Finite Group Databasebeta

← #16932068

Subgroups — #16932068

Order 4352.

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

normal subgroups only

This group has 610 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#14533251C211yesmaximal, characteristic
2#14528864C211yesmaximal, characteristic, complemented
3#14527312C212yesmaximal
4#14526636C212yesmaximal, complemented
5C17×#2313C211yesmaximal, characteristic, complemented
6C2×#10587615—21no—
7C2×#10587615—21no—
8C2×#10587616C2211yescharacteristic
9#11762773C2212yes—
10#11762744—22no—
11#11762710C412yescomplemented
12#11762678C412yescomplemented
13C4×C22⋊Dic17C2211yescharacteristic
14#11762578—22no—
15#11762542—24no—
16C22×C4⋊C4×C17C2211yescharacteristic
17C17×#387—24no—
18C17×#381C2212yes—
19C17×#380—21no—
20C4×C22⋊C4×C17—21no—
21C23×C4×C17D411yescharacteristic
22C23×C4×C17D411yescharacteristic
23C23×C4×C17C2311yescharacteristic
24C22×C22⋊D17—21no—
25C22×D68C2×C411yescharacteristic
26C2×C4⋊C4×C17—22no—
27C2×C4⋊C4×C17—22no—
28C2×C4⋊C4×C17—22no—
29C2×C4⋊C4×C17C2×C412yes—
30C2×C4⋊C4×C17D412yes—
31C2×C22⋊C4×C17—22no—
32C2×C42×C17—22no—
33C2×C22⋊Dic17—21no—
34#10587616—22no—
35#10587616—22no—
36#10587615—44no—
37#10587615—44no—
38C2×#63227—22no—
39C2×#63227—22no—
40C2×#63227—21no—
41C2×C4⋊Dic17C2×C411yescharacteristic
42C2×#63225—22no—
43C2×#63225—22no—
44C2×C4×Dic17C2×C412yes—
45C2×#63223—22no—
46C17×#95—22no—
47C17×#95—22no—
48C17×#95—22no—
49C17×#95D414yes—
50#10587502—21no—
51C22⋊#3622—22no—
52C22⋊#3622—22no—
53C24×C17C2×D411yescharacteristic
54C23×D17—21no—
55C22×C4×C17—21no—
56C22×C4×C17—21no—
57C22×C4×C17—21no—
58C22×C4×C17—21no—
59C22×C4×C17—24no—
60C22×C4×C17—42no—
61C22×C4×C17—42no—
62C22×C4×C17—42no—
63C22×C4×C17C22⋊C412yes—
64C22×C4×C17C22⋊C412yes—
65C22×C4×C17C22×C411yescharacteristic
66C22×C4×C17C2×D411yescharacteristic
67C22×C4×C17C2×D412yes—
68C22×C4×C17#4112yes—
69C2×C22⋊D17—42no—
70C2×C22⋊D17—42no—
71C2×C22⋊D17—41no—
72C2×C22⋊D17—41no—
73C22×Dic17—22no—
74C22×Dic17—21no—
75C2×D68—22no—
76C2×D68—42no—
77C4⋊C4×C17—24no—
78C4⋊C4×C17—24no—
79C4⋊C4×C17—82no—
80C22⋊C4×C17—44no—
81C42×C17—82no—
82C22⋊Dic17—24no—
83#63227—44no—
84#63227—44no—
85#63227—42no—
86C4⋊Dic17—22no—
87#63225—44no—
88#63225—44no—
89C4×Dic17—42no—
90#63223—44no—
91C2×#3622—42no—
92#9464—171nomaximal, Sylow
93C23×C17—22no—
94C23×C17—22no—
95C23×C17—41no—
96C23×C17C2×C22⋊C411yescharacteristic
97C23×C17#12011yescharacteristic
98C23×C17#12311yescharacteristic
99C22×D17—41no—
100C22×D17—41no—
101C22×D17—42no—
102C22×D17—41no—
103C22×D17—42no—
104C2×C4×C17—22no—
105C2×C4×C17—22no—
106C2×C4×C17—22no—
107C2×C4×C17—22no—
108C2×C4×C17—42no—
109C2×C4×C17—42no—
110C2×C4×C17—42no—
111C2×C4×C17—42no—
112C2×C4×C17—42no—
113C2×C4×C17—42no—
114C2×C4×C17—44no—
115C2×C4×C17—44no—
116C2×C4×C17—44no—
117C2×C4×C17—81no—
118C2×C4×C17—81no—
119C2×C4×C17C4×D412yes—
120C2×C4×C17#12112yes—
121C22⋊D17—82no—
122C22⋊D17—82no—
123C2×Dic17—24no—
124C2×Dic17—42no—
125C2×Dic17—42no—
126C2×Dic17—41no—
127C2×Dic17—41no—
128C2×Dic17—41no—
129D68—82no—
130#3622—82no—
131#2501—171no—
132#2313D1711yescharacteristic, complemented
133#1821—171no—
134#1510—172no—
135#1284—172no—
136C22×C17—21no—
137C22×C17—21no—
138C22×C17—21no—
139C22×C17—21no—
140C22×C17—21no—
141C22×C17—21no—
142C22×C17—21no—
143C22×C17—44no—
144C22×C17#38611yescharacteristic
145C22×C17#41012yes—
146C22×C17#45312yes—
147C2×D17—82no—
148C2×D17—82no—
149C2×D17—81no—
150C2×D17—81no—
151C2×D17—81no—
152C4×C17—22no—
153C4×C17—22no—
154C4×C17—42no—
155C4×C17—42no—
156C4×C17—84no—
157Dic17—44no—
158Dic17—81no—
159C2×#123—341no—
160C2×#123—341no—
161C2×#121—171no—
162C22×C4⋊C4C2×D1711yescharacteristic
163#407—172no—
164#399—342no—
165#390—172no—
166#387—24no—
167#384—172no—
168#381C2×D1712yes—
169#380—21no—
170C4×C22⋊C4—21no—
171C4×C22⋊C4—171no—
172#343—342no—
173#327—344no—
174C2×C17—22no—
175C2×C17—22no—
176C2×C17—41no—
177C2×C17#188711yescharacteristic
178C2×C17#189911yescharacteristic
179C2×C17#191211yescharacteristic
180D17—161no—
181C22×D4—171no—
182C22×D4—341no—
183C23×C4C22⋊D1711yescharacteristic
184C23×C4C22⋊D1711yescharacteristic
185C23×C4C22×D1711yescharacteristic
186C2×M16—342no—
187#123—684no—
188#123—684no—
189#121—342no—
190#121—342no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—171no—
195C2×C4⋊C4—342no—
196C2×C4⋊C4—342no—
197C2×C4⋊C4C4×D1712yes—
198C2×C4⋊C4D6812yes—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—341no—
201C2×C22⋊C4—341no—
202C2×C22⋊C4—342no—
203C2×C22⋊C4—342no—
204C2×C42—22no—
205C2×C42—172no—
206C22⋊C8—342no—
207C22⋊C8—342no—
208#95—22no—
209#95—22no—
210#95—22no—
211#95—341no—
212#95C22⋊D1714yes—
213C17#946411yescharacteristic, complemented, Sylow
214C24—341no—
215C24C2×C22⋊D1711yescharacteristic
216C2×D4—342no—
217C2×D4—682no—
218C2×D4—682no—
219C2×D4—682no—
220C2×D4—681no—
221C2×D4—681no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—24no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—342no—
231C22×C4—341no—
232C22×C4#6322712yes—
233C22×C4#6322712yes—
234C22×C4C2×C4×D1711yescharacteristic
235C22×C4C2×D6811yescharacteristic
236C22×C4#6325212yes—
237C22×C4C2×C22⋊D1712yes—
238M16—684no—
239C2×C8—682no—
240C4⋊C4—24no—
241C4⋊C4—24no—
242C4⋊C4—82no—
243C4⋊C4—342no—
244C4⋊C4—684no—
245C4⋊C4—684no—
246C22⋊C4—44no—
247C22⋊C4—344no—
248C22⋊C4—684no—
249C22⋊C4—684no—
250C22⋊C4—682no—
251C42—82no—
252C42—682no—
253C23—22no—
254C23—22no—
255C23—41no—
256C23—681no—
257C23—681no—
258C23—682no—
259C23—681no—
260C23—682no—
261C23C2×#6322711yescharacteristic
262C23#1058761511yescharacteristic
263C23#1058763911yescharacteristic
264D4—1362no—
265D4—1362no—
266D4—1362no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—44no—
278C2×C4—44no—
279C2×C4—44no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4—344no—
283C2×C4—682no—
284C2×C4—682no—
285C2×C4—681no—
286C2×C4—681no—
287C2×C4—681no—
288C2×C4C4×C22⋊D1712yes—
289C2×C4#1058761612yes—
290C8—1362no—
291C22—21no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—21no—
297C22—21no—
298C22—44no—
299C22—1362no—
300C22—1362no—
301C22—1361no—
302C22—1361no—
303C22—1361no—
304C22#1176268811yescharacteristic
305C22#1176278212yes—
306C22#1176297712yes—
307C4—22no—
308C4—22no—
309C4—42no—
310C4—42no—
311C4—84no—
312C4—684no—
313C4—1361no—
314C2—22no—
315C2—22no—
316C2—41no—
317C2—2721no—
318C2#1452918211yescharacteristic
319C2#1452924211yescharacteristic
320C2#1452932111yescharacteristic
subgroupG/H
centerC22#11762688
commutatorC22×C4×C17C22×C4
FrattiniC23×C4C22×D17
FittingC17×#2313C2
socleC22×C17#386
prime radicalC2×#10587616C22
cosocleC23×C4×C17C23
hypercenter#2313D17
hypocenterC17#9464
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2313

Complements

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

NH ≅ G/Ntype
2. #14528864C2semidirect
4. #14526636C2semidirect
5. C17×#2313C2semidirect
11. #11762710C4semidirect
12. #11762678C4semidirect
132. #2313D17semidirect
213. C17#9464semidirect

Maximal subgroups

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

subgroupindex
#145332512
#145288642
#145273122
#145266362
C17×#23132
#946417