Finite Group Databasebeta

← #16944933

Subgroups — #16944933

Order 4352.

590 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 590 conjugacy classes of subgroups — too many to draw. Only its 95 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
1C4×#10587505C211yesmaximal, characteristic
2#14527125C211yesmaximal, characteristic
3#14526805C214yesmaximal, complemented
4C17×#2409C211yesmaximal, characteristic
5#11762866—21no—
6C2×#10587505—21no—
7C4×C22⋊Dic17—21no—
8#11762576C2212yes—
9#11762533—21no—
10#11762533C2212yes—
11C23×Q8×C17C411yescharacteristic, complemented
12C2×C17×C22⋊Q8—21no—
13C2×C4×D4×C17C2211yescharacteristic
14C17×#395C412yescomplemented
15C17×#393C2212yes—
16C17×#391—21no—
17C17×#386—21no—
18C17×#385—21no—
19C17×#380C411yescharacteristic, complemented
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×C17C2×C411yescharacteristic
35C2×C22⋊C4×C17—21no—
36C2×C22⋊C4×C17—22no—
37C2×C22⋊C4×C17D411yescharacteristic
38C2×C22⋊C4×C17C2311yescharacteristic
39C2×C42×C17D411yescharacteristic
40C2×C22⋊Dic17—22no—
41C2×C4×Dic17—41no—
42C17×#95—22no—
43C17×#95—22no—
44C17×#95C2×C412yes—
45#10587505—24no—
46#10587502—41no—
47C22⋊#3622—22no—
48C4⋊#3622—41no—
49#10587474—41no—
50C24×C17C22⋊C411yescharacteristic
51C24×C17C2×D411yescharacteristic
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—22no—
62C2×D4×C17—22no—
63C2×D4×C17—41no—
64C22×C4×C17—22no—
65C22×C4×C17—21no—
66C22×C4×C17—21no—
67C22×C4×C17—21no—
68C22×C4×C17—21no—
69C22×C4×C17—22no—
70C22×C4×C17—44no—
71C22×C4×C17—42no—
72C22×C4×C17—42no—
73C22×C4×C17—42no—
74C22×C4×C17C22⋊C411yescharacteristic
75C22×C4×C17C22⋊C411yescharacteristic
76C22×C4×C17C22⋊C411yescharacteristic
77C22×C4×C17C22×C411yescharacteristic
78C22×C4×C17C2×D411yescharacteristic
79C22×Dic17—42no—
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—
89C22⋊Dic17—24no—
90C22⋊Dic17—42no—
91C4×Dic17—82no—
92C2×#3622—42no—
93#11785—171nomaximal, Sylow
94C23×C17—21no—
95C23×C17—21no—
96C23×C17—21no—
97C23×C17—21no—
98C23×C17—41no—
99C23×C17—41no—
100C23×C17—41no—
101C23×C17—41no—
102C23×C17#10412yes—
103C23×C17C2×C22⋊C411yescharacteristic
104C23×C17#13712yes—
105Q8×C17—44no—
106Q8×C17—44no—
107Q8×C17—44no—
108Q8×C17—44no—
109D4×C17—44no—
110C2×C4×C17—22no—
111C2×C4×C17—24no—
112C2×C4×C17—21no—
113C2×C4×C17—21no—
114C2×C4×C17—21no—
115C2×C4×C17—21no—
116C2×C4×C17—21no—
117C2×C4×C17—21no—
118C2×C4×C17—42no—
119C2×C4×C17—42no—
120C2×C4×C17—48no—
121C2×C4×C17—42no—
122C2×C4×C17—42no—
123C2×C4×C17—42no—
124C2×C4×C17—41no—
125C2×C4×C17—41no—
126C2×C4×C17—41no—
127C2×C4×C17—42no—
128C2×C4×C17—42no—
129C2×C4×C17—42no—
130C2×C4×C17—81no—
131C2×C4×C17#9911yescharacteristic
132C2×C4×C17#9911yescharacteristic
133C2×Dic17—44no—
134C2×Dic17—82no—
135C2×Dic17—82no—
136#3622—82no—
137#2409D1711yescharacteristic
138C4×#99—171no—
139#1466—171no—
140#1351—174no—
141C22×C17—21no—
142C22×C17—21no—
143C22×C17—22no—
144C22×C17—22no—
145C22×C17—42no—
146C22×C17—42no—
147C22×C17—42no—
148C22×C17—41no—
149C22×C17—41no—
150C22×C17—41no—
151C22×C17—41no—
152C22×C17—81no—
153C22×C17C2×#9911yescharacteristic
154C22×C17#41911yescharacteristic
155C22×C17C2×#10411yescharacteristic
156C4×C17—21no—
157C4×C17—44no—
158C4×C17—41no—
159C4×C17—41no—
160C4×C17—41no—
161C4×C17—41no—
162C4×C17—82no—
163C4×C17AΣL(1,𝔽16)12yes—
164Dic17—84no—
165C23×Q8Dic1711yescharacteristic, complemented
166C2×C22⋊Q8—21no—
167C2×C4×D4C2×D1711yescharacteristic
168#432—341no—
169C2×#99—341no—
170#395Dic1712yescomplemented
171#393C2×D1712yes—
172#391—21no—
173#386—21no—
174#385—21no—
175#380Dic1711yescharacteristic, complemented
176C4×C22⋊C4—341no—
177#342—172no—
178#323—172no—
179#323—341no—
180C2×C17—21no—
181C2×C17—21no—
182C2×C17—42no—
183C2×C17—42no—
184C2×C17#150811yescharacteristic
185C2×C17C2×AΣL(1,𝔽16)11yescharacteristic
186C2×C17#212911yescharacteristic
187C22×Q8—21no—
188C22×Q8—41no—
189C22×Q8—41no—
190C22×Q8—44no—
191C22×Q8C2×Dic1712yes—
192C22×D4C22⋊D1711yescharacteristic
193C23×C4—21no—
194C23×C4C2×Dic1711yescharacteristic
195C23×C4C22⋊D1711yescharacteristic
196C22⋊Q8—44no—
197C4×D4—41no—
198C4×D4—41no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4C2×Dic1711yescharacteristic
202C2×C22⋊C4—21no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—342no—
205C2×C22⋊C4C22⋊D1711yescharacteristic
206C2×C22⋊C4C22×D1711yescharacteristic
207C2×C42—681no—
208C2×C42C22⋊D1711yescharacteristic
209C4⋊C8—681no—
210#99—344no—
211C22⋊C8—342no—
212#97—681no—
213#95—22no—
214#95—22no—
215#95—681no—
216#95C2×Dic1712yes—
217C17#1178511yescharacteristic, complemented, Sylow
218C24C22⋊Dic1711yescharacteristic
219C24C2×C22⋊D1711yescharacteristic
220C2×Q8—24no—
221C2×Q8—44no—
222C2×Q8—42no—
223C2×Q8—42no—
224C2×Q8—44no—
225C2×Q8—48no—
226C2×Q8—44no—
227C2×Q8—41no—
228C2×Q8—41no—
229C2×D4—22no—
230C2×D4—22no—
231C2×D4—41no—
232C22×C4—22no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—22no—
238C22×C4—44no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—682no—
243C22×C4C22⋊Dic1711yescharacteristic
244C22×C4C22⋊Dic1711yescharacteristic
245C22×C4C22⋊Dic1711yescharacteristic
246C22×C4C22×Dic1711yescharacteristic
247C22×C4C2×C22⋊D1711yescharacteristic
248C2×C8—682no—
249C4⋊C4—41no—
250C4⋊C4—42no—
251C4⋊C4—44no—
252C22⋊C4—41no—
253C22⋊C4—41no—
254C22⋊C4—44no—
255C22⋊C4—42no—
256C22⋊C4—344no—
257C22⋊C4—682no—
258C42—21no—
259C42—21no—
260C42—1362no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23#1058750812yes—
270C23#1058762812yes—
271C23C2×C22⋊Dic1711yescharacteristic
272Q8—44no—
273Q8—44no—
274Q8—44no—
275Q8—44no—
276D4—44no—
277C2×C4—22no—
278C2×C4—24no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—48no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—81no—
298C2×C4—684no—
299C2×C4—1362no—
300C2×C4—1362no—
301C2×C4#1058750511yescharacteristic
302C2×C4#1058750511yescharacteristic
303C8—1362no—
304C22—21no—
305C22—21no—
306C22—22no—
307C22—22no—
308C22—42no—
309C22—42no—
310C22—42no—
311C22—41no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—81no—
316C22C2×#1058750511yescharacteristic
317C22#1176281811yescharacteristic
318C22C2×#1058750811yescharacteristic
319C4—21no—
320C4—41no—
321C4—44no—
322C4—41no—
323C4—41no—
324C4—41no—
325C4—82no—
326C4—1364no—
327C4#1176259612yes—
328C2—21no—
329C2—21no—
330C2—42no—
331C2—42no—
332C2#1452730411yescharacteristic
333C2C2×#1176259611yescharacteristic
334C2#1453045911yescharacteristic
subgroupG/H
centerC22C2×#10587505
commutatorC22×C4×C17C22×C4
FrattiniC2×C22⋊C4C22×D17
FittingC17×#2409C2
socleC22×C17C2×#99
prime radicalC22×D4×C17D4
cosocleC2×C22⋊C4×C17C23
hypercenter#2409D17
hypocenterC17#11785
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. #14526805C2semidirect
11. C23×Q8×C17C4semidirect
14. C17×#395C4semidirect
19. C17×#380C4semidirect
165. C23×Q8Dic17semidirect
170. #395Dic17semidirect
175. #380Dic17semidirect
217. C17#11785semidirect

Maximal subgroups

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

subgroupindex
C4×#105875052
#145271252
#145268052
C17×#24092
#1178517