Finite Group Databasebeta

← #16932153

Subgroups — #16932153

Order 4352.

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

normal subgroups only

This group has 614 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#14533058C211yesmaximal, characteristic
2#14528884C212yesmaximal, complemented
3#14527111C211yesmaximal, characteristic
4#14526637C212yesmaximal, complemented
5C17×#2312C211yesmaximal, characteristic
6C2×#10587612C2211yescharacteristic
7#11762773C2212yes—
8#11762745C412yescomplemented
9#11762734C412yescomplemented
10#11762697—22no—
11#11762682—21no—
12#11762672C2211yescharacteristic
13#11762665—21no—
14#11762569—24no—
15#11762533—22no—
16C22×C22⋊C4×C17C2211yescharacteristic
17C17×#388—24no—
18C17×#380—21no—
19C17×#379—21no—
20C4×C22⋊C4×C17C2212yes—
21C25×C17D411yescharacteristic
22C23×C4×C17D411yescharacteristic
23C23×C4×C17C2311yescharacteristic
24C2×C4⋊C4×C17—22no—
25C2×C22⋊C4×C17—22no—
26C2×C22⋊C4×C17—22no—
27C2×C22⋊C4×C17—22no—
28C2×C22⋊C4×C17—22no—
29C2×C22⋊C4×C17—22no—
30C2×C22⋊C4×C17C2×C412yes—
31C2×C22⋊C4×C17D412yes—
32C2×C42×C17D414yes—
33C2×C22⋊Dic17—22no—
34C2×C22⋊Dic17—21no—
35C2×C22⋊Dic17—21no—
36C2×C22⋊Dic17—22no—
37#10587612—22no—
38#10587612—22no—
39C2×C4⋊Dic17C2×C411yescharacteristic
40C2×C4⋊Dic17C2×C411yescharacteristic
41C2×#63225—22no—
42C2×C4×Dic17—21no—
43C2×#63223—22no—
44C17×#95—22no—
45C17×#95—22no—
46#10587502—22no—
47#10587502—21no—
48#10587502C2×C412yes—
49C22⋊#3622—22no—
50C22⋊#3622—22no—
51C24×C17—21no—
52C24×C17—21no—
53C24×C17—42no—
54C24×C17—42no—
55C24×C17—81no—
56C24×C17C22⋊C412yes—
57C24×C17C2×D411yescharacteristic
58C22×C4×C17—21no—
59C22×C4×C17—21no—
60C22×C4×C17—24no—
61C22×C4×C17—42no—
62C22×C4×C17—81no—
63C22×C4×C17C22⋊C412yes—
64C22×C4×C17C22×C411yescharacteristic
65C22×C4×C17C2×D411yescharacteristic
66C22×C4×C17C2×D412yes—
67C22×C4×C17#4112yes—
68C22×Dic17—22no—
69C22×Dic17—21no—
70C22×Dic17—21no—
71C4⋊C4×C17—82no—
72C22⋊C4×C17—24no—
73C22⋊C4×C17—24no—
74C22⋊C4×C17—44no—
75C22⋊C4×C17—82no—
76C22⋊C4×C17—82no—
77C42×C17—44no—
78C22⋊Dic17—42no—
79C22⋊Dic17—44no—
80C22⋊Dic17—41no—
81C22⋊Dic17—41no—
82C22⋊Dic17—82no—
83C4⋊Dic17—22no—
84C4⋊Dic17—22no—
85#63225—82no—
86C4×Dic17—42no—
87#63223—44no—
88C2×#3622—42no—
89#9481—171nomaximal, Sylow
90C23×C17—22no—
91C23×C17—22no—
92C23×C17—22no—
93C23×C17—22no—
94C23×C17—22no—
95C23×C17—41no—
96C23×C17—42no—
97C23×C17—42no—
98C23×C17—42no—
99C23×C17—42no—
100C23×C17—42no—
101C23×C17—42no—
102C23×C17—81no—
103C23×C17—81no—
104C23×C17—88no—
105C23×C17C2×C22⋊C411yescharacteristic
106C23×C17#12011yescharacteristic
107C23×C17#12311yescharacteristic
108C2×C4×C17—24no—
109C2×C4×C17—22no—
110C2×C4×C17—41no—
111C2×C4×C17—41no—
112C2×C4×C17—42no—
113C2×C4×C17—42no—
114C2×C4×C17—42no—
115C2×C4×C17—44no—
116C2×C4×C17—44no—
117C2×C4×C17—44no—
118C2×C4×C17—81no—
119C2×C4×C17C4×D412yes—
120C2×C4×C17#12112yes—
121C2×Dic17—42no—
122C2×Dic17—42no—
123C2×Dic17—42no—
124C2×Dic17—41no—
125C2×Dic17—41no—
126C2×Dic17—42no—
127C2×Dic17—42no—
128#3622—82no—
129#2482—171no—
130#2312D1711yescharacteristic
131#1826—172no—
132#1462—171no—
133#1284—172no—
134C22×C17—21no—
135C22×C17—21no—
136C22×C17—21no—
137C22×C17—21no—
138C22×C17—21no—
139C22×C17—21no—
140C22×C17—21no—
141C22×C17—44no—
142C22×C17—44no—
143C22×C17—42no—
144C22×C17—42no—
145C22×C17—42no—
146C22×C17—82no—
147C22×C17—84no—
148C22×C17—84no—
149C22×C17#38611yescharacteristic
150C22×C17#42112yes—
151C22×C17#45812yes—
152C4×C17—22no—
153C4×C17—22no—
154C4×C17—42no—
155C4×C17—84no—
156Dic17—82no—
157Dic17—81no—
158Dic17—81no—
159C2×#117—171no—
160C22×C22⋊C4C2×D1711yescharacteristic
161#407—172no—
162#399—172no—
163#397—172no—
164#388—24no—
165#388—342no—
166#385—341no—
167#381—171no—
168#380—21no—
169#379—21no—
170#379—341no—
171C4×C22⋊C4C2×D1712yes—
172#339—344no—
173#323—342no—
174C2×C17—22no—
175C2×C17—22no—
176C2×C17—41no—
177C2×C17—82no—
178C2×C17#189011yescharacteristic
179C2×C17#189711yescharacteristic
180C2×C17#191111yescharacteristic
181C25C22⋊D1711yescharacteristic
182C23×C4C22⋊D1711yescharacteristic
183C23×C4C22×D1711yescharacteristic
184C2×M16—342no—
185#117—342no—
186#117—342no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—171no—
189C2×C4⋊C4—171no—
190C2×C4⋊C4—342no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—341no—
197C2×C22⋊C4—341no—
198C2×C22⋊C4—342no—
199C2×C22⋊C4—342no—
200C2×C22⋊C4C4×D1712yes—
201C2×C22⋊C4D6812yes—
202C2×C42—341no—
203C2×C42C22⋊D1714yes—
204C22⋊C8—342no—
205C22⋊C8—342no—
206#95—22no—
207#95—22no—
208#95—172no—
209#95—341no—
210#95—342no—
211C17#948111yescharacteristic, complemented, Sylow
212C24—21no—
213C24—21no—
214C24—42no—
215C24—42no—
216C24—81no—
217C24#6322712yes—
218C24C2×C22⋊D1711yescharacteristic
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—24no—
222C22×C4—42no—
223C22×C4—81no—
224C22×C4—342no—
225C22×C4—341no—
226C22×C4—341no—
227C22×C4#6322712yes—
228C22×C4C2×C4×D1711yescharacteristic
229C22×C4C2×D6811yescharacteristic
230C22×C4#6325212yes—
231C22×C4C2×C22⋊D1712yes—
232M16—684no—
233C2×C8—682no—
234C4⋊C4—82no—
235C4⋊C4—342no—
236C4⋊C4—342no—
237C4⋊C4—1362no—
238C22⋊C4—24no—
239C22⋊C4—24no—
240C22⋊C4—44no—
241C22⋊C4—82no—
242C22⋊C4—82no—
243C22⋊C4—682no—
244C22⋊C4—684no—
245C22⋊C4—681no—
246C22⋊C4—681no—
247C22⋊C4—1362no—
248C42—44no—
249C42—682no—
250C23—22no—
251C23—22no—
252C23—22no—
253C23—22no—
254C23—22no—
255C23—41no—
256C23—42no—
257C23—42no—
258C23—42no—
259C23—42no—
260C23—42no—
261C23—42no—
262C23—81no—
263C23—81no—
264C23—88no—
265C23C2×#6322711yescharacteristic
266C23#1058761511yescharacteristic
267C23#1058763911yescharacteristic
268C2×C4—24no—
269C2×C4—22no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—44no—
276C2×C4—44no—
277C2×C4—44no—
278C2×C4—81no—
279C2×C4—682no—
280C2×C4—682no—
281C2×C4—682no—
282C2×C4—681no—
283C2×C4—681no—
284C2×C4—682no—
285C2×C4—682no—
286C2×C4C4×C22⋊D1712yes—
287C2×C4#1058761612yes—
288C8—1362no—
289C22—21no—
290C22—21no—
291C22—21no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—44no—
297C22—44no—
298C22—42no—
299C22—42no—
300C22—42no—
301C22—82no—
302C22—84no—
303C22—84no—
304C22#1176268811yescharacteristic
305C22#1176282712yes—
306C22#1176299912yes—
307C4—22no—
308C4—22no—
309C4—42no—
310C4—84no—
311C4—1362no—
312C4—1361no—
313C4—1361no—
314C2—22no—
315C2—22no—
316C2—41no—
317C2—82no—
318C2#1452919711yescharacteristic
319C2#1452923211yescharacteristic
320C2#1452931611yescharacteristic
subgroupG/H
centerC22#11762688
commutatorC22×C4×C17C22×C4
FrattiniC23×C4C22×D17
FittingC17×#2312C2
socleC22×C17#386
prime radicalC25×C17D4
cosocleC23×C4×C17C23
hypercenter#2312D17
hypocenterC17#9481
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2312

Complements

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

NH ≅ G/Ntype
2. #14528884C2semidirect
4. #14526637C2semidirect
8. #11762745C4semidirect
9. #11762734C4semidirect
211. C17#9481semidirect

Maximal subgroups

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

subgroupindex
#145330582
#145288842
#145271112
#145266372
C17×#23122
#948117