Finite Group Databasebeta

← #16945473

Subgroups — #16945473

Order 4352.

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

normal subgroups only

This group has 568 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#14528776C211yesmaximal, characteristic
2#14527102C211yesmaximal, characteristic
3#14526814C214yesmaximal, complemented
4C17×#2409C211yesmaximal, characteristic
5#11762833—21no—
6C2×#10587504—21no—
7C2×C22⋊#3622—21no—
8#11762578C2212yes—
9#11762535C2212yes—
10#11762533—21no—
11C23×Q8×C17C411yescharacteristic
12C2×C17×C22⋊Q8—21no—
13C2×C4×D4×C17C2211yescharacteristic
14C17×#395C2212yes—
15C17×#393C412yes—
16C17×#391—21no—
17C17×#386—21no—
18C17×#385—21no—
19C17×#380C411yescharacteristic
20C22×Q8×C17—21no—
21C22×Q8×C17—41no—
22C22×Q8×C17—41no—
23C22×Q8×C17—44no—
24C22×Q8×C17C2×C412yes—
25C22×D4×C17D411yescharacteristic
26C23×C4×C17—21no—
27C23×C4×C17C2×C411yescharacteristic
28C23×C4×C17D411yescharacteristic
29C17×C22⋊Q8—44no—
30C4×D4×C17—41no—
31C4×D4×C17—41no—
32C2×C4⋊C4×C17—21no—
33C2×C4⋊C4×C17—22no—
34C2×C4⋊C4×C17C2311yescharacteristic
35C2×C22⋊C4×C17—22no—
36C2×C22⋊C4×C17—21no—
37C2×C22⋊C4×C17C2×C411yescharacteristic
38C2×C22⋊C4×C17D411yescharacteristic
39C2×C42×C17D411yescharacteristic
40C2×#63223—22no—
41C22×#3622—41no—
42C17×#95—22no—
43C17×#95—22no—
44C17×#95C2×C412yes—
45#10587504—42no—
46C22⋊#3622—22no—
47C22⋊#3622—41no—
48C4⋊#3622—41no—
49C4⋊#3622—41no—
50C24×C17C22⋊C411yescharacteristic
51C24×C17C22⋊C411yescharacteristic
52C2×Q8×C17—24no—
53C2×Q8×C17—44no—
54C2×Q8×C17—42no—
55C2×Q8×C17—42no—
56C2×Q8×C17—44no—
57C2×Q8×C17—48no—
58C2×Q8×C17—44no—
59C2×Q8×C17—41no—
60C2×Q8×C17—41no—
61C2×D4×C17—41no—
62C2×D4×C17—41no—
63C2×D4×C17SD1612yes—
64C2×D4×C17Q1612yes—
65C22×C4×C17—22no—
66C22×C4×C17—21no—
67C22×C4×C17—21no—
68C22×C4×C17—21no—
69C22×C4×C17—21no—
70C22×C4×C17—22no—
71C22×C4×C17—44no—
72C22×C4×C17—42no—
73C22×C4×C17—42no—
74C22×C4×C17—42no—
75C22×C4×C17C22⋊C411yescharacteristic
76C22×C4×C17C22⋊C411yescharacteristic
77C22×C4×C17C22×C411yescharacteristic
78C22×C4×C17C2×D411yescharacteristic
79C22×C4×C17C2×D411yescharacteristic
80C4⋊C4×C17—41no—
81C4⋊C4×C17—42no—
82C4⋊C4×C17—44no—
83C22⋊C4×C17—41no—
84C22⋊C4×C17—41no—
85C22⋊C4×C17—44no—
86C22⋊C4×C17—42no—
87C42×C17—21no—
88C42×C17—21no—
89#63223—42no—
90#63223—42no—
91C2×#3622—42no—
92C2×#3622—42no—
93C2×#3622—82no—
94#11904—171nomaximal, Sylow
95C23×C17—21no—
96C23×C17—21no—
97C23×C17—21no—
98C23×C17—21no—
99C23×C17—41no—
100C23×C17—41no—
101C23×C17—41no—
102C23×C17—41no—
103C23×C17#10312yes—
104C23×C17#10312yes—
105C23×C17C2×C22⋊C411yescharacteristic
106Q8×C17—44no—
107Q8×C17—44no—
108Q8×C17—44no—
109Q8×C17—44no—
110D4×C17—42no—
111D4×C17—42no—
112C2×C4×C17—21no—
113C2×C4×C17—21no—
114C2×C4×C17—22no—
115C2×C4×C17—24no—
116C2×C4×C17—21no—
117C2×C4×C17—21no—
118C2×C4×C17—21no—
119C2×C4×C17—42no—
120C2×C4×C17—42no—
121C2×C4×C17—48no—
122C2×C4×C17—42no—
123C2×C4×C17—42no—
124C2×C4×C17—42no—
125C2×C4×C17—41no—
126C2×C4×C17—41no—
127C2×C4×C17—41no—
128C2×C4×C17—42no—
129C2×C4×C17—42no—
130C2×C4×C17—42no—
131C2×C4×C17—81no—
132C2×C4×C17#9911yescharacteristic
133C2×C4×C17#9911yescharacteristic
134C2×C4×C17C2×SD1611yescharacteristic
135C2×C4×C17C2×Q1611yescharacteristic
136#3622—82no—
137#3622—82no—
138#2409D1711yescharacteristic
139#1800—171no—
140#1459—171no—
141#1354—174no—
142C22×C17—21no—
143C22×C17—21no—
144C22×C17—22no—
145C22×C17—22no—
146C22×C17—42no—
147C22×C17—42no—
148C22×C17—42no—
149C22×C17—41no—
150C22×C17—41no—
151C22×C17—41no—
152C22×C17—41no—
153C22×C17—81no—
154C22×C17C2×#9911yescharacteristic
155C22×C17C2×#10311yescharacteristic
156C22×C17#42111yescharacteristic
157C4×C17—21no—
158C4×C17—21no—
159C4×C17—21no—
160C4×C17—41no—
161C4×C17—44no—
162C4×C17—41no—
163C4×C17—41no—
164C4×C17—82no—
165C4×C17AΣL(1,𝔽16)12yes—
166C23×Q8Dic1711yescharacteristic
167C2×C22⋊Q8—21no—
168C2×C4×D4C2×D1711yescharacteristic
169#423—341no—
170C2×#100—341no—
171C2×C22⋊C8—341no—
172#395C2×D1712yes—
173#393Dic1712yes—
174#391—21no—
175#386—21no—
176#385—21no—
177#380Dic1711yescharacteristic
178#343—172no—
179#324—172no—
180#323—341no—
181C2×C17—21no—
182C2×C17—21no—
183C2×C17—42no—
184C2×C17—42no—
185C2×C17#152011yescharacteristic
186C2×C17C2×AΣL(1,𝔽16)11yescharacteristic
187C2×C17#213511yescharacteristic
188C22×Q8—21no—
189C22×Q8—41no—
190C22×Q8—41no—
191C22×Q8—44no—
192C22×Q8C2×Dic1712yes—
193C22×D4C22⋊D1711yescharacteristic
194C23×C4—21no—
195C23×C4C2×Dic1711yescharacteristic
196C23×C4C22⋊D1711yescharacteristic
197C2×M16—342no—
198C22×C8—681no—
199C22⋊Q8—44no—
200C4×D4—41no—
201C4×D4—41no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4C22×D1711yescharacteristic
205C2×C22⋊C4—22no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4C2×Dic1711yescharacteristic
208C2×C22⋊C4C22⋊D1711yescharacteristic
209C2×C42C22⋊D1711yescharacteristic
210C4⋊C8—681no—
211C4⋊C8—681no—
212#100—682no—
213C22⋊C8—342no—
214C22⋊C8—681no—
215#95—22no—
216#95—22no—
217#95C2×Dic1712yes—
218C17#1190411yescharacteristic, complemented, Sylow
219C24C22⋊Dic1711yescharacteristic
220C24C22⋊Dic1711yescharacteristic
221C2×Q8—24no—
222C2×Q8—44no—
223C2×Q8—42no—
224C2×Q8—42no—
225C2×Q8—44no—
226C2×Q8—48no—
227C2×Q8—44no—
228C2×Q8—41no—
229C2×Q8—41no—
230C2×D4—41no—
231C2×D4—41no—
232C2×D4#6323012yes—
233C2×D4#6323112yes—
234C22×C4—22no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—22no—
240C22×C4—44no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4C22⋊Dic1711yescharacteristic
245C22×C4C22⋊Dic1711yescharacteristic
246C22×C4C22×Dic1711yescharacteristic
247C22×C4C2×C22⋊D1711yescharacteristic
248C22×C4C2×C22⋊D1711yescharacteristic
249M16—682no—
250M16—682no—
251C2×C8—682no—
252C2×C8—682no—
253C2×C8—1362no—
254C4⋊C4—41no—
255C4⋊C4—42no—
256C4⋊C4—44no—
257C22⋊C4—41no—
258C22⋊C4—41no—
259C22⋊C4—44no—
260C22⋊C4—42no—
261C42—21no—
262C42—21no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—41no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23#1058750612yes—
272C23#1058750612yes—
273C23C2×C22⋊Dic1711yescharacteristic
274Q8—44no—
275Q8—44no—
276Q8—44no—
277Q8—44no—
278D4—42no—
279D4—42no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—22no—
283C2×C4—24no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—48no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—81no—
300C2×C4#1058750511yescharacteristic
301C2×C4#1058750511yescharacteristic
302C2×C4C2×#6323011yescharacteristic
303C2×C4C2×#6323111yescharacteristic
304C8—1362no—
305C8—1362no—
306C22—21no—
307C22—21no—
308C22—22no—
309C22—22no—
310C22—42no—
311C22—42no—
312C22—42no—
313C22—41no—
314C22—41no—
315C22—41no—
316C22—41no—
317C22—81no—
318C22C2×#1058750511yescharacteristic
319C22C2×#1058750611yescharacteristic
320C22#1176282611yescharacteristic
321C4—21no—
322C4—21no—
323C4—21no—
324C4—41no—
325C4—44no—
326C4—41no—
327C4—41no—
328C4—82no—
329C4#1176259612yes—
330C2—21no—
331C2—21no—
332C2—42no—
333C2—42no—
334C2#1452735611yescharacteristic
335C2C2×#1176259611yescharacteristic
336C2#1453047911yescharacteristic
subgroupG/H
centerC22C2×#10587505
commutatorC22×C4×C17C22×C4
FrattiniC2×C4⋊C4C22×D17
FittingC17×#2409C2
socleC22×C17C2×#99
prime radicalC22×D4×C17D4
cosocleC2×C4⋊C4×C17C23
hypercenter#2409D17
hypocenterC17#11904
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2409

Complements

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

NH ≅ G/Ntype
3. #14526814C2semidirect
218. C17#11904semidirect

Maximal subgroups

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

subgroupindex
#145287762
#145271022
#145268142
C17×#24092
#1190417