Finite Group Databasebeta

← #16927230

Subgroups — #16927230

Order 4352.

486 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 101 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
1C2×#11762872C211yesmaximal, characteristic
2#14528820C212yesmaximal, complemented
3#14527120C212yesmaximal
4#14526661C211yesmaximal, characteristic, complemented
5C17×#1293C211yesmaximal, characteristic, complemented
6C23×#3622C2211yescharacteristic
7C22×C8×D17C411yescharacteristic
8C2×C4×C22⋊D17C2211yescharacteristic
9#11762872—22no—
10#11762872—22no—
11C2×#10587493—22no—
12C2×#10587485C411yescharacteristic
13#11762773C2212yes—
14C2×#10587491C411yescharacteristic
15C2×C8×Dic17C411yescharacteristic
16#11762573—22no—
17#11762534—22no—
18#11762532—22no—
19C17×#407C2212yes—
20C2×C17×C22⋊C8C2211yescharacteristic
21C23×C4×C17C2311yescharacteristic
22C22×C22⋊D17Q811yescharacteristic
23C22×C4×D17C2×C411yescharacteristic
24C2×M16×C17—22no—
25C22×C8×C17C2×C411yescharacteristic
26C22×C8×C17C2×C411yescharacteristic
27C2×C22⋊Dic17Q811yescharacteristic
28C4×C22⋊D17—22no—
29C4×C22⋊D17—22no—
30C2×#63227D411yescharacteristic
31C2×#63225D411yescharacteristic
32C2×C4×Dic17C2×C411yescharacteristic
33C2×#63223—22no—
34C22×#3622—42no—
35C22×#3622C2×C411yescharacteristic
36C22×#3622C2×C411yescharacteristic
37C22×#3622D412yes—
38C22×#3622D412yes—
39C2×C8×D17—21no—
40C2×C8×D17—21no—
41C2×C8×D17—42no—
42C17×C22⋊C8—22no—
43C17×C22⋊C8—22no—
44C17×C22⋊C8—22no—
45C22⋊#3622—22no—
46C22⋊#3622—22no—
47#10587493—44no—
48#10587491—22no—
49#10587485—22no—
50C8×Dic17—21no—
51C8×Dic17—21no—
52C24×C17C2×Q811yescharacteristic
53C23×D17C4⋊C411yescharacteristic
54C22×C4×C17—41no—
55C22×C4×C17—41no—
56C22×C4×C17C22×C411yescharacteristic
57C22×C4×C17C2×D412yes—
58C22×C4×C17C2×D411yescharacteristic
59C22×C4×C17#4112yes—
60C2×C22⋊D17—22no—
61C2×C22⋊D17—22no—
62C2×C22⋊D17—22no—
63C22×Dic17C4⋊C411yescharacteristic
64C22×Dic17C4⋊C411yescharacteristic
65C22×Dic17C4⋊C411yescharacteristic
66C2×C4×D17—21no—
67C2×C4×D17—21no—
68C2×C4×D17—41no—
69C2×C4×D17—41no—
70M16×C17—42no—
71M16×C17—42no—
72C2×C8×C17—21no—
73C2×C8×C17—21no—
74C2×C8×C17—41no—
75C2×C8×C17—41no—
76C2×C8×C17—42no—
77C22⋊Dic17—22no—
78#63227—22no—
79#63225—22no—
80C4×Dic17—21no—
81C4×Dic17—21no—
82#63223—42no—
83#63223—42no—
84C2×#3622—24no—
85C2×#3622—22no—
86C2×#3622—41no—
87C2×#3622—42no—
88C2×#3622—42no—
89C2×#3622C22⋊C411yescharacteristic
90C2×#3622C22⋊C412yes—
91C2×#3622C22⋊C411yescharacteristic
92C8×D17—42no—
93C8×D17—42no—
94#8611—171nomaximal, Sylow
95C23×C17—21no—
96C23×C17—21no—
97C23×C17—41no—
98C23×C17—41no—
99C23×C17C2×C4⋊C411yescharacteristic
100C23×C17C22⋊Q812yes—
101C22×D17—21no—
102C22×D17—21no—
103C22×D17—41no—
104C22×D17—41no—
105C22×D17#10812yescomplemented
106C2×C4×C17—21no—
107C2×C4×C17—22no—
108C2×C4×C17—21no—
109C2×C4×C17—22no—
110C2×C4×C17—41no—
111C2×C4×C17—42no—
112C2×C4×C17C2×C22⋊C411yescharacteristic
113C2×C4×C17#11711yescharacteristic
114C2×C4×C17#12112yes—
115C22⋊D17—28no—
116C2×Dic17—21no—
117C2×Dic17—21no—
118C2×Dic17—21no—
119C2×Dic17—21no—
120C2×Dic17—41no—
121C2×Dic17—41no—
122C2×Dic17#10812yescomplemented
123C4×D17—42no—
124C4×D17—41no—
125C4×D17—41no—
126C8×C17—41no—
127C8×C17—41no—
128C8×C17—82no—
129#3622—21no—
130#3622—22no—
131#3622—82no—
132#3622#9912yes—
133C2×C8×D4—171no—
134#1810—172no—
135#1465—172no—
136#1293—171no—
137#1293D1711yescharacteristic, complemented
138C22×C17—21no—
139C22×C17—21no—
140C22×C17—22no—
141C22×C17—22no—
142C22×C17—22no—
143C22×C17—42no—
144C22×C17—42no—
145C22×C17#38011yescharacteristic
146C22×C17C2×#10811yescharacteristic
147C22×C17C2×#10811yescharacteristic
148C2×D17—22no—
149C2×D17—42no—
150C2×D17—41no—
151C2×D17—41no—
152C2×D17—42no—
153C4×C17—22no—
154C4×C17—21no—
155C4×C17C2×#9911yescharacteristic
156C4×C17#41011yescharacteristic
157Dic17—22no—
158Dic17—41no—
159Dic17—41no—
160C23×C8—171no—
161C23×C8—171no—
162C2×C4×D4—171no—
163C8×D4—342no—
164C8×D4—342no—
165C2×#108—342no—
166C2×C4⋊C8—171no—
167#407—172no—
168#407C2×D1712yes—
169C2×C22⋊C8—171no—
170C2×C22⋊C8C2×D1711yescharacteristic
171C2×C4×C8—171no—
172#341—342no—
173#324—342no—
174#323—342no—
175C2×C17—21no—
176C2×C17—21no—
177C2×C17—42no—
178C2×C17#179911yescharacteristic
179C2×C17#181011yescharacteristic
180C2×C17#181311yescharacteristic
181D17—42no—
182C22×D4—171no—
183C23×C4—171no—
184C23×C4C22×D1711yescharacteristic
185C2×M16—22no—
186C2×M16—342no—
187C22×C8—171no—
188C22×C8—172no—
189C22×C8—171no—
190C22×C8—172no—
191C22×C8—341no—
192C22×C8—341no—
193C22×C8—682no—
194C22×C8—682no—
195C22×C8C4×D1711yescharacteristic
196C22×C8C4×D1711yescharacteristic
197C4×D4—342no—
198C4×D4—342no—
199C2×C4⋊C4—171no—
200C2×C22⋊C4—171no—
201C2×C22⋊C4—171no—
202C2×C42—171no—
203#108—684no—
204C4⋊C8—342no—
205C22⋊C8—22no—
206C22⋊C8—22no—
207C22⋊C8—22no—
208C22⋊C8—342no—
209C22⋊C8—342no—
210C22⋊C8—342no—
211C4×C8—341no—
212C4×C8—341no—
213C17#861111yescharacteristic, complemented, Sylow
214C24—171no—
215C24Q8×D1711yescharacteristic
216C2×D4—342no—
217C2×D4—342no—
218C2×D4—342no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—171no—
222C22×C4—171no—
223C22×C4—171no—
224C22×C4—341no—
225C22×C4—341no—
226C22×C4—681no—
227C22×C4—681no—
228C22×C4C2×C4×D1711yescharacteristic
229C22×C4#6325212yes—
230C22×C4D4×D1712yes—
231C22×C4D4×D1711yescharacteristic
232M16—42no—
233M16—42no—
234M16—682no—
235M16—682no—
236C2×C8—21no—
237C2×C8—21no—
238C2×C8—41no—
239C2×C8—41no—
240C2×C8—42no—
241C2×C8—171no—
242C2×C8—171no—
243C2×C8—172no—
244C2×C8—344no—
245C2×C8—342no—
246C2×C8—682no—
247C2×C8—682no—
248C2×C8—682no—
249C2×C8—681no—
250C2×C8—682no—
251C4⋊C4—342no—
252C22⋊C4—342no—
253C22⋊C4—342no—
254C42—341no—
255C42—341no—
256C23—21no—
257C23—21no—
258C23—41no—
259C23—41no—
260C23—172no—
261C23—341no—
262C23—341no—
263C23—681no—
264C23—681no—
265C23C4⋊C4×D1711yescharacteristic
266C23#1058758212yes—
267D4—348no—
268C2×C4—21no—
269C2×C4—22no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—42no—
273C2×C4—41no—
274C2×C4—172no—
275C2×C4—341no—
276C2×C4—341no—
277C2×C4—341no—
278C2×C4—341no—
279C2×C4—682no—
280C2×C4—681no—
281C2×C4—681no—
282C2×C4—681no—
283C2×C4—681no—
284C2×C4#1058755811yescharacteristic
285C2×C4C22⋊C4×D1711yescharacteristic
286C2×C4#1058757012yes—
287C8—41no—
288C8—41no—
289C8—82no—
290C8—172no—
291C8—341no—
292C8—342no—
293C8—1362no—
294C22—21no—
295C22—21no—
296C22—22no—
297C22—22no—
298C22—22no—
299C22—42no—
300C22—42no—
301C22—342no—
302C22—682no—
303C22—682no—
304C22—681no—
305C22—681no—
306C22#1176266811yescharacteristic
307C22#108×D1711yescharacteristic
308C22#108×D1711yescharacteristic
309C4—22no—
310C4—21no—
311C4—342no—
312C4—681no—
313C4—681no—
314C4#99×D1711yescharacteristic
315C4#1176278111yescharacteristic
316C2—21no—
317C2—21no—
318C2—42no—
319C2—682no—
320C2#1452877111yescharacteristic
321C2#1452882111yescharacteristic
322C2#1452883311yescharacteristic
subgroupG/H
centerC22#11762668
commutatorC22×C4×C17C22×C4
FrattiniC23×C4C22×D17
FittingC17×#1293C2
socleC22×C17#380
prime radicalC22×C22⋊D17Q8
cosocleC23×C4×C17C23
hypercenter#1293D17
hypocenterC17#8611
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1293

Complements

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

NH ≅ G/Ntype
2. #14528820C2semidirect
4. #14526661C2semidirect
5. C17×#1293C2semidirect
105. C22×D17#108semidirect
122. C2×Dic17#108semidirect
137. #1293D17semidirect
213. C17#8611semidirect

Maximal subgroups

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

subgroupindex
C2×#117628722
#145288202
#145271202
#145266612
C17×#12932
#861117