Finite Group Databasebeta

← #16944934

Subgroups — #16944934

Order 4352.

590 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 590 conjugacy classes of subgroups — too many to draw. Only its 87 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#14532149C211yesmaximal, characteristic
2C4×#10587505C211yesmaximal, characteristic
3#14526806C214yesmaximal, complemented
4C17×#1466C211yesmaximal, characteristic
5C23×Dic34C411yescharacteristic, complemented
6C2×#10587610—21no—
7C2×#10587505—21no—
8#11762729C412yescomplemented
9#11762721C2212yes—
10#11762715—21no—
11#11762689—21no—
12#11762681—21no—
13#11762669C411yescharacteristic, complemented
14C4×C22⋊Dic17—21no—
15#11762576C2212yes—
16C2×C4×D4×C17C2211yescharacteristic
17C17×#432—21no—
18C17×#323—21no—
19C17×#323C2212yes—
20C22×D4×C17D411yescharacteristic
21C23×C4×C17C2×C411yescharacteristic
22C23×C4×C17D411yescharacteristic
23C23×Dic17—21no—
24C22×Dic34—21no—
25C22×Dic34—44no—
26C22×Dic34—41no—
27C22×Dic34—41no—
28C22×Dic34C2×C412yes—
29C4×D4×C17—41no—
30C4×D4×C17—41no—
31C2×C4⋊C4×C17C2×C411yescharacteristic
32C2×C22⋊C4×C17D411yescharacteristic
33C2×C22⋊C4×C17C2311yescharacteristic
34C2×C42×C17D411yescharacteristic
35C2×C22⋊Dic17—22no—
36C2×C22⋊Dic17—21no—
37C2×C22⋊Dic17—22no—
38C2×C4⋊Dic17—21no—
39#10587610—44no—
40C2×#63225—22no—
41C2×C4×Dic17—41no—
42C17×C4⋊C8—41no—
43C17×C22⋊C8—22no—
44C17×#97—41no—
45#10587505—24no—
46#10587502—22no—
47#10587502—22no—
48#10587502—41no—
49#10587502C2×C412yes—
50C24×C17C22⋊C411yescharacteristic
51C24×C17C2×D411yescharacteristic
52C2×D4×C17—22no—
53C2×D4×C17—22no—
54C2×D4×C17—41no—
55C22×C4×C17—21no—
56C22×C4×C17—21no—
57C22×C4×C17—21no—
58C22×C4×C17—21no—
59C22×C4×C17—42no—
60C22×C4×C17—42no—
61C22×C4×C17C22⋊C411yescharacteristic
62C22×C4×C17C22⋊C411yescharacteristic
63C22×C4×C17C22⋊C411yescharacteristic
64C22×C4×C17C22×C411yescharacteristic
65C22×C4×C17C2×D411yescharacteristic
66C22×Dic17—22no—
67C22×Dic17—22no—
68C22×Dic17—44no—
69C22×Dic17—42no—
70C22×Dic17—42no—
71C2×Dic34—24no—
72C2×Dic34—42no—
73C2×Dic34—48no—
74C2×Dic34—44no—
75C2×Dic34—41no—
76C2×Dic34—41no—
77C2×Dic34—42no—
78C2×Dic34—44no—
79C2×Dic34—44no—
80C2×C8×C17—42no—
81C4⋊C4×C17—41no—
82C22⋊C4×C17—41no—
83C22⋊C4×C17—41no—
84C42×C17—21no—
85C42×C17—21no—
86C22⋊Dic17—24no—
87C22⋊Dic17—44no—
88C22⋊Dic17—42no—
89C22⋊Dic17—42no—
90C4⋊Dic17—42no—
91#63225—44no—
92C4×Dic17—82no—
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—
105D4×C17—44no—
106C2×C4×C17—22no—
107C2×C4×C17—21no—
108C2×C4×C17—21no—
109C2×C4×C17—21no—
110C2×C4×C17—21no—
111C2×C4×C17—21no—
112C2×C4×C17—21no—
113C2×C4×C17—42no—
114C2×C4×C17—42no—
115C2×C4×C17—42no—
116C2×C4×C17—41no—
117C2×C4×C17—41no—
118C2×C4×C17—41no—
119C2×C4×C17—81no—
120C2×C4×C17#9911yescharacteristic
121C2×C4×C17#9911yescharacteristic
122C2×Dic17—24no—
123C2×Dic17—42no—
124C2×Dic17—48no—
125C2×Dic17—42no—
126C2×Dic17—44no—
127C2×Dic17—42no—
128C2×Dic17—42no—
129C2×Dic17—42no—
130C2×Dic17—82no—
131C2×Dic17—82no—
132Dic34—44no—
133Dic34—44no—
134Dic34—44no—
135Dic34—44no—
136C8×C17—82no—
137#2409—171no—
138C4×#99—171no—
139#1466D1711yescharacteristic
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—41no—
158C4×C17—41no—
159C4×C17—41no—
160C4×C17—41no—
161C4×C17AΣL(1,𝔽16)12yes—
162Dic17—44no—
163Dic17—82no—
164Dic17—84no—
165C23×Q8—171no—
166C2×C22⋊Q8—341no—
167C2×C4×D4C2×D1711yescharacteristic
168#432—21no—
169C2×#99—341no—
170#395—172no—
171#393—172no—
172#391—341no—
173#386—341no—
174#385—341no—
175#380—171no—
176C4×C22⋊C4—341no—
177#342—172no—
178#323—21no—
179#323C2×D1712yes—
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—172no—
188C22×Q8—341no—
189C22×Q8—681no—
190C22×Q8—681no—
191C22×Q8—684no—
192C22×D4D6811yescharacteristic
193C23×C4—341no—
194C23×C4C4×D1711yescharacteristic
195C23×C4D6811yescharacteristic
196C22⋊Q8—684no—
197C4×D4—41no—
198C4×D4—41no—
199C2×C4⋊C4—342no—
200C2×C4⋊C4—341no—
201C2×C4⋊C4C4×D1711yescharacteristic
202C2×C22⋊C4—342no—
203C2×C22⋊C4—342no—
204C2×C22⋊C4—341no—
205C2×C22⋊C4C22⋊D1711yescharacteristic
206C2×C22⋊C4C22×D1711yescharacteristic
207C2×C42—681no—
208C2×C42C22⋊D1711yescharacteristic
209C4⋊C8—41no—
210#99—344no—
211C22⋊C8—22no—
212#97—41no—
213#95—172no—
214#95—342no—
215#95—342no—
216#95—681no—
217C17#1178511yescharacteristic, complemented, Sylow
218C24#6322711yescharacteristic
219C24C2×D6811yescharacteristic
220C2×Q8—344no—
221C2×Q8—682no—
222C2×Q8—682no—
223C2×Q8—684no—
224C2×Q8—684no—
225C2×Q8—688no—
226C2×Q8—684no—
227C2×Q8—681no—
228C2×Q8—681no—
229C2×D4—22no—
230C2×D4—22no—
231C2×D4—41no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—342no—
239C22×C4—342no—
240C22×C4—684no—
241C22×C4—682no—
242C22×C4—682no—
243C22×C4#6322711yescharacteristic
244C22×C4#6322711yescharacteristic
245C22×C4#6322711yescharacteristic
246C22×C4C2×C4×D1711yescharacteristic
247C22×C4C2×C22⋊D1711yescharacteristic
248C2×C8—42no—
249C4⋊C4—41no—
250C4⋊C4—682no—
251C4⋊C4—684no—
252C22⋊C4—41no—
253C22⋊C4—41no—
254C22⋊C4—344no—
255C22⋊C4—684no—
256C22⋊C4—682no—
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#1058749612yes—
270C23#1058759512yes—
271C23C2×#6322711yescharacteristic
272Q8—684no—
273Q8—684no—
274Q8—684no—
275Q8—684no—
276D4—44no—
277C2×C4—22no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—81no—
291C2×C4—344no—
292C2×C4—682no—
293C2×C4—688no—
294C2×C4—682no—
295C2×C4—684no—
296C2×C4—682no—
297C2×C4—682no—
298C2×C4—682no—
299C2×C4—1362no—
300C2×C4—1362no—
301C2×C4#1058747711yescharacteristic
302C2×C4#1058747711yescharacteristic
303C8—82no—
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×#1058747711yescharacteristic
317C22#1176281911yescharacteristic
318C22C2×#1058749611yescharacteristic
319C4—21no—
320C4—41no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—684no—
325C4—1362no—
326C4—1364no—
327C4#1176259712yes—
328C2—21no—
329C2—21no—
330C2—42no—
331C2—42no—
332C2#1452730511yescharacteristic
333C2C2×#1176259711yescharacteristic
334C2#1453045811yescharacteristic
subgroupG/H
centerC22C2×#10587477
commutatorC22×C4×C17C22×C4
FrattiniC2×C22⋊C4C22×D17
FittingC17×#1466C2
socleC22×C17C2×#99
prime radicalC22×D4×C17D4
cosocleC2×C22⋊C4×C17C23
hypercenter#1466D17
hypocenterC17#11785
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1466

Complements

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

NH ≅ G/Ntype
3. #14526806C2semidirect
5. C23×Dic34C4semidirect
8. #11762729C4semidirect
13. #11762669C4semidirect
217. C17#11785semidirect

Maximal subgroups

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

subgroupindex
#145321492
C4×#105875052
#145268062
C17×#14662
#1178517