Finite Group Databasebeta

← #16932276

Subgroups — #16932276

Order 4352.

682 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 682 conjugacy classes of subgroups — too many to draw. Only its 103 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#14537825C211yesmaximal, characteristic
2#14532149C211yesmaximal, characteristic
3#14528707C212yesmaximal, complemented
4#14526691C212yesmaximal, complemented
5C17×#1465C211yesmaximal, characteristic
6C23×Dic34C2211yescharacteristic
7C2×#10587610—21no—
8#11762887—24no—
9#11762862—21no—
10C2×#10587495—24no—
11#11762773—21no—
12#11762773C2212yes—
13#11762729—21no—
14#11762721—21no—
15#11762715—21no—
16#11762689C412yes—
17#11762681C412yes—
18#11762669C2211yescharacteristic
19C2×C4×D4×C17C2211yescharacteristic
20C17×#407C2212yes—
21C17×#324—21no—
22C17×#323—21no—
23C22×D4×C17C2×C411yescharacteristic
24C23×C4×C17D411yescharacteristic
25C23×C4×C17C2311yescharacteristic
26C23×Dic17C2×C412yes—
27C22×Dic34—21no—
28C22×Dic34—22no—
29C22×Dic34—21no—
30C22×Dic34—22no—
31C22×Dic34—44no—
32C2×M16×C17—22no—
33C4×D4×C17—22no—
34C4×D4×C17—22no—
35C2×C4⋊C4×C17D411yescharacteristic
36C2×C22⋊C4×C17C2×C411yescharacteristic
37C2×C22⋊C4×C17D411yescharacteristic
38C2×C42×C17D411yescharacteristic
39C2×C22⋊Dic17—22no—
40C2×C22⋊Dic17C2×C412yes—
41C2×C4⋊Dic17—21no—
42#10587610—44no—
43C2×#63225—22no—
44C2×#63223—22no—
45C2×#63223D414yes—
46C17×C22⋊C8—22no—
47C17×C22⋊C8—22no—
48#10587502—22no—
49#10587502—22no—
50#10587502—21no—
51C22⋊#3622—22no—
52C22⋊#3622—22no—
53C22⋊#3622—22no—
54#10587495—48no—
55C4⋊#3622—22no—
56C4×#3622—22no—
57C24×C17C22⋊C411yescharacteristic
58C24×C17C22×C411yescharacteristic
59C2×D4×C17—22no—
60C2×D4×C17—22no—
61C2×D4×C17—41no—
62C22×C4×C17—21no—
63C22×C4×C17—21no—
64C22×C4×C17—42no—
65C22×C4×C17—42no—
66C22×C4×C17C22⋊C411yescharacteristic
67C22×C4×C17C22⋊C411yescharacteristic
68C22×C4×C17C22⋊C411yescharacteristic
69C22×C4×C17C2×D411yescharacteristic
70C22×C4×C17C2×D412yes—
71C22×C4×C17C2×D411yescharacteristic
72C22×C4×C17#4112yes—
73C22×Dic17—22no—
74C22×Dic17—24no—
75C22×Dic17—22no—
76C22×Dic17—44no—
77C2×Dic34—28no—
78C2×Dic34—24no—
79C2×Dic34—44no—
80C2×Dic34—48no—
81C2×Dic34—44no—
82C2×Dic34—42no—
83C2×Dic34—41no—
84C2×Dic34—42no—
85C2×Dic34—41no—
86M16×C17—44no—
87C2×C8×C17—42no—
88C4⋊C4×C17—22no—
89C22⋊C4×C17—22no—
90C22⋊C4×C17—22no—
91C42×C17—21no—
92C42×C17—21no—
93C22⋊Dic17—44no—
94C22⋊Dic17—42no—
95C4⋊Dic17—42no—
96#63225—44no—
97#63223—44no—
98#63223—82no—
99C2×#3622—24no—
100#9506—171nomaximal, Sylow
101C23×C17—21no—
102C23×C17—21no—
103C23×C17—21no—
104C23×C17—21no—
105C23×C17—21no—
106C23×C17—41no—
107C23×C17—41no—
108C23×C17—41no—
109C23×C17—41no—
110C23×C17C2×C22⋊C411yescharacteristic
111C23×C17C4×D412yes—
112D4×C17—44no—
113C2×C4×C17—21no—
114C2×C4×C17—21no—
115C2×C4×C17—24no—
116C2×C4×C17—22no—
117C2×C4×C17—21no—
118C2×C4×C17—21no—
119C2×C4×C17—41no—
120C2×C4×C17—42no—
121C2×C4×C17—42no—
122C2×C4×C17—41no—
123C2×C4×C17—41no—
124C2×C4×C17—81no—
125C2×C4×C17#10112yescomplemented
126C2×C4×C17#12011yescharacteristic
127C2×C4×C17#12112yes—
128C2×C4×C17#12311yescharacteristic
129C2×Dic17—24no—
130C2×Dic17—28no—
131C2×Dic17—42no—
132C2×Dic17—44no—
133C2×Dic17—42no—
134C2×Dic17—42no—
135C2×Dic17—42no—
136C2×Dic17—42no—
137Dic34—48no—
138Dic34—44no—
139Dic34—44no—
140C8×C17—82no—
141#3622—44no—
142#2975—171no—
143#2409—171no—
144#1784—172no—
145#1465D1711yescharacteristic
146#1305—172no—
147C22×C17—21no—
148C22×C17—21no—
149C22×C17—22no—
150C22×C17—22no—
151C22×C17—22no—
152C22×C17—42no—
153C22×C17—42no—
154C22×C17—42no—
155C22×C17—41no—
156C22×C17—41no—
157C22×C17—41no—
158C22×C17—81no—
159C22×C17#38611yescharacteristic
160C22×C17C2×#10111yescharacteristic
161C22×C17#41311yescharacteristic
162C4×C17—22no—
163C4×C17—21no—
164C4×C17—22no—
165C4×C17—41no—
166C4×C17—41no—
167C4×C17B(4,𝔽2)12yes—
168Dic17—44no—
169Dic17—82no—
170C23×Q8—171no—
171C2×C22⋊Q8—341no—
172C2×C4×D4C2×D1711yescharacteristic
173#436—344no—
174#431—341no—
175C2×#101—344no—
176#407—172no—
177#407—341no—
178#407C2×D1712yes—
179#395—341no—
180#393—341no—
181#391—341no—
182#386—172no—
183#385—172no—
184#380—171no—
185#324—21no—
186#323—21no—
187C2×C17—21no—
188C2×C17—21no—
189C2×C17—42no—
190C2×C17—81no—
191C2×C17#188911yescharacteristic
192C2×C17#190111yescharacteristic
193C2×C17#191511yescharacteristic
194C22×Q8—341no—
195C22×Q8—342no—
196C22×Q8—342no—
197C22×Q8—341no—
198C22×Q8—684no—
199C22×D4C4×D1711yescharacteristic
200C23×C4—172no—
201C23×C4D6811yescharacteristic
202C23×C4C22×D1711yescharacteristic
203C2×M16—22no—
204C2×M16—174no—
205C2×M16—342no—
206C22⋊Q8—684no—
207C4×D4—22no—
208C4×D4—22no—
209C2×C4⋊C4—341no—
210C2×C4⋊C4—342no—
211C2×C4⋊C4C22⋊D1711yescharacteristic
212C2×C22⋊C4—172no—
213C2×C22⋊C4—342no—
214C2×C22⋊C4C4×D1711yescharacteristic
215C2×C22⋊C4C22⋊D1711yescharacteristic
216C2×C42D6811yescharacteristic
217C4⋊C8—342no—
218#101—688no—
219C22⋊C8—22no—
220C22⋊C8—22no—
221C22⋊C8—342no—
222C22⋊C8—342no—
223C22⋊C8—342no—
224C4×C8—342no—
225#95—341no—
226#95—342no—
227#95—342no—
228C17#950611yescharacteristic, complemented, Sylow
229C24#6322711yescharacteristic
230C24C2×C4×D1711yescharacteristic
231C2×Q8—348no—
232C2×Q8—344no—
233C2×Q8—682no—
234C2×Q8—684no—
235C2×Q8—688no—
236C2×Q8—684no—
237C2×Q8—681no—
238C2×Q8—681no—
239C2×Q8—682no—
240C2×D4—22no—
241C2×D4—22no—
242C2×D4—41no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—344no—
248C22×C4—342no—
249C22×C4—342no—
250C22×C4—684no—
251C22×C4#6322711yescharacteristic
252C22×C4#6322711yescharacteristic
253C22×C4#6322711yescharacteristic
254C22×C4C2×D6811yescharacteristic
255C22×C4D4×D1712yes—
256C22×C4#6325412yes—
257C22×C4C2×C22⋊D1711yescharacteristic
258M16—44no—
259M16—684no—
260M16—1362no—
261C2×C8—42no—
262C2×C8—344no—
263C4⋊C4—22no—
264C4⋊C4—682no—
265C4⋊C4—684no—
266C22⋊C4—22no—
267C22⋊C4—22no—
268C22⋊C4—684no—
269C22⋊C4—682no—
270C42—21no—
271C42—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23#1058756712yes—
282C23C2×#6322711yescharacteristic
283Q8—688no—
284Q8—684no—
285Q8—684no—
286D4—44no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—22no—
290C2×C4—24no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—42no—
294C2×C4—41no—
295C2×C4—42no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—81no—
299C2×C4—344no—
300C2×C4—348no—
301C2×C4—682no—
302C2×C4—684no—
303C2×C4—682no—
304C2×C4—682no—
305C2×C4—682no—
306C2×C4—682no—
307C2×C4#1058749512yescomplemented
308C2×C4#1058756811yescharacteristic
309C2×C4#1058757211yescharacteristic
310C2×C4#1058762512yes—
311C8—82no—
312C8—684no—
313C22—21no—
314C22—21no—
315C22—22no—
316C22—22no—
317C22—22no—
318C22—42no—
319C22—42no—
320C22—42no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—81no—
325C22#1176268611yescharacteristic
326C22C2×#1058749511yescharacteristic
327C22#1176279111yescharacteristic
328C4—22no—
329C4—21no—
330C4—22no—
331C4—41no—
332C4—41no—
333C4—684no—
334C4—1362no—
335C4#1176299712yes—
336C2—21no—
337C2—21no—
338C2—42no—
339C2—81no—
340C2#1452919011yescharacteristic
341C2#1452925011yescharacteristic
342C2#1452933411yescharacteristic
subgroupG/H
centerC22#11762686
commutatorC24×C17C22×C4
FrattiniC23×C4C22×D17
FittingC17×#1465C2
socleC22×C17#386
prime radicalC22×D4×C17C2×C4
cosocleC23×C4×C17C23
hypercenter#1465D17
hypocenterC17#9506
LCSG ▷ C24×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1465

Complements

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

NH ≅ G/Ntype
3. #14528707C2semidirect
4. #14526691C2semidirect
125. C2×C4×C17#101semidirect
228. C17#9506semidirect
307. C2×C4#10587495semidirect

Maximal subgroups

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

subgroupindex
#145378252
#145321492
#145287072
#145266912
C17×#14652
#950617