Finite Group Databasebeta

← #16926082

Subgroups — #16926082

Order 4352.

738 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 738 conjugacy classes of subgroups — too many to draw. Only its 151 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#14528780C211yesmaximal, characteristic, complemented
2#14527465C212yesmaximal, complemented
3#14527209C212yesmaximal
4#14527108C211yesmaximal, characteristic, complemented
5C2×C17×#556C211yesmaximal, characteristic
6#11762862C2212yes—
7#11762833C2212yes—
8C2×#10587507—21no—
9#11762578—21no—
10#11762558—24no—
11D4⋊#3622—24no—
12#11762535—22no—
13C17×#556—24no—
14C17×#556—24no—
15C2×C17×C4⋊Q8C2211yescharacteristic
16C2×C17×#126—22no—
17C2×C17×#124C412yes—
18C2×C17×C22⋊Q8—21no—
19C2×C4×Q8×C17C412yes—
20C2×C4×Q8×C17C2211yescharacteristic
21C2×C4×D4×C17C2211yescharacteristic
22C2×C17×#117—21no—
23C22×Q8×C17C2×C412yes—
24C22×Q8×C17D411yescharacteristic
25C22×D4×C17C2×C411yescharacteristic
26C23×C4×C17—21no—
27C17×C4⋊Q8—22no—
28C17×C4⋊Q8D414yes—
29C17×#126—216no—
30C17×#124—24no—
31C17×#124—24no—
32C17×C22⋊Q8—28no—
33C4×Q8×C17—24no—
34C4×Q8×C17—24no—
35C4×Q8×C17—22no—
36C4×Q8×C17—22no—
37C4×D4×C17—22no—
38C4×D4×C17D414yes—
39C17×#117—28no—
40C2×C4⋊C4×C17—22no—
41C2×C4⋊C4×C17—21no—
42C2×C4⋊C4×C17—21no—
43C2×C4⋊C4×C17C2×C411yescharacteristic
44C2×C4⋊C4×C17D411yescharacteristic
45C2×C4⋊C4×C17D412yes—
46C2×C22⋊C4×C17—22no—
47C2×C22⋊C4×C17—22no—
48C2×C22⋊C4×C17—21no—
49C2×C42×C17—22no—
50C2×C42×C17C2×C412yes—
51C2×C42×C17C2311yescharacteristic
52C2×#63223—22no—
53#10587507—42no—
54C22⋊#3622—22no—
55C4⋊#3622—22no—
56C4⋊#3622—22no—
57C4⋊#3622—22no—
58C4×#3622—22no—
59C24×C17—21no—
60C2×Q8×C17—22no—
61C2×Q8×C17—22no—
62C2×Q8×C17—22no—
63C2×Q8×C17—21no—
64C2×Q8×C17—21no—
65C2×Q8×C17—44no—
66C2×Q8×C17C22⋊C414yes—
67C2×D4×C17—21no—
68C2×D4×C17—42no—
69C2×D4×C17C22⋊C412yes—
70C22×C4×C17—21no—
71C22×C4×C17—22no—
72C22×C4×C17—24no—
73C22×C4×C17—42no—
74C22×C4×C17—41no—
75C22×C4×C17—41no—
76C22×C4×C17C22⋊C412yes—
77C22×C4×C17C22⋊C412yes—
78C22×C4×C17C22×C411yescharacteristic
79C22×C4×C17C2×D411yescharacteristic
80C22×C4×C17C2×D411yescharacteristic
81C4⋊C4×C17—28no—
82C4⋊C4×C17—21no—
83C4⋊C4×C17—24no—
84C4⋊C4×C17—24no—
85C4⋊C4×C17—24no—
86C4⋊C4×C17—22no—
87C4⋊C4×C17C22⋊C412yes—
88C22⋊C4×C17—28no—
89C22⋊C4×C17—28no—
90C22⋊C4×C17—24no—
91C42×C17—28no—
92C42×C17—22no—
93C42×C17—22no—
94C42×C17C2×D412yes—
95C42×C17C2×D412yes—
96#63223—42no—
97#63223—42no—
98C2×#3622—42no—
99C2×#3622—42no—
100#8394—171nomaximal, Sylow
101C23×C17—22no—
102C23×C17—42no—
103C23×C17—42no—
104C23×C17—41no—
105C23×C17—41no—
106C23×C17C2×C22⋊C411yescharacteristic
107Q8×C17—44no—
108Q8×C17—44no—
109Q8×C17—42no—
110Q8×C17—42no—
111D4×C17—24no—
112D4×C17—42no—
113C2×C4×C17—22no—
114C2×C4×C17—22no—
115C2×C4×C17—22no—
116C2×C4×C17—24no—
117C2×C4×C17—22no—
118C2×C4×C17—22no—
119C2×C4×C17—22no—
120C2×C4×C17—41no—
121C2×C4×C17—44no—
122C2×C4×C17—41no—
123C2×C4×C17—41no—
124C2×C4×C17—44no—
125C2×C4×C17—44no—
126C2×C4×C17—42no—
127C2×C4×C17C2×C22⋊C411yescharacteristic
128C2×C4×C17C2×C22⋊C411yescharacteristic
129C2×C4×C17#12012yes—
130C2×C4×C17#12012yes—
131C2×C4×C17#13512yes—
132C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
133C2×C4×C17#13711yescharacteristic
134#3622—82no—
135#3622—82no—
136C2×#556D1711yescharacteristic
137#1801—171no—
138#1548—172no—
139#1486—172no—
140#1461—171no—
141C22×C17—21no—
142C22×C17—21no—
143C22×C17—42no—
144C22×C17—42no—
145C22×C17—42no—
146C22×C17—82no—
147C22×C17—82no—
148C22×C17#37911yescharacteristic
149C22×C17#41611yescharacteristic
150C22×C17#41711yescharacteristic
151C4×C17—22no—
152C4×C17—21no—
153C4×C17—42no—
154C4×C17—44no—
155C4×C17—42no—
156C4×C17—42no—
157C4×C17#41312yes—
158C4×C17#44912yes—
159C4×C17#45212yes—
160#556—24no—
161#556—24no—
162C2×C4⋊Q8C2×D1711yescharacteristic
163C2×#126—22no—
164C2×#124Dic1712yes—
165C2×C22⋊Q8—21no—
166C2×C4×Q8Dic1712yes—
167C2×C4×Q8C2×D1711yescharacteristic
168C2×C4×D4C2×D1711yescharacteristic
169C2×#117—21no—
170#431—172no—
171#423—172no—
172C2×#101—341no—
173#343—341no—
174#333—344no—
175D4⋊C8—344no—
176#324—342no—
177C2×C17—21no—
178C2×C17—21no—
179C2×C17—82no—
180C2×C17#178511yescharacteristic
181C2×C17#178811yescharacteristic
182C2×C17#179611yescharacteristic
183C22×Q8C2×Dic1712yes—
184C22×Q8C22⋊D1711yescharacteristic
185C22×D4C2×Dic1711yescharacteristic
186C23×C4—21no—
187C2×M16—342no—
188C4⋊Q8—22no—
189C4⋊Q8C22⋊D1714yes—
190#126—216no—
191#124—24no—
192#124—24no—
193C22⋊Q8—28no—
194C4×Q8—24no—
195C4×Q8—24no—
196C4×Q8—22no—
197C4×Q8—22no—
198C4×D4—22no—
199C4×D4C22⋊D1714yes—
200#117—28no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4C2×Dic1711yescharacteristic
205C2×C4⋊C4C22⋊D1711yescharacteristic
206C2×C4⋊C4C22⋊D1712yes—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—21no—
210C2×C42—22no—
211C2×C42C2×Dic1712yes—
212C2×C42C22×D1711yescharacteristic
213C4⋊C8—342no—
214C4⋊C8—342no—
215C4⋊C8—342no—
216#101—682no—
217C22⋊C8—342no—
218C4×C8—342no—
219C17#839411yescharacteristic, complemented, Sylow
220C24—21no—
221C2×Q8—21no—
222C2×Q8—22no—
223C2×Q8—21no—
224C2×Q8—22no—
225C2×Q8—22no—
226C2×Q8—44no—
227C2×Q8C22⋊Dic1714yes—
228C2×D4—21no—
229C2×D4—42no—
230C2×D4C22⋊Dic1712yes—
231C22×C4—21no—
232C22×C4—22no—
233C22×C4—24no—
234C22×C4—41no—
235C22×C4—42no—
236C22×C4—41no—
237C22×C4C22⋊Dic1712yes—
238C22×C4C22⋊Dic1712yes—
239C22×C4C22×Dic1711yescharacteristic
240C22×C4C2×C22⋊D1711yescharacteristic
241C22×C4C2×C22⋊D1711yescharacteristic
242M16—682no—
243M16—682no—
244C2×C8—682no—
245C2×C8—682no—
246C4⋊C4—28no—
247C4⋊C4—21no—
248C4⋊C4—24no—
249C4⋊C4—24no—
250C4⋊C4—22no—
251C4⋊C4—24no—
252C4⋊C4C22⋊Dic1712yes—
253C22⋊C4—28no—
254C22⋊C4—28no—
255C22⋊C4—24no—
256C42—28no—
257C42—22no—
258C42—22no—
259C42C2×C22⋊D1712yes—
260C42C2×C22⋊D1712yes—
261C23—22no—
262C23—42no—
263C23—42no—
264C23—41no—
265C23—41no—
266C23C2×C22⋊Dic1711yescharacteristic
267Q8—44no—
268Q8—42no—
269Q8—42no—
270Q8—44no—
271D4—24no—
272D4—42no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—24no—
279C2×C4—22no—
280C2×C4—41no—
281C2×C4—44no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—44no—
285C2×C4—44no—
286C2×C4—42no—
287C2×C4#1058763511yescharacteristic
288C2×C4#1058763612yes—
289C2×C4#1058763711yescharacteristic
290C2×C4C2×C22⋊Dic1711yescharacteristic
291C2×C4C2×C22⋊Dic1711yescharacteristic
292C2×C4#1058763912yes—
293C2×C4#1058763912yes—
294C8—1362no—
295C8—1362no—
296C22—21no—
297C22—21no—
298C22—42no—
299C22—42no—
300C22—42no—
301C22—82no—
302C22—82no—
303C22#1176266511yescharacteristic
304C22#1176280511yescharacteristic
305C22#1176281011yescharacteristic
306C4—21no—
307C4—22no—
308C4—44no—
309C4—42no—
310C4—42no—
311C4—42no—
312C4#1176279312yes—
313C4#1176295412yes—
314C4#1176297112yes—
315C2—21no—
316C2—21no—
317C2—82no—
318C2#1452871211yescharacteristic
319C2#1452872411yescharacteristic
320C2#1452875911yescharacteristic
subgroupG/H
centerC22#11762665
commutatorC22×C4×C17C22×C4
FrattiniC2×C42C22×D17
FittingC2×C17×#556C2
socleC22×C17#379
prime radicalC22×D4×C17C2×C4
cosocleC2×C42×C17C23
hypercenterC2×#556D17
hypocenterC17#8394
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#556

Complements

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

NH ≅ G/Ntype
1. #14528780C2semidirect
2. #14527465C2semidirect
4. #14527108C2semidirect
219. C17#8394semidirect

Maximal subgroups

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

subgroupindex
#145287802
#145274652
#145272092
#145271082
C2×C17×#5562
#839417