Finite Group Databasebeta

← #16931674

Subgroups — #16931674

Order 4352.

446 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 99 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#14534482C211yesmaximal, characteristic
2#14527385C212yesmaximal, complemented
3#14527118C211yesmaximal, characteristic
4#14526678C212yesmaximal, complemented
5C17×#2921C211yesmaximal, characteristic
6C2×#10587556C2211yescharacteristic
7#11762866C2212yes—
8#11762748C411yescharacteristic
9#11762745C411yescharacteristic
10#11762739C411yescharacteristic
11#11762734C411yescharacteristic
12#11762704—21no—
13#11762703—21no—
14#11762693—21no—
15#11762682—21no—
16#11762655C2211yescharacteristic
17#11762550—24no—
18#11762535—21no—
19#11762533—21no—
20C2×C17×#117C2211yescharacteristic
21C17×#433—24no—
22C17×#423—21no—
23C17×#407—21no—
24C4×M16×C17C2212yes—
25C23×C4×C17D411yescharacteristic
26C2×M16×C17—22no—
27C17×#117—22no—
28C17×#117—22no—
29C2×C4⋊C4×C17C2×C411yescharacteristic
30C2×C4⋊C4×C17D411yescharacteristic
31C2×C22⋊C4×C17C2×C411yescharacteristic
32C2×C22⋊C4×C17D411yescharacteristic
33C2×C42×C17D411yescharacteristic
34C2×C42×C17C2311yescharacteristic
35C2×C22⋊Dic17—21no—
36C2×C22⋊Dic17—21no—
37C2×C4⋊Dic17C2×C411yescharacteristic
38C2×C4⋊Dic17C2×C411yescharacteristic
39C2×#63225—21no—
40C2×#63225—21no—
41C2×#63225—21no—
42C2×#63225—21no—
43C2×C4×Dic17—21no—
44#10587556—22no—
45#10587556—22no—
46C17×C4⋊C8—24no—
47C17×C22⋊C8—24no—
48C17×#97D414yes—
49C4×C8×C17—22no—
50#10587502—21no—
51#10587502—21no—
52#10587502—21no—
53#10587502—21no—
54#10587502—21no—
55#10587502C2×C411yescharacteristic
56#10587502C2×C411yescharacteristic
57C22⋊#3622—22no—
58C4⋊#3622—22no—
59#10587474—22no—
60C24×C17C22⋊C411yescharacteristic
61C22×C4×C17—21no—
62C22×C4×C17—21no—
63C22×C4×C17—41no—
64C22×C4×C17—41no—
65C22×C4×C17C22⋊C411yescharacteristic
66C22×C4×C17C22⋊C411yescharacteristic
67C22×C4×C17C22⋊C411yescharacteristic
68C22×C4×C17C22×C411yescharacteristic
69C22×C4×C17C2×D411yescharacteristic
70C22×C4×C17C2×D411yescharacteristic
71C22×Dic17—21no—
72C22×Dic17—21no—
73C22×Dic17—21no—
74C22×Dic17—21no—
75M16×C17—42no—
76M16×C17—42no—
77C2×C8×C17—24no—
78C4⋊C4×C17—22no—
79C4⋊C4×C17—22no—
80C22⋊C4×C17—22no—
81C22⋊C4×C17—22no—
82C42×C17—21no—
83C42×C17—21no—
84C42×C17C2×D412yes—
85C42×C17#4112yes—
86C22⋊Dic17—42no—
87C22⋊Dic17—42no—
88C4⋊Dic17—22no—
89C4⋊Dic17—22no—
90#63225—42no—
91#63225—42no—
92#63225—41no—
93#63225—42no—
94#63225—41no—
95C4×Dic17—42no—
96C2×#3622—42no—
97#9390—171nomaximal, Sylow
98C23×C17—21no—
99C23×C17—21no—
100C23×C17—21no—
101C23×C17—41no—
102C23×C17—41no—
103C23×C17C2×C22⋊C411yescharacteristic
104C2×C4×C17—21no—
105C2×C4×C17—21no—
106C2×C4×C17—21no—
107C2×C4×C17—21no—
108C2×C4×C17—21no—
109C2×C4×C17—21no—
110C2×C4×C17—21no—
111C2×C4×C17—41no—
112C2×C4×C17—41no—
113C2×C4×C17—42no—
114C2×C4×C17—42no—
115C2×C4×C17—41no—
116C2×C4×C17—41no—
117C2×C4×C17C4×D412yes—
118C2×C4×C17#12011yescharacteristic
119C2×C4×C17#12112yes—
120C2×C4×C17#12311yescharacteristic
121C2×Dic17—22no—
122C2×Dic17—41no—
123C2×Dic17—41no—
124C2×Dic17—41no—
125C2×Dic17—41no—
126C2×Dic17—41no—
127C2×Dic17—41no—
128C2×Dic17—41no—
129C2×Dic17—41no—
130C2×Dic17—41no—
131C2×Dic17—41no—
132C2×Dic17—41no—
133C8×C17—44no—
134#3622—82no—
135#2921D1711yescharacteristic
136#2611—171no—
137#1528—172no—
138#1464—171no—
139#1300—172no—
140C22×C17—21no—
141C22×C17—21no—
142C22×C17—42no—
143C22×C17—41no—
144C22×C17—41no—
145C22×C17—41no—
146C22×C17—81no—
147C22×C17#38611yescharacteristic
148C22×C17#42111yescharacteristic
149C22×C17#42111yescharacteristic
150C4×C17—22no—
151C4×C17—21no—
152C4×C17—21no—
153C4×C17—22no—
154C4×C17—41no—
155C4×C17—41no—
156C4×C17—41no—
157Dic17—42no—
158Dic17—81no—
159Dic17—81no—
160Dic17—81no—
161C2×#125—171no—
162C2×#117C2×D1711yescharacteristic
163#433—24no—
164#432—172no—
165#423—21no—
166#407—21no—
167C4×M16C2×D1712yes—
168#400—171no—
169#399—171no—
170#398—171no—
171#397—171no—
172#389—341no—
173#388—341no—
174#387—341no—
175#385—341no—
176#376—171no—
177#330—344no—
178#324—341no—
179#323—341no—
180C2×C17—21no—
181C2×C17—21no—
182C2×C17—81no—
183C2×C17#188411yescharacteristic
184C2×C17#189711yescharacteristic
185C2×C17#191511yescharacteristic
186C23×C4C22⋊D1711yescharacteristic
187C2×M16—22no—
188#125—342no—
189#125—342no—
190#117—22no—
191#117—22no—
192C2×C4⋊C4—171no—
193C2×C4⋊C4—171no—
194C2×C4⋊C4—341no—
195C2×C4⋊C4—341no—
196C2×C4⋊C4—341no—
197C2×C4⋊C4—341no—
198C2×C4⋊C4C4×D1711yescharacteristic
199C2×C4⋊C4D6811yescharacteristic
200C2×C22⋊C4—341no—
201C2×C22⋊C4—341no—
202C2×C22⋊C4C4×D1711yescharacteristic
203C2×C22⋊C4D6811yescharacteristic
204C2×C42—341no—
205C2×C42C22⋊D1711yescharacteristic
206C2×C42C22×D1711yescharacteristic
207C4⋊C8—24no—
208C4⋊C8—342no—
209C22⋊C8—24no—
210C22⋊C8—342no—
211#97—342no—
212#97C22⋊D1714yes—
213C4×C8—22no—
214#95—171no—
215#95—171no—
216#95—341no—
217#95—341no—
218#95—341no—
219#95—341no—
220#95—341no—
221C17#939011yescharacteristic, complemented, Sylow
222C24#6322711yescharacteristic
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—341no—
228C22×C4—341no—
229C22×C4—341no—
230C22×C4—341no—
231C22×C4#6322711yescharacteristic
232C22×C4#6322711yescharacteristic
233C22×C4#6322711yescharacteristic
234C22×C4C2×C4×D1711yescharacteristic
235C22×C4C2×D6811yescharacteristic
236C22×C4C2×C22⋊D1711yescharacteristic
237M16—42no—
238M16—42no—
239C2×C8—24no—
240C2×C8—682no—
241C4⋊C4—22no—
242C4⋊C4—22no—
243C4⋊C4—342no—
244C4⋊C4—342no—
245C4⋊C4—682no—
246C4⋊C4—682no—
247C4⋊C4—681no—
248C4⋊C4—682no—
249C4⋊C4—681no—
250C22⋊C4—22no—
251C22⋊C4—22no—
252C22⋊C4—682no—
253C22⋊C4—682no—
254C42—21no—
255C42—21no—
256C42—682no—
257C42#6325212yes—
258C42C2×C22⋊D1712yes—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—41no—
263C23—41no—
264C23C2×#6322711yescharacteristic
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—342no—
279C2×C4—681no—
280C2×C4—681no—
281C2×C4—681no—
282C2×C4—681no—
283C2×C4—681no—
284C2×C4—681no—
285C2×C4—681no—
286C2×C4—681no—
287C2×C4—681no—
288C2×C4—681no—
289C2×C4—681no—
290C2×C4C4×C22⋊D1712yes—
291C2×C4#1058761511yescharacteristic
292C2×C4#1058761612yes—
293C2×C4#1058763911yescharacteristic
294C8—44no—
295C8—1362no—
296C22—21no—
297C22—21no—
298C22—42no—
299C22—41no—
300C22—41no—
301C22—41no—
302C22—81no—
303C22#1176268811yescharacteristic
304C22#1176282711yescharacteristic
305C22#1176282711yescharacteristic
306C4—22no—
307C4—22no—
308C4—21no—
309C4—21no—
310C4—41no—
311C4—41no—
312C4—41no—
313C4—682no—
314C4—1361no—
315C4—1361no—
316C4—1361no—
317C2—21no—
318C2—21no—
319C2—81no—
320C2#1452916711yescharacteristic
321C2#1452923211yescharacteristic
322C2#1452933611yescharacteristic
subgroupG/H
centerC22#11762688
commutatorC22×C4×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#2921C2
socleC22×C17#386
prime radicalC24×C17C22⋊C4
cosocleC2×C42×C17C23
hypercenter#2921D17
hypocenterC17#9390
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2921

Complements

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

NH ≅ G/Ntype
2. #14527385C2semidirect
4. #14526678C2semidirect
221. C17#9390semidirect

Maximal subgroups

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

subgroupindex
#145344822
#145273852
#145271182
#145266782
C17×#29212
#939017