Finite Group Databasebeta

← #16945218

Subgroups — #16945218

Order 4352.

382 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 79 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#14528861C211yesmaximal, characteristic
3#14526809C212yesmaximal
4#14526785C212yesmaximal
5C17×#1458C211yesmaximal, characteristic
6C2×#10587556C411yescharacteristic, complemented
7#11762748C411yescharacteristic, complemented
8#11762745C2211yescharacteristic
9#11762739C411yescharacteristic, complemented
10#11762734C2211yescharacteristic
11#11762704—21no—
12#11762703—21no—
13#11762693—21no—
14#11762682—21no—
15#11762672—21no—
16#11762655C411yescharacteristic, complemented
17#11762576—21no—
18#11762576C2212yes—
19C2×C17×#117C2211yescharacteristic
20C17×#431—21no—
21C17×#324—21no—
22C17×#323C2212yes—
23C23×C4×C17D411yescharacteristic
24C17×#117—41no—
25C17×#117—41no—
26C2×C4⋊C4×C17C2×C411yescharacteristic
27C2×C4⋊C4×C17D411yescharacteristic
28C2×C22⋊C4×C17D411yescharacteristic
29C2×C22⋊C4×C17C2311yescharacteristic
30C2×C42×C17C2×C411yescharacteristic
31C2×C42×C17D411yescharacteristic
32C2×C22⋊Dic17—21no—
33C2×C22⋊Dic17—21no—
34C2×C22⋊Dic17—22no—
35C2×C4⋊Dic17C2×C411yescharacteristic
36C2×C4⋊Dic17C2×C411yescharacteristic
37C2×#63225—21no—
38C2×#63225—21no—
39C2×#63225—21no—
40C2×#63225—21no—
41C2×C4×Dic17—21no—
42#10587556—41no—
43#10587556—41no—
44C17×C4⋊C8—41no—
45C17×C22⋊C8—22no—
46C4×C8×C17—41no—
47#10587502—21no—
48#10587502—21no—
49#10587502—21no—
50#10587502—21no—
51#10587502—21no—
52#10587502—41no—
53#10587502—41no—
54#10587502C2×C411yescharacteristic
55#10587502C2×C411yescharacteristic
56C24×C17C2×D411yescharacteristic
57C22×C4×C17—22no—
58C22×C4×C17—21no—
59C22×C4×C17—21no—
60C22×C4×C17—41no—
61C22×C4×C17C22⋊C411yescharacteristic
62C22×C4×C17C22⋊C411yescharacteristic
63C22×C4×C17C22⋊C411yescharacteristic
64C22×C4×C17C22⋊C411yescharacteristic
65C22×C4×C17C22×C411yescharacteristic
66C22×C4×C17C2×D411yescharacteristic
67C22×Dic17—21no—
68C22×Dic17—21no—
69C22×Dic17—21no—
70C22×Dic17—21no—
71C22×Dic17—42no—
72C2×C8×C17—42no—
73C4⋊C4×C17—41no—
74C4⋊C4×C17—41no—
75C22⋊C4×C17—41no—
76C22⋊C4×C17—41no—
77C42×C17—21no—
78C42×C17—21no—
79C42×C17—21no—
80C42×C17—21no—
81C22⋊Dic17—42no—
82C22⋊Dic17—42no—
83C22⋊Dic17—42no—
84C22⋊Dic17—42no—
85C4⋊Dic17—41no—
86C4⋊Dic17—41no—
87#63225—42no—
88#63225—42no—
89#63225—42no—
90#63225—41no—
91#63225—41no—
92C4×Dic17—42no—
93#11846—171nomaximal, Sylow
94C23×C17—21no—
95C23×C17—21no—
96C23×C17—41no—
97C23×C17—41no—
98C23×C17C2×C22⋊C411yescharacteristic
99C23×C17#13512yes—
100C2×C4×C17—21no—
101C2×C4×C17—21no—
102C2×C4×C17—21no—
103C2×C4×C17—21no—
104C2×C4×C17—21no—
105C2×C4×C17—21no—
106C2×C4×C17—21no—
107C2×C4×C17—21no—
108C2×C4×C17—21no—
109C2×C4×C17—42no—
110C2×C4×C17—42no—
111C2×C4×C17—41no—
112C2×C4×C17—41no—
113C2×C4×C17—41no—
114C2×C4×C17—41no—
115C2×C4×C17#9911yescharacteristic
116C2×C4×C17#9911yescharacteristic
117C2×Dic17—22no—
118C2×Dic17—41no—
119C2×Dic17—41no—
120C2×Dic17—41no—
121C2×Dic17—41no—
122C2×Dic17—41no—
123C2×Dic17—41no—
124C2×Dic17—41no—
125C2×Dic17—41no—
126C2×Dic17—41no—
127C2×Dic17—41no—
128C2×Dic17—41no—
129C2×Dic17—82no—
130C2×Dic17—82no—
131C2×Dic17—82no—
132C8×C17—82no—
133#2611—171no—
134#1820—171no—
135#1458D1711yescharacteristic
136#1352—172no—
137#1344—172no—
138C22×C17—21no—
139C22×C17—21no—
140C22×C17—22no—
141C22×C17—42no—
142C22×C17—41no—
143C22×C17—41no—
144C22×C17—81no—
145C22×C17C2×#9911yescharacteristic
146C22×C17#41611yescharacteristic
147C22×C17#42111yescharacteristic
148C4×C17—21no—
149C4×C17—21no—
150C4×C17—41no—
151C4×C17—41no—
152C4×C17—41no—
153C4×C17—41no—
154C4×C17—41no—
155Dic17—42no—
156Dic17—81no—
157Dic17—81no—
158Dic17—81no—
159Dic17—162no—
160C2×#125—171no—
161C2×#117C2×D1711yescharacteristic
162#431—21no—
163#400—171no—
164#399—171no—
165#398—171no—
166#397—171no—
167#389—341no—
168#388—341no—
169#387—341no—
170#385—341no—
171#381—341no—
172#376—171no—
173#342—172no—
174#342—341no—
175#324—21no—
176#323C2×D1712yes—
177C2×C17—21no—
178C2×C17—21no—
179C2×C17—42no—
180C2×C17#151611yescharacteristic
181C2×C17#212311yescharacteristic
182C2×C17#212611yescharacteristic
183C23×C4D6811yescharacteristic
184#125—681no—
185#125—681no—
186#117—41no—
187#117—41no—
188C2×C4⋊C4—171no—
189C2×C4⋊C4—171no—
190C2×C4⋊C4—341no—
191C2×C4⋊C4—341no—
192C2×C4⋊C4—341no—
193C2×C4⋊C4—341no—
194C2×C4⋊C4C4×D1711yescharacteristic
195C2×C4⋊C4C22⋊D1711yescharacteristic
196C2×C22⋊C4—341no—
197C2×C22⋊C4—341no—
198C2×C22⋊C4—342no—
199C2×C22⋊C4D6811yescharacteristic
200C2×C22⋊C4C22×D1711yescharacteristic
201C2×C42—341no—
202C2×C42C4×D1711yescharacteristic
203C2×C42C22⋊D1711yescharacteristic
204C4⋊C8—41no—
205C22⋊C8—22no—
206C4×C8—41no—
207#95—171no—
208#95—171no—
209#95—341no—
210#95—341no—
211#95—341no—
212#95—341no—
213#95—341no—
214#95—681no—
215#95—681no—
216C17#1184611yescharacteristic, complemented, Sylow
217C24C2×D6811yescharacteristic
218C22×C4—22no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—41no—
222C22×C4—341no—
223C22×C4—341no—
224C22×C4—341no—
225C22×C4—341no—
226C22×C4—682no—
227C22×C4#6322711yescharacteristic
228C22×C4#6322711yescharacteristic
229C22×C4#6322711yescharacteristic
230C22×C4#6322711yescharacteristic
231C22×C4C2×C4×D1711yescharacteristic
232C22×C4C2×C22⋊D1711yescharacteristic
233C2×C8—42no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—681no—
237C4⋊C4—681no—
238C4⋊C4—682no—
239C4⋊C4—682no—
240C4⋊C4—682no—
241C4⋊C4—681no—
242C4⋊C4—681no—
243C22⋊C4—41no—
244C22⋊C4—41no—
245C22⋊C4—682no—
246C22⋊C4—682no—
247C22⋊C4—682no—
248C22⋊C4—682no—
249C42—21no—
250C42—21no—
251C42—21no—
252C42—21no—
253C42—682no—
254C23—21no—
255C23—21no—
256C23—41no—
257C23—41no—
258C23#1058759012yes—
259C23C2×#6322711yescharacteristic
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—342no—
276C2×C4—681no—
277C2×C4—681no—
278C2×C4—681no—
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—1362no—
288C2×C4—1362no—
289C2×C4—1362no—
290C2×C4#1058747711yescharacteristic
291C2×C4#1058747711yescharacteristic
292C8—82no—
293C22—21no—
294C22—21no—
295C22—22no—
296C22—42no—
297C22—41no—
298C22—41no—
299C22—81no—
300C22C2×#1058747711yescharacteristic
301C22#1176280611yescharacteristic
302C22#1176282511yescharacteristic
303C4—21no—
304C4—21no—
305C4—41no—
306C4—41no—
307C4—41no—
308C4—41no—
309C4—41no—
310C4—2722no—
311C4—682no—
312C4—1361no—
313C4—1361no—
314C4—1361no—
315C2—21no—
316C2—21no—
317C2—42no—
318C2#1452734111yescharacteristic
319C2#1453043911yescharacteristic
320C2#1453044711yescharacteristic
subgroupG/H
centerC22C2×#10587477
commutatorC22×C4×C17C22×C4
FrattiniC2×C22⋊C4C22×D17
FittingC17×#1458C2
socleC22×C17C2×#99
prime radicalC24×C17C2×D4
cosocleC2×C22⋊C4×C17C23
hypercenter#1458D17
hypocenterC17#11846
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1458

Complements

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

NH ≅ G/Ntype
6. C2×#10587556C4semidirect
7. #11762748C4semidirect
9. #11762739C4semidirect
16. #11762655C4semidirect
216. C17#11846semidirect

Maximal subgroups

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

subgroupindex
#145344822
#145288612
#145268092
#145267852
C17×#14582
#1184617