1712yes—
187#342—21no—
188#340—344no—
189C2×C17—21no—
190C2×C17—22no—
191C2×C17—21no—
192C2×C17—22no—
193C2×C17—81no—
194C2×C17#189011yescharacteristic
195C2×C17#189511yescharacteristic
196C2×C17#190911yescharacteristic
197C22×D4C22⋊D1711yescharacteristic
198C23×C4—341no—
199C23×C4D6811yescharacteristic
200C23×C4C22×D1711yescharacteristic
201C2×M16—342no—
202C22×C8—174no—
203#123—684no—
204C4×D4—22no—
205C4×D4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—172no—
208C2×C4⋊C4—342no—
209C2×C4⋊C4—342no—
210C2×C4⋊C4—341no—
211C2×C4⋊C4—341no—
212C2×C4⋊C4—342no—
213C2×C4⋊C4C4×D1711yescharacteristic
214C2×C22⋊C4—22no—
215C2×C22⋊C4—341no—
216C2×C22⋊C4—342no—
217C2×C22⋊C4C4×D1711yescharacteristic
218C2×C22⋊C4C22⋊D1711yescharacteristic
219C2×C42D6811yescharacteristic
220C4⋊C8—342no—
221#99—42no—
222C22⋊C8—342no—
223C22⋊C8—342no—
224C22⋊C8—342no—
225#97—342no—
226#95—22no—
227#95—172no—
228#95—342no—
229#95—342no—
230C17#947411yescharacteristic, complemented, Sylow
231C24#6322711yescharacteristic
232C24C2×C22⋊D1711yescharacteristic
233C2×D4—21no—
234C2×D4—22no—
235C2×D4—21no—
236C2×D4—22no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—342no—
243C22×C4—342no—
244C22×C4—681no—
245C22×C4—682no—
246C22×C4—681no—
247C22×C4—682no—
248C22×C4#6322711yescharacteristic
249C22×C4#6322711yescharacteristic
250C22×C4#6322711yescharacteristic
251C22×C4C2×C4×D1711yescharacteristic
252C22×C4C2×D6811yescharacteristic
253C22×C4D4×D1712yes—
254C22×C4#6325412yes—
255M16—1362no—
256C2×C8—344no—
257C2×C8—684no—
258C4⋊C4—22no—
259C4⋊C4—44no—
260C4⋊C4—344no—
261C4⋊C4—684no—
262C4⋊C4—682no—
263C4⋊C4—681no—
264C4⋊C4—681no—
265C22⋊C4—22no—
266C22⋊C4—22no—
267C22⋊C4—42no—
268C22⋊C4—42no—
269C22⋊C4—684no—
270C22⋊C4—682no—
271C42—21no—
272C42—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23C2×#6322711yescharacteristic
281C23#1058761811yescharacteristic
282C23#1058762512yes—
283C23#1058762811yescharacteristic
284C23#1058763512yes—
285D4—42no—
286D4—42no—
287C2×C4—24no—
288C2×C4—22no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—41no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—81no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—82no—
303C2×C4—682no—
304C2×C4—682no—
305C2×C4—682no—
306C2×C4—682no—
307C2×C4—682no—
308C2×C4—682no—
309C2×C4—1362no—
310C2×C4—1362no—
311C2×C4#1058756712yes—
312C2×C4#1058756811yescharacteristic
313C2×C4#1058757211yescharacteristic
314C8—684no—
315C22—21no—
316C22—22no—
317C22—21no—
318C22—22no—
319C22—41no—
320C22—41no—
321C22—42no—
322C22—42no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—81no—
327C22#1176268611yescharacteristic
328C22#1176280811yescharacteristic
329C22#1176281311yescharacteristic
330C22#1176305712yes—
331C22#1176306012yes—
332C4—21no—
333C4—22no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—162no—
338C4—1362no—
339C4—1362no—
340C4#1176300012yes—
341C2—22no—
342C2—21no—
343C2—21no—
344C2—22no—
345C2—81no—
346C2#1452919511yescharacteristic
347C2#1452922011yescharacteristic
348C2#1452930411yescharacteristic
subgroupG/H
centerC22#11762686
commutatorC22×C4×C17C22×C4
FrattiniC23×C4C22×D17
FittingC17×#1799C2
socleC22×C17#386
prime radicalC22×D4×C17D4
cosocleC23×C4×C17C23
hypercenter#1799D17
hypocenterC17#9474
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. #14528868C2semidirect
4. #14526651C2semidirect
12. #11762739C4semidirect
13. #11762734C4semidirect
230. C17#9474semidirect

Maximal subgroups

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

subgroupindex
#145378132
#145330922
#145288682
#145266512
C17×#17992
#947417