Finite Group Databasebeta

← #16927809

Subgroups — #16927809

Order 4352.

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

normal subgroups only

This group has 450 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#14531268C211yesmaximal, characteristic
2#14527118C211yesmaximal, characteristic
3#14526641C212yesmaximal, complemented
4#14526639C212yesmaximal, complemented
5C17×#1464C211yesmaximal, characteristic
6C2×#10587612C411yescharacteristic
7C22×C4×Dic17C411yescharacteristic
8#11762866C2212yes—
9#11762708—21no—
10#11762708—21no—
11#11762684—21no—
12#11762684—21no—
13C4×C4⋊Dic17C411yescharacteristic
14C4×C4⋊Dic17C411yescharacteristic
15C4×C4⋊Dic17C2211yescharacteristic
16C4×C4⋊Dic17C2211yescharacteristic
17#11762535—21no—
18#11762533—21no—
19C2×C17×#117C2211yescharacteristic
20C17×#432C2212yes—
21C17×#324—21no—
22C17×#323—21no—
23C23×C4×C17C2×C411yescharacteristic
24C23×Dic17—21no—
25C17×#117—22no—
26C17×#117—22no—
27C2×C4⋊C4×C17D411yescharacteristic
28C2×C4⋊C4×C17D411yescharacteristic
29C2×C22⋊C4×C17D411yescharacteristic
30C2×C22⋊C4×C17D411yescharacteristic
31C2×C42×C17C2×C411yescharacteristic
32C2×C42×C17C2311yescharacteristic
33C2×C22⋊Dic17—21no—
34#10587612—42no—
35C2×C4⋊Dic17—21no—
36C2×C4⋊Dic17—21no—
37C2×C4⋊Dic17—21no—
38C2×C4⋊Dic17—21no—
39C2×C4⋊Dic17—21no—
40C2×C4×Dic17—21no—
41C2×C4×Dic17—42no—
42C2×C4×Dic17C2×C411yescharacteristic
43C2×C4×Dic17C2×C411yescharacteristic
44C2×C4×Dic17C2×C411yescharacteristic
45C2×C4×Dic17C2×C411yescharacteristic
46C17×C4⋊C8—22no—
47C17×C22⋊C8—22no—
48C17×#97—22no—
49#10587502—21no—
50#10587502—21no—
51#10587502—21no—
52#10587502—21no—
53C22⋊#3622—22no—
54C4⋊#3622—22no—
55#10587474—22no—
56C24×C17C22⋊C411yescharacteristic
57C22×C4×C17—22no—
58C22×C4×C17—22no—
59C22×C4×C17—21no—
60C22×C4×C17—21no—
61C22×C4×C17C22⋊C411yescharacteristic
62C22×C4×C17C22⋊C411yescharacteristic
63C22×C4×C17C22⋊C411yescharacteristic
64C22×C4×C17C22×C411yescharacteristic
65C22×C4×C17C2×D411yescharacteristic
66C22×C4×C17C2×D411yescharacteristic
67C22×Dic17—21no—
68C22×Dic17—21no—
69C22×Dic17—21no—
70C22×Dic17—21no—
71C22×Dic17—42no—
72C22×Dic17—42no—
73C22×Dic17—42no—
74C2×C8×C17—42no—
75C4⋊C4×C17—22no—
76C4⋊C4×C17—22no—
77C22⋊C4×C17—22no—
78C22⋊C4×C17—22no—
79C42×C17—21no—
80C42×C17—21no—
81C42×C17#4112yes—
82C42×C17#4112yes—
83C22⋊Dic17—42no—
84C4⋊Dic17—42no—
85C4⋊Dic17—42no—
86C4⋊Dic17—42no—
87C4⋊Dic17—42no—
88C4⋊Dic17—42no—
89C4×Dic17—42no—
90C4×Dic17—41no—
91C4×Dic17—41no—
92C4×Dic17—41no—
93C4×Dic17—41no—
94C2×#3622—42no—
95#8742—171nomaximal, Sylow
96C23×C17—21no—
97C23×C17—21no—
98C23×C17—41no—
99C23×C17—41no—
100C23×C17#10012yes—
101C23×C17C2×C22⋊C411yescharacteristic
102C2×C4×C17—24no—
103C2×C4×C17—24no—
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—41no—
111C2×C4×C17—41no—
112C2×C4×C17—41no—
113C2×C4×C17—41no—
114C2×C4×C17#10112yes—
115C2×C4×C17#11711yescharacteristic
116C2×C4×C17#11711yescharacteristic
117C2×C4×C17#12312yes—
118C2×C4×C17#12312yes—
119C2×Dic17—22no—
120C2×Dic17—22no—
121C2×Dic17—41no—
122C2×Dic17—41no—
123C2×Dic17—48no—
124C2×Dic17—41no—
125C2×Dic17—41no—
126C2×Dic17—41no—
127C2×Dic17—41no—
128C2×Dic17—41no—
129C2×Dic17—41no—
130C2×Dic17—41no—
131C2×Dic17—41no—
132C8×C17—82no—
133#3622—82no—
134#2296—171no—
135#1464—171no—
136#1464D1711yescharacteristic
137#1285—172no—
138#1285—172no—
139C22×C17—21no—
140C22×C17—21no—
141C22×C17—22no—
142C22×C17—42no—
143C22×C17—42no—
144C22×C17—41no—
145C22×C17—41no—
146C22×C17#38111yescharacteristic
147C22×C17C2×#10011yescharacteristic
148C22×C17C2×#10111yescharacteristic
149C4×C17—22no—
150C4×C17—21no—
151C4×C17—21no—
152C4×C17—22no—
153C4×C17—22no—
154C4×C17—41no—
155C4×C17—41no—
156Dic17—42no—
157Dic17—42no—
158Dic17—81no—
159Dic17—81no—
160C2×#117—171no—
161C2×#117C2×D1711yescharacteristic
162C22×C42—171no—
163#432—172no—
164#432C2×D1712yes—
165#389—341no—
166#389—341no—
167#385—341no—
168#385—341no—
169C4×C4⋊C4—171no—
170C4×C4⋊C4—171no—
171C4×C4⋊C4—171no—
172C4×C4⋊C4—171no—
173#324—21no—
174#324—341no—
175#323—21no—
176#323—341no—
177C2×C17—21no—
178C2×C17—21no—
179C2×C17—42no—
180C2×C17#182111yescharacteristic
181C2×C17#182711yescharacteristic
182C2×C17#182711yescharacteristic
183C23×C4—341no—
184C23×C4C4×D1711yescharacteristic
185#117—22no—
186#117—22no—
187#117—682no—
188C2×C4⋊C4—341no—
189C2×C4⋊C4—341no—
190C2×C4⋊C4—341no—
191C2×C4⋊C4—341no—
192C2×C4⋊C4—341no—
193C2×C4⋊C4D6811yescharacteristic
194C2×C4⋊C4C22⋊D1711yescharacteristic
195C2×C22⋊C4—341no—
196C2×C22⋊C4D6811yescharacteristic
197C2×C22⋊C4C22⋊D1711yescharacteristic
198C2×C42—171no—
199C2×C42—171no—
200C2×C42—171no—
201C2×C42—171no—
202C2×C42—341no—
203C2×C42—682no—
204C2×C42C4×D1711yescharacteristic
205C2×C42C22×D1711yescharacteristic
206C4⋊C8—22no—
207C4⋊C8—342no—
208C22⋊C8—22no—
209C22⋊C8—342no—
210#97—22no—
211#97—342no—
212#95—341no—
213#95—341no—
214#95—341no—
215#95—341no—
216C17#874211yescharacteristic, complemented, Sylow
217C24#6322711yescharacteristic
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—341no—
223C22×C4—341no—
224C22×C4—341no—
225C22×C4—341no—
226C22×C4—682no—
227C22×C4—682no—
228C22×C4—682no—
229C22×C4#6322711yescharacteristic
230C22×C4#6322711yescharacteristic
231C22×C4#6322711yescharacteristic
232C22×C4C2×C4×D1711yescharacteristic
233C22×C4C2×D6811yescharacteristic
234C22×C4C2×C22⋊D1711yescharacteristic
235C2×C8—42no—
236C2×C8—682no—
237C4⋊C4—22no—
238C4⋊C4—22no—
239C4⋊C4—682no—
240C4⋊C4—682no—
241C4⋊C4—682no—
242C4⋊C4—682no—
243C4⋊C4—682no—
244C22⋊C4—22no—
245C22⋊C4—22no—
246C22⋊C4—682no—
247C42—21no—
248C42—21no—
249C42—681no—
250C42—681no—
251C42—681no—
252C42—681no—
253C42—682no—
254C42#6325412yes—
255C42#6325412yes—
256C23—21no—
257C23—21no—
258C23—41no—
259C23—41no—
260C23#1058749412yes—
261C23C2×#6322711yescharacteristic
262C2×C4—24no—
263C2×C4—24no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—342no—
275C2×C4—342no—
276C2×C4—681no—
277C2×C4—681no—
278C2×C4—688no—
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#1058749512yes—
288C2×C4#1058756311yescharacteristic
289C2×C4#1058756311yescharacteristic
290C2×C4#1058757212yes—
291C2×C4#1058762112yes—
292C8—82no—
293C8—1362no—
294C22—21no—
295C22—21no—
296C22—22no—
297C22—42no—
298C22—42no—
299C22—41no—
300C22—41no—
301C22#1176267011yescharacteristic
302C22C2×#1058749411yescharacteristic
303C22C2×#1058749511yescharacteristic
304C4—22no—
305C4—21no—
306C4—21no—
307C4—22no—
308C4—22no—
309C4—41no—
310C4—41no—
311C4—682no—
312C4—682no—
313C4—1361no—
314C4—1361no—
315C2—21no—
316C2—21no—
317C2—42no—
318C2#1452886211yescharacteristic
319C2#1452888511yescharacteristic
320C2#1452888911yescharacteristic
subgroupG/H
centerC22#11762670
commutatorC22×C4×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#1464C2
socleC22×C17#381
prime radicalC24×C17C22⋊C4
cosocleC2×C42×C17C23
hypercenter#1464D17
hypocenterC17#8742
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1464

Complements

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

NH ≅ G/Ntype
3. #14526641C2semidirect
4. #14526639C2semidirect
216. C17#8742semidirect

Maximal subgroups

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

subgroupindex
#145312682
#145271182
#145266412
#145266392
C17×#14642
#874217